Dynamical Systems in Neuroscience

Computational Neuroscience
Terrence J. Sejnowski and Tomaso A. Poggio, editors
Neural Nets in Electric Fish, Walter Heiligenberg, 1991
The Computational Brain, Patricia S. Churchland and Terrence J. Sejnowski, 1992
Dynamic Biological Networks: The Stomatogastric Nervous System, edited by Ronald M.
Harris-Warrick, Eve Marder, Allen I. Selverston, and Maurice Maulins, 1992
The Neurobiology of Neural Networks, edited by Daniel Gardner, 1993
Large-Scale Neuronal Theories of the Brain, edited by Christof Koch and Joel L. Davis, 1994
The Theoretical Foundations of Dendritic Function: Selected Papers of Wilfrid Rall with
Commentaries, edited by Idan Segev, John Rinzel, and Gordon M. Shepherd, 1995
Models of Information Processing in the Basal Ganglia, edited by James C. Houk, Joel L.
Davis, and David G. Beiser, 1995
Spikes: Exploring the Neural Code, Fred Rieke, David Warland, Rob de Ruyter van Stevenick,and William Bialek, 1997
Neurons, Networks, and Motor Behavior, edited by Paul S. Stein, Sten Grillner, Allen I.
Selverston, and Douglas G. Stuart, 1997
Methods in Neuronal Modeling: From Ions to Networks, second edition, edited by Christof
Koch and Idan Segev, 1998
Fundamentals of Neural Network Modeling: Neuropsychology and Cognitive Neuroscience,edited by Randolph W. Parks, Daniel S. Levin, and Debra L. Long, 1998
Neural Codes and Distributed Representations: Foundations of Neural Computation, editedby Laurence Abbott and Terrence J. Sejnowski, 1999
Unsupervised Learning: Foundations of Neural Computation, edited by Geoffrey Hinton andTerrence J. Sejnowski, 1999
Fast Oscillations in Cortical Circuits, Roger D. Traub, John G.R. Jefferys, and Miles Al
Whittington, 1999
Computational Vision: Information Processing in Perception and Visual Behavior, Hanspeter
A. Mallot, 2000
Graphical Models: Foundations of Neural Computation, edited by Michael I. Jordan andTerrence J. Sejnowski, 2001
Self-Organizing Map Formation: Foundation of Neural Computation, edited by Klaus Obermayer and Terrence J. Sejnowski, 2001
Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems,
Peter Dayan and L. F. Abbott, 2001
Neural Engineering: Computation, Representation, and Dynamics in Neurobiological Systems, Chris Eliasmith and Charles H. Anderson, 2003
The Computational Neurobiology of Reaching and Pointing, edited by Reza Shadmehr andSteven P. Wise, 2005
Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, Eugene M.
Izhikevich, 2007

Dynamical Systems in Neuroscience:
The Geometry of Excitability and Bursting

Eugene M. Izhikevich

The MIT Press
Cambridge, Massachusetts
London, England© 2007 Massachusetts Institute of Technology
All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage
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Library of Congress Cataloging-in-Publication Data
Izhikevich, Eugene M., 1967–
Dynamical systems in neuroscience: the geometry of excitability and bursting /Eugene M. Izhikevich.
p. cm. — (Computational neuroscience)
Includes bibliographical references and index.
ISBN 978-0-262-09043-8 (hc. : alk. paper)
1. Neural networks (Neurobiology) 2. Neurons - computer simulation. 3. Dynamical systems. 4. Computational neuroscience. I. Izhikevich, E. M. II Title. III.
Series.
QP363.3.I94 2007
573.8’01’13—DC21

10 9 8 7 6 5 4 3 22006040349

To my beautiful daughters, Liz and Kate.

Contents
Prefacexv

1 Introduction
1.1 Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 What Is a Spike? . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Where Is the Threshold? . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Why Are Neurons Different, and Why Do We Care? . . . . . . .
1.1.4 Building Models . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Phase Portraits . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Hodgkin Classification . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Neurocomputational properties . . . . . . . . . . . . . . . . . .
1.2.5 Building Models (Revisited) . . . . . . . . . . . . . . . . . . . .
Review of Important Concepts . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
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2 Electrophysiology of Neurons
2.1 Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Nernst Potential . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Ionic Currents and Conductances . . . . . . . . . . . . . . . . .
2.1.3 Equivalent Circuit . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Resting Potential and Input Resistance . . . . . . . . . . . . . .
2.1.5 Voltage-Clamp and I-V Relation . . . . . . . . . . . . . . . . . .
2.2 Conductances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Voltage-Gated Channels . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Activation of Persistent Currents . . . . . . . . . . . . . . . . .
2.2.3 Inactivation of Transient Currents . . . . . . . . . . . . . . . . .
2.2.4 Hyperpolarization-Activated Channels . . . . . . . . . . . . . .
2.3 The Hodgkin-Huxley Model . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Hodgkin-Huxley Equations . . . . . . . . . . . . . . . . . . . . .
2.3.2 Action Potential . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Propagation of the Action Potentials . . . . . . . . . . . . . . .

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CONTENTS
2.3.4 Dendritic Compartments . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Summary of Voltage-Gated Currents . . . . . . . . . . . . . . .
Review of Important Concepts . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43
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49
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50

3 One-Dimensional Systems
3.1 Electrophysiological Examples . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 I-V Relations and Dynamics . . . . . . . . . . . . . . . . . . . .
3.1.2 Leak + Instantaneous INa,p . . . . . . . . . . . . . . . . . . . .
3.2 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Geometrical Analysis . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Unstable Equilibria . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.6 Attraction Domain . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.7 Threshold and Action Potential . . . . . . . . . . . . . . . . . .
3.2.8 Bistability and Hysteresis . . . . . . . . . . . . . . . . . . . . .
3.3 Phase Portraits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Topological Equivalence . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Local Equivalence and the Hartman-Grobman Theorem . . . . .
3.3.3 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Saddle-Node (Fold) Bifurcation . . . . . . . . . . . . . . . . . .
3.3.5 Slow Transition . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.6 Bifurcation Diagram . . . . . . . . . . . . . . . . . . . . . . . .
3.3.7 Bifurcations and I-V Relations . . . . . . . . . . . . . . . . . . .
3.3.8 Quadratic Integrate-and-Fire Neuron . . . . . . . . . . . . . . .
Review of Important Concepts . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Two-Dimensional Systems
4.1 Planar Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Nullclines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Relaxation Oscillators . . . . . . . . . . . . . . . . . . . . . . .
4.2 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Local Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . .
4.2.4 Local Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS
4.2.5 Classification of Equilibria . . . . . . . . . . . . . . . . . . . . .
4.2.6 Example: FitzHugh-Nagumo Model . . . . . . . . . . . . . . . .
4.3 Phase Portraits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Bistability and Attraction Domains . . . . . . . . . . . . . . . .
4.3.2 Stable/Unstable Manifolds . . . . . . . . . . . . . . . . . . . . .
4.3.3 Homoclinic/Heteroclinic Trajectories . . . . . . . . . . . . . . .
4.3.4 Saddle-Node Bifurcation . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Andronov-Hopf Bifurcation . . . . . . . . . . . . . . . . . . . .
Review of Important Concepts . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix
103
106
108
108
109
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113
116
121
122
122

5 Conductance-Based Models and Their Reductions
127
5.1 Minimal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.1.1 Amplifying and Resonant Gating Variables . . . . . . . . . . . . 129
5.1.2 INa,p +IK -Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.1.3 INa,t -model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.1.4 INa,p +Ih -Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.1.5 Ih +IKir -Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.1.6 IK +IKir -Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.1.7 IA -Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.1.8 Ca2+ -Gated Minimal Models . . . . . . . . . . . . . . . . . . . . 147
5.2 Reduction of Multidimensional Models . . . . . . . . . . . . . . . . . . 147
5.2.1 Hodgkin-Huxley model . . . . . . . . . . . . . . . . . . . . . . . 147
5.2.2 Equivalent Potentials . . . . . . . . . . . . . . . . . . . . . . . . 151
5.2.3 Nullclines and I-V Relations . . . . . . . . . . . . . . . . . . . . 151
5.2.4 Reduction to Simple Model . . . . . . . . . . . . . . . . . . . . 153
Review of Important Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6 Bifurcations
159
6.1 Equilibrium (Rest State) . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.1.1 Saddle-Node (Fold) . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.1.2 Saddle-Node on Invariant Circle . . . . . . . . . . . . . . . . . . 164
6.1.3 Supercritical Andronov-Hopf . . . . . . . . . . . . . . . . . . . . 168
6.1.4 Subcritical Andronov-Hopf . . . . . . . . . . . . . . . . . . . . . 174
6.2 Limit Cycle (Spiking State) . . . . . . . . . . . . . . . . . . . . . . . . 178
6.2.1 Saddle-Node on Invariant Circle . . . . . . . . . . . . . . . . . . 180
6.2.2 Supercritical Andronov-Hopf . . . . . . . . . . . . . . . . . . . . 181
6.2.3 Fold Limit Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.2.4 Homoclinic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.3 Other Interesting Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

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CONTENTS
6.3.1 Three-Dimensional Phase Space . . . . . . . . . . . . . . . . . .
6.3.2 Cusp and Pitchfork . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.3 Bogdanov-Takens . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.4 Relaxation Oscillators and Canards . . . . . . . . . . . . . . . .
6.3.5 Bautin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.6 Saddle-Node Homoclinic Orbit . . . . . . . . . . . . . . . . . . .
6.3.7 Hard and Soft Loss of Stability . . . . . . . . . . . . . . . . . .
Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

190
192
194
198
200
201
204
205
210

7 Neuronal Excitability
215
7.1 Excitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.1.1 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7.1.2 Hodgkin’s Classification . . . . . . . . . . . . . . . . . . . . . . 218
7.1.3 Classes 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
7.1.4 Class 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7.1.5 Ramps, Steps, and Shocks . . . . . . . . . . . . . . . . . . . . . 224
7.1.6 Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7.1.7 Class 1 and 2 Spiking . . . . . . . . . . . . . . . . . . . . . . . . 228
7.2 Integrators vs. Resonators . . . . . . . . . . . . . . . . . . . . . . . . . 229
7.2.1 Fast Subthreshold Oscillations . . . . . . . . . . . . . . . . . . . 230
7.2.2 Frequency Preference and Resonance . . . . . . . . . . . . . . . 232
7.2.3 Frequency Preference in Vivo . . . . . . . . . . . . . . . . . . . 237
7.2.4 Thresholds and Action Potentials . . . . . . . . . . . . . . . . . 238
7.2.5 Threshold manifolds . . . . . . . . . . . . . . . . . . . . . . . . 240
7.2.6 Rheobase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
7.2.7 Postinhibitory Spike . . . . . . . . . . . . . . . . . . . . . . . . 242
7.2.8 Inhibition-Induced Spiking . . . . . . . . . . . . . . . . . . . . . 244
7.2.9 Spike Latency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.2.10 Flipping from an Integrator to a Resonator . . . . . . . . . . . . 248
7.2.11 Transition Between Integrators and Resonators . . . . . . . . . 251
7.3 Slow Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
7.3.1 Spike Frequency Modulation . . . . . . . . . . . . . . . . . . . . 255
7.3.2 I-V Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
7.3.3 Slow Subthreshold Oscillation . . . . . . . . . . . . . . . . . . . 258
7.3.4 Rebound Response and Voltage Sag . . . . . . . . . . . . . . . . 259
7.3.5 AHP and ADP . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
Review of Important Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 264
Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

CONTENTS

xi

8 Simple Models
267
8.1 Simplest Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
8.1.1 Integrate-and-Fire . . . . . . . . . . . . . . . . . . . . . . . . . 268
8.1.2 Resonate-and-Fire . . . . . . . . . . . . . . . . . . . . . . . . . 269
8.1.3 Quadratic Integrate-and-Fire . . . . . . . . . . . . . . . . . . . . 270
8.1.4 Simple Model of Choice . . . . . . . . . . . . . . . . . . . . . . 272
8.1.5 Canonical Models . . . . . . . . . . . . . . . . . . . . . . . . . . 278
8.2 Cortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
8.2.1 Regular Spiking (RS) Neurons . . . . . . . . . . . . . . . . . . . 282
8.2.2 Intrinsically Bursting (IB) Neurons . . . . . . . . . . . . . . . . 288
8.2.3 Multi-Compartment Dendritic Tree . . . . . . . . . . . . . . . . 292
8.2.4 Chattering (CH) Neurons . . . . . . . . . . . . . . . . . . . . . 294
8.2.5 Low-Threshold Spiking (LTS) Interneurons . . . . . . . . . . . . 296
8.2.6 Fast Spiking (FS) Interneurons . . . . . . . . . . . . . . . . . . 298
8.2.7 Late Spiking (LS) Interneurons . . . . . . . . . . . . . . . . . . 300
8.2.8 Diversity of Inhibitory Interneurons . . . . . . . . . . . . . . . . 301
8.3 Thalamus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
8.3.1 Thalamocortical (TC) Relay Neurons . . . . . . . . . . . . . . . 305
8.3.2 Reticular Thalamic Nucleus (RTN) Neurons . . . . . . . . . . . 306
8.3.3 Thalamic Interneurons . . . . . . . . . . . . . . . . . . . . . . . 308
8.4 Other Interesting Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
8.4.1 Hippocampal CA1 Pyramidal Neurons . . . . . . . . . . . . . . 308
8.4.2 Spiny Projection Neurons of Neostriatum and Basal Ganglia . . 311
8.4.3 Mesencephalic V Neurons of Brainstem . . . . . . . . . . . . . . 313
8.4.4 Stellate Cells of Entorhinal Cortex . . . . . . . . . . . . . . . . 314
8.4.5 Mitral Neurons of the Olfactory Bulb . . . . . . . . . . . . . . . 316
Review of Important Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
9 Bursting325
9.1 Electrophysiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
9.1.1 Example: The INa,p +IK +IK(M) -Model . . . . . . . . . . . . . . . 327
9.1.2 Fast-Slow Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 329
9.1.3 Minimal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
9.1.4 Central Pattern Generators and Half-Center Oscillators . . . . . 334
9.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
9.2.1 Fast-Slow Bursters . . . . . . . . . . . . . . . . . . . . . . . . . 336
9.2.2 Phase Portraits . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
9.2.3 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
9.2.4 Equivalent Voltage . . . . . . . . . . . . . . . . . . . . . . . . . 341
9.2.5 Hysteresis Loops and Slow Waves . . . . . . . . . . . . . . . . . 342
9.2.6 Bifurcations “Resting ↔ Bursting ↔ Tonic Spiking” . . . . . . 344

xii

CONTENTS
9.3

Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.1 Fold/Homoclinic . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.2 Circle/Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.3 SubHopf/Fold Cycle . . . . . . . . . . . . . . . . . . . . . . . .
9.3.4 Fold/Fold Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.5 Fold/Hopf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.6 Fold/Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Neurocomputational Properties . . . . . . . . . . . . . . . . . . . . . .
9.4.1 How to Distinguish? . . . . . . . . . . . . . . . . . . . . . . . .
9.4.2 Integrators vs. Resonators . . . . . . . . . . . . . . . . . . . . .
9.4.3 Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.4 Bursts as a Unit of Neuronal Information . . . . . . . . . . . . .
9.4.5 Chirps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.6 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . .
Review of Important Concepts . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

347
350
354
359
364
365
366
367
367
368
368
371
372
373
375
376
378

10 Synchronization
385
10.1 Pulsed Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
10.1.1 Phase of Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . 386
10.1.2 Isochrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
10.1.3 PRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
10.1.4 Type 0 and Type 1 Phase Response . . . . . . . . . . . . . . . . 391
10.1.5 Poincare Phase Map . . . . . . . . . . . . . . . . . . . . . . . . 394
10.1.6 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
10.1.7 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
10.1.8 Phase-Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
10.1.9 Arnold Tongues . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
10.2 Weak Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
10.2.1 Winfree’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . 400
10.2.2 Kuramoto’s Approach . . . . . . . . . . . . . . . . . . . . . . . 402
10.2.3 Malkin’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . 403
10.2.4 Measuring PRCs Experimentally . . . . . . . . . . . . . . . . . 405
10.2.5 Phase Model for Coupled Oscillators . . . . . . . . . . . . . . . 405
10.3 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
10.3.1 Two Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
10.3.2 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
10.3.3 Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
10.3.4 Mean-Field Approximations . . . . . . . . . . . . . . . . . . . . 416
10.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
10.4.1 Phase Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . 417
10.4.2 SNIC Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . 418

CONTENTS
10.4.3 Homoclinic Oscillators . . . . . . . . . . . . . . . . . . . . . . .
10.4.4 Relaxation Oscillators and FTM . . . . . . . . . . . . . . . . . .
10.4.5 Bursting Oscillators . . . . . . . . . . . . . . . . . . . . . . . . .
Review of Important Concepts . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii
424
426
428
430
431

Solutions to Exercises

441

References

481

Index

497

Preface
Historically, much of theoretical neuroscience research concerned neuronal circuits andsynaptic organization. The neurons were divided into excitatory and inhibitory types,
but their electrophysiological properties were largely neglected or taken to be identicalto those of Hodgkin-Huxley’s squid axon. The present awareness of the importance ofthe electrophysiology of individual neurons is best summarized by David McCormickin the fifth edition of Gordon Shepherd’s book The Synaptic Organization of the Brain:
Information-processing depends not only on the anatomical substrates of synaptic circuits but also on the electrophysiological properties of neurons... Even if
two neurons in different regions of the nervous system possess identical morphological features, they may respond to the same synaptic input in very different
manners because of each cell’s intrinsic properties.
McCormick (2004)

Much of present neuroscience research concerns voltage- and second-messengergated currents in individual cells, with the goal of understanding the cell’s intrinsic
neurocomputational properties. It is widely accepted that knowing the currents suffices
to determine what the cell is doing and why it is doing it. This, however, contradicts a
half-century–old observation that cells having similar currents can nevertheless exhibit
quite different dynamics. Indeed, studying isolated axons having presumably similarelectrophysiology (all are from the crustacean Carcinus maenas), Hodgkin (1948) injected a DC-current of varying amplitude, and discovered that some preparations could
exhibit repetitive spiking with arbitrarily low frequencies, while the others discharged
in a narrow frequency band. This observation was largely ignored by the neuroscience
community until the seminal paper by Rinzel and Ermentrout (1989), who showed thatthe difference in behavior is due to different bifurcation mechanisms of excitability.
Let us treat the amplitude of the injected current in Hodgkin’s experiments as abifurcation parameter: When the amplitude is small, the cell is quiescent; when theamplitude is large, the cell fires repetitive spikes. When we change the amplitude of theinjected current, the cell undergoes a transition from quiescence to repetitive spiking.
From the dynamical systems point of view, the transition corresponds to a bifurcationfrom equilibrium to a limit cycle attractor. The type of bifurcation determines the mostfundamental computational properties of neurons, such as the class of excitability, theexistence or nonexistence of threshold, all-or-none spikes, subthreshold oscillations,the ability to generate postinhibitory rebound spikes, bistability of resting and spiking
states, whether the neuron is an integrator or a resonator, and so on.
This book is devoted to a systematic study of the relationship between electrophysiology, bifurcations, and computational properties of neurons. The reader will learn why
cells having nearly identical currents may undergo distinct bifurcations, and hence they
will have fundamentally different neurocomputational properties. (Conversely, cells
xv

xvi

Prefacehaving quite different currents may undergo identical bifurcations, and hence they will
have similar neurocomputational properties.) The major message of the book can besummarized as follows (compare with the McCormick statement above):
Information-processing depends not only on the electrophysiological propertiesof neurons but also on their dynamical properties. Even if two neurons in thesame region of the nervous system possess similar electrophysiological features,
they may respond to the same synaptic input in very different manners becauseof each cell’s bifurcation dynamics.

Nonlinear dynamical system theory is a core of computational neuroscience research,but it is not a standard part of the graduate neuroscience curriculum. Neither is ittaught in most math/physics departments in a form suitable for a general biologicalaudience. As a result, many neuroscientists fail to grasp such fundamental concepts as
equilibrium, stability, limit cycle attractor, and bifurcations, even though neuroscientists constantly encounter these nonlinear phenomena.
This book introduces dynamical systems starting with simple one- and two-dimensional spiking models and continuing all the way to bursting systems. Each chapter
is organized from simple to complex, so everybody can start reading the book; only
the reader’s background will determine where he or she stops. The book emphasizes
the geometrical approach, so there are few equations but a lot of figures. Half of them
are simulations of various neural models, so there are hundreds of possible exercises
such as “Use MATLAB (GENESIS, NEURON, XPPAUT, etc.) and parameters in thecaption of figure X to simulate the figure.” Additional problems are provided at theend of each chapter; the reader is encouraged to solve at least some of them and tolook at the solutions of the others at the end of the book. Problems marked [M.S.] or[Ph.D.] are suggested thesis topics.

Acknowledgments. I thank the scientists who reviewed the first draft of thebook: Pablo Achard, Jose M. Amigo, Vlatko Becanovic, Brent Doiron, George Bard
Ermentrout, Richard FitzHugh, David Golomb, Andrei Iacob, Paul Kulchenko, Maciej
Lazarewicz, Georgi Medvedev, John Rinzel, Anil K. Seth, Gautam C Sethia, Arthur
Sherman, Klaus M. Stiefel, and Takashi Tateno. I also thank the anonymous referees who peer-reviewed the book and made quite a few valuable suggestions insteadof just rejecting it. Special thanks go to Niraj S. Desai, who made most of the invitro recordings used in the book (the data are available on the author’s Web pagewww.izhikevich.com), and to Bruno van Swinderen, who drew the cartoons. I enjoyed the hospitality of The Neurosciences Institute – a monastery of interdisciplinaryscience – and I benefited greatly from the expertise and support of its fellows.
Finally, I thank my wife, Tatyana, and my wonderful daughters, Elizabeth andKate, for their support and patience during the five-year gestation of this book.
Eugene M. Izhikevich
San Diego, California

www.izhikevich.com
December 19, 2005

Chapter 1
Introduction
This chapter highlights some of the most important concepts developed in the book.
First, we discuss several common misconceptions regarding the spike generation mechanism of neurons. Our goal is to motivate the reader to think of a neuron not onlyin terms of ions and channels, as many biologists do, and not only in terms of an input/output relationship, as many theoreticians do, but also as a nonlinear dynamicalsystem that looks at the input through the prism of its own intrinsic dynamics. We
ask such questions as “What makes a neuron fire?” or “Where is the threshold?”, andthen outline the answers, using the geometrical theory of dynamical systems.
From a dynamical systems point of view, neurons are excitable because they arenear a transition, called bifurcation, from resting to sustained spiking activity. While
there is a huge number of possible ionic mechanisms of excitability and spike generation, there are only four bifurcation mechanisms that can result in such a transition.
Considering the geometry of phase portraits at these bifurcations, we can understandmany computational properties of neurons, such as the nature of threshold and all-ornone spiking, the coexistence of resting and spiking states, the origin of spike latencies,
postinhibitory spikes, and the mechanism of integration and resonance. Moreover, we
can understand how these properties are interrelated, why some are equivalent, andwhy some are mutually exclusive.

1.1

Neurons

If somebody were to put a gun to the head of the author of this book and ask him toname the single most important concept in brain science, he would say it is the conceptof a neuron. There are only 1011 or so neurons in the human brain, much fewer than
the number of non-neural cells such as glia. Yet neurons are unique in the sense that
only they can transmit electrical signals over long distances. From the neuronal level
we can go down to cell biophysics and to the molecular biology of gene regulation.
From the neuronal level we can go up to neuronal circuits, to cortical structures, tothe whole brain, and finally to the behavior of the organism. So let us see how much
we understand of what is going on at the level of individual neurons.
1

2

Introduction

recordingelectrodesoma

membrane potential, mV

apical dendrites

basal dendrites+35 mVspike
40 ms

synapse

0.1 mm-60 mVtime, msaxon

Figure 1.1: Two interconnected cortical pyramidal neurons (hand drawing) and in vitro
recorded spike.

1.1.1

What Is a Spike?

A typical neuron receives inputs from more than 10, 000 other neurons through the contacts on its dendritic tree called synapses; see Fig.1.1. The inputs produce electricaltransmembrane currents that change the membrane potential of the neuron. Synaptic
currents produce changes, called postsynaptic potentials (PSPs). Small currents produce small PSPs; larger currents produce significant PSPs that can be amplified by thevoltage-sensitive channels embedded in the neuronal membrane and lead to the generation of an action potential or spike – an abrupt and transient change of membrane
voltage that propagates to other neurons via a long protrusion called an axon.
Such spikes are the main means of communication between neurons. In general,
neurons do not fire on their own; they fire as a result of incoming spikes from otherneurons. One of the most fundamental questions of neuroscience is What, exactly,
makes neurons fire? What is it in the incoming pulses that elicits a response in one
neuron but not in another? Why can two neurons have different responses to exactly
the same input and identical responses to completely different inputs? To answer thesequestions, we need to understand the dynamics of spike generation mechanisms ofneurons.
Most introductory neuroscience books describe neurons as integrators with a threshold: neurons sum incoming PSPs and “compare” the integrated PSP with a certain
voltage value, called the firing threshold. If it is below the threshold, the neuron remains quiescent; when it is above the threshold, the neuron fires an all-or-none spike,as in Fig.1.3, and resets its membrane potential. To add theoretical plausibility to thisargument, the books refer to the Hodgkin-Huxley model of spike generation in squidIntroduction

3

Figure 1.2: What makes a neuron fire?
giant axons, which we study in chapter 2. The irony is that the Hodgkin-Huxley model
does not have a well-defined threshold; it does not fire all-or-none spikes; and it is not
an integrator, but a resonator (i.e., it prefers inputs having certain frequencies that
resonate with the frequency of subthreshold oscillations of the neuron). We consider
these and other properties in detail in this book.

1.1.2

Where Is the Threshold?

Much effort has been spent trying to experimentally determine the firing thresholdsof neurons. Here, we challenge the classical view of a threshold. Let us consider two
typical experiments, depicted in Fig.1.4, that are designed to measure the threshold.
in Fig.1.4a, we shock a cortical neuron (i.e., we inject brief but strong pulses of currentof various amplitudes to depolarize the membrane potential to various values). Is therea clear-cut voltage value, as in Fig.1.3, above which the neuron fires but below which
no spikes occur? If you find one, let the author know! In Fig.1.4b we inject long but
weak pulses of current of various amplitudes, which results in slow depolarization anda spike. The firing threshold, if it exists, must be somewhere in the shaded region, butwhere? Where does the slow depolarization end and the spike start? Is it meaningful
to talk about firing thresholds at all?
all-or-none
spikesthresholdrestingno spike

Figure 1.3: The concept of a firing threshold.

4Introduction
spikes(a)(b)

spikes cut

threshold?

injected pulses of current

1 ms

20 mV

-40 mVsubthreshold response

15 ms

injected pulses of current

Figure 1.4: Where is the firing threshold? Shown are in vitro recordings of two layer 5
rat pyramidal neurons. Notice the differences of voltage and time scales.
(a)

(b)
20 mV

20 mV
5 ms

100 ms-60 mV

Figure 1.5: Where is the rheobase (i.e., the minimal current that fires the cell)? (a)in vitro recordings of the pyramidal neuron of layer 2/3 of a rat’s visual cortex show
increasing latencies as the amplitude of the injected current decreases. (b) Simulationof the INa,p +IK – model (pronounced: persistent sodium plus potassium model) shows
spikes of graded amplitude.
Perhaps, we should measure current thresholds instead of voltage thresholds. The
current threshold (i.e., the minimal amplitude of injected current of infinite durationneeded to fire a neuron) is called the rheobase. In Fig.1.5 we decrease the amplitudes
of injected pulses of current to find the minimal one that still elicits a spike or the
maximal one that does not. In Fig.1.5a, progressively weaker pulses result in longer
latencies to the first spike. Eventually the neuron does not fire because the latency islonger than the duration of the pulse, which is 1 second in the figure. Did we reallymeasure the neuronal rheobase? What if we waited a bit longer? How long is longenough? In Fig.1.5b the latencies do not grow, but the spike amplitudes decrease until
the spikes do not look like spikes at all. To determine the current threshold, we needto draw the line and separate spike responses from “subthreshold” ones. How can wedo that if the spikes are not all-or-none? Is the response denoted by the dashed line a
spike?
Risking adding more confusion to the notion of a threshold, consider the following. If excitatory inputs depolarize the membrane potential (i.e., bring it closer toIntroduction

5

10 mV
10 ms

Figure 1.6: In vitro recording of rebound spikesof a rat’s brainstem mesV neuron in response to abrief hyperpolarizing pulse of current.

-45 mV

0 pA-100 pA

10ms
5ms
non-resonant burst

10ms
resonant burst

15ms
non-resonant burst

inhibitory burst

Figure 1.7: Resonant response of the mesencephalic V neuron of a rat’s brainstem topulses of injected current having a 10 ms period (in vitro).
the “firing threshold”), and inhibitory inputs hyperpolarize the potential and move itaway from the threshold, then how can the neuron in Fig.1.6 fire in response to theinhibitory input? This phenomenon, also observed in the Hodgkin-Huxley model, iscalled anodal break excitation, rebound spike, or postinhibitory spike.
Many biologists say that rebound responses are due to the activation and inactivation of certain
slow currents, which bring the membrane potential over the threshold or, equivalently,
lower the threshold upon release from the hyperpolarization – a phenomenon called alow-threshold spike in thalamocortical neurons. The problem with this explanation isthat neither the Hodgkin-Huxley model nor the neuron in Fig.1.6 has these currents,and even if they did, the hyperpolarization is too short and too weak to affect thecurrents.
Another interesting phenomenon is depicted in Fig.1.7. The neuron is stimulatedwith brief pulses of current mimicking an incoming burst of three spikes. When thestimulation frequency is high (5 ms period), presumably reflecting a strong input,
the neuron does not fire at all. However, stimulation with a lower frequency (10ms period) that resonates with the frequency of subthreshold oscillation of the neuron
evokes a spike response, regardless of whether the stimulation is excitatory or inhibitory.
Stimulation with even lower frequency (15 ms period) cannot elicit spike response again.
Thus, the neuron is sensitive only to the inputs having resonant frequency. The same
pulses applied to a cortical pyramidal neuron evoke a response only in the first case(small period), but not in the other cases.

6

1.1.3

Introduction

Why Are Neurons Different, and Why Do We Care?

Why would two neurons respond completely differently to the same input? A biologist
would say that the response of a neuron depends on many factors, such as the typeof voltage- and Ca2+ -gated channels expressed by the neuron, the morphology of its
dendritic tree, the location of the input, and other factors. These factors are indeed
important, but they do not determine the neuronal response per se. Rather they
determine the rules that govern dynamics of the neuron. Different conductances andcurrents can result in the same rules, and hence in the same responses; conversely,
similar currents can result in different rules and in different responses. The currents
define what kind of dynamical system the neuron is.
We study ionic transmembrane currents in chapter 2. In subsequent chapters weinvestigate how the types of currents determine neuronal dynamics. We divide all currents into two major classes: amplifying and resonant, with the persistent Na+ current
INa,p and the persistent K+ current IK being the typical examples of the former andthe latter, respectively. Since there are tens of known currents, purely combinatorialargument implies that there are millions of different electrophysiological mechanismsof spike generation. We will show later that any such mechanism must have at least
one amplifying and one resonant current. Some mechanisms, called minimal in thisbook, have one resonant and one amplifying current. They provide an invaluable tool
in classifying and understanding the electrophysiology of spike generation.
Many illustrations in this book are based on simulations of the reduced INa,p + IK model (pronounced persistent sodium plus potassium model), which consists of a fastpersistent Na+ (amplifying) current and a slower persistent K+ (resonant) current. It
is equivalent to the famous and widely used Morris-Lecar ICa +IK -model (Morris andLecar 1981). We show that the model exhibits quite different dynamics, depending on
the values of the parameters, e.g., the half-activation voltage of the K+ current: in one
case, it can fire in a narrow frequency range, it can exhibit coexistence of resting andspiking states, and it has damped subthreshold oscillations of membrane potential. In
another case, it can fire in a wide frequency range and show no coexistence of restingand spiking and no subthreshold oscillations. Thus, seemingly inessential differencesin parameter values could result in drastically distinct behaviors.

1.1.4

Building Models

To build a good model of a neuron, electrophysiologists apply different pharmacological blockers to tease out the currents that the neuron has. Then they apply different
stimulation protocols to measure the kinetic parameters of the currents, such as theBoltzmann activation function, time constants, and maximal conductances. We consider all these functions in chapter 2. Next, they create a Hodgkin-Huxley-type modeland simulate it using the NEURON, GENESIS, or XPP environment or MATLAB (thefirst two are invaluable tools for simulating realistic dendritic structures).
The problem is that the parameters are measured in different neurons and then put
together into a single model. As an illustration, consider two neurons having the sameIntroduction

7

Figure 1.8: Neurons are dynamical systems.

currents, say INa,p and IK , and exhibiting excitable behavior; that is, both neurons arequiescent but can fire a spike in response to a stimulation. Suppose the second neuron
has stronger INa,p , which is balanced by stronger IK . If we measure Na+ conductanceusing the first neuron and K+ conductance using the second neuron, the resulting
INa,p + IK -model will have an excess of K+ current and probably will not be able to fire
spikes at all. Conversely, if we measure Na+ and K+ conductances using the second
neuron and then the first neuron, respectively, the model would have too much Na+
current and probably would exhibit sustained pacemaking activity. In any case, themodel fails to reproduce the excitable behavior of the neurons whose parameters wemeasured.
Some of the parameters cannot be measured at all, so many arbitrary choices are
made via a process called “fine-tuning”. Navigating in the dark, possibly with the helpof some biological intuition, the researcher modifies parameters, compares simulations
with experiment, and repeats this trial-and-error procedure until he or she is satisfied
with the results. Since seemingly similar values of parameters can result in drasticallydifferent behaviors, and quite different parameters can result in seemingly similar behaviors, how do we know that the resulting model is correct? How do we know that its
behavior is equivalent to that of the neuron we want to study? And what is equivalent
in this case? Now, you are primed to consider dynamical systems. If not, see Fig.1.8.

8

Introduction

1.2

Dynamical Systems

In chapter 2 we introduce the Hodgkin-Huxley formalism to describe neuronal dynamics
in terms of activation and inactivation of voltage-gated conductances. An importantresult of the Hodgkin-Huxley studies is that neurons are dynamical systems, so theyshould be studied as such. Below we mention some of the important concepts ofdynamical systems theory. The reader does not have to follow all the details of thissection because the concepts are explained in greater detail in subsequent chapters.
A dynamical system consists of a set of variables that describe its state and a
law that describes the evolution of the state variables with time (i.e., how the stateof the system in the next moment of time depends on the input and its state in theprevious moment of time). The Hodgkin-Huxley model is a four-dimensional dynamicalsystem because its state is uniquely determined by the membrane potential, V , and socalled gating variables n, m, and h for persistent K+ and transient Na+ currents. The
evolution law is given by a four-dimensional system of ordinary differential equations.
Typically, all variables describing neuronal dynamics can be classified into fourclasses, according to their function and the time scale.
1. Membrane potential.
2. Excitation variables, such as activation of a Na+ current. These variables are
responsible for the upstroke of the spike.
3. Recovery variables, such as inactivation of a Na+ current and activation of a fast
K+ current. These variables are responsible for the repolarization (downstroke)of the spike.
4. Adaptation variables, such as activation of slow voltage- or Ca2+ -dependent currents. These variables build up during prolonged spiking and can affect excitability in the long run.
The Hodgkin-Huxley model does not have variables of the fourth type, but manyneuronal models do, especially those exhibiting bursting dynamics.

1.2.1

Phase Portraits

The power of the dynamical systems approach to neuroscience, as well as to manyother sciences, is that we can tell something, or many things, about a system without
knowing all the details that govern the system evolution. We do not even use equations
to do that! Some may even wonder why we call it a mathematical theory.
As a start, let us consider a quiescent neuron whose membrane potential is resting. From the dynamical systems point of view, there are no changes of the statevariables of such a neuron; hence it is at an equilibrium point. All the inward currents
that depolarize the neuron are balanced, or equilibrated, by the outward currents that
hyperpolarize it. If the neuron remains quiescent despite small disturbances and membrane noise, as in Fig.1.9a (top), then we conclude that the equilibrium is stable. Isn’tmembrane potential, V(t)Introduction

9(a) resting(b) excitable
spike

PSP

PSP
time, tstimulus
K+ activation gate, n(c) periodic spiking

A B
stimulispikeequilibriumPSP

periodicorbit

ABmembrane potential, V

Figure 1.9: Resting, excitable, and periodic spiking activity correspond to a stableequilibrium (a and b) or limit cycle (c), respectively.
it amazing that we can reach such a conclusion without knowing the equations that
describe the neuron’s dynamics? We do not even know the number of variables needed
to describe the neuron; it could be infinite, for all we care.
In this book we introduce the notions of equilibria, stability, threshold, and attraction domains using one- and two-dimensional dynamical systems, e.g., the INa,p +IK model with instantaneous Na+ kinetics. The state of this model is described by themembrane potential, V , and the activation variable, n, of the persistent K+ current, so
it is a two-dimensional vector (V, n). Instantaneous activation of the Na+ current is afunction of V , so it does not result in a separate variable of the model. The evolutionof the model is a trajectory (V (t), n(t)) on the V × n – plane. Depending on the initialpoint, the system can have many trajectories, such as those depicted in Fig.1.9a (bottom). Time is not explicitly present in the figure, but units of time may be thoughtof as plotted along each trajectory. All of the trajectories in the figure are attracted
to the stable equilibrium denoted by the black dot, called an attractor. The overall
qualitative description of dynamics can be obtained through the study of the phase
portrait of the system, which depicts certain special trajectories (equilibria, separatrices, limit cycles) that determine the topological behavior of all the other trajectories in
the phase space. Probably 50 percent of illustrations in this book are phase portraits.
A fundamental property of neurons is excitability, illustrated in Fig.1.9b. The neuron is resting, i.e., its phase portrait has a stable equilibrium. Small perturbations,such as A, result in small excursions from the equilibrium, denoted as PSP (postsynaptic potential). Larger perturbations, such as B, are amplified by the neuron’s intrinsic
dynamics and result in the spike response. To understand the dynamic mechanism ofsuch amplification, we need to consider the geometry of the phase portrait near the
resting equilibrium, i.e., in the region where the decision to fire or not to fire is made.

10

Introduction
spiking
mode

resting mode

Figure 1.10: Rhythmic transitions between resting and spiking modes result in bursting
behavior.

layer 5 pyramidal cellbrainstem mesV celltransitiontransition
20 mV-50 mV-60 mV
3000 pA

200 pA
0 pA

0 pA
500 ms

500 msFigure 1.11: As the magnitude of the injected current slowly increases, the neurons
bifurcate from resting (equilibrium) mode to tonic spiking (limit cycle) mode.

If we inject a sufficiently strong current into the neuron, we bring it to a pacemaking
mode, so that it exhibits periodic spiking activity, as in Fig.1.9c. From the dynamicalsystems point of view, the state of such a neuron has a stable limit cycle, also known as
a stable periodic orbit. The electrophysiological details of the neuron (i.e., the number
and the type of currents it has, their kinetics, etc.) determine only the location, theshape, and the period of the limit cycle. As long as the limit cycle exists, the neuron
can have periodic spiking activity. Of course, equilibria and limit cycles can coexist,
so a neuron can be switched from one mode to another by a transient input. The
famous example is the permanent extinguishing of ongoing spiking activity in the squid
giant axon by a brief transient depolarizing pulse of current applied at a proper phase(Guttman et al. 1980) – a phenomenon predicted by John Rinzel (1978) purely on
the basis of theoretical analysis of the Hodgkin-Huxley model. The transition between
resting and spiking modes could be triggered by intrinsic slow conductances, resulting
in the bursting behavior in Fig.1.10.

Introduction

1.2.2

11

Bifurcations

Now suppose that the magnitude of the injected current is a parameter that we cancontrol, e.g., we can ramp it up, as in Fig.1.11. Each cell in the figure is quiescentat the beginning of the ramps, so its phase portrait has a stable equilibrium and it
may look like the one in Fig.1.9a or Fig.1.9b. Then it starts to fire tonic spikes, so
its phase portrait has a limit cycle attractor and it may look like the one in Fig.1.9c,
with a white circle denoting an unstable resting equilibrium. Apparently there is someintermediate level of injected current that corresponds to the transition from resting
to sustained spiking, i.e., from the phase portrait in Fig.1.9b to Fig.1.9c. What does
the transition look like?
From the dynamical systems point of view, the transition corresponds to a bifurcation of neuron dynamics, i.e., a qualitative change of phase portrait of the system. For
example, there is no bifurcation going from the phase portrait in Fig.1.9a to that inFig.1.9b, since both have one globally stable equilibrium; the difference in behavior isquantitative but not qualitative. In contrast, there is a bifurcation going from Fig.1.9b
to Fig.1.9c, since the equilibrium is no longer stable and another attractor, limit cycle,
has appeared. The neuron is not excitable in Fig.1.9a but it is in Fig.1.9b, simply
because the former phase portrait is far from the bifurcation and the latter is near.
In general, neurons are excitable because they are near bifurcations from restingto spiking activity, so the type of the bifurcation determines the excitable propertiesof the neuron. Of course, the type depends on the neuron’s electrophysiology. An
amazing observation is that there could be millions of different electrophysiologicalmechanisms of excitability and spiking, but there are only four – yes, four – different
types of bifurcations of equilibrium that a system can undergo without any additional
constraints, such as symmetry. Thus, considering these four bifurcations in a general
setup, we can understand excitable properties of many models, even those that have notbeen invented yet. What is even more amazing, we can understand excitable propertiesof neurons whose currents are not measured and whose models are not known, provided
we can experimentally identify which of the four bifurcations the resting state of theneuron undergoes.
The four bifurcations are summarized in Fig.1.12, which plots the phase portrait
before (left), at (center), and after (right) a particular bifurcation occurs. Mathematicians refer to these bifurcations as being of codimension-1 because we need to vary only
one parameter, e.g., the magnitude of the injected DC current I, to observe the bifurcations reliably in simulations or experiments. There are many more codimension-2, 3,(etc.), bifurcations, but they need special conditions to be observed. We discuss thesein chapter 6.
Let us consider the four bifurcations and their phase portraits in Fig.1.12. The
horizontal and vertical axes are the membrane potential with instantaneous activation
variable and a recovery variable, respectively. At this stage, the reader is not required
to fully understand the intricacies of the phase portraits in the figure, since they will
be explained systematically in later chapters.

12

Introduction

node

recovery

spik
ing
lim

(a)

cle
cy
it

potential

saddle-nodesaddlesaddle-node bifurcation

in

(b)

node

saddle

ircle
nt c
ria
a
v

saddle-nodesaddle-node on invariant circle (SNIC) bifurcation

ble
sta

ctor
ttra

limit cycle
king
a
spi
un

(c)

subcritical Andronov-Hopf bifurcation(d)supercritical Andronov-Hopf bifurcation

Figure 1.12: Four generic (codimension-1) bifurcations of an equilibrium state leading
to the transition from resting to periodic spiking behavior in neurons.

Introduction

13• Saddle-node bifurcation. As the magnitude of the injected current or any otherbifurcation parameter changes, a stable equilibrium corresponding to the restingstate (black circle marked “node” in Fig.1.12a) is approached by an unstableequilibrium (white circle marked “saddle”); they coalesce and annihilate eachother, as in Fig.1.12a (middle). Since the resting state no longer exists, the trajectory describing the evolution of the system jumps to the limit cycle attractor,
indicating that the neuron starts to fire tonic spikes. Notice that the limit cycle, or some other attractor, must coexist with the resting state in order for thetransition resting → spiking to occur.
• Saddle-node on invariant circle bifurcation is similar to the saddle-node bifurcation except that there is an invariant circle at the moment of bifurcation, whichthen becomes a limit cycle attractor, as in Fig.1.12b.
• Subcritical Andronov-Hopf bifurcation. A small unstable limit cycle shrinks toa stable equilibrium and makes it lose stability, as in Fig.1.12c. Because of
instabilities, the trajectory diverges from the equilibrium and approaches a largeamplitude spiking limit cycle or some other attractor.
• Supercritical Andronov-Hopf bifurcation. The stable equilibrium loses stability
and gives birth to a small-amplitude limit cycle attractor, as in Fig.1.12d. As
the magnitude of the injected current increases, the amplitude of the limit cycle
increases and it becomes a full-size spiking limit cycle.
Notice that there is a coexistence of resting and spiking states in the case of saddlenode and subcritical Andronov-Hopf bifurcations, but not in the other two cases. Sucha coexistence reveals itself via a hysteresis behavior when the injected current slowly
increases and then decreases past the bifurcation value, because the transitions “resting→ spiking” and “spiking → resting” occur at different values of the current. In addition,brief stimuli applied at the appropriate times can switch the activity from spiking to
resting and back. There are also spontaneous noise-induced transitions between thetwo modes that result in the stuttering spiking that, for instance, is exhibited by
the so-called fast spiking (FS) cortical interneurons when they are kept close to thebifurcation (Tateno et al. 2004). Some bistable neurons have a slow adaptation current
that activates during the spiking mode and impedes spiking, often resulting in bursting
activity.
Systems undergoing Andronov-Hopf bifurcations, whether subcritical or supercritical, exhibit damped oscillations of membrane potential, whereas systems near saddlenode bifurcations, whether on or off an invariant circle, do not. The existence ofsmall amplitude oscillations creates the possibility of resonance to the frequency of theincoming pulses, as in Fig.1.7, and other interesting features.
We refer to neurons with damped subthreshold oscillations as resonators and to
those that do not have this property as integrators. We refer to the neurons that exhibit the coexistence of resting and spiking states, at least near the transition from

14

Introduction

NO(monostable)

saddle-nodesaddle-node on
invariant circle

supercritical
Andronov-Hopf

Class 1 excitability

40

30

20

F-I curve
10

0

0

100

200

300

injected dc-current, I (pA)

Figure 1.13: Classification of neurons intomonostable/bistable integrators/resonators
according to the bifurcation of the restingstate in Fig.1.12.
asymptotic firing frequency, Hz

YES(bistable)

subcritical
Andronov-Hopf

asymptotic firing frequency, Hz

YESNO(resonator) (integrator)

subthreshold oscillations

coexistence of resting and spiking states

Class 2 excitability

250
200

F-I curve

150
100
50
0

0

500

1000

1500

injected dc-current, I (pA)

Figure 1.14: Frequency-current (F-I) curves of cortical pyramidal neuron and brainstemmesV neuron from Fig.7.3. These are the same neurons used in the ramp experimentin Fig.1.11.
resting to spiking, as bistable, and to those that do not, monostable. The four bifurcations in Fig.1.12 are uniquely defined by these two features. For example, a bistable
resonator is a neuron undergoing subcritical Andronov-Hopf bifurcation, and a monostable integrator is a neuron undergoing saddle-node on invariant circle bifurcation (seeFig.1.13). Cortical fast spiking (FS) and regular spiking (RS) neurons, studied inchapter 8, are typical examples of the former and the latter, respectively.

1.2.3

Hodgkin Classification

Hodgkin (1948) was the first to study bifurcations in neuronal dynamics, years before
the mathematical theory of bifurcations was developed. He stimulated squid axons
with pulses of various amplitudes and identified three classes of responses:• Class 1 neural excitability. Action potentials can be generated with arbitrarily
low frequency, depending on the strength of the applied current.
• Class 2 neural excitability. Action potentials are generated in a certain frequency
band that is relatively insensitive to changes in the strength of the applied current.

Introduction

15• Class 3 neural excitability. A single action potential is generated in response to apulse of current. Repetitive (tonic) spiking can be generated only for extremely
strong injected currents or not at all.
The qualitative distinction between the classes is that the frequency-current relation(the F-I curve in Fig.1.14) starts from zero and continuously increases for Class 1
neurons, is discontinuous for Class 2 neurons, and is not defined at all for Class 3
neurons.
Obviously, neurons belonging to different classes have different neurocomputational
properties. Class 1 neurons, which include cortical excitatory pyramidal neurons,
can smoothly encode the strength of the input into the output firing frequency, as
in Fig.1.11 (left). In contrast, Class 2 neurons, such as fast-spiking (FS) cortical inhibitory interneurons, cannot do that; instead, they fire in a relatively narrow frequency
band, as in Fig.1.11 (right). Class 3 neurons cannot exhibit sustained spiking activity, so Hodgkin regarded them as “sick” or “unhealthy”. There are other distinctions
between the classes, which we discuss later.
Different classes of excitability occur because neurons have different bifurcationsof resting and spiking states – a phenomenon first explained by Rinzel and Ermentrout (1989). If ramps of current are injected to measure the F-I curves, then Class
1 excitability occurs when the neuron undergoes the saddle-node bifurcation on an
invariant circle depicted in Fig.1.12b. Indeed, the period of the limit cycle attractor
is infinite at the bifurcation point, and then it decreases as the bifurcation parameter– say, the magnitude of the injected current – increases. The other three bifurcations
result in Class 2 excitability. Indeed, the limit cycle attractor exists and has a finite
period when the resting state in Fig.1.12 undergoes a subcritical Andronov-Hopf bifurcation, so emerging spiking has a nonzero frequency. The period of the small limit
cycle attractor appearing via supercritical Andronov-Hopf bifurcation is also finite, so
the frequency of oscillations is nonzero, but their amplitudes are small. In contrast
to the common and erroneous folklore, the saddle-node bifurcation (off-limit cycle)
also results in Class 2 excitability because the limit cycle has a finite period at thebifurcation. There is a considerable latency (delay) to the first spike in this case,but the subsequent spiking has nonzero frequency. Thus, the simple scheme “Class 1= saddle-node, Class 2 = Hopf” that permeates many publications is unfortunately
incorrect.
When pulses of current are used to measure the F-I curve, as in Hodgkin’s experiments, the firing frequency depends on factors besides the type of the bifurcation of
the resting state. In particular, low-frequency firing can be observed in systems near
Andronov-Hopf bifurcations, as we show in chapter 7. To avoid possible confusion, we
define the class of excitability only on the basis of slow ramp experiments.
Hodgkin’s classification has an important historical value, but it is of little use forthe dynamic description of a neuron, since naming a class of excitability of a neuron
does not tell much about the bifurcations of the resting state. Indeed, it says onlythat saddle-node on invariant circle bifurcation (Class 1) is different from the otherthree bifurcations (Class 2), and only when ramps are injected. Dividing neurons into

16

Introduction

integrators and resonators with bistable or monostable activity is more informative,
so we adopt the classification in Fig.1.13 in this book. In this classification, a Class 1
neuron is a monostable integrator, whereas a Class 2 neuron can be a bistable integrator
or a resonator.

1.2.4

Neurocomputational properties

Using the same arrangement as in Fig.1.13, we depict typical geometry of phase
portraits near the four bifurcations in Fig.1.15. Let us use the portraits to explain
what happens “near the threshold”, i.e., near the place where the decision to fire or
not to fire is made. To simplify our geometrical analysis, we assume here that neurons
receive shock inputs, i.e., brief but strong pulses of current that do not change thephase portraits, but only push or reset the state of the neuron into various regions of
the phase space. We consider these and other cases in detail in chapter 7.
The horizontal axis in each plot in Fig.1.15 corresponds to the membrane potential
V with instantaneous Na+ current, and the vertical axis corresponds to a recovery variable, say activation of K+ current. Black circles denote stable equilibria corresponding
to the neuronal resting state. Spiking limit cycle attractors correspond to sustained
spiking states, which exist in the two cases depicted in the left half of the figure corresponding to the bistable dynamics. The limit cycles are surrounded by shaded regions– their attraction domains. The white region is the attraction domain of the equilibrium. To initiate spiking, the external input should push the state of the system into
the shaded region, and to extinguish spiking, the input should push the state back into
the white region.
There are no limit cycles in the two cases depicted in the right half of the figure,so the entire phase space is the attraction domain of the stable equilibrium, and the
dynamics are monostable. However, if the trajectory starts in the shaded region, it
makes a large-amplitude rotation before returning to the equilibrium – a transient
spike. Apparently, to elicit such a spike, the input should push the state of the system
into the shaded region.
Now let us contrast the upper and lower halves of the figure, corresponding tointegrators and resonators, respectively. We distinguish these two modes of operationon the basis of the existence of subthreshold oscillations near the equilibrium.
First, let us show that inhibition impedes spiking in integrators, but can promote itin resonators. In the integrator, the shaded region is in the depolarized voltage range,i.e., to the right of the equilibrium. Excitatory inputs push the state of the system
toward the shaded region, while inhibitory inputs push it away. In resonators, both
excitation and inhibition push the state toward the shaded region, because the region
wraps around the equilibrium and can be reached along any direction. This explains
the rebound spiking phenomenon depicted in Fig.1.6.
Integrators have all-or-none spikes; resonators may not. Indeed, any trajectory
starting in the shaded region in the upper half of Fig.1.15 has to rotate around thewhite circle at the top that corresponds to an unstable equilibrium. Moreover, theIntroduction

17

ctor
tra
at

g

je c
tra

to r

y

thresh
old

thre
sho
ld
spikin
g lim
it c
yc
le

potential

saddle-node on invariant circle bifurcation

sp
iki
n

recoverysaddle-node bifurcation

inh

exc

2

1

subcritical Andronov-Hopf bifurcation

supercritical Andronov-Hopf bifurcation

tor
rac
att
e
l
yc
tc

half-amplitude
spike

spik
ing
l

im
i

half-amplitude
spike
spike

spike
PSP
1

PSP

inh

2

3

exc

Figure 1.15: The geometry of phase portraits of excitable systems near four bifurcations
can explain many neurocomputational properties (see section 1.2.4 for details).

state of the system is quickly attracted to the spiking trajectory and moves along that
trajectory, thereby generating a stereotypical spike. A resonator neuron also can firelarge amplitude spikes when its state is pushed to or beyond the trajectory denoted“spike”. Such neurons generate subthreshold responses when the state slides along
the smaller trajectory denoted PSP; they also can generate spikes of an intermediate
amplitude when the state is pushed between the PSP and “spike” trajectories, which
explains the partial-amplitude spiking in Fig.1.5b or in the squid axon in Fig.7.26. The
set of initial conditions corresponding to such spiking is quite small, so typical spikes
have large amplitudes and partial spikes are rare.
Integrators have well-defined thresholds; resonators may not. The white circles near
the resting states of integrators in Fig.1.15 are called saddles. They are stable along thevertical direction and unstable along the horizontal direction. The two trajectories

18

Introductionthat lead to the saddle along the vertical direction are called separatrices because
they separate the phase space into two regions – in this case, white and shaded. The
separatrices play the role of thresholds since only those perturbations that push thestate of the system beyond them result in a spike. The closer the state of the system
is to the separatrices, the longer it takes to converge and then diverge from the saddle,
resulting in a long latency to the spike. Notice that the threshold is not a point, but
a tilted curve that spans a range of voltage values.
Resonators have a well-defined threshold in the case of subcritical Andronov-Hopfbifurcation: it is the small unstable limit cycle that separates the attraction domainsof stable equilibrium and spiking limit cycle. Trajectories inside the small cycle spiral toward the stable equilibrium, whereas trajectories outside the cycle spiral awayfrom it and eventually lead to sustained spiking activity. When a neuronal model isfar from the subcritical Andronov-Hopf bifurcation, its phase portrait may look similar to the one corresponding to the supercritical Andronov-Hopf bifurcation. The
narrow shaded band in the figure is not a threshold manifold but a fuzzy threshold set called “quasi-threshold” by FitzHugh (1955). Many resonators, including the
Hodgkin-Huxley model, have quasi-thresholds instead of thresholds. The width of thequasi-threshold in the Hodgkin-Huxley model is so narrow that for all practical reasons
it may be assumed to be just a curve.
Integrators integrate, resonators resonate. Now consider inputs consisting of multiple pulses, e.g., a burst of spikes. Integrators prefer high-frequency inputs; the higher
the frequency, the sooner they fire. Indeed, the first spike of such an input, marked“1” in the top-right phase portrait in Fig.1.15, increases the membrane potential andshifts the state to the right, toward the threshold. Since the state of the system isstill in the white area, it slowly converges back to the stable equilibrium. To crossthe threshold manifold, the second pulse must arrive shortly after the first one. The
reaction of a resonator to a pair of pulses is quite different. The first pulse initiates adamped subthreshold oscillation of the membrane potential, which looks like a spiralin the bottom-right phase portrait in Fig.1.15. The effect of the second pulse depends
on its timing. If it arrives after the trajectory makes half a rotation, marked “2” in the
figure, it cancels the effect of the first pulse. If it arrives after the trajectory makes afull rotation, marked “3” in the figure, it adds to the first pulse and either increases theamplitude of subthreshold oscillation or evokes a spike response. Thus, the responseof the resonator neuron depends on the frequency content of the input, as in Fig.1.7.
Integrators and resonators constitute two major modes of activity of neurons. Most
cortical pyramidal neurons, including the regular spiking (RS), intrinsically bursting(IB), and chattering (CH) types considered in Chap. 8, are integrators. So are thalamocortical neurons in the relay mode of firing, and neostriatal spiny projection neurons.
Most cortical inhibitory interneurons, including the FS type, are resonators. So arebrainstem mesencephalic V neurons and stellate neurons of the entorhinal cortex. Somecortical pyramidal neurons and low-threshold spiking (LTS) interneurons can be at theborder of transition between integrator and resonator modes. Such a transition corresponds to another bifurcation, which has codimension-2, and hence it is less likely toIntroduction

19

spike

Figure 1.16: Phase portrait of a system near a
Bogdanov-Takens bifurcation that corresponds tothe transition from integrator to resonator mode.
be encountered experimentally. We consider this and other uncommon bifurcations indetail later. The phase portrait near the bifurcation is depicted in Fig.1.16, and it is agood exercise for the reader to explain why such a system has damped oscillations andpostinhibitory responses, yet a well-defined threshold, all-or-none spikes, and possibly
long latencies.
Of course, figures 1.15 and 1.16 cannot encompass all the richness of neuronal behavior, otherwise this book would be only 19pages long (this book is actually quite short;most of the space is taken by figures, exercises, and solutions). Many aspects of neuronal dynamics depend on other bifurcations, e.g., those corresponding to appearance
and disappearance of spiking limit cycles. These bifurcations describe the transitions
from spiking to resting, and they are especially important when we consider bursting
activity. In addition, we need to take into account the relative geometry of equilibria,
limit cycles, and other relevant trajectories, and how they depend on the parameters of
the system, such as maximal conductances, and activation time constants. We explore
all these issues systematically in subsequent chapters.
In chapter 2 we review some of the most fundamental concepts of neuron electrophysiology, culminating with the Hodgkin-Huxley model. This chapter is aimed
at mathematicians learning neuroscience. In chapters 3 and 4 we use one- and twodimensional neuronal models, respectively, to review some of the most fundamental
concepts of dynamical systems, such as equilibria, limit cycles, stability, attraction
domain, nullclines, phase portrait, and bifurcation. The material in these chapters,
aimed at biologists learning the language of dynamical systems, is presented with theemphasis on geometrical rather than mathematical intuition. In fact, the spirit of
the entire book is to explain concepts by using pictures, not equations. Chapter 5
explores phase portraits of various conductance-based models and the relations between ionic currents and dynamic behavior. In Chapter 6 we use the INa,p +IK -model
to systematically introduce the geometric bifurcation theory. Chapter 7, probably themost important chapter of the book, applies the theory to explain many computational
properties of neurons. In fact, all the material in the previous chapters is given so thatthe reader can understand this chapter. In chapter 8 we use a simple phenomenologicalmodel to simulate many cortical, hippocampal, and thalamic neurons. This chapter

20

Introduction

contains probably the most comprehensive up-to-date review of various firing patterns
exhibited by mammalian neurons. In chapter 9 we introduce the electrophysiologicaland topological classification of bursting dynamics, as well as some useful methods tostudy the bursters. Finally, the last and the most mathematically advanced chapterof the book, Chap. 10, deals with coupled neurons. There we show how the details of
the spike generation mechanism of neurons affect neurons’ collective properties, suchas synchronization.

1.2.5

Building Models (Revisited)

To have a good model of a neuron, it is not enough to put the right kind of currents
together and tune the parameters so that the model can fire spikes. It is not evenenough to reproduce the right input resistance, rheobase, and firing frequencies. The
model has to reproduce all the neurocomputational features of the neuron, starting withthe coexistence of resting and spiking states, spike latencies, subthreshold oscillations,and rebound spikes, among others.
A good way to start is to determine what kind of bifurcations the neuron underconsideration undergoes and how the bifurcations depend on neuromodulators andpharmacological blockers. Instead of or in addition to measuring neuronal responses
to get the kinetic parameters, we need to measure them to get the right bifurcation
behavior. Only in this case we can be sure that the behavior of the model is equivalent
to that of the neuron, even if we omitted a current or guessed some of the parameters
incorrectly.
Implementation of this research program is still a pipe dream. The people who
understand the mathematical aspects of neuron dynamics – those who see beyond
conductances and currents – usually do not have the opportunity to do experiments.
Conversely, those who study neurons in vitro or in vivo on a daily basis – those who see
spiking, bursting, and oscillations; those who can manipulate the experimental setup
to test practically any aspect of neuronal activity – do not usually see the value ofstudying phase portraits, bifurcations, and nonlinear dynamics in general. One of thegoals of this book is to change this state and bring these two groups of people closer
together.

Introduction

21

Review of Important Concepts• Neurons are dynamical systems.
• The resting state of neurons corresponds to a stable equilibrium; thetonic spiking state corresponds to a limit cycle attractor.
• Neurons are excitable because the equilibrium is near a bifurcation.
• There are many ionic mechanisms of spike generation, but only four
generic bifurcations of equilibria.
• These bifurcations divide neurons into four categories: integrators
or resonators, monostable or bistable.
• Analyses of phase portraits at bifurcations explain why some neurons have well-defined thresholds, all-or-none spikes, postinhibitory
spikes, frequency preference, hysteresis, and so on, while others do
not.
• These features, and not ionic currents per se, determine the neuronal
responses, i.e., the kind of computations neurons do.
• A good neuronal model must reproduce not only electrophysiology
but also the bifurcation dynamics of neurons.

Bibliographical Notes
Richard FitzHugh at the National Institutes of Health (NIH) pioneered the phase planeanalysis of neuronal models with the view to understanding their neurocomputational
properties. He was the first to analyze the Hodgkin-Huxley model (FitzHugh 1955;
years before they received the Nobel Prize) and to prove that it has neither threshold
nor all-or-none spikes. FitzHugh (1961) introduced the simplified model of excitability(see Fig.1.18) and showed that one can get the right kind of neuronal dynamics in models lacking conductances and currents. Nagumo et al. (1962) designed a corresponding
tunnel diode circuit, so the model is called the FitzHugh-Nagumo oscillator. Chapter 8
deals with such simplified models. The history of the development of FitzHugh-Nagumo
model is reviewed by Izhikevich and FitzHugh (2006).
FitzHugh’s research program was further developed by John Rinzel and G. Bard
Ermentrout (see Fig.1.19 and Fig.1.20). In their 1989 seminal paper, Rinzel and Ermentrout revived Hodgkin’s classification of excitability and pointed out the connection
between the behavior of neuronal models and the bifurcations they exhibit. (They also
referred to the excitability as “type I” or “type II”). Unfortunately, many people treat

22

Introduction

Figure 1.17: Richard FitzHugh in 1984.

absolutely refractory

1

i ve
act

r e lati v e ly

cto
refra

lar
i
de

po

erp
ola
hy
p

0.5

ze

d

d

NO MAN'S LAND

riz
e

0

ry

recovery variable, W

0.5

erative
regen

)/.8

ol d
sh
e
r
h
t
self-excitatoryq u a si-

W=

I

(V

+.7

V3 /3+-1

W=V-

resting

-2.5

-2

-1.5

-1

-0.5
0
0.5
membrane potential, V

1

1.5

2

Figure 1.18: Phase portrait and physiological state diagram of FitzHugh-Nagumo
model V̇ = V − V 3 /3 − W + I, Ẇ = 0.08(V + 0.7 − 0.8W ). The meaning of curves
and trajectories is explained in chapter 4. (Reproduced from Izhikevich and FitzHugh(2006) with permission.)

Introduction

23

Figure 1.19: John Rinzel in 2004. Depicted on his T-shirt is the cover of the first issueof Journal of Computational Neuroscience, in which the Pinsky-Rinzel (1994) model
appeared.

Figure 1.20: G. Bard Ermentrout (G. stands for George) with his parrot, Junior, in
1983.

24

Introduction

the connection in a simpleminded fashion and incorrectly identify “type I = saddlenode, type II = Hopf”. If only life were so simple!
The geometrical analysis of neuronal models was further developed by, among others, Izhikevich (2000a), who stressed the integrator and resonator modes of operationand made connections to other neurocomputational properties.
The neuroscience and mathematics parts of this book are standard, though many
connections are new. The literature sources are listed at the end of each chapter.
Among many outstanding books on computational neuroscience, the author especially
recommends Spikes, Decisions, and Actions by Wilson (1999), Biophysics of Computation by Koch (1999), Theoretical Neuroscience by Dayan and Abbott (2001), andFoundations of Cellular Neurophysiology by Johnston and Wu (1995). The present volume complements these excellent books in the sense that it is more ambitious, focused,
and thorough in dealing with neurons as dynamical systems. Though its views may
be biased by the author’s philosophy and taste, the payoffs in understanding neuronal
dynamics are immense, provided the reader has enough patience and perseverance tofollow the author’s line of thought.
The NEURON simulation environment is described by Hines (1989) and Carnevaleand Hines (2006) (http://www.neuron.yale.edu); the GENESIS environment, byBower and Beeman (1995) (http://www.genesis-sim.org); the XPP environment,
by Ermentrout (2002). The author of this book uses MATLAB, which has become astandard computational tool in science and engineering. MATLAB is the registeredtrademark of The MathWorks, Inc. (http://www.mathworks.com).

Chapter 2
Electrophysiology of Neurons
In this chapter we remind the reader of some fundamental concepts of neuronal electrophysiology that are necessary to understand the rest of the book. We start with ionsand currents, and move quickly toward the dynamics of the Hodgkin-Huxley model.
If the reader is already familiar with the Hodgkin-Huxley formalism, this chapter canbe skipped. Our exposition is brief, and it cannot substitute for a good introductory
neuroscience course or the reading of such excellent textbooks as Theoretical Neuroscience by Dayan and Abbott (2001), Foundations of Cellular Neurophysiology by
Johnston and Wu (1995), Biophysics of Computation by Koch (1999), or Ion Channelsof Excitable Membranes by Hille (2001).

2.1

Ions

Electrical activity in neurons is sustained and propagated via ionic currents throughneuron membranes. Most of these transmembrane currents involve one of four ionicspecies: sodium (Na+ ), potassium (K+ ), calcium (Ca2+ ), or chloride (Cl− ). The first
three have a positive charge (cations) and the fourth has a negative charge (anion). The
concentrations of these ions are different on the inside and the outside of a cell, which
creates electrochemical gradients – the major driving forces of neural activity. The
extracellular medium has a high concentration of Na+ and Cl− (salty, like seawater)
and a relatively high concentration of Ca2+ . The intracellular medium has high concentrations of K+ and negatively charged molecules (denoted by A− ), as we illustrate
in Fig.2.1.
The cell membrane has large protein molecules forming channels through whichions (but not A− ) can flow according to their electrochemical gradients. The flow of
Na+ and Ca2+ ions is not significant, at least at rest, but the flow of K+ and Cl− ionsis. This, however, does not eliminate the concentration asymmetry for two reasons.
• Passive redistribution. The impermeable anions A− attract more K+ into the cell
(opposites attract) and repel more Cl− out of the cell, thereby creating concentration gradients.
• Active transport. Ions are pumped in and out of the cell via ionic pumps. For
example, the Na+ -K+ pump depicted in Fig.2.1 pumps out three Na+ ions forevery two K+ ions pumped in, thereby maintaining concentration gradients.
25

26

Electrophysiology of Neurons
OutsideNa+ (145 mM)
K + (5 mM)
Cl - (110 mM)
Ca 2+(2.5-5 mM)
A- (25 mM)

ActiveTransport
Na +Na+ Na +
Pump
K+ K+

InsideEquilibrium Potentials
Na+

Na + (5-15 mM)
K + (140 mM)
Cl - (4 mM)
Ca 2+(0.1 µ M)
A- (147 mM)

62 log 145
15 = 61 mVK+
A-

K + A-

62 log 145= 90 mV
5K+

562 log 140= −90 mVCl−

−62 log 110= −89 mV4

Cl -Passive
Redistribution

Ca2+ 31 log 102.5−4 = 136 mV
31 log 105−4 = 146 mV

Na + Cl -Figure 2.1: Ion concentrations and Nernst equilibrium potentials (2.1) in a typicalmammalian neuron (modified from Johnston and Wu 1995). A− are membraneimpermeant anions. Temperature T = 37◦ C (310◦ K).

2.1.1

Nernst Potential

There are two forces that drive each ion species through the membrane channel: concentration and electric potential gradients. First, the ions diffuse down the concentration
gradient. For example, the K+ ions depicted in Fig.2.2a diffuse out of the cell because
K+ concentration inside is higher than that outside. While exiting the cell, K+ ionscarry a positive charge and leave a net negative charge inside the cell (consisting mostlyof impermeable anions A− ), thereby producing the outward current. The positive andnegative charges accumulate on the opposite sides of the membrane surface, creating
an electric potential gradient across the membrane – transmembrane potential or membrane voltage. This potential slows the diffusion of K+ , since K+ ions are attracted
to the negatively charged interior and repelled from the positively charged exterior of
the membrane, as we illustrate in Fig.2.2b. At some point an equilibrium is achieved:
the concentration gradient and the electric potential gradient exert equal and opposite
forces that counterbalance each other, and the net cross-membrane current is zero, as
in Fig.2.2c. The value of such an equilibrium potential depends on the ionic species,
and it is given by the Nernst equation (Hille 2001):
Eion =

RT [Ion]outln,zF[Ion]in(2.1)

where [Ion]in and [Ion]out are concentrations of the ions inside and outside the cell,respectively; R is the universal gas constant (8, 315 mJ/(K◦ ·Mol)); T is temperature
in degrees Kelvin (K◦ = 273.16+C◦ ); F is Faraday’s constant (96, 480 coulombs/Mol),Electrophysiology of Neurons
InsideOutside27
InsideOutsideNa +Inside

K+

Na +

K+

A-

Cl -

Cl Na+

Na+

A-

A-

Cl -

DiffusionK+

Diffusion

A-

A-

K+

A-

K+
K+

A-

A-

K+
AA-

Na+

Electric
Potential

Na +

K+

A-

K+
Na +

Cl -

A-

A-

K+

Electric
Potential

A-

A-

K+

A-

K+

Na +

K+

Cl -

K+
Na +

K+

K+

Na +

K+

A-

Cl -

Na +

A-

K+

AK+

Cl -

Cl -

K+

A-

Cl -

Na +

K+

A-

Na +

Cl Na +

K+

Diffusion
K+

K+

K+

A-

K+

A-

Cl -

A-

A-

K+

K+
Na +

A-

Cl A-

Cl -

K+

OutsideNa +Cl Na +

Cl -

A-

K+

A-

Na +

A-

Cl -

A-

Cl -

Cl -

K+

a

c

b

Figure 2.2: Diffusion of K+ ions down the concentration gradient though the membrane(a) creates an electric potential force pointing in the opposite direction (b) until the
diffusion and electrical forces counter each other (c). The resulting transmembranepotential (2.1) is referred to as the Nernst equilibrium potential for K+ .
z is the valence of the ion (z = 1 for Na+ and K+ ; z = −1 for Cl− ; and z = 2 for
Ca2+ ). Substituting the numbers, taking log10 instead of natural ln and using bodytemperature T = 310◦ K (37◦ C) results inEion ≈ 62 log[Ion]out[Ion]in(mV)

for monovalent (z = 1) ions. Nernst equilibrium potentials in a typical mammalian
neuron are summarized in Fig.2.1.

2.1.2

Ionic Currents and Conductances

In the rest of the book V denotes the membrane potential and ENa , ECa , EK , and ECldenote the Nernst equilibrium potentials. When the membrane potential equals theequilibrium potential, say EK , the net K+ current, denoted as IK (µA/cm2 ), is zero
(this is the definition of the Nernst equilibrium potential for K+ ). Otherwise, the net
K+ current is proportional to the difference of potentials; that is,
IK = gK (V − EK ) ,where the positive parameter gK (mS/cm2 ) is the K+ conductance and (V − EK ) is theK+ driving force. The other major ionic currents,
INa = gNa (V − ENa ) ,ICa = gCa (V − ECa ) ,ICl = gCl (V − ECl ) ,28Electrophysiology of Neurons
outside

I Na

I Ca

IK

I Cl

g

g

g

g

Na

E Na

Ca

E Ca

K

EK

Cl

E Cl

CV
C

Figure 2.3: Equivalent circuit representation of a patch of cell membrane.

insidecould also be expressed as products of nonlinear conductances and corresponding driving forces. A better description of membrane currents, especially Ca2+ current, isprovided by the Goldman-Hodgkin-Katz equation (Hille 2001), which we do not use inthis book.
When the conductance is constant, the current is said to be Ohmic. In general,ionic currents in neurons are not Ohmic, since the conductances may depend on time,
membrane potential, and pharmacological agents, e.g., neurotransmitters, neuromodulators, second-messengers, etc. It is the time-dependent variation in conductances thatallows a neuron to generate an action potential, or spike.

2.1.3

Equivalent Circuit

It is traditional to represent electrical properties of membranes in terms of equivalentcircuits similar to the one depicted in Fig.2.3. According to Kirchhoff’s law, the total
current, I, flowing across a patch of a cell membrane is the sum of the membrane
capacitive current C V̇ (the capacitance C ≈ 1.0 µF/cm2 in the squid axon) and all theionic currents
I = C V̇ + INa + ICa + IK + ICl ,where V̇ = dV /dt is the derivative of the voltage variable V with respect to time t.
The derivative arises because it takes time to charge the membrane. This is the first
dynamic term in the book! We write this equation in the standard “dynamical system”
form
C V̇ = I − INa − ICa − IK − ICl(2.2)
or
C V̇ = I − gNa (V − ENa ) − gCa (V − ECa ) − gK (V − EK ) − gCl (V − ECl ) .

(2.3)

If there are no additional current sources or sinks, such as synaptic current, axial
current, or tangential current along the membrane surface, or current injected via anElectrophysiology of Neurons

29

electrode, then I = 0. In this case, the membrane potential is typically bounded by
the equilibrium potentials in the order (see Fig.2.4)
EK < ECl < V(at rest) < ENa < ECa ,so that INa , ICa < 0 (inward currents) and IK , ICl > 0 (outward currents). From (2.2)
it follows that inward currents increase the membrane potential, that is, make it morepositive (depolarization), whereas outward currents decrease it, that is, make it morenegative (hyperpolarization). Note that ICl is called an outward current even though
the flow of Cl− ions is inward; the ions bring negative charge inside the membrane,
which is equivalent to positively charged ions leaving the cell, as in IK .

2.1.4

Resting Potential and Input ResistanceIf there were only K+ channels, as in Fig.2.2, the membrane potential would quickly
approach the K+ equilibrium potential, EK , which is around −90 mV. Indeed,
C V̇ = −IK = −gK (V − EK )in this case. However, most membranes contain a diversity of channels. For example,
Na+ channels would produce an inward current and pull the membrane potential toward
the Na+ equilibrium potential, ENa , which could be as large as +90 mV. The value ofthe membrane potential at which all inward and outward currents balance each other sothat the net membrane current is zero corresponds to the resting membrane potential.
It can be found from (2.3) with I = 0, by setting V̇ = 0. The resulting expression,
Vrest =

gNa ENa + gCa ECa + gK EK + gCl ECl
gNa + gCa + gK + gCl(2.4)

has a nice mechanistic interpretation: Vrest is the center of mass of the balance depicted
in Fig.2.4. Incidentally, the entire equation (2.3) can be written in the form
C V̇ = I − ginp (V − Vrest ) ,

(2.5)

where
ginp = gNa + gCa + gK + gCl
is the total membrane conductance, called input conductance. The quantity Rinp =1/ginp is the input resistance of the membrane, and it measures the asymptotic sensitivity of the membrane potential to injected or intrinsic currents. Indeed, from (2.5) it
follows that
V → Vrest + IRinp ,(2.6)
so greater values of Rinp imply greater steady-state displacement of V due to theinjection of DC current I.
A remarkable property of neuronal membranes is that ionic conductances, and hence
the input resistance, are functions of V and time. We can use (2.6) to trace an action

30Electrophysiology of Neurons
Rest
g

g g
K

Na

gCa

Cl

-100

E K ECl-50 mV

0 mV

50 mV

ENa

100 mV

ECa

Vrest

Action Potential
g

Na

gCa

g g
K

Cl

-100

EK ECl-50 mV

0 mV

ENa

50 mV

100 mV

ECa

Vrest

Figure 2.4: Mechanistic interpretation of the resting membrane potential (2.4) as thecenter of mass. Na+ conductance increases during the action potential.
potential in a quasi-static fashion, i.e., assuming that time is frozen. When a neuron isquiescent, Na+ and Ca2+ conductances are relatively small, Vrest is near EK and ECl ,as in Fig.2.4 (top), and so is V . During the upstroke of an action potential, the Na+
or Ca2+ conductance becomes very large; Vrest is near ENa , as in Fig.2.4 (bottom), andV increases, trying to catch Vrest . This event is, however, quite brief, for the reasons
explained in subsequent sections.

2.1.5

Voltage-Clamp and I-V Relation

In section 2.2 we will study how the membrane potential affects ionic conductancesand currents, assuming that the potential is fixed at certain value Vc controlled by anexperimenter. To maintain the membrane potential constant (clamped), one inserts
a metallic conductor to short-circuit currents along the membrane (space-clamp), andthen injects a current proportional to the difference Vc − V (voltage-clamp), as in
Fig.2.5. From (2.2) and the clamp condition V̇ = 0, it follows that the injected current
I equals the net current generated by the membrane conductances.
In a typical voltage-clamp experiment the membrane potential is held at a certain
resting value Vc and then reset to a new value Vs , as in Fig.2.6a. The injected membranecurrent needed to stabilize the potential at the new value is a function of time, thepre-step holding potential Vc , and the step potential Vs . First, the current jumps toa new value to accommodate the instantaneous voltage change from Vc to Vs . From
(2.5) we find that the amplitude of the jump is ginp (Vs − Vc ). Then, time- and voltage-

Electrophysiology of Neurons

31
Vc

DI=(Vc-V)/R

I

V

I(Vc)

axon

analysis

injectFigure 2.5: Two-wire voltage-clamp experiment on the axon. The top wire is used tomonitor the membrane potential V . The bottom wire is used to inject the current I,
proportional to the difference Vc − V , to keep the membrane potential at Vc .
prestep
potentialVc(b)step potentials
Vs

600

instantaneous I-V
steady-state I-V

current, I (pA)

2000
1500
1000

I (Vs)

500
0current, I (pA)
inwardoutward(a)

I (Vs)

400
200
0
-200

I0(Vc,Vs)-400

I0(Vc,Vs)

- 500
0

1

-100
2

time, ms

3

4

5

Vs
-50

0

50

membrane potential, V (mV)

Figure 2.6: Voltage-clamp experiment to measure instantaneous and steady-state I-V
relation. Shown are simulations of the INa +IK -model (see Fig.4.1b); the continuous
curves are theoretically found I-V relations.
dependent processes start to occur and the current decreases and then increases. The
value at the negative peak, marked by the open circle “o” in Fig.2.6, depends only on Vc
and Vs , and it is called the instantaneous current-voltage (I-V) relation, or I0 (Vc , Vs ).
The asymptotic (t → ∞) value depends only on Vs and it is called the steady-state
current-voltage (I-V) relation, or I∞ (Vs ).
Both relations, depicted in Fig.2.6b, can be found experimentally (black circles) or
theoretically (curves). The instantaneous I-V relation usually has a non-monotone Nshape reflecting nonlinear autocatalytic (positive feedback) transmembrane processes,
which are fast enough on the time scale of the action potential that they can be assumed
to have instantaneous kinetics. The steady-state I-V relation measures the asymptotic
values of all transmembrane processes, and it may be monotone (as in the figure) ornot, depending on the properties of the membrane currents. Both I-V relations provide
invaluable quantitative information about the currents operating on fast and slow time

32

Electrophysiology of Neurons

Figure 2.7: To tease out neuronal currents, biologists employ an arsenal of sophisticated“clamp” methods, such as current-, voltage-, conductance-, and dynamic-clamp.
scales, and both are useful in building mathematical models of neurons. Finally, whenI∞ (V ) = 0, the net membrane current is zero, and the potential is at rest or equilibrium,
which may still be unstable, as we discuss in the next chapter.

2.2

Conductances

Ionic channels are large transmembrane proteins having aqueous pores through whichions can flow down their electrochemical gradients. The electrical conductance of individual channels may be controlled by gating particles (gates), which switch the channels
between open and closed states. The gates may be sensitive to the following factors:• Membrane potential. Example: voltage-gated Na+ or K+ channels• Intracellular agents (second-messengers). Example: Ca2+ -gated K+ channels• Extracellular agents (neurotransmitters and neuromodulators). Examples: AMPA,
NMDA, or GABA receptors.
Despite the stochastic nature of transitions between open and closed states in individual
channels, the net current generated by a large population or ensemble of identicalchannels can reasonably be described by the equation
I = ḡ p (V − E) ,(2.7)

where p is the average proportion of channels in the open state, ḡ is the maximal
conductance of the population, and E is the reverse potential of the current, i.e., theElectrophysiology of Neurons

33
Na+

Extracellularvoltagesensormembrane+++channelproteinselectivity
filter

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+

activation
gate

Intracellular

inactivation
gate

Closed(not activated)Open(activated)

Closed(inactivated)

Figure 2.8: Structure of voltage-gated ion channels. Voltage sensors open an activationgate and allow selected ions to flow through the channel according to their electrochemical gradients. The inactivation gate blocks the channel. (Modified from Armstrongand Hille 1998.)
potential at which the current reverses its direction.
If the channels are selective
for a single ionic species, then the reverse potential E equals the Nernst equilibrium
potential (2.1) for that ionic species (see exercise 2).

2.2.1

Voltage-Gated Channels

When the gating particles are sensitive to the membrane potential, the channels are
said to be voltage-gated. The gates are divided into two types: those that activate oropen the channels, and those that inactivate or close them (see Fig.2.8). According to
the tradition initiated in the middle of the twentieth century by Hodgkin and Huxley,
the probability of an activation gate being in the open state is denoted by the variablem (sometimes the variable n is used for K+ and Cl− channels). The probability of an
inactivation gate being in the open state is denoted by the variable h. The proportionof open channels in a large population isp = ma hb ,(2.8)

where a is the number of activation gates and b is the number of inactivation gates
per channel.
The channels can be partially (0 < m < 1) or completely activated(m = 1); not activated or deactivated (m = 0); inactivated (h = 0); released frominactivation or deinactivated (h = 1). Some channels do not have inactivation gates(b = 0), hence p = ma . Such channels do not inactivate, and they result in persistentcurrents. In contrast, channels that do inactivate result in transient currents.
Below we describe voltage- and time-dependent kinetics of gates. This description
is often referred to as the Hodgkin-Huxley gate model of membrane channels.

34

Electrophysiology of Neurons
1.6

1.0

1.4
1.2

τ (V) (ms )

m (V)

0.8
0.6
0.4

1.0
0.8
0.6

0.2

0.4

0.0

0.2
-80

-40

0
V (mV)

40

80

-120

-80

-40
0
V (mV)

40

80

Figure 2.9: The activation function m∞ (V ) and the time constant τ (V ) of the fast transient K+ current in layer 5 neocortical pyramidal neurons. (Modified from Korngreen
and Sakmann 2000.)

2.2.2

Activation of Persistent Currents

The dynamics of the activation variable m is described by the first-order differential
equationṁ = (m∞ (V ) − m)/τ (V ) ,(2.9)

where the voltage-sensitive steady-state activation function m∞ (V ) and the time constant τ (V ) can be measured experimentally. They have sigmoid and unimodal shapes,
respectively, as in Fig.2.9 (see also Fig.2.20). The steady-state activation function
m∞ (V ) gives the asymptotic value of m when the potential is fixed (voltage-clamp).
Smaller values of τ (V ) result in faster dynamics of m.
In Fig.2.10 we depict a typical experiment to determine m∞ (V ) of a persistentcurrent, i.e., a current having no inactivation variable. Initially we hold the membranepotential at a hyperpolarized value V0 so that all activation gates are closed and I ≈0. Then we step-increase V to a greater value Vs (s = 1, . . . , 7; see Fig.2.10a) andhold it there until the current is essentially equal to its asymptotic value, which isdenoted here as Is (s stands for “step”; see Fig.2.10b). Repeating the experimentfor various stepping potentials Vs , one can easily determine the corresponding Is , andhence the entire steady-state I-V relation, which we depict in Fig.2.10c. According to(2.7), I(V ) = ḡm∞ (V )(V − E), and the steady-state activation curve m∞ (V ) depicted
in Fig.2.10d is I(V ) divided by the driving force (V − E) and normalized so thatmax m∞ (V ) = 1. To determine the time constant τ (V ), one needs to analyze theconvergence rates. In exercise 6 we describe an efficient method to determine m∞ (V )and τ (V ).

Electrophysiology of Neurons

35
0-30 mVVs(a)current, I (pA)

-60 mV
step potentials
prestep
potential V0-20-100 mV

(c)

-40
-60

I(V)

Is

-80

-5 pA

I(t)

(b)
ac
on

ms

gating varitable, m

atio
act
iv

0.5

0

m (V)

(d)

-85 pA

n

(e)

-60
-40
-20
membrane voltage, V (mV)

1

ati

tiv

Is

1

Vs

-100
-80

0.8
0.6

ms

0.4
0.2

0

m(t)
0

2

4
6
time (ms)

8

10

0
-80

Vs
-60
-40
-20
membrane voltage, V (mV)0

Figure 2.10: An experiment to determine m∞ (V ). Shown are simulations of the persistent Na+ current in Purkinje cells (see section 2.3.5).

2.2.3

Inactivation of Transient Currents

The dynamics of the inactivation variable h can be described by the first-order differential equationḣ = (h∞ (V ) − h)/τ (V ) ,(2.10)
where h∞ (V ) is the voltage-sensitive steady-state inactivation function depicted in
Fig.2.11. In Fig.2.12 we present a typical voltage-clamp experiment to determineh∞ (V ) in the presence of activation m∞ (V ). It relies on the observation that inactivation kinetics is usually slower than activation kinetics. First, we hold the membranepotential at a certain pre-step potential Vs for a long enough time that the activationand inactivation variables are essentially equal to their steady-state values m∞ (Vs ) andh∞ (Vs ), respectively, which have yet to be determined. Then we step-increase V toa sufficiently high value V0 , chosen so that m∞ (V0 ) ≈ 1. If activation is much fasterthan inactivation, m approaches 1 after the first few milliseconds, while h continues
to be near its asymptotic value hs = h∞ (Vs ), which can be found from the peak valueof the current Is ≈ ḡ · 1 · hs (V0 − E). Repeating this experiment for various pre-step
potentials, one can determine the steady-state inactivation curve h∞ (V ) in Fig.2.11.
In exercise 6 we describe a better method to determine h∞ (V ) that does not rely on
the difference between the activation and inactivation time scales.

36Electrophysiology of Neurons

1
0.8

m (V)
hs

0.6

Figure 2.11: Steady-state activation function
m∞ (V ) from Fig.2.10, and inactivation function
h∞ (V ) and values hs from Fig.2.12. Their overlap(shaded region) produces a noticeable, persistent“window” current.

0.4
h (V)

Vs

0.2
0
-80

-60
-40
-20
membrane voltage, V (mV)pre-step
potentials

0step potential V0-20 mVVs-80 mVI(t)peak I s

inactivation

1
m(t)
0.5

instantaneous
activation

0
1
h(t)

hs

0.5
0
0

10

20

30
40
time (ms)

50

60

70

Figure 2.12: Dynamics of the current(I), activation (m), and inactivation(h) variables in the voltage-clamp
experiment aimed at measuring h∞ (V )in Fig.2.11.

The voltage-sensitive steady-state activation and inactivation functions overlap in
a shaded window depicted in Fig.2.11. Depending on the size of the shaded area in the
figure, the overlap may result in a noticeable “window” current.

2.2.4

Hyperpolarization-Activated Channels

Many neurons in various parts of the brain have channels that are opened by hyperpolarization. These channels produce currents that are turned on by hyperpolarizationand turned off by depolarization. Biologists refer to such currents as “exceptional” or“weird”, and denote them as IQ (queer), If (funny), Ih (hyperpolarization-activated),
or IKir (K+ inward rectifier). (We will consider the last two currents in detail in thenext chapter). Most neuroscience textbooks classify these currents in a special category– hyperpolarization-activated currents. However, from the theoretical point of view, it
is inconvenient to create special categories. In this book we treat these currents asElectrophysiology of Neurons

37“normal” transient currents with the understanding that they are always activated(either a = 0 or variable m = 1 in (2.8)), but can be inactivated by depolarization(variable h → 0) or deinactivated by hyperpolarization (variable h → 1). Moreover,there is biophysical evidence suggesting that closing/opening of IKir is indeed related
to the inactivation/deinactivation process (Lopatin et al. 1994).

2.3

The Hodgkin-Huxley Model

In section 2.1 we studied how the membrane potential depends on the membrane currents, assuming that ionic conductances are fixed. In section 2.2 we used the HodgkinHuxley gate model to study how the conductances and currents depend on the membrane potential, assuming that the potential is clamped at different values. In thissection we put it all together and study how the potential ↔ current nonlinear interactions lead to many interesting phenomena, such as generation of action potentials.

2.3.1

Hodgkin-Huxley Equations

One of the most important models in computational neuroscience is the HodgkinHuxley model of the squid giant axon. Using pioneering experimental techniques of thattime, Hodgkin and Huxley (1952) determined that the squid axon carries three major
currents: voltage-gated persistent K+ current with four activation gates (resulting in
the term n4 in the equation below, where n is the activation variable for K+ ); voltagegated transient Na+ current with three activation gates and one inactivation gate (theterm m3 h below), and Ohmic leak current, IL , which is carried mostly by Cl− ions.
The complete set of space-clamped Hodgkin-Huxley equations is
I

C V̇
ṅ
ṁ
ḣ

=
=
=
=

I

I

Na
K
L
}|
{
z
}|
{
z
z
}|
{
I − ḡK n4 (V − EK ) − ḡNa m3 h(V − ENa ) − gL (V − EL )αn (V )(1 − n) − βn (V )nαm (V )(1 − m) − βm (V )mαh (V )(1 − h) − βh (V )h ,where
10 − V,exp( 10−V
)−1
10


−V
βn (V ) = 0.125 exp,
80
25 − V
αm (V ) = 0.1,
exp( 25−V
)−1
10


−V
βm (V ) = 4 exp,
18
αn (V ) = 0.01

38Electrophysiology of Neuronsαh (V ) = 0.07 exp
βh (V ) =

−V
20



1
)+1
exp( 30−V
10,.

These parameters, provided in the original Hodgkin and Huxley paper, correspondto the membrane potential shifted by approximately 65 mV, so that the resting potential is at V ≈ 0. Hodgkin and Huxley did that for the sake of convenience, but
the shift has led to a lot of confusion over the years. The shifted Nernst equilibrium
potentials are
EK = −12 mV ,ENa = 120 mV ,EL = 10.6 mV;(see also exercise 1). Typical values of maximal conductances areḡK = 36 mS/cm2 ,ḡNa = 120 mS/cm2 ,gL = 0.3 mS/cm2 .

C = 1 µF/cm2 is the membrane capacitance and I = 0 µA/cm2 is the applied current.
The functions α(V ) and β(V ) describe the transition rates between open and closed
states of the channels. We present this notation only for historical reasons. In the restof the book, we use the standard formṅ = (n∞ (V ) − n)/τn (V ) ,ṁ = (m∞ (V ) − m)/τm (V ) ,ḣ = (h∞ (V ) − h)/τh (V ) ,where
n∞ = αn /(αn + βn ) ,m∞ = αm /(αm + βm ) ,h∞ = αh /(αh + βh ) ,τn = 1/(αn + βn ) ,τm = 1/(αm + βm ) ,τh = 1/(αh + βh )as depicted in Fig.2.13. These functions can be approximated by the Boltzmann andGaussian functions; see Ex. 4. We also shift the membrane potential back to its true
value, so that the resting state is near -65 mV.
The membrane of the squid giant axon carries only two major currents: transient+
Na and persistent K+ . Most neurons in the central nervous system have additional
currents with diverse activation and inactivation dynamics, which we summarize insection 2.3.5. The Hodgkin-Huxley formalism is the most accepted model to describe
their kinetics.
Since we are interested in geometrical and qualitative methods of analysis of neuronal models, we assume that all variables and parameters have appropriate scales
and dimensions, but we do not explicitly state them. An exception is the membranepotential V , whose mV scale is stated in every figure.

Electrophysiology of Neurons

39

1
8

h (V)

τ h (V)

m (V)

n (V)

τ n (V)
τm(V)

0
-40

0

V (mV)

100

0
-40

0

V (mV)

100

Figure 2.13: Steady-state (in)activation functions (left) and voltage-dependent time
constants (right) in the Hodgkin-Huxley model.

Figure 2.14: Studies of spike-generation mechanism in “giant squid” axons won Alan
Hodgkin and Andrew Huxley the 1963 Nobel Prize for physiology or medicine (sharedwith John Eccles). See also Fig. 4.1 in Keener and Sneyd (1998).

40Electrophysiology of Neurons

E

Na

Membrane voltage (mV)depolarization
rest
hyperpolarization

100

Excited(regenerative)
Repolarization

50

small
depolarization

Upstrokelargedepolarization
Actionpotential(spike)

E

L

Rest

0

Repolarization

Absolute
refractory

Relativerefractory

After-hyperpolarization

E

V(t)

K

activation

deactivation

1

Gating variables

h(t)
n(t)
m(t)

0.5
0

inactivation
Conductances (mS/cm 2)

20

deinactivation

g (t)
Na

g (t)
K

0
2
500 Currents (µ A/cm )

I (t)
K

0

I Na+IK +I L

-500

I (t)
Na

Applied current (µ A/cm 2 )
20

I(t)

0
0

2

4

6

8

10

12

14

16

time (ms)Figure 2.15: Action potential in the Hodgkin-Huxley model.

18

20

Electrophysiology of Neurons

Depolarization

Increase in g

41

Na

Increasein gNa

Depolarization
Na+
Inflow

Increase
in g
K

Repolarization
Hyperpolarization

Figure 2.16: Positive and negative feedback loops resulting in excited (regenerative)
behavior in neurons.

2.3.2

Action Potential

Recall that when V = Vrest , which is 0 mV in the Hodgkin-Huxley model, all inward
and outward currents balance each other so the net current is zero, as in Fig.2.15. The
resting state is stable: a small pulse of current applied via I(t) produces a small positive
perturbation of the membrane potential (depolarization), which results in a small net
current that drives V back to resting (repolarization).
However, an intermediatesize pulse of current produces a perturbation that is amplified significantly because
membrane conductances depend on V . Such a nonlinear amplification causes V todeviate considerably from Vrest – a phenomenon referred to as an action potential or
spike.
In Fig.2.15 we show a typical time course of an action potential in the HodgkinHuxley system. Strong depolarization increases activation variables m and n and decreases inactivation variable h. Since τm (V ) is relatively small, variable m is relatively
fast. Fast activation of Na+ conductance drives V toward ENa , resulting in further
depolarization and further activation of gNa . This positive feedback loop, depicted in
Fig.2.16, results in the upstroke of V . While V moves toward ENa , the slower gating
variables catch up. Variable h → 0, causing inactivation of the Na+ current, and variable n → 1, causing slow activation of the outward K+ current. The latter and theleak current repolarize the membrane potential toward Vrest .
When V is near Vrest , the voltage-sensitive time constants τn (V ) and τh (V ) are
relatively large, as one can see in Fig.2.13. Therefore, recovery of variables n and h isslow. In particular, the outward K+ current continues to be activated (n is large) evenafter the action potential downstroke, thereby causing V to go below Vrest toward EK– a phenomenon known as afterhyperpolarization.
In addition, the Na+ current continues to be inactivated (h is small) and not available for any regenerative function. The Hodgkin-Huxley system cannot generate another action potential during this absolute refractory period. While the current deinactivates, the system becomes able to generate an action potential, provided the stimulus
is relatively strong (relative refractory period).
To study the relationship between these refractory periods, we stimulate the HodgkinHuxley model with 1-ms pulses of current having various amplitudes and latencies. The

Electrophysiology of Neurons
memrane potential (mV)

42
100

50

0

Ap

pulse size, Ap, needed to
produce 2nd spike (µA/cm2)

tp

20
absolute
refractory

relativerefractory

100

hyperexcitability
0

5

10

15

20

25

time, tp, after 1st spike (ms)Figure 2.17: Refractory periods in the Hodgkin-Huxley model with I = 3.
minimal amplitude of the stimulation needed to evoke a second spike in the model isdepicted in Fig.2.17 (bottom). Notice that around 14 ms after the first spike, the model
is hyper-excitable, that is, the stimulation amplitude is less than the baseline amplitude Ap ≈ 6 needed to evoke a spike from the resting state. This occurs because the
Hodgkin-Huxley model exhibits damped oscillations of membrane potential (discussed
in chapter 7).

2.3.3

Propagation of the Action Potentials

The space-clamped Hodgkin-Huxley model of the squid giant axon describes nonpropagating action potentials since V (t) does not depend on the location, x, along
the axon. To describe propagation of action potentials (pulses) along the axon having potential V (x, t), radius a (cm), and intracellular resistivity R (Ω·cm), the partial
derivative Vxx is added to the voltage equation to account for axial currents along themembrane. The resulting nonlinear parabolic partial differential equation
C Vt =aVxx + I − IK − INa − IL2R

is often referred to as the Hodgkin-Huxley cable or propagating equation. Its important
type of solution, a traveling pulse, is depicted in Fig.2.18. Studying this equation goesbeyond the scope of this book; the reader can consult Keener and Sneyd (1998) andreferences therein.

Electrophysiology of Neurons

V(x,t 2)V(x,t 3)V(x,t 4)100

Traveling pulse

Membrane voltage (mV)V(x,t 1)

43

0
x

Axon

Figure 2.18: Traveling pulse solution of the Hodgkin-Huxley cable equation at four
successive moments.

2.3.4

Dendritic Compartments

Modifications of the Hodgkin-Huxley model, often called Hodgkin-Huxley-type models
or conductance-based models, can describe the dynamics of spike-generation of many,
if not all, neurons recorded in nature. However, there is more to the computational
property of neurons than just the spike-generation mechanism. Many neurons havean extensive dendritic tree that can sample the synaptic input arriving at different
locations and integrate it over space and time.
Many dendrites have voltage-gated currents, so the synaptic integration is nonlinear, sometimes resulting in dendritic spikes that can propagate forward to the somaof the neuron or backward to distant dendritic locations. Dendritic spikes are prominent
in intrinsically bursting (IB) and chattering (CH) neocortical neurons considered inchapter 8. In that chapter we also model regular spiking (RS) pyramidal neurons,
the most numerous class of neurons in mammalian neocortex, and show that their
spike-generation mechanism is one of the simplest. The computation complexity of RS
neurons must be hidden, then, in the arbors of their dendritic trees.
It is not feasible at present to study the dynamics of membrane potential in dendritic
trees either analytically or geometrically (i.e., without resort to computer simulations),
unless dendrites are assumed to be passive (linear) and semi-infinite, and to satisfy
Rall’s branching law (Rall 1959). Much of the insight can be obtained via simulations,which typically replace the continuous dendritic structure in Fig.2.19a with a networkof discrete compartments in Fig.2.19b. Dynamics of each compartment is simulated bya Hodgkin-Huxley-type model, and the compartments are coupled via conductances.
For example, if Vs and Vd denote the membrane potential at the soma and in thedendritic tree, respectively, as in Fig.2.19c, thenCs V̇s = −Is (Vs , t) + gs (Vd − Vs ) ,and

Cd V̇d = −Id (Vd , t) + gd (Vs − Vd ) ,44Electrophysiology of Neurons
(a)

(b)

(c)

(d)
neuron 1
neuron 2

dendrite

Vd

soma

Vs

Figure 2.19: A dendritic tree of a neuron (a) is replaced by a network of compartments (b), each modeled by a Hodgkin-Huxley-type model. The two-compartmentneuronal model (c) may be equivalent to two neurons coupled via gap junctions (electrical synapse) (d).
where each I(V, t) represents the sum of all voltage-, Ca2+ -, and time-dependent currents in the compartment, and gs and gd are the coupling conductances that dependon the relative sizes of dendritic and somatic compartments. One can obtain many
spiking and bursting patterns by changing the conductances and keeping all the otherparameters fixed (Pinsky and Rinzel 1994, Mainen and Sejnowski 1996).
Once we understand how to couple two compartments, we can do it for hundreds or
thousands of compartments. GENESIS and NEURON simulation environments could
be useful here, especially since they contain databases of dendritic trees reconstructedfrom real neurons.
Interestingly, the somatic-dendritic pair in Fig.2.19c is equivalent to a pair of neurons in Fig.2.19d coupled via gap-junctions. These are electrical contacts that allow
ions and small molecules to pass freely between the cells. Gap junctions are often
called electrical synapses, because they allow potentials to be conducted directly from
one neuron to another.
Computational study of multi-compartment dendritic processing is outside of thescope of this book. We consider multi-compartment models of cortical pyramidal neurons in chapter 8 and gap-junction coupled neurons in chapter 10 (which is on theauthor’s webpage).

2.3.5

Summary of Voltage-Gated Currents

Throughout this book we model kinetics of various voltage-sensitive currents using the
Hodgkin-Huxley gate model
I = ḡ ma hb (V − E)Electrophysiology of Neurons

τ (V)

pe

m (V)

Cbase+Camp

slo

1

45

Camp

0.5

Cbase

σ

Camp/e

V

0

V
Vmax

2k V1/2

Figure 2.20: Boltzmann (2.11) and Gaussian (2.12) functions and geometrical interpretations of their parameters.
where
I - current , (µA/cm2 ),V - membrane voltage, (mV),E - reverse potential, (mV),ḡ - maximal conductance, (mS/cm2 ),m - probability of activation gate to be open,
h - probability of inactivation gate to be open,a - the number of activation gates per channel,b - the number of inactivation gates per channel.
The gating variables m and n satisfy linear first-order differential equations (2.9) and(2.10), respectively. We approximate the steady-state activation curve m∞ (V ) by the
Boltzmann function depicted in Fig.2.20,
m∞ (V ) =1
1 + exp {(V1/2 − V )/k}(2.11)The parameter V1/2 satisfies m∞ (V1/2 ) = 0.5, and k is the slope factor (negative for theinactivation curve h∞ (V )). Smaller values of |k| result in steeper m∞ (V ).
The voltage-sensitive time constant τ (V ) can be approximated by the Gaussianfunction−(Vmax − V )2,
(2.12)
τ (V ) = Cbase + Camp exp
σ2
see Fig.2.20. The graph of the function is above Cbase with amplitude Camp . The
maximal value is achieved at Vmax . The parameter σ measures the characteristic widthof the graph, that is, τ (Vmax ± σ) = Cbase + Camp /e. The Gaussian description is often
not adequate, so we replace it with other functions whenever appropriate.
Below is the summary of voltage-gated currents whose kinetics were measured experimentally. The division into persistent and transient is somewhat artificial, sincemost “persistent” currents can still inactivate after seconds of prolonged depolarization.
Hyperpolarization-activated currents, such as the h-current or K+ inwardly rectifying
current, are mathematically equivalent to currents that are always activated, but canbe inactivated by depolarization. To avoid possible confusion, we mark these currents“opened by hyperpolarization”.

46

Electrophysiology of Neurons

Na+ currents
V1/2
Fast transient 1
activation
inactivation
Fast transient 2
activation
inactivation
Fast transient 3
activation
inactivation
Fast persistent 4,aactivation
inactivation
Fast persistent 5,aactivation
inactivation

Eq. (2.11)
kParameters (Fig.2.20)
Eq. (2.12)
Vmax σCamp

Cbase

INa,t = ḡ m3 h(V − ENa )
−40
−62

−38
−67

15
−7

30
20

0.46
7.4

0.04
1.2

INa,t = ḡ m∞ (V )h(V − ENa )
−30
−70

5.5
−5.8

−
−
−
−
τh (V ) = 3 exp((−40 − V )/33)

INa,t = ḡ m∞ (V )h(V − ENa )
−28
−66

6.7
−6

−
−
−
−
τh (V ) = 4 exp((−30 − V )/29)

INa,p = ḡ m∞ (V )h(V − ENa )
−50
−49

4
−10

−
−66

−
35

−
4.5 sec−
2 sec

INa,p = ḡ m∞ (V )(0.14 + 0.86h)(V − ENa )
−50
−56

6
−7

−
−
−
−
τh (V ) = 63.2 + 25 exp(−V /25.5)

Fast persistent 2

INa,p = ḡ m(V − ENa )activation
Fast persistent 6−54
9INa,p = ḡ m(V − ENa )

−

−

−

0.8

−42

−

−

−

0.8

activation

4

1. Squid giant axon (Hodgkin and Huxley 1952); see exercise 4.
2. Thalamocortical neurons in rats (Parri and Crunelli 1999).
3. Thalamocortical neurons in cats (Parri and Crunelli 1999).
4. Layer-II principal neurons in entorhinal cortex (Magistretti and Alonso 1999).
5. Large dorsal root ganglion neurons in rats (Baker and Bostock 1997, 1998).
6. Purkinje cells (Kay et al. 1998).
a Very slow inactivation.

Electrophysiology of Neurons

K+ currents
V1/2
Delayed rectifier 1
activation
Delayed rectifier 2,4
activation
inactivation
M current 3
activation
Transient 4
activation
inactivation
Transient 5
activation
inactivation

47

Parameters (Fig.2.20)
Eq. (2.12)
Eq. (2.11)kVmax σ Camp

Cbase

IK = ḡ n4 (V − EK )
−53

15

−79

50

4.7

1.1

−50
−50

30
50

47
1000

5
360

−50

25

320

20

−71
−73

60
23

0.92
50

0.34
8

−
−

−
−

−
−

−
15.5

IK = ḡ mh(V − EK )
−3
−51

10
−12

IK(M) = ḡ m(V − EK )−44

8

IA = ḡ mh(V − EK )
−3
−66

20
−10

IA = ḡ mh(V − EK )
−26
−72

20
−9.6

Transient 6
IA = ḡ m4 h (V − EK )Fast component (60% of total conductance)
activation
−60
8.5
−58
25
2
0.37
inactivation
−78
−6
−78
25
45
19
Slow component (40% of total conductance)
activation
−36
20
−58
25
2
0.37
inactivation
−78
−6
−78
25
45
19
τh (V ) = 60 when V > −73
Inward rectifier 7
IKir = ḡ h∞ (V )(V − EK )(opened by hyperpolarization )inactivation

−80

−12

−

−

−

<1

1. Squid giant axon (Hodgkin and Huxley 1952); see exercise 4.
2. Neocortical pyramidal neurons (Bekkers 2000).
3. Rodent neuroblastoma-glioma hybrid cells (Robbins et al. 1992).
4. Neocortical pyramidal neurons (Korngreen and Sakmann 2000).
5. Hippocampal mossy fiber boutons (Geiger and Jonas 2000).
6. Thalamic relay neurons (Huguenard and McCormick 1992).
7. Horizontal cells in catfish retina (Dong and Werblin 1995); AP cell of leech (Wessel et al.
1999); rat locus coeruleus neurons (Williams et al. 1988, V1/2 = EK ).

48

Electrophysiology of Neurons
Parameters (Fig.2.20)
Eq. (2.12)
Eq. (2.11)kVmax σ Camp

Cation currents
V1/2

Ih current 1
Ih = ḡ h (V − Eh ),(opened by hyperpolarization )−75

inactivation

Cbase

Eh = −43 mV

−5.5

−75

15

1000

100

Ih current 2
inact. (soma)
inact. (dendrite)

Ih = ḡ h (V − Eh ),
−82
−9
−90
−8.5

Eh = −1 mV
−75
20
−75
20

50
40

10
10

Ih current 3
fast inact. (65%)
slow inact. (35%)

Ih = ḡ h (V − Eh ),
−67
−12
−58
−9

Eh = −21 mV
−75
30
50
−65
30 30020100

1. Thalamic relay neurons (McCormick and Pape 1990; Huguenard and McCormick 1992).
2. Hippocampal pyramidal neurons in CA1 (Magee 1998).
3. Entorhinal cortex layer II neurons (Dickson et al. 2000).
voltage-gated currents

activation
inactivation

1000

Ih

slow

Ih

IK(M)
IK(M)

time constant, (ms)

100

IA

IA
Ih

IK(M)

Ih
IK

10

INat

INat
IK

INat

IA

IA

1

IA

fast

IKir
-100

low-threshold

INap

INap
-50

INat
high-threshold0

half-voltage, V1/2 (mV)

Figure 2.21: Summary of current kinetics. Each oval (rectangle) denotes the voltage
and temporal scale of activation (inactivation) of a current. Transient currents are
represented by arrows connecting ovals and rectangles.

Electrophysiology of Neurons

49

Figure 2.22: Alan Hodgkin (right) and Andrew Huxley (left) in their Plymouth Marine
Lab in 1949. (Photo provided by National Marine Biological Library, Plymouth, UK).

Review of Important Concepts• Electrical signals in neurons are carried by Na+ , Ca2+ , K+ , and Cl−
ions, which move through membrane channels according to theirelectrochemical gradients.
• The membrane potential V is determined by the membrane conductances gi and corresponding reversal potentials Ei :X
C V̇ = I −gi · (V − Ei ) .
i• Neurons are excitable because the conductances depend on the membrane potential and time.
• The most accepted description of kinetics of voltage-sensitive conductances is the Hodgkin-Huxley gate model.
• Voltage-gated activation of inward Na+ or Ca2+ current depolarizes(increases) the membrane potential.
• Voltage-gated activation of outward K+ or Cl− current hyperpolarizes (decreases) the membrane potential.
• An action potential or spike is a brief regenerative depolarization of
the membrane potential followed by its repolarization and possibly
hyperpolarization, as in Fig.2.16.

50

Electrophysiology of Neurons

Bibliographical Notes
Our summary of membrane electrophysiology is limited: we present only those concepts that are necessary to understand the Hodgkin-Huxley description of generationof action potentials. We have omitted such important topics as the Goldman-HodgkinKatz equation, cable theory, dendritic and synaptic function, although some of thosewill be introduced later in the book.
The standard textbook on membrane electrophysiology is the second edition of
Ion Channels of Excitable Membranes by B. Hille (2001). An excellent introductory
textbook with an emphasis on the quantitative approach is Foundations of Cellular
Neurophysiology by D. Johnston and S. Wu (1995). A detailed introduction to mathematical aspects of cellular biophysics can be found in Mathematical Physiology by
J. Keener and J. Sneyd (1998). The latter two books complement rather than repeat each other. Biophysics of Computation by Koch (1999) and chapters 5 and 6 ofTheoretical Neuroscience by Dayan and Abbott (2001) provide a good introduction tobiophysics of excitable membranes.
The first book devoted exclusively to dendrites is Dendrites by Stuart et al. (1999).
It emphasizes the active nature of dendritic dynamics. Arshavsky et al. (1971; Russian
language edition, 1969) make the first, and probably still the best, theoretical attempt
to understand the neurocomputational properties of branching dendritic trees endowedwith voltage-gated channels and capable of generating action potentials. Had they
published their results in the 1990s, they would have been considered classics in thefield. Unfortunately, the computational neuroscience community of the 1970s was
not ready to accept the “heretic” idea that dendrites can fire spikes, that spikes canpropagate backward and forward along the dendritic tree, that EPSPs can be scaled-up
with distance, that individual dendritic branches can perform coincidence detection andbranching points can perform nonlinear summation, and that different and independent
computations can be carried out at different parts of the neuronal dendritic tree. We
touch on some of these issues in chapter 8.

Exercises
1. Determine the Nernst equilibrium potentials for the membrane of the squid giant
axon using the following data:
K+
Na+
Cl−

Inside (mM)
430
50
65

Outside (mM)
20
440
560

and T = 20◦ C.
2. Show that a nonselective cation current
I = ḡNa p (V − ENa ) + ḡK p (V − EK )

Electrophysiology of Neurons
voltage steps

51
0 mV

-100 mV-10 mV

currents-60 mV

time (ms)
0

1

2

3

4

Figure 2.23: Current traces
corresponding to voltage stepsof various amplitudes; see
exercise 6.

5can be written in the form (2.7) withḡ = ḡNa + ḡK

and

E=

ḡNa ENa + ḡK EK.
ḡNa + ḡK

3. Show that applying a DC current I in the neuronal model
C V̇ = I − gL (V − EL ) − Iother (V )is equivalent to changing the leak reverse potential EL .
4. Steady-state (in)activation curves and voltage-sensitive time constants can beapproximated by the Boltzmann (2.11) and Gaussian (2.12) functions, respectively, depicted in Fig.2.20. Explain the meaning of the parameters V1/2 , k,Cbase , Camp , Vmax , and σ and find their values that provide satisfactory fit near
the rest state V = 0 for the Hodgkin-Huxley functions depicted in Fig.2.13.
5. (Willms et al. 1999) Consider the curve mp∞ (V ), where m∞ (V ) is the Boltzmann
function with parameters V1/2 and k, and p > 1. This curve can be approximatedby another Boltzmann function with some parameters Ṽ1/2 and k̃ (and p = 1).
Find the formulas that relate Ṽ1/2 and k̃ to V1/2 , k, and p.
6. (Willms et al. 1999) Write a MATLAB program that determines activationand inactivation parameters via a simultaneous fitting of current traces from a
voltage-clamp experiment similar to the one in Fig.2.23. Assume that the valuesof the voltage pairs – e.g., −60, −10; −100, 0 (mV) – are in the file v.dat. The
values of the current (circles in Fig.2.23) are in the file current.dat, and thesampling times – e.g., 0, 0.25, 0.5, 1, 1.5, 2, 3, 5 (ms) – are in the file times.dat.
7. Modify the MATLAB program from exercise 6 to handle multi-step (Fig.2.24)
and ramp protocols.
8. [M.S.] Find the best sequence of step potentials that can determine activation and inactivation parameters (a) in the shortest time, (b) with the highest
precision.

52

Electrophysiology of Neurons+50 mV-10 mV-20 mV-40 mV-60 mV-80 mV-100 mV

10 ms

Figure 2.24: Multiple voltage steps are often needed to determine time constants ofinactivation; see exercise 7.

9. [M.S.] Modify the MATLAB program from exercise 6 to handle multiple currents.
10. [M.S.] Add a PDE solver to the MATLAB program from exercise 6 to simulate
poor space and voltage clamp conditions.
11. [Ph.D.] Introduce numerical optimization into the dynamic clamp protocol toanalyze experimentally in real time the (in)activation parameters of membranecurrents.
12. [Ph.D.] Use new classification of families of channels (Kv3,1, Nav 1.2, etc.; seeHille 2001) to determine the kinetics of each subgroup, and provide a complete
table similar to those in section 2.3.5.

Chapter 3
One-Dimensional Systems
In this chapter we describe geometrical methods of analysis of one-dimensional dynamical systems, i.e., systems having only one variable. An example of such a system isthe space-clamped membrane having Ohmic leak current IL :C V̇ = −gL (V − EL ) .

(3.1)

Here the membrane voltage V is a time-dependent variable, and the capacitance C,leak conductance gL , and leak reverse potential EL are constant parameters describedin chapter 2. We use this and other one-dimensional neural models to introduce andillustrate the most important concepts of dynamical system theory: equilibrium, stability, attractor, phase portrait, and bifurcation.

3.1

Electrophysiological Examples

The Hodgkin-Huxley description of dynamics of membrane potential and voltage-gatedconductances can be reduced to a one-dimensional system when all transmembrane
conductances have fast kinetics. For the sake of illustration, let us consider a spaceclamped membrane having leak current and a fast voltage-gated current Ifast with only
one gating variable p,
Ifast
Leak IL
}|
{ z
}|
{
z
C V̇ = − gL (V − EL ) − g p (V − E)ṗ = (p∞ (V ) − p)/τ (V ) ,

(3.2)
(3.3)

with dimensionless parameters C = 1, gL = 1, and g = 1. Suppose that the gating
kinetics (3.3) is much faster than the voltage kinetics (3.2), which means that thevoltage-sensitive time constant τ (V ) is very small, that is, τ (V ) ≪ 1 in the entire
biophysical voltage range. Then the gating process may be treated as being instantaneous, and the asymptotic value p = p∞ (V ) may be used in the voltage equation (3.2)
53

54

One-Dimensional Systems
1

20

p(t)

0.8

Activation Variable

V(t)

10

Membrane Voltage (mV)

0.9

0.7

τ (V) < 0.50τ (V) < 0.1

-10

0.6

τ (V) < 0.01

-20

0.5

Instantaneous

-30

0.4

-40

0.3

-50

0.2

-60

0.1

-70

0

-80
0

1

2

3

4

5

6

7

8

9

10

0

1

Time (ms)

2

3

4

5

6

7

8

9

10

Time (ms)Figure 3.1: Solution of the full system (3.2, 3.3) converges to that of the reduced
one-dimensional system (3.4) as τ (V ) → 0.
to reduce the two-dimensional system (3.2, 3.3) to a one-dimensional equation:
instantaneous Ifast}|{z
C V̇ = −gL (V − EL ) − g p∞ (V ) (V − E) .

(3.4)

This reduction introduces a small error of the order τ (V ) ≪ 1, as one can see in Fig.3.1.
Since the hypothetical current Ifast can be either inward (E > EL ) or outward(E < EL ), and the gating process can be either activation (p is m, as in the HodgkinHuxley model) or inactivation (p is h), there are four fundamentally different choices
for Ifast (V ), which we summarize in Fig.3.2 and elaborate on below.

GatingCurrent
inward outward
activation

I Na,p

IK

inactivation

Ih

I Kir

3.1.1

Figure 3.2: Four fundamental examples of voltagegated currents with one gating variable. In thisbook we treat “hyperpolarization-activated” currents Ih and IKir as inactivating currents, which
are turned off (inactivated via h) by depolarization and turned on (deinactivated) by hyperpolarization (see discussion in section 2.2.4).

I-V Relations and Dynamics

The four choices in Fig.3.2 result in four simple one-dimensional models of the form
(3.4):
INa,p -model ,IK -model ,Ih -model , and IKir -model .
These models might seem too simple to biologists, who can easily understand their
behavior just by looking at the I-V relations of the currents depicted in Fig.3.3 without

One-Dimensional Systems

55
currents

activation, m
inactivation, hgatinginwardINa,poutward
I(V)

IK

I(V)

V
V

negative
conductance

Ih

V

IKir

negative
conductance
V

I(V)

I(V)

Figure 3.3:
Typical currentvoltage (I-V) relations of the four
currents considered in this chapter. Shaded boxes correspond tononmonotonic I-V relations having a region of negative conductance (I ′ (V ) < 0) in the biophysically relevant voltage range.

using any dynamical systems theory. The models might also appear too simple tomathematicians, who can easily understand their dynamics just by looking at thegraphs of the right-hand side of (3.4) without using any electrophysiological intuition.
In fact, the models provide an invaluable learning tool, since they establish a bridge
between electrophysiology and dynamical systems.
In Fig.3.3 we plot typical steady-state current-voltage (I-V) relations of the four
currents considered above. Note that the I-V curve is nonmonotonic for INa,p and IKir
but monotonic for IK and Ih , at least in the biophysically relevant voltage range. This
subtle difference is an indication of the fundamentally different roles these currents
play in neuron dynamics. The I-V relation in the first group has a region of “negative
conductance” (i.e., I ′ (V ) < 0), which creates positive feedback between the voltage andthe gating variable (Fig.3.4), and plays an amplifying role in neuron dynamics. We
refer to such currents as amplifying currents. In contrast, the currents in the second
group have negative feedback between voltage and gating variable, and they often
result in damped oscillation of the membrane potential, as we show in chapter 4. We
refer to such currents as resonant currents. Most neural models involve a combinationof at least one amplifying and one resonant current, as we discuss in chapter 5. The
way these currents are combined determines whether the neuron is an integrator or aresonator.

3.1.2

Leak + Instantaneous INa,pTo ease our introduction into dynamical systems, we will use the INa,p -model
instantaneous INa,p}|{z
C V̇ = I − gL (V − EL ) − gNa m∞ (V ) (V − ENa ) ,called persistent sodium model, withm∞ (V ) = 1/(1 + exp {(V1/2 − V )/k})

(3.5)

56

One-Dimensional Systems
currents

activation, m
inactivation, hgatinginwarddepolarization++

inwardcurrent+

hyperpolarization-

inwardcurrent+--

outward+increasein mdepolarization-

outwardcurrent++
increasein h-hyperpolarization+++
increasein m+increasein hFigure 3.4:
Feedback loops
between voltage and gating
variables in the four models presented above (see also Fig.5.2).

outwardcurrent+

negative feedback,resonant current

feedback,+ positive
amplifying current

throughout the rest of this chapter. (Some biologists refer to transient Na+ currents
with very slow inactivation as being persistent, since the current does not change muchon the time scale of 1 sec.) We obtain the experimental parameter values
C = 10 pF,
gNa = 74 nS,

I = 0 pA,
V1/2 = 1.5 mV,

EL = −67 mV,
ENa = 60 mV

gL = 19 nS,
k = 16 mV,using whole-cell patch-clamp recordings of a layer 5 pyramidal neuron in the visualcortex of a rat at room temperature. We prove in exercise 3.3.8 and illustrate in
Fig.3.15 that the model approximates the action potential upstroke dynamics of thisneuron.

current (nA)1derivative of membrane potential (mV/ms)

100
IL(V)

I(V)

0

-1
INa,p(V)-2-60-40 -2002040membrane potential (mV)a60

V=F(V)

50

F(V)=-I(V)/C
0

-50

-60

-40 -2002040membrane potential, V (mV)60bFigure 3.5: (a) I-V relations of the leak current IL , fast Na+ current INa , and combined
current I(V ) = IL (V ) + INa (V ) in the INa,p -model (3.5). Dots denote I0 (V ) data fromlayer 5 pyramidal cell in rat visual cortex. (b) The right-hand side of the INa,p -model(3.5).

One-Dimensional Systems

57

bistability (I=0)

monostability (I=60)

20
0

V(t)

-20
-40
-60

40

excitedresting0

1

2
3
time (ms)45membrane potential, V (mV)membrane potential, V (mV)40

excited

20
0

V(t)

-20
-40
-60

0

1

2
3
time (ms)

a

4

5

b

Figure 3.6: Typical voltage trajectories of the INa,p -model (3.5) having different valuesof I.
The model’s I-V relation, I(V ), is depicted in Fig.3.5a. Due to the negative conductance region in the I-V curve, this one-dimensional model can exhibit a numberof interesting nonlinear phenomena, such as bistability, i.e. coexistence of resting andexcited states. From a mathematical point of view, bistability occurs because theright-hand-side function in the differential equation (3.5), depicted in Fig.3.5b, is not
monotonic. In Fig.3.6 we depict typical voltage time courses of the model (3.5) withtwo values of injected DC current I and 16 different initial conditions. The qualitative behavior in Fig.3.6a is clearly bistable: depending on the initial condition, thetrajectory of the membrane potential goes either up to the excited state or down to
the resting state. In contrast, the behavior in Fig.3.6b is monostable, since the restingstate does not exist. The goal of the dynamical system theory reviewed in this chapter
is to understand why and how the behavior depends on the initial conditions and theparameters of the system.

3.2

Dynamical Systems

In general, dynamical systems can be continuous or discrete, depending on whether
they are described by differential or difference equations. Continuous one-dimensional
dynamical systems are usually written in the formV̇ = F (V ) ,V (0) = V0 ∈ R .

(3.6)

For example,
V̇ = −80 − V ,V (0) = −20 ,where V is a scalar time-dependent variable denoting the current state of the system,V̇ = Vt = dV /dt is its derivative with respect to time t, F is a scalar function (itsoutput is one-dimensional) that determines the evolution of the system, e.g., the right-

58

One-Dimensional Systems

V0 V(0)
membrane potential, V

V(h)
V(2h)
V(3h)

- g Lt/C

V(t)=EL+(V0 -E L)e

Figure 3.7:
Explicit analytical solution(V (t) = EL + (V0 − EL )e−gL t/C ) of the linear
equation (3.1) and corresponding numericalapproximation (dots) using Euler’s method (3.7).

ELtime, thand side of (3.5) divided by C; see Fig.3.5b. V0 ∈ R is an initial condition, and R is
the real line, that is, a line of real numbers (Rn would be the n-dimensional real space).
In the context of dynamical systems, the real line R is called the phase line orstate line (phase space or state space for Rn ) to stress the fact that each point in R
corresponds to a certain, possibly inadmissible state of the system, and each state of
the system corresponds to a certain point in R. For example, the state of the Ohmic
membrane (3.1) is its membrane potential V ∈ R. The state of the Hodgkin-Huxleymodel (see section 2.3) is the four-dimensional vector (V, m, n, h) ∈ R4 . The state of
the INa,p -model (3.5) is its membrane potential V ∈ R, because the value m = m∞ (V )is unequivocally defined by V .
When all parameters are constant, the dynamical system is called autonomous.
When at least one of the parameters is time-dependent, the system is nonautonomous,
denoted as V̇ = F (V, t).
“To solve equation (3.6)” means to find a function V (t) whose initial value is V (0) =V0 and whose derivative is F (V (t)) at each moment t ≥ 0. For example, the function
V (t) = V0 + at is an explicit analytical solution to the dynamical system V̇ = a. The
exponentially decaying function V (t) = EL + (V0 − EL )e−gL t/C depicted in Fig.3.7, asolid curve, is an explicit analytical solution to the linear equation (3.1). (Check bydifferentiating).
Finding explicit solutions is often impossible even for such simple systems as (3.5),
so quantitative analysis is carried out mostly via numerical simulations. The simplest
procedure to solve (3.6) numerically, known as the first-order Euler method, replaces(3.6) with the discretized system[V (t + h) − V (t)]/h = F (V (t)) ,where t = 0, h, 2h, 3h, . . . , is the discrete time and h is a small time step. Knowing thecurrent state V (t), we can find the next state point via
V (t + h) = V (t) + hF (V (t)) .

(3.7)

Iterating this difference equation starting with V (0) = V0 , we can approximate theanalytical solution of (3.6) (see the dots in Fig.3.7). The approximation has a noticeable error of order h, so scientific software packages, such as MATLAB, use moresophisticated high-precision numerical methods.

One-Dimensional Systems

59

Ileak-modelINa,p-model
100

ILgraph of F(V)=Vgraph of F(V)=V

100
50
0

V>0

-50

V=0

V<0

stableequilibrium

80

IL+INa

60
40
20
0

-100
-100
-50
0
membrane potential, V (mV)

-50
0
50
membrane potential, V (mV)040stable equilibrium

30

V(t)

-20
-30
-40
-50
-60

stable equilibrium-70

membrane potential, V (mV)membrane potential, V (mV)

-10

20
10
V(t)
0
-10
-20
-30

-80

-40

-90

-50

-100
0

-60
0

unstable equilibrium

stable equilibriumtime (ms)5time (ms)aFigure 3.8: Graphs of the righthand side functions of equations(3.1) and (3.5), and corresponding numerical solutions startingfrom various initial conditions.

5b

In many cases, however, we do not need exact solutions, but qualitative understanding of the behavior of (3.6) and how it depends on parameters and the initial state V0 .
For example, we might be interested in the number of equilibrium (rest) points thesystem could have, whether the equilibria are stable, their attraction domains, etc.

3.2.1

Geometrical Analysis

The first step in the qualitative geometrical analysis of any one-dimensional dynamicalsystem is to plot the graph of the function F , as in Fig.3.8 (top). Since F (V ) = V̇ ,at every point V where F (V ) is negative, the derivative V̇ is negative, and hence thestate variable V decreases. In contrast, at every point where F (V ) is positive, V̇ ispositive, and the state variable V increases; the greater the value of F (V ), the faster
V increases. Thus, the direction of movement of the state variable V , and hence theevolution of the dynamical system, is determined by the sign of the function F (V ).
The right-hand side of the Ileak -model (3.1) or the INa,p -model (3.5) in Fig.3.8 is thesteady-state current-voltage (I-V) relation, IL (V ) or IL (V )+INa,p (V ) respectively, takenwith the minus sign, see Fig.3.5. Positive values of the right-hand-side F (V ) mean
negative I-V, corresponding to a net inward current that depolarizes the membrane.
Conversely, negative values mean positive I-V, corresponding to a net outward current
that hyperpolarizes the membrane.

60

3.2.2

One-Dimensional Systems

Equilibria

The next step in the qualitative analysis of any dynamical system is to find its equilibria
or rest points, that is, the values of the state variable where
F (V ) = 0(V is an equilibrium).

At each such point V̇ = 0, the state variable V does not change. In the context ofmembrane potential dynamics, equilibria correspond to the points where the steadystate I-V curve passes zero. At each such point there is a balance of the inward andoutward currents so that the net transmembrane current is zero, and the membrane
voltage does not change. (Incidentally, the part libra in the Latin word aequilibriummeans balance).
The IK - and Ih -models mentioned in section 3.1 can have only one equilibrium
because their I-V relations I(V ) are monotonic increasing functions. The corresponding
functions F (V ) are monotonic decreasing and can have only one zero.
In contrast, the INa,p - and IKir -models can have many equilibria because their I-V
curves are not monotonic, and hence there is a possibility for multiple intersections
with the V -axis. For example, there are three equilibria in Fig.3.8b corresponding to
the resting state (around −53 mV), the threshold state (around −40 mV), and theexcited state (around 30 mV). Each equilibrium corresponds to the balance of theoutward leak current and partially (rest), moderately (threshold), or fully (excited)
activated persistent Na+ inward current. Throughout this book we denote equilibria
as small open or filled circles, depending on their stability, as in Fig.3.8.

3.2.3

Stability

If the initial value of the state variable is exactly at equilibrium, then V̇ = 0 and thevariable will stay there forever. If the initial value is near the equilibrium, the statevariable may approach the equilibrium or diverge from it. Both cases are depicted in
Fig.3.8. We say that an equilibrium is asymptotically stable if all solutions starting
sufficiently near the equilibrium will approach it as t → ∞.
Stability of an equilibrium is determined by the signs of the function F around
it. The equilibrium is stable when F (V ) changes the sign from “plus” to “minus” as
V increases, as in Fig.3.8a. Obviously, all solutions starting near such an equilibrium
converge to it. Such an equilibrium “attracts” all nearby solutions, so it is called an
attractor. A stable equilibrium point is the only type of attractor that can exist in onedimensional continuous dynamical systems defined on a state line R. Multidimensional
systems can have other attractors, e.g., limit cycles.
The differences between stable, asymptotically stable, and exponentially stableequilibria are discussed in exercise 18 at the end of the chapter. The reader is also
encouraged to solve exercise 4 (piecewise continuous F (V )).

One-Dimensional Systems

61

stableequilibrium

unstableequilibrium

F(V)

F(V)

V

V

negative slope

positive slope

F (V)<0

F (V)>0

Figure 3.9: The sign of the slope,
λ = F ′ (V ), determines the stabilityof the equilibrium.

F(V)>0
?
+

-

? +

-

+

-

V

?
?
F(V)<0Figure 3.10: Two stable equilibrium points must be separated by at least one unstableequilibrium point because F (V ) has to change the sign from “minus” to “plus”.

3.2.4

Eigenvalues

A sufficient condition for an equilibrium to be stable is that the derivative of thefunction F with respect to V at the equilibrium is negative, provided the function isdifferentiable. We denote this derivative here by
λ = F ′ (V ) ,(V is an equilibrium; that is, F (V ) = 0)

and note that it is the slope of the graph of F at the point V (see Fig.3.9). Obviously,
when the slope, λ, is negative, the function changes the sign from “plus” to “minus”,
and the equilibrium is stable. Positive slope λ implies instability. The parameter λdefined above is the simplest example of an eigenvalue of an equilibrium. We introduce
eigenvalues formally in chapter 4 and show that they play an important role in defining
the types of equilibria of multidimensional systems.

3.2.5

Unstable Equilibria

If a one-dimensional system has two stable equilibrium points, then they must be
separated by at least one unstable equilibrium point, as we illustrate in Fig.3.10. (Thismay not be true in multidimensional systems.) Indeed, a continuous function F has
to change the sign from “minus” to “plus” somewhere in between those equilibria;that is, it has to cross the V -axis at some point, as in Fig.3.8b. This point would
be an unstable equilibrium, since all nearby solutions diverge from it. In the context

62

One-Dimensional Systems

V

F(V)
xxxxx
xxxxx
xxxxx
xxxxx
xxxxx
- F(v)dv
xxxxx
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xxxxx
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Figure 3.11: Mechanistic interpretation of stable and unstableequilibria. A massless (inertiafree) ball moves toward energy
minima with the speed proportional to the slope.
A onedimensional system V̇ = F (V )has the energy landscape E(V ) =RV− −∞ F (v) dv (see exercise 17).
Zeros of F (V ) with negative (positive) slope correspond to minima(maxima) of E(V ).

of neuronal models, unstable equilibria lie in the region of the steady-state I-V curvewith negative conductance. (Please, check that this is in accordance with the fact that
F (V ) = −I(V )/C; see Fig.3.5.) An unstable equilibrium is sometimes called a repeller.
Attractors and repellers have a simple mechanistic interpretation depicted in Fig.3.11.
If the initial condition V0 is set to an unstable equilibrium point, then the solution
will stay at this unstable equilibrium; that is, V (t) = V0 for all t, at least in theory. In
practice, the location of an equilibrium point is known only approximately. In addition,
small noisy perturbations that are always present in biological systems can make V (t)deviate slightly from the equilibrium point. Because of instability, such deviations will
grow, and the state variable V (t) will eventually diverge from the repelling equilibrium
the same way that the ball set at the top of the hill in Fig.3.11 will eventually rolldownhill. If the level of noise is low, it could take a long time to diverge from therepeller.

3.2.6

Attraction Domain

Even though unstable equilibria are hard to see experimentally, they still play an important role in dynamics, since they separate attraction domains. Indeed, the ball in
Fig.3.11 could go left or right, depending on which side of the hilltop it is on initially.
Similarly, the state variable of a one-dimensional system decreases or increases, depending on which side of the unstable equilibrium the initial condition is, as one canclearly see in Fig.3.8b.
In general, the basin of attraction or attraction domain of an attractor is the setOne-Dimensional Systems

63
Unstable
Equilibrium
V
F(V)

Attraction Domain

Attraction Domain

Figure 3.12: Two attraction domains in a one-dimensional system are separated by theunstable equilibrium.
of all initial conditions that lead to the attractor. For example, the attraction domainof the equilibrium in Fig.3.8a is the entire voltage range. Such an attractor is called
global. In Fig.3.12 we plot attraction domains of two stable equilibria. The middle
unstable equilibrium is always the boundary of the attraction domains.

3.2.7

Threshold and Action Potential

Unstable equilibria play the role of thresholds in one-dimensional bistable systems, i.e.,
in systems having two attractors. We illustrate this in Fig.3.13, which is believed todescribe the essence of the mechanism of bistability in many neurons. Suppose the statevariable is initially at the stable equilibrium point marked “state A” in the figure, andsuppose that perturbations can kick it around the equilibrium. Small perturbations
may not kick it over the unstable equilibrium so that the state variable continues to bein the attraction domain of “state A”. We refer to such perturbations as subthreshold.
In contrast, we refer to perturbations as superthreshold (or suprathreshold) if they
are large enough to push the state variable over the unstable equilibrium so that it
becomes attracted to the “state B”. We see that the unstable equilibrium acts as a
threshold that separates two states.
The transition between two stable states separated by a threshold is relevant to themechanism of excitability and generation of action potentials in many neurons, which
is illustrated in Fig.3.14. In the INa,p -model (3.5) with the I-V relation in Fig.3.5 theexistence of the resting state is largely due to the leak current IL , while the existenceof the excited state is largely due to the persistent inward Na+ current INa,p . Small(subthreshold) perturbations leave the state variable in the attraction domain of therest state, while large (superthreshold) perturbations initiate the regenerative process –the upstroke of an action potential – and the voltage variable becomes attracted to theexcited state. Generation of the action potential must be completed via repolarization,

64

One-Dimensional Systems

V
state A

thresholdAttraction domainof state A

state B

Attraction domainof state B

F(V)

xxxxx
xxxxx
xxxxx
xxxxx
subthreshold
xxxxx
xxxxx
perturbation
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Figure 3.13: Unstable equilibrium plays the role of a threshold that separates two
attraction domains.

One-Dimensional Systems

65

repolarization(another variable)

e
tiv
ra
ne e
ge ok
re pstr
u

xxxx
xxxx
superthreshold
xxxx
xxxx
xxxx
perturbation
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IK
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Figure 3.14: Mechanistic illustration of the mechanism ofgeneration of an action potential.

which moves V back to the resting state. Typically, repolarization occurs because ofa relatively slow inactivation of Na+ current and/or slow activation of an outward K+
current, which are not taken into account in the one-dimensional system (3.5). To
account for such processes, we consider two-dimensional systems in the next chapter.
Recall that the parameters of the INa,p -model (3.5) were obtained from a corticalpyramidal neuron. In Fig.3.15 (left), we stimulate (in vitro) the cortical neuron by
short (0.1 ms) strong pulses of current to reset its membrane potential to various
initial values, and interpret the results using the INa,p -model. Since activation of theNa+ current is not instantaneous in real neurons, we allow variable m to converge tom∞ (V ), and ignore the 0.3-ms transient activity that follows each pulse. We also ignore
40

20 mV

stable equilibrium

30

1 ms

20
10
+30 mVactionpotentials-40 mVmembrane potential (mV)membrane potential (mV)strongdepolarization

0
-10
-20
-30
unstable equilibrium

-40
-50

pulses of current-60

stable equilibrium
0

0.2

0.4

0.6
0.8
time (ms)1

1.2

Figure 3.15: Upstroke dynamics of layer 5 pyramidal neuron in vitro (compare with
the INa,p -model (3.5) in Fig.3.8b).

66

One-Dimensional Systems

membranepotential

15 mV-59 mV-75 mV
100 pA

2s

Figure 3.16: Membrane potential bistability in a cat TC neuron in the presence of
ZD7288 (pharmacological blocker of Ih . (Modified from Fig. 6B of Hughes et al. 1999).
(a)

(d)
(a)
(b)

(b)
(b)
(c)
I+F(V)
I

(c)

V

V

Figure 3.17: Bistability and hysteresis loop as I changes.
the initial segment of the downstroke of the action potential, and plot the magnificationof the voltage traces in Fig.3.15 (right). Comparing this figure with Fig.3.8b we see that
the INa,p -model is a reasonable one-dimensional approximation of the action potential
upstroke dynamics. It predicts the value of the resting (−53 mV), the instantaneousthreshold (−40 mV), and the excited (+30 mV) states of the cortical neuron.

3.2.8

Bistability and Hysteresis

Systems having two (many) coexisting attractors are called bistable (multistable). Many
neurons and neuronal models, such as the Hodgkin-Huxley model, exhibit bistability
between resting (equilibrium) and spiking (limit cycle) attractors. Some neurons can
exhibit bistability of two stable resting states in the subthreshold voltage range, forexample, −59 mV and −75 mV in the thalamocortical neurons (Hughes et al. 1999)
depicted in Fig.3.16, or −50 mV and −60 mV in mitral cells of the olfactory bulb
(Heyward et al. 2001), or −45 mV and −60 mV in Purkinje neurons. Brief inputs
can switch such neurons from one state to the other, as in Fig.3.16. Though theionic mechanisms of bistability are different in the three neurons, the mathematicalmechanism is the same.
Consider a one-dimensional system V̇ = I + F (V ) with function F (V ) having aOne-Dimensional Systems

67

cubic N-shape. Injection of a DC current I shifts the function I + F (V ) up or down.
When I is negative, the system has only one equilibrium, depicted in Fig.3.17a. As weremove the injected current I, the system becomes bistable, as in Fig.3.17b, but its stateis still at the left equilibrium. As we inject positive current, the left stable equilibrium
disappears via another saddle-node bifurcation, and the state of the system jumps tothe right equilibrium, as in Fig.3.17c. But as we slowly remove the injected current
that caused the jump and go back to Fig.3.17b, the jump to the left equilibrium does
not occur until a much lower value corresponding to Fig.3.17a is reached. The failureof the system to return to the original value when the injected current is removed iscalled hysteresis. If I were a slow V -dependent variable, then the system could exhibit
relaxation oscillations depicted in Fig.3.17d and described in the next chapter.

3.3

Phase Portraits

An important component in the qualitative analysis of any dynamical system is reconstruction of its phase portrait. It depicts all stable and unstable equilibria (as black
and white circles, respectively), representative trajectories, and corresponding attraction domains in the system’s state/phase space, as we illustrate in Fig.3.18. The phase
portrait is a geometrical representation of system dynamics. It depicts all possible
evolutions of the state variable and how they depend on the initial state. Looking at
the phase portrait, one immediately gets all important information about the system’squalitative behavior without even knowing the equation for F .

+

-

- +

+

-

- +

+

-

- +

+

-

V

Function F(V)

Attraction domains

V
Phase Portrait

Figure 3.18: Phase portrait of a one-dimensional system V̇ = F (V ).

68

One-Dimensional Systems
F1(V)

V

V

Figure 3.19: Two “seemingly different”
dynamical systems, V̇ = F1 (V ) andV̇ = F2 (V ), are topologically equivalent. Hence they have qualitatively
similar dynamics.

F2(V)

F1(V)
V

?

F2(V)
V

3.3.1

Figure 3.20: Two “seemingly alike”
dynamical systems V̇ = F1 (V ) andV̇ = F2 (V ) are not topologicallyequivalent, hence they do not havequalitatively similar dynamics. (The
first system has three equilibria, while
the second system has only one.)

Topological Equivalence

Phase portraits can be used to determine qualitative similarity of dynamical systems.
In particular, two one-dimensional systems are said to be topologically equivalent when
the phase portrait of one of them, treated as a piece of rubber, can be stretched or
shrunk to fit the other one, as in Fig.3.19. Topological equivalence is a mathematicalconcept that clarifies the imprecise notion of “qualitative similarity”, and its rigorous
definition is provided, for instance, by Guckenheimer and Holmes (1983).
The stretching and shrinking of the “rubber” phase space are topological transformations that do not change the number of equilibria or their stability. Thus, twosystems having different numbers of equilibria cannot be topologically equivalent and,hence, they have qualitatively different dynamics, as we illustrate in Fig.3.20. Indeed,
the top system is bistable because it has two stable equilibria separated by an unstable one. The evolution of the state variable depends on which attraction domain theinitial condition is in initially. Such a system has “memory” of the initial condition.
Moreover, sufficiently strong perturbations can switch it from one equilibrium state toanother. In contrast, the bottom system in Fig.3.20 has only one equilibrium, which isa global attractor, and the state variable converges to it regardless of the initial condition. Such a system has quite primitive dynamics, and it is topologically equivalent
to the linear system (3.1).

One-Dimensional Systems

69

λ(V-Veq)

V
Veq
F(V)

3.3.2

Figure 3.21: Hartman-Grobman theorem: The nonlinear system V̇ = F (V )is topologically equivalent to the linear
one V̇ = λ(V − Veq ) in the local (shaded)neighborhood of the hyperbolic equilibrium Veq .

Local Equivalence and the Hartman-Grobman Theorem

In computational neuroscience, we usually face quite complicated systems describing
neuronal dynamics. A useful strategy is to replace such systems with simpler ones
having topologically equivalent phase portraits. For example, both systems in Fig.3.19
are topologically equivalent to V̇ = V − V 3 (readers should check this), which is easier
to deal with analytically.
Quite often we cannot find a simpler system that is topologically equivalent to ourneuronal model on the entire state line R. In this case, we make a sacrifice: we restrict
our analysis to a small neighborhood of the line R (e.g., a neighborhood of the restingstate or of the threshold), and study behavior locally in this neighborhood.
An important tool in the local analysis of dynamical systems is the HartmanGrobman theorem, which says that a nonlinear one-dimensional systemV̇ = F (V )sufficiently near an equilibrium V = Veq is locally topologically equivalent to the linear
systemV̇ = λ(V − Veq ) ,(3.8)
provided the eigenvalue
λ = F ′ (Veq )at the equilibrium is nonzero, that is, the slope of F (V ) is nonzero. Such an equilibrium
is called hyperbolic. Thus, nonlinear systems near hyperbolic equilibria behave as if
they were linear, as in Fig.3.21.
It is easy to find the exact solution of the linearized system (3.8) with an initialcondition V (0) = V0 . It is V (t) = Veq + eλt (V0 − Veq ) (readers should check by differentiating). If the eigenvalue λ < 0, then eλt → 0 and V (t) → Veq as t → ∞, sothat the equilibrium is stable. Conversely, if λ > 0, then eλt → ∞ meaning that theinitial displacement, V0 − Veq , grows with time and the equilibrium is unstable. Thus,
the linearization predicts qualitative dynamics at the equilibrium, and the quantitative
rate of convergence/divergence to/from the equilibrium.
If the eigenvalue λ = 0, then the equilibrium is non-hyperbolic, and analysis of
the linearized system V̇ = 0 cannot describe the behavior of the nonlinear system.
Typically, non-hyperbolic equilibria arise when the system undergoes a bifurcation,i.e., a qualitative change of behavior, which we consider next. To study stability, weneed to consider higher-order terms of the Taylor series of F (V ) at Veq .

70

One-Dimensional Systems

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xxxxx
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3.3.3

Figure 3.22: Mechanistic illustration of a bifurcation as a changeof the landscape.

Bifurcations

The final and most advanced step in the qualitative analysis of any dynamical systemis the bifurcation analysis. In general, a system is said to undergo a bifurcation whenits phase portrait changes qualitatively. For example, the energy landscape in Fig.3.22
changes so that the system is no longer bistable. The precise mathematical definitionof a bifurcation will be given later.
Qualitative change of the phase portrait may or may not necessarily reveal itself
in a qualitative change of behavior, depending on the initial conditions. For example,
there is a bifurcation in Fig.3.23 (left), but no change of behavior, because the ballremains in the attraction domain of the right equilibrium. To see the change, weneed to drop the ball at different initial conditions and observe the disappearance ofthe left equilibrium. In the same veins, there is no bifurcation in Fig.3.23 (middle
and right), – the phase portraits in each column are topologically equivalent, but theapparent change of behavior is caused by the expansion of the attraction domain of
the left equilibrium or by the external input. Dropping the ball at different locations
would result in the same qualitative picture – two stable equilibria whose attraction
domains are separated by the unstable equilibrium. When mathematicians talk aboutbifurcations, they assume that all initial conditions could be sampled, in which casebifurcations do result in a qualitative change of behavior of the system as a whole.
To illustrate the importance of sampling all initial conditions, let us consider the inOne-Dimensional Systems

71

bifurcation
but no change of behavior

change of behavior but no bifurcation

pulsedinput

Figure 3.23: Bifurcations are not equivalent to qualitative change of behavior if thesystem is started with the same initial condition or subject to external input.
vitro recordings of a pyramidal neuron in Fig.3.24. We inject 0.1-ms strong pulses ofcurrent of various amplitudes to set the membrane potential to different initial values.
Right after each pulse, we inject a 4-ms step of DC current of amplitude I = 0, I = 16,
or I = 60 pA. The case of I = 0 pA is the same as in Fig.3.15, so some initial conditions
result in upstroke of the action potential, while others do not. When I = 60 pA, allbistablebistablemonostable

10 mV
membrane potential, mV

1 ms-40 mVpulsesdc-current I=60

dc-current I=16

dc-current I=0
pulsespulsesFigure 3.24: Qualitative change of the upstroke dynamics of a layer 5 pyramidal neuron
from rat visual cortex (the same neuron as in Fig.3.15).

72

One-Dimensional Systems
I=0

80
60
40
20
0

rest

thresholdbifurcation

100

I=16

80
60
40
20
0

tangent pointmonostability

100
F(V)
80
60
40
20
0

-50
0
50
membrane potential, V (mV)

excited state

20
0
-20
threshold
rest

-40
-60

40
membrane potential, V (mV)

F(V)

V(t)

40
membrane potential, V (mV)

F(V)

V(t)

excited state

20
0
-20
-40
-60

I=60

V(t)

40
membrane potential, V (mV)bistability
100

excited state

20
0
-20
-40
-60

0

1

2
3
time (ms)45Figure 3.25: Bifurcation in the INa,p -model (3.5): The resting state and the thresholdstate coalesce and disappear when the parameter I increases.
initial conditions result in the generation of an action potential. Clearly, a change of
qualitative behavior occurs for some I between 0 and 60.
To understand the qualitative dynamics in Fig.3.24, we consider the one-dimensional
INa,p -model (3.5) having different values of the parameter I and depict its trajectories
in Fig.3.25. One can clearly see that the qualitative behavior of the model depends
on whether I is greater or less than 16. When I = 0 (top of Fig.3.25), the system
is bistable. The resting and the excited states coexist. When I is large (bottom of
Fig.3.25) the resting state no longer exists because the leak outward current cannot
balance the large injected DC current I and the inward Na+ current.
What happens when we change I past 16? The answer lies in the details of thegeometry of the right-hand-side function F (V ) of (3.5) and how it depends on theOne-Dimensional Systems

73
I=18

F(V)

no equilibria

I=17
I=16

bifurcation

I=15
I=14
I=13
I=12

two equilibria

I=11
tangent

V

point

stable equilibria

unstable equilibria

Figure 3.26: Saddle-node bifurcation: As the graph of the function F (V ) is lifted up,
the stable and unstable equilibria approach each other, coalesce at the tangent point,
and then disappear.
parameter I. Increasing I elevates the graph of F (V ). The higher the graph of F (V ) is,
the closer its intersections with the V -axis are, as we illustrate in Fig.3.26, which depicts
only the low-voltage range of the system. When I approaches 16, the distance between
the stable and unstable equilibria vanishes; the equilibria coalesce and annihilate eachother. The value I = 16, at which the equilibria coalesce, is called the bifurcation
value. This value separates two qualitatively different regimes. When I is near to but
less than 16, the system has three equilibria and bistable dynamics. The quantitative
features, such as the exact locations of the equilibria, depend on the particular valuesof I, but the qualitative behavior remains unchanged no matter how close I is to thebifurcation value. In contrast, when I is near to but greater than 16, the system has
only one equilibrium and monostable dynamics.
In general, a dynamical system may depend on a vector of parameters, say p. A
point in the parameter space, say p = a, is said to be a regular or non-bifurcation
point, if the system’s phase portrait at p = a is topologically equivalent to the phase
portrait at p = c for any c sufficiently close to a. For example, the value I = 13 inFig.3.26 is regular, since the system has topologically equivalent phase portraits for all
I near 13. Similarly, the value I = 18 is also regular. Any point in the parameter spacethat is not regular is called a bifurcation point. Namely, a point p = b is a bifurcation
point if the system’s phase portrait at p = b is not topologically equivalent to the phase
portrait at a point p = c no matter how close c is to b. The value I = 16 in Fig.3.26 isa bifurcation point. It corresponds to the saddle-node (also known as fold or tangent)bifurcation for reasons described later. It is one of the simplest bifurcations consideredin this book.

74

One-Dimensional Systems
saddle-node

not saddle-nodeFV
non-hyperbolic

hyperbolic

hyperbolic

non-degenerate

degenerate

degeneratetransversal

not transversal

not transversal

Figure 3.27: Geometrical illustration of the three conditions defining saddle-node bifurcations. Arrows denote the direction of displacement of the function F (V, I) as thebifurcation parameter I changes.

3.3.4

Saddle-Node (Fold) Bifurcation

In general, a one-dimensional systemV̇ = F (V, I),
having an equilibrium point V = Vsn for some value of the parameter I = Isn (i.e.,
F (Vsn , Isn ) = 0), is said to be at a saddle-node bifurcation (sometimes called a foldbifurcation) if the following mathematical conditions, illustrated in Fig.3.27, are satisfied:• Non-hyperbolicity. The eigenvalue λ at Vsn is zero; that is,
λ = FV (V, Isn ) = 0(at V = Vsn ),where FV denotes the derivative of F with respect to V , that is, FV = ∂F/∂V .
Equilibria with zero or pure imaginary eigenvalues are called non-hyperbolic.
Geometrically, this condition implies that the graph of F has horizontal slope atthe equilibrium.
• Non-degeneracy. The second-order derivative with respect to V at Vsn is nonzero;that is,FV V (V, Isn ) ̸= 0
(at V = Vsn ).
Geometrically, this means that the graph of F looks like the square parabola V 2
in Fig.3.27.

One-Dimensional Systems

75• Transversality. The function F (V, I) is non-degenerate with respect to the bifurcation parameter I; that is,
FI (Vsn , I) ̸= 0

(at I = Isn ),where FI denotes the derivative of F with respect to I. Geometrically, thismeans that as I changes past Isn , the graph of F approaches, touches, and thenintersects the V -axis.
Saddle-node bifurcation results in appearance or disappearance of a pair of equilibria,
as in Fig.3.26. None of the six examples on the right-hand side of Fig.3.27 can undergoa saddle-node bifurcation because at least one of the conditions above is violated.
The number of conditions involving strict equality (“=”) is called the codimension of a bifurcation. The saddle-node bifurcation has codimension-1 because there is
only one condition involving “=”; the other two conditions involve inequalities (“̸=”).
Codimension-1 bifurcations can be reliably observed in systems with one parameter.
It is an easy exercise to check that the one-dimensional systemV̇ = I + V 2(3.9)

is at saddle-node bifurcation when V = 0 and I = 0 (readers should check all three
conditions). This system is called the topological normal form for saddle-node bifurcation. The phase portraits of this system are topologically equivalent to those depicted
in Fig.3.26, except that the bifurcation occurs at I = 0, and not at I = 16.

3.3.5

Slow Transition

All physical, chemical, and biological systems near saddle-node bifurcations possesscertain universal features that do not depend on particulars of the systems. Consequently, all neural systems near such a bifurcation share common neurocomputational
properties, which we will discuss in detail in chapter 7. Here we take a look at one suchproperty – slow transition through the ruins (or ghost) of the resting state attractor,
which is relevant to the dynamics of many neocortical neurons.
In Fig.3.28 we show the function F (V ) of the system (3.5) with I = 30 pA, which isgreater than the bifurcation value 16 pA, and the corresponding behavior of a corticalneuron (compare with Fig.3.15). The system has only one attractor, the excited state,
and any solution starting from an arbitrary initial condition should quickly approach
this attractor. However, the solutions starting from the initial conditions around −50
mV do not seem to hurry. Instead, they slow down near −46 mV and spend a considerable amount of time in the voltage range corresponding to the resting state, as if
the state were still present. The closer I is to the bifurcation value, the more time themembrane potential spends in the neighborhood of the resting state. Obviously, sucha slow transition cannot be explained by a slow activation of the inward Na+ current,
since Na+ activation in a cortical neuron is practically instantaneous.

76

One-Dimensional Systems
10 mV

0.5 msF(V)ExcitedAttractor
ruins

ExcitedV-40 mVslow transition

membrane potential (mV)

Attractor
ruins

0

Figure 3.28: Slow transition through the ghost of the resting state attractor in a corticalpyramidal neuron with I = 30 pA (the same neuron as in Fig.3.15). Even though theresting state has already disappeared, the function F (V ), and hence the rate of change,V̇ , is still small when V ≈ −46 mV.
20 mV
100 msslow transition-60 mV43.1 pA
0 pA

Figure 3.29: A 400-ms latency in a layer 5 pyramidal neuron of rat visual cortex.

The slow transition occurs because the neuron or the system (3.5) in Fig.3.28 isnear a saddle-node bifurcation. Even though I is greater than the bifurcation value,
and the resting state attractor is already annihilated, the function F (V ) is barely above
the V -axis at the “annihilation site”. In other words, the resting state attractor has
already been ruined, but its “ruins” (or its “ghost”) can still be felt, because
V̇ = F (V ) ≈ 0(at attractor ruins, V ≈ −46 mV),

as one can see in Fig.3.28. In chapter 7 we will show how this property explains theability of many neocortical neurons, such as the one in Fig.3.29, to generate repetitive
action potentials with small frequency, and how it predicts that all such neurons,considered as dynamical systems, reside near saddle-node bifurcations.

One-Dimensional Systems

77

-40

membrane potential, V (mV)unsta

ble equilibria-45
saddle-node(fold) bifurcation-50
stable

-55

0

ibria
equil

5

10
15
injected dc-current I, (pA)

20

25

Figure 3.30: Bifurcation diagram of the system in Fig.3.26.

3.3.6

Bifurcation Diagram

The final step in the geometrical bifurcation analysis of one-dimensional systems is theanalysis of bifurcation diagrams, which we do in Fig.3.30 for the saddle-node bifurcation
shown in Fig.3.26. To draw the bifurcation diagram, we determine the locations of thestable and unstable equilibria for each value of the parameter I and plot them as
white or black circles in the (I, V ) plane in Fig.3.30. The equilibria form two branches
that join at the fold point corresponding to the saddle-node bifurcation (hence thealternative name fold bifurcation). The branch corresponding to the unstable equilibria
is dashed to stress its instability. As the bifurcation parameter I varies from left toright, passing through the bifurcation point, the stable and unstable equilibria coalesce
and annihilate each other. As the parameter varies from right to left, two equilibria– one stable and one unstable – appear from a single point. Thus, depending on
the direction of movement of the bifurcation parameter, the saddle-node bifurcation
explains disappearance or appearance of a new stable state. In any case, the qualitative
behavior of the systems changes exactly at the bifurcation point.

3.3.7

Bifurcations and I-V Relations

In general, determining saddle-node bifurcation diagrams of neurons may be a daunting
mathematical task. However, it is a trivial exercise when the bifurcation parameter isthe injected DC current I. In this case, the bifurcation diagram, such as the one in
Fig.3.30, is the steady-state I-V relation I∞ (V ) plotted on the (I, V )-plane. Indeed,
the equation
C V̇ = I − I∞ (V ) = 0

78

One-Dimensional Systems

steady-state current (pA)200

I-V relation

0

I=16
I=-100

-200
-400

I

(V)

-600

Figure 3.31: Equilibria are intersections of the steady-state I-V curve
I∞ (V ) and a horizontal line I = const.

-800
-100

-80

-60

-40
-20
0
20
membrane potential, V (mV)40states that V is an equilibrium if and only if the net membrane current, I − I∞ (V ), iszero. For example, equilibria of the INa,p -model are solutions of the equation
I∞ (V )z}|{
0 = I − (gL (V − EL ) + gNa m∞ (V )(V − ENa )) ,which follows directly from (3.5). In Fig.3.31 we illustrate how to find the equilibria
geometrically: We plot the steady-state I-V curve I∞ (V ) and draw a horizontal linewith altitude I. Any intersection satisfies the equation I = I∞ (V ), and hence is anequilibrium (stable or unstable). Obviously, when I increases past 16, the saddle-nodebifurcation occurs.
Note that the equilibria are points on the curve I∞ (V ), so flipping and rotating the
curve by 90◦ , as we do in Fig.3.32 (left), results in a complete saddle-node bifurcation
diagram. The diagram conveys all important information about the qualitative behavior of the INa,p -model in a very condensed manner. The three branches of the S-shaped
40

excited200saddle-node(fold) bifurcation

thre

sho
ld s
tate

s

-20
-40

statessaddle-node(fold) bifurcation

-60
-80

-100

es

tat

ts

res

I

20
0
-20
-40
-60
-80

16

-120
-1000

(V)

membrane potential, V (mV)membrane potential, V (mV)

40

-100
-890

-500
0
injected dc-current, I (pA)

-120
-1000

-500
0
injected dc-current, I (pA)

Figure 3.32: Bifurcation diagram of the INa,p -model (3.5).

One-Dimensional Systems

79
I=-400V(t)Bistability

Threshold
F(V)

I=-890
Bifurcation

Rest

Tangent Point

F(V)

Rest

F(V)

membrane potential, V

Rest

V(t)
membrane potential, V

Monostability

Rest

V(t)

I=-1000

0

Thresholdmembrane potential, V

0

Excitedmembrane potential, V

0

Excited

Rest

Rest
time

Figure 3.33: Bifurcation in the INa,p -model (3.5). The excited state and the thresholdstate coalesce and disappear when the parameter I is sufficiently small.
curve, which is the 90◦ -rotated and flipped copy of the N-shaped I-V curve, correspondto the resting, threshold, and excited states of the model. Each slice I = const represents the phase portrait of the system, as we illustrate in Fig.3.32 (right). Each point
where the branches fold (max or min of I∞ (V )) corresponds to a saddle-node bifurcation. Since there are two such folds, at I = 16 pA and at I = −890 pA, there are two
saddle-node bifurcations in the system. The first one, studied in Fig.3.25, corresponds
to the disappearance of the resting state. The other one, illustrated in Fig.3.33, corresponds to the disappearance of the excited state. It occurs because I becomes so
negative that the Na+ inward current is no longer strong enough to balance the leakoutward current and the negative injected DC current to keep the membrane in the
depolarized (excited) state.
Below, the reader can find more examples of bifurcation analysis of the INa,p - andIKir -models, which have nonmonotonic I-V relations and can exhibit multistability ofstates. The IK - and Ih -models have monotonic I-V relations, and hence only one
equilibrium state. These models cannot have saddle-node bifurcations, as the reader
is asked to prove in exercise 13 and 14.

One-Dimensional Systems

steady-state current (pA)

80
(Vsn, Isn)

20
0

I-V relation
-20
-40
-60

Isn-k(V-Vsn)2
I

(V)

-60
-40
membrane potential, V (mV)

-20

Figure 3.34: Magnification of the I-V curve in Fig.3.31 at the left knee shows that it
can be approximated by a square parabola.

3.3.8

Quadratic Integrate-and-Fire Neuron

Let us consider the topological normal form for the saddle-nodepp bifurcation (3.9). From2
0 = I + V we find that there are two equilibria, Vrest = − |I| and Vthresh = + |I|
when I < 0. The equilibria approach and annihilate each other via saddle-node bifurcation when I = 0, so there are no equilibria when I > 0. In this case, V̇ ≥ I and V (t)increases to infinity. Because of the quadratic term, the rate of increase also increases,
resulting in a positive feedback loop corresponding to the regenerative activation of theNa+ current. In exercise 15 we show that V (t) escapes to infinity in a finite time, whichcorresponds to the upstroke of the action potential. The same upstroke is generated
when I < 0, if the voltage variable is pushed beyond the threshold value Vthresh .
Considering infinite values of the membrane potential may be convenient from apurely mathematical point of view, but this has no physical meaning and there is no
way to simulate it on a digital computer. Instead, we fix a sufficiently large constant
Vpeak and say that (3.9) generated a spike when V (t) reached Vpeak . After the peak of
the spike is reached, we reset V (t) to a new value Vreset . The topological normal form
for the saddle-node bifurcation with the after-spike resetting,V̇ = I + V 2 ,if V ≥ Vpeak , then V ← Vreset(3.10)

is called the quadratic integrate-and-fire neuron. It is the simplest model of a spiking
neuron. The name stems from its resemblance to the leaky integrate-and-fire neuron
V̇ = I − V considered in chapter 8. In contrast to the common folklore, the leaky
neuron is not a spiking model because it does not have a spike generation mechanism,i.e., a regenerative upstroke of the membrane potential, whereas the quadratic neuron
does. We discuss this and other issues in detail in chapter 8.
In general, the quadratic integrate-and-fire model could be derived directly fromthe equation C V̇ = I − I∞ (V ) through approximating the steady-state I-V curve near

81

0

rest
Vreset
I<0

reset

I+V2
rest
0

Vreset
I>0

reset

I+V2

0

Vreset0
membrane potential, v

membrane potential, V

threshold

reset

I+V2

membrane potential, V

I<0membrane potential, V

One-Dimensional Systems
excitability

Vpeak
thresholdrest

0Vresetinput

bistability

0

Vpeak

Vreset
input

Vpeak

0
Vresettime

Figure 3.35: Quadratic integrate-and-fire neuron (3.10) with time-dependent input.
the resting state by the square parabola I∞ (V ) ≈ Isn − k(V − Vsn )2 , where k > 0 andthe peak of the curve (Vsn , Isn ) could easily be found experimentally (see Fig.3.34).
Approximating the I-V curve by other functions – for example I∞ (V ) = gleak (V −Vrest ) − kepV , results in other forms of the model, such as the exponential integrateand-fire model (Fourcaud-Trocme et al. 2003), which has certain advantages overthe quadratic form. Unfortunately, the model is not solvable analytically, and it isexpensive to simulate. The form I∞ (V ) = gleak (V − Vleak ) − k(V − Vth )2+ , where x+ = xwhen x > 0 and x+ = 0 otherwise, combines the advantages of both models. The
parameters Vpeak and Vreset are derived from the shape of the spike. Normalization ofvariables and parameters results in the form (3.10) with Vpeak = 1.
In Fig.3.35 we simulate the quadratic integrate-and-fire neuron to illustrate a number of its features, which will be described in detail in subsequent chapters using
conductance-based models. First, the neuron is an integrator; each input pulse in
Fig.3.35 (top), pushes V closer to the threshold value; the higher the frequency of the
input, the sooner V reaches the threshold and starts the upstroke of a spike. The neuron is monostable when Vreset ≤ 0 and can be bistable otherwise. Indeed, the first spikein Fig.3.35 (middle) is evoked by the input, but the subsequent spikes occur because
the reset value is superthreshold.
The neuron can be Class 1 or Class 2 excitable, depending on the sign of Vreset .

82

One-Dimensional Systems

Suppose the injected current I slowly ramps up fromp a negative to a positive value.
The membrane potential follows the resting state − |I| in a quasi-static fashion until
the bifurcation point I = 0 is reached. At this moment, the neuron starts to fire tonic
spikes. In the monostable case Vreset < 0 in Fig.3.35 (bottom), the membrane potential
is reset to the left of the ghost of the saddle-node point (see section 3.3.5), thereby
producing spiking with an arbitrary small frequency, and hence Class 1 excitability.
Because of the recurrence, such a bifurcation is called saddle-node on invariant circle.
Many pyramidal neurons in mammalian neocortex exhibit such a bifurcation. In contrast, in the bistable case Vreset > 0, not shown in the figure, the membrane potential
is reset to the right of the ghost, no slow transition is involved, and the tonic spiking
starts with a nonzero frequency. (As an exercise, explain why there is a noticeable
latency [delay] to the first spike right after the bifurcation.) This type of behavior istypical in spiny projection neurons of neostriatum and basal ganglia, as we show inchapter 8.

Review of Important Concepts• The one-dimensional dynamical system V̇ = F (V ) describes how
the rate of change of V depends on V . Positive F (V ) means V
increases; negative F (V ) means V decreases.
• In the context of neuronal dynamics, V is often the membrane potential, and F (V ) is the steady-state I-V curve taken with the minussign.
• A zero of F (V ) corresponds to an equilibrium of the system. (Indeed,
if F (V ) = 0, then the state of the system, V , neither increases nor
decreases.)• An equilibrium is stable when F (V ) changes the sign from “plus” to“minus”. A sufficient condition for stability is that the eigenvalue
λ = F ′ (V ) at the equilibrium be negative.
• A phase portrait is a geometrical representation of the system’s dynamics. It depicts all equilibria, their stability, representative trajectories, and attraction domains.
• A bifurcation is a qualitative change of the system’s phase portrait.
• The saddle-node (fold) is a typical bifurcation in one-dimensional
systems. As a parameter changes, a stable and an unstable equilibrium approach, coalesce, and then annihilate each other.

One-Dimensional Systems

83

Bibliographical Notes
There is no standard textbook on dynamical systems theory. The classic book Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by Guckenheimer and Holmes (1983) plays the same role in the dynamical systems community
as the book Ion Channels of Excitable Membranes by Hille (2001) plays in the neuroscience community. A common feature of these books is that they are not suitable forbeginners.
Most textbooks on differential equations, such as Differential Equations and Dynamical Systems by Perko (1996), develop the theory starting with a comprehensive
analysis of linear systems, then applying it to local analysis of nonlinear systems, andthen discussing global behavior. To get to bifurcations, the reader has to go througha lot of daunting math, which is fun only for mathematicians. Here we follow an approach similar to that in Nonlinear Dynamics and Chaos by Strogatz (1994). Insteadof going from linear to nonlinear systems, we go from one-dimensional nonlinear systems (this chapter) to two-dimensional nonlinear systems (next chapter). Rather thanburdening the theory with a lot of mathematics, we use the geometrical approach tostimulate the reader’s intuition. (There is plenty of fun math in exercises and in later
chapters.)

Exercises
1. Consider a neuron having a Na+ current with fast activation kinetics. Assume
that inactivation of this current, as well as (in)activations of the other currents
in the neuron are much slower. Prove that the initial segment of action potential upstroke of this neuron can be approximated by the INa,p -model (3.5). Use
Fig.3.15 to discuss the applicability of this approximation.
2. Draw phase portraits of the systems in Fig.3.36. Clearly mark all equilibria, theirstability, attraction domains, and direction of trajectories. Determine the signsof eigenvalues at each equilibrium.
F(V)

F(V)

F(V)
V

V
a

b

V
c

Figure 3.36: Draw a phase portrait of the system V̇ = F (V ) with shown F (V ).
3. Draw phase portraits of the following systems:(a) ẋ = −1 + x2 ,84

One-Dimensional Systems

V

F1 (V)F1 (V)F1 (V)F2 (V)

V

V

d

V

V

c

b

V

F2 (V)F2 (V)F2 (V)a
F1 (V)V

V

F1 (V)F1 (V)VV

F2 (V)F2 (V)Ve

V

f

Figure 3.37: Which of the pairs correspond to topologically equivalent dynamical systems? (All intersections with the V -axis are marked as dots.)(b) ẋ = x − x3 .
Determine the eigenvalues at each equilibrium.
4. Determine stability of the equilibrium x = 0 and draw phase portraits of thefollowing piecewise continuous systems:

2x, if x < 0(a)ẋ =x, if x ≥ 0 −1, if x < 00, if x = 0
(b)
ẋ =

1, if x > 0
−2/x, if x ̸= 0(c)ẋ =0, if x = 0
5. Draw phase portraits of the systems in Fig.3.37. Which of the pairs in the figure
correspond to topologically equivalent dynamical systems?
6. (Saddle-node bifurcation) Draw the bifurcation diagram and representative phase
portraits of the system ẋ = a + x2 , where a is a bifurcation parameter. Find theeigenvalues at each equilibrium.

One-Dimensional Systems

85

1
0.8

2

20

h (V)

F(V)

0

Current

0.6
0.4

-20

1

I L (V)

-60
0.2
0
-140

-80
-70

0

Membrane Voltage (mV)

0

-40

-100
-140

I Kir (V)
EK

-1

EL

-70

Membrane Voltage (mV)

-2
0

-100

-50

0

Membrane Voltage (mV)

Figure 3.38: The IKir -model having injected current (I), leak current (IL ), and instantaneous K+ inward rectifier current (IKir ) and described by (3.11). Inactivation curveh∞ (V ) is modified from Wessel et al. (1999). Parameters: C = 1, I = 6, gL = 0.2,
EL = −50, gKir = 2, EK = −80, V1/2 = −76, k = −12 (see Fig.2.20).
7. (Saddle-node bifurcation) Use definition in section 3.3.4 to find saddle-node bifurcation points in the following systems:(a) ẋ = a + 2x + x2 ,(b) ẋ = a + x + x2 ,(c) ẋ = a − x + x2 ,(d) ẋ = a − x + x3 (Hint: Verify the non-hyperbolicity condition first.),(e) ẋ = 1 + ax + x2 ,(f) ẋ = 1 + 2x + ax2 ,where a is the bifurcation parameter.
8. (Pitchfork bifurcation) Draw the bifurcation diagram and representative phase
portraits of the system ẋ = bx − x3 , where b is a bifurcation parameter. Find theeigenvalues at each equilibrium.
9. Draw the bifurcation diagram of the IKir -model
instantaneous IKir}|{z
C V̇ = I − gL (V − EL ) − gKir h∞ (V )(V − EK ) ,(3.11)

using parameters from Fig.3.38 and treating I as a bifurcation parameter.
10. Derive an explicit formula that relates the position of the equilibrium in the
Hodgkin-Huxley model to the magnitude of the injected DC current I. Are there
any saddle-node bifurcations?
11. Draw the bifurcation diagram of the INa,p -model (3.5), using parameters from
Fig.3.39 and treating

86

One-Dimensional Systems
1

50

100

m (V)

50

0.4

Current

0.8

I L (V)

0.2

-100

0
-100

-150
-100

0.6

F(V)

I Na,p (V)

0
0

-50

-50

0

50

100

Membrane Voltage (mV)

EL

E Na
-50

0

50

100

-50
-100

Membrane Voltage (mV)

-50

0

50

100

Membrane Voltage (mV)

Figure 3.39: The INa,p -model with leak current (IL ) and persistent Na+ current (INa,p ),described by (3.5) with the right-hand-side function F (V ). Parameters: C = 1, I = 0,gL = 1, EL = −80, gNa = 2.25, ENa = 60, V1/2 = −20, k = 15 (see Fig.2.20).
1

100

50

m (V)

F(V)
Current

0.8
0.6
0.4

I L (V)50

0

I K(V)
0

0.2
0
-120

-100

-80

-60

-40

Membrane Voltage (mV)

-20

0

-50
-120

EK EL
-100

-80

-60

-40

-20

Membrane Voltage (mV)

0

-50
-120

-100

-80

-60

-40

-20

0

Membrane Voltage (mV)

Figure 3.40: The IK -model with leak current (IL ) and persistent K+ current (IK ),described by (3.12). Parameters: C = 1, gL = 1, EL = −80, gK = 1, EK = −90,
V1/2 = −53, k = 15 (see Fig.2.20).
(a) gL as a bifurcation parameter, or(b) EL as a bifurcation parameter.
12. Draw the bifurcation diagram of the IKir -model (3.11), using parameters from
Fig.3.38 and treating(a) gL as a bifurcation parameter, or(b) gKir as a bifurcation parameter.
13. Show that the IK -model in Fig.3.40
instantaneous IKz}|{
4
C V̇ = −gL (V − EL ) − gK m∞ (V )(V − EK ) .

(3.12)

cannot exhibit saddle-node bifurcation for V > EK . (Hint: Show that F ′ (V ) ̸= 0
for all V > EK .)One-Dimensional Systems

87

1

50

0.8

100

I L (V)h (V)Current

50

0.6
0.4

0

I h (V)

-50

-50

0.2
0
-120

-100

-80

-60

-40

-20

0

F(V)

0

-100
-120

Membrane Voltage (mV)

EL
-100

-80

Eh
-60

-40

-20

0

-100
-120

Membrane Voltage (mV)

-100

-80

-60

-40

-20

0

Membrane Voltage (mV)

Figure 3.41: The Ih -model with leak current (IL ) and “hyperpolarization-activated”
inward current Ih , described by (3.13). Parameters: C = 1, gL = 1, EL = −80, gh = 1,
Eh = −43, V1/2 = −75, k = −5.5 (Huguenard and McCormick 1992).
14. Show that the Ih -model in Fig.3.41,
instantaneous Ih}|{z
C V̇ = −gL (V − EL ) − gh h∞ (V )(V − Eh ) ,(3.13)

cannot exhibit saddle-node bifurcation for any V < Eh .
15. Prove that the upstroke of the spike in the quadratic integrate-and-fire neuron(3.9) has the asymptote 1/(c − t) for some c > 0.
16. (Cusp bifurcation) Draw the bifurcation diagram and representative phase portraits of the system ẋ = a + bx − x3 , where a and b are bifurcation parameters.
Plot the bifurcation diagram in the (a, b, x)-space and on the (a, b)-plane.
17. (Gradient systems) An n-dimensional dynamical system ẋ = f (x), with x =(x1 , . . . , xn ) ∈ Rn is said to be gradient when there is a potential (energy) function
E(x) such thatẋ = − grad E(x) ,wheregrad E(x) = (Ex1 , . . . , Exn )is the gradient of E(x). Show that all one-dimensional systems are gradient.
(Hint: See Fig.3.11.) Find potential (energy) functions for the following onedimensional systems(a) V̇ = 0 ,(d) V̇ = −1 + V 2 ,(b) V̇ = 1 ,(e) V̇ = V − V 3 ,(c) V̇ = −V ,(f) V̇ = − sin V .

18. Consider a dynamical system ẋ = f (x) , x(0) = x0 .
(a) Stability. An equilibrium y is stable if any solution x(t) with x0 sufficiently
close to y remains near y for all time. That is, for all ε > 0 there exists
δ > 0 such that if |x0 − y| < δ then |x(t) − y| < ε for all t ≥ 0.

88

One-Dimensional Systems(b) Asymptotic stability. A stable equilibrium y is asymptotically stable if all
solutions starting sufficiently close to y approach it as t → ∞. That is, if
δ > 0 can be chosen from the definition above so that limt→∞ x(t) = y.
(c) Exponential stability. A stable equilibrium y is said to be exponentiallystable when there is a constant a > 0 such that |x(t) − y| < exp(−at) for all
x0 near y and all t ≥ 0.
Prove that (c) implies (b), and (b) implies (a). Show that (a) does not imply(b) and (b) does not imply (c). That is, present a system having stable but not
asymptotically stable equilibrium, and a system having asymptotically but notexponentially stable equilibrium.

19. (INMDA -model) Show that voltage-dependent activation of NMDA synaptic receptors in a passive dendritic tree with a constant concentration of glutamate ismathematically equivalent to the INa,p -model.

Chapter 4
Two-Dimensional Systems
In this chapter we introduce methods of phase plane analysis of two-dimensional systems. Most concepts will be illustrated using the INa,p +IK -model in Fig.4.1:
instantaneous INa,p
IK
leak I}|{ z}|{z }| L { zC V̇ = I − gL (V −EL ) − gNa m∞ (V ) (V −ENa ) − gK n (V −EK ) ,ṅ = (n∞ (V ) − n)/τ (V ) ,

(4.1)
(4.2)

having leak current IL , persistent Na+ current INa,p with instantaneous activation kinetic and a relatively slower persistent K+ current IK with either high (Fig.4.1a) or low(Fig.4.1b) threshold (the two choices result in fundamentally different dynamics). The
state of the INa,p +IK -model is a two-dimensional vector (V, n) ∈ R2 on the phase plane
R2 . New types of equilibria, orbits, and bifurcations can exist on the phase plane thatcannot exist on the phase line R. Many interesting features of single neuron dynamics
can be illustrated or explained using two-dimensional systems. Even neuronal bursting,which occurs in multidimensional systems, can be understood via bifurcation analysisof two-dimensional systems.
This model is equivalent in many respects to the well-known and widely used
ICa +IK -model proposed by Morris and Lecar (1981) to describe voltage oscillationsin the barnacle giant muscle fiber.

4.1

Planar Vector Fields

Two-dimensional dynamical systems, also called planar systems, are often written in
the formẋ = f (x, y) ,ẏ = g(x, y) ,where the functions f and g describe the evolution of the two-dimensional state variable(x(t), y(t)). For any point (x0 , y0 ) on the phase plane, the vector (f (x0 , y0 ), g(x0 , y0 ))89

90

Two-Dimensional Systems

high-threshold IK

I

low-threshold IK

current (pA)

100

0

INa,p

IK
neuron-100

I(V)=IL+INa,p+IK

IL-100 EK-50

0

I(V)=IL+INa,p+IK
50 E
Na

membrane potential, V (mV)a0-100 E
K

-50
0
50 E
Na
membrane potential, V (mV)b

Figure 4.1: The INa,p +IK -model (4.1, 4.2). Parameters in (a): C = 1, I = 0, EL = −80
mV, gL = 8, gNa = 20, gK = 10, m∞ (V ) has V1/2 = −20 and k = 15, n∞ (V ) has
V1/2 = −25 and k = 5, and τ (V ) = 1, ENa = 60 mV and EK = −90 mV. Parametersin (b) as in (a) except EL = −78 mV and n∞ (V ) has V1/2 = −45; see section 2.3.5.

Figure 4.2: Harold Lecar (back), Richard
FitzHugh (front), and Cathy Morris at
NIH Biophysics Lab, summer of 1983.

Two-Dimensional Systems

91
10

8

8

6

6

4

4

2

2

0

y 0

-2

-2

-4

-4

-6

-6

y

-8

-8

-10

-10
-10

-10

-8

-6

-4

-2

0

2

4

6

8

10

-8

-6

-4

-2

0

2

x

x

4

6

8

b

a
10

10

8

8

6

6

4

4
2

2

y0

y0

-2

-2

-4

-4
-6

-6
-8

-8
-10

-10
-10

-8

-6

-4

-2

0

x

2

4

6

8

10

-10

-8

-6

-4

-2

0

2

4

6

8

10

x

d

c

Figure 4.3: Examples of vector fields.
indicates the direction of change of the state variable. For example, negative f (x0 , y0 )and positive g(x0 , y0 ) imply that x(t) decreases and y(t) increases at this particularpoint. Since each point on the phase plane (x, y) has its own vector (f, g), the system
above is said to define a vector field on the plane, also known as a direction field ora velocity field, see Fig.4.3. Thus, the vector field defines the direction of motion;
depending on where you are, it tells you where you are going.
Let us consider a few examples. The two-dimensional systemẋ = 1 ,ẏ = 0defines a constant horizontal vector field in Fig.4.3a because each point has a horizontal
vector (1, 0) attached to it. (Of course, we depict only a small sample of vectors.)
Similarly, the systemẋ = 0 ,ẏ = 1
defines a constant vertical vector field depicted in Fig.4.3b. The systemẋ = −x ,ẏ = −y92

Two-Dimensional Systems

defines a vector field that points to the origin (0, 0), as in Fig.4.3c, and the systemẋ = −y ,ẏ = −x

(4.3)
(4.4)

defines a saddle vector field, as in Fig.4.3d. Vector fields provide geometrical information about the joint evolution of state variables. For example, the vector field in
Fig.4.3d is directed rightward in the lower half-plane and leftward in the upper halfplane. Therefore the variable x(t) increases when y < 0 and decreases otherwise, which
obviously follows from equation (4.3). Quite often, however, geometrical analysis ofvector fields can provide information about the behavior of the system that may not
be obvious from the form of the functions f and g.

4.1.1

Nullclines

The vector field in Fig.4.3d is directed rightward (x increases) or leftward (x decreases)in different regions of the phase plane. The set of points where the vector field changes
its horizontal direction is called the x-nullcline, and it is defined by the equation
f (x, y) = 0. Indeed, at any such point x neither increases nor decreases becauseẋ = 0. The x-nullcline partitions the phase plane into two regions where x moves in
opposite directions. Similarly, the y-nullcline is defined by the equation g(x, y) = 0,
and it denotes the set of points where the vector field changes its vertical direction.
This nullcline partitions the phase plane into two regions where y either increases ordecreases. The x- and y-nullclines partition the phase plane into four different regions:(a) x and y increase, (b) x decreases and y increases, (c) x and y decrease, and (d) xincreases and y decreases, as we illustrate in Fig.4.4.
Each point of intersection of the nullclines is an equilibrium point, since f (x, y) =g(x, y) = 0, and hence ẋ = ẏ = 0. Conversely, every equilibrium of a two-dimensional
system is the point of intersection of its nullclines. Because nullclines are so important,
we consider two examples in detail below (the reader is urged to solve exercise 1 at theend of this chapter).
Let us determine nullclines of the system (4.3, 4.4) with the vector field shown in
Fig.4.3d. From (4.3) it follows that the x-nullcline is the horizontal line y = 0, andfrom (4.4) it follows that the y-nullcline is the vertical line x = 0. These nullclines(dashed lines in Fig.4.3d) partition the phase plane into four quadrants, in each ofwhich the vector field has a different direction. The intersection of the nullclines isthe equilibrium (0, 0). Later in this chapter we will study how to determine stabilityof equilibria in two-dimensional systems, though in this particular case one can easily
guess that the equilibrium is not stable.
As another example, let us determine the nullclines of the INa,p +IK -model (4.1,
4.2). The V -nullcline is given by the equation
I − gL (V −EL ) − gNa m∞ (V ) (V −ENa ) − gK n (V −EK ) = 0 ,Two-Dimensional Systems

93

0.7

(b)

0.6

e
n-nullclin

absolute
refractory

0.5

peak
of spike

0.4

0.3

V-n

(a)

(d)

ullc

K+ activation variable, n

(c)

spike
downstroke

relativerefractory

line

spike
upstroke(regenerative)

0.2

0.1

0

resting
-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

Figure 4.4: Nullclines of
the INa,p +IK -model (4.1,
4.2) with low-thresholdK+ current in Fig.4.1b.
(The vector field is slightly
distorted for the sake ofclarity of illustration).

membrane voltage, V (mV)which has the solutionn=I − gL (V −EL ) − gNa m∞ (V ) (V −ENa )gK (V − EK )(V -nullcline) ,depicted in Fig.4.4. It typically has the form of a cubic parabola. The equationn∞ (V ) − n = 0
defines the n-nullclinen = n∞ (V )(n-nullcline),

which coincides with the K+ steady-state activation function n∞ (V ), though only an
initial segment of this curve fits in Fig.4.4. It is easy to see how the V - and n-nullclines
partition the phase plane into four regions, in each of which the vector field has adifferent direction:(a) Both V and n increase. Both Na+ and K+ currents activate and lead to the
upstroke of the action potential.
(b) V decreases but n still increases. The Na+ current deactivates, but the slower
K+ current still activates and leads to the downstroke of the action potential.
(c) Both V and n decrease. Both Na+ and K+ currents deactivate while V is small,leading to a refractory period.
(d) V increases but n still decreases. Partial activation of the Na+ current combinedwith further deactivation of the residual K+ current leads to a relative refractory
period, then to an excitable period, and possibly to another action potential.

94

Two-Dimensional Systems

The intersection of the V - and n-nullclines in Fig.4.4 is an equilibrium corresponding
to the resting state. The number and location of equilibria may be difficult to infer
via analysis of equations (4.1, 4.2), but it is a trivial geometrical exercise once thenullclines are determined. Because nullclines are so useful and important in geometricalanalysis of dynamical systems, few scientists bother to plot vector fields. Following thistradition, we will not show vector fields in the rest of the book (except for this chapter).
Instead, we plot nullclines and representative trajectories, which we discuss next.

4.1.2

Trajectories

A vector function (x(t), y(t)) is a solution of the two-dimensional systemẋ = f (x, y) ,ẏ = g(x, y) ,starting with an initial condition (x(0), y(0)) = (x0 , y0 ) when dx(t)/dt = f (x(t), y(t))
and dy(t)/dt = g(x(t), y(t)) at each t ≥ 0. This requirement has a simple geometricalinterpretation: a solution is a curve (x(t), y(t)) on the phase plane R2 which is tangent
to the vector field, as we illustrate in Fig 4.5. Such a curve is often called a trajectoryor an orbit.
One can think of the vector field as a stationary flow of a fluid. Then a solution
is just a trajectory of a small particle dropped at a certain (initial) point and carriedby the flow. To study the flow, it is useful to drop a few particles and see where they
are going. Thus, to understand the geometry of a vector field, it is always useful toplot a few representative trajectories starting from various initial points, as we do in
Fig.4.6. Due to the uniqueness of the solutions, the trajectories cannot cross, so they
partition or foliate the phase space. This is an important step toward determining thephase portrait of a two-dimensional system.
Let us return to the INa,p +IK -model (4.1, 4.2) with low-threshold K+ current andexplain two odd phenomena discussed in chapter 1: Failure to generate all-or-none
action potentials (Fig.1.5b) and inability to have a fixed value of the threshold voltage.

(f(x(t),y(t)),g(x(t),y(t)))y(x(t),y(t))(f(x0,y0),g(x0,y0))(x0,y0)x

Figure 4.5: Solutions are trajectories tangent to the vector field.

Two-Dimensional Systems

95

10

8

6

4

2

y 0
-2

-4

-6

-8

-10
-10

-8

-6

-4

-2

0

2

4

6

8

10

Figure 4.6: Representative trajectories of the two-dimensional
system (4.3, 4.4).

x

Brief and strong current pulses in Fig.4.7 reset the value of the voltage variable V but
do not change the value of the K+ activation variable n. Thus, each voltage trace after
the pulse corresponds to a trajectory starting with different values of V0 but the samevalue n0 . We see that each trajectory makes a counterclockwise excursion and returns
to the resting state. However, the size of the excursion depends on the initial value ofthe voltage variable and can be small (subthreshold response), intermediate, or large(action potential). This phenomenon was considered theoretically by FitzHugh in theearly 1960s (see bibliography) and demonstrated experimentally by Cole et al. (1970),
using the squid giant axon at higher than normal temperatures.
0.7

0

ne

0.4

membrane voltage, V (mV)n-nullcli

0.5

ullc

V-n

0.3

line

K+ activation variable, n

0.6

actio
np
ote
nti
al

0.2

s

0.1

sho
thre ld
ub

-20

-40

intermediateamplitudeaction potentials
-60

n0
0

-80

-70

-60

-50

-40

-30

-20

-10

membrane voltage, V (mV)

0

10

20

-80
0

2

4

time (ms)Figure 4.7: Failure to generate all-or-none action potentials in the INa,p +IK -model.

6

96

Two-Dimensional Systems
0.7

n-nullcli
ne

0.5
0.4

al

0.3
d
reshol
bth
su

0.1
0

-80

-70

-60

-50

0

I=-10

-10

I=0

-20
-30
-40
-50
-60

I=+10

line
ullc

0.2

V-n

K+ activation variable, n0.6

membrane voltage, V (mV)

acti
on
po
ten
ti

-40

-30

-20

-10

membrane voltage, V (mV)

0

10

20

I=+10 pre-pulse
I=0
I=-10

1 ms

Figure 4.8: Failure to have a fixed value of threshold voltage in the INa,p +IK -model.
In Fig.4.8 we apply a long pre-pulse current of various amplitudes to reset theK activation variable n to various values, and then a brief strong pulse to reset V toexactly −48 mV. Each voltage trace after the pulse corresponds to a trajectory startingwith the same V0 = −48 mV, but different values of n0 . We see that some trajectories
return immediately to the resting state, while others do so after generating a transientaction potential. Therefore, V = −48 mV is a subthreshold value when n0 is large, anda superthreshold value otherwise. In particular, the system does not have a clear-cut
voltage threshold – a ubiquitous property of many neurons.
+

4.1.3

Limit Cycles

A trajectory that forms a closed loop is called a periodic trajectory or a periodic orbit(the latter is usually reserved for mappings, which we do not consider here). Sometimes periodic trajectories are isolated, as in Fig.4.9, and sometimes they are part of acontinuum, as in Fig.4.13 (left). An isolated periodic trajectory is called a limit cycle.

Figure 4.9: Limit
cycles (periodic
orbits).
stableunstableTwo-Dimensional Systems

97

0.7

n-nullcli
n

0.5

0.4

0.3

ine
ullcl
V-n

K+ activation variable, n

e

0.6

0.2

0.1

membrane voltage, V (mV)

0

-80

-70

-60

-50

-40

-30

-20

-10

membrane voltage, V (mV)

0

10

20

0

-20

-40

-60

-80

0

5

10

15

20

25

30

time (ms)Figure 4.10: Stable limit cycle in the INa,p +IK -model (4.1, 4.2) with low-threshold K+
current and I = 40.
The existence of limit cycles is a major feature of two-dimensional systems that cannot
exist in R1 . If the initial point is on a limit cycle, then the solution (x(t), y(t)) stays
on the cycle forever, and the system exhibits periodic behavior; that is,x(t) = x(t + T )andy(t) = y(t + T )(for all t)

for some T > 0. The minimal T for which this equality holds is called the periodof the limit cycle. A limit cycle is said to be asymptotically stable if any trajectorywith the initial point sufficiently near the cycle approaches the cycle as t → ∞. Such
asymptotically stable limit cycles are often called limit cycle attractors, since they
“attract” all nearby trajectories. The stable limit cycle in Fig.4.9 is an attractor.
The limit cycle in Fig.4.10 is also an attractor; it corresponds to the periodic (tonic)
spiking of the INa,p +IK -model (4.1, 4.2). The unstable limit cycle in Fig.4.9 is often
called a repeller, since it repels all nearby trajectories. Notice that there is always atleast one equilibrium inside any limit cycle on a plane.

98

Two-Dimensional Systems

derivative, V' (mV/ms)

cortical pyramidal neuron

cortical interneuron

brainstem neuron-80 -60 -40 -20 0 20 40
membrane potential, V (mV)

-80 -60 -40 -20 0 20 40
membrane potential, V (mV)

200
100
0
-100
-80 -60 -40 -20 0 20 40
membrane potential, V (mV)membranepotential (mV)

40
20
0
-20
-40
-60
0

50

100

0

20

40
time (ms)

60

0

10

20

Figure 4.11: Limit cycles corresponding to tonic spiking of three types of neurons
recorded in vitro.
In Fig.4.11 we depict limit cycles of three types of neurons recorded in vitro. Since
we do not know the state of the internal variables, such as the magnitude of theactivation and inactivation of Na+ and K+ currents, we plot the cycles on the (V, V ′ )plane, where V ′ is the time derivative of V . The cycles look jerky because of the poor
data sampling rate during each spike.

4.1.4

Relaxation Oscillators

Many models in science and engineering can be reduced to two-dimensional fast/slow
systems of the formẋ = f (x, y)ẏ = µg(x, y)(fast variable)(slow variable) ,where the small parameter µ describes the ratio of time scales of variables x and y.
Typically, the fast variable x has a cubic like nullcline that intersects the y-nullcline
somewhere in the middle branch, as in Fig.4.12a, resulting in relaxation oscillations.
The periodic trajectory of the system slides down along the left (stable) branch of the
cubic nullcline until it reaches the left knee, A. At this moment, it quickly jumps topoint B and then slowly slides up along the right (also stable) branch of the cubic
nullcline. Upon reaching the right knee, C, the system jumps to the left branch andstarts to slide down again, thereby completing one oscillation. Relaxation oscillations
are easy to grasp conceptually, but some of their features are quite difficult to study
mathematically. (We consider relaxation oscillations in detail in section 6.3.4).
Note that the jumps in Fig.4.12a are nearly horizontal – a distinctive signature of
relaxation oscillations that is due to the disparately different time scales in the system.

Two-Dimensional Systems
1.5

y=x-x3/3

99
2.5

x=0

(a)

1

(b)

B

B

2
1.5

D

C

C

C

1

0.5

0.5

y

x(t)

0

0
-0.5

-0.5

B

A

-1

A

A

-1.5

1

-2
-1.5

-2

-1

0

1

2

-2.5

D

D
0

100

x

200

300

400

time, tFigure 4.12: Relaxation oscillations in the van der Pol model ẋ = x − x3 /3 − y, ẏ = µxwith µ = 0.01.
Although many neuronal models have fast and slow time scales and could be reduced
to the fast/slow form above, they do not exhibit relaxation oscillations because theparameter µ is not small enough. Anybody who records from neurons would probably
notice the weird square shape of “spikes” in Fig.4.12b, something that most biologicalneurons do not exhibit. Nevertheless, relaxation oscillations in fast/slow systems areimportant when we consider neuronal bursting in chapter 9; the fast variable x is
two-dimensional there.

4.2

Equilibria

An important step in the analysis of any dynamical system is to find its equilibria,that is, points wheref (x, y) = 0 ,g(x, y) = 0(point (x, y) is an equilibrium).

As mentioned before, equilibria are intersections of nullclines. If the initial point (x0 , y0 )is an equilibrium, then ẋ = 0 and ẏ = 0, and the trajectory stays at equilibrium; thatis, x(t) = x0 and y(t) = y0 for all t ≥ 0. If the initial point is near the equilibrium,then the trajectory may converge to or diverge from the equilibrium, depending on its
stability.
From the electrophysiological point of view, any equilibrium of a neuronal modelis the zero crossing of its steady-state I-V relation I∞ (V ). For example, the INa,p +IK model (4.1, 4.2) with high-threshold K+ current has an I-V curve with three zeroes

100

Two-Dimensional Systems

Figure 4.13: Neutrally stable equilibria. Some trajectories neither converge to nor
diverge from the equilibria.

ab

Figure 4.14: Unstable equilibria.
(Fig.4.1a); hence it has three equilibria: around −66 mV, −56 mV, and −28 mV. In
contrast, the same model with low-threshold K+ current has a monotonic I-V curvewith only one zero (Fig.4.1b); hence it has a unique equilibrium, which is around −61
mV.

4.2.1

Stability

In chapter 3, exercise 18, we provide rigorous definitions of stability of equilibria in
one-dimensional systems. The same definitions apply to higher-dimensional systems.
Briefly, an equilibrium is stable if any trajectory starting sufficiently close to the equilibrium remains near it for all t ≥ 0. If, in addition, all such trajectories converge
to the equilibrium as t → ∞, the equilibrium is asymptotically stable, as in Fig.4.3c.
When the convergence rate is exponential or faster, then the equilibrium is said to
be exponentially stable. Note that stability does not imply asymptotic stability. For
example, all equilibria in Fig.4.13 are stable but not asymptotically stable. They areoften referred to as neutrally stable.
An equilibrium is called unstable, if it is not stable. Obviously, if all nearby trajectories diverge from the equilibrium, as in Fig.4.14a, then it is unstable. This, however,is an exceptional case. For instability it suffices to have at least one trajectory thatdiverges from the equilibrium no matter how close the initial condition is to the equilibrium, as in Fig.4.14b. Indeed, any trajectory starting in the shaded area (attraction

Two-Dimensional Systems

101

domain) converges to the equilibrium, but any trajectory starting in the white area
diverges from it, regardless of how close the initial point is to the equilibrium.
In contrast to the one-dimensional case, the stability of a two-dimensional equilibrium cannot be inferred from the slope of the steady-state I-V curve. For example,
the equilibrium around −28 mV in Fig.4.1a is unstable even though the I-V curve has
positive slope.
To determine the stability of an equilibrium, we need to look at the behavior of
the two-dimensional vector field in a small neighborhood of the equilibrium. Quite
often visual inspection of the vector field does not give conclusive information aboutstability. For example, is the equilibrium in Fig.4.4 stable? What about the equilibriumin Fig.4.10? The vector fields in the neighborhoods of the two equilibria exhibit subtle
differences that are difficult to spot without the help of analytical tools, which we
discuss next.

4.2.2

Local Linear Analysis

Below we remind the reader of some basic concepts of linear algebra, assuming thathe or she has some familiarity with matrices, eigenvectors, and eigenvalues. Consider
a two-dimensional dynamical systemẋ = f (x, y)ẏ = g(x, y)

(4.5)
(4.6)

having an equilibrium point (x0 , y0 ). The nonlinear functions f and g can be linearizednear the equilibrium; that is, written in the formf (x, y) = a(x − x0 ) + b(y − y0 ) + higher-order terms,
g(x, y) = c(x − x0 ) + d(y − y0 ) + higher-order terms,
where higher-order terms include (x − x0 )2 , (x − x0 )(y − y0 ), (x − x0 )3 , and so on, and∂f(x0 , y0 ),∂x∂gc=(x0 , y0 ),
∂x

a=

∂f
(x0 , y0 ),∂y∂gd=(x0 , y0 )∂yb=

are the partial derivatives of f and g with respect to the state variables x and yevaluated at the equilibrium (x0 , y0 ). (First, evaluate the derivatives, then substitutex = x0 and y = y0 ; if you do in the opposite order, you will always get zero). Many
questions regarding the stability of the equilibrium can be answered by considering thecorresponding linear systemu̇ = au + bw ,ẇ = cu + dw ,

(4.7)
(4.8)

102

Two-Dimensional Systems

where u = x − x0 and w = y − y0 are the deviations from the equilibrium, and thehigher-order terms u2 , uw, w3 , and so on, are neglected. We can write this system inthe vector form

 


u̇
a b
u
=
.
ẇ
c d
w
The linearization matrix


L=

a b
c d



is called the Jacobian matrix of the system (4.5, 4.6) at the equilibrium (x0 , y0 ). For
example, the Jacobian matrix of the system (4.3, 4.4) at the origin is


0 −1
.
(4.9)
−1 0
It is important to remember that Jacobian matrices are defined for equilibria, andthat a nonlinear system can have many equilibria, and hence many different Jacobianmatrices.

4.2.3

Eigenvalues and Eigenvectors

A nonzero vector v ∈ R2 is said to be an eigenvector of the matrix L corresponding to
the eigenvalue λ if
Lv = λv(matrix notation) .
For example, the matrix (4.9) has two eigenvectors,
 


1
1
v1 =
and
v2 =
,
1
−1
corresponding to the eigenvalues λ1 = −1 and λ2 = 1, respectively. Any textbook on
linear algebra explains how to find eigenvectors and eigenvalues of an arbitrary matrix.
It is important for the reader to get comfortable with these notions, since they are used
extensively in the rest of the book.
Eigenvalues play an important role in the analysis of stability of equilibria. To findthe eigenvalues of a 2×2-matrix L, one solves the characteristic equation


a−λ
b
det
=0.
c
d−λ
This equation can be written in the polynomial form (a − λ)(d − λ) − bc = 0 or
λ2 − τ λ + ∆ = 0 ,whereτ = tr L = a + d

and∆ = det L = ad − bcTwo-Dimensional Systems

103

are the trace and the determinant of the matrix L, respectively. Such a quadratic
polynomial has two solutions of the form√√τ − τ 2 − 4∆
τ + τ 2 − 4∆
and
λ2 =
,
(4.10)λ1 =2
2and they are either real (when τ 2 − 4∆ ≥ 0) or complex-conjugate (when τ 2 − 4∆ < 0).
What can you say about the case τ 2 = 4∆?
In general, 2 × 2 matrices have two eigenvalues with distinct (independent) eigenvectors. In this case a general solution of the linear system has the formu(t)= c1 eλ1 t v1 + c2 eλ2 t v2 ,w(t)where c1 and c2 are constants that depend on the initial condition. This formula is valid
for real and complex-conjugate eigenvalues. When both eigenvalues are negative (orhave negative real parts), u(t) → 0 and w(t) → 0, meaning x(t) → x0 and y(t) → y0 ,so that the equilibrium (x0 , y0 ) is exponentially (and hence asymptotically) stable. It is
unstable when at least one eigenvalue is positive or has a positive real part. We denotestable equilibria by filled circles • and unstable equilibria by open circles ◦ throughout
the book.

4.2.4

Local Equivalence

An equilibrium whose Jacobian matrix does not have zero eigenvalues or eigenvalues
with zero real parts is called hyperbolic. Such an equilibrium can be stable or unstable.
The Hartman-Grobman theorem states that the vector field, and hence the dynamics
of a nonlinear system, such as (4.5, 4.6), near such a hyperbolic equilibrium is topologically equivalent to that of its linearization (4.7, 4.8). That is, the higher-order terms
that are neglected when (4.5, 4.6) is replaced by (4.7, 4.8) do not play any qualitativerole. Thus, understanding and classifying the geometry of vector fields of linear systems provides an exhaustive description of all possible behaviors of nonlinear systems
near hyperbolic equilibria.
A zero eigenvalue (or eigenvalues with zero real parts) arises when the equilibrium
undergoes a bifurcation, as in Fig.4.14b; such equilibria are called non-hyperbolic.
Linear analysis cannot answer the question of stability of a nonlinear system in thiscase, since small nonlinear (high-order) terms play a crucial role here. We denoteequilibria undergoing a bifurcation by half-filled circles, .

4.2.5

Classification of Equilibria

Besides defining the stability of an equilibrium, the eigenvalues also define the geometryof the vector field near the equilibrium, as we illustrate in Fig.4.15, and as the reader
is asked to prove in exercise 4. (The proof is a straightforward consequence of (4.10).)
There are three major types of equilibria.

104

Two-Dimensional Systems

eigenvalues

saddle-node bifurcation

τ

saddle(real eigenvalues, different signs)

saddle-node bifurcation

0(real positive eigenvalues)
unstable node

τ2 − 4∆=0

unstable focus(complex eigenvalues,
positive real part)
Andronov-Hopf bifurcation∆

stable focus(complex eigenvalues,
negative real part)
stable node(real negative eigenvalues)

τ 2 − 4∆=00

Figure 4.15: Classification of equilibria according to the trace (τ ) and the determinant(∆) of the Jacobian matrix L. The shaded region corresponds to stable equilibria.
Node (Fig.4.16). The eigenvalues are real and of the same sign. The node is stable
when the eigenvalues are negative, and unstable when they are positive. The
trajectories tend to converge to or diverge from the node along the eigenvector
corresponding to the eigenvalue having the smallest absolute value.
Saddle (Fig.4.17). The eigenvalues are real and of opposite signs. Saddles are always
unstable, since one of the eigenvalues is always positive. Most trajectories approach the saddle equilibrium along the eigenvector corresponding to the negative(stable) eigenvalue and then diverge from it along the eigenvector corresponding
to the positive (unstable) eigenvalue.
Focus (Fig.4.18). The eigenvalues are complex-conjugate. Foci are stable when the
eigenvalues have negative real parts, and unstable when the eigenvalues have positive real parts. The imaginary part of the eigenvalues determines the frequencyof rotation of trajectories around the focus equilibrium.
When the system undergoes a saddle-node bifurcation, one of the eigenvalues becomeszero and a mixed type of equilibrium occurs – saddle-node equilibrium, illustrated in
Fig.4.14b. There could be other types of mixed equilibria, such as saddle-focus or
focus-node, and so on, in dynamical systems having dimension 3 and higher.
Depending upon the value of the injected current I, the INa,p +IK -model (4.1, 4.2)
with a low-threshold K+ current has a stable focus (Fig.4.8) or an unstable focus(Fig.4.10) surrounded by a stable limit cycle. In Fig.4.19 we depict the vector fieldand nullclines of the same model with a high-threshold K+ current. As one expects
from the shape of the steady-state I-V curve in Fig.4.1a, the model has three equilibria:

Two-Dimensional Systems

105

v2

v2

v1

stable nodev1

unstable node

Figure 4.16: Node equilibrium occurs when both eigenvalues are real and have thesame sign, for example, λ1 = −1 and λ2 = −3 (stable) or λ1 = +1 and λ2 = +3
(unstable). Most trajectories converge to or diverge from the node along the eigenvector
v1 corresponding to the eigenvalue having the smallest absolute value.
v2

v2

v1
v1

saddlesaddleFigure 4.17: Saddle equilibrium occurs when two real eigenvalues have opposite signs,
such as λ1 = +1 and λ2 = −1. Most trajectories diverge from the equilibrium along
the eigenvector corresponding to the positive eigenvalue (in this case, v1 ).

stable focus

unstable focus

Figure 4.18: Focus equilibrium occurs when the eigenvalues are complex-conjugate, forinstance, λ = −3 ± i (stable) or λ = +3 ± i (unstable). The imaginary part (here, 1)
determines the frequency of rotation around the focus.

106

Two-Dimensional Systems

n-nullcli

ne

0.6

0.5

line
ullc

0.3

V-n

K+ activation variable, n

0.4

0.2

0.1

0

-0.1
-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

membrane voltage, V (mV)

Figure 4.19: Phase portrait of the INa,p +IK -model having high-threshold K+ current.
a stable node, a saddle, and an unstable focus. Notice that the third equilibrium is
unstable even though the I-V relation has a positive slope around it.
Also notice that the y-axis starts at the negative value -0.1. However, the gating
variable n represents the proportion (probability) of the K+ channels in the open state;hence a value less than zero has no physical meaning. So while we can happily calculate
the nullclines for the negative n, and even start the trajectory with the initial conditionn < 0, we cannot interpret the result. (As an exercise, prove that if all gating variablesof a model are initially in the range [0, 1], then they stay in the range for all t ≥ 0.)

4.2.6

Example: FitzHugh-Nagumo Model

The FitzHugh-Nagumo model (FitzHugh 1961; Nagumo et al. 1962; Izhikevich and
FitzHugh 2006)
V̇ = V (a − V )(V − 1) − w + I ,ẇ = bV − cw ,(4.11)(4.12)

imitates generation of action potentials by Hodgkin-Huxley-type models having cubic(N-shaped) nullclines, as in Fig.4.4. Here V mimics the membrane voltage and the“recovery” variable w mimics activation of an outward current. Parameter I mimics
the injected current, and for the sake of simplicity we set I = 0 in our analysis below.
Parameter a describes the shape of the cubic parabola V (a−V )(V −1), and parameters

ullc
w-n

recovery, w

cline

0.2

w-n
ullc

V-null

line

107

line

Two-Dimensional Systems

00.5
membrane voltage, V

1

cline

V-null0?
00.5
membrane voltage, V

a

1

b

Figure 4.20: Nullclines in the FitzHugh-Nagumo model (4.11, 4.12). Parameters: I =0, b = 0.01, c = 0.02, a = 0.1 (left) and a = −0.1 (right).
b > 0 and c ≥ 0 describe the kinetics of the recovery variable w. When b and c aresmall, the model may exhibit relaxation oscillations.
The nullclines of the FitzHugh-Nagumo model have the cubic and linear formw = V (a − V )(V − 1) + I
w = b/c V(V -nullcline),(w-nullcline),

and they can intersect in one, two, or three points, resulting in one, two, or threeequilibria, all of which may be unstable. Below, we consider the simple case I = 0, sothat the origin, (0, 0), is an equilibrium. Indeed, the nullclines of the model, depicted
in Fig.4.20, always intersect at (0, 0) in this case. The intersection may occur on theleft (Fig.4.20a) or middle (Fig.4.20b) branch of the cubic V -nullcline, depending on thesign of the parameter a. Let us determine how the stability of the equilibrium (0, 0)
depends on the parameters a, b, and c.
There is a common dogma that the equilibrium in Fig.4.20a corresponding to a > 0
is always stable, the equilibrium in Fig.4.20b corresponding to a < 0 is always unstable,
and the loss of stability occurs “exactly” at a = 0, that is, at the bottom of the left
knee. Let us check that this is not necessarily true, at least when c ̸= 0. The Jacobianmatrix of the FitzHugh-Nagumo model (4.11,4.12) at the equilibrium (0, 0) has the
form


−a −1
L=
.
b −c
It is easy to check that
τ = tr L = −a − cand∆ = det L = ac + b .

Using Fig.4.15, we conclude that the equilibrium is stable when tr L < 0 and det L > 0,
which corresponds to the shaded region in Fig.4.21. Both conditions are always satisfied

108

Two-Dimensional Systems
c
tr L

=-

det L

stability

a-

c=

+b
=ac

tr L < 0
det L > 0

0

=0

b

b

0

a

Figure 4.21: Stability diagram of theequilibrium (0, 0) in the FitzHughNagumo model (4.11,4.12).

when a > 0; hence the equilibrium in Fig.4.20a is indeed stable. However, since bothconditions may also be satisfied for negative a, the equilibrium in Fig.4.20b may also
be stable. Thus, the equilibrium loses stability not at the left knee, but slightly to theright of it, so that a part of the “unstable branch” of the cubic nullcline is actuallystable. The part is small when b and c are small, i.e., when (4.11,4.12) is in a relaxation
regime.

4.3

Phase Portraits

An important step in geometrical analysis of dynamical systems is sketching their phase
portraits. The phase portrait of a two-dimensional system is a partitioning of thephase plane into orbits or trajectories. Instead of depicting all possible trajectories, it
usually suffices to depict some representative trajectories. The phase portrait contains
all important information about qualitative behavior of the dynamical system, suchas relative location and stability of equilibria, their attraction domains, separatrices,limit cycles, and other special trajectories that are discussed in this section.

4.3.1

Bistability and Attraction Domains

Nonlinear two-dimensional systems can have many coexisting attractors. For example,
the FitzHugh-Nagumo model (4.11,4.12) with nullclines depicted in Fig.4.22 has two
stable equilibria separated by an unstable equilibrium. Such a system is called bistable(multi-stable when there are more than two attractors). Depending on the initialconditions, the trajectory may approach the left or the right equilibrium. The shaded
area denotes the attraction domain of the right equilibrium; that is, the set of all
initial conditions that lead to this equilibrium. Since there are only two attractors,
the complementary white area denotes the attraction domain of the other equilibrium.
The domains are separated not by equilibria, as in the one-dimensional case, but by
special trajectories called separatrices, which we discuss in section 4.3.2.
Many neural models are bistable or can be made bistable when the parameters haveappropriate values. Often bistability results from the coexistence of an equilibriumTwo-Dimensional Systems

se
pa
ra

trix

recovery, w

cline

V-null

0.2

109

e
lclin

ul
w-n

0

se
-0.4

-0.2

at
par
0

attraction domain

ri x
0.2

0.4

0.6

0.8

1

membrane voltage, V

Figure 4.22: Bistability of two equilibrium attractors (black circles) in the FitzHughNagumo model (4.11,4.12). The shaded area – attraction domain of the right equilibrium. Parameters: I = 0, b = 0.01, a = c = 0.1.
attractor corresponding to the resting state and a limit cycle attractor corresponding
to the repetitive firing state. Figure 4.23 depicts one of many possible cases. Here
we use the INa,p +IK -model with a high-threshold fast K+ current. The resting state
exists due to the balance of partially activated Na+ and leak currents. The repetitive
spiking state persists because the K+ current deactivates too fast and cannot bring themembrane potential into the subthreshold voltage range. If the initial state is in theshaded area, which is the attraction domain of the limit cycle attractor, the trajectory
approaches the limit cycle attractor and the neuron fires an infinite train of action
potentials.

4.3.2

Stable/Unstable Manifolds

In contrast with one-dimensional systems, in two-dimensional systems unstable equilibria do not necessarily separate attraction domains. Nevertheless, they play an important role in defining the boundary of attraction domains, as in Fig.4.22 and Fig.4.23.
In both cases the attraction domains are separated by a pair of trajectories, called
separatrices, which converge to the saddle equilibrium. Such trajectories form thestable manifold of a saddle point. Locally, the manifold is parallel to the eigenvector
corresponding to the negative (stable) eigenvalue; see Fig.4.24. Similarly, the unstablemanifold of a saddle is formed by the two trajectories that originate exactly from theTwo-Dimensional Systems

n-null
cline

110

0.6

K+ activation variable, n

0.5

0.4

0.3

l
ullc
V-n

ara

trix

ine

0.2

0

rest

-80

membrane voltage, V (mV)

se

pa

ra

trix

sep

0.1

-70

attraction
domain
-60

-50

-40

-30

-20

membrane voltage, V (mV)

-10

0

10

20

-10

rest

spiking-70
05time, t (ms)1015

Figure 4.23: Bistability of rest and spiking states in the INa,p +IK -model (4.1, 4.2) withhigh-threshold fast (τ (V ) = 0.152) K+ current and I = 3. A brief strong pulse ofcurrent (arrow at t = 5 ms) brings the state vector of the system into the attraction
domain of the stable limit cycle.

ema n if o

ld

A

v1

to spiking
membrane voltage

to spiking

st
ab
le

unstab
l

ld

C

x)
tri
ra
a
ep
(s

m
an

v2

to rest

if o

B

A
C
B

separatrix
to saddle

dunstable manifol

to resttime

Figure 4.24: Stable and unstable manifolds to a saddle. The eigenvectors v1 and v2
correspond to positive and negative eigenvalues, respectively.

Two-Dimensional Systems

111

saddle (or approach the saddle if the time is reversed). Locally, the unstable manifold
is parallel to the eigenvector corresponding to the positive (unstable) eigenvalue.
The stable manifold of the saddle in Fig.4.23 plays the role of a threshold, since
it separates resting and spiking states. We illustrate this concept in Fig.4.24: if theinitial state of the system, denoted as A, is in the shaded area, the trajectory will
converge to the spiking attractor (right) no matter how close the initial condition is tothe stable manifold. In contrast, if the initial condition, denoted as B, is in the whitearea, the trajectory will converge to the resting attractor (left). If the initial condition
is precisely on the stable manifold (point C), the trajectory converges neither to resting
nor to spiking, but to the saddle equilibrium. Of course, this case is highly unstable,
and small perturbations will certainly push the trajectory to one side or the other.
The important message in Fig.4.24 is that a threshold is not a point, i.e., a single
voltage value, but a trajectory on the phase plane. (Find an exceptional case where
the threshold looks like a single voltage value. Hint: See Fig.4.17.)

4.3.3

Homoclinic/Heteroclinic Trajectories

Figure 4.24 shows that trajectories forming the unstable manifold originate from thesaddle. Where do they go? Similarly, the trajectories forming the stable manifold
terminate at the saddle. Where do they come from? We say that a trajectory isheteroclinic if it originates at one equilibrium and terminates at another equilibrium,
as in Fig.4.25. A trajectory is homoclinic if it originates and terminates at the sameequilibrium. These types of trajectories play an important role in geometrical analysis
of dynamical systems.
Heteroclinic trajectories connect unstable and stable equilibria, as in Fig.4.26, andthey are ubiquitous in dynamical systems having two or more equilibrium points. In
fact, there are infinitely many heteroclinic trajectories in Fig.4.26, since all trajectories
inside the bold loop originate at the unstable focus and terminate at the stable node.
(Find the exceptional trajectory that ends elsewhere.)
In contrast, homoclinic trajectories are rare. First, a homoclinic trajectory diverges
from an equilibrium, so the equilibrium must be unstable. Next, the trajectory makesa loop and returns to the same equilibrium, as in Fig.4.27. It needs to hit the unstable equilibrium precisely, since a small error would make it deviate from the unstableequilibrium. Though uncommon, homoclinic trajectories indicate that the system undergoes a bifurcation – appearance or disappearance of a limit cycle. The homoclinic
trajectory in Fig.4.27 indicates that the limit cycle in Fig.4.23 is about to (dis)appear

h ete

r o c li n i c o r b i t

h o m o c li n i c orbitFigure 4.25: A heteroclinic orbit starts
and ends at different equilibria. A homoclinic orbit starts and ends at thesame equilibrium.

112

Two-Dimensional Systems

n-nullcli

ne

0.6

0.5

line
ullc

0.3

V-n

K+ activation variable, n

0.4

0.2

0.1

oc
h e ter

linic

t
orbi

0

heteroclinic orbit
-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

membrane voltage, V (mV)

Figure 4.26: Two heteroclinic orbits (bold curves connecting stable and unstable equilibria) in the INa,p +IK -model with high-threshold K+ current.

n-null

cline

0.6

0.4

0.3

om

oc

li n

ic o

0.2

line
ullc

rbit

V-n

K+ activation variable, n

0.5

0.1

h

0

-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

membrane voltage, V (mV)

Figure 4.27: Homoclinic orbit (bold) in the INa,p +IK -model with high-threshold fast(τ (V ) = 0.152) K+ current.

Two-Dimensional Systems

113

n-nullcli
ne

0.6

0.5

line

0.3

V-n
ullc

K+ activation variable, n

0.4

0.2

0.1

h

c or
clini
o mo

bit

0

-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

membrane voltage, V (mV)

Figure 4.28: Homoclinic orbit (bold) to saddle-node equilibrium in the INa,p +IK -modelwith high-threshold K+ current and I = 4.51.
via saddle homoclinic orbit bifurcation. The homoclinic trajectory in Fig.4.28 indicates that a limit cycle is about to (dis)appear via saddle-node on invariant circle
bifurcation. We study these bifurcations in detail in chapter 6.

4.3.4

Saddle-Node Bifurcation

In Fig.4.29 we simulate the injection of a ramp current I into the INa,p +IK -model
having high-threshold K+ current. Our goal is to understand the transition from theresting state to repetitive spiking. When I is small, the phase portrait of the model
is similar to the one depicted in Fig.4.26 for I = 0. There are two equilibria in the
low-voltage range – a stable node corresponding to the resting state and a saddle. The
equilibria are the intersections of the cubic V -nullcline and the n-nullcline. Increasing
the parameter I changes the shape of the cubic nullcline and shifts it upward, but does
not change the n-nullcline. As a result, the distance between the equilibria decreases,
until they coalesce as in Fig.4.28 so that the nullclines touch each other only in the
low-voltage range. Further increase of I results in the disappearance of the saddle andnode equilibrium, and hence in the disappearance of the resting state. The new phase
portrait is depicted in Fig.4.30; it has only a limit cycle attractor corresponding torepetitive firing. We see that increasing I past the value I = 4.51 results in transition
from resting to periodic spiking dynamics. What kind of bifurcation occurs whenI = 4.51?

Two-Dimensional Systems

membrane voltage, V (mV)

114

20

spiking

0-20transition(bifurcation)

-40

rest

-60

injected current, I

-80
10

5

I=4.51
0

0

50

100

150

200

250

300

time (ms)Figure 4.29: Transition from resting state to repetitive spiking in the INa,p +IK -modelwith injected ramp current I (see also Fig.4.26, Fig.4.28, and Fig.4.30). Note that thefrequency of spiking is initially small, then increases as the amplitude of the injectedcurrent increases.

n-nullcli

ne

0.6

0.5

line
ullc

0.3

V-n

K+ activation variable, n

0.4

0.2

0.1

li

cle
m i t cy

attra

ct o r

0

-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

membrane voltage, V (mV)

Figure 4.30: Limit cycle attractor (bold) in the INa,p +IK -model when I = 10 (comparewith Fig.4.26 and 4.28).

Two-Dimensional Systems

115

eigenvalues:
λ1<0 λ2<0eigenvalues:
λ1>0 λ2<0

node

saddleeigenvalues:
λ1=0 λ2<0saddlenodeno equilibria

Figure 4.31: Saddle-node bifurcation: The saddle and node equilibria approach eachother, coalesce, and annihilate each other (shaded area is the basin of attraction of thestable node).

Those readers who did not skip section 3.3.3 in chapter 3 will immediately recognize
the saddle-node bifurcation, whose major stages are summarized in Fig.4.31. As abifurcation parameter changes, the saddle and the node equilibrium approach eachother, coalesce, and then annihilate each other so there are no equilibria left. Whenthey coalesce, the joint equilibrium is neither a saddle nor a node, but a saddle-node.
Its major feature is that it has precisely one zero eigenvalue, and it is stable on one sideof the neighborhood and unstable on the other side. In chapter 6 we will provide anexact definition of a saddle-node bifurcation in a multi-dimensional system, and willshow that there are two important subtypes of this bifurcation, resulting in slightly
different neurocomputational properties.
It is a relatively simple exercise to determine bifurcation diagrams for saddle-nodebifurcations in neuronal models. For this, we just need to determine all equilibria ofthe model and how they depend on the injected current I. Any equilibrium of the

116

Two-Dimensional Systems
-20

-50

sadd
le (
membrane voltage, V (mV)membrane voltage, V (mV)

-30

-40

-50

-60

-60

s t a bl

e no

st
rest
de (

ate

)

saddle-nodebifurcation

-70

-70

thre
sho
ld s
tate
)

4.51
-100

-50

0
50
injected dc-current I

100

-10

-5

0
5
injected dc-current I

10

Figure 4.32: Saddle-node bifurcation diagram of the INa,p +IK -model. The curve is
given by the equation (4.13).
INa,p +IK -model satisfies the one-dimensional equation
0 = I − gL (V −EL ) − gNa m∞ (V ) (V −ENa ) − gK n∞ (V ) (V −EK ) ,where n = n∞ (V ). Instead of solving this equation for V , we use V as a free parameterand solve it for I,
steady-state I∞ (V )z}|{
I = gL (V −EL ) + gNa m∞ (V ) (V −ENa ) + gK n∞ (V ) (V −EK ) ,(4.13)and then depict the solution as a curve in the (I, V ) plane in Fig.4.32. In the magnification (Fig.4.32, right) one can clearly see how two branches of equilibria approachand annihilate each other as I approaches the bifurcation value 4.51. (Is there anyother saddle-node bifurcation in the figure?)

4.3.5

Andronov-Hopf Bifurcation

In Fig.4.33 we repeat the current ramp experiment, using the INa,p +IK -model with alow-threshold K+ current. The phase portrait of such a model is simple – it has aunique equilibrium, as we illustrate in Fig.4.34. When I is small, the equilibrium is astable focus corresponding to the resting state. When I increases past I = 12, the focus
loses stability and gives birth to a small-amplitude limit cycle attractor. The amplitudeof the limit cycle grows as I increases. We see that increasing I beyond I = 12 resultsin the transition from resting to spiking behavior. What kind of bifurcation occurs
there?
Recall that stable foci have a pair of complex-conjugate eigenvalues with negative
real part. When I increases, the real part of the eigenvalues also increases until itmembrane voltage, V (mV)Two-Dimensional Systems

117

0

spiking-20transition(bifurcation)

-40

rest

-60

injected current, I

-80
30
20
10
0

I=12
0

10

20

30

40

50

60

70

80

90

100

time (ms)Figure 4.33: Transition from resting state to repetitive spiking in the INa,p +IK -modelwith ramp injected current I; see also Fig.4.34 (small-amplitude noise is added to themodel to mask the slow passage effect). Note that the frequency of spiking is relatively
constant for a wide range of injected current.

n-nullcli

0.6
0.5
0.4
0.3

ullc
line

0.2
0.1
-80

-60
-40
-20
0
membrane voltage, V (mV)

0.4
0.3
0.2
0.1
-80

-60
-40
-20
0
membrane voltage, V (mV)

20

0.7

I=20

0.6

K+ activation variable, nK+ activation variable, n

0.7

0.5
0.4
0.3
0.2
0.1
0

0.5

0

20

I=12

0.6

-80

-60
-40
-20
0
membrane voltage, V (mV)

20

I=40

0.6
0.5

le

0

K+ activation variable, n

ne

0.7

I=0

V-n

K+ activation variable, n

0.7

0.4

lim

it c

yc

0.3
0.2
0.1
0

-80

-60
-40
-20
0
membrane voltage, V (mV)

20

Figure 4.34: Supercritical Andronov-Hopf bifurcation in the INa,p +IK -model (4.1, 4.2)
with low-threshold K+ current when I = 12 (see also Fig.4.33).

118

Two-Dimensional Systems
0

0

-10

-10

-20

-20

-30

Andronov-Hopf bifurcation
is somewhere here

-40

I

(V)

-50

unstable

-80

-30

ble
sta

periodic orbits

supercritical
Andronov-Hopfbifurcation

-40

rium

uilib
unstable eq

-50
-60

-60
-70

membrane voltage, V

membrane voltage, V

max V(t)

t
res

-70

min V(t)
0

20
40
60
injected dc-current, I

a

80

100

-80

0

20
40
60
injected dc-current, I

80

100

b

Figure 4.35: Andronov-Hopf bifurcation diagram in the INa,p +IK -model with lowthreshold K+ current. a. Equilibria of the model (solution of (4.13)). b. Equilibria
and limit cycles of the model.
becomes zero (at I = 12) and then positive (when I > 12), meaning that the focus
is no longer stable. The transition from stable to unstable focus described above iscalled the Andronov-Hopf bifurcation. It occurs when the eigenvalues become purely
imaginary, as happens when I = 12. We will study Andronov-Hopf bifurcations indetail in chapter 6, where we will show that they can be supercritical or subcritical.
The former correspond to birth of a small-amplitude limit cycle attractor, as in Fig.4.34.
The latter correspond to the death of an unstable limit cycle.
In Fig.4.35a we plot the solution of (4.13) as an attempt to determine the bifurcation
diagram for the Andronov-Hopf bifurcation in the INa,p +IK -model. However, all we
can see is that the equilibrium persists as I increases, but there is no information on
its stability or on the existence of a limit cycle attractor. To study the limit cycle
attractor, we need to simulate the model with various values of parameter I. For
each I, we disregard the transient period and plot min V (t) and max V (t) on the(I, V )-plane, as in Fig.4.35b. When I is small, the solutions converge to the stableequilibrium, and both min V (t) and max V (t) are equal to the resting voltage. When
I increases past I = 12, the min V (t) and max V (t) values start to diverge, meaningthat there is a limit cycle attractor whose amplitude increases as I does. This method
is appropriate for analysis of supercritical Andronov-Hopf bifurcations, but it fails for
subcritical Andronov-Hopf bifurcations. (Why?)
Figure 4.36 depicts an interesting phenomenon observed in many biological neurons,
excitation block. Spiking activity of the layer 5 pyramidal neuron of rat’s visual cortex
is blocked by strong excitation (i.e., injection of strong depolarizing current). The

Two-Dimensional Systems

119
excitation blocksupercritical
Andronov-Hopfbifurcation-10 mV

20 mV
100 ms-60 mV180 pA

injected current

Figure 4.36: Excitation block in layer 5 pyramidal neuron of rat’s visual cortex as theamplitude of the injected current ramps up.

geometry of this phenomenon is illustrated in Fig.4.37 (bottom). As the magnitudeof the injected current increases, the unstable equilibrium, which is the intersection
point of the nullclines, moves to the right branch of the cubic V -nullcline and becomesstable. The limit cycle shrinks and the spiking activity disappears, typically but notnecessarily via the supercritical Andronov-Hopf bifurcation. Thus, the INa,p +IK -modelwith low-threshold K+ current can exhibit two such bifurcations in response to ramping
up of the injected current, one leading to the appearance of periodic spiking activity(Fig.4.34), and then one leading to its disappearance (Fig.4.37).

Supercritical and subcritical Andronov-Hopf bifurcations in neurons result in slightly
different neurocomputational properties. In contrast, the saddle-node and AndronovHopf bifurcations result in dramatically different neurocomputational properties. In
particular, neurons near a saddle-node bifurcation act as integrators – they preferhigh-frequency input. The higher the frequency of the input, the sooner they fire.
In contrast, neural systems near Andronov-Hopf bifurcations have damped oscillatory
potentials and act as resonators – they prefer oscillatory input with the same frequency
as that of damped oscillations. Increasing the frequency may delay or even terminate
their response. We discuss this and other neurocomputational properties in chapter 7.

120

Two-Dimensional Systems

membrane potential, V (mV)0excitation block
-10
-20
-30
-40
-50

supercritical
Andronov-Hopfbifurcation

-60
-70

I=400

injected
current

I=300
I=150
I=40
0

10

20

30

40

50
time, ms

60

70

80

90

0.6
0.4
0.2
0

it c
im

yc

I=150
K+ activation variable, n

0.8

le

l

K+ activation variable, n

I=40
1

100

1
0.8
0.6
0.4
0.2
0

-80 -60 -40 -20
0 20
membrane potential, V (mV)

-80 -60 -40 -20
0 20
membrane potential, V (mV)
excitation block

1
0.8
0.6
0.4
0.2
0

I=400
K+ activation variable, nK+ activation variable, n

I=300-80 -60 -40 -20
0 20
membrane potential, V (mV)

1
0.8
0.6
0.4
0.2
0

-80 -60 -40 -20
0 20
membrane potential, V (mV)

Figure 4.37: Excitation block in the INa,p +IK -model. As the magnitude of the injectedcurrent I ramps up, the spiking stops.

Two-Dimensional Systems

121

Review of Important Concepts• A two-dimensional system of differential equationsẋ = f (x, y)ẏ = g(x, y) ,describes joint evolution of state variables x and y, which often are the membrane voltage and a “recovery” variable.
• Solutions of the system are trajectories on the phase plane R2 that are tangent
to the vector field (f, g).
• The sets given by the equations f (x, y) = 0 and g(x, y) = 0 are the x- andy-nullclines, respectively, where trajectories change their x and y directions.
• Intersections of the nullclines are equilibria of the system.
• Periodic dynamics correspond to closed loop trajectories.
• Some special trajectories (e.g., separatrices) define thresholds and separate
attraction domains.
• An equilibrium or a periodic trajectory is stable if all nearby trajectories are
attracted to it.
• To determine the stability of an equilibrium, one needs to consider the Jacobianmatrix of partial derivatives


fx fy
.
L=
gx gy
• The equilibrium is stable when both eigenvalues of L are negative or havenegative real parts.
• The equilibrium is a saddle, a node, or a focus when L has real eigenvaluesof opposite signs, of the same signs, or complex-conjugate eigenvalues, respectively.
• When the equilibrium undergoes a saddle-node bifurcation, one of the eigenvalues becomes zero.
• When the equilibrium undergoes an Andronov-Hopf bifurcation (birth or deathof a small periodic trajectory), the complex-conjugate eigenvalues become
purely imaginary.
• The saddle-node and Andronov-Hopf bifurcations are ubiquitous in neuralmodels, and they result in different neurocomputational properties.

122

Two-Dimensional Systems

Bibliographical Notes
Among many textbooks on the mathematical theory of dynamical systems we recommend the following three.
• Nonlinear Dynamics and Chaos by Strogatz (1994) is suitable as an introductory book for undergraduate math or physics majors or graduate students in life
sciences. It contains many exercises and worked-out examples.
• Differential Equations and Dynamical Systems by Perko (1996, 3rd ed., 2000) is
suitable for math and physics graduate students, but may be too technical forlife scientists. Nevertheless, it should be a standard textbook for computational
neuroscientists.
• Elements of Applied Bifurcation Theory by Kuznetsov (1995, 3rd ed., 2004) issuitable for advanced graduate students in mathematics or physics and for computational neuroscientists who want to pursue bifurcation analysis of neural models.
The second edition of The Geometry of Biological Time by Winfree (2001) is a good
introduction to oscillations, limit cycles, and synchronization in biology. It requires little background in mathematics and can be suitable even for undergraduate life science
majors. Mathematical Biology by Murray (2nd ed., corr., 1993, 3rd ed., 2003) is an
excellent example of how dynamical system theory can solve many problems in population biology and shed light on pattern formation in biological systems. Most of thisbook is suitable for advanced undergraduate or graduate students in mathematics andphysics. Mathematical Physiology by Keener and Sneyd (1998) is similar to Murray’sbook, but is more focused on neural systems. Spikes, Decisions, and Actions by H. R.
Wilson (1999) is a short introduction to dynamical systems with many neuroscienceexamples.

Exercises
1. Use a pencil (as in Fig.4.39) to sketch the nullclines of the vector fields depicted
in figures 4.40 through 4.44.
2. Assume that the continuous curve is the x-nullcline and the dashed curve is they-nullcline in Fig.4.38, and that ẋ or ẏ changes sign when (x, y) passes throughthe corresponding nullcline. The arrow indicates the direction of the vector field
in one region. Determine the approximate directions of the vector field in theother regions of the phase plane.
3. Use a pencil (as in Fig.4.39) to sketch phase portraits of the vector fields depicted
in figures 4.40 through 4.44. Clearly mark all equilibria, their stability, and their
attraction domains. Show directions of all homoclinic, heteroclinic, and periodic

Two-Dimensional Systems

123

a

b

c

d

Figure 4.38: Determine the approximate direction of the vector field in each region
between the nullclines. Continuous (dashed) curve is the x-nullcline (y-nullcline), andthe direction of the vector field in one region is indicated by the arrow.
trajectories, as well as other representative trajectories. Estimate the signs ofeigenvalues at each equilibrium.
4. Prove the classification diagram in Fig.4.15.
5. (van der Pol oscillator) Determine nullclines and draw the phase portrait of thevan der Pol oscillator given in the Liénard (1928) formẋ = x − x3 /3 − y ,ẏ = bx ,where b > 0 is a parameter.
6. (Bonhoeffer–van der Pol oscillator) Determine the nullclines and sketch representative phase portraits of the Bonhoeffer–van der Pol oscillatorẋ = x − x3 /3 − y ,ẏ = b(x − a) − cy ,in the case of c = 0. Treat a and b > 0 as parameters.
7. (Hindmarsh-Rose spiking neuron) The following system is a generalization of the
FitzHugh-Nagumo model (Hindmarsh and Rose 1982):ẋ = f (x) − y + I ,ẏ = g(x) − y ,124

Two-Dimensional Systems

Figure 4.39: Phase portrait of a vector field. Use pencil to draw phase portraits infigures 4.40 through 4.44.

Figure 4.40: Use a pencil to draw a phase portrait, as in Fig.4.39.

Two-Dimensional Systems

Figure 4.41: Use a pencil to draw a phase portrait, as in Fig.4.39.

Figure 4.42: Use a pencil to draw a phase portrait, as in Fig.4.39.

Figure 4.43: Use a pencil to draw a phase portrait, as in Fig.4.39.

125

126

Two-Dimensional Systems

Figure 4.44: Use a pencil to draw a phase portrait, as in Fig.4.39.
where f (x) = −ax3 + bx2 , g(x) = −c + dx2 , and a, b, c, d, and I are parameters.
Suppose (x̄, ȳ) is an equilibrium. Determine its type and stability as a functionof f ′ = f ′ (x̄) and g ′ = g(x̄); that is, plot a diagram similar to the one in Fig.4.15,
with f ′ and g ′ as coordinates.
8. (IK -model) Show that the unique equilibrium in the IK -model
C V̇ = −gL (V − EL ) − gK m4 (V − EK ) ,ṁ = (m∞ (V ) − m)/τ (V ) .

(4.14)(4.15)

discussed in chapter 3 (see Fig.3.40), is always stable, at least when EL > EK .
(Hint: Look at the signs of the trace and the determinant of the Jacobian matrix).
9. (Ih -model) Show that the unique equilibrium in the full Ih -model
C V̇ = −gL (V − EL ) − gh h(V − Eh ) ,ḣ = (h∞ (V ) − h)/τ (V ) ,discussed in chapter 3 is always stable.
10. (Bendixson’s criterion) If the divergence of the vector field∂f (x, y) ∂g(x, y)+∂x∂yof a two-dimensional dynamical system is not identically zero, and does notchange sign on the plane, then the dynamical system cannot have limit cycles.
Use this criterion to show that the IK -model and the Ih -model cannot oscillate.
11. Determine the stability of equilibria in the modelẋ = a + x2 − y ,ẏ = bx − cy ,where a ∈ R, b ≥ 0, and c > 0 are some parameters.

Chapter 5
Conductance-Based Models andTheir Reductions
In this chapter we present examples of geometrical phase plane analysis of various
two-dimensional neural models. In particular, we consider minimal models, i.e., thosehaving minimal sets of currents that enable the models to generate action potentials.
The remarkable fact is that all these models can be reduced to planar systems having
N-shaped V -nullclines. We will see that the behavior of the models depends not somuch on the ionic currents as on the relationship between (in)activation curves and thetime constants. That is, models involving completely different currents can have identical dynamics and, conversely, models involving similar currents can have completely
different dynamics.

5.1

Minimal Models

There are a few dozen known voltage- and Ca2+ -gated currents having diverse activation and inactivation dynamics, and this number grows every year. Some of them aresummarized in section 2.3.5. Almost any combination of the currents would result ininteresting nonlinear behavior, such as excitability. Therefore, there are billions (morethan 230 ) of different electrophysiological models of neurons. Here we say that twomodels are “different” if, for example, one has the h-current Ih and the other does not,without even considering how much of the Ih there is. How can we classify all suchmodels?
Let us do the following thought experiment. Consider a conductance-based model
capable of exhibiting periodic spiking, that is, having a limit cycle attractor. Let uscompletely remove a current or one of its gating variables, then ask the question “Doesthe reduced model have a limit cycle attractor, at least for some values of parameters?”
If it does, we remove one more gating variable or current, and proceed until we arrive
at a model that satisfies the following two properties:• It has a limit cycle attractor, at least for some values of parameters.
127

128

Conductance-Based Models• If one removes any current or gating variable, the model has only equilibrium
attractors for any values of parameters.

We refer to such a model as being minimal or irreducible for spiking. Thus, minimalmodels can exhibit periodic activity, even if it is of small amplitude, but their reductions
cannot. According to this definition, any space-clamped conductance-based modelof a neuron either is a minimal model or could be reduced to a minimal model ormodels by removing gating variables. This will be the basis for our classification ofelectrophysiological mechanisms in neurons.
For example, the Hodgkin-Huxley model considered in section 2.3 is not minimal
for spiking. Recall that the model consists of three currents: leakage IL , transientsodium INa,t (gating variables m and h), and persistent potassium IK (gating variablen); see Fig.5.1. Which of these currents are responsible for excitability and spiking?
We can remove the leakage current and the gating variable, h, for the inactivationof the sodium current: The resulting INa,p +IK -model
INa,p
IK
}|
{
z
}|
{
z
C V̇ = I − gK n4 (V − EK ) − gNa m3 (V − ENa ) ,ṅ = (n∞ (V ) − n)/τn (V ) ,ṁ = (m∞ (V ) − m)/τm (V ) ,Hodgkin-HuxleyTransient Na Current (m,h)
Persistent K Current (n)Leak Current
Remove hand LeakRemove nTransient Na Current (m,h)

Persistent Na Current (m)Leak Current

Persistent K Current (n)Minimal Models

Leak Current

h

m

n

Gating for
Inactivationof Na Current

Gating for
Activationof Na Current

Gating for
Activationof K Current

Gating variables

Figure 5.1: The Hodgkin-Huxley model (top box) is a combination of minimal models(shaded boxes on second level). Each minimal model can oscillate for at least somevalues of its parameters.

Conductance-Based Models

129

was considered in the previous chapter where we have shown that it could oscillate dueto the interplay between the activation of persistent sodium and potassium currents.
Alternatively, we can remove the K+ current from the Hodgkin-Huxley model, yet the
new INa,t -model
INa,t
IL
}|
{
z
z
}|
{
C V̇ = I − gNa m3 h(V − ENa ) − gL (V − EL ) ,ṁ = (m∞ (V ) − m)/τm (V ) ,ḣ = (h∞ (V ) − h)/τh (V ) ,can still oscillate via the interplay between activation and inactivation of the Na+
current, as we will see later in this chapter. Both models are minimal, because removalof any other gating variable results in the INa,p -, IK -, or Ih -models, none of which canhave a limit cycle attractor, as the reader is asked to prove at the end of chapter 4.
We see that the Hodgkin-Huxley model is not minimal by itself, but is a combinationof two minimal models. Minimal models are appealing because they are relatively
simple; each individual variable has an established electrophysiological meaning, and its
role in dynamics can be easily identified. As we show below, many minimal models canbe reduced to planar systems, which are amenable to analysis using geometrical phaseplane methods. In section 5.2 we discuss other methods of reducing multidimensionalmodels, e.g., the Hodgkin-Huxley model, to planar systems.
There are only few minimal models, and understanding their dynamics can shedlight on dynamics of more complicated electrophysiological models. However, thereader should be aware of the limitations of such an approach: Understanding minimalmodels cannot provide exhaustive information about all electrophysiological models(just as understanding the zeros of the equations y = x and y = x2 does not provide
complete information about the zeros of the equation y = x + x2 ).

5.1.1

Amplifying and Resonant Gating Variables

The definition of the minimal models involves a top-down approach: take a complicatedmodel and strip it down to minimal ones. It is unlikely that this could be done for all
230 or so electrophysiological models. Instead, we employ here a bottom-up approach,which is based on the following rule of thumb: a mixture of one amplifying and one
resonant (recovery) gating variable (plus an Ohmic leak current) results in a minimalmodel. Indeed, neither of the variables alone can produce oscillation, but together theycan (as we will see below).
The amplifying gating variable is the activation variable m for voltage-gated inward current or the inactivation variable h for voltage-gated outward current, as inFig.5.2. These variables amplify voltage changes via a positive feedback loop. Indeed,130

Conductance-Based Models

activation, m
inactivation, hgatingcurrents
inward(Na, Ca)outward(K, Cl)amplifying

resonant

reversepotential

resonant

reversepotential

reversepotential

amplifying

reversepotential

Figure 5.2: Gating variables may
be amplifying or resonant dependingon whether they represent activation/inactivation of inward/outward
currents (see also Fig.3.3 and 3.4).

a small depolarization increases m and decreases h, which in turn increases inwardand decreases outward currents and increases depolarization. Similarly, a small hyperpolarization decreases m and increases h, resulting in less inward and more outward
current, and hence in more hyperpolarization.
The resonant gating variable is the inactivation variable h for an inward current or
the activation variable n for an outward current. These variables resist voltage changes
via a negative feedback loop. A small depolarization decreases h and increases n, which
in turn decreases inward and increases outward currents and produces a net outward
current that resists the depolarization. Similarly, a small hyperpolarization produces
inward current and, possibly, rebound depolarization.
Currents with amplifying gating variables can result in bistability, and they behave
essentially like the INa,p -model or IKir -model considered in chapter 3. Currents withresonant gating variables have one stable equilibrium with possibly damped oscillation,and they behave essentially like the IK -model or the Ih -model (compare Fig.5.2 with
Fig.3.3). A typical neuronal model consists of at least one amplifying and at least one
resonant gating variable. (Amplifying and resonant gating variables for Ca2+ -sensitive
currents are discussed later in this chapter).
To get spikes in a minimal model, we need a fast positive feedback and a slower
negative feedback. Indeed, if an amplifying gating variable has a slow time constant,it would act more as a low-pass filter, hardly affecting fast fluctuations and amplifying
only slow fluctuations. If a resonant gating variable has a fast time constant, it will
act to damp input fluctuations (faster than they can be amplified by the amplifying
variable), resulting in stability of the resting state. Instead, the resonant variable
acts as a band-pass filter; it has no effect on oscillations with a period much smaller
than its time constant; it damps oscillations having a period much larger than its
time constant, because the variable oscillates in phase with the voltage fluctuations; itamplifies oscillations with a period that is about the same as its time constant because

Conductance-Based Models

131

resonant gating variables

activation of
inward current
inactivation ofoutward current

amplifying gating variables

inactivation of
inward current

activation ofoutward current

INa,t-modelINa,p+IK -modelINa,p+Ih -model

IA-modelIKir+Ih -model
IKir+IK -model

Figure 5.3: Any combination of one
amplifying variable and one resonant
gating variable results in a spiking
model.

the variable lags the voltage fluctuations.
Since the amplifying gating variable, say m, has relatively fast kinetics, it can be
replaced by its equilibrium (steady-state) value m∞ (V ). This allows us to reduce thedimension of the minimal models from 3 (say V , m, n) to 2 (V and n).
Two amplifying and two resonant gating variables produce four different combinations, depicted in Fig.5.3. However, the number of minimal models is not four, but six.
The additional models arise due to the fact that a pair of gating variables may describe
activation/inactivation properties of the same current or of two different currents. For
example, the activation and inactivation gating variables m and h may describe thedynamics of a transient inward current, such as INa,t , or the dynamics of a combinationof one persistent inward current, such as INa,p , and one “hyperpolarization-activated”
inward current, such as Ih . Hence this pair results in two models, INa,t and INa,p +Ih .
Similarly, the pair of activation and inactivation variables of an outward current may
describe the dynamics of the same transient current, such as IA , or the dynamics oftwo different outward currents; hence the two models, IA and IKir +IK .
Below we present the geometrical analysis of the six minimal voltage-gated models
shown in Fig.5.3. Though they are based on different ionic currents, the models have
many similarities from the dynamical systems point of view. In particular, all can
exhibit saddle-node and Andronov-Hopf bifurcations. For each model we first provide aword description of the mechanism of generation of sustained oscillations, and then use
phase plane analysis to provide a geometrical description. The first two, the INa,p +IK model and the INa,t -model, are common; they describe the mechanism of generation ofaction potentials or subthreshold oscillations by many cells. The other four models are
rare; they might even be classified as weird or bizarre by biologists, since they reveal
rather unexpected mechanisms for voltage oscillations. Nevertheless, it is educational
to consider all six models to see how the theory of dynamical systems works where
intuition and common sense fail.

132

Conductance-Based Models
1

1

I=0

llclin
e

I=0

0.4

15

0.4

ullc

V-n

V-n

0.2

line

line

ullc

0.2

0

0.6

line

0.6

n-nullc

15

K+ activation, n

0

0.8

n-nu

K+ activation, n

0.8

0

0
-60

-20
0
membrane voltage, V

-60

20

20
membrane voltage, V

1

1

I=40

llclin

e

I=10
0.8

0.4

0.6

0.4

line
ullc

li
ullc

V-n

0.2

15

line

0.6

0

n-nullc

15

K+ activation, n

n-nu

0

V-n

K+ activation, n

0.8

0.2

ne

0

0
-60

-20
0
membrane voltage, V

a

20

-60

-20
0
membrane voltage, V

20bFigure 5.4: Possible forms of nullclines in the INa,p +IK -model (parameters as in
Fig.4.1).

5.1.2

INa,p +IK -Model

One of the most fundamental models in computational neuroscience is the INa,p +IK model (pronounced persistent sodium plus potassium model), which consists of a fast
Na+ current and a relatively slower K+ current:
INa,pIK
leak Iz }| L { z
}|
{ z
}|
{
C V̇ = I − gL (V −EL ) − gNa m(V − ENa ) − gK n(V − EK ) ,ṁ = (m∞ (V ) − m)/τm (V ) ,ṅ = (n∞ (V ) − n)/τn (V ) .
This model is in many respects equivalent to the ICa +IK -model proposed by Morrisand Lecar (1981) to describe voltage oscillations in the barnacle giant muscle fiber.
A reasonable assumption based on experimental observations is that the Na+ gating
variable m(t) is much faster than the voltage variable V (t), so that m approaches theConductance-Based Models

133

asymptotic value m∞ (V ) practically instantaneously. In this case we can substitutem = m∞ (V ) into the voltage equation and reduce the three-dimensional system above
to a planar system,
instantaneous INa,p
IK
leak I}|{ z}|{z }| L { zC V̇ = I − gL (V −EL ) − gNa m∞ (V ) (V −ENa ) − gK n (V −EK )ṅ = (n∞ (V ) − n)/τ (V ) ,

(5.1)
(5.2)

which was considered in detail in chapter 4. In Fig.5.4 we summarize the dynamic
repertoire of the model. A striking observation is that the other minimal models canhave similar nullclines and similar dynamic repertoire, even though they consist ofquite different ionic currents.

5.1.3

INa,t -model

An interesting example of a spiking mechanism, implicitly present in practically every
biological neuron, is given by the INa,t -model (pronounced transient sodium model),INa,tleak IL
}|
{
z
}|
{
z
3
C V̇ = I − gL (V − EL ) − gNa m h(V − ENa ) ,ṁ = (m∞ (V ) − m)/τm (V ) ,ḣ = (h∞ (V ) − h)/τh (V ) ,consisting only of an Ohmic leak current and a transient voltage-gated inward Na+
current. How could such a model generate action potentials? The upstroke of anaction potential is generated because of the regenerative process involving the activation gate m. This mechanism is similar to that in the Hodgkin-Huxley model andthe INa,p +IK -model: increase of m results in an increase of the inward current; hence
more depolarization and further increase of m until the excited state is achieved. At
the excited state there is a balance of the Na+ inward current and the leak outward
current.
Since there is no IK , the downstroke from the excited state occurs via a different
mechanism. While in the excited state, the Na+ current inactivates (turns off) and the
Ohmic leak current slowly repolarizes the membrane potential toward the leak reverse
potential EL , which determines the resting state. While at rest, the Na+ current
deinactivates (i.e., becomes available), and the neuron is ready to generate anotheraction potential. This mechanism is summarized in Fig.5.5.
To study the dynamics of the INa,t -model, we first reduce it to a planar system.
Assuming that activation dynamics is instantaneous, we use m = m∞ (V ) in the voltage
equation and obtain

C V̇

INa,t , inst. activation
leak IL
z
}|
{
}|
{
z
3
= I − gL (V − EL ) − gNa m∞ (V )h(V − ENa ) ,134Conductance-Based Models

inactivationof INadeinactivationof INaactivationof INadepolarization

resting potential

leak current

Figure 5.5: Mechanism of generation of sustained oscillations in the INa,t -model.

injected
dc-currentḣ = (h∞ (V ) − h)/τh (V ) .

One can easily find the nullclinesh=I − gL (V −EL )gNa m3∞ (V )(V − ENa )(V -nullcline)andh = h∞ (V )(h-nullcline) .

The V -nullcline looks like a cubic parabola (flipped N-shape), and the h-nullcline hasa sigmoid shape. In Fig.5.6 we depict two typical cases (we invert the h-axis so thatthe vector field is directed counterclockwise, and this phase portrait is consistent withthe other phase portraits in this book).
When the inactivation curve h∞ (V ) has a high threshold (i.e., INa,t is a window
current), there are three intersections of the nullclines, and hence three equilibria,
as in Fig.5.6a. A stable node (filled circle) corresponds to the resting state, and a
nearby saddle corresponds to the threshold state. Another equilibrium, an unstablefocus denoted by a white circle at the top of the figure, determines the shape of theaction potential since all “spiking” trajectories have to go around it. Because of thehigh threshold of inactivation, the INa,t current is deinactivated at rest. Moreover,small fluctuations of V do not produce significant changes of the inactivation variableh because the h-nullcline is nearly horizontal at rest. Such a system does not perform
damped oscillations, and the nonlinear dynamics of the V near resting state can be
described by the one-dimensional system (where h = h∞ (V ))C V̇ = I − gL (V − EL ) − gNa m3∞ (V )h∞ (V )(V − ENa )studied in chapter 3. When I increases, the stable node and the saddle approach,coalesce, and annihilate each other via saddle-node bifurcation. When I = 0.5, there isa periodic trajectory with a long period (compare the time scales in the bottom insets
in Fig.5.6a and 5.6b).
When the Na+ inactivation curve h∞ (V ) has a low threshold, the nullclines have
only one intersection; hence there is only one equilibrium, as in Fig.5.6b. When I =0, the equilibrium (filled circle) is stable, and all trajectories converge to it. ThereConductance-Based Models
I=0

I=0

Na+ inactivation, h

0.2

Na+ inactivation, h

0.4

0.6

0

1

1

h-n

0.8

e

100

0.8

line

0.6

V-nullclin

h-null

cline

0.4

ullc

0.2

0

V-nullcline

0

135

50

-50

0
membrane voltage, V

I=4Na+ inactivation, h0.2

llclin

e

0

0.8

h-nu

h-null

0.6

V-nullclin

0.6

e

0.4

cline

0.4

300

0.8
0

1

50

0
I=0.5

Na+ inactivation, h0.20membrane voltage, V

V-nullcline

0

-50

100

0

-50

0
membrane voltage, V

a

50

1

-50

100

0
membrane voltage, V

50bFigure 5.6: Possible forms of nullclines in the INa,t -model. Notice that the h-axis
is inverted. Parameters for INa,t are as in the Hodgkin-Huxley model, except thatτh (V ) = 5 ms. ENa = 60 mV, EL = −70 mV, gL = 1, gNa = 15 (in b) and gNa = 10
and V1/2 = −42 mV (in a).

are damped oscillations near the equilibrium, though they can hardly be seen in the
figure. The oscillations occur because the INa,t current is partially inactivated at rest.
An increase of V leads to more inactivation, less inward current, and hence rebounddecrease of V , which in turn leads to partial deinactivation, more inward current, andto rebound increase of V . When the applied DC current I increases, the equilibrium
loses stability via Andronov-Hopf bifurcation. When I = 4, the equilibrium is an
unstable focus (white circle in the figure), and there is a stable limit cycle attractor
around it corresponding to periodic spiking.
We see that the INa,t -model exhibits essentially the same dynamic repertoire as theINa,p +IK -model, even though the models are quite different from the electrophysiological point of view.

Conductance-Based Models
leak current
or injecteddc-current
inactivation
deactivation
of Ih
of I

5.1.4

Na,p

deinactivation of Ih

activationof INa,pdepolarization

hyperpolarization

136

Figure 5.7: Mechanism of generation of sustained voltage oscillations in the INa,p +Ih model.

INa,p +Ih -Model

The system (pronounced persistent sodium plus h-current model)
INa,pI
leak I}|{ z}|h{z }| L { zC V̇ = I − gL (V −EL ) − gNa m(V − ENa ) − gh h(V − Eh ) ,ṁ = (m∞ (V ) − m)/τm (V ) ,ḣ = (h∞ (V ) − h)/τh (V ) ,is believed to describe the essence of the mechanism of slow subthreshold voltage oscillations in some cortical, thalamic, and hippocampal neurons, which we summarize
in Fig.5.7. Like any other minimal model in this section, it consists of one amplifying(INa,p ) and one resonant (Ih ) current. Both currents may be partially active at resting voltage. Recall that we treat the h-current as an inward current that is always
activated (its activation variable m = 1 all the time), but can be inactivated (turnedoff) by depolarization and deinactivated (turned on) by hyperpolarization. At resting
voltage this current is usually inactivated (turned off). A sufficient hyperpolarizationof V deinactivates (turns on) the h-current, resulting in rebound depolarization. While
depolarized, the h-current inactivates (turns off), and the leak current repolarizes themembrane potential toward the resting state. Without the persistent Na+ current,or some other amplifying current, these oscillations always subside, as the reader was
asked to prove in chapter 4, exercise 10. However, they may become sustained when
INa,p is involved.
To study dynamics of the INa,p +Ih -model, we assume that the activation kineticsof the Na+ current is instantaneous, and use m = m∞ (V ) in the voltage equation to
obtain a two-dimensional systeminstantaneous INa,pleak I
I
}|
{ z
}|h
{
z }| L { zC V̇ = I − gL (V −EL ) − gNa m∞ (V )(V − ENa ) − gh h(V − Eh ) ,ḣ = (h∞ (V ) − h)/τh (V ) .

Conductance-Based Models
0

-50

e
lin

-60

0.4

0

lc
ul

n
h-

-70
0

0.6

time, mseinactivation, h

I=0V-nullclin

0.2

137

1000

0.8
0.06

-70

-60

-50

-70

-60

-50

1

0

-50
-60

0.4

0

lc

ul
-n

h

-70
0

0.6

e
lin

e
V-nullclin

inactivation, h

0.2

I=-1

time, ms

1000

0.8
0.06

1
-100

-90

-80

-70
membrane voltage, V

-60

-50

Figure 5.8: Rest and sustained subthreshold oscillations in the INa,p +Ih -model. Parameters for currents are as in thalamocortical neurons, ENa = 20 mV, Eh = −43 mV,
EL = −80 mV, gL = 1.3, gNa = 0.9, and gh = 3.
The nullclines of this system,h=I − gL (V −EL ) − gNa m∞ (V )(V − ENa )gh (V − Eh )(V -nullcline)andh = h∞ (V )(h-nullcline),

have the familiar N- and sigmoid shapes depicted in Fig.5.8. We take the parameters for both currents from the experimental studies of thalamic relay neurons (seesection 2.3.5). This choice results in one intersection of the nullclines in the relevant
voltage range, which corresponds to only one equilibrium. This equilibrium is a stable
resting state when no current is injected, i.e., when I = 0. In Fig.5.8 (top), one canclearly see that h ≈ 0; that is, the h-current is inactivated (turned off). The restingstate is due to the balance of the inward persistent Na+ current and the Ohmic leakcurrent. A small hyperpolarization deactivates the fast Na+ current and shifts thebalance toward the leak current, which brings V closer to Eleak . This, in turn, results

138

Conductance-Based Models

deinactivationof IKir

inactivationof Ih

deinactivation of Ih

inactivationof IKir

depolarization

hyperpolarization

leak current

Figure 5.9: Mechanism of generation of sustained voltage oscillations in the Ih +IKir model.

in slow deinactivation (turning on) of the h-current, which produces a strong inward
current and brings the membrane voltage back to the resting state.
Negative injected current (case I = −1 in Fig.5.8) destroys the balance of inward(INa,p ) and outward (Ileak ) currents at rest, and makes the resting state unstable. As
a result, the model exhibits sustained subthreshold oscillations of membrane potential. Indeed, prolonged hyperpolarization turns on a strong h-current and produces
prolonged depolarization. Such a depolarization turns off the h-current, and the negative injected current hyperpolarizes the membrane potential again. As a result, themodel exhibits sustained oscillations in the voltage range of −55 mV to −65 mV. The
frequency of such oscillations depends on the parameters of the voltage equation andthe time constant τ (V ) of the h-current; it is near 4 Hz in Fig.5.8.

5.1.5

Ih +IKir -Model

The persistent Na+ current, which amplifies the damped oscillations in the INa,p +Ih model, can be replaced by the K+ inwardly rectifying current IKir to achieve the same
amplifying effect. The resulting Ih +IKir -model (pronounced h-current plus inwardly
rectifying potassium model)
IKir
I
leak I}|{ z}|h{z }| L { zC V̇ = I − gL (V −EL ) − gKir hKir (V − EK ) − gh h(V − Eh ) ,ḣKir = (hKir,∞ (V ) − hKir )/τKir (V ) ,ḣ = (h∞ (V ) − h)/τh (V ) ,can exhibit sustained subthreshold oscillations of membrane voltage via a rather weird
mechanism illustrated in Fig.5.9. The inwardly rectifying K+ current IKir behaves like
Ih , except that the former is an outward current. A brief hyperpolarization deinactivates (turns on) the fast outward current IKir and produces more hyperpolarization
via a positive feedback loop. Such a regenerative process results in a prolonged hyperpolarization that deinactivates (turns on) the slower inward current Ih and produces
a rebound depolarization. This depolarization is enhanced by inactivation (turningoff) of the fast IKir . However, the membrane potential cannot hold long in the depolarized state because of the slow depolarization-triggered decrease of Ih , and the leak

Conductance-Based Models

139

0

0
I=10

I=9.75-70-65

time, s

2

-60
-55
-50
membrane voltage, V

ine
ullc
l

0.3

-60
0

time, s

2

e

0

-50

h-n

ine
ullc
l
h-n

-60

0.2

e

0.3

-50

V-nullclin

0.2

inactivation, h0.1

V-nullclin

inactivation, h

0.1

-45

-70

-65

-60
-55
-50
membrane voltage, V-45

Figure 5.10: Resting and sustained subthreshold oscillations in the Ih +IKir -model.
Parameters: EK = −80 mV, Eh = −43 mV, EL = −50 mV, gKir = 4, gh = 0.5,
gL = 0.44. The h-current is the same as in section 5.1.4, except V1/2 = −65 mV.
Instantaneous IKir has V1/2 = −76 mV and k = −11.
current repolarizes the membrane potential. The repolarization is enhanced by thedeinactivation of IKir and becomes a hyperpolarization again, leading to the oscillations summarized in Fig.5.9.
Since the kinetics of IKir is practically instantaneous, one can use hKir = hKir,∞ (V )in the voltage equation above and consider the two-dimensional systeminstantaneous IKir
I
leak I}|{ z}|h{z }| L { zC V̇ = I − gL (V −EL ) − gKir hKir,∞ (V )(V − EK ) − gh h(V − Eh ) ,ḣ = (h∞ (V ) − h)/τh (V ) .

One can easily find the nullclines of this system,h=I − gL (V −EL ) − gKir hKir,∞ (V )(V − EK )gh (V − Eh )(V -nullcline)andh = h∞ (V )(h-nullcline),

which have the familiar form depicted in Fig.5.10. Most values of the parameters
result in a phase portrait similar to the one depicted in Fig.5.10 (left). The V -nullcline
is a monotonic curve that intersects the h-nullcline in one point corresponding to astable resting state. An injected DC current I shifts the resting state, but does notchange its stability: Voltage perturbations always subside, resulting only in damped
oscillations. There is, however, a narrow region in parameter space (it took the authora few hours to find that region) that produces just the right relationship between
inactivation curves and conductances so that the V -nullcline becomes N-shaped andthe subthreshold oscillations become sustained, as in Fig.5.10 (right).

Conductance-Based Models
activationof IK
deactivationof IK

inactivationof IKir
injected
dc-current

5.1.6

depolarization

hyperpolarization

140

Figure 5.11: Mechanism of generation of sustained voltage oscillations in the IK +IKir model.

IK +IKir -Model

The last two minimal models consist exclusively of outward K+ currents, yet they can
exhibit sustained oscillations of membrane voltage. The models defy the imaginationof many biologists: How can a neuron with only outward K+ currents and no inward
Na+ or Ca2+ currents fire action potentials?
In the IK +IKir -model (pronounced persistent plus inwardly rectifying potassiummodel)
IK
IKir
}|
{ z
}|
{
z
C V̇ = I − gKir h(V − EK ) − gK n(V − EK ) ,ṅ = (n∞ (V ) − n)/τn (V ) ,ḣ = (h∞ (V ) − h)/τh (V ) ,the amplifying current is IKir with inactivation gating variable h, and the resonantcurrent is IK with activation variable n. The mechanism of generation of action potentials is summarized in Fig.5.11. A strong injected current depolarizes the membranepotential and inactivates (turns off) IKir , which amplifies the depolarization. While
depolarized, the slower K+ current IK activates and brings the potential down withpossible hyperpolarization, which is amplified by the deinactivation of IKir . While themembrane potential is hyperpolarized, IK deactivates and the strong injected current
brings the potential up again. Thus, the upstroke of the action potential is due exclusively to the injected DC current I, while the downstroke is due to the persistentoutward K+ current.
To perform the geometrical phase plane analysis of the model, we take advantage ofthe same observation as before: the kinetics of the amplifying current IKir is relatively
fast, so that h = h∞ (V ) can be used in the voltage equation to reduce the threedimensional system above to the two-dimensional systeminstantaneous IKir
IK
z
}|
{ z
}|
{
C V̇ = I − gKir h∞ (V )(V − EK ) − gK n(V − EK ) ,ṅ = (n∞ (V ) − n)/τn (V ) .

Conductance-Based Models

50

n-null
c

line

0.4

0.2

0.6

0.4

0.2

0

0
-40
-20
0
membrane voltage, V

-80

20

V-nullcline

0.8

I=68
0.8

n-null

cline

0.4

0.2

0.6

0.4

0.2

0

0
-80

-60

20

I=73

100

K+ activation, n

0

0.6

-40
-20
0
membrane voltage, V

V-nullcline

1

1

-60

50

cline

-60

0

n-null-80

K+ activation, n

I=70

cline

0

0.6

K+ activation, n0.8

n-nullV-nullcline

0.8

I=65

V-nullcline

1

1

K+ activation, n

141

-40
-20
0
membrane voltage, V

20

-80

0

-60

-40
-20
0
membrane voltage, V

a

100

20

b

Figure 5.12: Possible intersections of nullclines in the IK +IKir -model. Parameters:
EK = −80 mV, gKir = 20, gK = 2. Instantaneous IKir with V1/2 = −80 mV andk = −12. Slower IK with k = 5, τ (V ) = 5 ms, and V1/2 = −40 mV (in a) or
V1/2 = −55 mV (in b).
It is an easy exercise to find the nullclines
n = I/{gK (V − EK )} − gKir h∞ (V )/gK

(V -nullcline)

andn = n∞ (V )(n-nullcline) ,which we depict in Fig.5.12. There are two interesting cases corresponding to highthreshold (Fig.5.12a) and low-threshold (Fig.5.12b) K+ current IK .
When the IK has a low threshold, it is partially activated at resting potential. In
this case, the resting state corresponds to the balance of partially activated IK , partially inactivated IKir , and a strong injected DC current I. (Without the DC current
the membrane voltage would converge to EK = −80 mV and stay there forever.) A142

Conductance-Based Models

small depolarization partially inactivates fast IKir but leaves the slower IK relatively
unchanged. This results in an imbalance of the inward DC current I and all outward currents, and the net inward current further depolarizes the membrane voltage.
Depending on the size of the depolarization, the model may generate a subthreshold response or an action potential, as one can see in Fig.5.12b (top). During the generationof the action potential, the persistent K+ current activates and causes afterhyperpolarization. During the afterhyperpolarization, the persistent K+ current deactivates
below the resting level. This lets the injected DC current I depolarize the membranepotential again, provided that I is strong enough, as in Fig.5.12b (bottom).
In Fig.5.12a we leave all parameters unchanged except that we increase the halfvoltage activation V1/2 of IK by 15 mV, and decrease I to compensate for the deficitof outward current. Now, the resting state corresponds to the balance of IKir andI, because the high-threshold persistent K+ current is completely deactivated in thisvoltage range. The behavior near the resting state is determined by the interplay
between instantaneous IKir and I, and it was studied in chapter 3 (see exercises, IKir model). There are two equilibria: a stable node corresponding to the resting state
and a saddle corresponding to the threshold state. A sufficiently strong perturbation
can push V beyond the saddle equilibrium, as in Fig.5.12a (top), and can cause theminimal model to fire an action potential. If we increase I, the node and the saddle
approach, coalesce, and annihilate each other via a saddle-node bifurcation, and themodel starts to fire action potentials periodically.
We see that IK +IKir -model has essentially the same dynamic repertoire as the moreconventional INa,p +IK -model or the INa,t -model, despite the fact that it is based on arather bizarre ionic mechanism for excitability and spiking.

5.1.7

IA -Model

The last minimal voltage-gated model has only one transient K+ current, often referred
to as the A-current IA , yet it can also generate sustained oscillations. In some sense, themodel is similar to the INa,t -model. Indeed, each consists of only one transient current
and an Ohmic leak current. The only difference is that the A-current is outward, andas a result the action potentials are fired downward; see Fig.5.14 and Fig.5.15.
The A-current has activation and inactivation variables m and h, respectively, andthe IA -model (pronounced transient potassium model or A-current model) has the formIA
leak IL
}|
{
z
}|
{
z
C V̇ = I − gL (V − EL ) − gA mh(V − EK )ṁ = (m∞ (V ) − m)/τm (V )ḣ = (h∞ (V ) − h)/τh (V ) .

The mechanism of generation of downward action potentials is summarized in Fig.5.13.
Due to a strong injected DC current, the resting state is in the depolarizing voltage

Conductance-Based Models

143

deactivationof IA

activation +inactivationof IA

injected
dc-current

resting potential

hyperpolarization

de-inactivationof IA

Figure 5.13: Mechanism of generation of sustained voltage oscillations in the IA -model.

range, and it corresponds to the balance of the partially activated, partially inactivated
A-current, the leak outward current, and the injected DC current. A small hyperpolarization can simultaneously deactivate and deinactivate the A-current, that is, decrease
variable m and increase variable h. Depending on the relationship between the activation and inactivation time constants, this may result in an increase of the A-current
conductance, which is proportional to the product mh. More outward current produces
more hyperpolarization and even more outward current. As a result of this regenerative
process, the membrane voltage produces a sudden downstroke. While hyperpolarized,
the A-current deactivates (variable m → 0), and the injected DC current slowly brings
the membrane potential toward the resting state, resulting in a slow upstroke. A fast
downstroke and a slower upstroke from a depolarized resting state look like an action
potential pointing downward.
If activation kinetics is much faster than inactivation kinetics, we can substitutem = m∞ (V ) into the voltage equation above and reduce the IA -model to a twodimensional system, which hopefully would have the right kind of nullclines and alimit cycle attractor. After all, this is what we have done with previous minimalmodels, and it has always worked. As the reader is asked to prove in exercise 1, theIA -model cannot have a limit cycle attractor when the A-current activation kinetics
is instantaneous. Oscillations are possible only when the activation and inactivation
kinetics have comparable time constants or inactivation is much faster than activation.
Even though none of the experimentally measured A-currents show fast inactivation
and a relatively slower activation, this case is still interesting from the pure theoreticalpoint of view, since it shows how a single K+ current can give rise to oscillations.
Assuming instantaneous inactivation and using h = h∞ (V ) in the voltage equation, we
obtain a two-dimensional system,

IA , inst. inactivation
leak IL
}|
{
z
}|
{
z
C V̇ = I − gL (V − EL ) − gA m h∞ (V )(V − EK ) ,ṁ = (m∞ (V ) − m)/τm (V ) ,144

Conductance-Based Models

whose nullclines can easily be found:m=

I − gL (V −EL )gA h∞ (V )(V − EK )(V -nullcline)andm = m∞ (V )(m-nullcline) .

Two typical cases are depicted in Fig.5.14a and 5.14b. We start with the simpler casein Fig.5.14b.
Figure 5.14b depicts nullclines when the A-current has a low activation threshold.
There is only one intersection of the nullclines; hence there is only one equilibrium,
which is a stable focus when the injected DC current I is not strong enough (Fig.5.14b,
top). Increasing I makes the equilibrium lose stability via a supercritical AndronovHopf bifurcation that gives birth to a small amplitude limit cycle attractor (not shownin the figure). A further increase of I increases the amplitude of oscillations (e.g., whenI = 10 in the middle of Fig.5.14b), and the attractor corresponds to periodic firing of action potentials. When I = 10.5, the attractor disappears and the equilibrium becomesstable (via Andronov-Hopf bifurcation) again. The model, however, becomes excitable.
A small hyperpolarization does not significantly change the A-current, and the voltage
returns to resting state, resulting in a “subthreshold response”. A sufficiently large
hyperpolarization deinactivates enough IA to open the K+ current and hyperpolarize
the membrane even further. This regenerative process produces the downstroke andbrings V close to EK . During the state of hyperpolarization, the A-current deactivates(m → 0), and the DC current I brings V back to resting state. Notice that the action
potential is directed downward.
In Fig.5.14a we consider the IA -model with exactly the same parameters exceptthat we shift the half-voltage activation V1/2 of IA by 10 mV, so that the A-current has
a higher activation threshold. This does not greatly affect the behavior of the system
when I is small. However, when I ≈ 10.7, the spiking limit cycle attractor undergoes
another kind of bifurcation – saddle-node bifurcation – resulting in the appearanceof two new equilibria: a stable node and a saddle. If the reader looks at Fig.5.14a
upside-down, he or she will notice that this figure resembles figure 5.4a, 5.6a, or 5.12a,with all the consequences: the node corresponds to a resting state, and the saddle
corresponds to the threshold state. The large-amplitude trajectory that starts at thesaddle and terminates at the node corresponds to an action potential, though a weird
one. Thus, the behavior of this model is similar to the behavior of other models, withthe exception that the V -axis is reversed.
The existence of “upside-down” K+ spikes may (or, better say, does) look bizarre to
many researchers, even though “inverted” K+ and Cl− spikes were reported in many
preparations, including frog and toad axons, squid axons, lobster muscle fibers, anddog cardiac muscle, as reviewed by Reuben et al. (1961) and Grundfest (1971). Twosuch cases are depicted in Fig.5.15. Interestingly, Reuben et al. (1961) postulated,albeit reluctantly, that the inverted spikes are caused by the inactivation of the K+

Conductance-Based Models

145

1

1

I=6

0.2

-60
-40
-20
membrane voltage, V

m-

0
-80

0

1

-60
-40
-20
membrane voltage, V
I=10

0.8
time, ms

0.6

e

0.4
0.2

ine
llcl
nu
time, ms
0
m
0
-80
-60
-40
-20
membrane voltage, V

500

0.6
0.4

V-nullcline

K+ activation, m0V-nullclin

K+ activation, m

0.8

0.2
llcline

m-nu

500

0
-80

0

1

-60
-40
-20
membrane voltage, V
I=10.5

0.8
mnu

0.4
0.2

0

time, ms 200

0.6
0.4
V-nullcline

K+ activation, m

llcl
ine

0.8
V-nullcline

K+ activation, m01
I=10.8

0.2
0

0
-80

0

1
I=10.6

0.6

llcl
in
nu

0.4
0.2

ne

0
-80

time, ms 200

cline

0.4

0

0.6
ll
V-nu

m-

0.6

e

0.8
K+ activation, mtime, ms 200

nu

0

llcl
ine

0.8

llcli
V-nu

K+ activation, m

I=8time, ms

-60
-40
-20
membrane voltage, V

a

200

0

ne
ullcli

m-n
0
-80

-60
-40
-20
membrane voltage, V0b

Figure 5.14: Possible intersections of nullclines in the IA -model. Parameters: EK =−80 mV, EL = −60 mV, gA = 5, gL = 0.2. Instantaneous inactivation kinetics with
V1/2 = −66 mV and k = −10. Activation of the A-current with k = 10, τ (V ) = 20 ms,
and V1/2 = −45 mV (in a) or V1/2 = −35 mV (in b).
current. The reluctance was due to the fact that transient K+ IA was not known atthat time.
By now the reader must be convinced that quite different models can have practically identical dynamics. Conversely, the same model could have quite different behavior if only one parameter, e.g., V1/2 , is changed by as little as 10 mV. Such dramatic
conclusions emphasize the importance of geometrical phase plane analysis of neuronalmodels, since the conclusions can hardly be drawn from mere word descriptions of the
spiking mechanisms.

146

Conductance-Based Models
1 sec

100 nA

50 mV

20 mV

(b)

(a)

1 sec

Figure 5.15: Anomalous (upside-down) spikes in (a) lobster muscle fibers (modified
from Fig.2 of Reuben et al. 1961) and in (b) Ascaris esophageal cells (modified from
Fig.16 of del Castillo and Morales 1967; the cell is depolarized by injected DC current).
The voltage axis is not inverted.
Voltage-Gated
Currents

Activation

Inactivation

Ca2+-Gated
Activation

Inactivation

I leakInwardI Na,p I Ca

fast

I Na,t I Ca(T)

fast

fast

I Ca(P)

fast

slow

I Ca(L)fast

I Ca(N)
Ih
I CAN

fast

slow
mediummediummediumslow

Outward

IK

fast

I K(M)slow

IA
I K(D)

fast

fast

mediumslow

I K(Ca)
I AHP
I Kir

fast

fast

slow
fast

Figure 5.16: Some representative voltage- and Ca2+ -gated ionic currents (Johnston andWu 1995; Hille 2001; Shepherd 2004).

Conductance-Based Models

5.1.8

147

Ca2+ -Gated Minimal Models

So far, we have considered minimal models consisting of voltage-gated currents only.
However, there are many ionic currents that depend not only on the membrane potential but also on the concentration of intracellular ions, mostly Ca2+ . Such currents arereferred to as Ca2+ -gated, and they are summarized in Fig.5.16. In addition, there are
Cl− -gated, K+ -gated, and Na+ -gated currents, such as the SLO gene family of Cl− gated K+ currents discovered in C. elegans, and the related “slack and slick” family of
Na+ -gated K+ currents (Yuan et al. 2000, 2003). Considering minimal models involving these currents lies outside the scope of this book, but it could be a good intellectual
exercise for an expert reader (see also exercise 7 and 8).
Ca2+ -gated currents can also be divided into amplifying and resonant currents.
Ca2+ -activated inward currents, such as the cation nonselective ICAN , act as amplifying
currents. Indeed, activation of such a current leads to an influx of Ca2+ ions and to more
activation. Similarly, a hypothetical outward current inactivated by Ca2+ (not present
in the figure) might also act as an amplifying current. Indeed, a depolarization due tothe Ca2+ influx inactivates such a hypothetical outward current, thereby producing a
net shift toward inward currents and leading to more depolarization.
In contrast, Ca2+ -inactivating inward currents and Ca2+ -activating outward currents, such as ICa(L) and IAHP , respectively, act as resonant currents. Indeed, a depolarization due to the Ca2+ influx inactivates the inward current and activates theoutward current, and resists further depolarization.
Any combination of one voltage- or Ca2+ -gated amplifying current and one voltageor Ca2+ -gated resonant current leads to a minimal model for spiking. All such combinations are depicted in Fig.5.17. Here, I1 denotes a hypothetical Ca2+ -activatedvoltage-inactivated transient inward current. Though such a current is not presently
known, one can easily write a conductance-gated model for it. A biologist would treatsuch a current as hyperpolarization- and Ca2+ -activated. I2 is a hypothetical Ca2+ current that is inactivated by Ca2+ . I3 is a hypothetical voltage-inactivated Ca2+ current.
I4 is an outward Ca2+ -inactivated current.
We see that there are many minimal models in Fig.5.17. Six of them are purely
voltage-gated, and they have been investigated above. The others are mixed-mode or
purely Ca2+ -gated models. An interested reader can work out the details of their phase
portraits.

5.2

Reduction of Multidimensional Models

5.2.1

Hodgkin-Huxley model

Let us again consider the Hodgkin-Huxley model

C V̇

INa
IK
IL
}|
{
z
}|
{
z
z
}|
{
4
3
= I − gK n (V − EK ) − gNa m h(V − ENa ) − gL (V − EL ) ,148

Conductance-Based Models
resonant gate
Ca2+-gatedvoltage-gatedactivationof inward
current
inactivationof outward
current
activationof inward
current
inactivationof outward
current

voltage-gated
Ca2+-gatedamplifying gate

inactivation activation inactivation activationof inward of outward of inward of outward
current
current
current
current
INa,p+Ih

INa,p+IK

INa,t

IKir+Ih

ICa+I2ICa(L)

IKir+IK

ICa+IK(Ca)

IKir+I2IA

ICAN+Ih

ICAN+IK

ICAN+I2

I1

ICAN,t

I4+I3

I4+I2

ICAN+IK(Ca)

Figure 5.17: Voltage- and Ca2+ -gated minimal models. I1 , . . . , I4 are hypotheticalcurrents that could exist theoretically, but have never been recorded experimentally(see text).
ṅ = (n∞ (V ) − n)/τn (V ) ,ṁ = (m∞ (V ) − m)/τm (V ) ,ḣ = (h∞ (V ) − h)/τh (V ) ,with the original values of parameters presented in chapter 2. How can we understandthe qualitative dynamics of this model? One way, discussed above, is to throw away
the variable h or n and to reduce this model to the INa,p +IK -model or INa,t -model,
respectively. Although the reduced minimal models can tell a lot about the behaviorof the original model, they are not equivalent to the Hodgkin-Huxley model from theelectrophysiological or the dynamical systems point of view. Below we discuss anothermethod of reduction of multidimensional electrophysiological models to planar systems.
The Hodgkin-Huxley model has four independent variables. Early computer simulations by Krinskii and Kokoz (1973) have shown that there is a relationship between
the gating variables n(t) and h(t), namely,n(t) + h(t) ≈ 0.84 ,as shown in Fig.5.18. In fact, plotting the variables on the (n, h) plane, as we do in
Fig.5.19, reveals that the orbit is near the straight lineh = 0.89 − 1.1n .

Conductance-Based Models

149

100

V(t)
50
0

1

m(t)
n(t)
0.5

h(t)

0
2

n(t)+h(t)
0.84
0

0

10

20

30

40

50

60

70

80

90

100

time, t (ms)Figure 5.18: The sum n(t) + h(t) ≈ 0.84 in the Hodgkin-Huxley model. Parameters
are as in chapter 2 and I = 8.
We can use this relationship in the voltage equation to reduce the Hodgkin-Huxleymodel to a three-dimensional system. If, in addition, we assume that the activation
kinetics of the Na+ current is instantaneous, that is, m = m∞ (V ), then the HodgkinHuxley model can be reduced to the two-dimensional systeminstantaneous INa
IK
I
z
}|
{ z
}|
{ z }|L {4
3
C V̇ = I − gK n (V −EK ) − gNa m∞ (V )(0.89−1.1n)(V −ENa ) − gL (V −EL ) ,ṅ = (n∞ (V ) − n)/τn (V ) ,whose solutions retain qualitative and some quantitative agreement with the originalfour-dimensional Hodgkin-Huxley system (see Fig.5.20). The first step in the analysis
1

h=
n

.1

-1

89

0.

0.8

0.6

h
0.4

0.2

0
0

0.2

0.4

0.6

n

0.8

1

Figure 5.19: The relationship between n(t) andh(t) in the Hodgkin-Huxley model can be better
described by h = 0.89 − 1.1n.

150

Conductance-Based Models

original Hodgkin-Huxley model0000

reduced Hodgkin-Huxley model

10 mV

0

Figure 5.20: Action
potentials in the original (top) and reduced(bottom)
HodgkinHuxley model (I = 8).

1 ms

0
0
0

0

10

20
time (ms)30

40

of any two-dimensional system is to find its nullclines. The V -nullcline can be foundby numerically solving the equation
I − gK n4 (V −EK ) − gNa m3∞ (V )(0.89−1.1n)(V −ENa ) − gL (V −EL ) = 0
for n. The nullcline has the familiar N-shape depicted in Fig.5.21. Notice that it
has only one intersection with the n-nullcline n = n∞ (V ), hence there is only one
equilibrium, which is stable when I = 0. When the parameter I increases, the equilibrium loses stability via subcritical Andronov-Hopf bifurcation, as discussed in the next
chapter. When I is sufficiently large (e.g. I = 12 in Fig.5.21), there is a limit cycle
attractor corresponding to periodic spiking. In exercise 2 we discuss what happens
when I becomes very large.
I=0

I=12
0.8

line
n-n
ullc

K+ activation, n

n-n
ullc

V-nullcline

line

0.7
0.6
0.5

V-nullcline

0.4
0.3

0

20

40

60

80

membrane voltage, V

100

0.2

0

20

40

60

80

100

membrane voltage, V

Figure 5.21: Reduction of the Hodgkin-Huxley model to the (V, n) phase plane.

Conductance-Based Models

5.2.2

151

Equivalent Potentials

Inspired by the reduction idea of Krinskii and Kokoz (1973), Kepler et al. (1992) suggested a systematic method of reducing the complexity of conductance-based HodgkinHuxley-type models
C V̇ = I − I(V, x1 , . . . , xn )ẋi = (mi,∞ (V ) − xi )/τi (V ) ,i = 1, . . . , n ,where x1 , . . . , xn is a set of gating variables. The goal is to find certain patterns or
combinations of the gating variables that can be lumped to reduce the dimension of
the system. For example, we want to combine all resonant variables operating on asimilar time scale into a “master” recovery variable, then do the same for amplifying
variables.
Let us convert each variable xi (t) to the equivalent potential vi (t) that satisfies
xi = mi,∞ (vi ) .
In other words, the equivalent potential is the voltage which, in a voltage clamp, would
give the value xi when the model is at an equilibrium. Applying the chain rule tovi = m−1i,∞ (xi ), we express the model above in terms of equivalent potentials:
C V̇ = I − I(V, m1,∞ (v1 ), . . . , mn,∞ (vn )) ,v̇i = (mi,∞ (V ) − mi,∞ (vi ))/(τi (V ) m′i,∞ (vi )) .
Since the Boltzmann functions mi,∞ (V ) are invertible, the denominators do not vanish.
No approximations have been made yet; the new model is entirely equivalent to theoriginal one – it is just expressed in a different coordinate system. The new coordinates,
however, expose many patterns among the equivalent voltage variables that were notobvious in the original, gating coordinate system.
Kepler et al. (1992) developed an algorithm that replaces resonant and amplifying
variables with their weighted averages. The weights are found using Lagrange multipliers and strictly local criteria aimed at preserving the bifurcation structure of themodel. There is also a set of tests that informs the user when the method is likely
to fail. The method results in a lower-dimensional system that is easier to simulate,visualize, and understand.

5.2.3

Nullclines and I-V Relations

We saw that the form and the position of nullclines provided important information
about the neuron dynamics, that is, the number of equilibria, their stability, the existence of limit cycle attractors, and so on. The same information, in principle, can also
be obtained from the analysis of the neuronal current-voltage (I-V) relations. This is
not a coincidence, since there is a profound relationship between nullclines and experimentally measured I-V curves.

152

Conductance-Based Models
1000
outward

(V)

I0(V)

(V)

500
currentI0
inwardcurrent

I

I-V relations

250

0
I0(V)

-250

b

-50

-100
0

EK
time, ms

10

line

6

{I+I

4

(V)-I0(V)}/(V-EK){I-I0(V)}/(V-EK)

ullc

8

g-n

V

slow conductance, g0eV-nullclin

holding voltage, (mV)

10

2
EK
0
-100
-80

-60
-40
-20
0
membrane voltage, V (mV)

a

20

c

Figure 5.22: Voltage-clamp protocol to measure instantaneous (peak) and steady-state
current-voltage (I-V) relations. (Shown are simulations of the INa,p +IK -model from
Fig.5.4b.)
Let us illustrate the relationship by using the INa,p +IK -model, which we write in
the form
C V̇ = I − I0 (V ) − g(V − EK ) ,ġ = f (V, g) ,

(5.3)
(5.4)

where
I0 (V ) = gL (V −EL ) + gNa m∞ (V )(V − ENa )is the instantaneous (peak) current, and g = gK n is the slow conductance. The functionf (V, g) describes the dynamics of g, and its form is not important here. The method
described below is quite general, and it can be used in many circumstances when little
is known about the neuron’s electrophysiology.
In Fig.5.22 we describe a typical voltage-clamp experiment to measure the instantaneous (peak) and the steady-state I-V relations, denoted here as I0 (V ) and I∞ (V ),respectively. The holding voltage (Fig.5.22a, bottom) is kept at EK and then steppedto various values V . The recorded current (Fig.5.22a, top) typically consists of a fast(peak) component I0 (V ) that is due to the instantaneous activation of Na+ currents,
leak current, and other fast currents, and then it relaxes to the asymptotic steady-statevalue I∞ (V ). Repeating this experiment for various V , one can measure the I-V functions I0 (V ) and I∞ (V ) depicted in Fig.5.22b. Note that I0 (V ) has the N-shape with alarge region of negative slope. This region corresponds to the regenerative activation

Conductance-Based Models

153of the Na+ current, and it is responsible for the excitability property of the neuron. It
is also responsible for the N-shape of the V -nullcline, as we see next.
Once the I-V relations are found, we can find the nullclines of the system (5.3, 5.4).
From the equation
I − I0 (V ) − g(V − EK ) = 0
we can easily find the V -nullcline
g = {I − I0 (V )}/(V − EK )(V -nullcline) ,which has the inverted N-shape depicted in Fig.5.22c because I0 (V ) does. While measuring I∞ (V ), we hold V long enough so that all conductances reach their steady-statevalues. The steady-state value g = g∞ (V ) can be obtained from the equation
I − I0 (V ) − g(V − EK ) = −I∞ (V ) ,which says that the asymptotic steady-state current is the sum of the steady-state fastcurrent and steady-state slow current. Therefore,g = {I + I∞ (V ) − I0 (V )}/(V − EK )(g-nullcline)

depicted in Fig.5.22c. Since we used the INa,p +IK -model with parameters as in Fig.5.4b(top), we are not surprised that the V - and g-nullclines found here have the same shapeand relative position as those in Fig.5.4b (top). In exercise 5 we further explore therelationship between the I-V curves and neuronal dynamics.

5.2.4

Reduction to Simple ModelAll models discussed in this chapter can be reduced to two-dimensional systems having
a fast voltage variable, V , and a slower “recovery” variable, u, with N-shaped and sigmoidal nullclines, respectively. The decision to fire or not to fire is made at the restingstate, which is the intersection of the nullclines near the left knee, as we illustrate inFig.5.23a. To model the subthreshold behavior of such neurons and the initial segmentof the upstroke of an action potential, we need to consider only a small neighborhoodof the left knee confined to the shaded square in Fig.5.23. The rest of the phase spaceis needed only to model the peak and the downstroke of the action potential. If the
shape of the action potential is less important than the subthreshold dynamics leading
to this action potential, then we can retain detailed information about the left knee
and its neighborhood, and simplify the vector field outside the neighborhood. Thisapproach results in a simple model capable of exhibiting quite realistic dynamics, as
we will see in chapter 8.
Derivation via Nullclines
The fast nullcline in Fig.5.23b can be approximated by the quadratic parabolau = umin + p(V − Vmin )2 ,154

Conductance-Based Models
1

0.2

u-nullcline
V-nu
llclinerecovery variable, u

0.8

0.6

0.4

0.1
0.2

0

-80

-60
-40
-20
0
membrane potential, V (mV)

20

a

0
-70

-60
-50
-40
membrane potential, V (mV)b

Figure 5.23: Phase portrait (a) and its magnification (b) of a typical neuronal model
having voltage variable V and a recovery variable u.
where (Vmin , umin ) is the location of the minimum on the left knee, and p ≥ 0 is ascaling coefficient. Similarly, the slow nullcline can be approximated by the straightline
u = s(V − V0 ) ,where s is the slope and V0 is the V -intercept. All these parameters can easily be
determined geometrically or analytically.
Using these nullclines, we approximate the dynamics in the shaded region in Fig.5.23
by the systemV̇ = τf p(V − Vmin )2 − (u − umin ) ,u̇ = τs {s(V − V0 ) − u} ,where the parameters τf and τs describe the fast and slow time scales. Because of
the term (V − Vmin )2 , the variable V can escape to infinity in a finite time. This
corresponds to the firing of an action potential, more precisely, to its upstroke. To
model the downstroke, we assume that Vmax is the peak value of the action potential,
and we reset the state of the system(V, u) ← (Vreset , u + ureset ) ,when V = Vmax ,as if the spiking trajectory disappears at the right edge and appears at the left edgein Fig.5.23b. Here Vreset and ureset are parameters. Appropriate rescaling of variablestransforms the simple model into the equivalent formv̇ = I + v 2 − uu̇ = a(bv − u)having only four dimensionless parameters.

if v ≥ 1, thenv ← c, u ← u + d

(5.5)
(5.6)

Conductance-Based Models

155

current, I-k(v-vr)(v-vt)+b(v-vr)
I (V)-k(v-vr)(v-vt)
I0(V)

0

vr

vt

Figure 5.24: The relationship between
the parameters of the simple model(5.7, 5.8) and the instantaneous andsteady-state I-V relations, I0 (V ) andI∞ (V ), respectively.

membrane potential, v

Derivation via I-V Relations
The parameters of the simple model can be derived using instantaneous (peak) andsteady-state I-V relations. Let us represent the model in the equivalent form
C v̇ = k(v − vr )(v − vt ) − u + I
u̇ = a{b(v − vr ) − u}if v ≥ vpeak , thenv ← c, u ← u + d

(5.7)
(5.8)

where v is the membrane potential, u is the recovery current, and C is the membrane capacitance. The quadratic polynomial −k(v − vr )(v − vt ) approximates thesubthreshold part of the instantaneous I-V relation I0 (V ). Here, vr is the resting membrane potential, and vt is the instantaneous threshold potential, as in Fig.5.24. Thatis, instantaneous depolarizations above vt result in spike response. The polynomial−k(v − vr )(v − vt ) + b(v − vr ) approximates the subthreshold part of the steady-stateI-V relation I∞ (V ). When b < 0, its maximum approximates the rheobase current of
the neuron, i.e., the minimal amplitude of a DC current needed to fire a cell. Itsderivative with respect to v at v = vr , that is, b − k(vr − vt ), corresponds to the resting
input conductance, which is the inverse of the input resistance. Knowing both therheobase and the input resistance of a neuron, one can use the two equations above todetermine the parameters k and b. We do that in chapter 8 using recordings of real
neurons. This method does not work when b > 0.
The sum of all slow currents that modulate the spike generation mechanism is
combined in the phenomenological variable u, with outward currents taken with theplus sign. The form of (5.8) ensures that u = 0 at rest (i.e., when I = 0 and v = vr ).
The sign of b determines whether u is an amplifying (b < 0) or a resonant (b > 0)
variable. In the latter case, the neuron sags in response to hyperpolarized pulses ofcurrent, peaks in response to depolarized subthreshold pulses, and produces rebound(postinhibitory) responses. The recovery time constant is a. The spike cutoff value isvpeak , and the voltage reset value is c. The parameter d describes the total amount ofoutward minus inward currents activated during the spike and affecting the after-spikebehavior. All these parameters can easily be fit to any particular neuron type, as weshow in chapter 8.

156

Conductance-Based Models

Review of Important Concepts• Amplifying gating variables describe activation of an inward current
or inactivation of an outward current. They amplify voltage changes.
• Resonant gating variables describe inactivation of an inward current
or activation of an outward current. They resist voltage changes.
• To exhibit excitability, it is enough to have one amplifying and one
resonant gating variable in a neuronal model.
• Many models can be reduced to two-dimensional systems with one
equation for voltage and instantaneous amplifying currents, and one
equation for a resonant gating variable.
• The behavior of a two-dimensional model depends on the positionof its nullclines. Many models have an N-shaped V -nullcline and asigmoid-shaped nullcline for the gating variable.
• There is a relationship between nullclines and I-V curves.
• Quite different electrophysiological models can have similar nullclines, and hence essentially the same dynamics.
• The spike generation mechanism of detailed electrophysiologicalmodels depends on the dynamics near the left knee of the fast V nullcline, and it can be captured by a simple model (5.5, 5.6).

Bibliographical Notes
Richard FitzHugh pioneered the use of phase planes and nullclines to study the HodgkinHuxley model (FitzHugh 1955). First, he used an analog computer, consisting of operational amplifiers, function generators, and vacuum tubes, to simulate the model.
According to FitzHugh (see Izhikevich and FitzHugh 2006), the tubes were continually failing, and he had to find and replace several tubes a week. The heat fromall these tubes overloaded the air conditioning, so that on hot summer days he had
to take off his shirt and wear shorts to be comfortable. Not surprisingly, FitzHughcame up with a simple model with N-shaped cubic V -nullcline and a straight-line slow
nullcline, known as the FitzHugh-Nagumo model, to illustrate the mechanism of excitability of the Hodgkin-Huxley system. However, it was Krinskii and Kokoz (1973)
who first discovered the relationship n(t) + h(t) ≈ const, and thus were able to reduce the four-dimensional Hodgkin-Huxley model to a two-dimensional system. Since
then, the phase plane analysis of neuronal models has become standard, at least in
Russian-language literature.

Conductance-Based Models

157

Current awareness of the geometrical methods of phase plane analysis of neuronalmodels is mostly due to the seminal paper by John Rinzel and Bard Ermentrout Analysis of Neural Excitability and Oscillations, published as a chapter in Koch and Segev’sbook Methods in Neuronal Modeling (1989, 2nd ed., 1999). Not only did they introduce
the geometrical methods to a wide computational neuroscience audience, but they also
were able to explain a number of outstanding problems, such as the origin of Class 1
and 2 excitability observed by Hodgkin in 1948.
Rinzel and Ermentrout illustrated most of the concepts using the Morris-Lecar(1981) model, which is a ICa + IK -minimal voltage-gated model equivalent of the INa,p+ IK -model considered above. Due to its simplicity, the Morris-Lecar model is widelyused in computational neuroscience research. This is the reason we use its analogue,
the INa,p + IK -model, throughout the book.
Hutcheon and Yarom (2000) suggested classifying all currents into amplifying andresonant. There have been no attempts to classify various electrophysiological mechanisms of excitability in neurons, though minimal models, such as the INa,t -model orthe ICa +IK(C) -model, would not surprise most researchers. The other models would
probably look bizarre to classical electrophysiologists, though they provide a good
opportunity to practice geometrical phase plane analysis and support FitzHugh’s observation that an N-shaped V -nullcline is the key characteristic of neuronal dynamics.
Izhikevich (2003) took advantage of this observation and suggested the simple model(5.5, 5.6) that captures the spike generation mechanism of many known neuronal types(see chapter 8).

Exercises
1. Show that the IA -model cannot have a limit cycle attractor when IA has instantaneous activation kinetics. (Hint: Use the Bendixson criterion.)
2. When the injected DC current I or the Na+ maximal conductance gNa in theINa,p +IK -model has a large value, the excited state (V ≈ −20 mV) becomesstable. Sketch possible intersections of nullclines of the model.
3. Using I as a bifurcation parameter, determine the saddle-node bifurcation diagram of• The INa,t -model with parameters as in Fig.5.6a.
• The IA -model with parameters as in Fig.5.14a.
4. Why is g in Fig.5.22c negative when V is hyperpolarized?
5. In Fig.5.25 we plot the currents that constitute the right-hand side of the voltage
equation (5.3),
I − Ifast (V )andIslow (V ) = g(V − EK ) ,158

Conductance-Based Models

500

500

Islow(V) = g(V-EK)

Islow(V) = g(V-EK)

300

400

0

membrane current, I

membrane current, I

400

15

200
I-Ifast(V)
100

300

200

0

15

I-Ifast(V)

100

0

0

-100 EK-60-200membrane voltage, V (mV)

20

EK

-100

500

-60
-20
0
membrane voltage, V (mV)

500
Islow(V) = g(V-EK)

Islow(V) = g(V-EK)

200

400

0

membrane current, I

membrane current, I

400

300

20

15

I-Ifast(V)

100

300

200

0

15

I-Ifast(V)

100

0

0

-100 EK-60-200membrane voltage, V (mV)

20-100 EK-60-200membrane voltage, V (mV)

20

Figure 5.25:
Exercise 5: The (V, I)phase plane of theINa,p +IK -model (compare with Fig.5.4).

on the (V, I) plane. The curves define fast and slow movements of the state of
the system. Interpret the figure. (Hint: Treat the curves as “sort-of” nullclines.)
6. Show that the ICl +IK -model can have oscillations. (Hint: Inject negative DC
current so that the voltage-gated Cl− current becomes inward/amplifying).
7. (NMDA+IK -model) Show that a neuronal model consisting of an NMDA current
and a resonant current, (say, IK ) can exhibit excitability and periodic spiking.
8. The Nernst potential of an ion is a function of its concentration inside/outside thecell membrane, which may change. Consider the INa,p +ENa ([Na+ ]in/out )-modeland show that it can exhibit excitability and oscillations on a slow time scale.
9. Determine when the IA -model has a limit cycle attractor without assumingτh (V ) ≪ τm (V ).
10. [Ph.D.] There are Na+ -gated and Cl− -gated currents in addition to the Ca2+ gated currents considered in this book. In addition, the Nernst potentials maychange as concentrations of ions inside/outside the cell membrane change. This
may lead to new minimal models. Classify and study all these models.

Chapter 6
Bifurcations
Neuronal models can be excitable for some values of parameters, and fire spikes periodically for other values. These two types of dynamics correspond to a stable equilibrium
and a limit cycle attractor, respectively. When the parameters change, e.g., the injected
DC current in Fig.6.1 ramps up, the models can exhibit a bifurcation – a transition
from one qualitative type of dynamics to another. We consider transitions away fromequilibrium point in section 6.1 and transitions away from a limit cycle in section 6.2.
All these transitions can be reliably observed when only one parameter, in our case I,
changes. Mathematicians refer to such transitions as bifurcations of codimension-1. In
this chapter we provide definitions and examples of all codimension-1 bifurcations of an
equilibrium and a limit cycle that can occur in two-dimensional systems. In section 6.3
we mention some codimension-1 bifurcations in high-dimensional systems, as well assome codimension-2 bifurcations. In chapter 7 we discuss how the type of bifurcation
determines a cell’s neurocomputational properties.

6.1

Equilibrium (Rest State)

A neuron is excitable because its resting state is near a bifurcation, i.e., near a transition from quiescence to periodic spiking. Typically, such a bifurcation can be revealed
by injecting a ramp current, as we do in Fig.6.1. The four bifurcations in the figure havequalitatively different properties, summarized in Fig.6.2. In this section we use analytical and geometrical tools to understand what the differences among the bifurcations
are.
Recall (see chapter 4) that an equilibrium of a dynamical system is stable if all the
eigenvalues of the Jacobian matrix at the equilibrium have negative real parts. Whena parameter, say I, changes, one of two events can happen:
1. A negative eigenvalue increases and becomes 0. This happens at the saddle-nodebifurcation, and the equilibrium disappears.
2. Two complex-conjugate eigenvalues with negative real parts approach the imaginary axis and become purely imaginary. This happens at the Andronov-Hopfbifurcation, and the equilibrium loses stability but does not disappear.
159

160

Bifurcations

saddle-node bifurcation

1 ms

saddle-node on invariant circle bifurcation

20 mV
100 ms-60 mV

60 pA

supercritical Andronov-Hopf bifurcation
5 ms

subcritical Andronov-Hopf bifurcation
20 mV
100 ms-60 mV

2 nA

Figure 6.1: Transitions from resting to tonic (periodic) spiking occur via bifurcations ofequilibrium (marked by arrows). Saddle-node on invariant circle bifurcation: in vitro
recording of pyramidal neuron of rat’s primary visual cortex. Subcritical AndronovHopf bifurcation: in vitro recording of brainstem mesencephalic V neuron. The othertwo traces are simulations of the INa,p +IK -model.

Bifurcations

161

Fast
Bifurcation of an equilibrium subthresholdoscillations

Amplitudeof spikes

Frequencyof spikes

saddle-nodenononzerononzerosaddle-node on invariant circle

nononzero

√A I−Ib → 0

supercritical Andronov-Hopfyes√
A I−Ib → 0

nonzerosubcritical Andronov-HopfyesnonzerononzeroFigure 6.2: Summary of codimension-1 bifurcations of an equilibrium. I denotes theamplitude of the injected current, Ib is the bifurcation value, and A is a parameterthat depends on the biophysical details.

Thus, there are only two qualitative events that can happen with a stable equilibriumin a dynamical system of arbitrary dimension: it can either disappear or lose stability.
Of course, there could also be a third event: all eigenvalues continue to have negative
real parts, in which case the equilibrium remains stable.
Since any equilibrium of a neuronal model is the zero of the steady-state I-V curve
I∞ (V ) (the net current at the equilibrium must be zero), analysis of the shape of theI-V curve can provide invaluable information about possible bifurcations of the restingstate.
Two typical steady-state I-V curves are depicted in Fig.6.3. The I-V curve in
Fig.6.3a has a region with a negative slope and thus may have three equilibria: theleft equilibrium is probably stable (though it might be unstable; see exercise 8), themiddle is unstable, and the right equilibrium can be stable or unstable, depending on
the kinetics of the gating variables (it is stable in the one-dimensional case, i.e., whengating variables have instantaneous kinetics). The I-V curve in Fig.6.3b is monotone.
A positive (inward) injected DC current I shifts the I-V curves down. This leads to thedisappearance of the equilibrium in Fig.6.3a, but not in Fig.6.3b. Therefore, Fig.6.3acorresponds to the saddle-node bifurcation and Fig.6.3b to the Andronov-Hopf bifurcation. Exactly when the equilibrium loses stability in Fig.6.3b cannot be inferredfrom the I-V relations (for this, we need to consider the full neuronal model). But
what we can infer is that the bifurcation cannot be of the saddle-node type. Surprisingly, nonmonotonic I-V curves result in saddle-node bifurcations but do not exclude
Andronov-Hopf bifurcations, as the reader is asked to demonstrate in exercise 8. Thisphenomenon is relevant to the cortical pyramidal neurons considered in chapter 8.

162

Bifurcations
saddle-node bifurcation

rest state

Andnronov-Hopf bifurcation

I(V)

I(V)

rest state00noequilibria

new equilibrium
I=0
I>0

I=0
I>0
-100

-50

0

-100

-50

membrane voltage, V (mV)membrane voltage, V (mV)

(a)

(b)

0

Figure 6.3: Steady-state I-V curves of the INa,p +IK -model with high-threshold (left)
and low-threshold (right) K+ current (parameters as in Fig.4.1).

6.1.1

Saddle-Node (Fold)We provided the definition of a saddle-node bifurcation in one-dimensional systems
in section 3.3.4, and the reader is encouraged to look at that section and at Fig.4.31
before proceeding further.
A k-dimensional dynamical systemẋ = f (x, b) ,x ∈ Rkhaving an equilibrium point xsn for some value of the bifurcation parameter bsn (i.e.,f (xsn , bsn ) = 0) exhibits saddle-node (also known as fold) bifurcation if the equilibrium
is non-hyperbolic with a simple zero eigenvalue, the function f is non-degenerate, and
it is transversal with respect to b. The first condition is easy to check:• Non-hyperbolicity. The Jacobian k ×k matrix of partial derivatives at the equilibrium (see section 4.2.2) has exactly one zero eigenvalue, and the other eigenvalues
have nonzero real parts.
In general, the remaining two conditions have complicated forms, since they involve
projections of the vector field on the center manifold, which is tangent to the eigenvector
corresponding to the zero eigenvalue of the Jacobian matrix. However, there is a
shortcut for conductance-based neuronal models.
Let I(V, b) denote the steady-state I-V relation, which can be measured experimentally, divided by the membrane capacitance C. For example, I(V, I) = {I − I∞ (V )}/C
when the injected DC current I is used as a bifurcation parameter. We replace themulti-dimensional neuronal model with the one-dimensional system V̇ = I(V, b). From
I(V, b) = 0 (equilibrium condition) we find b = I∞ (V ). Non-hyperbolicity implies
IV (V, b) = 0, so that the bifurcation occurs at the local maxima and minima of I∞ (V ).
We considered all these properties in chapter 3.

Bifurcations

163• Non-degeneracy. The second-order derivative of I(V, bsn ) with respect to V isnonzero, that is,1 ∂ 2 I(V, bsn )̸= 0(at V = Vsn ) .
(6.1)
2 ∂V 2That is, the piece of the I-V curve, I∞ (V ), at the bifurcation point, Vsn , looks
like the square parabola.
a=• Transversality. Function I(V, b) is non-degenerate with respect to the bifurcation
parameter b; that is,∂I(Vsn , b)
̸= 0
(at b = bsn ) .
c=∂bThis condition is always satisfied when the injected DC current I is the bifurcation
parameter, because ∂I/∂b = ∂I/∂I = 1/C.
The saddle-node bifurcation has codimension-1 because only one condition (non-hyperbolicity) involves strict equality (“=”), and the other two involve inequalities (“̸=”).
The dynamics of multi-dimensional neuronal systems near a saddle-node bifurcation
can be reduced to that of the topological normal formV̇ = c(b − bsn ) + a(V − Vsn )2 ,(6.2)

where V is the membrane voltage, and a and c are defined above. In the context ofneuronal models, this equation with an after-spike resetting is called the quadratic
integrate-and-fire neuron, which we discuss in chapters 3 and 8.
Example: The INa,p +IK -model
Let us use the INa,p +IK -model (4.1, 4.2) with a high-threshold K+ current to illustratethese conditions. The saddle-node bifurcation occurs when the V -nullcline touches then-nullcline, as in Fig.6.4. Solving the equations numerically, we find that this occurs
when Isn = 4.51 and (Vsn , nsn ) = (−60.935, 0.0007). The Jacobian matrix at theequilibrium,
0.0435 −290
L=,0.00015−1
has two eigenvalues, λ1 = 0 and λ2 = −0.9565, with corresponding eigenvectors




1
1
v1 =
and
v2 =
,
0.00015
0.0034
depicted in the inset in Fig.6.4. (It is easy to check that Lv1 = 0 and Lv2 = −0.9565v2 .)The non-degeneracy and transversality conditions yield a = 0.1887 and c = 1, so thatthe topological normal form for the INa,p +IK -model isV̇ = (I − 4.51) + 0.1887(V + 61)2 ,(6.3)

which can be solved analytically. The corresponding bifurcation diagrams are depicted
in Fig.6.5. It is no surprise that there is a fairly good match when I is near thebifurcation value.

BifurcationsneV-nullcli

164

0.5

K+ activation, n

0.4

v2

v1

0.001

0.3
0
-63

0.2

-62

-61

-60

-59

-58

0.1
n-nullcline

0

-80

-70

-60
-50
-40
-30
membrane voltage, V (mV)

-20

-10

0

Figure 6.4: Saddle-node bifurcation in the INa,p +IK -model (4.1, 4.2) with highthreshold K+ current (parameters as in Fig.4.1a) and I = 4.51.
-50

membrane voltage, V (mV)

-55

-60

-65

-70

Figure 6.5:
Bifurcation
diagrams of the topologicalnormal form (6.3) and theINa,p +IK -model (4.1, 4.2).

l form
norma

el

-mod
I Na+I K

-75
-15

-10

-5

0

5

10

injected dc-current, I

6.1.2

Saddle-Node on Invariant Circle

As its name indicates, saddle-node on invariant circle bifurcation (also known as SNICor SNLC bifurcation) is a standard saddle-node bifurcation described above with anadditional condition: it occurs on an invariant circle, compare Fig.6.6a and 6.6b.
Here, the invariant circle consists of two trajectories, called heteroclinic trajectories
connecting the node and the saddle. It is called invariant because any solution startingon the circle remains on the circle. As the saddle and node coalesce, the small trajectoryshrinks and the large heteroclinic trajectory becomes a homoclinic invariant circle, i.e.,Bifurcations

165(a) saddle-node bifurcation
cle
it cy
lim

node

saddle-nodesaddle(b) saddle-node on invariant circle (SNIC) bifurcationi

circle
ant
i
r
a
nv

node

saddlesaddle-nodeFigure 6.6: Two types of saddle-node bifurcation.
originating and terminating at the same point. When the point disappears, the circle
becomes a limit cycle.
Both types of the bifurcation can occur in the INa,p +IK -model, as we show in
Fig.6.7. The difference between the top and the bottom of the figure is the time
constant τ (V ) of the K+ current. Since the K+ current has a high threshold, thetime constant does not affect dynamics at rest, but it makes a huge difference whenan action potential is generated. If the current is fast (top), it activates during the
upstroke, thereby decreasing the amplitude of the action potential, and deactivates
during the downstroke, thereby resulting in overshoot and another action potential.
In contrast, the slower K+ current (bottom) does not have time to deactivate during
the downstroke, thereby resulting in undershoot (short afterhyperpolarization), with
V going below the resting state.
From the geometrical point of view, the phase portraits in Fig.6.6b and in Fig.6.7(bottom), have the same topological structure: there is a homoclinic trajectory (aninvariant circle) that originates at the saddle-node point, leaves its small neighborhood(to fire an action potential), then reenters the neighborhood, and terminates at thesaddle-node point. This homoclinic trajectory is a limit cycle attractor with infinite
period, which corresponds to firing with zero frequency. This and other neurocomputational features of saddle-node bifurcations are discussed in the next chapter. Below,
we only explore how the frequency of oscillation depends on the bifurcation parameter,e.g., on the injected DC current I.

166

Bifurcations

cline

fast K+ current

n-null

0.6

0.4

0.3

ullc

V-n

0.2

line

K+ activation variable, n

0.5

0.1

0

saddle-node

0.6

n-nullclineslow K+ current
0.5

line
ullc

0.3

V-n

K+ activation variable, n

0.4

0.2

0.1

i

ircle
nt c
a
i
r
n va

0

saddle-node
on invariant circle
-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

membrane voltage, V (mV)

Figure 6.7: Saddle-node bifurcation in the INa,p +IK -model with a high-threshold K+
current can be off the limit cycle (top) or on the invariant circle (bottom). Parameters
are as in Fig.4.1a with τ (V ) = 0.152 (top) or τ (V ) = 1 (bottom).

Bifurcations

167

K+ gating variable, n

80
0.3

2

10
00
/T

100

0.4

frequency, Hz

V-nullcline

0.5

a
retic
theo
l
erica
num

l

60
40

0.2
20
0
4.5

0.1

5
injected dc-current, IA

5.5

n-nullcline

0

B

-80

-70

-60

-50

-40
-30
-20
membrane voltage, V (mV)

-10

0

10

20

90

100

-30

membrane voltage, V (mV)

cut spikecut spike

-40

B

-50

tan t
-60

-70

A

-80

T2

T1
0

10

20

30

40

50
60
time, t (ms)7080

Figure 6.8: The INa,p +IK -model can fire a periodic train of action potentials with
arbitrary small frequency when it is near a saddle-node on invariant circle bifurcation.
The trajectory moves fast from point B to A (a spike) and slowly in the shaded region
from point A to B. The segment labeled “tan t” behaves like tan t in the limit of
infinitesimal proximity to the bifurcation.

168

Bifurcations

A remarkable fact is that one can estimate the frequency of the large-amplitude limit
cycle attractor by considering a small neighborhood of the saddle-node point. Indeed,a trajectory on the limit cycle generates a fast spike from point B to A in Fig.6.8 andthen slowly moves from A to B (shaded region in the figure) because the vector field(the velocity) in the neighborhood between A and B is very small. The duration of thestereotypical action potentials, denoted here as T1 , is relatively constant and does notdepend much on the injected current I. In contrast, the time spent in the neighborhood(A, B) depends significantly on I. Since the behavior in the neighborhood is describedby the topological normal form (6.2), we can estimate the time the trajectory spends
there in terms of the parameters a, b, and c (see exercise 3). This yields
T2 = pπac(b − bsn ),where the parameters a, b, and c are those defined in the previous section. Thus, theperiod of one oscillation is T = T1 + T2 .
Figure 6.8 (top) illustrates the accuracy of this estimation, using the INa,p +IK model, whose topological normal form (6.3) was derived earlier. The duration of theaction potential is T1 = 4.7 ms, and the length of time the voltage variable spends inthe shaded neighborhood (A,B) (here −61 ± 11 mV) is approximated by
T2 = pπ0.1887(I − 4.51)(ms).

The analytical curveω=

1000
T1 + T2(Hz)

matches the numerically found frequency of oscillation (Fig.6.8, top) in a fairly broadfrequency range. For comparison, we plot the curve 1000/T2 to show that neglecting
the duration of the spike, T1 , can be justified only when I is very close to the bifurcation
point.

6.1.3

Supercritical Andronov-Hopf

If a neuronal model has a monotonic steady-state I-V relation, a saddle-node bifurcation cannot occur. The resting state in such a model does not disappear, but it loses
stability, typically via an Andronov-Hopf (sometimes called Hopf) bifurcation. The
loss of stability is accompanied either by the appearance of a stable limit cycle (supercritical Andronov-Hopf) or by the disappearance of an unstable limit cycle (subcritical
Andronov-Hopf).
Let us consider a two-dimensional systemv̇ = F (v, u, b)u̇ = G(v, u, b)(6.4)

Bifurcations

169

and suppose that (v, u) = (0, 0) is an equilibrium when the bifurcation parameterb = 0, that is, F (0, 0, 0) = G(0, 0, 0) = 0. This system undergoes an Andronov-Hopfbifurcation at the equilibrium if the following three conditions are satisfied:• Non-hyperbolicity. The Jacobian 2 × 2 matrix of partial derivatives at the equilibrium (see section 4.2.2),


Fv Fu
L=
,
Gv Gu
has a pair of purely imaginary eigenvalues, ±iω ∈ C with ω ̸= 0. That is,
tr L = Fv + Gu = 0 and ω 2 = det L = Fv Gu − Fu Gv > 0 at v = u = b = 0.
The linear change of variables
v=x

and

Fu u = −Fv x − ωy(6.5)

converts (6.4) into the formẋ = −ωy + f (x, y)ẏ = ωx+ g(x, y) ,(6.6)

where the functions
f (x, y) = F (v, u) + ωyandg(x, y) = −(Fv F (v, u) + Fu G(v, u))/ω − ωxhave no linear terms in x and y. Now we are ready to state the other two conditions:• Non-degeneracy. The parametera=1
1{fxxx + fxyy + gxxy + gyyy }+{fxy (fxx + fyy ) − gxy (gxx + gyy ) − fxx gxx + fyy gyy }
16
16ω
(6.7)

is nonzero.
• Transversality. Let c(b) ± iω(b) denote the complex-conjugate eigenvalues of theJacobian matrix of (6.4) for b near 0, with c(0) = 0 and ω(0) = ω. The real part,c(b), must be non-degenerate with respect to b, that is, c′ (0) ̸= 0.
The codimension of Andronov-Hopf bifurcation is one, since only one condition involves
strict equality (tr L = 0), and the other two involve inequalities (“̸=”).
The sign of a determines the type of the Andronov-Hopf bifurcation, depicted in
Fig.6.9:• Supercritical Andronov-Hopf bifurcation occurs when a < 0. It corresponds to astable limit cycle appearing from a stable equilibrium.
• Subcritical Andronov-Hopf bifurcation occurs when a > 0. It corresponds to an
unstable limit cycle shrinking to a stable equilibrium.

170

Bifurcationsr

unstableit cyclele limstabrstable equilibrium

unstable equilibrium

stableequilibrium

limit cycle

unstableit cyclele limstab

limit cycle

cSupercritical (a<0)

unstableequilibriumcFigure 6.9: AndronovHopf bifurcation: A
stableequilibrium
becomes unstable inthe system (6.8, 6.9).

Subcritical (a>0)

r
0ϕFigure 6.10: Polar coordinates: r is the amplitude(radius) and φ is the phase (angle) of oscillation.

Finding a in applications can be challenging. A few useful examples are considered in
exercises 14–18.
Any system undergoing an Andronov-Hopf bifurcation can be reduced to the topological normal form by a change of variables (see also exercise 4)ṙ = c(b)r + ar3 ,φ̇ = ω(b) + dr2 ,

(6.8)
(6.9)

where r ≥ 0 is the amplitude (radius), and φ is the phase (angle) of oscillation, as inFig.6.10, and a, b, c(b), and ω(b) are as above.
The function c(b) in the normal form (6.8, 6.9) determines the stability of theequilibrium r = 0 corresponding to a non-oscillatory state (stable for c < 0 and unstablefor c > 0, regardless of the value of a). The function ω(b) determines the frequency of
damped or sustained oscillations around this state. The parameter d describes how thefrequency of oscillation depends on its amplitude. A state-dependent change of time
can remove the term dr2 from (6.9) (Kuznetsov 1995), so many assume d = 0 to startwith.

Bifurcations

171

n

V

stableI=40

cles

limit cystableunstableI

max/min of oscillations ofmembrane potential, mV

I=30

0
-10

stable cycles
supercritical
Andronov-Hopfbifurcation

-20
-30

I=25

-40

bria
unstable equili

-50

le
stab

-60
-70

I=20

-80
0

20

40

60

80

injected DC current, I

I=15

I=10

I=0

I=-10
0.7
0.6

C
imaginary part, ω

0.4

V-nullcline

0.5

supercritical Andronov-Hopfbifurcation at I=14.66

cline

0.3

n-null

0.2
0.1
0

-80

-60

-40

-20

0

membrane voltage, V (mV)20

eigenvalues c + iω0unstablestable0real part, cFigure 6.11: Supercritical Andronov-Hopf bifurcation in the INa,p +IK -model with lowthreshold K+ current: As the bifurcation parameter I increases, the equilibrium loses
stability and gives birth to a stable limit cycle with growing amplitude. Parameters
are as in Fig.4.1b.

172

Bifurcations
15
membrane voltage, (mV)ω(I)
2eigenvalues = c(I) + ω(I)i

1

c(I)
0

izationlinear
al
eric
num

-1
0

10

14.66
20

30

10

al

frequency, 2π/period (rads/ms)

2.5

nu
th
m
eo
er
re
ica
tic
l

3

theoreticalnumerical

2

1.5

amplitude5

frequency

1

0.5

0

0

injected dc-current, I

10

20

30

14.66
0

0

injected dc-current, I

10

20

30

injected dc-current, I

Figure 6.12: Supercritical Andronov-Hopf bifurcation in the INa,p +IK -model with lowthreshold K+ current (see Fig.6.11). Dots represent numerical simulation of the fullmodel, continuous curves represent analytical results using the topological normal form
(6.8, 6.9).
Example: The INa,p +IK -Model
Let us use the INa,p +IK -model with low-threshold K+ current in Fig.6.11 to illustrate
the three conditions above. As the magnitude of the injected DC current I increases,
the equilibrium loses stability and gives birth to a stable limit cycle with growing
amplitude. Using simulations we find that the bifurcation occurs when Iah = 14.66
and (Vah , nah ) = (−56.5, 0.09). The Jacobian matrix at the equilibrium,
1 −335
L=,
0.0166−1
has a pair of complex conjugate eigenvalues ±2.14i, so the non-hyperbolicity conditionis satisfied. Next, we find numerically (in Fig.6.12 or analytically in exercise 9) thatthe eigenvalues at the equilibrium can be approximated byc(I) + ω(I)i ≈ 0.03{I − 14.66} ± (2.14 + 0.04{I − 14.66})i
in a neighborhood of the bifurcation point I = 14.66. Since the slope of c(I) is nonzero,
the transversality condition is also satisfied. Using exercise 17 we find that a = −0.0026
and d = −0.0029, so that the non-degeneracy condition is also satisfied, and thebifurcation is of the supercritical type. The corresponding topological normal form isṙ = 0.03{I − 14.66}r − 0.0026r3 ,φ̇ = (2.14 + 0.04{I − 14.66}) − 0.0029r2 .
To analyze the normal form, we consider the r-equation and neglect the phase variable
φ. Fromr(c(b) + ar2 ) = 0

Bifurcations

173
r'

r'
4

c(b)<0c(b)>0

0

-4
-1

0

1

radius, r

-1

0

1

radius, r

yy
yyy
yyyyy
supercritical Andronov-Hopf bifurcation

Figure 6.13: Supercritical Andronov-Hopf bifurcation in (6.8, 6.9) with c = ±1 anda = −1 (negative values of the radial polar coordinate r do not have any physicalmeaning).
we conclude that r = 0 is an equilibrium for any value of c(b). Since(c(b)r + ar3 )r = c(b)at r = 0,

the equilibrium is stable for c(b) < 0 and unstable for c(b) > 0, as we illustrate inFig.6.13. Indeed, the resting state in the INa,p +IK -model is stable when I < 14.66 andunstable when I > 14.66.
When c(b) > 0, the normal form has a family of stable periodic solutions withamplitudepr = c(b)/|a|and(frequency) = ω(b) + d c(b)/|a| .
Hence, the INa,p +IK -model has a family of periodic attractors withpr = 0.03{I − 14.66}/0.0026
and(frequency) = (2.14 + 0.04{I − 14.66}) − 0.0029 · 0.03{I − 14.66}/0.0026 ,174

Bifurcations

c
cy

le

n-nu

sta
ble

llclin

e

it

e

un
sta

t cycle
imi

lin

el
bl

llc

0.5

nu
V-

K+ activation variable, nlim

1stable equilibrium (rest)

0
-80

-70

-60

-50

-40

-30

-20

-10

0

10

Figure 6.14: Phase portrait of
the INa,p +IK -model: An unstable
limit cycle (dashed circle) is
often surrounded by a stable one(solid circle) in two-dimensional
neuronal models.

membrane voltage, V (mV)depicted in Fig.6.12. We see that the topological normal form describes the fullINa,p +IK -model near the Andronov-Hopf bifurcation not only qualitatively but also
quantitatively.

6.1.4

Subcritical Andronov-Hopf

Neuronal models with monotonic steady-state I-V relations can often exhibit subcritical
Andronov-Hopf bifurcations, as we illustrate in Fig.6.16, using the INa,p +IK -model having a low-threshold K+ current and a steep activation curve for the Na+ current. The
stable equilibrium in such a system is surrounded by an unstable limit cycle (dashedcircle), which is often surrounded by another stable cycle, as in Fig.6.14 (not depicted
in Fig.6.16 for clarity). As the magnitude of the injected DC current I increases, theunstable cycle shrinks to the stable equilibrium and makes it lose stability. Systems
undergoing such a bifurcation satisfy the same three conditions – non-hyperbolicity,
non-degeneracy, and transversality – presented in the previous section, and they canbe reduced to the topological normal form (6.8, 6.9) with positive a.
Analysis of the normal form shows that the stability of the non-oscillatory equilibrium r = 0 depends on the sign of c(b):p• When c(b) < 0 (see Fig.6.15, left), there is a pair of equilibria, r = ± |c(b)|/a
corresponding to an unstable periodic solution that shrinks to r = 0 as c(b) → 0
and makes the stable equilibrium r = 0 lose its stability.
• When c(b) > 0 (see Fig.6.15, right), the non-oscillatory state r = 0 is unstable,
and all trajectories diverge from it.
This behavior can be clearly seen in Fig.6.16.
Finally, note that there is always a bistability (co-existence) of the resting attractor and some other attractor near a subcritical Andronov-Hopf bifurcation in two-Bifurcations

175

r'

r'
c(b)<0

4

0

0

-4

-4

-1

0

c(b)>0

4

1

radius, r

-1

0

1

radius, r

yyy
yyy
yyy
yyy
yyyyyysubcritical Andronov-Hopf bifurcation

Figure 6.15: Subcritical AndronovHopf bifurcation in (6.8, 6.9) withc = ±1 and a = 1.

dimensional conductance-based models, as in Fig.6.14 (in non-neural models, the trajectories can go to infinity and there need not be bistability). The bistability mustalso be present at the saddle-node bifurcation of an equilibrium, but may or may not
be present at the saddle-node on invariant circle or at a supercritical Andronov-Hopfbifurcation.
Delayed Loss of Stability
In Fig.6.17a we inject a ramp of current into the INa,p +IK -model to drive it slowly
through the subcritical Andronov-Hopf bifurcation point I ≈ 48.75 (see Fig.6.16). We
choose the ramp so that the bifurcation occurs exactly at t = 100. Even though thefocus equilibrium is unstable for t > 100, the membrane potential remains near -50
mV, as if the equilibrium were still stable. This phenomenon, discovered by Shishkova(1973), is called delayed loss of stability. It is ubiquitous in simulations of smooth
dynamical systems near subcritical or supercritical Andronov-Hopf bifurcations.
The mechanism of delayed loss of stability is quite simple. The state of the system
is attracted to the stable focus while t < 100. Even though the focus loses stability
at t = 100, the state of the system is infinitesimally close to the equilibrium, so ittakes a long time to diverge from it. The longer the convergence to the equilibrium,
the longer the divergence from it; hence the noticeable delay. The delay has an upperbound that depends on the smoothness of the dynamical system (Nejshtadt 1985).
It can be shortened or even reversed (advanced loss of stability) by weak noise thatis always present in neurons. This may explain why the delay has never been seen
experimentally despite the fact that it is practically unavoidable in simulations.

176

Bifurcations
unstable

n

I=49

V

limit cycles

stableunstable

I

0
-20
-40

un

max/min of oscillations ofmembrane potential, mV

I=48.75s ta

b l e listablemit cycl
es

I=48

unstable

-60
-80
45
50
injected current, I

55

I=47

I=45

subcritical Andronov-Hopfbifurcation at I=48.8

C

I=43

V-nullcline

0.8

imaginary part, ω1

0.6

eigenvalues c + iω0unstablestable0.4

0real part, c

0.2

0

e
n-nullc li n
-80
-60
-40
-20
membrane voltage, V (mV)020

Figure 6.16: Subcritical Andronov-Hopf bifurcation in the INa,p +IK -model. As the bifurcation parameter I increases, an unstable limit cycle (dashed circle; see also Fig.6.14)
shrinks to an equilibrium and makes it lose stability. Parameters are as in Fig.4.1b,
except gL = 1, gNa = gK = 4, and the Na+ activation function has V1/2 = −30 mV andk = 7.

177(a)limit cycle
stableunstab leli m i
t cy
cle

0

-20

subcritical
Andronov-Hopfbifurcation-40
stableunstabledelayedloss of stability

-60

(b)
limit cycle
stableunstab leli m it cynoise-inducedclesustained oscillations

0-20

subcritical
Andronov-Hopfbifurcation

-40

-60

injected
current, I(t)membrane potential, V (mV)membrane potential, V (mV)

Bifurcations

60

40
0

20

40

60

80

100
time, t

120

140

160

180

200

Figure 6.17: Delayed loss of stability (a) and noise-induced sustained oscillations (b)near subcritical Andronov-Hopf bifurcation. Shown are simulations of the INa,p +IK model with parameters as in Fig.6.16 and the same initial conditions. Small conductance noise is added in (b) to unmask oscillations.
Unmasking of Oscillations by NoiseIn Fig.6.17b we repeat the same simulation as in Fig.6.17a except that we add a weak
conductance noise to the INa,p +IK -model. Starting with the same initial conditions,the system converges to the stable focus equilibrium, as expected, exhibiting damped
oscillations of membrane potential. After a while, however, it diverges from the equilibrium and exhibits sustained waxing and waning oscillations, as if there were a small
amplitude limit cycle attractor with a variable amplitude. The oscillations persist until
the state of the system escapes from the attraction domain of the stable focus, which
is bounded by the unstable limit cycle, to the attraction domain of the large-amplitude
stable limit cycle.

178

Bifurcations

Let us explain how weak noise unmasks damped oscillations and makes them sustained. It is convenient to treat noise as a series of perturbations that push the membrane potential in random directions, often away from the resting state. Each suchperturbation evokes a damped oscillation toward the resting state. Superposition ofmany such damped oscillations, occurring at different times, results in the waxing
and waning rhythmic activity seen in the figure (see also exercise 3 in chapter 7). In
chapters 8 and 9 we present many examples of noise-induced sustained oscillations inbiological neurons, and in chapter 7 we study their neurocomputational properties.

6.2

Limit Cycle (Spiking State)

In section 6.1 we considered all codimension-1 bifurcations of equilibria, which typicallycorrespond to transitions from resting to spiking states in neuronal models. Below
we consider all codimension-1 bifurcations of limit cycle attractors on a phase plane.
These bifurcations typically correspond to transitions from repetitive spiking to resting
behavior, as we illustrate in Fig.6.19, and they will be important in chapter 9 where
we consider bursting dynamics.
In Fig.6.18 we summarize how the bifurcations affect a periodic attractor. Saddlenode on invariant circle and saddle homoclinic orbit bifurcations involve homoclinic
trajectories having an infinite period (zero frequency). They result in oscillations withdrastically increasing interspike intervals as the system approaches the bifurcation state(see Fig.6.19).
In contrast, supercritical Andronov-Hopf bifurcation results in oscillations withvanishing amplitude, as one can clearly see in Fig.6.19. If neither the frequency nor
the amplitude vanishes, then the bifurcation is of the fold limit cycle type. Indeed,

Bifurcation of a limit cycle attractor

Amplitude

Frequency

saddle-node on invariant circle

nonzero

√A I−Ib → 0

supercritical Andronov-Hopf√
A I−Ib → 0

nonzerofold limit cycle

nonzerononzerosaddle homoclinic orbit

nonzero−A→0ln |I−Ib |Figure 6.18: Summary of codimension-1 bifurcations of a limit cycle attractor on aplane. I denotes the amplitude of the injected current, Ib is the bifurcation value, andA is a parameter that depends on the biophysical details.

Bifurcations

179

saddle-node on invariant circle bifurcation

10 mV
50 ms
-30 mV
700 pAsupercritical Andronov-Hopf bifurcation-10 mV

20 mV
100 ms180 pA-60 mV

fold limit cycle bifurcation
20 mV
10 ms
-40 mVsaddle homoclinic orbit bifurcation
20 mV
50 ms

Figure 6.19: Transitions from tonic (periodic) spiking to resting occur via bifurcations of limit cycle attractors (marked by arrows). Saddle-node on invariant circle
bifurcation: recording of layer 5 pyramidal neuron in rat’s visual cortex. Supercritical
Andronov-Hopf bifurcation: excitation block in pyramidal neuron of rat’s visual cortex. Fold limit cycle bifurcation: brainstem mesencephalic V neuron of rat. Saddle
homoclinic orbit bifurcation: neuron in pre-Boltzinger complex of rat brainstem. (Dataprovided by C. A. Del Negro and J. L. Feldman.)

180

Bifurcations

the amplitude and the interspike period are constant before the arrow in Fig.6.19
corresponding to the fold limit cycle bifurcation. Damped small-amplitude oscillation
after the arrow occurs because of oscillatory convergence to the equilibrium.
We start with a brief review of the saddle-node on invariant circle and the supercritical Andronov-Hopf bifurcations, which we considered in detail in section 6.1. Thesebifurcations can explain not only transitions from rest to spiking but also transitions
from spiking to resting states. Then we consider fold limit cycle and saddle homoclinic
orbit bifurcations.

6.2.1

Saddle-Node on Invariant Circle

A stable limit cycle can disappear via a saddle-node on invariant circle bifurcation as
depicted in Fig.6.20. The necessary condition for such a bifurcation is that the steadystate I-V relation is not monotonic. We considered this bifurcation in section 6.1.2
as a bifurcation from an equilibrium to a limit cycle; that is, from left to right in
Fig.6.20. Now consider it from right to left. As a bifurcation parameter changes,
e.g., the injected DC current I decreases, a stable limit cycle (circle in Fig.6.20, right)
disappears because there is a saddle-node bifurcation (Fig.6.20, center) that breaks thecycle and gives birth to a pair of equilibria– stable node and unstable saddle (Fig.6.20,
left). After the bifurcation, the limit cycle becomes an invariant circle consisting of aunion of two heteroclinic trajectories.
Depending on the direction of change of a bifurcation parameter, the saddle-node
on invariant circle bifurcation can explain either appearance or disappearance of alimit cycle attractor. In either case, the amplitude of the limit cycle remains relatively
constant, but its period becomes infinite at the bifurcation point because the cycle
becomes a homoclinic trajectory to the saddle-node equilibrium (Fig.6.20, center).
√ As
we showed in section 6.1.2 (see Fig.6.8), the frequency of oscillation scales as I − Ib
when the bifurcation parameter approaches the bifurcation value Ib .
bifurcation parameter changes
an
ari
inv

node

cle
t cir

saddle

an
ari
inv

cle
t cir

t
limi
ble
a
t
s

cycle

saddle-nodeFigure 6.20: Saddle-node on invariant circle (SNIC) bifurcation of a limit cycle attractor.

Bifurcations

181
bifurcation parameter changes
le limstabe
it cyclstable equilibrium

unstable equilibrium

Figure 6.21: Supercritical Andronov-Hopf bifurcation of a limit cycle attractor.

6.2.2

Supercritical Andronov-Hopffoimit cycle
le l
b
ta
limit cycle
ble

l
ld

cycle
imit

un
sta

s

A stable limit cycle can shrink to a point via supercritical Andronov-Hopf bifurcationin Fig.6.21, which we considered in section 6.1.3. Indeed, as the bifurcation parameterchanges, e.g., the injected DC current I in Fig.6.11 decreases, the amplitude of the limit
cycle attractor vanishes, and the cycle becomes a stable√ equilibrium. As we showed
in section 6.1.3 (see Fig.6.12), the amplitude scales as I − Ib when the bifurcation
parameter approaches the bifurcation value Ib .

Figure 6.22: Fold limit cycle bifurcation: a stable and an unstable limit cycle approachand annihilate each other.

6.2.3

Fold Limit Cycle

A stable limit cycle can appear (or disappear) via the fold limit cycle bifurcation
depicted in Fig.6.22. Consider the figure from left to right, which corresponds to thedisappearance of the limit cycle, and hence to the disappearance of periodic spiking
activity. As the bifurcation parameter changes, the stable limit cycle is approached byan unstable one; they coalesce and annihilate each other. At the point of annihilation,
there is a periodic orbit, but it is neither stable nor unstable. More precisely, it is stablefrom the side corresponding to the stable cycle (outside in Fig.6.22), and unstable fromthe other side (inside in Fig.6.22). This periodic orbit is referred to as a fold (also known

182

Bifurcations

as a saddle-node) limit cycle, and it is analogous to the fold (saddle-node) equilibrium
studied in section 6.1. Considering Fig.6.22 from right to left explains how a stable
limit cycle can appear seemingly out of nowhere: As a bifurcation parameter changes,
a fold limit cycle appears and then bifurcates into a stable limit cycle and an unstable
one.
Fold limit cycle bifurcation can occur in the INa,p +IK -model having low-threshold
K current, as we demonstrate in Fig.6.23. The top phase portrait, corresponding to
I = 43, is the same as the one in Fig.6.16. In that figure we studied how the equilibrium loses stability via subcritical Andronov-Hopf bifurcation, which occurs when an
unstable limit cycle shrinks to a point. We never questioned where the unstable limit
cycle came from. Neither were we concerned with the existence of a large-amplitude
stable limit cycle corresponding to the periodic spiking state. In Fig.6.23 we study
this problem. We decrease the bifurcation parameter I to see what happens with thelimit cycles. As I approaches the bifurcation value 42.18, the unstable and stable limit
cycles approach and annihilate each other. When I is less than the bifurcation value,
there are no periodic orbits, only one stable equilibrium corresponding to the restingstate.
+

Notice that the fold limit cycle bifurcation explains how (un)stable limit cycles
appear or disappear, but it does not explain the transition from resting to periodic
spiking behavior. Indeed, let us start with I = 42 in Fig.6.23 and slowly increase
the parameter. The state of the INa,p +IK -model is at the stable equilibrium. WhenI passes the bifurcation value, a large-amplitude stable limit cycle corresponding toperiodic spiking appears, yet the model is still quiescent, because it is still at the stableequilibrium. Thus, the limit cycle is just a geometrical object in the phase space that
corresponds to spiking behavior. However, for it to actually exhibit spiking, the stateof the system must somehow be pushed into the attraction domain of the cycle, say byexternal stimulation. This issue is related to the computational properties of neurons,
and it is discussed in detail in the next chapter.
In Fig.6.24 we depict the bifurcation diagram of the INa,p +IK -model. For each valueof I, we simulate the model forward (t → ∞) to find the stable limit cycle and backward(t → −∞) to find the unstable limit cycle. Then we plot their amplitudes (maximal
voltage minus minimal voltage along the limit cycle) on the (I, V )-plane. One canclearly see that there is a fold limit cycle bifurcation (left) and a subcritical AndronovHopf bifurcation (right). The left part of the bifurcation diagram looks exactly like theone for saddle-node bifurcation, which explains why the fold limit cycle bifurcation isoften referred to as fold or saddle-node of periodics.
The similarity of the fold limit cycle bifurcation and the saddle-node bifurcation
is not a coincidence. Stability of limit cycles can be studied using Floquet theory or
Poincare cross-section maps (Kuznetsov 1995), or by brute force (e.g., by reducing
the model to an appropriate polar coordinate system). When a limit cycle attractor
undergoes fold limit cycle bifurcation, its radius undergoes saddle-node bifurcation.
(This is a hint for exercise 10.)

Bifurcations

183

stable
lim
it

I=43

s
cle
cy

b le
sta
un

I=42.5

s
cycle
limit

fold limit cycle

I=42.18

ycle
it c

fold
l

im

I=42.35

I=42

-20

un

le limit c
ycles
stable equilibria

40

45

line
ullc

0.5

stab

bifurcationn-ne

le limit cycles
stab

0

-40

V-nullclin

min/max of oscillation of
membrane potential, mV

1

-60
-80

injected current, I

50

0
-80

-60
-40
-20
membrane voltage, mV

0

Figure 6.23: Fold limit cycle bifurcation in the INa,p +IK -model. As the bifurcation
parameter I decreases, the stable and unstable limit cycles approach and annihilate
each other. Parameters are as in Fig.6.16.

184

Bifurcations

amplitude (max-min), mV

80

stable limit cycles

70
60
50

fold limit cycle
bifurcation

40
30

unsta

ble lim20

it cycles

subcritical
Andronov-Hopfbifurcation

10
0
40

41

42

43

44

45

46

47

48

49

50

injected dc-current, I

bitsaddlelemoclinic orho

it cycl
lim

e

st
ab

Figure 6.24: Bifurcation diagram of the INa,p +IK -model. Parameters are as in Fig.6.16.

saddlesaddlea. supercritical saddle homoclinic orbit bifurcation

saddlesaddlesaddleb. subcritical saddle homoclinic orbit bifurcation

Figure 6.25: Saddle homoclinic orbit bifurcation.

letlimi cyc
le

un
sta
b

bit

moclinic orhoBifurcations

185

c orbit
lini
oc
m
ho

saddle homoclinic orbit
bifurcation

clin
mo
o
h

bit
ic or

saddle-node on invariant circle
bifurcation

Figure 6.26: Two bifurcations involving homoclinic trajectories to an equilibrium.

6.2.4

Homoclinic

A limit cycle can appear or disappear via a saddle homoclinic orbit bifurcation, as
depicted in Fig.6.25. As the bifurcation parameter changes, the cycle becomes a homoclinic orbit to the saddle equilibrium, and its period becomes infinite. After thebifurcation, the cycle no longer exists. A necessary condition for such a bifurcation isthat the steady-state I-V relation is not monotonic.
One should be careful to distinguish the saddle homoclinic orbit bifurcation fromthe saddle-node on invariant circle bifurcation depicted in Fig.6.26. Indeed, it might
be easy to confuse these bifurcations, since both involve an equilibrium and a largeamplitude homoclinic trajectory that becomes a limit cycle. The key difference is thatthe equilibrium is a saddle in the former and a saddle-node in the latter. The saddle
equilibrium persists as the bifurcation parameter changes, whereas the saddle-nodeequilibrium disappears or bifurcates into two points, depending on the direction ofchange of the bifurcation parameter.
Recall that a saddle on a plane has two real eigenvalues of opposite signs. Theirsum, λ1 + λ2 , is called the saddle quantity.
• If λ1 + λ2 < 0, then the saddle homoclinic orbit bifurcation is supercritical, whichcorresponds to the (dis)appearance of a stable limit cycle.
• If λ1 + λ2 > 0, then the saddle homoclinic orbit bifurcation is subcritical, whichcorresponds to the (dis)appearance of an unstable limit cycle.
Thus, the saddle quantity plays the same role as the parameter a in the Andronov-Hopfbifurcation. The supercritical saddle homoclinic orbit bifurcation is more common inneuronal models than the subcritical one for the reason explained in section 6.3.6.
Hence, we consider only the supercritical case below, and we drop the word “supercritical” for the sake of brevity.

186

Bifurcations

?

d

in

ld

sta
ble
ma
nif
ol

out

saddle

un

l
sta b

a
em

ni

fo

Figure 6.27: Saddle homoclinic orbit bifurcation
occurs when the stable and unstable submanifolds ofthe saddle make a loop.

A useful way to look at the bifurcation is to note that the saddle has one stable andone unstable direction on a phase plane. There are two orbits associated with these
directions, called the stable and unstable submanifolds, depicted in Fig.6.27. Typically,
the submanifolds miss each other, that is, the unstable submanifold goes either insideor outside the stable one. This could happen for two different values of the bifurcation
parameter. One can imagine that as the bifurcation parameter changes continuously
from one value to the other, the submanifolds join at some point and form a single
homoclinic trajectory that starts and ends at the saddle.
The saddle homoclinic orbit bifurcation is ubiquitous in neuronal models, and it caneasily be observed in the INa,p +IK -model with fast K+ conductance, as we illustrate
in Fig.6.28. Let us start with I = 7 (top of Fig.6.28) and decrease the bifurcation
parameter I. First, there is only a stable limit cycle corresponding to periodic spiking
activity. When I decreases, a stable equilibrium and an unstable equilibrium appear
via saddle-node bifurcation (not shown in the figure), but the state of the model is stillon the limit cycle attractor. Further decrease of I moves the saddle equilibrium closer
to the limit cycle (case I = 4 in the figure), until the cycle becomes an infinite period
homoclinic orbit to the saddle (case I ≈ 3.08), and then disappears (case I = 1). At
this moment, the state of the system approaches the stable equilibrium, and the tonic
spiking stops.
Similar to the fold limit cycle bifurcation, the saddle homoclinic orbit bifurcation
explains how the limit cycle attractor corresponding to periodic spiking behavior appears and disappears. However, it does not explain the transition to periodic spiking
behavior. Indeed, when I = 4 in Fig.6.28, the limit cycle attractor exists, yet the neuron may still be quiescent because its state may be at the stable node. The periodic
spiking behavior appears only after external perturbations push the state of the system
into the attraction domain of the limit cycle attractor, or I increases further and thestable node disappears via a saddle-node bifurcation.
We can use linear theory to estimate the frequency of the limit cycle attractor
near the saddle homoclinic orbit bifurcation. Because the vector field is small near
the equilibrium, the periodic trajectory passes slowly through a small neighborhoodof the equilibrium, then quickly makes a rotation and returns to the neighborhood,as we illustrate in Fig.6.29. Let T1 denote the time required to make one rotation

Bifurcations

187

0.8

I

ycl
es

limi

t cy
cle

0
-80

V

it c

sta
ble

e
V-nullclin

0.2

n

lim

0.6

0.4

I=7

n-nullcline

-60
-40
-20
membrane voltage, mV

sad

I=4

0

dle

nod

saddleit

I=3.08min/max of oscillations ofmembrane potential, mV

node

hom

ocli

hom
oclin

nic

orb

ic or
bit

node

sta

stable

ble

limit

limi

t cy

cycle

cle

e

saddleI=10-20

homoclinic
bifurcation-40

limit cycles

saddles

-60

nodes

-80

0

node

saddle2
46injected dc-current, I

Figure 6.28: Saddle homoclinic orbit bifurcation in the INa,p +IK -model with parameters
as in Fig.4.1a and fast K+ current (τ (V ) = 0.16). As the bifurcation parameter I
decreases, the stable limit cycle becomes a homoclinic orbit to a saddle.

188

Bifurcations
-10

0.8

0.7

0.04
T2

-20

0.02

-55

-50

membrane voltage, V (mV)

-60

line

T1

saddlen-nullc

0
0.5

-30

it c
y

llcline

0.2

-40

lim

0.3

cle

0.4

V-nu

K+ activation gate, n

0.6

-50
T1

0.1

T2

T1
T2

0
-80

-60
-70

-60

-50

-40

-30

-20

-10

0

10

membrane voltage, V (mV)

0

1

2

3

4

5

6

time, ms

Figure 6.29: The period of the limit cycle is T = T1 + T2 with T2 → ∞ as the cycle
approaches the saddle equilibrium. Shown is the INa,p +IK -model with I = 3.5.
(dashed part of the limit cycle in the figure) and T2 denote the time spent in thesmall neighborhood of the saddle equilibrium (continuous part of the limit cycle in the
shadowed region), so that the period of the limit cycle is T = T1 + T2 . While T1 isrelatively constant, T2 → ∞ as I approaches the bifurcation value Ib = 3.08, and thelimit cycle approaches the saddle. In exercise 11 we show that
T2 = −

1ln{τ (I − Ib )} ,λ1

where λ1 is the positive (unstable) eigenvalue of the saddle, and τ is a parameter that
depends on the size of the neighborhood, global features of the vector field, and so on.
We can represent the period, T , in the form
T (I) = −1
ln{τ1 (I − Ib )} ,λ1

where a single parameter τ1 = τ e−λ1 T1 accounts for all global features of the model,
including the width of the action potential and the shape of the limit cycle. One caneasily determine τ1 if the eigenvalue λ1 and the period of the limit cycle are known for
at least one value of I. The INa,p +IK -model has τ1 = 0.2, as we show in Fig.6.30. Notethat the theoretical frequency 1000/T (I) matches the numerically found frequency ina broad range. Also note how imprecise the numerical results are (see inset in the
figure).
Both the saddle-node on invariant circle bifurcation and the saddle homoclinic orbit
bifurcation result in spiking with decreasing frequency, so that their frequency-current

Bifurcations

189

frequency (Hz)400lnumerica
theoretical

300
200

200
100

1/ln

100
0
3.08

0

3

3.1

3.2
3.3
3.4
injected dc-current, I

3.09

3.5

3.6

big
ho

Figure 6.30: Frequency of spiking in the INa,p +IK -model with parameters as inFig.6.28 near a saddle homoclinic orbit bifurcation. Dots are numerical results; thecontinuous curveis ω(I) = 1000λ(I)/{− ln(0.2(I − 3.0814))}, where the eigenvalue√λ(I) = 0.87 4.51 − I was obtained from the normal form (6.3). The inset shows amagnified region near the bifurcation value I = 3.0814.
it
c orb
lini
c
o
m

ycle
it c
il m

Figure 6.31: Big saddle homoclinic orbit bifurcation.
(F-I)√ curves go continuously to zero. The key difference is that the former asymptotes
as I − Ib , and the latter as 1/ ln(I −Ib ). The striking feature of the logarithmic decay
in Fig.6.30 is that the frequency is greater than 100 Hz and the theoretical curve does
not seem to go to zero for all I except those in an infinitesimal neighborhood of thebifurcation value Ib . Such a neighborhood is almost impossible to catch numerically,
let alone experimentally in real neurons.
Many neuronal models, and even some cortical pyramidal neurons (see Fig.7.42)
exhibit a saddle homoclinic orbit bifurcation depicted in Fig.6.31. Here, the unstablemanifold of a saddle returns to the saddle along the opposite side, thereby making a
big loop; hence the name big saddle homoclinic orbit bifurcation. This kind of bifurcation often occurs when an excitable system is near a codimension-2 Bogdanov-Takens
bifurcation considered in section 6.3.3, and it has the same properties as the “small”
homoclinic orbit bifurcation considered above: it can be subcritical or supercritical,
depending on the saddle quantity; it results in a logarithmic F-I curve; and it implies
the coexistence of attractors. All methods of analysis of excitable systems near “small”
saddle homoclinic orbit bifurcations can also be applied to the case in Fig.6.31.

190

Bifurcations

heteroclinic orbit

Figure 6.32: Heteroclinic orbit bifurcation does not change the existence or stability of
any equilibrium or periodic orbit.

6.3

Other Interesting Cases

Saddle-node and Andronov-Hopf bifurcations of equilibria, combined with fold limit
cycle, homoclinic orbit bifurcation, and heteroclinic orbit bifurcation (see Fig.6.32),
exhaust all possible bifurcations of codimension-1 on a plane. These bifurcations canalso occur in higher-dimensional systems. Below we discuss additional codimension-1
bifurcations in three-dimensional phase space, and then we consider some codimension2 bifurcations that play an important role in neuronal dynamics. The beginning reader
may read only section 6.3.6 and skip the rest.

6.3.1

Three-Dimensional Phase SpaceSo far we have considered four bifurcations of equilibria and four bifurcations of limit
cycles on a phase plane. These eight bifurcations can appear in multidimensional
systems. Below we briefly discuss the kinds of bifurcations that are possible in athree-dimensional phase space but cannot occur on a plane.
First, there are no new bifurcations of equilibria in multidimensional phase space.
Indeed, what could possibly happen with the Jacobian matrix of an equilibrium of amultidimensional dynamical system? A simple zero eigenvalue would result in a saddlenode bifurcation, and a simple pair of purely imaginary complex-conjugate eigenvalues
would result in an Andronov-Hopf bifurcation. Both are exactly the same as in thelower-dimensional systems already considered. Thus, adding dimensions to a dynamicalsystem does not create new possibilities for bifurcations of equilibria.
In contrast, adding the third dimension to a planar dynamical system creates new
possibilities for bifurcations of limit cycles, some of which are depicted in Fig.6.33.
Below we briefly describe these bifurcations.
The saddle-focus homoclinic orbit bifurcation in Fig.6.33 is similar to the saddle
homoclinic orbit bifurcation considered in section 6.2.4, except that the equilibrium
has a pair of complex-conjugate eigenvalues and a nonzero real eigenvalue. The homoclinic orbit originates in the subspace spanned by the eigenvector corresponding to
the real eigenvalue (as in Fig.6.33) and terminates along the subspace spanned by the
eigenvectors corresponding to the complex-conjugate pair. The reverse direction is also
possible. Depending on the direction and the relative magnitude of the eigenvalues,
this bifurcation can result in the (dis)appearance or a stable (supercritical) or unstable(subcritical) twisted large-period orbit.

Bifurcations

191

Saddle-Focus Homoclinic Orbit Bifurcation

Subcritical Flip Bifurcation

Subcritical Neimark-Sacker Bifurcation

Blue-Sky Catastrophe

Fold Limit Cycle on Homoclinic Torus Bifurcation

Figure 6.33: Some codimension-1 bifurcations of limit cycles in three-dimensional phasespace (modified from Izhikevich 2000a).

192

Bifurcations

The subcritical flip bifurcation in Fig.6.33 occurs when a stable periodic orbit is surrounded by an unstable orbit of twice the period. The unstable periodic orbit shrinks
to the stable one and makes it lose stability. This bifurcation is similar to the pitchfork bifurcation studied below, except that it has codimension-1 (pitchfork bifurcation
has infinite codimension unless one considers dynamical systems with symmetry). A
supercritical flip bifurcation is similar, except that an unstable cycle is surrounded bya stable double-period cycle.
The subcritical Neimark-Sacker bifurcation in Fig.6.33 occurs when a stable periodic orbit is surrounded by an unstable invariant torus. The latter shrinks and makes
the periodic orbit lose its stability. In some sense, which we will not elaborate on here,
this bifurcation is similar to the subcritical Andronov-Hopf bifurcation of an equilibrium. The supercritical Neimark-Sacker bifurcation occurs when an unstable orbit issurrounded by a stable invariant torus.
The blue-sky catastrophe in Fig.6.33 occurs when a small-amplitude stable limit
cycle disappears and a large-amplitude large-period stable orbit appears out of nowhere(from the blue sky). The orbit has an infinite period at the bifurcation, yet it is
not homoclinic to any equilibrium. A careful analysis shows that the large orbit ishomoclinic to the small limit cycle at the moment the cycle disappears. In some sense,
which we elaborate on later, this bifurcation is similar to the saddle-node on invariant
circle bifurcation (see exercise 20). In particular, both bifurcations share the same
asymptotics.
The fold limit cycle on homoclinic torus bifurcation in Fig.6.33 is similar to theblue-sky catastrophe except that the disappearance of the small periodic orbit results
in a large-amplitude torus (quasi-periodic) attractor.

6.3.2

Cusp and Pitchfork

Recall that an equilibrium xb of a one-dimensional system ẋ = f (x, b) is at a saddlenode bifurcation when fx = 0 (first derivative of f ) but fxx ̸= 0 (second derivativeof f ) at the equilibrium. The latter is called the non-degeneracy condition, and it
guarantees that the system dynamics is equivalent to that of ẋ = c(b) + x2 .
If fx = 0 and fxx = 0, but fxxx ̸= 0, then the equilibrium is at the codimension-2
cusp bifurcation, and the behavior of the system near the equilibrium can be describedby the topological normal formẋ = c1 (b) + c2 (b)x + ax3 ,where
c1 (b) = f (xb , b) ,c2 (b) = fx (xb , b) ,a = fxxx /6 ̸= 0 ,in particular, c1 = c2 = 0 at the cusp point. The cusp bifurcation is supercritical whena < 0 and subcritical otherwise. It is explained by the shape of the surface
c1 + c2 x + ax3 = 0 ,depicted in Fig.6.34.

Bifurcations

193

x
c1
c2

Figure 6.34: Cusp surface.
Let us treat c1 and c2 as independent parameters, and check that there are saddlenode bifurcations in any neighborhood of the cusp point. The bifurcation sets of the
topological normal form can easily be found. Differentiating c1 + c2 x + ax3 with respect
to x gives c2 + 3ax2 . Equating both of these expressions to zero and eliminating x gives
two saddle-node bifurcation curves
2  c2 3/2,
c1 = ± p|a| 3
depicted at the bottom of Fig.6.34. Since c1 = c1 (b) and c2 = c2 (b), varying thebifurcation parameter b results in a path on the (c1 , c2 )-plane. Depending on the shapeand location of this path, one can get many one-dimensional bifurcation diagrams. A
summary of some special cases is depicted in Fig.6.35, showing that there can be many
interesting dynamical regimes in the vicinity of a cusp bifurcation point.

c2(b)

x

x

b

b

x

x
x

b

x

b

b
x

b
x

x
b

b

c1(b)
x

b

b

Figure 6.35: Summary of special cases for the supercritical cusp bifurcation. Dottedsegments are paths c1 = c1 (b), c2 = c2 (b), where b is a one-dimensional bifurcation
parameter. The corresponding bifurcation diagrams are depicted in boxes. Continuous
curves represent stable solutions, dashed curves represent unstable solutions. (Modified
from Hoppensteadt and Izhikevich 1997.)

194

Bifurcations

x
+ b/a

x

unsta
b

e

stabl

le

stableunstablebstableunstable+ b/|a|
b

yyy
yyy
yy
yyy
yyyyyyyy
yyy
yyyyyy- b/a unstablesubcritical pitchfork, a > 0

stable- b/|a|supercritical pitchfork, a < 0

Figure 6.36: Pitchfork bifurcation diagrams.

An important special case is when c1 = 0 and c2 (b) = b, so that the topologicalnormal form isẋ = bx + ax3 .
This form corresponds to a pitchfork bifurcation, whose diagram is depicted in Fig.6.36(see also the bottom bifurcation diagram in Fig.6.35). This bifurcation has an infinite
codimension unless one considers dynamical systems with symmetry, such as, ẋ =f (x, b) with f (−x, b) = −f (x, b) for all x and b.

6.3.3

Bogdanov-Takens

Can an equilibrium undergo Andronov-Hopf and saddle-node bifurcations simultaneously? There are two possibilities, illustrated in Fig.6.37:• Fold-Hopf. The Jacobian matrix at the equilibrium has a pair of pure imaginary complex-conjugate eigenvalues (Andronov-Hopf bifurcation) and one zero
eigenvalue (saddle-node bifurcation). In this case the two bifurcations occur indifferent subspaces.
• (Bogdanov-Takens) The Jacobian matrix has two zero eigenvalues. In this casethe two bifurcations occur in the same subspace.
The fold-Hopf bifurcations occur in systems having dimension 3 and up, while the
Bogdanov-Takens bifurcation can occur in two-dimensional systems. Both bifurcations
have codimension-2; that is, they require two bifurcation parameters. Note that theBifurcations

195

eigenvalues
Hopf
fold

fold-Hopf bifurcation

eigenvalues

Bogdanov-Takens bifurcation

Figure 6.37: Two ways an equilibrium can undergo a saddle-node (fold) and an
Andronov-Hopf bifurcation simultaneously.
fold-Hopf bifurcation has three eigenvalues with zero real part, whereas the BogdanovTakens bifurcation has only two zero eigenvalues. This bifurcation can, on the one
hand, be viewed as a saddle-node bifurcation in which another (negative) eigenvalue
gets arbitrarily close to zero, and, on the other hand, as an Andronov-Hopf bifurcationin which the imaginary part of the complex-conjugate eigenvalues goes to zero.
The Jacobian matrix of an equilibrium at the Bogdanov-Takens bifurcation satisfies
two conditions: det L = 0 (saddle-node bifurcation) and tr L = 0 (Andronov-Hopfbifurcation). For example, it can have the form


0 1
L=
.
(6.10)
0 0
Because of these two conditions, the codimension of this bifurcation is 2. There arealso certain non-degeneracy and transversality conditions (see Kuznetsov 1995). The
corresponding topological normal form,u̇ = v ,v̇ = a + bu + u2 + σuv ,(6.11)

has two bifurcation parameters, a and b, and the parameter σ = ±1 determines whether
the bifurcation is subcritical or supercritical. This parameter depends on the combination of the second-order partial derivatives with respect to the first variable, and it
is nonzero because of the non-degeneracy conditions (Kuznetsov 1995). The bifurcation diagram and representative phase portraits for various a, b, and σ are depicted in
Fig.6.38 (the case σ > 0 can be reduced to σ < 0 by the substitutions t → −t andv → −v). A remarkable fact is that the saddle-node and the Andronov-Hopf bifurcations do not occur alone. There is also a saddle homoclinic orbit bifurcation always
appearing near the Bogdanov-Takens point.
Bogdanov-Takens bifurcation often occurs in neuronal models with nullclines intersecting as in Fig.6.39a. We show in the next chapter that this bifurcation separates

196

Bifurcations

1

2

1

2

3 AH

4

b
SN2

1
3 AH

4

SHO

4

BT

2

AH

3

a

SN1

SN1

SN2

SN1

SN2

SHO

BT

SHO

BT

supercritical, s<0

subcritical, s>0

Figure 6.38: Bogdanov-Takens (BT) bifurcation diagram of the topological normal form
(6.11). Abbreviations: AH, Andronov-Hopf bifurcation; SN, saddle-node bifurcation;
SHO, saddle homoclinic orbit bifurcation.

integrators from resonators, and it can occur in some layer 5 pyramidal neurons of rat
visual cortex, as we discuss in section 7.2.11 and 8.2.1. The two equilibria in the lower(left) knee of the fast nullcline in Fig.6.39b are not necessarily a saddle and a stablenode, but can be a saddle and an (un)stable focus, as in the phase portraits in Fig.6.38.
Interestingly, the global vector field structure of neuronal models with nullclines
as in Fig.6.39a results in the birth of a spiking limit cycle attractor via a big saddle
homoclinic orbit bifurcation, so the neuronal model undergoes a cascade of bifurcations,depicted in Fig.6.40, as the amplitude of the injected current I increases. The local
phase portraits corresponding to I0 , I1 , and I2 are topologically equivalent to the phase
portrait “1” in Fig.6.38 (right). (The equivalence is local near the left knee; there is
no global equivalence because of the extra equilibrium in Fig.6.40 and because of the
big homoclinic or periodic orbit.) As I increases, a stable large-amplitude spiking
limit cycle appears via a big supercritical homoclinic orbit bifurcation at some I1 . It
coexists with the stable resting state for all I1 < I < I5 . At some point I2 , thesaddle quantity, i.e., the sum of its eigenvalues, changes from negative to positive(it is zero at the Bogdanov-Takens bifurcation), so another saddle homoclinic orbit

197

s lo
wn
ullc
lin

line
nullc
fast

e

Bifurcations

Bogdanov-Takens
point

(a)

(b)

Figure 6.39: Intersection of nullclines of a two-dimensional system, resulting in
Bogdanov-Takens bifurcation.

bifurcation (at some I3 ) occurs, which is subcritical, giving birth to an unstable limit
cycle. The phase portrait at I3 is locally topologically equivalent to the one marked
SHO in Fig.6.38. Similarly, the phase portrait at I4 is locally equivalent to the one
labeled “2” in Fig.6.38. The unstable cycle shrinks to the equilibrium and makes it losestability via a subcritical Andronov-Hopf bifurcation at some I5 , which corresponds tocase AH in Fig.6.38. Further increase of I converts the unstable focus into an unstablenode, which approaches the saddle and disappears via the saddle-node bifurcation SN1
in Fig.6.38 (not shown in Fig.6.40).

I1

I0=0

b ig

I3

I4

small
homoclinic orbit

rbit
oclinic o
hom

I2

I5

subcritical
Andronov-Hopfbifurcation

Figure 6.40: Transformations of phase portraits of a neuronal model near the subcritical Bogdanov-Takens bifurcation point as the magnitude of the injected current I
increases (here Ik+1 > Ik ). Shaded regions are the attraction domains of the equilibrium corresponding to the resting state.

198

Bifurcations

(a)

(b)

(c)

(d)

(f)

(g)

(h)

g=0

f=0

(e)

Figure 6.41: Canard (French duck) limit cycles in a relaxation oscillator (hand drawing).

6.3.4

Relaxation Oscillators and Canards

Let us consider a relaxation oscillatorẋ = f (x, y, b)ẏ = µg(x, y, b)(fast variable)(slow variable)with fast and slow nullclines, as in Fig.6.41a, and µ ≪ 1. Suppose that there is astable equilibrium, as in Fig.6.41a, for some values of the bifurcation parameter b < 0,and a stable limit cycle, as in Fig.6.41h, for some other values b > 0. What kind of
bifurcation of the equilibrium occurs when b increases from negative to positive, andthe slow nullcline passes the left knee of the fast N-shaped nullcline?
The Jacobian matrix at the equilibrium has the form


fx fy
L=
µgx µgy
Since fx = 0 at the knee (prove this), but fy typically does not, the Jacobian matrix
resembles the one for the Bogdanov-Takens bifurcation (6.10) in the limit µ = 0.
However, the resemblance is only superficial, since the relaxation oscillator does notsatisfy the non-degeneracy conditions. In particular, second-order partial derivativesof µg(x, y, b) vanish in the limit µ → 0, resulting in σ = 0 and in the disappearance ofthe term u2 from the topological normal form (6.11).
A purely geometrical consideration confirms that the transition from Fig.6.41a
to Fig.6.41h cannot be of the Bogdanov-Takens type, since there is a unique equilibrium and no possibility for a saddle-node bifurcation, which always accompanies
the Bogdanov-Takens bifurcation. Actually, the equilibrium loses stability via an
Andronov-Hopf bifurcation that occurs whentr L = fx + µgy = 0

anddet L = µ(fx gy − fy gx ) > 0 .

Bifurcations

199

The loss of stability typically happens not at the left knee, where fx = 0, but a little to
the right of the knee, where fx = −µgy > 0 (because gy < 0 in neuronal models). We
saw this phenomenon in section 4.2.6 when we considered FitzHugh-Nagumo model.
An interesting observation is that the period of damped or sustained oscillations√
near the Andronov-Hopf bifurcation point in Fig.6.41b is of the order 1/ µ, because√√
the frequency ω = det L ≈ µ, whereas the period of large-amplitude relaxation
oscillation is of the order 1/µ, because it takes 1/µ units of time to slide up anddown along the branches of the fast nullcline in Fig.6.41h. Thus, the period of small
subthreshold oscillations of a neural model may have no relation to the period of
spiking, if the model has many time scales.
The Andronov-Hopf bifurcation can be supercritical or subcritical, depending on thefunctions f and g; see exercise 14 and exercise 18. Figure 6.41 depicts the supercriticalcase. In the subcritical case, stable and unstable limit cycles are typically born via
fold limit cycle bifurcation; then the unstable limit cycle goes through the shapes as
in Fig.6.41g, f, e, d, c, and b, and finally shrinks to a point.
Canards
The distinctive feature of limit cycles in Fig.6.41c–6.41g is that they follow the unstablebranch (dashed curve) of the fast nullcline before jumping to the left or to the right(stable) branch. Due to the relaxation nature of the system, the vector field is horizontal outside the N-shape fast nullcline, so any transition from Fig.6.41b to Fig.6.41h
must gradually go through the stages in Fig.6.41c–6.41g. Because the cycle in Fig.6.41f
resembles a duck, at least in the eyes of the French mathematicians E. Benoit, J.-L.
Callot, F. Diener, and M. Diener, who discovered this phenomenon in 1977, it is often
called a canard (French duck) cycle.
In general, any trajectory that follows the unstable branch is called a canard trajectory. Canard trajectories play an important role in defining thresholds for resonator
neurons, as we discuss in section 7.2.5. It takes on the order of 1/µ units of time toslide along the unstable branch of the fast nullcline. A small perturbation to the left or
to the right can result in an immediate jump to the corresponding stable branch of thenullcline. Hence, the initial conditions should be specified with an unrealistic precisionof the order of e−1/µ to follow the unstable branch, which explains why the canard
trajectories are difficult to catch numerically, let alone experimentally. Consequently,
the canard cycles, though stable, exists in an exponentially small region of values ofthe parameter b. A typical simulation shows a sudden explosion of a stable limit cyclefrom small (Fig.6.41b) to large (Fig.6.41h) as the parameter b is slowly varied. In summary, canard cycles in two-dimensional relaxation oscillators play an important role ofthresholds, but they are fragile and rather exceptional.
In contrast, canard trajectories in three-dimensional relaxation oscillators (one fast
and two slow variables) are generic in the sense that they exist in a wide range of parameter values. A simple way to see this is to treat b as the second slow variable. Then
there is a set of initial conditions corresponding to the canard trajectories. Studying
canards in R3 goes beyond the scope of this book (see bibliographical notes).

200

Bifurcations
Supercritical Andronov-Hopf
Bifurcation

c =0
a <0

BautinBifurcation 0

a 2 - 4c a 2= 0

Fold
Limit Cycle
Bifurcation

c

c =0
a >0

a
Subcritical Andronov-Hopf
Bifurcation

Figure 6.42: Supercritical Bautin bifurcation in (6.12); see also Fig.9.42 (left).

6.3.5

BautinWhat happens when a subcritical Andronov-Hopf bifurcation becomes supercritical,
that is, when the parameter a in the topological normal form for Andronov-Hopf bifurcation (6.8, 6.9) changes sign? The bifurcation becomes degenerate when a = 0,
and the behavior of the system is described by the topological normal form for Bautinbifurcation, which we write here in the complex formż = (c + iω)z + az|z|2 + a2 z|z|4 ,(6.12)

where z ∈ C is a complex variable, and c, a, and a2 are real parameters. The parameters a and a2 are called the first and second Liapunov (often spelled Lyapunov)
coefficients. The Bautin bifurcation occurs when a = c = 0 and a2 ̸= 0, and hence it
has codimension-2. It is subcritical when a2 > 0 and supercritical otherwise. If a2 = 0,then one needs to consider the next term a3 z|z|6 , not shown in the normal form (6.12),
to get a bifurcation of codimension-3, and so on.
We can easily determine bifurcations of the topological normal form. First of all,
(6.12) undergoes Andronov-Hopf bifurcation when c = 0, which is supercritical fora < 0 and subcritical otherwise. Moreover, if a and a2 have different signs, then (6.12)
undergoes fold limit cycle bifurcation whena2 − 4ca2 = 0 ,Bifurcations201

as we illustrate in Fig.6.42. Thus, both Andronov-Hopf and fold limit cycle bifurcations
occur simultaneously at the Bautin point a = c = 0. Many two-dimensional neuronalmodels, such as the INa,p +IK -model with low-threshold K+ current, are relatively near
this bifurcation, which explains why the unstable limit cycle involved in the subcritical
Andronov-Hopf bifurcation is usually born via fold limit cycle bifurcation. There issome evidence that rodent trigeminal interneurons, dorsal root ganglion neurons, andmes V neuron in the brainstem are also near this bifurcation; see section 9.3.3.

6.3.6

Saddle-Node Homoclinic Orbith

Let us compare the saddle-node on invariant circle bifurcation and the saddle homoclinic orbit bifurcation depicted in Fig.6.43 (top). In both cases there is a homoclinic
orbit (i.e., a trajectory that originates and terminates at the same equilibrium). However, the equilibria are of different types, and the orbit returns to them along different
directions. Now suppose a system undergoes both bifurcations simultaneously, as we
illustrate in Fig.6.43 (bottom). Such a bifurcation, called saddle-node homoclinic orbit
bifurcation, has codimension-2, since two strict conditions must be satisfied. First, theequilibrium must be at the saddle-node bifurcation point, i.e., must have the eigenvalue λ1 = 0. Second, the homoclinic trajectory must return to the equilibrium along
the noncentral direction, i.e., along the stable direction corresponding to the negative eigenvalue λ2 . Since the saddle-node quantity, λ1 + λ2 , is always negative, thisbifurcation always results in the (dis)appearance of a stable limit cycle.

inic
ocl
m
o

orbit

clinic orbitmoohclinic orbitmohosaddle-node on invariant circle
bifurcation

saddle homoclinic orbit
bifurcation

saddle-node homoclinic orbit
bifurcation

Figure 6.43: Saddle-node homoclinic orbit bifurcation occurs when a system undergoes
a saddle-node on invariant circle and saddle homoclinic orbit bifurcations simultaneously.

202

Bifurcations
saddle-node on invariant
circle bifurcation

0.18

rest (excitable)

saddle-nodehomoclinic orbit bifurcation

oc

lin
i

co

rbi

tb

0.165

ho

0.16

l

e

sa

dd

bistability

saddle-node bifurcation

if u rc a ti o

n

0.17

m

K+ conductance time constant

0.175

periodic spiking

0.155

0

1

2

3

4 4.51 5

6

7

8

9

injected dc-current I

Figure 6.44: Unfolding of saddle-node homoclinic orbit bifurcation in the INa,p +IK model with parameters as in Fig.6.28.
In Fig.6.44 we illustrate the saddle-node homoclinic orbit bifurcation using theINa,p +IK -model with two bifurcation parameters: the injected DC current I and theK+ time constant τ . The bifurcation occurs at the point (I, τ ) = (4.51, 0.17). Note that
there are three other codimension-1 bifurcation curves converging to this codimension-2
point, as we illustrate in Fig.6.45. Since the model undergoes a saddle-node bifurcation
at I = 4.51 and any τ , the straight vertical line I = 4.51 is the saddle-node bifurcation
curve. The point τ = 0.17 on this line separates two cases. When τ > 0.17, theactivation and deactivation of the K+ current is sufficiently slow that the membranepotential V undershoots the equilibrium, resulting in the saddle-node on invariant circle
bifurcation. When τ < 0.17, deactivation of the K+ current is fast, and V overshootsthe saddle-node equilibrium, resulting in the saddle-node off limit cycle bifurcation.
Shaded triangular areas in the figures denote the parameter region corresponding
to the bistability of stable equilibrium and a limit cycle attractor (resting and spiking states). Let us decrease the parameter I and cross such a region from right toleft. When I = 4.51, a saddle equilibrium and a node equilibrium appear. Further

203

saddle-nodehomoclinic orbit
bifurcation

o
nic
ocli
m
o
h

rb

if
it b

ur

ca

saddle-node bifurcation

ti o

n

saddle-node on
invariant circle bifurcation

Bifurcations

Figure 6.45: Unfolding of saddle-node homoclinic orbit bifurcation.
decreasing I moves the saddle equilibrium rightward and the limit cycle leftward, until
they merge. This occurs on the saddle homoclinic orbit bifurcation curve, which isdetermined numerically in Fig.6.44.
Neuronal models exhibiting saddle-node homoclinic bifurcations can be reduced toa topological normal formV̇ = c(b − bsn ) + a(V − Vsn )2 ,if V (t) = Vmax , then V (t) ← Vreset ,(6.13)

which is similar to that for saddle-node bifurcation (6.2), except that there is a reset
V ← Vreset when the membrane voltage reaches a certain large value Vmax . Any sufficiently large Vmax will work equally well, even Vmax = +∞, because V reaches +∞ in a
finite time (see exercise 3). Using Vmax = 30 and results of section 6.1.1, the topologicalnormal form for the INa,p +IK -model isV̇ = (I − 4.51) + 0.1887(V + 61)2 ,if V (t) = 30, then V (t) ← Vreset .

The saddle-node homoclinic bifurcation occurs when I = 4.51 and Vreset = −61. This
normal form is called the quadratic integrate-and-fire neuron; see chapters 3 and 8.
The topological normal form (6.13) is a useful equation, as will be seen in the restof the book. It describes quantitative and qualitative features of neuronal dynamics
remarkably well, yet it has only one non-linear term. This makes it suitable for realtime simulations of huge numbers of neurons. Its bifurcation structure is studied in
exercise 12 (see also Fig.8.3), and the reader should at least look at the solution at theend of the book.

204

Bifurcations

6.3.7

Hard and Soft Loss of Stability

Bifurcation is a qualitative change of the phase portrait of a system. Not all changes
are equally dramatic, however. Some are hardly noticeable. For example, consider
an equilibrium undergoing a supercritical Andronov-Hopf bifurcation: as a bifurcation
parameter changes, the equilibrium loses stability and a small-amplitude stable limit
cycle appears, as in Fig.6.11. The state of the system remains near the equilibrium; itjust exhibits small-amplitude oscillations around it. We can change the parameter inthe opposite direction, and then the limit cycle shrinks to a point and the system returns
to the equilibrium. In neurons, such a bifurcation does not lead to an immediate spike.
The neuron remains quiescent; it just exhibits subthreshold small-amplitude sustained
oscillations. Such a loss of stability is called soft: the equilibrium is no longer stable,
but its small neighborhood remains attractive. Supercritical pitchfork, cusp, and flip
bifurcations correspond to soft loss of stability.
In contrast, if the equilibrium loses stability via subcritical Andronov-Hopf bifurcation, the state of the system diverges from it, which results in an immediate spike orsome kind of large-amplitude jump. Such a loss of stability is called hard: neither theequilibrium nor its neighborhood is attractive. The hard loss of stability usually leads
to noticeable or catastrophic changes in systems behavior, and the stability boundaryis called dangerous (Bautin 1949). Changing the bifurcation parameter in the opposite
direction will make the equilibrium stable again, but may not bring the state of thesystem back to equilibrium. Saddle-node bifurcation is hard unless it is on an invariant circle. In this case, the loss of stability is catastrophic, i.e., leading to noticeable
spikes, but reversible. Saddle homoclinic orbit bifurcation is hard. In general, mostbifurcations in neurons, or at least in neuronal models, are hard.

Review of Important Concepts• Stable equilibrium (resting state) in a typical neuronal model can– disappear via saddle-node bifurcation, which can be off or on invariant circle or– lose stability via Andronov-Hopf bifurcation, which can be supercritical or subcritical.
These four cases are summarized in Fig.6.46.
• Stable limit cycle (periodic spiking state) in a typical twodimensional neuronal model can– be cut by saddle-node on invariant circle bifurcation– shrink to a point via supercritical Andronov-Hopf bifurcation– disappear via fold limit cycle bifurcation or– disappear via saddle homoclinic orbit bifurcation.
These four cases are summarized in Fig.6.47.
• Some atypical (codimension-2) bifurcations may play importantroles in neuronal dynamics.
• Bogdanov-Takens bifurcation separates integrators from resonators.

Bifurcations

205

Bibliographical Notes
Though bifurcation theory can be traced back to Poincare and Andronov, it is a relatively new branch of mathematics. The first attempt to apply it to neuroscience wasin 1955, when Richard FitzHugh concluded his paper on mathematical modeling ofthreshold phenomena by saying that many neuronal properties. . . are invariant under continuous, one-to-one transformations of the coordinates ofphase space and fall within the domain of topology, a branch of mathematics which
may be intrinsically better fitted for the preliminary description and classification of
biological systems than analysis, which includes differential equations. This suggestion
is of little practical value at present, since too little is known of the topology of vector
fields in many-dimensional spaces, at least to those interested in theoretical biology.
Nevertheless, the most logical procedure in the description of a complex biologicalsystem might be to characterize the topology of its phase space, then to establish
a set of physically identifiable coordinates in the space, and finally to fit differential
equations to the trajectories, instead of trying to reach this final goal at one leap.

It is remarkable that FitzHugh was explicitly talking about topological equivalenceand bifurcations, though he never called them such, years before these mathematical
notions were firmly established. This book continues the line of research initiated by
FitzHugh and further developed by Rinzel and Ermentrout (1989).
In this chapter we provided a fairly detailed exposition of bifurcation theory. What
we covered should be sufficient not only for understanding the rest of the book, butalso for navigating through bifurcation papers concerned with computational neuroscience. More bifurcation theory, including bifurcations in mappings xn+1 = f (xn , b),
can be found in the excellent book Elements of Applied Bifurcation Theory by Yuri
Kuznetsov (1995; new ed., 2004), which, however, might be a bit technical for a nonmathematician. Some of the bifurcations considered in this chapter, such as the bluesky catastrophe, are classified as “exotic” by Kuznetsov (1995), though the catastrophe
was recently found in a model of a leech heart interneuron (Shilnikov and Cymbalyuk2005).
There is no unified naming scheme for the bifurcations, mostly because they werediscovered and rediscovered independently in many fields and in many countries. For
example, the Andronov-Hopf bifurcation was known to A. Poincare, so some scientists
refer to it as the Poincare-Andronov-Hopf bifurcation. Many refer to it as just the Hopf
bifurcation due to the fault of the famous Russian mathematician Vladimir Igorevich
Arnold and the famous French mathematician Rene Thom. According to Arnold’s
account, he was visited by Thom in the 1960s. While there were discussing various
bifurcations, Arnold put too much emphasis on Hopf’s “recent” (1942) paper. As
a result of Arnold’s misattribution, Thom popularized the bifurcation as the Hopf
bifurcation. In Fig.6.49 we provide some common alternative names for the bifurcations
considered in this chapter. The complete list of names of known bifurcations is very
long, and it resembles the list of faculty members of the Department of Radiophysics
at Gorky State University, in what is now Nizhnii Novgorod, Russia. The department
was founded by A. A. Andronov in 1945 (see Fig. 6.50).

206

Bifurcations

node

saddlesaddle-nodesaddle-node bifurcation

an
ari
v
n
i

node

saddleclet cir

saddle-nodesaddle-node on invariant circle (SNIC) bifurcation

supercritical Andronov-Hopf bifurcation

subcritical Andronov-Hopf bifurcation

Figure 6.46: Summary of all codimension-1 bifurcations of a stable equilibrium (restingstate).

Bifurcations

207

circle
ant
i
r
a
inv

node

saddle-nodesaddlesaddle-node on invariant circle (SNIC) bifurcationfo

limit cyclebletalimit cycle
ble

l
ld

cycle
imit

un
sta

s

supercritical Andronov-Hopf bifurcationt

saddleleclinic orbmoiho

it cycl
lim

e

st
ab

fold limit cycle bifurcation

saddlesaddlesaddle homoclinic orbit bifurcation

Figure 6.47: Summary of all codimension-1 bifurcations of a stable limit cycle (tonic
spiking state) on a plane.

208

Bifurcations

Figure 6.48: Richard FitzHugh with analog computer at the National Institute of
Health, Bethesda, Maryland, ca. 1960. (Photograph provided by R. FitzHugh).

bifurcation

alternative names

saddle-node

fold, limit point, saddle-node off limit cycle

saddle-node on invariant circle

SNIC, saddle-node on limit cycle (SNLC), circle,
saddle-node homoclinic, saddle-node central homoclinic,
saddle-node infinite period (SNIPer), homoclinic

Andronov-Hopf
saddle homoclinic orbit

Hopf, Poincare-Andronov-Hopf
homoclinic, saddle-loop, saddle separatrix loop, Andronov-Leontovich

fold limit cycle

saddle-node of limit cycles, double limit cycle, fold cycle,
saddle-node (fold) of periodics

saddle-node homoclinic orbit

saddle-node noncentral homoclinic, saddle-node separatrix-loop

Bogdanov-Takens

Takens-Bogdanov, double-zero

Bautindegenerate Hopf, generalized Hopf

flip

period doubling

Figure 6.49: Popular alternative names for some of the bifurcations considered in thischapter.

Bifurcations

209

Figure 6.50: The founder of the Russian school of nonlinear dynamics, Aleksander Aleksandrovich Andronov(1901–1952) in 1950. (Picture provided by M. I. Rabinovich.)
x

b

x=
ble

sta
stable0

unstable

b

ble

ta
ns

u

Figure 6.51: Transcritical bifurcation in ẋ = x(b − x).
The division of bifurcations into subcritical and supercritical may be confusing toa novice. For example, some scientists erroneously think that supercritical bifurcations result in the appearance of attractors (stable equilibria, limit cycles, etc.), andsubcritical bifurcations result in their disappearance. Let us emphasize here that theappearance or disappearance of an equilibrium or a limit cycle depends on the direction of change of the bifurcation parameter. For example, the subcritical pitchfork
bifurcation in Fig.6.36 can result in the appearance of a stable equilibrium x = 0 if
b decreases past 0. Our classification of bifurcations into subcritical and supercriticalis consistent with the following widely accepted rule: let the bifurcation parameterchange in the direction leading to the increase in the number of objects (equilibria,
limit cycles). The bifurcation is supercritical if stable objects appear, subcritical if
unstable objects appear, and transcritical (as in Fig.6.51) if equal numbers of stableand unstable objects appear or disappear. The condition for supercritical (subcritical)
Andronov-Hopf bifurcation, (6.7), is taken from Guckenheimer and Holmes (1983).
Delayed loss of stability was first described by Shishkova (1973), and studied indetail by Nejshtadt (1985) (many find his paper difficult to read). An alternativedescription is given by Arnold et al. (1994) and Baer et al. (1989).

210

Bifurcations

Canard (French duck) solutions were reported by Benoit et al. (1981). Due to therecent political climate in the USA, some refer to “French ducks” as “freedom ducks”,
probably to emphasize that “French = freedom”. Canards in R3 were studied by Benoit(1984), Samborskij (1985; in Rn ), and more recently by Szmolyan and Wechselberger(2001, 2004) and Wechselberger (2005).

Exercises
1. (Transcritical bifurcation) Justify the bifurcation diagram in Fig.6.51.
2. Show that the non-degeneracy and transversality conditions are necessary forthe saddle-node bifurcation. That is, present a system that does not exhibit
saddle-node bifurcation, but satisfies(a) the non-hyperbolicity and non-degeneracy conditions or(b) the non-hyperbolicity and transversality conditions.
3. Consider the modelV̇ = c(b − bsn ) + a(V − Vsn )2 ,with positive a and c, and b > bsn . Show that the sojourn time in a boundedneighborhood of the point V = Vsn scales asπ
T =pac(b − bsn )when b is near bsn . (Hint: Find the solution that starts at −∞ and terminates
at +∞.)
4. Show that the two-dimensional systemu̇ = c(b)u − ω(b)v + (au − dv)(u2 + v 2 ) ,v̇ = ω(b)u + c(b)v + (du + av)(u2 + v 2 ) ,the complex-valued systemż = (c(b) + iω(b))z + (a + id)z|z|2 ,and the polar-coordinate systemṙ = c(b)r + ar3 ,φ̇ = ω(b) + dr2
are equivalent.

(6.14)(6.15)

Bifurcations

211
600

llcline

non-monotone
I-V relation

0.3

400

current, I

n-nu

0.4

ine
llcl
nu
V-

K+ activation, n

0.5

200

0.2
0

l
tentia
o
p
n
actio

0.1

0
-80

-60

-40

-20

membrane voltage, V (mV)

0

-200

-80

-60

-40

-20

0

membrane voltage, V (mV)

Figure 6.52: exercise 8: This INa,p +IK -model has a non-monotonic I-V relation, yet
the resting state becomes unstable via Andronov-Hopf bifurcation before disappearing
via saddle-node bifurcation. Parameters are as in Fig.4.1a, except that Eleak = −78mV and n∞ (V ) has k = 12 mV.
5. Show that the non-degeneracy and transversality conditions are necessary for the
Andronov-Hopf bifurcation. That is, present a system that does not exhibit theAndronov-Hopf bifurcation, but satisfies(a) the non-hyperbolicity and non-degeneracy conditions or(b) the non-hyperbolicity and transversality conditions.
6. Show that the system (6.14, 6.15) with c(b) = b, ω(b) = 1, a ̸= 0 and d = 0
exhibits Andronov-Hopf bifurcation. Check all three conditions.
7. Determine the stability of the limit cycle
p near an Andronov-Hopf bifurcation.
(Hint: Consider the equilibrium r =|c/a| in the topological normal form
(6.8)).
8. The model in Fig.6.52 has a non-monotonic I-V relation. Nevertheless, the resting state loses stability via Andronov-Hopf bifurcation before disappearing via
saddle-node bifurcation. Draw representative phase portraits of the model. Is
the system near Bogdanov-Takens bifurcation?
9. Consider a generic two-dimensional conductance-based modelV̇ = I − I(V, x) ,ẋ = (x∞ (V ) − x)/τ (V ) ,(6.16)(6.17)

212

Bifurcations
v2
a
1

1

-1

v1

-1

Figure 6.53: See exercise 11.
where V and x are the membrane voltage and a gating variable, respectively, I is
the injected DC current, and I(V, x) is the instantaneous I-V relation, which ofcourse depends on the gating variable x. Here the membrane capacitance C = 1
for the sake of simplicity. Show that the eigenvalues at an equilibrium c ± ω aregiven byc = (IV (V, x) + 1/τ (V ))/2
and
ω=

p
′ (V )/τ (V )c2 − I∞

where I∞ (V ) = I(V, x∞ (V )) is the steady-state I-V relation of the model. In
particular, the frequency at the Andronov-Hopf bifurcation isp′ (V )/(Cτ (V )) ,(frequency) = I∞
where C is the membrane capacitance.
10. Determine when the systemz ′ = (a + ω i)z + z|z|2 − z|z|4 ,z∈Cundergoes fold limit cycle bifurcation.
11. Consider a square neighborhood of a saddle equilibrium in Fig.6.53 (comparewith the inset in Fig.6.29). Here v1 and v2 are eigenvectors with eigenvalues
λ2 < 0 < λ1 . Suppose the limit cycle enters the square at the point a = τ (I − Ib ),where τ > 0 is some parameter. Determine the amount of time the trajectory
spends in the square as a function of I.
12. Determine the bifurcation diagram of the topological normal form (6.13) for
saddle-node homoclinic bifurcation.

Bifurcations

213

13. Prove that the systemv̇ = I + v 2 − u ,u̇ = a(bv − u)with a > 0 undergoes• saddle-node bifurcation when b2 = 4I,• Andronov-Hopf bifurcation when a < b and a2 − 2ab + 4I = 0,√
• Bogdanov-Takens bifurcation when a = b = 2 I.
Use the results of exercise 15 to prove that the Andronov-Hopf bifurcation in themodel above is always subcritical.
14. Use (6.7) to prove that the relaxation oscillatorv̇ = f (v) − uu̇ = µ(v − b)with an N-shaped fast nullcline u = f (v) undergoes Andronov-Hopf bifurcation
when f ′ (b) = 0 (i.e., exactly at the knee; what is so special about this model?).
Show that the bifurcation is supercritical when f ′′′ (b) < 0 and subcritical whenf ′′′ (b) > 0.
15. Prove that the Andronov-Hopf bifurcation point inv̇ = F (v) − uu̇ = µ(bv − u)satisfies F ′ = µ and b > µ. Use (6.7) to show thata = {F ′′′ + (F ′′ )2 /(b − µ)}/16 .
16. Prove that the Andronov-Hopf bifurcation point inv̇ = F (v) − uu̇ = µ(G(v) − u)satisfies F ′ = µ and G′ > µ. Use (6.7) to show thata = {F ′′′ + F ′′ (F ′′ − G′′ )/(G′ − µ)}/16 .
17. Prove that the Andronov-Hopf bifurcation point inv̇ = F (v) − (v + 1)uu̇ = µ(G(v) − u)satisfies F ′ = µ and G′ > µ. Use (6.7) to show thata = {F ′′′ + µ − (F ′′ − µ)(1 + µ[G′′ − F ′′ + 2µ]/ω 2 )}/16 .

214

Bifurcations

18. Use (6.7) to show that a two-dimensional relaxation oscillatorv̇ = F (v, u)u̇ = µG(v, u)
at an Andronov-Hopf bifurcation point hasFvv Gu − Fu Gvv Fvu
1√−
Fvvv + Fvv+ O( µ) .
a=
16
Fu Gv
Fu
19. [M.S.] A leaky integrate-and-fire model has the same asymptotic firing rate
(1/ln) as a system near saddle homoclinic orbit bifurcation. Explore the possibility that integrate-and-fire models describe neurons near such a bifurcation.
20. [M.S.] (blue-sky catastrophe) Prove thatφ̇ = ω,ẋ = a + x2 ,if x = +∞, then x ← −∞,and φ ← 0 ,is the canonical model (see section 8.1.5) for blue-sky catastrophe. This model
without the reset of φ is canonical for the fold limit cycle on homoclinic torus
bifurcation. The model with the reset x ← b+sin φ is canonical for the LukyanovShilnikov bifurcation of a fold limit cycle with non-central homoclinics (Shilnikov
and Cymbalyuk 2004, Shilnikov et al. 2005). Here, φ is the phase variable on
the unit circle and a and b are bifurcation parameters.
21. [M.S.] Define topological equivalence and the notion of a bifurcation for piecewise
continuous flows.
22. [Ph.D.] Use the definition in the exercise above to classify codimension-1 bifurcations in piecewise continuous flows.
23. [M.S.] The bifurcation sequence in Fig.6.40 seems to be typical in two-dimensional
neuronal models. Develop the theory of Bogdanov-Takens bifurcation with a
global reentrant orbit.
24. [Ph.D.] Develop an automated dynamic clamp protocol (Sharp et al. 1993) thatanalyzes bifurcations in neurons in vitro, similar to what AUTO, XPPAUT, and
MATCONT do in models.

Chapter 7
Neuronal Excitability
Neurons are excitable in the sense that they are typically at rest but can fire spikes in
response to certain forms of stimulation. What kind of stimulation is needed to fire agiven neuron? What is the evoked firing pattern? These are the questions concerning
the neuron’s computational properties, e.g., whether they are integrators or resonators,
their firing frequency range, the spike latencies (delays), the coexistence of resting andspiking states, etc. From the dynamical systems point of view, neurons are excitablebecause they are near a bifurcation from resting to spiking activity. The type ofbifurcation, and not the ionic currents per se, determines the computational propertiesof neurons. In this chapter we continue our effort to understand the relationshipbetween bifurcations of the resting state and the neurocomputational properties of
excitable systems.

7.1

Excitability

A textbook definition of neuronal excitability is that a “subthreshold” synaptic input
evokes a small graded postsynaptic potential (PSP), while a “superthreshold” input
evokes a large all-or-none action potential, which is an order of magnitude larger than
the amplitude of the subthreshold response. Unfortunately, we cannot adopt thisdefinition to define excitability of dynamical systems because many systems, including
some neuronal models discussed in chapter 4, have neither all-or-none action potentials
nor firing thresholds. Instead, we employ a purely geometrical definition.
From the geometrical point of view, a dynamical system with a stable equilibrium is
excitable if there is a large-amplitude piece of trajectory that starts in a small neighborhood of the equilibrium, leaves the neighborhood, and then returns to the equilibrium,
as we illustrate in Fig.7.1 (left).
In the context of neurons, the equilibrium corresponds to the resting state. Because
it is stable, all trajectories starting in a sufficiently small region of the equilibrium, muchsmaller than the shaded neighborhood in the figure, converge back to the equilibrium.
Such trajectories correspond to subthreshold PSPs. In contrast, the large trajectory in
the figure corresponds to firing a spike. Therefore, superthreshold synaptic inputs are
215

216

Excitability

spike
?

spike
rest

rest

excitable

oscillatory

Figure 7.1: Left: An abstract definition of excitability. There is a spike trajectory thatstarts near a stable equilibrium and returns to it. Right: Excitable systems are nearbifurcations. A modification of the vector field in the small shaded region can resultin a periodic trajectory.
those that push the state of the neuron to or near the beginning of the large trajectory(small square in Fig.7.1), thereby initiating the spike. These inputs can be injectedby an experimenter via an attached electrode, or they can represent the total synaptic
input from the other neurons in the network, or both.

7.1.1

Bifurcations

The definition in Fig.7.1 is quite general, and it does not make any assumptions regarding the details of the vector field inside or outside of the small shaded neighborhood.
Let us use the theory presented in chapter 6 to show that such an excitable system isnear a bifurcation from resting to oscillatory dynamics.
• Bifurcation of a limit cycle. The vector field in the small shaded neighborhoodof the equilibrium can be modified slightly so that the spike trajectory enters
the square and becomes periodic, as in Fig.7.1 (right). That is, the dynamicalsystem goes through a bifurcation resulting in the appearance of a limit cycle.
What happens to the stable equilibrium, denoted as “?” in the figure? Dependingon the type of the bifurcation of the limit cycle, the equilibrium may disappear or
may lose stability. This happens when the limit cycle appears via saddle-node on an
invariant circle or a supercritical Andronov-Hopf bifurcation, respectively. Both cases
are depicted in Fig.7.2.

Excitability
excitable

217
bistable

oscillatory

saddle-node on invariant
circle bifurcation

saddle-homoclinic
orbit bifurcation

saddlenodebifurcation

supercritical Andronov-Hopfbifurcation

fold limit cycle
bifurcation

subcritical
AndronovHopf
bifurcation

Figure 7.2: Excitable dynamical systems bifurcate into oscillatory ones either directly
or indirectly, via bistable systems.

218

Excitability

Alternatively, the equilibrium may remain stable and coexist with the newly born
limit cycle, as happens during saddle homoclinic orbit or fold limit cycle bifurcationsin Fig.7.2. The dynamical system is no longer excitable, but bistable, though many
scientists still treat bistable systems as excitable. An appropriate synaptic input canswitch the behavior from resting to spiking and back. Note that we considered only
bifurcations of a limit cycle so far.
• Bifurcation of the equilibrium. Suppose the system is bistable, as in Fig.7.2. Since
the equilibrium is near the cycle, a small modification of the vector field in theshaded neighborhood can make it disappear via saddle-node bifurcation, or losestability via subcritical Andronov-Hopf bifurcation.
In any case, excitable dynamical systems can bifurcate into oscillatory systems either
directly or indirectly through bistable systems. All these cases are summarized in
Fig.7.2.

7.1.2

Hodgkin’s Classification

As we mentioned in the introduction chapter, the first person to study bifurcation
mechanisms of excitability (years before mathematicians discovered such bifurcations)was Hodgkin (1948), who injected steps of currents of various amplitudes into excitable
membranes and looked at the resulting spiking behavior. We illustrate his experiments
in Fig.7.3, using recordings of rat neocortical and brainstem neurons. When the current
is weak, the neurons are quiescent. When the current is strong, the neurons fire trainsof action potentials. Depending on the average frequency of such firing, Hodgkin
identified two major classes of excitability:• Class 1 neural excitability. Action potentials can be generated with arbitrarily
low frequency, depending on the strength of the applied current.
• Class 2 neural excitability. Action potentials are generated in a certain frequency
band that is relatively insensitive to changes in the strength of the applied current.
Class 1 neurons, sometimes called type I neurons, fire with a frequency that may varysmoothly over a broad range of about 2 to 100 Hz or even higher. The important
observation here is that the frequency can be changed tenfold. In contrast, the frequency band of Class 2 neurons is quite limited, e.g., 150 − 200 Hz, but it can varyfrom neuron to neuron. The exact numbers are not important here. The qualitative
distinction between the classes noted by Hodgkin is that the frequency-current relation(the F-I curve in Fig.7.3, bottom) starts from zero and continuously increases for Class
1 neurons, but is discontinuous for Class 2 neurons.
Obviously, the two classes of excitability have different neurocomputational properties. Class 1 excitable neurons can smoothly encode the strength of an input, e.g.,
the strength of the applied DC current or the strength of the incoming synaptic bombardment, into the frequency of their spiking output. Class 2 neurons cannot do that.

Excitability219

Layer 5 pyramidal cell

Brainstem mesV cell

I=320

I=1000

I=80

I=600

I=60

I=500
100 ms

20 ms

40 mV

I=40

I=400

I=20

I=200

I (pA)I (pA)
0 pA

0 pA

F-I curve

30

20

10

0

0

100

200

Class 2 excitability

asymptotic firing frequency, Hz

asymptotic firing frequency, Hz

Class 1 excitability

40

20 mV

300

injected dc-current, I (pA)

F-I curve

250
200
150
100
50
0

0

500

1000

1500

injected dc-current, I (pA)

Figure 7.3: Top: Typical responses of membrane potentials of two neurons to steps of
DC current of various magnitudes I. Bottom: Corresponding frequency-current (F-I)
relations are qualitatively different. Shown are recordings of layer 5 pyramidal neurons
from rat primary visual cortex (left) and mesV neuron from rat brainstem (right). The
asymptotic frequency is 1000/T∞ , where T∞ is taken to be the interval between thelast two spikes in a long spike train.

220

Excitability
50 ms

20 mV

1 ms

-60 mV
1000 pA
0 pA

500 pA

Figure 7.4: Class 3 excitability of a mesV neuron of rat brainstem (contrast with
Fig.7.3).
20 mV
100 ms

-20mV

50 ms

3 mV

subthreshold oscillations

700 pA

100 pA

Figure 7.5: Class 3 excitability of a layer 5 pyramidal neuron of rat visual cortex. The
inset shows subthreshold oscillations of membrane potential.
Instead, they can act as threshold elements reporting when the strength of an input isabove a certain value. Both properties are important in neural computations.
Hodgkin also observed that axons left in oil or seawater for long periods exhibited• Class 3 neural excitability. A single action potential is generated in response to apulse of current. Repetitive (tonic) spiking can be generated only for extremely
strong injected currents or not at all.
Two examples of Class 3 excitable systems are depicted in Fig.7.4 and Fig.7.5. The
mesV neuron in the figure fires a phasic spike at the onset of the pulse of current, andthen remains quiescent. Even injecting pulses as high as 1000 pA, which result in spiketrains in another mesV neuron in Fig.7.3, cannot evoke multiple spikes in this neuron.
Similarly, the pyramidal neuron in Fig.7.5 cannot sustain tonic spiking even when theExcitability

221

injected current is ten times stronger than the neuron’s rheobase. Ironically, neurons
exhibiting such a behavior would most likely be discarded as “sick” or “unhealthy”,
though the neurons analyzed in the figures looked normal from any other point of view.
We will study the dynamic mechanism of this class of excitability and show that it may
have nothing to do with sickness.
It will shortly be clear that this classification is of limited value except that it
points to the fact that neurons should be distinguished according not only to ionicmechanisms of excitability but also to dynamic mechanisms, in particular to the typeof bifurcation of the resting state.

7.1.3

Classes 1 and 2

Let us consider the strength of the applied current in Hodgkin’s experiments as beinga bifurcation parameter. Instead of changing the parameter abruptly, as in Fig.7.3,
we change it slowly in Fig.7.6 (both figures show recordings of the same neurons). In
section 7.1.5 we explain the fundamental difference between these two protocols.
When the current ramps up, the resting potential increases until a bifurcation
occurs, resulting in loss of stability or disappearance of the equilibrium corresponding to
the resting state, and the neuron activity becomes oscillatory. Note that the pyramidal
neuron in Fig.7.6 starts to fire with a small frequency, which then increases according
to the F-I curve in Fig.7.3 (a slower current ramp is needed to span the entire frequency
range of the F-I curve). In contrast, the brainstem neuron starts to fire with a highfrequency that remains relatively constant even though the magnitude of the injectedcurrent increases.
Among all four codimension-1 bifurcations of equilibrium, discussed in chapter 6
and mentioned in Fig.7.2, only saddle-node on invariant circle bifurcation results in alayer 5 pyramidal cellbrainstem mesV celltransitiontransition
20 mV-50 mV-60 mV
3000 pA

200 pA
0 pA

0 pA
500 ms

500 msFigure 7.6: As the magnitude of injected DC current increases, the neurons bifurcatefrom resting to repetitive spiking behavior. Shown are recordings of the neurons in
Fig.7.3. Note that the ratio of the first and last interspike intervals of the pyramidal
cell is much greater than that of the mesV neuron.

222

Excitability
Class 3 excitable neuron

0 mV

10 mV
50 ms

4,000 pA

-60 mV
0 pA

Figure 7.7: A Class 3 excitable brainstem mesV neuron does not fire in response to aramp current, even though the injected current is stronger than the one in Fig.7.4.
limit cycle attractor with arbitrarily small frequency and continuous F-I curve. The
other three bifurcations result in limit cycle attractors with relatively large frequencies
and discontinuous F-I curves. Therefore,• Class 1 neural excitability corresponds to the resting state disappearing via
saddle-node on invariant circle bifurcation.
• Class 2 neural excitability corresponds to the resting state disappearing via
saddle-node (off invariant circle) bifurcation or losing stability via subcriticalor supercritical Andronov-Hopf bifurcations.
Of course, the resting state can lose stability or disappear via other bifurcations having higher codimension, sometimes leading to counterintuitive results (e.g., Class 1
excitability near Andronov-Hopf bifurcation; see exercise 6 and section 7.2.11). In thischapter we concentrate on the four bifurcations above because they have the lowest
codimension and hence are the most likely to be seen experimentally.

7.1.4

Class 3In Fig.7.7 we inject a slow ramp current into the Class 3 excitable system. In contrast
to Fig.7.6, no spiking and no bifurcation occur in this experiment, despite the fact thatthe membrane potential goes all the way to 0 mV. Therefore,• Class 3 neural excitability occurs when the resting state remains stable for any
fixed I in a biophysically relevant range.
Then why are there single spikes in Fig.7.4? Their existence in the figure and theirabsence in the ramp experiment are related to the phenomenon of accommodation that
we now describe.
Let us consider a neuron having a transient Na+ current with relatively fast inactivation. If a sufficiently slow ramp of current is injected, the current has enough timeto inactivate, and no action potentials can be generated. Such a neuron accommodates

223
w-nullcline

Excitability
0.2

0.15

V-nullclinew0.1

I=0.03
I=0
0.05

I=0

0.06
0.045
0.03
0.015
0.01

0
0.4

0.2

0

0.2

0.4

0.6

0.8

V

Figure 7.8: Class 3 excitability in FitzHugh-Nagumo model (4.11, 4.12) with a =0.1, b = 0.01, c = 0. The model fires a single spike for any pulse of current.
to the slow ramp. In contrast, a quick membrane depolarization due to a strong stepof current does not give enough time for Na+ inactivation, thereby resulting in a spike.
During the spike, the current inactivates quickly and precludes any further action potentials. Instead of inactivating the Na+ current, we could have used a low-thresholdpersistent K+ current, or any other resonant current, to illustrate the phenomenon ofaccommodation.
From the dynamical systems point of view, slow ramp results in quasi-static dynamics so that all gating variables follow their steady-state values, x = x∞ (V ), and themembrane potential follows its I-V curve. As long as the equilibrium corresponding to
the resting state is stable, the neuron is at rest. Even global bifurcations resulting in
the appearance of stable limit cycles do not change that. Only when the equilibrium
bifurcates (loses stability or disappears), does the neuron change its behavior, e.g.,
jumps to a limit cycle attractor and starts to fire spikes. Class 3 excitable systems do
not fire in response to slow ramps because the resting state does not bifurcate.
In contrast, a pulse of current changes the phase portrait in a rather abrupt manner, as we illustrate in Fig.7.8, using the FitzHugh-Nagumo model with vertical slow
nullcline. Injecting I shifts the fast nullcline upward. Though no bifurcation can occur
in the model, and the resting state is stable for any value of I, its location suddenly
shifts when I jumps. The trajectory from the old equilibrium, (0, 0), to the new one
goes through the right branch of the cubic V -nullcline, thereby resulting in a single
spike. Since the new equilibrium (0, 0.03) is a global attractor and no limit cycles
exist, periodic spiking cannot be generated. In exercise 7 we explore the relationshipbetween Class 3 excitability and Andronov-Hopf bifurcation (note the subthresholdoscillations of membrane potential of the pyramidal neuron in Fig.7.5). We see that
injecting ramps of current is not equivalent to injecting pulses of current. The system
goes through a bifurcation of the equilibrium in the former, but may bypass it andjump somewhere else in the latter.

224

Excitability

0.2

I0
0.1

0.05

0
-70

-60
-50
-40
membrane potential, V (mV)

0
-20
-40
-60
-80

I1

I1

K+ gating variable, n

I1

0.15

I0

-70

-60
-50
-40
membrane potential, V (mV)

-70

-60
-50
-40
membrane potential, V (mV)

V(t)

I(t)

I1

I1

I1

I0

I0

slow ramp from I0 to I1(a)step from I0 to I1

shock pulse at I1

(b)

(c)

Figure 7.9: The difference between ramp, step, and shock stimulations is in the resettingof initial condition.

7.1.5

Ramps, Steps, and Shocks

In Fig.7.9 we elaborate the differences among injecting slow ramps, steps, and shocks(i.e., brief pulses) of current. In the first two cases the magnitude of the injectedcurrent changes from I0 to I1 , while in the third case the current is I1 except for theinfinitesimally brief moment when it has an infinitely large strength. In all three cases
the dynamics of the model can be understood via analysis of its phase portrait at
I = I1 . The key difference among the stimulation protocols is how they reset theinitial condition.
At the beginning of the slow ramp in Fig.7.9a, the state of the neuron is at thestable equilibrium. As the current slowly increases, the equilibrium slowly moves, andthe trajectory follows it. When the current reaches I = I1 , the trajectory is at the new
equilibrium, so no response is evoked because the equilibrium is stable. In contrast,
when the current is stepped from I0 to I1 in Fig.7.9b, the location of the equilibriumchanges instantaneously, but the membrane potential and the gating variables do nothave time to catch up. To understand the response of the model to the step, we needto consider its dynamics at I = I1 with the initial condition set to the location of
the old equilibrium (marked by the white square in the figure). Such a step evokes aspike response even though the new equilibrium is stable. Finally, shocking the neuron

Excitability

225
frequency-current (F-I) relation

frequency

(a)

(c)

pulsesramp

Andronov-Hopf
homoclinic
3.88

5.25

current, I(b)subcritical
Andronov-Hopfbifurcation(d)

I=5.25

K+ gating variable, n

0.7

K+ gating variable, n0.7

near subcritical
Andronov-Hopfbifurcation

(e)

0.1

0.1

-80 membrane potential, V (mV) 0

I=3.8866
near saddle
homoclinic orbit
bifurcation-80 membrane potential, V (mV) 0

Figure 7.10: The INa,p +IK -model undergoes subcritical Andronov-Hopf bifurcation, yet
can exhibit low-frequency firing when steps (but not ramps) of current are injected.
Parameters: C = 1, I = 0, EL = −66.2, gL = 2, gNa = 5, gK = 4.5, m∞ (V ) has
V1/2 = −30 and k = 10, n∞ (V ) has V1/2 = −34 and k = 13, and τ (V ) = 1, ENa = 60mV and EK = −90 mV. The shaded region is the attraction domain of the restingstate. The inset shows a distorted drawing of the phase portrait.
results in an instantaneous increase of its membrane potential to a new value. (As an
exercise, prove that the magnitude of the increase equals the product of pulse width andpulse height divided by the membrane capacitance.) This shifts the initial condition
horizontally to a new point, marked by the white square in Fig.7.9c, and results in a
spike response.
Now, let us revisit the Hodgkin experiments and demonstrate the fundamental
difference between the stimulation protocols. In Fig.7.10a, b, and c we simulate theINa,p +IK -model and show that it is Class 2 excitable in response to ramps of current
but Class 1 excitable in response to steps of current. The apparent contradiction is
resolved in Fig.7.10d and e, where we consider the model’s phase portraits. Noticethe coexistence of the resting state and a limit cycle attractor. The resting state
loses stability via subcritical Andronov-Hopf bifurcation at I = 5.25, so the emerging

226

Excitability

cle

lim
it cy
sp

iki

ng

un
s

line

attraction
domain

exc.
pulse
n
V

e
llclin

e
ycl
it c
lim

n-nu

spi
kin
g

n-nullline

excitatory
pulse

ullc
V-n

inhibitory
pulse

subcritical Andronov-Hopf bifurcation
V-nullccline

saddle-node bifurcationbletainhibitory
pulse
attraction
domainof spiking
limit cycle

Figure 7.11: Coexistence of stable equilibrium and spiking limit cycle attractor in
the INa,p +IK -model. Left: The resting state is about to disappear via saddle-nodebifurcation. Right: The resting state is about to lose stability via subcritical AndronovHopf bifurcation. Right (left) arrows denote the location and the direction of an
excitatory (inhibitory) pulse that switches spiking behavior to resting behavior.
spiking has non-zero frequency at I ≈ 5.25. However, injecting steps of current resultsin transitions to the limit cycle even before the resting state loses its stability. The
limit cycle in the model appears via saddle homoclinic orbit bifurcation at I ≈ 3.8866,
and its period is quite large, resulting in the Class 1 response to steps of current.
The F-I curves for homoclinic bifurcations have logarithmic scaling, so small-frequency
oscillations are difficult to catch numerically, let alone experimentally.
The surprising discrepancy in Fig.7.10a occurs because the resting state of theINa,p +IK -model is near the Bogdanov-Takens bifurcation (i.e., the model is near atransition from resonator to integrator). Such a bifurcation was recorded, thoughindirectly, in some neocortical pyramidal neurons, as we will show later in this chapter
and in chapter 8. Another surprising example of Andronov-Hopf bifurcation with Class
1 excitability is presented in exercise 6. To avoid such surprises, we adopt the ramp
definition of excitability throughout the book.

7.1.6

Bistability

When transition from the resting to the spiking state occurs via saddle-node (off invariant circle) or subcritical Andronov-Hopf bifurcation, there is a coexistence of astable equilibrium and a stable limit cycle attractor just before the bifurcation, as we
illustrate in Fig.7.11. We refer to such systems as bistable. They have a remarkableneurocomputational property: bistable systems can be switched from one state to theother by an appropriately timed brief stimulus. Rinzel (1978) predicted such a behavior in the Hodgkin-Huxley model, and then bistability and hysteresis were foundExcitability

227

20 mV

F-I curve
Class 2 excitability

250

oise
ut n
ti ho
w

200

noise

150
100

w it haverage firing frequency, Hz

50 ms

50
0

0

500

1000

1500

injected DC current, I (pA)

Figure 7.12: Examples of noise-induced
low-frequency firingsof a Class 2 excitablesystem. The F-I curve
may look like theone for Class 1 excitability. Shown arerecordings of brainstem mesV neuron.

experimentally in the squid axon (Guttman et al. 1980). What was really surprising
for many neuroscientists is that neurons can be switched from repetitive spiking to
resting by brief depolarizing shock stimuli.
This phenomenon is illustrated in Fig.7.11. Each shaded area in the figure denotes
the attraction domain of a spiking limit cycle attractor. Obviously, the state of theresting neuron must be pushed into the shaded area to initiate periodic spiking. Similarly, the state of the periodically spiking neuron must be pushed out of the shaded
area to stop the spiking. As the arrows in the figure indicate, both excitatory and inhibitory stimuli can do that, depending on their timing relative to the phase of spiking
oscillation. This protocol can be used to test bistability experimentally. (As an exercise, use geometrical and electrophysiological arguments to explain why a system withhigh-threshold slow persistent inward current can be bistable but cannot be switchedfrom one mode to another by brief pulses of current.) Bistable behavior reveals itself indirectly when a neuron is kept close to the bifurcation, e.g., when the injected
DC current is just below the rheobase. Noisy perturbations can switch the neuron
from resting to spiking state, thereby creating an irregular spike train consisting of
short bursts of spikes. Such stuttering spiking have been observed in many neurons,
including some regular spiking (RS) and fast spiking (FS) neocortical neurons, as we
discuss in chapter 8. The mean firing frequency during stuttering is proportional to theamplitude of the injected current and it can be quite low even for a Class 2 excitablesystem, as we illustrate in Fig.7.12. Thus, caution should be used when experimentally
determining the class of excitability; only spike trains with regular interspike periods
should be accepted to measure the F-I relations.

228

Excitability
spiking(b)

spiking

Class 2

resting
Class 1

I=3.08I=4.51

spiking(c)
F-I curve

800
frequency, Hz(a) resting

Class 2(b)(a)

Class 1
spiking

resting

resting02

4current, I

6

Figure 7.13: (a) The frequency of emerging oscillations at the transition “resting →spiking” defines the class of excitability. (b) The frequency of disappearing oscillations
at the transition “spiking → resting” defines the class of spiking. (c) The INa,p +IK model with high-threshold K+ current exhibits class 2 excitability but class 1 spiking.
Its F-I curve has a hysteresis.

7.1.7

Class 1 and 2 Spiking

The class of excitability is determined by the frequency of emerging oscillations at
the transition “resting → spiking”, as in Fig.7.13a. Let us look at the frequency
of disappearing oscillations at the transition “spiking → resting”. To induce such atransition, we inject a strong pulse of DC current of slowly decreasing amplitude, as inFig.7.13b. Similarly to the Hodgkin classification of excitability, we say that a neuron
has Class 1 spiking if the frequency-current (F-I) curve at the transition “spiking →resting” decreases to zero, as in Fig.7.13c, and Class 2 spiking if it stops at a certain
non-zero value.
The class of excitability coincides with the class of spiking when the transitions“resting ↔ spiking” occur via saddle-node on invariant circle bifurcation or supercritical Andronov-Hopf bifurcation. Indeed, if the current ramps are sufficiently slow, theneuron as a dynamical system goes through the same bifurcation, but in the opposite
direction. The classes may differ when the bifurcation is of the saddle-node (off invariant circle) type or the subcritical Andronov-Hopf type because of the bistability of theresting and spiking states. Such a bistability results in the hysteresis behavior of thesystem when the injected current I increases and decreases slowly, which may resultin the hysteresis of the F-I curve. For example, the transition “resting → spiking” inFig.7.13a occurs via saddle-node bifurcation at I = 4.51, and the frequency of spiking
equals the frequency of the limit cycle attractor, which is non-zero at this value of I.
Decreasing I results in the transition “spiking → resting” via the saddle homoclinic
orbit bifurcation in Fig.7.13b, and in the oscillations with zero frequency at I = 3.08.
Thus, the F-I behavior of the model in this figure (and in Fig.7.10) exhibits Class 2
excitability but Class 1 spiking. Because of the logarithmic scaling of the F-I curveat the saddle homoclinic bifurcation (see section 6.2.4), experimentally estimating theExcitability

229

zero value of the F-I curves is challenging.
Interestingly, steps of injected DC current, as in Fig.7.10c, induce the transition“resting → spiking”. But because the model in the figure is near a codimension-2
Bogdanov-Takens bifurcation, the steps test the frequency of the limit cycle attractor
at the bifurcation “spiking → resting”, as in Fig.7.10e; that is, they test the class of
spiking! The F-I curve in response to steps in the figure is the same as the F-I curvein response to a slowly decreasing current ramp. (As an exercise, explain why this istrue for Fig.7.10 but not for Fig.7.13.)
To summarize, we define the class of excitability according to the frequency of
emerging spiking of a neuron in response to a slowly increasing current ramp. The classof excitability corresponds to a bifurcation of the resting state (equilibrium) resulting
in the transition “resting → spiking”. We define the class of spiking according to thefrequency of disappearing spiking of a neuron in response to a slowly decreasing current
ramp. The class of spiking corresponds to the bifurcation of the limit cycle, resulting
in the transition “spiking → resting”. Stimulating a neuron with ramps (and pulses) is
the first step in exploring the bifurcations in the neuron dynamics. Combined with thetest for the existence of subthreshold oscillations of the membrane potential, it tells
whether the neuron is an integrator or a resonator, and whether it is monostable orbistable, as we discuss next.

7.2

Integrators vs. Resonators

In this book we classify excitable systems based on two features: the coexistence ofresting and spiking states and the existence of subthreshold oscillations. The former
feature divides all systems into monostable and bistable. The latter feature divides
all systems into integrators (no oscillations) and resonators. These features uniquelydetermine the type of bifurcation of the resting state, as we summarize in Fig.7.14.
For example, a bistable integrator corresponds to a saddle-node bifurcation, whereas
monostable resonator corresponds to a supercritical Andronov-Hopf bifurcation. Integrators and resonators have drastically different neurocomputational properties, summarized in Fig.7.15 and discussed next (the I-V curves are discussed in chapter 6).

YES
NO(resonator) (integrator)

subthreshold oscillations

coexistence of resting and spiking states
YES(bistable)NO(monostable)

saddle-nodesaddle-node on
invariant circle

subcritical
Andronov-Hopf

supercritical
Andronov-Hopf

Figure 7.14: Classification of neurons intomonostable/bistable
integrators/resonators
according to the bifurcation of the resting state.

230

Excitability
properties

integrators

resonatorsbifurcation

saddle-node on
invariant circle

saddle-node

subcritical
Andronov-Hopf

supercritical
Andronov-Hopf

excitability

class 1class 2class 2class 2

oscillatory
potentials

no

yes

frequency
preference

no

yes

I-V relation
at restnon-monotone

monotonespike latency

large

small

thresholdand rheobase

well-defined

may not be defined

all-or-none
action potentials

yes

no

co-existence ofresting and spiking

no

yes

yes

no

post-inhibitory spikeor facilitation(brief stimuli)

no

yes

inhibition-induced
spikingnopossible

Figure 7.15: Summary of neurocomputational properties.

7.2.1

Fast Subthreshold Oscillations

According to the definition, resonators have oscillatory potentials, whereas integrators do not. This feature is so important that many of the other neuronal properties
discussed later are mere consequences of the existence or absence of such oscillations.
Fast subthreshold oscillations, as in Fig.7.16, are typically due to a fast lowthreshold persistent K+ current. At rest, there is a balance of all inward currents
and this partially activated K+ current. A brief depolarization further activates theK+ current and results in fast afterhyperpolarization. While the cell is hyperpolarized,
the current deactivates below its steady-state level, the balance is shifted toward theinward currents, and the membrane potential depolarizes again. And so on.
The existence of fast subthreshold oscillatory potentials is a distinguishable featureof neurons near an Andronov-Hopf bifurcation. Indeed, the resting state of such aneuron is a stable focus. When it is stimulated by a brief synaptic input or an injected
pulse of current, the state of the system deviates from the focus equilibrium, then returns to the equilibrium along a spiral trajectory, as depicted in Fig.7.16 (top), thereby
producing a damped oscillation. The frequency of such an oscillation is the imaginary
part of the complex-conjugate eigenvalues at the equilibrium (see section 6.1.3), and it
can be as large as 200 Hz in mammalian neurons.

Excitability

231

damped oscillations
e
V-

lc

n-n
ullc
lin

0.8

l
nu

0.6

e
li n

K+ activation, n

1

0.4
0.2
0
-80

-60

-40
-20
membrane potential, V (mV)0

Hodgkin-Huxley model

brainstem (mes V neurons)

12.5 ms
9 ms

1 mV

2 mV
-49 mVsustained noisy oscillations
mammalian cortical layer 4 cellbrainstem (mes V neurons)-46 mV-40 mV
10 mV

100 ms

10 ms-43 mV-50 mV-47 mV-68 mV1 mV-59 mV-62 mVthalamocortical neurons
5 mV

mitral cells in olfactory bulb

-65 mV
20 ms

Figure 7.16: Examples of fast damped (top) and sustained (bottom) subthresholdoscillations of membrane potential in neurons and their voltage dependence. (Modified
from Izhikevich et al. 2003).

232

Excitability

In exercise 3 we prove that noise can make such oscillations sustained. While thestate of the system is perturbed and returns to the focus equilibrium, another strong
random perturbation may push it away from the equilibrium, thereby starting a new
damped oscillation. As a result, persistent noisy perturbations create a random sequence of damped oscillations and do not let the neuron rest. The membrane potentialof such a neuron exhibits noisy sustained oscillations of small amplitude, depicted in
Fig.7.16 and discussed in section 6.1.4.
Injected DC current or background synaptic noise increases the resting potential,changes its eigenvalues, and hence changes the frequency and amplitude of noisy oscillations. Fig.7.16 depicts typical cases when the frequency and the amplitude increaseas the resting state becomes more depolarized.
One should be careful to distinguish fast and slow subthreshold oscillations of membrane potential. Fast oscillations, as in Fig.7.16, are those having a period comparablewith the membrane time constant or with the period of repetitive spiking. In contrast, some neurons found in entorhinal cortex, inferior olive, hippocampus, thalamus,
and many other brain regions can exhibit slow subthreshold oscillations with a period of 100 ms and more. These oscillations reflect the interplay between fast andslow membrane currents, such as Ih or IT , and may be irrelevant to the bifurcation
mechanism of excitability. We will discuss this issue in detail in section 7.3.3 and inchapter 9. Amazingly, such neurons still possess many neurocomputational properties of resonators, such as frequency preference and rebound spiking, but exhibit theseproperties on a slower time scale.

7.2.2

Frequency Preference and ResonanceA standard experimental procedure to test the propensity of a neuron to subthresholdoscillations is to stimulate it with a sinusoidal current having slowly increasing frequency (called a zap current), as in Fig.7.17. The amplitude of the evoked oscillationsof the membrane potential, normalized by the amplitude of the stimulating oscilla-

membrane potential response, mV-40 mV
10 Hz

5 mV

0 pA

200 pA

1 sec

140 Hzzap current, pA

200 Hzamplitude of the response

spikes cut
integrators

resonators

resonance
frequency of stimulation

Figure 7.17: Response of the mesV neuron to injected zap current sweeping through arange of frequencies. Integrators and resonators have different responses.

Excitability

233

-42

-42

-42

5 ms
0

10
20
time (ms)

-42

10 ms

-52
30

10
20
time (ms)

30

-42

5 ms

-52
0

10
20
time (ms)

10 ms

-52
30

0

10
20
time (ms)30-42potential (mV)potential (mV)resonators

0

15 ms

-52

potential (mV)-52

potential (mV)potential (mV)potential (mV)

integrators

0

10
20
time (ms)

15 ms

-52
30

0

10
20
time (ms)30

Figure 7.18: Responses of integrators (top) and resonators (bottom) to input pulses
having various inter-pulse periods.
tory current, is called the neuronal impedance – a frequency domain extension of theconcept of resistance. The impedance profile of integrators is decreasing while thatof resonators has a peak corresponding to the frequency of subthreshold oscillations,around 140 Hz in the mesV neuron in the figure. Thus, integrators act as low-passfilters while resonators act as band-pass filters to periodic signals.
Instead of sinusoidal stimulation, consider more biological stimulation with pulsesof current simulating synaptic bombardment. The response of any neuron to input
pulses depends on the frequency content of these pulses. In Fig.7.18 we use tripletswith various inter-pulse periods to illustrate the issue. The pulses may arrive fromthree different presynaptic neurons or from a single presynaptic neuron firing shortbursts.
In Fig.7.18 (top) we show that integrators prefer high-frequency inputs. The first
pulse in each triplet evokes a postsynaptic potential (PSP) that decays exponentially.
The PSP evoked by the second pulse adds to the first one, and so on. The dependenceof the combined PSP amplitude on the inter-pulse period is shown in Fig.7.19. Apparently, the integrator acts as a coincidence detector because it is most sensitive to thepulses arriving simultaneously.
Resonators also can detect coincidences, as one can see in Fig.7.19. In addition, they
can detect resonant inputs. Indeed, the first pulse in each triplet in Fig.7.18 (bottom),evokes a damped oscillation of the membrane potential, which results in an oscillation

234

Excitability
coincidence
detection

integrator
resonator

combined PSP amplitude(normalized)

1

0.3

resonant
frequency

0

3

6
9
12
interpulse period, ms

15

18

Figure 7.19: Dependence of combined PSP amplitude on the inter-pulse period; see
Fig.7.18.
of the firing probability. The natural period of such an oscillation is around 9 ms forthe mesencephalic V neuron used in the figure. The effect of the second pulse depends
on its timing relative to the first pulse: if the interval between the pulses is near thenatural period, which is 10 ms in Fig.7.18 and Fig.7.20, the second pulse arrives during
the rising phase of oscillation, and it increases the amplitude of oscillation even further.
In this case the effects of the pulses add up. The third pulse increases the amplitudeof oscillation even further, thereby increasing the probability of an action potential, as
in Fig.7.20.

5ms

10ms

15ms

non-resonant burst

resonant burst

non-resonant burst

Figure 7.20: Experimental observations of selective response to a resonant (10 ms
interspike period) burst in mesencephalic V neurons in brainstem having subthresholdmembrane oscillations with a natural period around 9 ms; see also Fig.7.18. Three
consecutive voltage traces are shown to demonstrate some variability of the result.
(Modified from Izhikevich et al. 2003).

Excitability

235

inhibitory input
5ms
non-resonant burst

15ms
non-resonant burst

10ms
resonant burst

Figure 7.21: Experimental observations of selective response to inhibitory resonant
burst in mesencephalic V neurons in brainstem having oscillatory potentials with thenatural period around 9 ms. (Modified from Izhikevich et al. 2003).
If the interval between pulses is near half the natural period, e.g., 5 ms in Fig.7.18
and Fig.7.20, the second pulse arrives during the falling phase of oscillation, and it
leads to a decrease in oscillation amplitude. The spikes effectively cancel each other inthis case. Similarly, the spikes cancel each other when the interpulse period is 15 ms,
which is 60 percent greater than the natural period. The same phenomenon occurs
for inhibitory synapses, as we illustrate in Fig.7.21. Here the second pulse increases(decreases) the amplitude of oscillation if it arrives during the falling (rising) phase.
We study the mechanism of such frequency preference in exercise 4, and present its
geometrical illustration in Fig.7.22. There, we depict a projection of the phase portrait
excitatory pulses

spike

inhibitory pulses
spike

non-resonant

e
spik

1

1 2

V
small
PSP

2

1
2

n+m+h

Hodgkin-Huxley model

resonant

spike

2
1

1

2

Figure 7.22: (Left) Projection of trajectories of the Hodgkin-Huxley model on a plane.
(Right) Phase portrait and typical trajectories during resonant and non-resonant response of the model to excitatory and inhibitory doublets of spikes. (Modified from
Izhikevich 2000a).

236

Excitability
Resonant for B

A

12 ms

Resonant for C

18 ms

Period
12 ms
2 mV

B

Period
18 ms
C

20 mV
2 mV

Figure 7.23: Selective communication via bursts. Neuron A sends bursts of spikes toneurons B and C, which have different natural periods (12 ms and 18 ms, respectively;
both are simulations of the Hodgkin-Huxley model). As a result of changing the interspike frequency, neuron A can selectively affect either B or C without changing theefficacy of synapses. (Modified from Izhikevich 2002).
of the Hodgkin-Huxley model having a stable focus equilibrium. The model does nothave a true threshold, as we discuss in section 7.2.4. To fire a spike, a perturbationmust push the state of the model beyond the shaded figure that is bounded by two
trajectories, one of which corresponds to a small postsynaptic potential (PSP), while
the other corresponds to a spike.
Figure 7.22 (right) depicts responses of the model to pairs of pulses, called doublets.
Pulse 1 in the excitatory doublet shifts the membrane potential from the equilibrium
to the right, thereby initiating a subthreshold oscillation. The effect of pulse 2 depends
on its timing: if it arrives when the trajectory finishes one full rotation around theequilibrium, it pushes the voltage variable even more to the right, beyond the shaded
area into the spiking zone, and the neuron fires an action potential. In contrast, if
it arrives too soon, the trajectory does not finish the rotation, and it is still to theleft of the equilibrium. In this case, pulse 2 pushes the state of the model closer tothe equilibrium, thereby canceling the effect of pulse 1. Similarly, the effect of an
inhibitory doublet depends on the interspike period between the inhibitory pulses. If
the interpulse period is near the natural period of damped oscillations, pulse 2 arrives
when the trajectory finishes one full rotation, and it adds to pulse 1, thereby firing theneuron. If it arrives too soon or too late, it cancels the effect of pulse 1.
Quite often, the frequency of subthreshold oscillations depends on their amplitudes,
for instance, oscillations in the Hodgkin-Huxley model slow down as they become larger.
In this case, the optimal input is a resonant burst with a slowly decreasing (adapting)interspike frequency. We will see many examples of such bursts in chapter 9.

Excitability

237

The fact that resonator neurons prefer inputs with “resonant” frequencies is notinteresting by itself. What makes it interesting is the observation that the same input
can be resonant for one neuron and non-resonant for another, depending on theirnatural periods. For example, in Fig.7.23 neurons B and C have different periods ofsubthreshold oscillations: 12 and 18 ms, respectively. By sending a burst of spikes withan interspike interval of 12 ms, neuron A can elicit a response in neuron B, but not inneuron C. Similarly, the burst with an interspike interval of 18 ms elicits a response inneuron C, but not in neuron B. Thus, neuron A can selectively affect either neuron Bor neuron C merely by changing the intra-burst (interspike within a burst) frequency
without changing the efficacy of synaptic connections. In contrast, integrators do nothave this property.

7.2.3

Frequency Preference in Vivo

Figures 7.20 and 7.21 convincingly demonstrate the essence of frequency preference
and resonance phenomenon in vitro, when the neuron is quiescent and “waiting” forthe resonant burst to come. What if the neuron is under a constant bombardment ofsynaptic input, as happens in vivo, firing ten or so spikes per second? Would it be able
to tell the difference between the resonant and non-resonant inputs?
To address this question, we performed a frozen-noise experiment pioneered by
Bryant and Segundo (1976) and depicted in Fig.7.24. We generated a noisy signal(frozen noise in Fig.7.24a) and saved it into the memory of the program that injects
current into a neuron. Then we injected the stored signal into the neuron 50 times tosee how reliable its spike response was. Despite the in vivo-like activity in Fig.7.24b,
the spike raster in Fig.7.24c shows vertical clusters indicating that the neuron prefers
to fire at certain “scheduled” moments of time corresponding to certain features of thefrozen noise input.
In Fig.7.24d–g, we added bursts of three spikes to the frozen noise. The amplitudesof the bursts were constant (less than 10 percent of the frozen noise amplitude), but
the interspike periods were different. The idea was to see whether the response of theneuron would be any different when the burst period was near the neuronal intrinsic
period of 6.7 ms (see the inset in Fig.7.24b). As one would expect, the non-resonant
bursts with 4 ms and 9 ms periods remained undetected by the neuron, since the spikerasters in Fig.7.24d and g are essentially the same as in Fig.7.24c. The resonant burstwith 7 ms period in Fig.7.24f produced the most significant deviation from Fig.7.24c(marked by the black arrow), indicating that the neuron is most sensitive to the resonant input. Typically, the resonant burst does not make the neuron fire extra spikes,
but only changes the timing of “scheduled” spikes. Injecting resonant bursts at different moments results in other interesting phenomena, such as extra spikes and theomission of “scheduled” spikes (not shown here), or no effect at all. Finally, there is a
subtle but noticeable effect of the resonant (7 ms) and nearly resonant (6 ms) bursts
even 100 ms after the stimulation (white arrows in the figure), for which we have no
explanation.

238

Excitability
9 ms

(f)

7 ms

spike raster

(g)

(e)

6 ms
4 ms

(c)

no burst
input

(b)

membranepotential (mV)

(d)

2 mV

20
0

6.7 ms

-20
-40

(a)

injected
current (pA)

-60

200
0

frozen noise
burst input

50

100

150

200

250
300
time (ms)

350

400

450

500

Figure 7.24: Frozen noise experiments demonstrate frequency preference and resonanceto embedded bursts. (a) A random signal (frozen noise) is injected into a neuron (b)in vitro to simulate in vivo conditions. The neuron responds with some spike-timing
variability, depicted in (c). (d–g) Burst input is added to the frozen noise. Note thatthe neuron is most sensitive to the input having the resonant period 7 ms, which isnear the period of subthreshold oscillation (6.7 ms). Shown are in vitro responses ofmesencephalic V neuron of rat brainstem recorded by the author, Niraj S. Desai, andBetsy C. Walcott. The order of stimulation was the first line of c, d, e, f, g, then thesecond line of c, d, e, f, g, then the third line, and so on, to avoid slow artifacts.

7.2.4

Thresholds and Action Potentials

A common misconception is that all neurons have firing thresholds. Moreover, great
effort has been made to determine such thresholds experimentally. Typically, a neuron is stimulated with brief current pulses of various amplitudes to elicit various degrees of depolarization of the membrane potential, as we illustrate in Fig.7.25 using
the Hodgkin-Huxley model. Small “subthreshold” depolarizations decay while large“superthreshold” or “suprathreshold” depolarizations result in action potentials. The
maximal value of the subthreshold depolarization is taken to be the firing thresholdvalue for that neuron. Indeed, the neuron will fire a spike if depolarized just abovethat value.

Excitability

239

1ms

10 mV

threshold ?

Figure 7.25: Finding the threshold in the HodgkinHuxley model.

squid axon

model

0 mV

0 mV

-50 mV-50 mV

5 ms

5 ms

Figure 7.26: Variable-size action potentials in squid giant axon and revised HodgkinHuxley model (Clay 1998) in response to brief steps of currents of variable magnitudes.
(Data provided by John Clay.)

The notion of a firing threshold is simple and attractive, especially when teachingneuroscience to undergraduates. Everybody, including the author of this book, uses
it to describe neuronal properties. Unfortunately, it is wrong. First, the problem isin the definition of an action potential. Are the two dashed curves in Fig.7.26 action
potentials? What about a curve in between (not shown in the figure)? Suppose wedefine an action potential to be any deviation from the resting potential, say by 20
mV. Is the concept of a firing threshold well defined in this case? Unfortunately, theanswer is still NO.
The membrane potential value that separates subthreshold depolarizations fromaction potentials (whatever the definition of an action potential is) depends on theprior activity of the neuron. For example, if a neuron having transient Na+ current
has just fired an action potential, the current is partially inactivated, and a subsequent
depolarization above the firing threshold may not evoke another action potential. Conversely, if the neuron was briefly hyperpolarized and then released from hyperpolarization, it could fire a rebound postinhibitory spike, as we discuss later in this chapter(see Fig.7.29). Apparently, releasing from hyperpolarization does not qualify as a superthreshold stimulation. Why, then, did the neuron fire?

240

Excitability(a) integrator

0.06

0.5

(b) resonator
thre
sh
old
m

e

small
EPSP

sp
ik

small
EPSP
spike

old

0

ho

ld

ma

nif

K+ activation

an

ld

i fo

th r

es

0.2

-70

-60

-50

-40

-70

-60

-50

-40

1

(d) resonatorspike(c) resonator

sm
all

0.8

ca
n

K+ activation

half-amplitude
spike
SP
EP

d
ar

to
jec
tra

P

ry

0.6

spikes

0.4

0.1

th r

0.2
e s h o ld s et

0
-60

-50

-40

membrane potential, mV

-30

-70

-60

-50

-40

-30

-20

membrane potential, mV

Figure 7.27: Threshold manifolds and sets in the INa,p +IK -model. Parameters in (a)are as in Fig.4.1a, and in (b), (c), and (d) as in Fig.6.16 with I = 45 (b) and I = 42(c and d).

7.2.5

Threshold manifolds

The problem of formulating a mathematical definition of firing thresholds was first
tackled by FitzHugh (1955). Using geometrical analysis of neural models, he noticed
that thresholds, if they exist, are never numbers but manifolds, e.g., curves in twodimensional systems. We illustrate his concept in Fig.7.27, using phase plane analysisof the INa,p +IK -model.
Integrators do have well-defined threshold manifolds. Since an integrator neuron isnear a saddle-node bifurcation, whether on or off an invariant circle, there is a saddle
point with its stable manifold (see Fig.7.27a). This manifold separates two regions of
the phase space, and for this reason is often called a separatrix. Depending on the prioractivity of the neuron and the size of the input, its state can end up in the shaded
area and generate a subthreshold potential, or in the white area and generate an action
potential. An intermediate-size input cannot reduce the size of the action potential; it
can only delay its occurrence. In the extreme case, a perturbation can put the statevector precisely on the threshold manifold, and the system converges to the saddle, atleast in theory. Since the saddle is unstable, small noise present in neurons pushes theExcitability

241state either to the left or to the right, resulting in either a long subthreshold potential
or a large-amplitude spike with a long latency, as we discuss in section 7.2.9 and show
in Fig.7.34. Finally, note that a neuron has a single threshold value of membranepotential only when its threshold manifold is a straight line orthogonal to the V axis.
Resonators may or may not have well-defined threshold manifolds, depending on
the type of bifurcation. Consider a resonator neuron in the bistable regime; that is,sufficiently near a subcritical Andronov-Hopf bifurcation with an unstable limit cycle
separating the resting and the spiking states, as in Fig.7.27b. Such an unstable cycle
acts as a threshold manifold. Any perturbation that leaves the state of the neuron
inside the attraction domain of the resting state, which is the shaded region boundedby the unstable cycle, results in subthreshold potentials. Any perturbation that pushes
the state of the neuron outside the shaded region results in an action potential. In theextreme case, a perturbation may put the state right on the unstable limit cycle. Then,the neuron exhibits unstable ”threshold” oscillations, at least in theory. In practice,such oscillations cannot be sustained because of noise, and they will either subside or
result in spikes.
The bistable regime near subcritical Andronov-Hopf bifurcation is the only case inwhich a resonator can have a well-defined threshold manifold. In all other cases, including the supercritical Andronov-Hopf bifurcation, resonators do not have well-defined
thresholds. We illustrate this in Fig.7.27c. A small deviation from the resting state
produces a trajectory corresponding to a “subthreshold” potential. A large deviation
produces a trajectory corresponding to an action potential. We refer to the shaded
region between the two trajectories as a threshold set. It consists of trajectories corresponding to partial-amplitude action potentials, such as those in Fig.7.26. No single
curve separates small potentials from action potentials, so there is no well-defined
threshold manifold.
FitzHugh (1955) noticed that the threshold set can be quite thin in some models,
including the Hodgkin-Huxley model. In particular, the difference between the trajectories corresponding to small potentials and action potentials can be as small as 0.0001
mV, which is smaller than the noisy fluctuations of the membrane potential. Thus,
to observe an intermediate-amplitude spike in such models, one needs to simulate themodels with accuracy beyond the limits of uncertainty that appear when the physicalinterpretation of the model is considered. As a result, for any practical purpose suchmodels exhibit all-or-none behavior, with the threshold set looking like a threshold
manifold. FitzHugh referred to this as being a quasi-threshold phenomenon.
Quasi-thresholds are related to the special canard trajectory depicted in Fig.7.27d.
The trajectory follows the unstable branch of the cubic nullcline all the way to the rightknee point P. The flow near the trajectory is highly unstable; any small perturbation
pushes the state of the system to the left or to the right, resulting in a “subthreshold”
or “superthreshold” response. The solutions depicted in Fig.7.26 (right) try to follow
such a trajectory. An easy way to compute the trajectory in two-dimensional relaxation
oscillators is to start with the point P and integrate the system backward (t → −∞).
We discuss canard solutions in detail in section 6.3.4.

242

Excitability(a) integrator(b) resonator
?

sh

re
th

et

B

old restingstate, I=0

spike

new resting
state

ds

A

PSP

h ol

old

spike

thr e snew restingstate

old restingstate, I=0

Figure 7.28: Integrators have a well-defined rheobase current, while resonators may
not.

7.2.6

Rheobase

The neuronal rheobase, that is, the minimal amplitude of a current of infinite durationthat makes the neuron fire, measures the “current threshold” of the neuron. Integrators have a well-defined rheobase, while resonators may not. To see this, consider
an integrator neuron in Fig.7.28a receiving a current step that instantly changes its
phase portrait. In particular, the current moves the equilibrium from the old location
corresponding to I = 0 (white square in the figure) to a new location (black circle).
Whether the neuron fires or not depends on the location of the old equilibrium relative
to the stable manifold to the saddle, which plays the role of the new threshold. Incase A the neuron does not fire; in case B it fires even though the resting state is still
stable. The neuronal rheobase is the amplitude of the current I that puts the thresholdexactly on the location of the old equilibrium. Such a value of I always exists, and it
often corresponds to the saddle-node bifurcation value. Note that the rheobase current
results in a spike with infinite latency, at least theoretically.
A resonator neuron may not have a well-defined rheobase simply because it may
not have well-defined threshold. Indeed, the dotted line in Fig.7.28b may correspond toa subthreshold or superthreshold response, depending on where it is in the thresholdset. Stimulating such a neuron with “rheobase” current produces spikes with finite
latencies but partial amplitudes. A bistable resonator (near subcritical Andronov-Hopfbifurcation) may have a well-defined rheobase because it has a well-defined threshold– the small-amplitude unstable limit cycle.

7.2.7

Postinhibitory Spike

Prolonged injection of a hyperpolarizing current and then sudden release from hyperpolarization can produce a rebound postinhibitory response in many neurons. The
hyperpolarizing current is often called an anodal current, release from the hyperpolarization is called anodal break, and rebound spiking is called anodal break excitation

Excitability

243

10 mV
10 ms
-45 mV

0 pA-100 pA

Figure 7.29: Rebound spikes in response toa brief hyperpolarizing pulse in a brainstemmesV neuron having fast subthreshold oscillations of membrane potential.

(FitzHugh 1976). Note that firing of a neuron follows a sudden increase of injectedcurrent, whether it is a positive step or a release from a negative step.
Often, postinhibitory responses are caused by the “hyperpolarization-activated” hcurrent, which slowly builds up and, upon termination of the hyperpolarization, drives
the membrane potential over the threshold manifold (or threshold set). Alternatively,
the rebound response can be caused by slow deinactivation of Na+ or Ca2+ currents,
or slow deactivation of a K+ current that is partially activated at rest and prevents
firing. In any case, such a rebound response relies on slow currents and long or strong
hyperpolarizing steps; it does not depend on the bifurcation mechanism of excitability,
and it can occur in integrators or resonators.
Some neurons can exhibit rebound spikes after short and relatively weak hyperpolarizing currents, as we illustrate in Fig.7.29. The negative pulse deactivates a fast
low-threshold resonant current (e.g., K+ current) that is partially activated at rest.
Upon release from the hyperpolarization, there is a deficit of the outward current andthe net membrane current results in rebound depolarization and possibly a spike. Sucha response occurs on the fast time scale, and it does depend on the bifurcation mechanism of excitability.
In Fig.7.30 we show why integrators cannot fire rebound spikes in response to shortstimulation, while resonators typically can. A brief excitatory pulse of current depolarizes the membrane and brings it closer to the threshold manifold, as in Fig.7.30a.
Consequently, an inhibitory pulse hyperpolarizes the membrane and increases the distance to the threshold manifold. The dynamics of such a neuron is consistent with theintuition that excitation facilitates spiking and inhibition prevents it.
Contrary to our intuition, however, inhibition can also facilitate spiking in resonator
neurons because the threshold set may wrap around the resting state, as in Fig.7.30b.
A sufficiently strong inhibitory pulse can push the state of the neuron beyond thethreshold set, thereby evoking a rebound action potential. If the inhibitory pulse is
not strong, it still can have an excitatory effect, since it brings the state of the systemcloser to the threshold set. For example, it can enhance the effect of subsequent
excitatory pulses, as we illustrate in Fig.7.31. The excitatory pulse here is subthreshold
if applied alone. However, it becomes superthreshold if preceded by an inhibitory pulse.
The timing of pulses is important here, as we discussed in section 7.2.2. John Rinzel
suggested calling this phenomenon a postinhibitory facilitation.

244

Excitability(b) resonatorK+ activation gate(a) integrator

inhibition
excitation

excitation

inhibition

-60
-40
-20
membrane potential, mV

-60

-40
-20
membrane potential, mV

K+ activation gate

Figure 7.30: Direction of excitatory and inhibitory input in integrators (a) and resonators (b).

10 mV
1 ms

inhibitory
pulse

excitatory
pulse
inhibitory
pulse

excitatory
pulse
-60

-40

-20

membrane potential, mV

Figure 7.31: Postinhibitory facilitation: A subthreshold excitatory pulse can becomesuperthreshold if it is preceded by an inhibitory pulse.

7.2.8

Inhibition-Induced Spiking

In Fig.7.32 (left) we use the INa,t -model introduced in chapter 5 to illustrate an interesting property of some resonators: inhibition-induced spiking. Recall that the model
consists of an Ohmic leak current and a transient Na+ current with instantaneous activation and relatively slow inactivation kinetics. It can generate action potentials dueto the interplay between the amplifying gate m and the resonant gate h.
We widened the activation function h∞ (V ) so that the Na+ current is largely inactivated at the resting state; see the inset in Fig.7.32 (right). Indeed, h = 0.27 whenI = 0. Even though such a system is excitable, it cannot fire repetitive action potentials when a positive step of current (e.g., I = 10) is injected. Depolarization produced

245
0

0
-20
-40
-60

I=-15
50

I=10

I=-10

0.5

100

150

time (ms)0.7

V-nullclines

I=-5

0.3
0.4

ne
lcli
ul

I=5
I=0

0.2

I=-15

I=-20

1
m (V)
h (V)
0
-80 mV rest
20 mV

0.6

I=10
0

0.1

hn

20
Na+ inactivation gate, hmembrane potential, V (mV)

Excitability

-60

-40
-20
0
membrane potential, V (mV)

20

40

Figure 7.32: Inhibition-induced spiking in the INa,t -model. Parameters are the sameas in Fig.5.6b, except gleak = 1.5 and m∞ (V ) has k = 27.

inactivationof INadeinactivationof INaspike

hyperpolarization

negative
injected
dc-current

resting
potential

excessof INaactivationof INaFigure 7.33: Mechanism of inhibitioninduced spiking in the INa,t -model.

by the injected current inactivates the Na+ current so much that no repetitive spikes
are possible. Such a system is Class 3 excitable.
Remarkably, injection of a negative step of current (e.g., I = −15 in the figure)
results in a periodic train of action potentials! How is it possible? Inhibition-induced
spiking or bursting is possible in neurons having slow h-current or T-current, such as
the thalamocortical relay neurons. (We discuss these and other examples in the next
chapter.) The INa,t -model does not have such currents, yet it can fire in response toinhibition.
Figure 7.33 summarizes the ionic mechanism of inhibition-induced spiking. The
resting state in the model corresponds to the balance of the outward leak current anda partially activated, partially inactivated inward Na+ current. When the membranepotential is hyperpolarized by the negative injected current, two processes take place:
the Na+ current both deinactivates (variable h increases) and deactivates (variable m =m∞ (V ) decreases). Since m∞ (V ) is flatter than h∞ (V ), deinactivation is stronger thandeactivation and the Na+ conductance, gNa mh, increases. This leads to an imbalanceof the inward current and to the generation of the first spike. During the spike, thecurrent inactivates completely, and the leak and negative injected currents repolarize
and then hyperpolarize the membrane. During the hyperpolarization, clearly seen in
the figure, the Na+ current deinactivates and is ready for the generation of the next
spike.

246

Excitability

20 mV
50 ms

-75 mV

20 pA

Figure 7.34: Long latencies
and threshold crossing oflayer 5 neuron recorded invitro of rat motor cortex.

0 pA

To understand the dynamic mechanism of such an inhibition-induced spiking, weneed to consider the geometry of the nullclines of the model, depicted in Fig.7.32(right). Note how the position of the V -nullcline depends on I. Negative I shifts thenullcline down and leftward so that the vertex of its left knee, marked by a dot, moves
to the left. As a result, the equilibrium of the system, which is the intersection of theV - and h-nullclines, moves toward the middle branch of the cubic V -nullcline. WhenI = −2, the equilibrium loses stability via supercritical Andronov-Hopf bifurcation,and the model exhibits periodic activity.
Instead of the INa,t -model, we could have used the INa + IK -model or any othermodel with a low-threshold resonant gating variable. The key point here is not the ionicbasis of the spike generation mechanism, but its dynamic attribute – the AndronovHopf bifurcation. Even the FitzHugh-Nagumo model (4.11, 4.12) can exhibit thisphenomenon (see exercise 1).

7.2.9

Spike Latency

In Fig.7.34 we illustrate an interesting neuronal property - latency to first spike. A
barely superthreshold stimulation evokes action potentials with a significant delay,
which could be as large as a second in some cortical neurons. Usually, such a delay is
attributed to slow charging of the dendritic tree or to the action of the A-current, which
is a voltage-gated transient K+ current with fast activation and slow inactivation. The
current activates quickly in response to a depolarization and prevents the neuron fromimmediate firing. With time, however, the A-current inactivates and eventually allows
firing. (A slowly activating Na+ or Ca2+ current would achieve a similar effect.)
In Fig.7.35 we explain the latency mechanism from the dynamical systems pointof view. Long latencies arise when neurons undergo saddle-node bifurcation, depicted
in Fig.7.35 (left). When a step of current is delivered, the V -nullcline moves up sothat the saddle and node equilibria that existed when I = 0 coalesce and annihilate
each other. Although there are no equilibria, the vector field remains small in theshaded neighborhood, as if there were still a ghost of the resting equilibrium there (seesection 3.3.5). The voltage variable increases and passes that neighborhood.
As we√discussed in exercise 3 of chapter 6, the passage time scales as 1/ I − Ib , where Ib
is the bifurcation point (see Fig.6.8). Hence, the spike is generated with a significant

Excitability

247
saddle-node bifurcation
en I=
0

1

ine
wh
en
wh
I>
en
I
I=0

V-n
u

ll cl

ine

ll cl

line
ullc
0.5

V-n
u

in e

llc
l
nu

n-

n-n

K+ activation gate

llclin

e wh

b

V-nu

V-n

ullcl
ine
w

hen

I>I
b

0.1

Andronov-Hopf bifurcation

0

-70

-60
-50
membrane potential, mV

-40

-70

-60
-50
-40
-30
membrane potential, mV

Figure 7.35: Bifurcation mechanism of latency to first spike when the injected DC
current steps from I = 0 to I > Ib , where Ib is a bifurcation value. The shaded circle
denotes the region where vector field is small. Phase portraits of the INa,p +IK -model
are shown.
latency. If the bifurcation is on an invariant circle, then the state of the neuron returns
to the shaded neighborhood after each spike, resulting in firing with small frequency,a characteristic of Class 1 excitability (see Fig.7.3). In contrast, if the saddle-nodebifurcation is off an invariant circle, then the state does not return to the neighborhood,and the firing frequency can be large, as in Fig.7.34 or in the neostriatal and basal
ganglia neurons reviewed in section 8.4.2.
We see that the existence of long spike latencies is an innate neurocomputational
property of integrators. It is still not clear how or when the brain uses it. Two of
the most plausible hypotheses are 1) that neurons encode the strength of input into
spiking latency, and 2) that neuronal responses become less sensitive to noise, since
only prolonged inputs can cause spikes.
Interestingly, resonators do not exhibit long latencies even though there is a neighborhood where the vector field is small and even zero, as we show in Fig.7.35 (right).
When the current pulse is applied, the V -nullcline moves up and the voltage variable accelerates. However, it misses the shaded neighborhood, and the neuron fires an action
potential practically without any latency. In exercise 5 we discuss why some models near Andronov-Hopf bifurcation, including the Hodgkin-Huxley model in Fig.7.26,
seem to exhibit small but noticeable latencies. In section 8.2.7 we show that latencies
could result from slow charging of the dendritic compartment. In this case, integrator
neurons exhibit latency to the first spike, while resonator neurons may exhibit latency
to the second spike (after they fire the first, transient spike).

248

Excitability
20 mV

10 mV
100 ms200 msoscillations

resonator

up-state-60 mV

down-state

n-nu
llcli
ne

integrator

ll c
nu
V

e
lin

up-state(focus)down-state(node)

Figure 7.36: Bistability of the up-state and down-state of mitral cells in rat main
olfactory bulb. The cells are integrators in the down-state and resonators in the upstate. Membrane potential recordings are modified from Heyward et al. (2001). The
shaded area denotes the attraction domain of the up-state.

7.2.10Flipping from an Integrator to a Resonator

One of the reasons we provided so many examples of neuronal systems in chapter 5
was to convince the reader that all neuronal models can exhibit both saddle-nodeand Andronov-Hopf bifurcations, depending on the parameters describing the ioniccurrents. Since the kinetics of ionic currents in neurons can change during development
or due to the action of neuromodulators, neurons can switch from being integrators tobeing resonators.
In Fig.7.36 we illustrate an interesting case: mitral cells in rat main olfactory bulb
can exhibit bistability of membrane potential. That is, the potential can be in two
states: down-state around −60 mV, and up-state around −50 mV (Heyward et al.
2001). A sufficiently strong synaptic input can shift the cell between these states ina matter of milliseconds. An amazing observation is that the down-state is a stablenode and the up-state is a stable focus, as we illustrate at the bottom of the figure andstudy in detail in section 8.4.5. As a result, mitral cells can be quickly switched frombeing integrators to being resonators by synaptic input.

Excitability

249

1 mV
20 ms

fast oscillations-60 mV
5 mV

100 msFigure 7.37: Fast subthreshold oscillations during complex spikes of cerebellar Purkinje
neuron of a guinea-pig. (Data was by Yonatan Loewenstein.)?

resonator(Andronov-Hopf bifurcation)inV-nullcl
e

n-n
ullc
line

integrator(saddle-node bifurcation)?
Figure 7.38: Is there an intermediate mode between integrators and resonators?

A similar phenomenon was observed in a cerebellar Purkinje neuron (see Fig.7.37).
It acts as an integrator in the down-state, but has fast (> 100 Hz) subthreshold oscillations in the up-state, and hence can act as a resonator.
Cortical pyramidal neurons can also exhibit up- and down-states, though the states
are not intrinsic, but induced by the synaptic activity. Since the neurons are depolarized in the up-state, there is an interesting possibility that fast K+ conductances
are partially activated and the fast Na+ inactivation gate is partially inactivated sothat the neuron exhibits fast subthreshold oscillations and acts as a resonator. Thatis, integrator neurons can switch to the resonator mode when in the up-state. Thispossibility needs to be tested experimentally.

250

Excitability
Bogdanov-Takens bifurcation
line

0.05

0.4
0.3
0.2
0.1
0

-60

-40

-20

0.04
K+ activation gate, nn-null
c

V-n

ine
lcl
ul

K+ activation gate, n

0.5

0.03
0.02
0.01
0

0

-65

-60

-55

-50

-60

-55

-50

V-n

n-null

0.5
0.4
0.3
0.2

K+ activation gate, n

cline

0.05

ine
lcl
ul

K+ activation gate, nintegrator (near saddle-node bifurcation)

0.1
0

-60

-40

-20

0.04
0.03
0.02
0.01
0

0

threshold
-65

V-n

0.3
0.2
0.1
0

-60
-40
-20
membrane potential, V (mV)0

K+ activation gate, n0.4n-null

0.5

0.05

ine
lcl
ul

K+ activation gate, n

0.6

cline

resonator (near Andronov-Hopf bifurcation)

0.04
0.03
0.02
0.01
0

-65
-60
-55
membrane potential, V (mV)

-50

Figure 7.39: Bogdanov-Takens bifurcation in the INa + IK -model (4.1, 4.2). Parameters
are as in Fig.4.1a, except n∞ (V ) has k = 7 mV and V1/2 = −31.64 mV, Eleak = −79.42
and I = 5. Integrator: V1/2 = −31 mV and I = 4.3. Resonator: V1/2 = −34 mV andI = 7.

Excitability

7.2.11251

Transition Between Integrators and Resonators

Consider the INa +IK -model or any other minimal model from chapter 5 that can exhibit
saddle-node or Andronov-Hopf bifurcation, depending on the parameter values. Let us
start with the INa + IK -model near saddle-node bifurcation, and hence in the integrator
mode. The intersection of its nullclines at the left knee is similar to the one in Fig.7.38(left). Now, slowly change the parameters toward the values corresponding to theAndronov-Hopf bifurcation with the nullclines intersecting as in Fig.7.38 (right). At
some point, the behavior of the model must change from integrator to resonator mode.
Is the change sudden, or is it gradual?
Any qualitative change of the behavior of the system is a bifurcation. Such a bifurcation should somehow combine the saddle-node and the Andronov-Hopf cases. Thatis, it should have a zero eigenvalue, and a pair of complex-conjugate eigenvalues withzero real part. Since the INa + IK -model is two-dimensional, these two conditions are
satisfied only when the model undergoes the Bogdanov-Takens bifurcation consideredin section 6.3.3. This bifurcation has codimension-2, that is, it can be reliably observed
when two parameters are changed – in this case, Eleak and the half-voltage, V1/2 , ofn∞ (V ).
The top of Fig.7.39 depicts the phase portrait of the INa +IK -model at the BogdanovTakens bifurcation. Note that the nullclines are tangent near the left knee, but thetangency is degenerate. A small change of the parameter V1/2 can result in either asaddle and a node (middle of the figure) or a focus equilibrium (bottom of the figure).
The neuron acts as an integrator in the former case and as a resonator in the latter
case.
Due to the proximity to a codimension-2 bifurcation, the behavior of the INa + IK model is quite degenerate. That is, it can exhibit features that are normally notobserved. For example, the integrator can exhibit postinhibitory spiking, as in Fig.7.40.
This occurs because the shaded region in the figure, bounded by the stable manifold

-40
-60
-80
I=-10
0

10

0.04

n-n
ullc
line

K+ activation gate, nmembrane potential, V (mV)-20ellcl in
V-nu

0.05
0

0.03
0.02
I=-10
0.01

I=4.3
20
time (ms)

30

40

50

0

I=4.3
-65

-60

-55

-50

membrane potential, V (mV)

Figure 7.40: Postinhibitory spike of an integrator neuron near Bogdanov-Takens bifurcation; see Fig.7.39.

252

Excitability
integrator

node

2

1

spi
k

spi

e

ke

resonator

2

1

focus

1
2

Figure 7.41: Postinhibitory facilitation – enhancement of subthreshold depolarizing
pulse (2) by preceding inhibitory pulse (1) – can occur in integrator neurons near
Bogdanov-Takens bifurcation.
of the saddle, goes to the left of the resting state. An inhibitory pulse of current that
hyperpolarizes the membrane potential to V < −65 mV and deactivates the K+ current
to n < 0.005 pushes the point (V, n) to the shaded region (i.e., beyond the threshold).
Upon release from inhibition, the integrator neuron produces a rebound spike and thenreturns to the resting state.
Integrator neurons can also exhibit frequency preference and resonance, as illustrated in Fig.7.41. The postinhibitory facilitation in resonator neurons in Fig.7.41a
was described in section 7.2.7. It may occur in integrator neurons when the node equilibrium has nearly equal eigenvalues and nearly parallel eigenvectors, as in Fig.7.41b.
The former are about to become complex-conjugate resulting in rotation of the vector
field around the equilibrium, and hence in the postinhibitory rebound response to thefirst (inhibitory) pulse.
Resonator neurons near a Bogdanov-Takens bifurcation can fire spikes with noticeable latencies. This occurs because the V -nullcline follows the n-nullcline at the focus
equilibrium in Fig.7.39 (bottom). Such a proximity creates a “tunnel” with a small
vector field that slows the spiking trajectory. Finally, the neuron can exhibit an oscillation (marked P in Fig.7.42, bottom) before firing a spike in response to a pulse ofcurrent. Of course, these behaviors are difficult to catch experimentally, because thesystem must be near a codimension-2 bifurcation.

7.3

Slow Modulation

So far we have considered neuronal models having voltage- or Ca2+ -gated conductances
operating on a fast time scale comparable with the duration of a spike. Such conductances participate directly or indirectly in the generation of each spike and subsequent
repolarization of the membrane potential. In addition, neurons have dendritic trees
and some slow conductances and currents that may not be involved in the spike generation mechanism directly, but rather may modulate it. For example, some cortical

Excitability

253

20 mV
200 ms
S
S
A

A

P

S
P

S
A
A

Figure 7.42: Proximity to Bogdanov-Takens bifurcation in layer 5 pyramidal neuron of
rat primary visual cortex results in slow subthreshold oscillation before a spike. Shown
are a hand-drawn phase portrait and in vitro recordings obtained while an automatedprocedure was testing the neuronal rheobase.
pyramidal neurons have Ih , and all thalamocortical neurons have Ih and ICa(T) . Activation and inactivation kinetics of these currents are too slow to participate in thegeneration of the upstroke or downstroke of a spike, but the currents can modulate thespiking pattern (e.g., they can transform it into bursting).
To illustrate the phenomenon of slow modulation, we use the INa,p +IK +IK(M) model
IK(M)INa,p +IK -model
}|
{ z
}|
{
z
C V̇ = I −gL (V −EL )−gNa m∞ (V )(V −ENa )−gK n(V −EK ) − gM nM (V −EK )ṅ= (n∞ (V ) − n)/τ (V )ṅM = (n∞,M (V ) − nM )/τM (V )(slow K+ M-current),(7.1)

whose excitable and spiking properties are similar to those of the INa,p +IK -submodel
on a short time scale. However, the long-term behavior of the two models may be quite
different. For example, the K+ M-current may result in frequency adaptation during along train of action potentials. It can change the shape of the I-V relation of the modeland result in slow oscillations, postinhibitory spikes, and other resonator properties
even when the INa,p +IK -submodel is an integrator. All these interesting phenomena
are discussed in this section.

Excitability

(a)

40

T0

T1

T3

T4

T5

20
0
-20
-40

40

-60

Ti 20

I=300 pA

0

0 pA

-80
0

50

time (ms)(b)

150interspike period, Ti (ms)

T2

100

ramp

T0

50

0

T
T3
T2
T
T0 1

0

100
200
300
injected current, I (pA)interspike frequency, Fi=1000/Ti (Hz)membrane potential (mV)

254

0

5

10

spike number, i

15

100

150

(c)

120

F0

100
80

F1

60
F2
F3,4,5
F

40
20
0

ramp

0

100
200
300
injected current, I (pA)

Figure 7.43: Spike frequency adaptation in layer 5 pyramidal cell (see Fig.7.3). Ramp
data are from Fig.7.6.
In general, models having fast and slow currents, such as (7.1), can be written in
the fast-slow formẋ = f (x, u)u̇ = µg(x, u)(fast spiking),(slow modulation),(7.2)

where the vector x ∈ Rm describes fast variables responsible for spiking. It includes themembrane potential V and activation and inactivation gating variables for fast currents,
among others. The vector u ∈ Rk describes relatively slow variables that modulate fast
spiking, e.g., the gating variable of a slow K+ current, the intracellular concentrationof Ca2+ ions, etc. The small parameter µ represents the ratio of time scales between
spiking and its modulation. Such systems often result in bursting activity, and westudy them in detail in chapter 9.

Excitability

255
15

0
-20

10

-40

5

-80

interspike intervals (ms)membrane potential (mV)

-60

0
-20
-40
-60
-80
0
-20

10
5
0
15
10

-40

5

-60
-80

0
15

0

50

100
time (ms)

150

200

0

0

10
20
spike number

30

Figure 7.44: Spike-frequency acceleration of a cortical fast spiking (FS) interneuron.
Data kindly provided by Barry Connors.

7.3.1

Spike Frequency Modulation

Slow currents can modulate the instantaneous spiking frequency of a long train ofaction potentials, as we illustrate in Fig.7.43a, using recordings of a layer 5 pyramidal
neuron. The neuron generates a train of spikes with increasing interspike interval (seeinset in the figure) in response to a long pulse of injected DC current. In Fig.7.43b
we plot the instantaneous interspike intervals Ti , that is, the time intervals between
spikes i and i + 1, as a function of the magnitude of injected current I. Notice that
Ti (I) < Ti+1 (I), meaning that the intervals increase with each spike. The function
T0 (I) describes the latency of the first spike, and T∞ (I) describes the steady-state(asymptotic) interspike period. The instantaneous frequencies, defined as Fi (I) =1000/Ti (I) (Hz), are depicted in Fig.7.43c. Since the neuron is Class 1 excitable, theF-I curves are square-root parabolas (see section 6.1.2). Note that F0 (I) is nearly astraight line, probably reflecting the passive charging of the dendritic tree.
Decrease of the instantaneous spiking frequency, as in Fig.7.43, is referred to as
spike-frequency adaptation. This is a prominent feature of cortical pyramidal neuronsof the regular spiking (RS) type (Connors and Gutnick 1990), as well as of manyother types of neurons discussed in chapter 8. In contrast, cortical fast spiking (FS)interneurons (Gibson et al. 1999) exhibit spike frequency acceleration, depicted in
Fig.7.44, that is, the instantaneous interspike intervals decrease, and the frequency
increases with each spike.

256

Excitability

Whether a neuron exhibits spike frequency adaptation or acceleration depends on
the nature of the slow current or currents and how they affect the spiking limit cycleof the fast subsystem. At first glance, a resonant slow current (e.g., a slowly activating
K+ or slowly inactivating Na+ current) builds up during each spike and provides anegative feedback that should slow spiking of the fast subsystem. Buildup of a slow
amplifying current (e.g., a slowly activating Na+ or inactivating K+ current) or slow
charging of the dendritic tree should have the opposite effect. In chapter 9, devoted
to bursting, we will show that this simple rule works for many models, but there arealso many exceptions. To understand how the slow subsystem modulates repetitive
spiking, we need to consider bifurcations of the fast subsystem in (7.2), treating theslow variable u as a bifurcation parameter.

7.3.2

I-V Relation

Slow currents and conductances, though not responsible for the generation of spikes,
can mask the true I-V relation of the fast subsystem in (7.2) responsible for spiking.
Take, for example, the INa,p +IK -model with parameters as in Fig.4.1a (high-thresholdK+ current) that has nonmonotonic I-V curve with a region of negative slope, depicted
in Fig.7.45a. Such a system is near saddle-node on invariant circle bifurcation and it
acts as an integrator. Now add a slow K+ M-current with an I-V relation depicted asa dashed curve in the figure and a time constant τM = 100 ms. The spike generating
mechanism of the combined INa,p +IK +IK(M) -model is described by the fast INa,p +IK submodel, so that the neuron continues to have integrator properties, at least on the(a) 150(b) 300

IK(M)

50

200

I

current, I

current, I

100

(V)

0
-50

-100
-150
-100

I

(V)

0
-100
-200

Ifast(V)=IL+INa,p+IK-50
membrane potential, V (mV)100

Ifast(V)=IL+INa,p+IK

0

-300
-100

-50
0
membrane potential, V (mV)

50

Figure 7.45: Slow conductances can mask the true I-V relation of the spike generating
mechanism. (a) The INa,p +IK -model with parameters as in Fig.4.1a has a nonmonotonic I-V curve Ifast (V ). Addition of the slow K+ M-current with parameters as insection 2.3.5 and gM = 5 (dashed curve) makes the asymptotic I-V relation, I∞ (V ),of the full INa,p +IK +IK(M) -model monotonic. (b) Addition of a slow inactivation gate
to the K+ current of the INa,p +IK -model with parameters as in Fig.4.1b results in anonmonotonic asymptotic I-V relation of the full INa,p +IA -model.

Excitability

257

millisecond time scale. However, the asymptotic I-V relation I∞ (V ) is dominated by
the strong IK(M) (V ) and is monotonic, as if the INa,p +IK +IK(M) -model were a resonator.
The model can indeed exhibit some resonance properties, such as postinhibitory (rebound) responses, but only on the long time scale of hundreds of milliseconds, whichis the time scale of the slow K+ M-current.
Similarly, we can take a resonator model with a monotonic I-V relation and add aslow amplifying current or a gating variable to get a non-monotonic I∞ (V ), as if themodel becomes an integrator. For example, in Fig.7.45b we use the INa,p +IK -modelwith parameters as in Fig.4.1b (low-threshold K+ current) and adds an inactivationgate to the persistent K+ current, effectively transforming it into transient A-current.
If the inactivation kinetics is sufficiently slow, the INa,p +IA -model retains resonatorproperties on the millisecond time scale, which is the time scale of individual spikes.
However, its asymptotic I-V relation, depicted in Fig.7.45b, becomes non-monotonic.
Besides spike-frequency acceleration, the model acquires another interesting property:
bistability. A single spike does not inactivate IA significantly. A burst of spikes caninactivate the K+ A-current to such a degree that repetitive spiking becomes sustained.
Slow inactivation of the A-current is believed to facilitate the transition from downstates to up-states in neocortical and neostriatal neurons.
When a neuronal model consists of conductances operating on drastically different
time scales, it has multiple I-V relations, one for each time scale. We illustrate thisphenomenon in Fig.7.46, using the INa,p +IK +IK(M) -model with an activation time constant of 0.01 ms for INa,p , 1 ms for IK , and 100 ms for IK(M) . The upstroke of an action
potential is described only by leak and persistent Na+ currents, since the K+ currents
do not have enough time to activate during such a short event. During the upstroke,
the model can be reduced to a one-dimensional system (see chapter 3) with instantaneous I-V relation I0 (V ) = Ileak + INa,p (V ) depicted in Fig.7.46a. The dynamics during
and immediately after the action potential is described by the fast INa,p +IK -subsystemwith its I-V relation Ifast (V ) = I0 (V ) + IK (V ). Finally, the asymptotic I-V relation,
I∞ (V ) = Ifast (V ) + IK(M) (V ), takes into account all currents in the model.
The three I-V relations determine the fast, medium, and asymptotic behavior ofa neuron in a voltage-clamp experiment. If the time scales are well separated (theyare in Fig.7.46), all three I-V relations can be measured from a simple voltage-clamp
experiment, depicted in Fig.7.46b. We hold the model at V = −70 mV and step
the command voltage to various values. The values of the current, taken at t = 0.05
ms, t = 5 ms, and t = 500 ms in Fig.7.46b, result in the instantaneous, fast, andsteady-state I-V curves, respectively. Notice that the data in Fig.7.46b are plottedon the logarithmic time scale. Various magnifications using the linear time scale are
depicted in Fig.7.46c,d, and e. Numerically obtained values of the three I-V relations
are depicted as dots in Fig.7.46a. They approximate the theoretical values quite wellbecause there is a 100 -fold separation of time scales in the model.

258

Excitability

(a)

(b)
300

2000

I0(V)

I

(V)

0
Ifast(V)

-100

500
0

-300
-100
-50
0
50
membrane potential, V (mV)

(c)

-500

(d)

instantaneous I-V

1000

-2

-1
0
1
2
time, (logarithmic scale, log10ms)(e)fast I-V relation

1500

3

steady-state I-V

2000

I

Ifast(V)

I0(V)

(V)

1500

1000
500

0

0

0.1

time (ms)0.2

current, I

1000

current, I

current, I

(V)

1000

I0(V)

-200

-500

I

Ifast(V)

1500

100

current, I

current, I

200

current response to voltage-clamp steps

500
0

-500

0

10

time (ms)

20

500
0

-500

0

500

1000

time (ms)Figure 7.46: (a) The INa,p +IK +IK(M) -model in Fig.7.45a has three I-V relations: Instantaneous I0 (V ) = Ileak (V ) + INa,p (V ) describes spike upstroke dynamics. The curve
Ifast (V ) = I0 (V ) + IK (V ) is the I-V relation of the fast INa,p +IK -subsystem responsible
for the spike generation mechanism. The curve I∞ (V ) = Ifast (V ) + IK(M) (V ) is thesteady-state (asymptotic) I-V relation of the full model. Dots denote values obtained
from a simulated voltage-clamp experiment in (b); note the logarithmic time scale.
Magnifications of current responses are shown in (c – e). Simulated time constants:
τNa,p (V ) = 0.01 ms, τK (V ) = 1 ms, τM (V ) = 100 ms.

7.3.3

Slow Subthreshold Oscillation

Interactions between fast and slow conductances can result in low-frequency subthreshold oscillation of membrane potential, such as the one in Fig.7.47, even when the fastsubsystem is near a saddle-node bifurcation, acts as an integrator, and cannot havesubthreshold oscillations. The oscillation in Fig.7.47 is caused by the interplay between
activation and inactivation of the slow Ca2+ T-current and the inward h-current, and
it is a precursor of bursting activity (which we consider in detail in chapter 9).
There are at least three different mechanisms of slow subthreshold oscillations ofmembrane potential of a neuron.
• The fast subsystem responsible for spiking has a small-amplitude subthreshold
limit cycle attractor. The period of the limit cycle may be much larger than theExcitability

259

50 mV
300 msFigure 7.47: Slow subthreshold oscillation of membrane potential of cat thalamocorticalneuron evoked by slow hyperpolarization. (Modified from Roy et al. 1984.)
time scale of the slowest variable of the fast subsystem when the cycle is near asaddle-node on invariant circle, saddle homoclinic orbit bifurcation, or BogdanovTakens bifurcation (considered in chapter 6). In this case, no slow currents or
conductances modulating the fast subsystem are involved. However, such a cycle
must be near the bifurcation; hence the low-frequency subthreshold oscillation
exists in a narrow parameter range and it is difficult to be seen experimentally.
• The I-V relation of the fast subsystem has an N-shape in the subthreshold voltage
range, so that there are two stable equilibria corresponding to two resting states(as, e.g., in Fig.7.36). A slow resonant variable switches the fast subsystem
between those two states via a hysteresis loop, resulting in a subthreshold slow
relaxation oscillation.
• If the fast subsystem has a monotonic I-V relation, then a stable subthresholdoscillation can result from the interplay between slow variables.
The first case does not need any slow variables; the second case needs only one slow
variable; and the third case may need at least two.

7.3.4

Rebound Response and Voltage Sag

A slow resonant current can cause a neuron to fire a rebound spike or a burst in response to a sufficiently long hyperpolarizing current, even when the spike generating
mechanism of the neuron is near a saddle-node bifurcation and hence has the neurocomputational properties of an integrator. For example, the cortical pyramidal neuron
in Fig.7.48a has a slow resonant current Ih , which is opened by hyperpolarization. A
short pulse of current does not open enough of Ih and results only in a small subthreshold rebound potential. In contrast, a long pulse of current opens enough Ih , resulting
in a strong inward current that produces the voltage sag and, upon termination of
stimulation, drives the membrane potential over the threshold.
Similarly, the thalamocortical neuron in Fig.7.48b has a low-threshold Ca2+ Tcurrent ICa(T) that is partially activated but completely inactivated at rest. A negative
pulse of current hyperpolarizes the neuron and deinactivates the T-current, thereby
making it available to generate a spike. Note that there is no voltage sag in Fig.7.48b
because the T-current is deactivated at low membrane potentials. Upon termination of
the long pulse of current, the membrane potential returns to the resting state around260

Excitability

Figure 7.48: Rebound responses to long inhibitory pulses in (a) pyramidal neuron of
sensorimotor cortex of juvenile rat (modified from Hutcheon et al. 1996) and (b) rat
auditory thalamic neurons. (Modified from Tennigkeit et al. 1997.)
(a)

(b)
25 mV

-50 mV

100 mssuperthreshold

subthreshold
100 pA

0 pA

0 pA

Figure 7.49: Postinhibitory facilitation (a) and post-excitatory depression (b) in a layer
5 pyramidal neuron (IB type) of rat visual cortex recorded in vitro in response to along hyperpolarizing pulse.
−68 mV, and the Ca2+ T-current activates (but does not have time to inactive) anddrives the neuron over the threshold. A distinctive feature of thalamocortical neurons
is that they fire a rebound burst of spikes in response to strong negative currents.
Even when the rebound depolarization is not strong enough to elicit a spike, it
may increase the excitability of the neuron, so that it fires a spike to an otherwise
subthreshold stimulus, as in Fig.7.49a. This type of postinhibitory facilitation relieson the slow currents, and not on the resonant properties of the spike generation mechanism (as in Fig.7.31). Figure 7.49b demonstrates the inverse property, post-excitatory
depression, that is, a decreased excitability after a transient depolarization. In thisseemingly counterintuitive case, a superthreshold stimulation becomes subthresholdwhen it is preceded by a depolarized pulse, because the pulse partially inactivates theNa+ current and/or activates the K+ current.

7.3.5

AHP and ADP

The membrane potential may undergo negative and positive deflections right after
the spike, as illustrated in Fig.7.50 and Fig.7.51. These are known as afterhyperpolarizations (AHP) and afterdepolarizations (ADP). The latter are sometimes called

Excitability

261

20 mV
100 msADP

AHP-60 mV

Figure 7.50: Afterhyperpolarizations (AHP) and afterdepolarizations (ADP) in intrinsically bursting (IB) pyramidal neurons of the rat motor cortex, recorded in vitro.

Figure 7.51: Rebound spikes and afterdepolarization(marked ADP) at the break of hyperpolarizing current in thalamocortical neurons of the cat dorsal lateral geniculate nucleus. (Data modified from Pirchio
et al. 1997; resting potential is −56 mV, holding potential is −67 mV.)
depolarizing afterpotentials (DAP). A great effort is usually made to determine theionic basis of AHPs and ADPs, since it is often implicitly assumed that they are generated by slow currents that turn on right after the spike, such as the slow Ca2+ -gated
K+ current IAHP or the slow persistent Na+ current, respectively. Below we discussthese and other mechanisms.
Let us consider the AHP first. Each spike in the initial burst in Fig.7.50 presumably
activates a slow voltage- or Ca2+ -dependent outward K+ current, which eventuallystops the burst and hyperpolarizes the membrane potential. During the AHP period,
the slow outward current deactivates, and the neuron can fire again. The neuron canswitch from bursting to tonic spiking mode due to the incomplete deactivation of theslow current. The same explanation holds if we replace “activation of outward” with“inactivation of inward” current.
Similarly, slow inactivation of the transient Ca2+ T-current explains the reboundresponse and the long afterdepolarization (marked ADP) in Fig.7.51: The current
was deinactivated by the preceding hyperpolarization, so upon release from the hyperpolarization, it quickly activates and slowly inactivates, thereby producing a slow
depolarizing wave on which fast spikes can ride. The ADP seen in the figure is the tailof the wave.
Probably the most common mechanism of ADPs is due to the dendritic spikes,
at least in pyramidal neurons of neocortex considered in chapter 8. In Fig.7.52a we262

Excitability

(a)

(b)

(c)
10 mV

dendritic
spike

10 ms
ADP
20 mV

-40 mV

spikes cut at 0 mV
10 mV
10 ms

ADP

ADPs
-40 mV

25 ms
ADP

120 pA
0 pA

60 pA

Figure 7.52: (a) A somatic spike evokes a dendritic spike, which in turn produces
afterdepolarization (ADP) in the soma of the pyramidal neuron of rat somatosensory
cortex (in vitro recording was provided Greg Stuart and Maarten Kole). (b) and (c)Increased level of depolarization in another neuron (the same as in Fig.7.49) converts
ADP to a second spike.
depict a dual somatic/dendritic recording of the membrane potential of a pyramidal
neuron. The somatic spike backpropagates into the dendritic tree, activates voltagegated conductances there, and results in a slower dendritic spike. The latter depolarizes
the soma and produces a noticeable ADP. Recordings of another neuron in Fig.7.52b
and c show that if there is an additional source of depolarization, such as the injected
DC current, the ADPs can grow and result in a second spike. This may evoke anotherdendritic spike, another ADP or spike, and so on. Such a somatic-dendritic ping-pong(Wang 1999; Doiron et al. 2002) results in a bursting activity discussed in section 8.2.2.
In contrast, adult CA1 pyramidal neurons in hippocampus generate ADP and bursting even when their apical dendrites are cut (Yue et al. 2005; Golomb et al. 2006).
The ADP there is caused by the slow deactivation of somatic persistent Na+ current.
Slow ADPs can also be generated by a nonlinear interplay of fast currents responsible for spiking, rather than by slow currents or dendritic spikes. One obvious example
is the damped oscillation of membrane potential of the INa,p +IK -model in Fig.7.53 immediately after the spike, with the trough and the peak corresponding to an AHP and
an ADP, respectively. Note that the duration of the ADP is ten times the duration of
the spike even though the model does not have any slow currents. Such a long-lasting
effect appears because the trajectory follows the separatrix, comes close to the saddle
point, and spends some time there before returning to the stable resting state.
An example in Fig.7.54 shows the membrane potential of a model neuron slowly
passing through a saddle-node on invariant circle bifurcation. Because the vector fieldis small at the bifurcation, which takes place around t = 70 ms, the membrane potential is slowly increasing along the limit cycle and then slowly decreasing along thelocus of stable node equilibria, thereby producing a slow ADP. In chapter 9 we willshow that such ADPs exist in 4 out of 16 basic types of bursting neurons, including
thalamocortical relay neurons and R15 bursting cells in the abdominal ganglion of themollusk Aplysia.

263

membrane potential, V (mV)

n-n
ullc
l
saddle

0.1

ADP
n
V-

lc

ul

K+ activation gate, nine

Excitability

li n

e

0.05 separ a tri x
-65

0
-20
ADP
-40
-60
-80

-60
-55
-50
-45
membrane potential, V (mV)

0

10

20
time (ms)30

40

Figure 7.53: A long afterdepolarization (ADP) in the INa,p +IK -model without any slow
currents. Parameters are as in Fig.6.52.

-10saddle-node on invariant circle
bifurcation-20

saddleinvariant circle

membrane potential, V (mV)

0

-30

ADP

-40

node

-50
-60
-70
-80

0

20

40

60

80

100
time (ms)

120

140

160

180

200

Figure 7.54: Afterdepolarization in the INa,p +IK -model passing slowly through saddlenode on invariant circle bifurcation, as the magnitude of the injected current ramps
down.

264

Excitability

Review of Important Concepts• A neuron is excitable because, as a dynamical system, it is near abifurcation from resting to spiking activity.
• The type of bifurcation determines the neuron’s computational properties, summarized in Fig.7.15.
• Saddle-node on invariant circle bifurcation results in Class 1 excitability: the neuron can fire with arbitrarily small frequency andencode the strength of input into the firing rate.
• Saddle-node off invariant circle and Andronov-Hopf bifurcations result in Class 2 excitability: the neuron can fire only within a certain
frequency range.
• Neurons near saddle-node bifurcation are integrators: they preferhigh-frequency excitatory input, have well-defined thresholds, andfire all-or-none spikes with some latencies.
• Neurons near Andronov-Hopf bifurcation are resonators: they have
oscillatory potentials, prefer resonant-frequency input, and can easily fire postinhibitory spikes.

Bibliographical Notes
There is no universally accepted definition of excitability. Our definition is consistent
with the one involving ε-pseudo orbits (Izhikevich 2000a). FitzHugh (1955, 1960, 1976)
pioneered geometrical analyses of phase portraits of neuronal models with the view tounderstanding their neurocomputational properties. It is amazing that such important neurocomputational properties as all-or-none action potentials, firing thresholds,
and integration of EPSPs are still introduced and illustrated using the Hodgkin- Huxley model, which according to FitzHugh, cannot have these properties. Throughout,this chapter follows Izhikevich (2000a) to compare and contrast neurocomputational
properties of integrators and resonators.
The frozen noise experiment in Fig.7.24 was pioneered by Bryant and Segundo(1976), but due to an interesting quirk of history, it is better known at present as theMainen-Sejnowski (1995) experiment (despite the fact that the latter paper refers to theformer). Postinhibitory facilitation was pointed out by Luk and Aihara (2000) and laterby Izhikevich (2001). John Rinzel suggested calling it “postinhibitory exaltation” (in asimilar vein; the phenomenon in Fig.7.49b may be called “post-excitatory hesitation”).

Excitability

265

Richardson et al. (2003) pointed out that frequency preference and resonance may
occur without subthreshold oscillations when the system is near the transition from an
integrator to a resonator.
The Hodgkin classification of neuronal excitability can be used to classify any rhythmic system, even contractions of the uterus during labor. Typically, the contractions
start with low frequency that gradually increases – Class 1 excitability. My wife had
to have labor induced pharmacologically, a typical medical intervention when the baby
is overdue. The contraction monitor showed a sinusoidal signal with constant period,
around 2 minutes, but slowly growing amplitude – Class 2 excitability via supercritical
Andronov-Hopf bifurcation! Since my wife has an advanced degree in applied mathematics, I waited for a 1-minute period of quiescence between the contractions andmanaged to explain to her the basic relationship between bifurcations and excitability.
Five years later, induced delivery of our second daughter resulted in the same supercritical Andronov-Hopf bifurcation. I recalled this for my wife and explained it to the
obstetrician minutes after the delivery.

Exercises
1. When can the FitzHugh-Nagumo model (4.11, 4.12) exhibit inhibition-induced
spiking, such as in Fig.7.32?
2. (Canards) Numerically investigate the quasi-threshold in the FitzHugh-Nagumo
model (4.11, 4.12). How is it related to the canard (French duck; see Eckhaus
1983) limit cycles discussed in section 6.3.4?
3. (Noise-induced oscillations) Consider the systemż = (−ε + iω)z + εI(t) ,

z∈C

(7.3)

which has a stable focus equilibrium z = 0 and is subject to a weak noisy input
εI(t). Show that the system exhibits sustained noisy oscillations with an average
amplitude |I ∗ (ω)|, where1
I (ω) = lim
T →∞ T∗Z T

e−iωt I(t) dt

0

is the Fourier coefficient of I(t) corresponding to the frequency ω.
4. (Frequency preference) Show that a system exhibiting damped oscillation withfrequency ω is sensitive to an input having frequency ω in its power spectrum.
(Hint: use exercise 3.)
5. (Rush and Rinzel 1995) Use the phase portrait of the reduced Hodgkin-Huxleymodel in Fig.5.21 to explain some small but noticeable latencies in Fig.7.26.

266

Excitability

Figure 7.55: exercise 6: Zero frequency firing near subcritical Andronov-Hopf bifurcation in the INa,p +IK -model with parameters as in Fig.6.16 and a high-threshold slow
K+ current (gK,slow = 25, τK,slow = 10 ms, n∞,slow (V ) has V1/2 = −10 mV and k = 5mV.)
6. The neuronal model in Fig.7.55 has a high-threshold slow persistent K+ current.
Its resting state undergoes a subcritical Andronov-Hopf bifurcation, yet it canfire low-frequency spikes, and hence exhibits Class 1 excitability. Explain. (Hint:
Show numerically that the model is near a certain codimension-2 bifurcation
involving a homoclinic orbit.)
7. Show that the resting state of a Class 3 excitable conductance-based model isnear an Andronov-Hopf bifurcation if some other variable, not I, is used as abifurcation parameter.

Chapter 8
Simple Models
The advantage of using conductance-based models, such as the INa +IK -model, is thateach variable and parameter has a well-defined biophysical meaning. In particular,
they can be measured experimentally. The drawback is that the measurement procedures may not be accurate: the parameters are usually measured in different neurons,
averaged, and then fine-tuned (a fancy term meaning “to make arbitrary choices”). As
a result, the model does not have the behavior that one sees in experiments. And even
if it “looks” right, there is no guarantee that the model is accurate from the dynamicalsystems point of view, that is, it exhibits the same kind of bifurcations as the type ofneuron under consideration.
Sometimes we do not need or cannot afford to have a biophysically detailed conductance-based model. Instead, we want a simple model that faithfully reproduces all theneurocomputational features of the neuron. In this chapter we review salient featuresof cortical, thalamic, hippocampal, and other neurons, and we present simple models
that capture the essence of their behavior from the dynamical systems point of view.

8.1

Simplest Models

Let us start with reviewing the simplest possible models of neurons. As one canguess from their names, the integrate-and-fire and resonate-and-fire neurons capture
the essence of integrators and resonators, respectively. The models are similar in many
respects: both are described by linear differential equations, both have a hard firing
threshold and a reset, and both have a unique stable equilibrium at rest. The only
difference is that the equilibrium is a node in the integrate-and-fire case, but a focus
in the resonate-and-fire case. One can model the former using only one equation, andthe latter using only two equations, though multi-dimensional extensions are straightforward. Both models are useful from the analytical point of view, that is, to prove
theorems.
Many scientists, including myself, refer to these neural models as “spiking models”.
The models have a threshold, but they lack any spike generation mechanism, that is,they cannot produce a brief regenerative depolarization of membrane potential corre267

268

Simple Models

membrane potential, V

spike

EthreshEleak
EK

Figure 8.1: Leaky integrateand-fire neuron with noisy
input. The spike is added
manually for aesthetic purposes and to fool the reader
into believing that this is a
spiking neuron.

time

sponding to the spike upstroke. Therefore, they are not spiking models; the spikes infigures 8.1 and 8.2, as well as in hundreds of scientific papers devoted to these models, are drawn by hand. The quadratic integrate-and-fire model is the simplest truly
spiking model.

8.1.1

Integrate-and-FireThe leaky integrate-and-fire model (Lapicque 1907; Stein 1967; Tuckwell 1988) is an
idealization of a neuron having Ohmic leakage current and a number of voltage-gatedcurrents that are completely deactivated at rest. Subthreshold behavior of such aneuron can be described by the linear differential equation
Ohmic leakage}|{z
C V̇ = I − gleak (V − Eleak ),where all parameters have the same biophysical meanings as in the previous chapters. When the membrane potential V reaches the threshold value Ethresh , the voltagesensitive currents instantaneously activate, the neuron is said to fire an action potential,
and V is reset to EK , as in Fig.8.1. After appropriate rescaling, the leaky integrateand-fire model can be written in the formv̇ = b − v ,if v = 1, then v ← 0,

(8.1)

where the resting state is v = b, the threshold value is v = 1, and the reset value isv = 0. Apparently the neuron is excitable when b < 1 and fires a periodic spike train
when b > 1 with period T = − ln(1 − 1/b). (The reader should verify this.)
The integrate-and-fire neuron illustrates a number of important neurocomputational properties:• All-or-none spikes. Since the shape of the spike is not simulated, all spikes are
implicitly assumed to be identical in size and duration.
• Well-defined threshold. A stereotypical spike is fired as soon as V = Ethresh ,leaving no room for any ambiguity (see, however, exercise 1).

Simple Models

269• Relative refractory period. When EK < Eleak , the neuron is less excitable immediately after the spike.
• Distinction between excitation and inhibition. Excitatory inputs (I > 0) bringthe membrane potential closer to the threshold, and hence facilitate firing, while
inhibitory inputs (I < 0) do the opposite.
• Class 1 excitability. The neuron can continuously encode the strength of an input
into the frequency of spiking.
In summary, the neuron seems to be a good model for an integrator.
However, a closer look reveals that the integrate-and-fire neuron has flaws. The
transition from resting to repetitive spiking occurs neither via saddle-node nor via
Andronov-Hopf bifurcation, but via some other weird type of bifurcation that can beobserved only in piecewise continuous systems. As a result, the F-I curve has logarithmic scaling and not the expected square-root scaling of a typical Class 1 excitablesystem (see, however, exercise 19 in chapter 6). The integrate-and-fire model cannot
have spike latency to a transient input because superthreshold stimuli evoke immediate spikes without any delays (compare with Fig.8.8(I)). In addition, the model hassome weird mathematical properties, such as non-uniqueness of solutions, as we show
in exercise 1. Finally, the integrate-and-fire model is not a spiking model. Technically,
it did not fire a spike in Fig.8.1, it was only “said to fire a spike”, which was manually
added afterward to fool the reader.
Despite all these drawbacks, the integrate-and-fire model is an acceptable sacrifice
for a mathematician who wants to prove theorems and derive analytical expressions.
However, using this model might be a waste of time for a computational neuroscientist
who wants to simulate large-scale networks. At the end of this section we present
alternative models that are as computationally efficient as the integrate-and-fire neuron,
yet as biophysically plausible as Hodgkin-Huxley-type models.

8.1.2

Resonate-and-Fire

The resonate-and-fire model is a two-dimensional extension of the integrate-and-firemodel that incorporates an additional low-threshold persistent K+ current or h-current,or any other resonant current that is partially activated at rest. Let W denote themagnitude of such a current. In the linear approximation, the conductance-based
equations describing neuronal dynamics can be written in the form
C V̇ = I − gleak (V − Eleak ) − W ,Ẇ = (V − V1/2 )/k − W .
known as the Young (1937) model (see also equation 2-1 in FitzHugh 1969). Whenever
the membrane potential reaches the threshold value, Vthresh , the neuron is said to firea spike. Young did not specify what happens after the spike. The resonate-and-fire

270

Simple Models

1thresholdyyzreset

0
1
0
x

-1
-1

0

1

0

20

40

60
time, ms

80

100

Figure 8.2: Resonate-and-fire model with b = −0.05, ω = 0.25, and zreset = i. The
spike was added manually.
model is the Young model with the following resetting: if V ≥ Vthresh , then V ← Vreset
and W ← Wreset , where Vrest and Wreset are some parameters.
When the resting state is a stable focus, the model can be recast in complex coordinates asż = (b + iω)z + I ,where b + iω ∈ C is the complex eigenvalue of the resting state, and z = x + iy ∈ C isthe complex-valued variable describing damped oscillations with frequency ω aroundthe resting state. The real part, x, is a current-like variable. It describes the dynamics
of the resonant current and synaptic currents. The imaginary part, y, is a voltage-likevariable. The neuron is said to fire a spike when y reaches the threshold y = 1. Thus,
the threshold is a horizontal line on the complex plane that passes through i ∈ C, asin Fig.8.2, though other choices are also possible. After firing the spike, the variable z
is reset to zreset .
The resonate-and-fire model illustrates the most important features of resonators:
damped oscillations, frequency preference, postinhibitory (rebound) spikes, and Class 2
excitability. It cannot have sustained subthreshold oscillations of membrane potential.
Integrate-and-fire and resonate-and-fire neurons do not contradict, but complement,each other. Both are linear, and hence are useful when we prove theorems and derive
analytical expressions. They have the same flaws limiting their applicability, whichwere discussed earlier. In contrast, two simple models described below are difficult totreat analytically, but because of their universality they should be the models of choice
when large-scale simulations are concerned.

8.1.3

Quadratic Integrate-and-Fire

Replacing −v with +v 2 in (8.1) results in the quadratic integrate-and-fire modelv̇ = b + v 2 ,if v = vpeak , then v ← vreset ,(8.2)

which we considered in section 3.3.8. Here vpeak is not a threshold, but the peak (cutoff) of a spike, as we explain below. It is useful to use vpeak = +∞ in analytical studies.

Simple Models

271

vreset =|b|1/2
sadd
bistability
le
ho
mo
cli
ni
c
or
tion
rca
ifu

parameter vresettbbivresetvresetvreset0

excitablevresetvresetsaddle-node on invariant
circle bifurcation

saddle-node homoclinic
orbit bifurcation
vreset-1-1

saddle-node bifurcation

1

0
parameter b

periodic

Figure 8.3: Bifurcation
diagram of the quadratic
integrate-and-fire neuron (8.2).

vreset1

In simulations, the peak value is assumed to be large but finite, so it can be normalized
to vpeak = 1.
Note that v̇ = b + v 2 is a topological normal form for the saddle-node bifurcation.
That is, it describes dynamics of any Hodgkin-Huxley-type system near that bifurcation, as we discuss in chapter 3 and 6. There we derived the normal form (6.3) for theINa,p +IK -model and showed that the two systems agree quantitatively in a reasonably
broad voltage range. By resetting v to vreset , the quadratic integrate-and-fire model
captures the essence of recurrence when the saddle-node bifurcation is on an invariant
circle.
When b > 0, the right-hand side of the model is strictly positive, and the neuron
fires a periodic train of action potentials. Indeed, v increases, reaches the peak, resets
to vreset , and then increases again, as we show in Fig.3.35 (top). In exercise 3 we provethat the period of such spiking activity is


vpeak
vresetπ1atan √ − atan √
<√ ,
T =√
b
b
b
b
√
so that the frequency scales as b, as in Class 1 excitable systems.
pWhen b < 0, the parabola b + v 2 has two zeroes, ± |b|. One corresponds tothe stable node equilibrium (resting state), and the other corresponds to the unstablenode (threshold state); see exercise 2. Subthreshold perturbations are those that keep
v below the unstable node. Superthreshold perturbations are those that push v beyond
the unstable node, resulting in the initiation of an action potential, reaching the peak272

Simple Models
1

0.2

u-nullcline
V-nu
llclinerecovery variable, u

0.8

0.6

0.4

0.1
0.2

0

-80

-60
-40
-20
0
membrane potential, V (mV)

20

a

0
-70

-60
-50
-40
membrane potential, V (mV)b

Figure 8.4: Phase portrait (a) and its magnification (b) of a typical neuronal model
having voltage variable V and a recovery variable u.
pvalue vpeak , and then resetting to vreset . If, in addition, vreset > |b|, then there is acoexistence of resting and periodic spiking states, as in Fig.3.35 (middle). The periodof the spiking state is provided in exercise 4. A two-parameter bifurcation diagram of(8.2) is depicted in Fig.8.3.
Unlike its linear predecessor, the quadratic integrate-and-fire neuron is a genuine
integrator. It exhibits saddle-node bifurcation; it has a soft (dynamic) threshold; andit generates spikes with latencies, as many mammalian cells do. Besides, the modelis canonical in the sense that the entire class of neuronal models near saddle-node on
invariant circle bifurcation can be transformed into this model by a piecewise continuous change of variables (see section 8.1.5 and the Ermentrout-Kopell theorem in
Hoppensteadt and Izhikevich 1997). In conclusion, the quadratic, and not the leaky,
integrate-and-fire neuron should be used in simulations of large-scale networks of integrators. A generalization of this model is discussed next.

8.1.4

Simple Model of Choice

A striking similarity among many spiking models, discussed in chapter 5, is that they
can be reduced to two-dimensional systems having a fast voltage variable and a slower“recovery” variable, which may describe activation of the K+ current or inactivationof the Na+ current or their combination. Typically, the fast variable has an N-shaped
nullcline and the slower variable has a sigmoid-shaped nullcline. The resting state insuch models is the intersection of the nullclines near the left knee, as we illustrate
in Fig.8.4a. There, V and u denote the fast and the slow variable, respectively. In
chapter 7 we showed that many computational properties of biological neurons can be
explained by considering dynamics at the left knee.
In section 5.2.4 we derive a simple model that captures the subthreshold behavior

Simple Models

273
integrator
thresholdmanifoldresonator
u=I+v2

u=bv

u=I+v2

0.025
small
EPSP

u=bv
spike

0

recovery variable, u

0.1

spike
0

small
EPSP

threshold
set

-0.1
-0.2

-0.1
0
0.1
0.2
membrane potential, v

0.3

-0.5

-0.25
0
0.25
0.5
membrane potential, v

0.75

Figure 8.5: The simple model (8.3, 8.4) can be an integrator or a resonator. Comparewith Fig.7.27.
in a small neighborhood of the left knee confined to the shaded square in Fig.8.4 andthe initial segment of the upstroke of an action potential. In many cases, especiallyinvolving large-scale simulations of spiking models, the shape of the action potential isless important than the subthreshold dynamics leading to this action potential. Thus,
retaining detailed information about the left knee and its neighborhood and simplifying
the vector field outside the neighborhood is justified.
The simple modelv̇ = I + v 2 − uu̇ = a(bv − u)if v ≥ 1, thenv ← c, u ← u + d

(8.3)
(8.4)

has only four dimensionless parameters. Depending on the values of a and b, it canbe an integrator or a resonator, as we illustrate in Fig.8.5. The parameters c and d
do not affect steady-state subthreshold behavior. Instead, they take into account theaction of high-threshold voltage-gated currents activated during the spike, and affectonly the after-spike transient behavior. If there are many
P currents with diverse timescales, then u, a, b, and d are vectors, and (8.3) contains
u instead of u.
The simple model may be treated as a quadratic integrate-and-fire neuron withadaptation in the simplest case b = 0. When b < 0, the model can be treated as aquadratic integrate-and-fire neuron with a passive dendritic compartment (see exercise 10). When b > 0, the connection to the quadratic integrate-and-fire neuron is lost,
and the simple model represents a novel class of spiking models.
In the rest of this chapter we tune the simple model to reproduce spiking andbursting behavior of many known types of neurons. It is convenient to use it in theform
C v̇ = k(v − vr )(v − vt ) − u + I
u̇ = a{b(v − vr ) − u}if v ≥ vpeak , thenv ← c, u ← u + d

(8.5)
(8.6)

274

Simple Models

where v is the membrane potential, u is the recovery current, C is the membrane
capacitance, vr is the resting membrane potential, and vt is the instantaneous threshold
potential. Though the model seems to have ten parameters, but it is equivalent to (8.3,
8.4) and hence has only four independent parameters. As we described in section 5.2.4,
the parameters k and b can be found when one knows the neuron’s rheobase and input
resistance. The sum of all slow currents that modulate the spike generation mechanism
is combined in the phenomenological variable u with outward currents taken with theplus sign.
The sign of b determines whether u is an amplifying (b < 0) or a resonant (b > 0)
variable. In the latter case, the neuron sags in response to hyperpolarized pulses ofcurrent, peaks in response to depolarized subthreshold pulses, and produces rebound(postinhibitory) responses. The recovery time constant is a. The spike cutoff value isvpeak , and the voltage reset value is c. The parameter d describes the total amount ofoutward minus inward currents activated during the spike and affecting the after-spikebehavior. All these parameters can easily be fitted to any particular neuron type, as
we show in subsequent sections.
Implementation and Phase Portrait
The following MATLAB code simulates the model and produces Fig.8.6a.
C=100; vr=-60; vt=-40; k=0.7;
a=0.03; b=-2; c=-50; d=100;
vpeak=35;% parameters used for RS% neocortical pyramidal neurons% spike cutoff

T=1000; tau=1;% time span and step (ms)n=round(T/tau);% number of simulation steps
v=vr*ones(1,n); u=0*v;% initial values
I=[zeros(1,0.1*n),70*ones(1,0.9*n)];% pulse of input DC current
for i=1:n-1% forward Euler methodv(i+1)=v(i)+tau*(k*(v(i)-vr)*(v(i)-vt)-u(i)+I(i))/C;u(i+1)=u(i)+tau*a*(b*(v(i)-vr)-u(i));
if v(i+1)>=vpeak% a spike is fired!
v(i)=vpeak;% padding the spike amplitudev(i+1)=c;% membrane voltage resetu(i+1)=u(i+1)+d;% recovery variable update
end;
end;
plot(tau*(1:n), v);% plot the result

Note that the spikes were padded to vpeak to avoid amplitude jitter associated withthe finite simulation time step tau=1 ms. In Fig.8.6b we magnify the simulated voltage
trace and compare it with a recording of a neocortical pyramidal neuron (dashed curve).
There are two discrepancies, marked by arrows: In the first, the pyramidal neuron has(1) a sharper spike upstroke and (2) a smoother spike downstroke. The first discrepancy

membrane potential, v (mV)

Simple Models
40

275

(a)

vpeak

(b)

20
20 ms

0
upstroke-20

downstroke(reset)-40
vreset

-60
0

200

I=0

400
600
time (ms)

800

AHP

1000

I=70

(c)

AHP
0

resting

u-n
u

llcl

-50

-60
vr

v-nullcline

recovery variable, u

50
after

-spik

set

spiking limit cycle attractor

ine

vreset

e re

-40
vt

-20

0

membrane potential, v (mV)

20

35
vpeak

Figure 8.6: (a) Output of the MATLAB code simulating the simple model (8.5, 8.6).
(b) Comparison of the simulated (solid curve) and experimental (dashed curve) voltage
traces shows two major discrepancies, marked by arrows. (c) Phase portrait of themodel.
can be removed by assuming that the coefficient k of the square polynomial in (8.5)
is voltage-dependent (e.g., k = 0.7 for v ≤ vt and k = 7 for v > vt ), or by using themodification of the simple model presented in exercise 13 and exercise 17. The second
discrepancy results from the instantaneous after-spike resetting, and it less importantbecause it does not affect the decision whether or when to fire. However, the slopeof the downstroke may become important in studies of gap-junction-coupled spiking
neurons.
The phase portrait of the simple model is depicted in Fig.8.6c. Injection of the stepof DC current I = 70 pA shifts the v-nullcline (square parabola) upward and makes
the resting state, denoted by a black square, disappear. The trajectory approaches the
spiking limit cycle attractor, and when it crosses the cutoff vertical line vpeak = 35 mV,
it is reset to the white square, resulting in periodic spiking behavior. Note the slow
afterhyperpolarization (AHP) following the reset that is due to the dynamics of therecovery variable u. Depending on the parameters, the model can have other types ofphase portraits, spiking, and bursting behavior, as we demonstrate later.
In Fig.8.7 we illustrate the difference between the integrate-and-fire neuron and thesimple model. The integrate-and-fire model is said to fire spikes when the membranepotential reaches a preset threshold value. The potential is reset to a new value, and276

Simple Models
simple model

integrate-and-fire model

spikesare drawn
by handthresholdresting

reset
input

spike cutoff(not a threshold)spikes are
generated

reset

threshold ?
resting
input

Figure 8.7: Voltage reset in the integrate-and-fire model and in the simple model.

the spikes are drawn by hand. In contrast, the simple model generates the upstrokeof the spike due to the intrinsic (regenerative) properties of the voltage equation. The
voltage reset occurs not at the threshold, but at the peak, of the spike. In fact, thefiring threshold in the simple model is not a parameter, but a property of the bifurcation
mechanism of excitability. Depending on the bifurcation of equilibrium, the model may
not even have a well-defined threshold, a situation similar to many conductance-basedmodels.
When numerically implementing the voltage reset, whether at the threshold or at
the peak of the spike, one needs to be aware of the numerical errors, which translate
into the errors of spike timing. These errors are inversely proportional to the slope ofthe voltage trace (i.e. v̇) at the reset value. The slope is small in the integrate-and-firemodel, so clever numerical methods are needed to catch the exact moment of thresholdcrossing (Hansel et al. 1998). In contrast, the slope is nearly infinite in the simple
model, hence the error is infinitesimal, and no special methods are needed to identify
the peak of the spike.
In Fig.8.8 we used the model to reproduce the 20 of the most fundamental neurocomputational properties of biological neurons. Let us check that the model is thesimplest possible system that can exhibit the kind of behavior in the figure. Indeed, it
has only one nonlinear term, v 2 . Removing the term makes the model linear (between
the spikes) and equivalent to the resonate-and-fire neuron. Removing the recovery
variable u makes the model equivalent to the quadratic integrate-and-fire neuron withall its limitations, such as the inability to burst or to be a resonator. In summary, we
found the simplest possible model capable of spiking, bursting, being an integrator ora resonator, and it should be the model of choice in simulations of large-scale networksof spiking neurons.

Simple Models(A) tonic spiking

277(B) phasic spiking(C) tonic bursting(D) phasic bursting(E) mixed mode(F) spike frequency
adaptation(G) Class 1 excitable(H) Class 2 excitable(I) spike latency(J) subthresholdoscillations(K) resonator(L) integrator(M) rebound spike(N) rebound burst(O) thresholdvariability(P) bistability(Q) depolarizing
after-potential(R) accommodation(S) inhibition-induced
spiking(T) inhibition-induced
bursting

input dc-current
20 ms

DAP

Figure 8.8: Summary of neurocomputational properties exhibited by the simple model;
see exercise 11. The figure is reproduced, with permission, from www.izhikevich.com.
(An electronic version of the figure, the MATLAB code that generates the voltage
responses, and reproduction permissions are available at www.izhikevich.com.)278

Simple Models

Figure 8.9: A real conversation between the author of this book and his boss.

8.1.5

Canonical Models

It is quite rare, if ever possible, to know precisely the parameters describing dynamics
of a neuron (many erroneously think that the Hodgkin-Huxley model of the squidaxon is an exception). Indeed, even if all ionic channels expressed by the neuron are
known, the parameters describing their kinetics are usually obtained via averaging overmany neurons; there are measurement errors; the parameters change slowly, and soon. Thus, we are forced to consider families of neuronal models with free parameters(e.g. the family of INa +IK -models). It is more productive, from the computational
neuroscience point of view, to consider families of neuronal models having a common
property, e.g., the family of all integrators, the family of all resonators, or the familyof “fold/homoclinic” bursters considered in the next chapter. How can we study the
behavior of the entire family of neuronal models if we have no information about mostof its members?
The canonical model approach addresses this issue. Briefly, a model is canonical fora family if there is a piecewise continuous change of variables that transforms any modelfrom the family into this one, as we illustrate in Fig.8.10. The change of variables does
not have to be invertible, so the canonical model is usually lower-dimensional, simple,
and tractable. Nevertheless, it retains many important features of the family. For
example, if the canonical model has multiple attractors, then each member of thefamily has multiple attractors. If the canonical model has a periodic solution, theneach member of the family has a periodic (quasi-periodic or chaotic) solution. If thecanonical model can burst, then each member of the family can burst. The advantageof this approach is that we can study universal neurocomputational properties thatare shared by all members of the family because all such members can be put into thecanonical form by a change of variables. Moreover, we need not actually present suchSimple Models

279

neuronfamily of models

x=f1(x)h1

x=f2(x)x=f3(x)h2h3

x=f4(x)h4

canonical

y=g(y) model

Figure 8.10: Dynamical system ẏ = g(y) is a canonical model for the family{f1 , f2 , f3 , f4 } of neural models ẋ = f (x) because each such model can be transformed
into the form ẏ = g(y) by the piecewise continuous change of variables hi .

a change of variables explicitly, so derivation of canonical models is possible even when
the family is so broad that most of its members are given implicitly (e.g., the familyof “all resonators”).
The process of deriving canonical models is more an art than a science, since a
general algorithm for doing this is not known. However, much success has been achievedin some important cases. The canonical model for a system near an equilibrium isthe topological normal form at the equilibrium (Kuznetsov 1995). Such a canonicalmodel is local, but it can be extended to describe global dynamics. For example, thequadratic integrate-and-fire model with a fixed vreset < 0 is a global canonical model
for all Class 1 excitable systems, that is, systems near saddle-node on invariant circle
bifurcation. The same model with variable vreset is a global canonical model for all
systems near saddle-node homoclinic orbit bifurcation (considered in section 6.3.6).
The phase model ϑ̇ = 1 derived in chapter 10 is a global canonical model for thefamily of nonlinear oscillators having exponentially stable limit cycle attractors. Other
examples of canonical models for spiking and bursting can be found in subsequent
chapters of this book.
The vector-field of excitable conductance-based models in the subthreshold region
and in the region corresponding to the upstroke of the spike can be converted into thesimple form (8.3, 8.4), possibly with u being a vector. Therefore, the simple model(8.3, 8.4) is a local canonical model for the spike generation mechanism and the spike
upstroke of Hodgkin-Huxley-type neuronal models. It is not a global canonical model
because it ignores the spike downstroke. Nevertheless, it describes remarkably well the
spiking and bursting dynamics of many biological neurons, as we demonstrate next.

280

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excitatory
regular spiking (RS)chattering (CH)

300 pA

600 pA

100 pA

300 pA

35 pA

200 pAintrinsically bursting (IB)50 mV
100 ms600 pA

500 pA

400 pA

inhibitory
fast spiking (FS)

low threshold spiking (LTS)

500 pA

200 pA

100 pA

300 pAlate spiking (LS)400 pA

175 pA

150 pA

delay

Figure 8.11: The six most fundamental classes of firing patterns of neocortical neurons
in response to pulses of depolarizing DC current. RS and IB are in vitro recordingsof pyramidal neurons of layer 5 of primary visual cortex of a rat; CH was recorded invivo in cat visual cortex (area 17; data provided by D. McCormick). FS was recordedin vitro in rat primary visual cortex, LTS was recorded in vitro in layer 4 or 6 of rat
barrel cortex (data provided by B. Connors). LS was recorded in layer 1 of rat’s visualcortex (data provided by S. Hestrin). All recordings are plotted on the same voltage
and time scale, and the data are available at www.izhikevich.com.

Simple Models

8.2

281

CortexIn this section we consider the six most fundamental classes of firing patterns observedin the mammalian neocortex and depicted in Fig.8.11 (Connors and Gutnick 1990;
Gray and McCormick 1996; Gibson et al. 1999). Though most biologists agree withthe classification in the figure, many would point out that it is greatly oversimplified(Markram et al. 2004), that the distinction between the classes is not sharp, that there
are subclasses within each class (Nowak et al. 2003; Toledo-Rodriguez et al. 2004), andthat neurons can change their firing class depending on the state of the brain (Steriade2004).
• (RS) Regular spiking neurons fire tonic spikes with adapting (decreasing) frequency in response to injected pulses of DC current. Most of them have Class 1
excitability in the sense that the interspike frequency vanishes when the amplitude of the injected current decreases. Morphologically, these neurons are spiny
stellate cells in layer 4 and pyramidal cells in layers 2, 3, 5, and 6.
• (IB) Intrinsically bursting neurons generate a burst of spikes at the beginningof a strong depolarizing pulse of current, then switch to tonic spiking mode.
They are excitatory pyramidal neurons found in all cortical layers, but are mostabundant in layer 5.
• (CH) Chattering neurons fire high-frequency bursts of spikes with relatively shortinterburst periods; hence another name, FRB (fast rhythmic bursting). Output
of such a cell fed to the loudspeaker “sounds a lot like a helicopter – cha, cha,cha – real fast”, according to Gray and McCormick (1996). CH neurons were
found in visual cortex of adult cats, and morphologically they are spiny stellate
or pyramidal neurons of layers 2 - 4, mainly layer 3.
• (FS) Fast spiking interneurons fire high-frequency tonic spikes with relatively
constant period. They exhibit Class 2 excitability (Tateno et al. 2004). When themagnitude of the injected current decreases below a certain critical value, they
fire irregular spikes, switching randomly between resting and spiking states. Morphologically, FS neurons are sparsely spiny or aspiny nonpyramidal cells (basket
or chandelier; see Kawaguchi and Kubota 1997) providing local inhibition along
the horizontal (intra-laminar) direction of the neocortex (Bacci et al. 2003).
• (LTS) Low-threshold spiking neurons fire tonic spikes with pronounced spikefrequency adaptation and rebound (postinhibitory) spikes, often called “lowthreshold spikes” by biologists (hence the name). They seem to be able to fire
low-frequency spike trains, though their excitability class has not yet been determined. Morphologically, LTS neurons are nonpyramidal interneurons providing
local inhibition along the vertical (inter-laminar) direction of the neocortex (Bacci
et al. 2003).

282

Simple Models• (LS) Late spiking neurons exhibit voltage ramp in response to injected DC
current near the rheobase, resulting in delayed spiking with latencies as longas 1 sec. There is a pronounced subthreshold oscillation during the ramp, but
the discharge frequency is far less than that of FS neurons. Morphologically,
LS neurons are nonpyramidal interneurons (neurogliaform; see Kawaguchi andKubota 1997) found in all layers of neocortex (Kawaguchi 1995), especially inlayer 1 (Chu et al. 2003).

Our goal is to use the simple model (8.5, 8.6) presented in section 8.1.4 to reproduce
each of the firing types. We want to capture the dynamic mechanism of spike generationof each neuron, so that the model reproduces the correct responses to many types of
the inputs, and not only to the pulses of DC current. We strive to have not only
qualitative but also quantitative agreement with the published data on neurons’ resting
potential, input resistance, rheobase, F-I behavior, the shape of the upstroke of theaction potential, and so on, though this is impossible in many cases, mostly because
the data are contradictory. To fine-tune the model, we use recordings of real neurons.
We consider the tuning successful when the quantitative difference between simulated
and recorded responses is smaller than the difference between the responses of two“sister” neurons recorded in the same slice. We do not want to claim that the simple
model explains the mechanism of generation of any of the firing patterns recorded inreal neurons (simply because the mechanism is usually not known). Although in many
instances we must resist the temptation to use the Wolfram (2002) new-kind-of-science
criterion: “If it looks the same, it must be the same”.

8.2.1

Regular Spiking (RS) Neurons

Regular spiking neurons are the major class of excitatory neurons in the neocortex.
Many are Class 1 excitable, as we show in Fig.7.3, using in vitro recordings of a layer
5 pyramidal cell of rat primary visual cortex (see also Tateno et al. 2004). RS neurons
have a transient K+ current IA , whose slow inactivation delays the onset of the first
spike and increases the interspike period, and a persistent K+ current IM , which isbelieved to be responsible for the spike frequency adaptation seen in Fig.7.43. Let us
use the simple model (8.5, 8.6) to capture qualitative and some quantitative featuresof typical RS neurons.
We assume that the resting membrane potential is vr = −60 mV and the instantaneous threshold potential is vt = −40 mV; that is, instantaneous depolarizations above−40 mV cause the neuron to fire, as in Fig.3.15. Assuming that the rheobase is 50 pA
and the input resistance is 80 MΩ, we find k = 0.7 and b = −2. We take the membrane
capacitance C = 100 pF, which yields a membrane time constant of 8 ms.
Since b < 0, depolarizations of v decrease u as if the major slow current is theinactivating K+ current IA . The inactivation time constant of IA is around 30 ms inthe subthreshold voltage range; hence one takes a = 0.03 ≈ 1/30. The membranepotential of a typical RS neuron reaches the peak value vpeak = +35 mV during a
spike (the precise value has little effect on dynamics) and then repolarizes to c = −50

Simple Models

283

layer 5 neuron

simple model
I=100 pA

I=85 pA

I=70 pA+35 mV100

200 msreset-60 mV
AHP
input

input

recovery, u

reset

I>0
I=0

I=60 pA

50
AHP
0

-50

rest
-60

spike
-40
-20
0
20
membrane potential, v (mV)

35

Figure 8.12: Comparison of in vitro recordings of a regular spiking (RS) pyramidal
neuron with simulations of the simple model 100v̇ = 0.7(v + 60)(v + 40) − u + I,u̇ = 0.03{−2(v + 60) − u}, if v ≥ +35, then v ← −50, u ← u + 100.
mV or below, depending on the firing frequency. The parameter d describes the total
amount of outward minus inward currents activated during the spike and affecting theafter-spike behavior. Trying different values, we find that d = 100 gives a reasonable
F-I relationship, at least in the low-frequency range.
As follows from exercise 10, one can also interpret u as the membrane potential ofa passive dendritic compartment, taken with the minus sign. Thus, when b < 0, thevariable u represents the combined action of slow inactivation of IA and slow charging
of the dendritic tree. Both processes slow the frequency of somatic spiking.
Note that we round up all the parameters, that is, we use d = 100 and not 93.27.
Nevertheless, the simulated voltage responses in Fig.8.12 agree quantitatively with thein vitro recordings of the layer 5 pyramidal neuron used in Fig.7.3. Tweaking the parameters, considering multidimensional u, or adding multiple dendritic compartments,
one can definitely improve the quantitative correspondence between the model and thein vitro data of that particular neuron, but this is not our goal here. Instead, we wantto understand the qualitative dynamics of RS neurons, using the geometry of theirphase portraits.

284

Simple Models-30 mV-30 mVS

5 mV
100 ms

P

rest

rest

b<0

b>0
u

u

I>0

I=0

I>0
I=0
P

rest
u=bv

bv

rest

u=

v

v

u

u

I>0

I=0

I>0
I=0
S

rest
u=bv

v

bv
u=

rest
v

Figure 8.13:
Two
types
of
qualitative
behaviorof
RS neurons.
Someexhibitmonotone
responses to weak
injected currents (caseb < 0); others exhibit
non-monotone overshooting
responses(case b > 0). Shownare in vitro recordingsof two RS neurons
from the same sliceof rat primary visualcortex while an automated procedure wastrying to determine
the neurons’ rheobase.
Phase portraits are
drawn by hand andillustrate a possible
dynamic mechanismof the phenomenon.

Phase Plane Analysis
Figure 8.13 shows recordings of two pyramidal RS neurons from the same slice while
an automated procedure injects pulses of DC current to determine their rheobase. The
neuron on the left exhibits monotonically increasing (ramping) or decreasing responsesof membrane potential to weak input pulses, long latencies of the first spike, and norebound spikes, whereas the neuron on the right exhibits non-monotone overshooting
responses to positive pulses, sags and rebound spikes to negative pulses (as in Fig.7.48),
relatively short latencies of the first spike, and other resonance phenomena. The evenmore extreme example in Fig.7.42 shows a pyramidal neuron executing a subthresholdoscillation before switching to a tonic spiking mode.
The difference between the types in Fig.8.13 can be explained by the sign of theparameter b in the simple model (8.5, 8.6), which depends on the relative contributionsof amplifying and resonant slow currents and gating variables. When b < 0 (or b ≈ 0;e.g., b = 0.5 in Fig.8.14), the neuron is a pure integrator near saddle-node on invariant
circle bifurcation. Greater values of b > 0 put the model near the transition from an
integrator to a resonator the via codimension-2 Bogdanov-Takens bifurcation studiedin section 6.3.3 and 7.2.11.

Simple Models

285
saddle-node on
invariant circle bifurcation

b=-2
40

I=52

I=60

ll cl
ine

I=40

v-n
u

20

0

u-n

u ll crecovery variable, u-20

li n espike-40

saddle-node on
invariant circle bifurcation

b=0.5
40
I=70

I=75

I=80

30
20
10

spike

0
-10
-20
-60

-55

-50

-45

-40

-60

-55
-50
-45
-40
membrane potential, v (mV)

-60

-55

-50

-45

-40

Figure 8.14: Saddle-node on invariant circle bifurcations in the RS neuron model as
the magnitude of the injected current I increases.
big saddle homoclinic
orbit bifurcation

100
I=120

I=124.5

I=125

90

t cy
limi

or b

cl e

ic
clin

60
50

it

recovery variable, u

in g

ke
spi

om o

70

s pik

big h

80

40

100

saddle homoclinic
orbit bifurcation

subcritical Andronov-Hopfbifurcation

I=126.5

I=127

I=127.5

90
80
70
60
50
40
-60

-55

-50

-45

-40

-60

-55

-50

-45

-40

-60

-55

-50

-45

-40

membrane potential, v (mV)

Figure 8.15: The sequence of bifurcations of the RS model neuron (8.5, 8.6) in resonatorregime. Parameters are as in Fig.8.12 and b = 5; see also Fig.6.40.

286

Simple Models
20 mV

in g
li m i

49 pA

s pik

cle
t cy

recovery variable

100 msB
A

A

B-60 mVmembrane potential

Figure 8.16: Stuttering behavior of an RS neuron. (Data provided by Dr. Klaus M.
Stiefel: P28-36 adult mouse, coronal slices, 300µm, layer II/III pyramid, visual cortex.)
The sequence of bifurcations when b > 0 is depicted in Fig.8.15. Injection of
depolarizing current below the neuron’s rheobase transforms the resting state intoa stable focus and results in damped oscillations of the membrane potential. The
attraction domain of the focus (shaded region in the figure) is bounded by the stablemanifold of the saddle. As I increases, the stable manifold makes a loop and becomesa big homoclinic orbit giving birth to a spiking limit cycle attractor. When I = 125,
stable resting and spiking states coexist, which plays an important role in explaining
the paradoxical stuttering behavior of some neocortical neurons discussed later. As
I increases, the saddle quantity (i.e., the sum of eigenvalues of the saddle) becomes
positive. When the stable manifold makes another, smaller loop, it gives birth to an
unstable limit cycle, which then shrinks to the resting equilibrium and results in a
subcritical Andronov-Hopf bifurcation.
What is the excitability class of the RS model neuron in Fig.8.15? If a slow ramp of
current is injected, the resting state of the neuron becomes a stable focus and then loses
stability via a subcritical Andronov-Hopf bifurcation. Hence the neuron is a resonator
exhibiting Class 2 excitability. Now suppose steps of DC current of amplitude I = 125
pA or less are injected. The trajectory starts at the initial point (v, u) = (−60, 0),
which is the resting state when I = 0, and then approaches the spiking limit cycle.
Because the limit cycle was born via a homoclinic bifurcation to the saddle, it has
a large period, and hence the neuron is Class 1 excitable. Thus, depending on thenature of stimulation, that is, ramps vs. pulses, we can observe small or large spiking
frequencies, at least in principle.
In practice, it is quite difficult to catch homoclinic orbits to saddles because they
are sensitive to noise. Injection of a constant current just below the neuron’s rheobase
in Fig.8.15 would result in random transitions between the resting state and a periodic
spiking state. Indeed, the two attractors coexist and are near each other, so weakmembrane noise can push the trajectory in and out of the shaded region, resultingSimple Models

287

(a)
simple model
layer 5 neuron (in vitro)
20 mV
20 ms

response

(b)

gGABA

input

32 nS
8 nS

gAMPA

(c)
layer 5 neuron(in vitro)

1 sec

simple model

Figure 8.17: (a) Comparison of responses of a rat motor cortex layer 5 pyramidal neuronof RS type and the simple model (8.5, 8.6) to in vivo-like stochastic input (8.7) withthe random conductances in (b). Part (a) is a magnification of a small region in (c).
Shown are simulations of 30v̇ = 3(v+55)(v+42)−u+I(t), u̇ = 0.01{−0.25(v+55)−u},
if v ≥ +10, then v ← −40, u ← u + 90. (Data provided by Niraj S. Desai and Betsy
C. Walcott.)in a stuttering spiking (illustrated in Fig.8.16) mingled with subthreshold oscillations.
Such behavior is also exhibited by FS interneurons, studied later in this section.

In Vivo-like Conditions
In Fig.8.17a (dashed curve) we show the response of in vitro recorded layer 5 pyramidal
neuron of rat motor cortex to fluctuating in vivo-like input. First, random excitatory
and inhibitory conductances, gAMPA (t) and gGABA (t) (Fig.8.17b), were generated using
the Ornstein-Uhlenbeck stochastic process (Uhlenbeck and Ornstein 1930), which was
originally developed to describe Brownian motion, but can equally well describe invivo-like fluctuating synaptic conductances produced by random firings (Destexhe etal. 2001). Let EAMPA = 0 mV and EGABA = −65 mV denote the reverse potentials of
excitatory and inhibitory synapses, respectively. The corresponding current
excitatory input
inhibitory input
}|
{
}|
{
z
z
I(t) = gAMPA (t)(EAMPA − V (t)) + gGABA (t)(EGABA − V (t)) ,

(8.7)

288

Simple Models

was injected into the neuron, using the dynamic clamp protocol (Sharp et al. 1993),
where V (t) denotes the instantaneous membrane potential of the neuron. The same
conductances were injected into the simple model (8.5, 8.6), whose parameters were
adjusted to fit this particular neuron. The superimposed voltage traces, depicted in
Fig.8.17a, show a reasonable fit. The simple model predicts more than 90 percent of
spikes of the in vitro neuron, often with a submillisecond precision (see Fig.8.17c). Of
course, we should not expect to get a total fit, since we do not explicitly model thesources of intrinsic and synaptic noise present in the cortical slice. In fact, presentationof the same input to the same neuron a few minutes later produces a response withspike jitter, missing spikes, and extra spikes (as in Fig.7.24) comparable with those in
the simulated response.

8.2.2

Intrinsically Bursting (IB) Neurons

The class of intrinsically bursting (IB) neurons forms a continuum of cells that differ
in their degree of “burstiness”, and it probably should consist of subclasses. At one
extreme, responses of IB neurons to injected pulses of DC current have initial stereotypical bursts (Fig.8.18a) of high-frequency spikes followed by low-frequency tonic spiking.
Many IB neurons burst even when the current is barely superthreshold and not strong
enough to elicit a sustained response (as in Fig.8.21, bottom traces). At the other extreme, bursts can be seen only in response to sufficiently strong current, as in Fig.8.11
or Fig.9.1b. Weaker stimulation elicits regular spiking responses. In comparison withtypical RS neurons, the regular spiking response of IB neurons has lower firing frequency and higher rheobase (threshold) current, and exhibits shorter latency to thefirst spike and noticeable afterdepolarizations (ADPs) (Compare RS and IB cells in
Fig.8.11.)
Magnifications of the responses of two IB neurons in Fig.8.18b and 8.18c showthat the interspike intervals within the burst may be increasing or decreasing, possiblyreflecting different ionic mechanisms of burst generation and termination. In any case,the initial high-frequency spiking is caused by the excess of the inward current or the
deficit of the outward current needed to repolarize the membrane potential below thethreshold. As a result, many spikes are needed to build up outward current to terminate
the high-frequency burst. After the neuron recovers, it fires low-frequency tonic spikes
because there is a residual outward current (or residual inactivation of inward current)
that prevents the occurrence of another burst. Many IB neurons can fire two or morebursts before they switch into tonic spiking mode, as in Fig.8.18a. Below, we present
two models of IB neurons, one relying on the interplay of voltage-gated currents, andthe other relying on the interplay of fast somatic and slow dendritic spikes.
Let us use the available data on the IB neuron in Fig.8.11 to build a simple onecompartment model (8.5, 8.6) exhibiting IB firing patterns. The neuron has a restingstate at vr = −75 mV and an instantaneous threshold at vt = −45 mV. Its rheobase
is 350 pA, and the input resistance is around 30 MΩ, resulting in k = 1.2 and b = 5.
The peak of the spike is at +50 mV, and the after-spike resetting point is aroundSimple Models

289
bursting(a)spiking
ADP

20 mV
100 ms(b)

increasing ISIs(c)

500 pA

20 mV

20 mV

100 ms

100 msdecreasing ISIs

600 pA

Figure 8.18: (a) bursting and spiking in an IB neuron (layer 5 of somatosensory cortex
of a four week old rat at 35◦ C; (data provided by Greg Stuart and Maarten Kole).
Note the afterdepolarization (ADP). (b) IB neuron of a cat (modified from figure 2 of
Timofeev et al. 2000). (c) pyramidal neuron of rat visual cortex. Note that IB neurons
may exhibit bursts with increasing or decreasing inter-spike intervals (ISIs).
c = −56 mV. The parameters a = 0.01 and d = 130 give a reasonable fit of theneuron’s current-frequency relationship.
The phase portraits in Fig.8.19 explain the mechanism of firing of IB patterns inthe simple model. When I = 0, the model has an equilibrium at −75 mV, whichis the intersection of the v-nullcline (dashed parabola) and the u-nullcline (straightline). Injection of a depolarizing current moves the v-nullcline upward. The pulse ofcurrent of magnitude I = 300 pA is below the neuron’s rheobase, so the trajectory
moves from the old resting state (black square) to the new one (black circle). Sinceb > 0, the trajectory overshoots the new equilibrium. The pulse of magnitude I = 370
pA is barely above the rheobase, so the model exhibits low-frequency tonic firing withsome spike frequency adaptation. Elevating the fast nullcline by injecting I = 500
pA transforms the first spike into a doublet. Indeed, the after-the-first-spike resetting
point (white square marked “1”) is below the parabola, so the second spike is fired
immediately. Similarly, injection of an even stronger current of magnitude I = 550 pA
transforms the doublet into a burst of three spikes, each raising the after-spike resetting
point. Once the resetting point is inside the parabola, the neuron is in tonic spiking
mode.
Figure 8.20a shows simultaneous recording of somatic and dendritic membrane potentials of a layer 5 pyramidal neuron. The somatic spike backpropagates into the290

Simple Models
layer 5 neuron

simple model

AHP

550 pA

3
4
2
1

AHP
500 pA
2
1-50 mV

370 pAreset
2
1

200 ms1

2
spike-75 mVv-nul lllclin cline
e, I=
0

peak
recovery, u

400-75 mV

input

input

200
0

rest
-80

peak

300 pAv-nu-60 mV-60-40-20
membrane potential, v (mV)0

Figure 8.19: Comparison of in vitro recordings of an intrinsically bursting (IB) neuron
with the simple model 150v̇ = 1.2(v + 75)(v + 45) − u + I, u̇ = 0.01{5(v + 75) − u},if v ≥ +50, then v ← −56, u ← u + 130. White squares denote the reset points,
numbered according to the spike number.
dendrite, activates voltage-gated dendritic Na+ and Ca2+ currents (Stuart et al. 1999;
Hausser et al. 2000), and results in a slower dendritic spike (clearly seen in the figure).
The slow dendritic spike depolarizes the soma, resulting in an ADP, which is typicalin many IB cells. Depending on the strength of the injected dc current and the stateof the neuron, the ADP can be large enough to cause another somatic spike, as illustrated in Fig.7.52. The somatic spike may initiate another dendritic spike, and so on,
resulting in a burst in Fig.8.20b. This mechanism is known as the dendritic-somatic
Ping-Pong (Wang 1999), and it occurs in the Pinsky-Rinzel (1994) model of the hippocampal CA3 neuron, the sensory neuron of weakly electric fish (Doiron et al. 2002),
and in chattering neurons considered below.
Let us build a two-compartment simple model that simulates the somatic and dendritic spike generation of IB neurons. Since we do not know the rheobase, input

Simple Models

291

(a)

(b)
soma
dendrite

20 mV
dendritic
spike

25 ms

recorded (in vitro)

ADP

(d)

4
5

simulated
1

6

nu
u d-

ne

llcli

3
5
6
2

resting1

evoked bysomatic spikepeak of spike

2

reset
4

line
ull c
v d-n

3

dendritic recovery variable, ud(c)dendritic membrane potential, vd

Figure 8.20: Somatic and dendritic spike (a) and burst (b) in an IB neuron. The dendritic spike in (a) is simulated in (c), using the simple model described in Fig.8.21. The
phase portrait (d) describes the geometry of the dendritic spike generation mechanism.
Recordings are from layer 5 of the somatosensory cortex of a four-week-old rat at 35◦ C;
the dendritic electrode is 0.43mm from the soma. (Data provided by Greg Stuart andMaarten Kole.)
resistance, and resting and instantaneous threshold potentials of the dendritic tree ofIB neurons, we cannot determine parameters of the dendritic compartment. Instead,
we feed the somatic recording V (t) in Fig.8.20a into the model dendritic compartment
and fine-tune the parameters so that the simulated dendritic spike in Fig.8.20c “looks
like” the recorded one in Fig.8.20a.
The phase portrait in Fig.8.20d explains the peculiarities of the shape of the simulated dendritic spike. The recorded somatic spike quickly depolarizes the dendritic
membrane potential from point 1 to point 2, and starts the regenerative process – theupstroke of a spike. Upon reaching the peak of the spike (3), the dendritic membranepotential and the recovery variable are reset by the action of fast voltage-gated K+
currents, which are not modeled here explicitly. The reset point (4) is near the stablemanifold of the saddle, so the membrane potential slowly repolarizes (5) and returns
to the resting state (6).
In Fig.8.21 we put the somatic and dendritic compartments together, adjust someof the parameters, and simulate the response of the IB neuron to pulses of current ofvarious amplitudes. Note that the model correctly reproduces the transient burst of292

Simple Models
layer 5 neuron

simple model

I=700 pA

I=560 pA
+50 mV
20 mV
100 ms
I=400 pA

Figure 8.21: Comparison of in vitro recordings of an intrinsically bursting (IB) neuron(layer 5 of somatosensory cortex of a four-week-old rat at 35◦ C; data provided by Greg
Stuart and Maarten Kole) with the two-compartment simple model. Soma: 150v̇ =3(v + 70)(v + 45) + 50(vd − v) − u + I, u̇ = 0.01{5(v + 70) − u}, if v ≥ +50, thenv ← −52, u ← u + 240. Active dendrite: 30v̇d = (vd + 50)2 + 20(v − vd ) − ud ,u̇d = 3{15(vd + 50) − ud }, if vd ≥ +20, then vd ← −20, ud ← ud + 500.
two closely spaced spikes when stimulation is weak, and the rhythmic bursting withdecreasing number of spikes per burst when stimulation is strong. Using this approach,
one can build models of pyramidal neurons having multiple dendritic compartments,
as we do next.

8.2.3

Multi-Compartment Dendritic TreeIn Fig.8.22 we simulate an IB pyramidal neuron having 47 compartments (Fig.8.22a,b), each described by a simple model with parameters provided in the caption of
the figure. Our goal is to illustrate a number of interesting phenomena that occur inneuronal models having active dendrites, that is, dendrites capable of generating action
potentials.
In Fig.8.22c we inject a current into compartment 2 on the apical dendrite to
evoke an excitatory postsynaptic potential (EPSP) of 4 mV, which is subthresholdfor the spike generation mechanism. This depolarization produces a current that passively spreads to neighboring compartments, and eventually into the somatic compartment. However, the somatic EPSP is much weaker, only 1 mV, reflecting the
distance-dependent attenuation of dendritic synaptic inputs. Note also that somaticSimple Models

293

(a)

(b)
7
6
4 5

C4
C3

3

1

C1

0

C0

(e) C6+C6-60 mVEPSP = 1 mV
(soma, C0)(f) C6+C6+Iall

synaptic input
to compartment C2(g) soma(h) soma+Iall

C3
C2
C1
20 ms

50 mV

back-propagating action potential

failure to propagate

C4

failure to propagate

EPSP=12 mV

forward-propagating action potential

EPSP=12 mVC5C0

EPSP = 4 mV(compartment C2)

10 ms(d) C6+C7

C6

(c)

C2

2

C7

C5

C7
C6

soma

Figure 8.22: (a) Hand drawing and (b) a 47-compartment representation of a layer
5 pyramidal neuron. (c) Injection of current (simulating excitatory synaptic input)
into compartment 2 evokes a large excitatory postsynaptic potential (EPSP) in thatcompartment, but a much smaller EPSP in the somatic compartment. (d) Synaptic
inputs to compartments 6 and 7 result in large EPSPs there, but no dendritic spike.
(e) The same synaptic inputs into compartment 6 result in a dendritic spike, whichfails to propagate forward to the soma. (f) The same input combined with backgroundexcitation Iall = 60 pA to all compartments results in forward-propagating dendritic
spikes. (g) Strong synaptic input to the soma results in a spike that fails to propagate
into the dendritic tree. (h) The same input combined with injection of Iall = 70
pA to all compartments (to simulate in vivo tonic background input) promotes backpropagation of spike into the dendritic tree. Each compartment is simulated by thesimple model 100v̇ = 3(v + 60)(v + 50) − u + I, u̇ = 0.01{5(v + 60) − u}. Soma: ifv ≥ +50, then v ← −55, u ← u + 500. Dendrites: if v ≥ +10, then v ← −35,u ← u + 1000. The conductance between any two adjacent compartments is 70 nS.

294

Simple Models

EPSP is delayed and has a wider time course, which is the result of dendritic low-passfiltering, or smoothing, of subthreshold neuronal signals. The farther the stimulationsite is from the soma, the weaker, more delayed, and longer lasting the somatic EPSP
is. For many years, dendrites were thought to be passive conductors whose sole purpose
is to collect and low-pass filter the synaptic input.
Now, we explore the active properties of dendrites and their dependence on thelocation, timing, and strength of synaptic input. First, we stimulate two synapses thatinnervate two sister dendritic compartments, e.g., compartments 6 and 7 in Fig.8.22dthat could interact via their mother compartment 5. Each synaptic input evokes astrong EPSP of 12 mV, but due to their separation, the EPSPs do not add up and no
dendritic spike is fired. The resulting somatic EPSP is only 0.15 mV due to the passive
attenuation. In Fig.8.22e we provide exactly the same synaptic input, but into the same
compartment, i.e., compartment 6. The EPSPs add up, and result in a dendritic spike,which propagates into the mother compartment 5 and then into the sister compartment
7 (which was not stimulated), but it fails to propagate along the apical dendrite into
the soma. Nevertheless, the somatic compartment exhibits an EPSP of 1.5 mV, hardly
seen in the figure. Thus, the location of synaptic stimulation, all other conditions
being equal, made a difference. In Fig.8.22f we combine the synaptic stimulation tocompartment 6 with injection of a weak current, Iall , to all compartments of the neuron.
This current represents a tonic background excitation to the neuron that is always
present in vivo. It depolarizes the membrane potential by 2.5 mV and facilitates the
propagation of the dendritic spike along the apical dendrite all the way into the soma.
The same effect could be achieved by an appropriately timed excitatory synaptic input
arriving at an intermediate compartment, e.g., compartment 3 or 2. Not surprisingly,
an appropriately timed inhibitory input to an intermediate compartment on the apicaldendrite could stop the forward-propagating dendritic spike in Fig.8.22f.
In Fig.8.22g and 8.22h we illustrate the opposite phenomenon – back-propagating
spikes from soma to dendrites. A superthreshold stimulation of the somatic compartment evokes a burst of three spikes, which fails to propagate along the apical dendrites
by itself, but can propagate if combined with a tonic depolarization of the dendritictree.
We see that dendritic trees can do more than just averaging and low-pass filtering of
distributed synaptic inputs. Separate parts of the tree can perform independent local
signal processing and even fire dendritic spikes. Depending on the synaptic inputs
to other parts of the tree, the spikes can be localized or they can forward-propagate
into the soma, causing the cell to fire. Spikes at the soma can backpropagate into thedendrites, triggering spike-time-dependent processes such as synaptic plasticity.

8.2.4

Chattering (CH) Neurons

Chattering neurons, also known as fast rhythmic bursting (FRB) neurons, generate
high-frequency repetitive bursts in response to injected depolarizing currents. The
magnitude of the DC current determines the interburst period, which could be as long

Simple Models

295

chattering neuron (in vivo)

simple model

I=600 pA

Figure 8.23:
Comparison ofin vivo recordings from cat
primary visual cortex with simulations of the simple model
50v̇ = 1.5(v + 60)(v + 40) − u + I,u̇ = 0.03{(v + 60) − u}, ifv ≥ +25, then v ← −40,u ← u + 150. (Data provided by
D. McCormick.)

I=400 pA

I=300 pA
+25 mV
50 ms-40 mV
I=200 pA

as 100 ms or as short as 15 ms, and the number of spikes within each burst, typically
two to five, as we illustrate in Fig.8.23, using in vivo recordings of a pyramidal neuronof cat visual cortex.
An RS model neuron, as shown in Fig.8.12, can easily be transformed into a CH
neuron by increasing the after-spike reset voltage to c = −40 mV, mimicking decreased
K+ and increased Na+ currents activated during each spike. The phase portrait inFig.8.24 explains the mechanism of chattering of the simple model (8.5, 8.6). A step
1000
v- nu
12 3 4 5

550

5

pA

4

AHP

l l c li n

2

e, I=

1

0p

200

AHP

3

v- nu

recovery variable, u

=
e, I

600

400

10 ms

l l c li n

800

Au-nullcline

0

spike

-200
-70

-60

-50

-40
-30
-20
-10
0
membrane potential, v (mV)

10

20

Figure 8.24: Phase portraitof the simple model in
Fig.8.23 exhibiting CH
firing pattern.

296

Simple Models

of depolarizing current shifts the fast quadratic nullcline upward and the trajectory
quickly moves rightward to fire a spike. The after-spike reset point (white square
marked “1” in the figure) is outside the parabola nullcline, so another spike is fired
immediately, and so on, until the total amount of outward current is large enough tostop the burst; that is, until the variable u moves the reset point (the white square
marked “5”) inside the quadratic parabola. The trajectory makes a brief excursion
to the left knee (afterhyperpolarization) and then moves rightward again, initiating
another burst. Since the second burst starts with an elevated value of u, it has fewer
spikes – a phenomenon exhibited by many CH neurons.

8.2.5

Low-Threshold Spiking (LTS) Interneurons

Low-threshold spiking interneurons behave like RS excitatory neurons (b > 0) in thesense that they exhibit regular spiking patterns in response to injected pulses of current(some call them regular spiking non-pyramidal (RSNP) neurons). There are some
subtle differences, however. The response of an LTS cell to a weak depolarizing current
consists of a phasic spike or a doublet with a relatively short latency followed by lowfrequency (less than 10 Hz) subthreshold oscillation of membrane potential. Stronger
pulses elicit tonic spikes with slow frequency adaptation, decreasing amplitudes, anddecreasing after-hyperpolarizations, as one can see in Fig.8.11.
LTS neurons have more depolarized resting potentials, lower threshold potentials,
and lower input resistances than RS neurons. To match the in vitro firing patternsof the LTS interneuron of rat barrel cortex in Fig.8.25, we take the simple model ofthe RS neuron and adjust the resting and instantaneous threshold potentials vr = −56mV and vt = −42 mV, and the values k = 1 and b = 8, resulting in the rheobase
current of 120 pA and the input resistance of 50 MΩ. To model the decreasing natureof the spike and AHP amplitudes, we assume that the peak of the spike and the afterspike resetting point depend on the value of the recovery variable u. This completely
unnecessary cosmetic adjustment has a mild effect on the quantitative behavior of themodel but gives a more “realistic” look to the simulated voltage traces in Fig.8.25.
The class of excitability of LTS neurons has not been studied systematically, though
the neurons seem to be able to fire periodic spike trains with a frequency as low as thatof RS neurons (Beierlein et al. 2003; Tateno and Robinson, personal communication).
The conjecture that they are near saddle-node on invariant circle bifurcation, andhence are Class 1 excitable integrators, seems to be at odds with the observationthat their membrane potential exhibits slow damped oscillation and that they can fire
postinhibitory rebound spikes (Bacci et al. 2003a), called low-threshold spikes (hence
the name – LTS neurons). They are better characterized as being at the transition
from integrators to resonators, with phase portraits as in Fig.8.15.
A possible explanation for the subthreshold oscillations in LTS (and some RS)
neurons is given in Fig.8.13, case b > 0. The resting state is a stable node when I = 0,
but it becomes a stable focus when the magnitude of the injected current is near theneuron’s rheobase. After firing a phasic spike, the trajectory spirals into the focus

Simple Models

297
simple model

I=300 pA

ike
sp
ak

pe

rese

t

LTS neuron (in vitro)decreasing AHPs
decreasing amplitudes

I=200 pA

e

ik
sp
pe

u

pe

v-nu
l

ike

v-n
ullcl

ine
lcline, I=0

ak
I=125 pA

lin

sp

reset

20 mV

e

lc
ul
-n

ak

100 msrest-56 mV

I=100 pA

attraction
domain

ike
sp

200

pe

dampedoscillation

spike

100
0

ak

recovery, u

300

focus
-60

-40

-20

0

20

40

membrane potential, v (mV)

Figure 8.25: Comparison of in vitro recordings of a low threshold spiking (LTS) interneuron (rat barrel cortex; data provided by B. Connors) with simulations of thesimple model 100v̇ = (v + 56)(v + 42) − u + I, u̇ = 0.03{8(v + 56) − u}, if v ≥ 40 − 0.1u,
then v ← −53 + 0.04u, u ← min{u + 20, 670}.

exhibiting damped oscillation. Its frequency is the imaginary part of the complexconjugate eigenvalues of the equilibrium, and it is small because the system is near
Bogdanov-Takens bifurcation.
A possible explanation for the rebound spike in LTS (or some RS) neurons is given
in Fig.8.26. The shaded region is the attraction domain of the resting state (black
circle), which is bounded by the stable manifold of the saddle (white circle). A sufficiently strong hyperpolarizing pulse moves the trajectory to the new, hyperpolarized
equilibrium (black square), which is outside the attraction domain. Upon release fromthe hyperpolarization, the trajectory fires a phasic spike and then returns to the resting state. Some LTS interneurons fire bursts of spikes, and for that reason are called
burst-spiking non-pyramidal (BSNP) neurons.

298

Simple Models

C
restingD-60

resting

recovery variable, u

membrane potential, v (mV)

-40

0

-100

, I= 0-20

100

I=-1 nA

I=0
I=-1 nA
injected current

0

saddlecline

I=0v-null200

20v-nullcline,

40

D

reset
restingrebound
spike

A
sag B
C

A
-80

B

0

-200

hyperpolarized state

sag
100

200

hyperpolarized state

300

-80

time (ms)-60-40
membrane potential, v (mV)

-20

Figure 8.26: The mechanism of sag and rebound spike of the model in Fig.8.25.

8.2.6

Fast Spiking (FS) Interneurons

Fast spiking neurons fire “fast” tonic spike trains of relatively constant amplitudeand frequency in response to depolarized pulses of current. In a systematic study,
Tateno et al. (2004) have shown that FS neurons have Class 2 excitability in the sensethat their frequency-current (F-I) relation has a discontinuity around 20 Hz. Whenstimulated with barely superthreshold current, such neurons exhibit irregular firing,randomly switching between spiking and fast subthreshold oscillatory mode (Kubota
and Kawaguchi 1999; Tateno et al. 2004).
The absence of spike frequency adaptation in FS neurons is mostly due to thefast K+ current that activates during the spike, produces deep AHP, completely deinactivates the Na+ current, and thereby facilitates the generation of the next spike.
Blocking the K+ current by TEA (Erisir et al. 1999) removes AHP, leaves residual
inactivation of the Na+ current, and slows the spiking, essentially transforming the FS
firing pattern into LTS.
The existence of fast subthreshold oscillations of membrane potential suggests thatthe resting state of the FS neurons is near the Andronov-Hopf bifurcation. Stuttering
behavior at the threshold currents points to the coexistence of resting and spiking
states, as in Fig.8.16, and suggests that the bifurcation is of the subcritical type.
However, FS neurons do not fire postinhibitory (rebound) spikes – the feature used
to distinguish them experimentally from LTS types. Thus, we cannot use the simple
model (8.5, 8.6) in its present form to simulate FS neurons because the model withlinear slow nullcline would fire rebound spikes according to the mechanism depicted in
Fig.8.26. In addition, the simple model has a non-monotone I-V relation, whereas FS
neurons have a monotone relation.
The absence of rebound responses in FS neurons means that the phenomenologicalrecovery variable (activation of the fast K+ current) does not decrease significantly
below the resting value when the membrane potential is hyperpolarized. That is, theslow u-nullcline becomes horizontal in the hyperpolarized voltage range. Accordingly,

Simple Models

299

FS neuron (in vitro)

simple model

2

3

1

I=400 pA

2

1

I=200 pA

23
1

AHP
1 2

recovery variable, u

AHP
400

I=100 pA

300
200
100
0

ng
li m

I=73.2 pA
ne

cy

it

20
0

-10

li
llc
nu

-55 mV

i ki

reset

40

-50
-40
-30
-20
membrane potenrial, v (mV)
v-

40 ms

sp

20 mV

recovery variable, u

-60

ike

sp

cle

u-nullclin e

rest
-55

-50
-45
membrane potenrial, v (mV)-40

Figure 8.27: Comparison of in vitro recordings of a fast spiking (FS) interneuron of layer
5 rat visual cortex with simulations of the simple model 20v̇ = (v + 55)(v + 40) − u + I,u̇ = 0.2{U (v) − u}, if v ≥ 25, then v ← −45 mV. Slow nonlinear nullcline U (v) = 0
when v < vb and U (v) = 0.025(v − vb )3 when v ≥ vb with vb = −55 mV. Shaded area
denotes the attraction domain of the resting state.
we simulate the FS neuron in Fig.8.27, using the simple model (8.5) with nonlinear
u-nullcline.
The phase portraits and bifurcation diagram of the FS neuron model are qualitatively similar to the fast subsystem of a “subHopf/fold cycle” burster: injection of DC
current I creates a stable and an unstable limit cycle via fold limit cycle bifurcation.
The frequency of the newborn stable cycle is around 20 Hz; hence the discontinuityof the F-I curve and Class 2 excitability. There is a bistability of resting and spiking
states, as in Fig.8.27 (bottom right), so that noise can switch the state of the neuron
back and forth, and result in irregular stuttering spiking with subthreshold oscillationsin the 10 − 40 Hz range between the spike trains. Further increase of I shrinks theamplitude of the unstable limit cycle, results in the subcritical Andronov-Hopf bifurcation of the resting state, removes the coexistence of attractors, and leaves only thetonic spiking mode.

300

Simple Models
LS neuron (in vitro)

simple model
100 ms20 mV

I=200 pA

I=150 pA

I=125 pA

Figure 8.28: Comparison of in vitro recordings of a late spiking (LS) interneuron oflayer 1 rat neocortex with simulations of the simple two-compartment model. Soma:
20v̇ = 0.3(v + 66)(v + 40) + 1.2(vd − v) − u + I, u̇ = 0.17{5(v + 66) − u}, if v ≥ 30, thenv ← −45, u ← u + 100. Passive dendrite (dotted curve): v̇d = 0.01(v − vd ). Weaknoise was added to simulations to unmask the subthreshold oscillations. (Recordings
were provided by Zhiguo Chu, Mario Galarreta, and Shaul Hestrin; traces I = 125 andI = 150 are from one cell; trace I = 200 is from another cell.)

8.2.7

Late Spiking (LS) Interneurons

When stimulated with long pulses of DC current, late spiking neurons exhibit a long
voltage ramp, barely seen in Fig.8.28 (bottom), and then switch into a tonic firing mode.
A stronger stimulation may evoke an immediate (transient) spike followed by a longramp and a long latency to the second spike. There are pronounced fast subthresholdoscillations during the voltage ramp, indicating the existence of at least two time scales:(1) fast oscillations resulting from the interplay of amplifying and resonant currents,and (2) slow ramp resulting from the slow kinetic of an amplifying variable, such as
slow inactivation of an outward current (e.g., the K+ A-current) or slow activation of
an inward current, or both. In addition, the ramp could result from the slow charging
of the dendritic compartment of the neuron.
The exact mechanism responsible for the slow ramp in LS neurons is not knownat present. Fortunately, we do not need to know the mechanism in order to simulate
LS neurons using the simple model approach. Indeed, simple models with passiveSimple Models

301

dendrites are equivalent to simple models with linear amplifying currents. For example,
the model in Fig.8.28 consists of a two-dimensional system (v, u) responsible for the
spike generation mechanism at the soma and a linear equation for the passive dendritic
compartment vd .
When stimulated with the threshold current (i.e., just above the neuronal rheobase),
LS neurons often exhibit the stuttering behavior seen in Fig.8.28 (middle). Subthreshold oscillations, voltage ramps, and stuttering are consistent with the following geometrical picture. Abrupt onset of stimulation evokes a transient spike followed bybrief hyperpolarization and then sustained depolarization. While depolarized, the fastsubsystem affects the slow subsystem, e.g., slowly charges the dendritic tree or slowly
inactivates the K+ current. In any case, there is a slow decrease of the outward current
or, equivalently, a slow increase of the inward current that drives the fast subsystemthrough the subcritical Andronov-Hopf bifurcation. Because of the coexistence of resting and spiking states near the bifurcation, the neuron can be switched from one state
to the other by the membrane noise. Once the bifurcation is passed, the neuron is in
the tonic spiking mode. Overall, LS neurons can be thought of as being FS neurons
with a slow subsystem that damps any abrupt changes, delays the onset of spiking,and slows the frequency of spiking.

8.2.8

Diversity of Inhibitory Interneurons

In contrast to excitatory neocortical pyramidal neurons, which have stereotypical morphological and electrophysiological classes (RS, IB, CH), inhibitory neocortical interneurons have wildly diverse classes with various firing patterns that cannot beclassified as FS, LTS, or LS. Markram et al. (2004) reviewed recent results on the relationship between electrophysiology, pharmacology, immunohistochemistry, and gene
expression patterns of inhibitory interneurons. An extreme interpretation of their findings is that there is a continuum of different classes of interneurons rather than a setof three classes.
Figure 8.29 summarizes five of the most ubiquitous groups in the continuum:• (NAC) Non-accommodating interneurons fire repetitively without frequency adaptation in response to a wide range of sustained somatic current injections. Many
FS and LS neurons are of this type.
• (AC) Accommodating interneurons fire repetitively with frequency adaptationand therefore do not reach the high firing rates of NAC neurons. Some FS andLS cells, but mostly LTS cells, are of this type.
• (STUT) Stuttering interneurons fire high-frequency clusters of regular spikes
intermingled with unpredictable periods of quiescence. Some FS and LS cells
exhibit this firing type.
• (BST) Bursting interneurons fire a cluster of three to five spikes riding on a slow
depolarizing wave, followed by a strong slow AHP.

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Simple Models

Figure 8.29: An alternative classification of neocortical inhibitory interneurons (modified from Markram et al. 2004). Five major classes: non-accommodating (NAC),
accommodating (AC), stuttering (STUT), bursting (BST), and irregular spiking (IS).
Most classes contain subclasses: delay (d), classic (c), and burst (b). For bursting
interneurons, the three types are repetitive (r), initial (i), and transient (t). Subclassd-IS is not provided in the original picture by Markram et al.

Simple Models

303

a

35 mV
350 ms

c-NAC

b-NAC

d-NAC

c-AC

b-AC

d-AC

b

c

30 mV
1 sec

c-STUTb-STUT

d-STUT

i-BST

r-BST

t-BST

c-IS

b-IS

d-IS

d

e

Figure 8.30: Simulations of the simple model with various parameters can reproduce
all firing patterns of neocortical inhibitory interneurons in Fig.8.29.

304

Simple Models• (IS) Irregular spiking interneurons fire single spikes randomly with pronounced
frequency accommodation.

NAC and AC are the most common response types found in the neocortex. Each groupcan be divided into three subgroups depending on the type of the onset of the response
to a step depolarization:• (c) Classical response is when the first spike has the same shape as any otherspike in the response.
• (b) Burst response is when the first three or more spikes are clustered into aburst.
• (d) Delayed response is when there is noticeable delay before the onset of spiking.
The BST type has different subdivisions: repetitive (r), initial (i), and transient (t)bursting.
In Fig.8.30 we use the simple model (8.5, 8.6) to reproduce all firing patterns of theinterneurons, including the delayed irregular spiking (d-IS) pattern that was omitted
from Fig.8.29. We use one-size-fits-all set of parameters: C = 100, k = 1, vr = −60
mV, and vt = −40 mV. We vary the parameters a, b, c, and d. We do not strive toreproduce the patterns quantitatively, but only qualitatively.
The parameters for the NAC and AC cells were similar to those for RS neurons, withan additional passive dendritic compartment for the delayed response. The parameters
for the STUT and IS cells were similar to those of the LS interneuron, with someminor modifications that affect the initial burstiness and delays. Irregular stutteringin these types results from the coexistence of a stable resting equilibrium and a spiking
limit cycle attractor, as in the cases of FS and LS neurons considered above. The
level of intrinsic noise controls the probabilities of transitions between the attractors.
The parameters for the BST cells were similar to those of IB and CH pyramidal cells.
Varying the parameters a, b, c, and d, we indeed can get all the firing patterns in
Fig.8.29, plus many intermediate patterns, thereby creating a continuum of types ofinhibitory interneurons.

8.3

ThalamusThe thalamus is the major gateway to the neocortex in the sense that no sensory
signal, such as vision, hearing, touch, or taste, etc., can reach the neocortex without
passing through an appropriate thalamic nucleus. Anatomically, the thalamic system
consists of three major types of neurons: thalamocortical (TC) neurons, which relay
signals into the neocortex; reticular thalamic nucleus (RTN) neurons; and thalamicinterneurons, which provide local reciprocal inhibition (Shepherd 2004). The three
types have distinct electrophysiological properties and firing patterns.
There are undoubtedly subtypes within each type of thalamic neurons, but theclassification is not as elaborate as the one in the neocortex. This, and the differencesSimple Models

305

cat TC neuron
tonic mode

simple model

burst mode

tonic mode

burst mode

200 ms-60 mV-80 mV0

tonic mode
saddle

ullc

line

-100

hyperpolarized
state

re

-200

sp
ike
cu
to

ff

node

u-n

recovery variable, uclin eattraction
domain ofresting state

ll
v-nu

100

rebound burst mode

t
se

-80

-60

-40
-20
membrane potential, v (mV)

0

20

40

Figure 8.31: Comparison of in vitro recordings of a thalamocortical (TC) cell of cat
dorsal lateral geniculate nucleus with simulations of the simple
model 200v̇ = 1.6(v +60)(v + 50) − u + I,u̇ = 0.01{b(v +65)−u},b = 15 if v ≤ −65 andb = 0 otherwise. Whenv ≥ 35+0.1u, then v ←−60−0.1u, u ← u+10.
Injected current pulses
are in 50 pA increments. In burst mode,the cell was hyperpolarized to −80 mV prior to
injection of a depolarizing pulse of current.
(Data provided by C.
L. Cox and S. M. Sherman.)

between species, ages, and various thalamic nuclei, explain the contradictory reportsof different firing patterns in presumably the same types of thalamic neurons. Below
we use the simple model (8.5, 8.6) to simulate a “typical” TC, TRN, and interneuron.
The reader should realize, though, that our attempt is as incomplete as the attempt
to simulate a “typical” neocortical neuron ignoring the fact that there are RS, IB, CH,
FS, and other cells.

8.3.1

Thalamocortical (TC) Relay Neurons

Thalamocortical (TC) relay neurons, the type of thalamic neurons that project sensory
input to the cortex, have two prominent models of firing, illustrated in Fig.8.31: tonicmode and burst mode. Both modes are ubiquitous in vitro and in vivo, including
awake and behaving animals, and both represent different patterns of relay of sensory
information into the cortex (Sherman 2001). The transition between the firing modes
depends on the degree of inactivation of a low-threshold Ca2+ T-current (Jahnsen and
Llinas 1984; McCormick and Huguenard 1992), which in turn depends on the holding
membrane potential of the TC neuron.

306

Simple Models

In tonic mode, the resting membrane potential of a TC neuron is around −60 mV,
which is above the inactivation threshold of the T-current. The slow Ca2+ current isinactivated and is not available to contribute to spiking behavior. The neuron fires Na+ K+ tonic spikes with a relatively constant frequency that depends on the amplitude of
the injected current and could be as low as a few Hertz (Zhan et al. 1999). Such acell, illustrated in Fig.8.31, is a typical Class 1 excitable system near a saddle-node on
invariant circle bifurcation. It exhibits regular spiking behavior similar to that of RS
neocortical neurons. It relays transient inputs into outputs, and for this reason, manyrefer to the tonic mode as the relay mode of firing.
To switch a TC neuron into the burst mode, an injected DC current or inhibitory
synaptic input must hyperpolarize the membrane potential to around −80 mV for atleast 50 − 100 ms. While it is hyperpolarized, the Ca2+ T-current deinactivates andbecomes available. As soon as the membrane potential is returned to the resting or
depolarized state, there is an excess of the inward current that drives the neuron over
the threshold and results in a rebound burst of high-frequency spikes (as in Fig.8.31),
called a low-threshold (LT) spike or a Ca2+ spike.
In Fig.8.31 (right) we simulate the TC neuron, using the simple model (8.5, 8.6),
and treating u as the low-threshold Ca2+ current. Since the current is inactivated in
the tonic mode, that is, u ≈ 0, we take b = 0. The resting and threshold voltagesof the neuron in the figure are vr = −60 mV and vt = −50 mV. The value p = 1.6
results in a 40 pA rheobase current and a 60 MΩ input resistance, and the membrane
capacitance C = 200 pF gives the correct current-frequency (F-I) relationship. Thus,in the tonic mode, our model is essentially the quadratic integrate-and-fire neuron
200v̇ = 1.6(v + 60)(v + 50) + I with the after-spike reset from +35 mV to −60 mV.
To model slow Ca2+ dynamics in the burst mode, assume that hyperpolarizations
below the Ca2+ inactivation threshold of −65 mV decrease u, thereby creating inward
current. In the linear case, take u̇ = 0.01{b(v + 65) − u} with b = 0 when v ≥ −65, andb = 15 when v < −65, resulting in the piecewise linear u-nullcline depicted in Fig.8.31(bottom). Prolonged hyperpolarization below −65 mV decreases u and moves the trajectory outside the attraction domain of the resting state (shaded region in the figure).
Upon release from the hyperpolarization, the model fires a rebound burst of spikes;
the variable u → 0 (inactivation of Ca2+ ), and the trajectory reenters the attraction
domain of the resting state. Steps of depolarized current produce rebound bursts followed by tonic spiking with adapting frequency. A better quantitative agreement with
TC recordings can be achieved when two slow variables, u1 and u2 , are used.

8.3.2

Reticular Thalamic Nucleus (RTN) Neurons

Reticular thalamic nucleus (RTN) neurons provide reciprocal inhibition to TC relay
neurons. RTN and TC cells are similar in the sense that they have two firing modes,
illustrated in Fig.8.32: They fire trains of single spikes following stimulation fromresting or depolarized potentials in the tonic mode, as well as rebound bursts upon
release from hyperpolarized potentials in the burst mode.

Simple Models

307

rat RTN neuron
tonic mode

simple model

burst mode

tonic mode

burst mode

200 ms-65 mV-85 mV

Figure 8.32: Comparison of in vitro recordings of a reticular thalamic nucleus (RTN)
neuron of a rat with simulations of the simple model 40v̇ = 0.25(v + 65)(v + 45) − u + I,u̇ = 0.015{b(v + 65) − u}, b = 10 if v ≤ −65 and b = 2 otherwise. When v ≥ 0 (spike
cutoff), then v ← −55, u ← u + 50. Injected current pulses are 50, 70, and 110 pA. In
burst mode, the cell was hyperpolarized to −80 mV prior to injection of a depolarizing
pulse of current. (Data provided S. H. Lee and C. L. Cox.)

The parameters of the simple model in Fig.8.32 are adjusted to match the in vitro
recording of the RTN cell in the figure, and they differ from the parameters of the
TC model cell. Nevertheless, the mechanism of rebound bursting of the RTN neuron
is the same as that of the TC neuron in Fig.8.31 (bottom). In contrast, the tonicmode of firing is different. Since b > 0, the model neuron is near the transition froman integrator to a resonator; it can fire transient spikes followed by slow subthresholdoscillations of membrane potential; it has coexistence of stable resting and spiking
states, with the bifurcation diagram similar to the one in Fig.8.15, and it can stutter andproduce clustered spikes when stimulated with barely threshold current. Interestingly,
similar behavior of TC neurons was reported by Pirchio et al. (1997), Pedroarenaand Llinas (1997), and Li et al. (2003). We will return to the issue of subthresholdoscillations and stuttering spiking when we consider stellate cells of entorhinal cortex
in section 8.4.4.

308

Simple Models

cat thalamic interneuron

simple model

20 mV
50 ms

-60 mV

8.3.3

Figure 8.33: Comparison ofin vitro recordings of dorsal
lateral geniculate nucleus interneuron of a cat with simulations of the simple model
20v̇ = 0.5(v + 60)(v + 50) −u + I, u̇ = 0.05{7(v +60) − u}. When v ≥ 20 −0.08u (spike cutoff), v ←−65 + 0.08u, u ← min{u +50, 530}. Injected current
pulses are 50, 100, 200, and
250 pA. (Data provided by C.
L. Cox and S. M. Sherman.)Thalamic Interneurons

In contrast to TC and RTN neurons, thalamic interneurons do not have a prominent
burst mode, though they can fire rebound spikes upon release from hyperpolarization(Pape and McCormick 1995). They have action potentials with short duration, andthey are able to generate high-frequency trains of spikes reaching 800 Hz, as do cortical
FS interneurons. The simple model in Fig.8.33 reproduces all these features. Its phase
portrait and bifurcation diagram are similar to those in Fig.8.15, but its dynamics has
a much faster time scale.

8.4

Other Interesting Cases

The neocortical and thalamic neurons span an impressive range of dynamic behavior.
Many neuronal types found in other brain regions have dynamics quite similar to someof the types discussed above, while many do not.

8.4.1

Hippocampal CA1 Pyramidal Neurons

Hippocampal pyramidal neurons and interneurons are similar to those of the neocortex,
and hence could be simulated using the simple model presented in section 8.2. Let us
elaborate, using the pyramidal neurons of the CA1 region of the hippocampus as an
example.
Jensen et al. (1994) suggested classifying all CA1 pyramidal neurons according
to their propensity to fire bursts of spikes, often called complex spikes. The majority(more than 80 percent) of CA1 pyramidal neurons are non-bursting cells, whereas theSimple Models

309

Figure 8.34: Classification of hippocampal CA1 pyramidal neurons. A–E, in vitro
recordings from five different pyramidal neurons arranged according to a gradient of
increasing propensity to burst. The neurons were stimulated with current pulses of 200
ms duration and amplitude 50 pA and 100 pA (a), or brief (3–5 ms) superthreshold
pulses (b). The non-burster (NB) neuron fires tonic spikes in response to long pulses
and a single spike in response to brief pulses. The high-threshold burster (HTB) fires
bursts only in response to strong long pulses, and single spikes in response to weak orbrief pulses. The grade I low-threshold burster (LTB I) generates bursts in response tolong pulses of current, but single spikes in response to brief pulses. The grade II LTB(LTB II) fires bursts in response to both long (a) and brief (b) current pulses. The
grade III LTB (LTB III), in addition to firing bursts in response to long and brief pulsesof current (not shown), also fires spontaneous rhythmic bursts, shown in contracted
and expanded time scales. (Reproduced from Su et al. 2001 with permission.)
remaining ones exhibit some form of bursts, which are defined in this context as setsof three or more closely spaced spikes. There are five different classes:• (NB) Non-bursting cells generate accommodating trains of tonic spikes in response to depolarizing pulses of DC current, and a single spike in response to abrief superthreshold pulse of current, as in Fig.8.34A.
• (HTB) High-threshold bursters fire bursts only in response to strong long pulsesof current, but fire single spikes in response to weak or brief pulses of current, as310

Simple Models

A

B

NB
+40 mV

a

C

HTB

a

D

LTB I

a

E

LTB IILTB IIIa

I=100 pA

-60mV

500 ms

I=50 pA

100 ms
50 ms

b

b

b

b

10 mV

100

line

200

I=50 pA

I=50 pA

I=50 pA

I=50 pAv-nullc

recovery variable, u

10 ms

reset

e
u-nullclin

0 rest-60 -40 -20 0 20 40

00-60 -40 -20 0 20 40

0-60 -40 -20 0 20 40-60 -40 -20 0 20 40

membrane potential, v (mV)

Figure 8.35: Simulations of hippocampal CA1 pyramidal neurons (compare with
Fig.8.34) using simple model 50v̇ = 0.5(v+60)(v+45)−u+I, u̇ = 0.02{0.5(v+60)−u}.
When v ≥ 40 (spike cutoff), v ← c and u ← u + d. Here c = −50, −45, −40, −35 mV
and d = 50, 50, 55, 60 for A–D, respectively. Parameters in E are the same as in D, but
I = 33 pA.
in Fig.8.34B.
• (LTB I) Grade I low-threshold bursters fire bursts in response to long pulses, but
single spikes in response to brief pulses of current, as in Fig.8.34C.
• (LTB II) Grade II low-threshold bursters fire stereotypical bursts in response tobrief pulses, as in Fig.8.34D.
• (LTB III) Grade III low-threshold bursters fire rhythmic bursts spontaneously,
which are depicted in Fig.8.34E, using two time scales.
NB neurons are equivalent to neocortical pyramidal neurons of the RS type, whereas
HTB and LTB I neurons are equivalent to neocortical pyramidal neurons of the IB

Simple Models

311

type. The author is not aware of any systematic studies of the ability of IB neurons
to fire stereotypical bursts in response to brief pulses, as in Fig.8.34Db, or to have
intrinsic rhythmic activity, as in Fig.8.34E. Therefore, it is not clear whether there are
any analogues of LTB grades II and III neurons in the neocortex.
The classification of hippocampal CA1 pyramidal neurons into five different classes
does not imply a fundamental difference in the ionic mechanism of spike generation, but
only a quantitative difference. This follows from the observation that pharmacologicalmanipulations can gradually and reversibly transform an NB neuron into an LTB III
neuron, and vice versa, by elevating the extracellular concentration of K+ (Jensen et al.
1994), reducing extracellular Ca2+ (Su et al. 2001), blocking the K+ M-current (Yueand Yaari 2004), or manipulating Ca2+ current dynamics in apical dendrites (Magee
and Carruth 1999).
In Fig.8.35 we modify the simple model for the neocortical RS neuron to reproduce
firing patterns of hippocampal pyramidal cells. To get the continuum of responses from
NB to LTB II, fix all the parameters and vary only the after-spike reset parameter cby an increment of 5 mV, and the parameter d. These phenomenological parametersdescribe the effect of high-threshold inward and outward currents activated during each
spike and affecting the after-spike behavior. Increasing c corresponds to up-regulating
slow INa,p or down-regulating slow K+ currents, which leads to transition from NB
to LTB III in the CA1 slice (Su et al. 2001) and in the simple model in Fig.8.35.
Interestingly, the same procedure results in transitions from RS to IB and possibly to
CH classes in neocortical pyramidal neurons (Izhikevich 2003). This is consistent withthe observation by Steriade (2004) that many neocortical neurons can change theirfiring classes in vivo, depending on the state of the brain.

8.4.2

Spiny Projection Neurons of Neostriatum and Basal
Ganglia

Spiny projection neurons, the major class of neurons in neostriatum and basal ganglia,
display a prominent bistable behavior in vivo, shown in Fig.8.36 (Wilson and Groves

500 ms

10 mV
up-state
down-state-49 mV-63 mV

Figure 8.36: Neostriatal spiny neurons have two-state behavior in vivo. (Data provided
by Charles Wilson.)

312

Simple Models
simple model

500

I=640 pA

I>0
I=0

0

nu
u-

spike

e

lin

llc

ullc

-500

I=520 pA

reset

restingv-n

recovery variable, u

spiny neuron

li n
e

50 mV
100 ms
I=430 pA

I=640 pA

I=420 pA

+40 mV

recovery variable, u

500
ghost ofsaddle-node
0
reset
spike

-500

latency

I=400 pA

I=400 pA
-100

-80
-60
-40
-20
membrane potential, v (mV)0

Figure 8.37: Comparison of in vitro recordings of a rat neostriatal spiny projection
neuron with simulations of the simple model 50v̇ = (v + 80)(v + 25) − u + I, u̇ =0.01{−20(v + 80) − u}, if v ≥ 40, then v ← −55, u ← u + 150. (In vitro data provided
by C. Wilson.)

1981; Wilson 1993): they shift the membrane potential from hyperpolarized to depolarized states in response to synchronous excitatory synaptic input from cortex and/or
thalamus. In vitro studies of such neurons reveal a slowly inactivating K+ A-current,
which is believed to be responsible for the maintenance of the up-state and down-state,in addition to the synaptic input. Indeed, the K+ current is completely deinactivatedat the hyperpolarized potentials (down-state), and reduces the response of the neuron
to any synaptic input. In contrast, prolonged depolarization (up-state) inactivates thecurrent and makes the neuron more excitable and ready to fire spikes.
The most remarkable feature of neostriatal spiny neurons is depicted in Fig.8.37.
In response to depolarizing current pulses, the neurons display a prominent slowly
depolarizing (ramp) potential, and hence long latency to spike discharge (Nisenbaum
et al. 1994). The ramp is mostly due to the slow inactivation of the K+ A-current andslow charging of the dendritic tree. The delay to spike can be as long as 1 sec, but theSimple Models

313

subsequent spike train has a shorter, relatively constant period that depends on themagnitude of the injected current – a feature consistent with the saddle-node off limit
cycle bifurcation.
Let us use the simple model (8.5, 8.6) to simulate the responses of spiny neurons tocurrent pulses. The resting membrane potential of the neuron in Fig.8.37 is around vr =−80 mV, and we set vt = −25 mV, p = 1, and b = −20 to get 30 MΩ input resistanceand 300 pA rheobase current. We take a = 0.01 to reflect the slow inactivation of theK+ A-current in the subthreshold voltage range. The membrane potential in the figure
reaches the peak of +40 mV during the spike and then resets to −55 mV or lower,
depending on the firing frequency. The value d = 150 provides a reasonable matchof the interspike frequencies for all magnitudes of injected current. Note that b < 0,so u represents either slow inactivation of IA or slow charging of the passive dendritic
compartment, or both. In any case, it is a slow amplifying variable, which is consistent
with the observation that spiny neurons do not “sag” in response to hyperpolarizing
current pulses, do not “peak” in response to depolarizing pulses (Nisenbaum et al.
1994), and do not generate rebound (postinhibitory) spikes.
Injection of a depolarizing current shifts the v-nullcline of the simple model upward,
and the resting state disappears via saddle-node bifurcation. The trajectory slowly
moves through the ghost of the bifurcation point (shaded rectangle in the figure),
resulting in the long latency to the first spike. The spike resets the trajectory to a point(white square) below the ghost, resulting in significantly smaller delays to subsequent
spikes. Because the resetting point is so close to the saddle-node bifurcation point,
the simple model, and probably the spiny projection neuron in the figure, are near thecodimension-2 saddle-node homoclinic orbit bifurcation discussed in section 6.3.6.

8.4.3

Mesencephalic V Neurons of Brainstem

The best examples of resonators with fast subthreshold oscillations, Class 2 excitability,
rebound spikes, and so on, are mesencephalic V (mesV) neurons of the brainstem (Wuet al. 2001) and primary sensory neurons of the dorsal root ganglion (Amir et al. 2002;
Jian et al. 2004). MesV neurons of the brainstem have monotone I-V curves, whereas
the simple model with linear equation for u does not. In Fig.8.38 we use a modificationof the simple model to simulate the responses of a mesV neuron (data from Fig.7.3) topulses of depolarizing current.
The model’s phase portrait is qualitatively similar to that of the FS interneurons in
Fig.8.27. The resting state is a stable focus, resulting in damped or noise-induced sustained oscillations of the membrane potential. Their amplitude and frequency dependon I and can be larger than 5 mV and 100 Hz, respectively. The focus loses stability
via subcritical Andronov-Hopf bifurcation. Because of the coexistence of the restingand spiking states, the mesV neuron can burst, and so can the simple model if noiseor a slow resonant variable is added.

314

Simple Models
MesV neuron

simple model
reset
800
400

I=600 pA
0

I=500 pA

800

cle
cy
it

reset

I=500 pA

sp

400

l im

+10 mV

20 mV

I=470 pA

oscillations

rest

recovery variable, u-50 mV
I=400 pAu -n
uv-n llclin
u ll ec li ne

0

25 ms

I=200 pA

g
in
ik

800

e

ik
sp

400
0

rest

I=400 pA

-60
-40
-20
0
membrane potential, v (mV)

Figure 8.38: Comparison of in vitro recordings of rat brainstem mesV neuron (fromFig.7.3) with simulations of the simple model 25v̇ = (v + 50)(v + 30) − u + I, u̇ =0.5{U (v + 50) − u}, with cubic slow nullcline U (x) = 25x + 0.009x3 . If v ≥ 10, thenv ← −40.

8.4.4

Stellate Cells of Entorhinal Cortex

The entorhinal cortex occupies a privileged anatomical position that allows it to gate
the main flow of information into and out of the hippocampus. in vitro studies showthat stellate cells, a major class of neurons in the entorhinal cortex, exhibit intrinsic
subthreshold oscillations with a slow dynamics of the kind shown in Fig.8.39b (Alonsoand Llinas 1989; Alonso and Klink 1993; Klink and Alonso 1993; Dickson et al. 2000).
The oscillations are generated by the interplay between a persistent Na+ current andan h-current, and they are believed to set the theta rhythmicity in the entorhinalhippocampal network.
The caption of Fig.8.39 provides parameters of the simple model (8.5, 8.6) that
captures the slow oscillatory dynamics of an adult rat entorhinal stellate cell recordedin vitro. The cell sags to injected hyperpolarizing current in Fig.8.39a and then fires arebound spike upon release from hyperpolarization. From a neurophysiological point ofview, the sag and the rebound response are due to the opening of the h-current; from

Simple Models

315

(a)
stellate cellof entorhinal
cortex

sag

15 mV
1 sec

simple model-60 mV-500 pA(b)

100 pA

165 pA

stellate cell of entorhinal cortex

200 pA

simple model

I=173 pA

I=170 pA

I=167 pA

(c)

200

ike

sp
160

120

80

spi
ep

kes

s

recovery variable, u

I=165 pA

ar

-60

atr

I=165 pA

ix

-55

-50

-45

-40

membrane potential, V (mV)

I=173 pA

limit cycle
attractor
-35

-60

-55

-50

-45

-40

-35

membrane potential, V (mV)

Figure 8.39: Comparison of in vitro recordings of stellate neurons of rat entorhinal
cortex with simulations of the simple model 200v̇ = 0.75(v + 60)(v + 45) − u + I,u̇ = 0.01{15(v + 60) − u}, if v ≥ 30, then v ← −50. (a) Responses to steps of
DC current. (b) Subthreshold oscillations and occasional spikes at various levels of
injected DC current. (c) Phase portraits corresponding to two levels of injected DC
current. Weak noise was added to simulations to unmask subthreshold oscillations.
(Data provided by Brian Burton and John A. White. All recordings are from the sameneuron, except steps of −500 pA and 200 pA were recorded from a different neuron.
Spikes are cut at 0 mV.)

316

Simple Models

a theoretical point of view, they are caused by the resonant slow variable u, which
could also describe deinactivation of a transient Na+ current and deactivation of a
low-threshold K+ current. The geometrical explanation of these responses is similar tothe one provided for LTS interneurons in Fig.8.26. Positive steps of current evoke atransient or a sustained spiking activity. Note that the first spike is actually a doublet
in the recording and in the simulation in Fig.8.39a (I = 200 pA).
Stellate cells in the entorhinal cortex of adult animals can exhibit damped or sustained subthreshold oscillations in a frequency range from 5 to 15 Hz. The oscillations
can be seen clearly when the cell is depolarized by injected DC current, as in Fig.8.39b.
The stronger the current, the higher the amplitude and frequency of oscillations, whichoccasionally result in spikes or even bursts of spikes (Alonso and Klink 1993). The simple model also exhibits slow damped oscillations because its resting state is a stablefocus. The focus loses stability via subcritical Andronov-Hopf bifurcation, and hence it
coexists with a spiking limit cycle. To enable sustained oscillations and random spikes,
we add channel noise to the v-equation (White et al. 2000).
In Fig.8.39c we explain the mechanism of random transitions between subthresholdoscillations and spikes, which is similar to the mechanism of stuttering in RS andFS neurons. When weak DC current is injected (left), the attraction domain of theresting state (shaded region) is separated from the rest of the phase space by the stablemanifold to the saddle equilibrium (denoted separatrix). Noisy perturbations evoke
small, sustained noisy oscillations around the resting state with an occasional spike
when the separatrix is crossed. Increasing the level of injected DC current results in theseries of bifurcations similar to those in Fig.8.15. As a result, there is a coexistence of alarge amplitude (spiking) limit cycle attractor and a small unstable limit cycle, which
encompasses the attraction domain of the resting state (right). Noisy perturbations
can randomly switch the activity between these attractors, resulting in the random
bursting activity in Fig.8.39b.

8.4.5

Mitral Neurons of the Olfactory Bulb

Mitral cells recorded in slices of rat main olfactory bulb exhibit intrinsic bistability ofmembrane potentials (Heyward et al. 2001). They spontaneously alternate between
two membrane potentials separated by 10 mV: a relatively depolarized (up-state) andhyperpolarized (down-state). The membrane potential can be switched between thestates by a brief depolarizing or hyperpolarizing pulse of current, as we show in Fig.7.36.
In response to stimulation, the cells are more likely to fire in the up-state than in the
down-state.
Current-voltage (I-V) relations of such mitral cells have three zeros in the subthreshold voltage range confirming that there are three equilibria: two stable ones
corresponding to the up-state and the down-state, and one unstable, the saddle. There
are no subthreshold oscillations in the down-state, hence it is a node, and the cell is
an integrator. There are small-amplitude 40 Hz oscillations in the up-state; hence it isa focus, and the cell is a resonator.

Simple Models

317
rat's mitral cell (in vitro)

simple model
40 mV
100 msI=35pA

I=20 pA

up-state
I=15pAsoma
I= 10pA

dendrite

Figure 8.40: Comparison of in vitro recordings of mitral neurons of rat olfactory bulb
with simulations of the simple two-compartment model. Soma: 40v̇ = (v+55)(v+50)+
0.5(vd −v)−u+I, u̇ = 0.4{(U (v)−u} with U (v) = 0 when v < vb and U (v) = 20(v−vb )when v ≥ vb = −48 mV. If v ≥ 35, then v ← −50. Passive dendrite (dotted curve):v̇d = 0.0125(v − vd ). Weak noise was added to simulations to unmask subthresholdoscillations in the up-state. The membrane potential of the neuron is held at −75
mV by injecting a strong negative current, and then stimulated with steps of positive
current. (Data provided by Philip Heyward.)
To model the bistability, we use the simple model with a piecewise linear slow
nullcline that approximates nonlinear activation functions n∞ (v) near the “threshold”
of the current and a passive dendritic compartment. In many respects, the model issimilar to the one for late spiking (LS) cortical interneurons. In Fig.8.40 we fine-tune
the model to simulate responses of a rat mitral cell to pulses of current of various
amplitudes. To prevent noise-induced spontaneous transitions between the up-state
and the down-state, the cell in the figure was held at −75 mV by injection of a large
negative current. Its responses to weak positive pulses of current show a fast-risingphase followed by an abrupt step (arrow in the figure) to a constant value corresponding
to the up-state. Increasing the magnitude of stimulation elicits trains of spikes witha considerable latency whose cause has yet to be determined experimentally. The
latency could be the result of slow activation of an inward current, slow inactivationof an outward current (e.g., the K+ A-current), or just slow charging of the dendritic
compartment. All three cases correspond to an additional slow variable in the simple
model, which we interpret as a membrane potential of a passive dendritic compartment.

318

Simple Models

up-state (-46 mV)
down-state (-55 mV)
simple model

rat's mitral cell (in vitro)
spikes cut at -20 mV

200

saddle

I=0 pA

ike

sp

150

thr
esh
old

20 mV 100 ms

recovery variable, u

rat's mitral cell (in vitro)

100
50
0

up-state(focus)saddle200

I = 7 pA

saddleattraction domainof the up-state150

recovery variable, u

stimulation

down-state(node)

ike
sp

100

0

v-

nu

ll c

lin
e

up-stated50ol

simple modelt hru-nullcline

-60

-55

-50

es

h

-45

-40

-35

membrane potential, v (mV)

Figure 8.41: Voltage responses of a rat mitral cell and a simple model from Fig.8.40
at two different values of the holding current. Right: Phase portraits of somatic compartments show coexistence of stable node (down-state) and stable focus (up-state)equilibria. Spikes are emitted only from the up-state.

To understand the dynamics of the simple model, and hopefully of the mitral cell,
we simulate its responses in Fig.8.41 to the activation of the olfactory nerve (ON). At
the top of Fig.8.41, the cell is held at I = 0 pA. Its phase portrait clearly shows thecoexistence of a stable node and focus equilibria separated by a saddle. The shaded
region corresponds to the attraction domain of the focus equilibrium. To fire a spikefrom the up-state, noise or external stimulation must push the state of the system fromthe shaded region over the threshold to the right. The cell returns to the down-state
immediately after the spike. Much stronger stimulation is needed to fire the cell fromthe down-state. Typically, the cell is switched to the up-state first, spends some time
oscillating at 40 Hz, and then fires a spike (Heyward et al. 2001).
At the bottom of Fig.8.41, the cell is held at a slightly depolarizing current I =7 pA. The node equilibrium disappears via saddle-node bifurcation, so there is nodown-state, but only its ghost. Stimulation at the up-state results in a spike, afterhyperpolarization, and slow transition through the ghost of the down-state back to
the up-state. Further increasing the holding current results in the stable manifold tothe upper saddle (marked “threshold” in the figure) making a loop, then becoming ahomoclinic trajectory to the saddle, giving birth to an unstable limit cycle which shrinks

Simple Models319

to the focus and makes it lose stability via subcritical Andronov-Hopf bifurcation.
Note that this phase portrait and the bifurcation scenario are different from the one
in Fig.7.36. However, in both cases, the neuron is an integrator in the down-state anda resonator in the up-state! The same property is exhibited by cerebellar Purkinjecells (see Fig.7.37), and possibly by other neurons kept in the up-state (intrinsically or
extrinsically).

Review of Important Concepts• An integrate-and-fire neuron is a linear model having a stable nodeequilibrium, an artificial threshold, and a reset.
• A resonate-and-fire neuron is a linear model having a stable focus
equilibrium, an artificial threshold, and a rest.
• Though technically not spiking neurons, these models are useful for
analytical studies, that is, to prove theorems.
• The quadratic integrate-and-fire model captures the nonlinearity ofthe spike generation mechanism of real neurons having Class 1 excitability (saddle-node on invariant circle bifurcation).
• Its simple extension, model (8.5, 8.6), quantitatively reproduces subthreshold, spiking, and bursting activity of all known types of cortical and thalamic neurons in response to pulses of DC current.
• The simple model makes testable hypotheses on the dynamic mechanisms of excitability in these neurons.
• The model is especially suitable for simulations of large-scale modelsof the brain.

Bibliographical Notes
Many people have used the integrate-and-fire neuron, treating it as a folklore model. It
was Tuckwell’s Introduction to Theoretical Neurobiology (1988) that gave appropriate
credit to its inventor, Lapicque (1907). Although better models, such as the quadratic
integrate-and-fire model, are available now, many scientists continue to favor the leaky
integrate-and-fire neuron mostly because of its simplicity. Such an attitude is understandable when one wants to derive analytical results. However, purely computational
papers can suffer from using the model because of its weird properties, such as thelogarithmic F-I curve and fixed threshold.

320

Simple Models

Figure 8.42: Louis Lapicque, the discoverer of theintegrate-and-fire neuron.
The resonate-and-fire model was introduced by Izhikevich (2001a), and then by
Richardson, Brunel, and Hakim (2003) and Brunel, Hakim, and Richardson (2003).
These authors initially called the model “resonate-and-fire”, but then changed its nameto “generalized integrate-and-fire” (GIF), possibly to avoid confusion.
A better choice is the quadratic integrate-and-fire neuron in the normal form (8.2)
or in the ϑ-form (8.8); see exercise 7. The ϑ-form was first suggested in the context ofcircle/circle (parabolic) bursting by Ermentrout and Kopell (1986a,b). Later, Ermentrout (1996) used this model to generalize numerical results of Hansel et al. (1995) on
synchronization of Class 1 excitable systems, discussed in chapter 10. Hoppensteadt
and Izhikevich (1997) introduced the canonical model approach, provided many examples of canonical models, and proved that the quadratic integrate-and-fire modelis canonical in the sense that all Class 1 excitable systems can be transformed into
this model by a piecewise continuous change of variables. They also suggested callingthe model the “Ermentrout-Kopell canonical model” to honor its inventors, but mostscientists follow Ermentrout and call it the “theta-neuron”.
The model presented in section 8.1.4 was first suggested by Izhikevich (2000a; equations (4) and (5), with voltage reset discussed in Sect. 2.3.1) in the ϑ-form. The form
presented here first appeared in Izhikevich (2003). The representation of the function
I + v 2 in the form (v − vr )(v − vt ) was suggested by Latham et al. (2000).
We stress that the simple model is useful only when one wants to simulate largescale networks of spiking neurons. He or she still needs to use the Hodgkin-Huxley-typeconductance-based models to study the behavior of one neuron or a small network ofneurons. The parameter values that match firing patterns of biological neurons presented in this chapter are only educated guesses (the same is true for conductance-basedmodels). More experiments are needed to reveal the true spike generation mechanismof any particular neuron. An additional insight into the question, “which model ismore realistic?” is in Fig.1.8.
Looking at the simple model, one gets an impression that the spike generation
mechanism of RS neurons is the simplest in the neocortex. This is probably true;
however, the complexity of the RS neurons, most of which are pyramidal cells, is hiddenin their extensive dendritic trees having voltage- and Ca2+ -gated currents. Dendritic

Simple Models

321

dynamics is a subject for a 500-page book by itself, and we purposely omitted it. An
interested reader is recommended to study Dendrites by Stuart et al. (1999), recent
reviews by Hausser and Mel (2003) and Williams and Stuart (2003), and the seminal
paper by Arshavsky et al. (1971; Russian language edition, 1969).

Exercises
1. (Integrate-and-fire network) The simplest implementation of a pulse-coupled integrate-and-fire neural network has the formXv̇i = bi − vi +cij δ(t − tj ) ,j̸=iwhere tj is the moment of firing of the jth neuron, that is, the moment vj (tj ) =1. Thus, whenever the jth neuron fires, the membrane potentials of the otherneurons are instantaneously adjusted by cij , i ̸= j. Show that the same initialconditions may result in different solutions, depending on the implementation
details.
2. (Latham et al. 2000) Determine the relationship between the normal form forsaddle-node bifurcation (6.2) and the equationV̇ = a(V − Vrest )(V − Vthresh ) .
3. Show that the period of oscillations in the quadratic integrate-and-fire model(8.2) is


1
vpeak
vresetatan √ − atan √
T =√
b
b
b
when b > 0.
4. Show that the period of oscillations in the quadratic integrate-and-fire model(8.2) with vpeak = 1 ispp !
1 − |b|vreset − |b|1
p − ln
p
ln
T = p
2 |b|
1 + |b|vreset + |b|pwhen b < 0 and vreset > |b|.
5. Justify the bifurcation diagram in Fig.8.3.
6. Brizzi et al. (2004) have shown that shunting inhibition of cat motoneurons
raises the firing threshold, and the rheobase current, and shifts the F-I curve tothe right without changing the shape of the curve. Use the quadratic integrateand-fire model to explain the effect. (Hint: Consider v̇ = b − gv + v 2 with g ≥ 0,
vreset = −∞, and vpeak = +∞.)

322

Simple Models

7. (Theta neuron) Determine when the quadratic integrate-and-fire neuron (8.2) isequivalent to the theta neuronϑ̇ = (1 − cos ϑ) + (1 + cos ϑ)r ,(8.8)

where r is the bifurcation parameter and ϑ ∈ [−π, π] is a phase variable on theunit circle.
8. (Another theta neuron) Show that the quadratic integrate-and-fire neuron (8.2)
is equivalent toϑ̇ = ϑ2 + (1 − |ϑ|)2 r .
where ϑ ∈ [−1, 1] and r have the same meaning as in exercise 7. Are there anyother “theta-neurons”?
9. When is the linear version of (8.3, 8.4),v̇ = I − v − uu̇ = a(bv − u)if v = 1, thenv ← 0, u ← u + d,equivalent to the integrate-and-fire or resonate-and-fire model?
10. Show that the simple model (8.3, 8.4) with b < 0 is equivalent to the quadratic
integrate-and-fire neuron with a passive dendritic compartment.
11. All membrane potential responses in Fig.8.8 were obtained using model (8.3, 8.4)
with appropriate values of the parameters. Use MATLAB to experiment withthe model and reproduce the figure.
12. Simulate the FS spiking neuron in Fig.8.27, using the simple model (8.5, 8.6) withlinear equation for u. What can you say about its possible bifurcation structure?
13. Fit the recordings of the RS neuron in Fig.8.12, using the modelC v̇ = I − g(v − vr ) + p(v − vt )2+ − uu̇ = a(b(v − vr ) − u)if v = vpeak , thenv ← c, u ← u + d,

where x+ = x when x > 0 and x+ = 0 when otherwise. This model better fits
the upstroke of the action potential.
14. Explore numerically the model (8.3, 8.4) with a nonlinear after-spike reset v ←f (u), u ← g(u), where f and g are some functions.
15. [M.S.] Analyze the generalization of the system (8.3, 8.4)v̇ = I + v 2 + evu − uu̇ = a(bv − u)where e is another parameter.

if v = 1, thenv ← c, u ← u + d

Simple Models

323

16. [M.S.] Analyze the generalization of the following system, related to the exponential integrate-and-fire modelv̇ = I − v + kev − uu̇ = a(bv − u)if v = 1, thenv ← c, u ← u + dwhere k is another parameter.
17. [M.S.] Analyze the system2−uv̇ = I − v + kv+
u̇ = a(bv − u)if v = 1, thenv ← c, u ← u + dwhere v+ = v when v > 0 and v+ = 0 otherwise.
18. [M.S.] Find an analytical solution to the system (8.3, 8.4) with time-dependent
input I = I(t).
19. [M.S.] Determine the complete bifurcation diagram of the system (8.3, 8.4).

Chapter 9
Bursting
A burst is two or more spikes followed by a period of quiescence. Neurons can fire
single spikes or stereotypical bursts of spikes, depending on the nature of stimulationand the intrinsic neuronal properties. Typically, bursting occurs due to the interplayof fast currents responsible for spiking activity and slow currents that modulate theactivity. In this chapter we study this interplay in detail.
To understand the geometry of bursting, it is customary to assume that the fast andslow currents have drastically different time scales. In this case we can dissect a burster,that is, freeze its slow currents and use them as parameters that control the fast spiking
subsystem. During bursting, the slow parameters drive the fast subsystem throughbifurcations of equilibria and limit cycles. We provide a topological classification of
bursters based on these bifurcations, and show that different topological types havedifferent neurocomputational properties.

9.1

Electrophysiology

Many spiking neurons can exhibit bursting activity if manipulated, for instance, pharmacologically. In Fig.9.1 we depict a few well-known examples of neurons that burst
under natural conditions without any manipulation. Some require an injected DC current to bias the membrane potential, while others do not. One can only be amazedby the diversity of bursting patterns and time scales. In this chapter we consider
electrophysiological and bifurcation mechanisms responsible for the generation of thesepatterns.
Is a zebra a black animal with white stripes or a white animal with black stripes?
This seemingly silly question is pertinent to every bursting pattern: Does bursting
activity correspond to an infinite period of quiescence interrupted by groups of spikes, ordoes it correspond to an infinite spike train interrupted by short periods of quiescence?
Biologists are mostly concerned with the question of what makes the neuron fire thefirst spike in a burst and what keeps it in the spiking regime afterward. The questionof why the spiking stops is often forgotten. It turns out that to fully understand theionic mechanism of bursting, we need to concentrate on the second question, that is, we
325

326

Bursting(a) cortical chattering neuron(c) cortical intrinsically bursting neuron(b) cortical intrinsically bursting neuron(d) thalamic reticular neuron

500 ms50 ms(e) thalamocortical relay neuron(f) hippocampal pyramidal neuron

100 ms

10 mV

20 mV

50 ms

-66 mV(g) respiratory neuron in pre-Botzinger complex(h) rodent trigeminal neuron

4 sec

Figure 9.1: Examples of intrinsic bursters. (a) and (b) cat primary visual corticalneurons (modified from Nowak et al. 2003). (c) cortical neuron in anesthetized cat(modified from Timofeev et al. 2000). (d) thalamic reticular (RE) neuron (modified
from Steriade 2003). (e) Cat thalamocortical relay neuron (modified from McCormickand Pape 1990). (f) CA1 pyramidal neuron exhibiting grade II low-threshold bursting
pattern (modified from Su et al. 2001). (g) respiratory neuron in the pre-Botzinger
complex (modified from Butera et al. 1999). (h) Trigeminal interneuron from rat
brainstem (modified from Del Negro et al. 1998).

Bursting

327

Figure 9.2: Is bursting a spiking state interrupted by periods of quiescence, or is it aquiescent state interrupted by groups of spikes?
V(t)
20 mV

50 ms

I(t)

Figure 9.3: Forced bursting in the INa,p +IK -model with parameters as in Fig.4.1a andtime-dependent injected current I(t).
need to treat bursting as an infinite spike train that is chopped into short bursts by aslow (resonant) current that builds up during the spiking phase and recovers during thequiescent phase. Before proceeding to a general case, let us consider a simple example.

9.1.1

Example: The INa,p +IK +IK(M) -Model

Any model neuron capable of spiking can also burst, as, for instance, the INa,p +IK model in Fig.9.3. However, this example is not interesting because the neuron is forced
to burst by the time-dependent input I(t).
In contrast, a modification of the INa,p +IK -model in Fig.9.4 fires a burst of spikes
in response to a brief pulse of current. The first spike in the burst is caused by thestimulation, whereas the subsequent spikes are generated autonomously due to theintrinsic properties of the neuron, and they outlast the stimulation. Such a burst isBurstingslow K+ activation
gate, nslow

membranepotential (mV)

328

-20
-40
-60

I(t)

I(t)

0.06
0.04
0.02
0
0

20
time (ms)

a

40

0

0

50

100
time (ms)

150

200

b

Figure 9.4: Intrinsic bursting in the INa,p +IK +IK(M) -model (7.1), consisting of theINa,p +IK -model with parameters as in Fig.4.1a, and a fast K+ current (gK = 9, τ (V ) =0.152) and a slow K+ current with gslow = 5, V1/2 = −20 mV, k = 5 mV, and τslow (V ) =20 ms. (a) Burst excitability when I = 0. (b) Periodic bursting when I = 5.
stereotypical and fairly independent of the amplitude or the duration of the pulse thattriggered it.
To make the INa,p +IK -model burst, we took parameters as in Fig.6.7a, so that there
is a coexistence of the resting and spiking states. The brief pulse of current excites theneuron, that is, moves its state into the attraction domain of the spiking limit cycle
and initiates periodic activity. Without any other modification, the model will produce
an infinite spike train. To stop the train, we added a slower high-threshold persistentK+ current similar to IK(M) that provides a negative feedback. This M-current is
not activated at rest. However, during the active (spiking) phase, the current slowly
activates, as indicated by the slow buildup of its gating variable nslow in the figure. The
neuron becomes less and less excitable, and eventually cannot sustain spiking activity.
If, instead of a pulse of current, a constant current is applied, the neuron can burst
periodically, as in Fig.9.4b.
This model presents only one of many possible examples of bursters, which we study
in this chapter. However, it illustrates a number of important issues common to all
bursters. For instance, in contrast to the forced bursting in Fig.9.3, this bursting is
intrinsic or autonomous. This stereotypical bursting pattern results from the intrinsic
voltage-sensitive currents, and not from a time-dependent input. The behavior in
Fig.9.4a is called burst excitability to emphasize that the model is an excitable system,with the exception that superthreshold stimulation elicits a burst of spikes instead ofa single spike. Hippocampal pyramidal neurons that are “grade III bursters”, depicted
in Fig.8.34Eb, exhibit burst excitability.

Bursting

329
interburst period
quiescentperiodduty cycle =activephaseinterspike(intraburst)
periodactive phaseinterburst period

V(t)Figure 9.5: Basic characteristics of bursting dynamics.
Biologists sometimes refer to the bursting in Fig.9.4b as conditional, because repetitive bursting occurs when a certain condition is satisfied, for instance, positive I is
injected. From a mathematical point of view, every burster is conditional, since itexists for some values of the parameters but not for others.

9.1.2

Fast-Slow Dynamics

In general, every bursting pattern consists of oscillations with two time scales: a fast
spiking oscillation within a single burst (intraburst oscillation, or spiking), and one
modulated by a slow oscillation between the bursts (inter burst oscillation); see Fig.9.5.
Typically, though not necessarily (see exercises at the end of this chapter), two timescales result from two interacting processes involving fast and slow currents. For example, the spiking in Fig.9.4 is generated by the fast INa,p +IK -subsystem and modulatedby the slow IK(M) -subsystem.
There are two questions associated with each bursting pattern:• What initiates sustained spiking during the burst?
• What terminates sustained spiking (temporarily) and ends the burst?
The answer to the first question is relatively simple. Repetitive spiking is initiated andsustained by the positive injected current I or some other source of persistent inward
current that causes the neuron to fire (most biologists are interested in identifying thissource, and they would not consider this question trivial). Surprisingly, the second
question is the more important for building a model of bursting. While the neuron
fires, relatively slow processes somehow make it non-excitable and eventually terminate
the firing. Such slow processes result in a slow buildup of an outward current or in aslow decrease of an inward current needed to sustain the spiking. During the quiescentphase, the neuron slowly recovers and regains the ability to generate action potentials.
Let us discuss possible ionic mechanisms responsible for the termination of spiking
within a burst. Suppose we are given a neuronal model that is capable of sustaineddeinactivationof inward
current

inactivationof inward
current

inactivationof inward
current

Ca2+
bufferingdeinactivationof inward
current

Ca2+ influxand buildup

depolarization

repolarization

repolarization

deactivationof outward
current

activationof outward
current

depolarization

resting

depolarization

activationof outward
current

Ca2+
buffering
deactivationof outward
current

Ca2+ influxand buildup

spiking

repolarization

spiking

repolarization

restingCa2+-gatedspiking

voltage-gatedspiking

resting

activation ofoutward current

resting

Bursting

inactivation of
inward current

330

depolarization

spiking

restingbuildup of resonant gate or [Ca2+]in(activation of outward current)(inactivation of inward current)

recovery of resonant gate or [Ca2+]in(deactivation of outward current)(deinactivation of inward current)

Figure 9.6: Four major classes of bursting models are defined by the slow resonant
gating variables that modulate spiking activity.
spiking activity, at least when a positive I is injected. To transform an infinite spiketrain into a finite burst of spikes, it suffices to add a slow resonant current or gating
variable (see section 5.1.1) that modulates the spiking via a slow negative feedback. The
resonant gating variable can describe inactivation of an inward current or activationof an outward current, either voltage- or Ca2+ -dependent (see Fig.5.17). Hence, there
are four major classes of bursting models, summarized in Fig.9.6:• Voltage-gated inactivation of an inward current, e.g., slow inactivation of a persistent Na+ current or inactivation of a Ca2+ transient T-current, or inactivationof the h-current (most biologists refer to this as activation of the h-current byhyperpolarization). Repetitive spiking slowly inactivates (turns off) the inward
current, and makes the neuron less excitable and unable to sustain spiking ac-

Bursting331

tivity. After a while, the spiking stops and the membrane potential repolarizes.
The inward current slowly de-inactivates (turns on) and depolarizes the membrane potential, possibly resulting in a new burst.
• Voltage-gated activation of an outward current, e.g., slow activation of a persistent K+ current, such as the M-current. Repetitive spiking slowly activates theoutward current, which eventually terminates the spiking activity. While at rest,
the outward current slowly deactivates (turns off) and unmasks inward currents
that can depolarize the membrane potential, possibly initiating another burst.
• Ca2+ -gated inactivation of an inward current, e.g., slow inactivation of highthreshold Ca2+ -currents ICa(L) or ICa(N) . Entry of calcium during repetitive spiking leads to its intracellular accumulation and slow inactivation of Ca2+ -channels
that provide an inward current needed for repetitive spiking. As a result, theneuron cannot sustain spiking activity, and becomes quiescent. During this period, intracellular Ca2+ ions are removed, Ca2+ channels are de-inactivated, andthe neuron is primed to start a new burst.
• Ca2+ -gated activation of an outward current, e.g., slow activation of the Ca2+ dependent K+ -current IAHP . Calcium entry and buildup during repetitive spiking
slowly activate the outward current and make the neuron less and less excitable.
When the spiking stops, intracellular Ca2+ ions are removed, the Ca2+ -gated outward current deactivates (turns off), and the neuron is no longer hyperpolarized
and is ready to fire a new burst of spikes.
In addition, the slow process may include Na+ -, K+ -, or Cl− -gated currents, such as the“slack and slick” family of Na+ -gated K+ currents, or slow change of ionic concentrations in the vicinity of the cell membrane (the Hodgkin-Frankenhaeuser layer), whichleads to slow change of the Nernst potential for ionic species. We do not elaborate
these cases in this book.
Note that in some cases, the slow process modulates fast currents responsible forspiking, while in other cases it produces an independent slow current that impedes
spiking. In any case, the slow process is directly responsible for the termination ofcontinuous spiking, and indirectly responsible for its initiation and maintenance.
The four mechanisms in Fig.9.6 and their combinations are ubiquitous in neurons,
as we summarize in Fig.9.7. However, there could be other, less obvious burstingmechanisms. In exercises 8–10 we provide examples of bursters having slowly activating
persistent inward current, such as INa,p . These surprising examples show that buildupof the inward current (or any other amplifying gate) can also be responsible for thetermination of the active phase and for the repolarization of the membrane potential.
To understand these mechanisms, one needs to study the geometry of bursting.

332

Bursting
slow dynamics
Ca2+-gatedvoltage-gatedneuronreferences

activation inactivation activation inactivationof outward of inward of outward of inward
neocortical chattering
neurons

IK(M) IKslow

pre-Botzinger complex(respiratory rhythm)

IKslowthalamic relay neurons
thalamic reticular
neurons

Wang (1999)

Butera et al. (1999)

INaslow
ICa(T)

Huguenard andMcCormick (1992)

Ih

ICa(T)

Destexhe et al. (1994)

hippocampal CA3
neurons

IAHP

Traub et al. (1991)

subiculum burstingneurons

IAHP

Stanford et al. (1998)

midbrain dopaminergic
neurons

IK(Ca)

ICa(L)

Amini et al. (1999)

anterior bursting (AB)neuron in lobster
stomatogastric ganglion

IK(Ca)

ICa(L)

Harris-Warrick andFlamm (1987)

ICa(L)

Canavier et al. (1991)

Aplysia abdominal
ganglion R15 neuron

Figure 9.7: Slow dynamics in bursting neurons.

9.1.3

Minimal Models

Let us follow the ideas presented in section 5.1 and determine minimal models forbursting. That is, we are interested in classification of all fast-slow electrophysiologicalmodels that can exhibit sustained bursting activity, as in Fig.9.4b, at least for somevalues of parameters. A bursting model is minimal if removal of any current or gating
variable eliminates the ability to burst.
One way to build a fast-slow minimal model for bursting is to take a minimal model
for spiking, which consists of an amplifying gate and a resonant gate (see Fig.5.17)
and add another slow resonant gate. Since there are many minimal spiking models
in Fig.5.17 and four choices of slow resonant gates in Fig.9.6, there are quite a few
combinations, which fill the squares in Fig.9.8. We present only a few reasonablemodels in the figure; the reader is asked to fill in the blanks. Completing the table
is an excellent test of one’s knowledge and understanding of how different currents
interact to produce nontrivial firing patterns.

Bursting333
Ca2+-gatedvoltage-gatedresonant gateactivationof inward
current

voltage-gatedamplifying gate

ICa(N)

resonant gateactivationof inward
current

ICa(T)+IAHP

ICa+IK+IAHP

inactivationof outward
current

amplifying gate

INa,p+IK+IK(M)

voltage-gatedvoltage-gated
Ca2+-gatedinactivation activation inactivation activationof inward of outward of inward of outward
current
current
current
current

Ca2+-gatedinactivation activationof outwardof inward
current
current

activationof inward
current
inactivationof outward
current

voltage-gatedamplifying gate

Ca2+-gatedinactivation activationof outwardof inward
current
current

activation of outward current

resonant gate
voltage-gated
Ca2+-gatedinactivation activation inactivation activationof inward of outward of inward of outward
current
current
current
current

INa,t+IK(M)

ICa(L)+IK

inactivationof outward
current(fast and slow) INa,p+IK+Ih

Ca2+-gatedinactivation activationof outwardof inward
current
current

activationof inward
current

voltage-gatedvoltage-gated
Ca2+-gatedinactivation activation inactivation activationof inward of outward of inward of outward
current
current
current
current

INa,tslow+IK

inactivationof outward
current

amplifying gate

INa,t+Ih
INa,tCa2+-gatedinactivation activationof outwardof inward
current
current

inactivation of inward current

resonant gate
voltage-gated
Ca2+-gatedinactivation activation inactivation activationof inward of outward of inward of outward
current
current
current
current

Figure 9.8: Some minimal models for bursting.
Some of the minimal models for bursting might seem too bizarre at first glance.
Yet Fig.9.8, upon completion, might prove to be a valuable tool that could allow
experimenters to formulate various ionic hypotheses. For example, if one uses pharmacological agents (e.g., TEA or Ba2+ ) to block Ca2+ -gated K+ channels and shows thatbursting persists, then the possible electrophysiological mechanisms of bursting are
confined to the left column in Fig.9.8. Minimal models in this column would provide
testable hypotheses for the ionic basis of bursting, and they could guide novel experiments. If a block abolishes bursting, we cannot conclude that the blocked current
drives the bursting – it may merely be necessary for providing background stimulation.
Note that the INa,tslow + IK -model and the INa,t + IK(M) -model in the figure (seethe shaded rectangles) consist of the same gating variables: Na+ activation gate m,inactivation gate h, and K+ activation gate n. Both models are equivalent to the334

Bursting
slow inactivation of inward current

slow activation of outward current
120

100

V(t)

V(t)

100

80
80
60
60
40
40
20
20

0
0

500

1000

a

1500

2000

0
0

500

1000 1500 2000 2500 3000 3500b

Figure 9.9: Hodgkin-Huxley (1952) model with three gating variables is minimal forbursting. (Modified from Fig. 1.10 in Izhikevich 2001b.)
Hodgkin-Huxley model, the only difference being the choice of the slow gate. Thus, incontrast to the common biophysical folklore, the Hodgkin-Huxley model is a minimalmodel for bursting, and there are two fundamentally different ways in which one canmake it burst without any additional currents, as we show in Fig.9.9. Of course, one
may argue that the model in the figure is not Hodgkin-Huxley at all, since we changed
the kinetics of some currents by an order of magnitude.
Thinking in terms of minimal models, we can understand what is essential forspiking and bursting and what is not. In addition, we can clearly see that some wellknown conductance-based models form a partially ordered set. For example, the chainof neuronal models Morris-Lecar (ICa +IK ) ≺ Hodgkin-Huxley (INa,t +IK ) ≺ ButeraRinzel-Smith (INa,t +IK +IK,slow ) is obtained by adding a conductance or gating variable
to one model to get the next one. Here, A ≺ B means A is a subsystem of B.
Understanding the ionic bases of bursting is an important step in analysis of bursting dynamics. However, such an understanding may not provide sufficient information
on why the bursting pattern looks as it does, what the neurocomputational propertiesof the neuron are, and how they depend on the parameters of the system. Indeed, we
showed in chapter 5 that spiking models based on quite different ionic mechanisms canhave identical dynamics and vice versa. This is true for bursting models as well.

9.1.4

Central Pattern Generators and Half-Center Oscillators

Bursting can also appear in small circuits of coupled spiking neurons, such as the two
mutually inhibitory oscillators in Fig.9.10, called half-center oscillators. While one cell
fires, the other is inhibited; then they switch roles; and so on. Such small circuits,suggested by Brown (1911), are the building blocks of central pattern generators in thepyloric network of the lobster stomatogastric ganglion, the medicinal leech heartbeat,Bursting335
mutual inhibition

uncoupled
half-center
oscillator

Figure 9.10: Central pattern generation by mutually inhibitory oscillators.
fictive motor patterns, and the swimming patterns of many vertebrates and invertebrates (Marder and Bucher 2001). We show later that this bursting is of cycle-cycle
type.
What makes the oscillators in Fig.9.10 alternate? Wang and Rinzel (1992) suggested
two mechanisms, release and escape, which were later refined to intrinsic and synaptic
by Skinner et al. (1994):• Intrinsic release: The active cell stops spiking, terminates inhibition, and allows
the inhibited cell to fire.
• Intrinsic escape: The inhibited cell recovers, starts to fire, and shuts off the active
cell.
• Synaptic release: The inhibition weakens (e.g., due to spike frequency adaptationor short-term synaptic depression) and allows the inhibited cell to fire.
• Synaptic escape: The inhibited cell depolarizes above a certain threshold andstarts to inhibit the active cell.
All four mechanisms assume that in addition to fast variables responsible for spiking,there are slow adaptation variables responsible for slowing or termination of spiking,recovery, or synaptic depression. Thus, similar to the minimal models above, the circuit
has at least two time scales, that is, it is a fast-slow system.

9.2

Geometry

To understand the neurocomputational properties of bursters, we need to study thegeometry of their phase portraits. In general, it is quite a difficult task. However, it
can be accomplished in the special case of fast-slow dynamics.

Bursting

9.2.1

u
st

in

g

u(t)
Re

y
ilit
sta
b
Bi

Sp
i

kin

g

336

Figure 9.11: Parameter u can control spiking behavior of the fast subsystem in (9.1).
When u changes slowly, the model exhibits
bursting behavior.

Fast-Slow Bursters

We say that a neuron is a fast-slow burster if its behavior can be described by afast-slow system of the formẋ = f (x, u)u̇ = µg(x, u)(fast spiking),(slow modulation).

(9.1)

The vector x ∈ Rm describes fast variables responsible for spiking. It includes themembrane potential V , activation and inactivation gating variables for fast currents,and so on. The vector u ∈ Rk describes relatively slow variables that modulate fast
spiking (e.g., gating variable of a slow K+ current, an intracellular concentration of
Ca2+ ions, etc.). The small parameter µ represents the ratio of time scales between
spiking and modulation. When we analyze models, we assume that µ ≪ 1; that is, it
can be as small as we wish. The results obtained via such an analysis may not haveany sense when µ is of the order 0.1 or greater.
To analyze bursters, we first assume that µ = 0, so that we can consider the fastand slow systems separately.
This constitutes the method of dissection of neuralbursting pioneered by Rinzel (1985). In fact, we have done this many times in theprevious chapters when we substituted m = m∞ (V ) into the voltage equation. The
fast subsystem can be resting (but excitable), bistable, or spiking, depending on thevalue of u; see Fig.9.11. Bursting occurs when u visits the spiking and quiescent areas
periodically. Many important aspects of bursting behavior can be understood via phase
portrait analysis of the fast subsystemẋ = f (x, u),x ∈ Rm ,treating u ∈ Rk as a vector of slowly changing bifurcation parameters.
We say that the burster is of the “m+k” type when the fast subsystem is mdimensional and the slow subsystem is k-dimensional. There are some “1+1” and“2+0” bursters (see exercises 1–4), though they do not correspond to any known neuron. Most of the bursting models in this chapter are of the “2+1” or “2+2” type.

9.2.2

Phase Portraits

Since most bursting models are at least of the “2+1” type, their phase space is atleast three-dimensional. Analyzing and depicting multidimensional phase portraits
is challenging. Even understanding the geometry of the single bursting trajectory
depicted in Fig.9.12 is difficult unless one uses a stereoscope.

337

0

0

−10

−10
membrane potential, V (mV)membrane potential, V (mV)

Bursting

−20
−30
−40
−50
−60
−70

−20
−30
−40
−50
−60
−70

1

−80
−0.02

fast K+ 0.5
activation, n0 0.020.04 0.06
0.08
slow K activation, nslow
+

0

1

−80
−0.02

0 0.020.04 0.06
0.08+
slow K activation, nslow

0.5
fast K+ activation, n0

Figure 9.12: Stereoscopic image of a bursting trajectory of the INa,p +IK +IK(M) -modelin the three-dimensional phase space (V, n, nslow ) (for cross-eye viewing).
In Fig.9.13 we geometrically investigate the INa,p +IK +IK(M) -model, which is a fastslow burster of the “2+1” type. The naked bursting trajectory is shown in the lower
left corner. We set µ = 0 (i.e., τslow (V ) = +∞) and slice the three-dimensional spaceby planes nslow =const, shown in the top right corner. Phase portraits of the twodimensional fast subsystem with fixed nslow are shown in the middle of the figure. Note
how the limit cycle attractors and the equilibria of the fast subsystem depend on thevalue of nslow . Gluing the phase portraits together, we see that there is a manifold oflimit cycle attractors (shaded cylinder) that starts when nslow < 0 and ends in a saddle
homoclinic orbit bifurcation when nslow = 0.066. There is also a locus of stable andunstable equilibria that appears via a saddle-node bifurcation when nslow = 0.0033.
Once we understand the transitions from one phase portrait to another as the slow
variable changes, we can understand the geometry of the burster. Suppose µ > 0 (i.e.,
τslow (V ) = 20 ms), so that nslow can evolve according to its gating equation.
Let us start with the membrane potential at the stable equilibrium corresponding
to the resting state. The parameters of the INa,p +IK +IK(M) -model (see caption of
Fig.9.4) are such that the slow K+ M-current deactivates at rest, that is, nslow slowly
decreases, and the bursting trajectory slides along the bold half-parabola corresponding
to the locus of stable equilibria. After a while, the K+ current becomes so small thatit cannot hold the membrane potential at rest. This happens when nslow passes thevalue 0.0033, the stable equilibrium coalesces with an unstable equilibrium (saddle),
and they annihilate each other via saddle-node bifurcation. Since the resting state nolonger exists (see the phase portrait at the top left of Fig.9.13), the trajectory jumps up
to the stable limit cycle corresponding to repetitive spiking. This jumping corresponds
to the transition from resting to spiking behavior.

338

Bursting

nslow=-0.03

V- n u l l cmembrane potential, V (mV)

n-n

ull
cli

ne

line

nslow=0.0033

0
-20
-40
-60
1
-80
-0.03

0.5

0

slow K +0.03

gatingng

0.06

variab
le, n

ti
+ ga
K
t

0.09 0

,
ble
ria
a
v

n

fas

slow

nslow=0.03

saddlenodebifurcation

nslow=0.06

nslow=0.066

saddlehomoclinic orbit
bifurcation
spiking

nslow=0.09
1

fast K+ activation gate, n

0.8
0.6

V

thresholdrestin
g

t
fas n

0.4
0.2

nslow
0
-80

-60

-40

-20

0

membrane potential, V (mV)

Figure 9.13: Bursting trajectory of the INa,p +IK +IK(M) -model in three-dimensional
phase space and its slices nslow = const.

Bursting339

While the fast subsystem fires spikes, the K+ M-current slowly activates, that is,nslow slowly increases. The bursting trajectory winds up around the cylinder corresponding to the manifold of limit cycles. Each rotation corresponds to firing a spike.
After the ninth spike in the figure, the K+ current becomes so large that repetitive
spiking cannot be sustained. This happens when nslow passes the value 0.066, the limit
cycle becomes a homoclinic orbit to a saddle, and then disappears. The bursting trajectory jumps down to the stable equilibrium corresponding to the resting state. Thisjumping corresponds to the termination of the active phase of bursting and transition
to resting. While at rest, the K+ current deactivates, nslow decreases, and so on.
Figure 9.13 presents the inner structure of the geometrical mechanism of burstingof the INa,p +IK +IK(M) -model with parameters as in Fig.9.4. Other values of the parameters can result in different geometrical mechanisms, summarized in section 9.3. In
all cases, our approach is the same: freeze the slow subsystem by setting µ = 0; analyze phase portraits of the fast subsystem, treating the slow variable as a bifurcation
parameter; glue the phase portraits; let µ ̸= 0 but small; and see how the evolutionof the slow subsystem switches the fast subsystem between spiking and resting states.
The method usually breaks down if µ is not small enough, because evolution of the“slow” variable starts to interfere with that of the fast variable. How small is small
depends on the particulars of the equations describing bursting activity. One should
worry when µ is greater than 0.1.

9.2.3

Averaging

What governs the evolution of the slow variable u? To study this question, we describe
a well-known and widely used method that reduces the fast-slow system (9.1) to its
slow component. In fact, we have already used this method in chapters 3 and 4 toreduce the dimension of neuronal models via the substitution m = m∞ (V ). Usingessentially the same ideas, we take advantage of the two time scales in (9.1) and get
rid of the fast subsystem by means of the substitution x = x(u).
When the neuron is resting, its membrane potential is at an equilibrium and all
fast gating variables are at their steady-state values, so that x = xrest (u). Using thisfunction in the slow equation in (9.1), we obtainu̇ = µg(xrest (u), u)(reduced slow subsystem),(9.2)

which easily can be studied using the geometrical methods presented in Chapters 3
and 4.
Let us illustrate all the steps involved using the INa,p +IK +IK(M) -model, with nslow
being the gating variable of the slow K+ M-current. First, we freeze the slow subsystem,that is, set τslow (V ) = ∞ so that µ = 1/τslow = 0, and numerically determine the resting
potential Vrest as a function of the slow variable nslow . The function V = Vrest (nslow )is depicted in Fig.9.14 (top) and it coincides with the solid half-parabola in Fig.9.13.
Then, we use this function in the gating equation for the M-current to obtain (9.2)
ṅslow = (n∞,slow (Vrest (nslow )) − nslow )/τslow (Vrest (nslow )) = ḡ(nslow ),340

Bursting
0

membrane potential, V (mV)maxspiking-30

Vspike(t,nslow)

min

-60
Vrest(nslow)-65

averaged function g

-70
0.015
g(nslow)spiking

0.010
0.005
0

resting-0.005
-0.02

0

0.02
0.04
0.06
+
slow K gating variable, nslow

0.08

Figure 9.14: Spiking solutions V (t) = Vspike (t, uslow ),resting membrane potential
V = Vrest (nslow ), and the
reduced slow subsystemṅslow = ḡ(nslow ) of theINa,p +IK +IK(M) -model.
The reduction is not valid
in the shaded regions.

0.1

depicted in Fig.9.14 (bottom). Note that ḡ < 0, meaning that nslow decreases while
the fast subsystem rests. The rate of decrease is fairly small when nslow ≈ 0.
A similar method of reduction, with an extra step, can be used when the fastsubsystem fires spikes. Let x(t) = xspike (t, u) be a periodic function corresponding toan infinite spike train of the fast subsystem when u is frozen. Slices of this function are
shown in Fig.9.14 (top). Let T (u) be the period of spiking oscillation. The periodicallyforced slow subsystemu̇ = µg(xspike (t, u), u)(slow subsystem)(9.3)

can be averaged and reduced to a simpler model,ẇ = µḡ(w)(averaged slow subsystem),(9.4)

by a near-identity change of variables w = u + o(µ), where o(µ) denotes small terms oforder µ or less. Here,1ḡ(w) =T (w)Z T (w)g(xspike (t, w), w) dt
0

is the average of g, shown in Fig.9.14 (bottom), for the INa,p +IK +IK(M) -model. (Thereader should check that ḡ(w) = g(xrest (w), w) when the fast subsystem is resting.)membrane
slow K+ activation
potential, V (mV)gate, nslow

Bursting

341

0
-20
-40
-60
0.08

w (averaged)

0.04

u (original)

0
0

10

20

time

30

40

50

Figure 9.15: The INa,p +IK +IK(M) model burster with original andaveraged slow variable.

Limit cycles of the averaged slow subsystem correspond to bursting dynamics, whereas
equilibria correspond to either resting or periodic spiking states of the full system (9.1)– the result is known as the Pontryagin–Rodygin (1960) theorem. Interesting regimes
correspond to the coexistence of limit cycles and equilibria of the slow averaged system.
The main purpose of averaging consists in replacing the wiggle trajectory of u(t)with a smooth trajectory of w(t), as we illustrate in Fig.9.15. We purposely useda different letter, w, for the new slow variable to stress that (9.4) is not equivalent
to (9.3). Their solutions are o(µ)-close to each other only when certain conditions are
satisfied; see Guckenheimer and Holmes (1983) or Hoppensteadt and Izhikevich (1997).
In particular, this straightforward averaging breaks down when u slowly passes thebifurcation values. For example, the period, T (u), of xspike (t, u) may go to infinity, as
happens near saddle-node on invariant circle and saddle homoclinic orbit bifurcations,or transients may take as long as 1/µ, or the averaged system (9.4) is not smooth. All
these cases are encountered in bursting models. Thus, one can use the reduced slow
subsystem only when the fast subsystem is sufficiently far from a bifurcation, that is,away from the shaded regions in Fig.9.14.

9.2.4

Equivalent Voltage

Let us consider a “2+1” burster with a slow subsystem depending only on the slow variable and the membrane potential V , as in the INa,p +IK +IK(M) -model. The nonlinear
equationg(V, u) = ḡ(u)(9.5)
can be solved for V . The solution, V = Vequiv (u), is referred to as the equivalent
voltage (Kepler et al. 1992; Bertram et al. 1995) because it replaces the periodic
function xspike (t, u) in (9.3) with an “equivalent” value of the membrane potential, sothat the reduced slow subsystem (9.3) has the same form,u̇ = µg(Vequiv (u), u)(slow subsystem),(9.6)

as in (9.1). (The reader should check that Vequiv (u) = Vrest (u) when the fast subsystem
is resting.) An interesting mathematical possibility occurs when Vequiv during spiking

342

Bursting

membrane potential, V (mV)0-10l i mi t c y c l e

(max)

-20 V
equiv(nslow)
unstable equilb

-30
-40

n

l i mi t cslow(V)ycle (m

in)

-50
e
saddl

-60
Vequiv(nslow)

-70

0
0.05
K+ activation gate, nslow

nslow(V)

0.1

n ode

0
0.05
K+ activation gate, nslow

0.1

Figure 9.16: Projection of bursting trajectory of the INa,p +IK +IK(M) -model onto the(nslow , V ) plane.
is below Vrest , leading to bizarre bursters having amplifying slow currents, such as theone in exercise 10.
We depict the equivalent voltage of the INa,p +IK +IK(M) -model in Fig. 9.16 (left)(variable u corresponds to nslow ). In the same figure we depict the steady-state activation function n = n∞,slow (V ) (notice the flipped coordinate system). We interpretthe two curves as fast and slow nullclines of the reduced (V, nslow ) system. During theactive (spiking) phase of bursting, the reduced system slides along the upper branchof Vequiv (nslow ) to the right. When it reaches the end of the branch, it falls downward
to the lower branch corresponding to resting, and slides along this branch to the left.
When it reaches the left end of the lower branch, it jumps to the upper branch, andthereby closes the hysteresis loop. Figure 9.16 (right) summarizes all the informationneeded to understand the transitions between resting and spiking states in this model.
It depicts the bursting trajectory, loci of equilibria of the fast subsystem, and the voltage range of the spiking limit cycle as a function of the slow gate nslow . With someexperience, one can read this complicated figure and visualize the three-dimensional
geometry underlying bursting dynamics.

9.2.5

Hysteresis Loops and Slow Waves

Sustained bursting activity of the fast-slow system (9.1) corresponds to periodic (orchaotic) activity of the reduced slow subsystem (9.6). Depending on the dimension ofu, that is, on the number of slow variables, there could be two fundamentally different
ways the slow subsystem oscillates.
If the slow variable u is one-dimensional, then there must be a bistability of restingand spiking states of the fast subsystem so that u oscillates via a hysteresis loop. Thatis, the reduced equation (9.6) consists of two parts: one for Vequiv (u), corresponding

Bursting

343

Sp

iki

ng

activation of outward currents
inactivation of inward currents

u(t)

Re
s

tin

g

u

deactivation of outward currents
deinactivation of inward currents

Re

st

in
g

Sp
i

kin

g

Figure 9.17: Hysteresis loop periodic bursting.

Perturbation

Figure 9.18: Burst excitability: a perturbation causes a burst of spikes.
to spiking, and one for Vequiv (u), corresponding to resting of the fast subsystem, as in
Fig.9.16 (left). Such a hysteresis loop bursting can also occur when u is multidimensional, as we illustrate in Fig.9.17. The vector field on the top (spiking) leaf pushes u
outside the spiking area, whereas the vector field on the bottom (resting) leaf pushes u
outside the resting area. As a result, u visits the spiking and resting areas periodically,
and the model exhibits hysteresis loop bursting.
If resting x does not push u into the spiking area, but leaves it in the bistable area,
then the neuron exhibits burst excitability. It has quiescent excitable dynamics, but
its response to perturbations is not a single spike, rather, it is a burst of spikes, as
we illustrate in Fig.9.18. Grade III bursters of the hippocampus (Fig.8.34Eb) produce
such a response, often called a complex spike response, to brief stimuli. In general,many bistable models are bistable only because they neglect slow currents and otherhomeostatic processes present in real neurons. If the currents are taken into account,then the models become bistable on a short time scale and burst excitable on a longer
time scale. This justifies why many researchers refer to bistable systems as excitable,
implicitly assuming that the response to superthreshold perturbations is either a single
spike or a long train of spikes.
If the fast subsystem does not have a coexistence of resting and spiking states, then344

Bursting

I=0
25 ms

25 mV

I=4.54

I=5

I=7

I=7.6

I=7.7

I=8
I
0

Figure 9.19: Bifurcations of bursting solutions in the INa,p +IK +IK(M) -model as themagnitude of the injected DC current I changes.
the reduced slow subsystem (9.6) must be at least two-dimensional to exhibit sustained
autonomous oscillation (however, see exercise 6). Such an oscillation produces a depolarization wave that drives the fast subsystem to spiking and back, as in Fig.9.3. Werefer to such bursters as slow-wave bursters. Quite often, however, the slow subsystemof a slow-wave burster needs the feedback from the fast subsystem to oscillate. For
example, in section 9.3.2 we consider slow-wave bursting in the INa,p +IK +INa,slow +IK(M) -model, whose slow subsystem consists of two uncoupled equations, and hencecannot oscillate by itself unless the fast subsystem is present.

9.2.6

Bifurcations “Resting ↔ Bursting ↔ Tonic Spiking”

Switching between spiking and resting states during bursting occurs because the slow
variable drives the fast subsystem through bifurcations of equilibria and limit cycle
attractors. These bifurcations play an important role in the classification of bursters
and in understanding their neurocomputational properties. We discuss them in detailin section 9.3.
Since the fast subsystem goes through bifurcations, does this mean that the entire
system (9.1) undergoes bifurcations during bursting? The answer is NO. As long as
parameters of (9.1) are fixed, the system as a whole does not undergo any bifurcations,no matter how small µ is. The system can exhibit periodic, quasi-periodic, or evenchaotic bursting activity, but its (m + k)-dimensional phase portrait does not change.

Bursting345

The only way to make system (9.1) undergo a bifurcation is to change its parameters. For example, in Fig.9.19 we change the magnitude of the injected DC current
I in the INa,p +IK +IK(M) -model. Apparently, no bursting exists when I = 0. Then,
repetitive bursting appears with a large interburst period that decreases as I increases.
The value I = 5 was used to obtain bursting solutions in Fig.9.12 and Fig.9.13. Increasing I further increases the duration of each burst, until it becomes infinite, thatis, bursting turns into tonic spiking. When I > 8, the slow K+ current is not enoughto stop spiking.
In Fig.9.20 we depict the geometry of bursting in the INa,p +IK +IK(M) -model whenI = 3 (just before periodic bursting appears) and when I = 10 (just after burstingturns into tonic spiking).
When I = 3, the nullcline of the slow subsystem nslow = n∞,slow (V ) intersects thelocus of stable equilibria of the fast subsystem. The intersection point is a globallystable equilibrium of the full system (9.1). Small perturbations, whether in the V
direction, the n direction, or the nslow direction, subside, whereas a large perturbation(e.g., in the V direction) that moves the membrane potential to the open square in the
figure initiates a transient (phasic) burst of seven spikes. Increasing the magnitude of
the injected current I shifts the saddle-node parabola to the right. When I ≈ 4.54,
the nullcline of the slow subsystem does not intersect the locus of stable equilibria, andthe resting state no longer exists, as in Fig.9.16 (right). There is still a global steadystate, but it is not stable.
Further increase of the magnitude of the injected current I results in the intersection of the nullcline of the slow subsystem with the equivalent voltage function
Vequiv (nslow ). The intersection, marked by the black circle in Fig.9.20 (right), corresponds to a globally stable (spiking) limit cycle of the full system (9.1). A sufficiently
strong perturbation can push the state of the fast subsystem into the attraction domain of the stable (resting) equilibrium. While the fast subsystem is resting, the slow
variable decreases (i.e., the K+ current deactivates), the resting equilibrium disappears,
and repetitive spiking resumes.
Figures 9.19 and 9.20 illustrate possible transitions between bursting and resting,and bursting and tonic spiking. There could be other routes of emergence of bursting
solutions from resting or spiking; some of them are in Fig.9.21. Each such route corresponds to a bifurcation in the full system (9.1) with some µ > 0. For example, the casea → 0 corresponds to supercritical Andronov-Hopf bifurcation; the case c → ∞ corresponds to a saddle-node on invariant circle or saddle homoclinic orbit bifurcation; thecase d → ∞ corresponds to a periodic orbit with a homoclinic structure, e.g., blue-sky
catastrophe, fold limit cycle on homoclinic torus bifurcation, or something more complicated. The transitions bursting ↔ spiking often exhibit chaotic (irregular) activity,
so Fig.9.21 is probably a great oversimplification. Understanding and classifying all
possible bifurcations leading to bursting dynamics is an important but open problem;see exercise 27.

346

Bursting

I=10

0
-20
-40
-60
0

50
100
time (ms)150

membrane potential, V (mV)membrane potential, V (mV)

I=3
0
-20
-40
-60
0

50
100
time (ms)stable (spiking)limit cycle

spik

ing

V

150

t
fas n

threshold
nslow
rest

ing

0
-20
n

slow(V)-40
saddle

-60

node

0
0.05
0.1
slow K+ gating variable, nslow

membrane potential, V (mV)membrane potential, V (mV)globalrestingstate

0stable (spiking) limit cyclel i mi t c y c l e

-20

(max)

Vequiv(nslow)

l i mi t cycle (m

-40
-60

n

slow(V)in)saddle
node

0
0.05
0.1
slow K+ gating variable, nslow

Figure 9.20: Burst excitability (I = 3, left) and periodic spiking (I = 10, right) in theINa,p +IK +IK(M) -model.

Bursting

347

a

b
d

a

0

c

c

bursting
c0resting

b

0

d

0

d

spiking

Figure 9.21: Possible transitions between repetitive bursting and resting, and repetitive
bursting and repetitive spiking.

9.3

Classification

In Fig.9.22 we identify two important bifurcations of the fast subsystem that are associated with bursting activity in the fast-slow burster (9.1):• (resting → spiking). Bifurcation of an equilibrium attractor that results in transition from resting to repetitive spiking.
• (spiking → resting). Bifurcation of the limit cycle attractor that results in transition from spiking to resting.
The ionic basis of bursting, that is, the fine electrophysiological details, determines thekinds of bifurcations in Fig.9.22. The bifurcations, in turn, determine the neurocomputational properties of fast-slow bursters, discussed in section 9.4.
spiking state(limit cycle)

bifurcation of limit cycle(transition to resting)bifurcation of equilibrium(transition to spiking)resting state (equilibrium)

resting state (equilibrium)

Figure 9.22: Two important bifurcations associated with fast-slow bursting.

348

Bursting

bifurcations of equilibria

bifurcations of limit cycles
saddle-node
on invariant
circlesaddlehomoclinic
orbit

supercritical
AndronovHopf

fold
limit
cycle

saddle-node(fold)

fold/
circlefold/homoclinic

fold/
Hopf

fold/
fold cycle

saddle-node
on invariant
circlecircle/circlecircle/homoclinic

circle/Hopfcircle/fold cycle

supercritical
AndronovHopf

Hopf/
circleHopf/homoclinic

Hopf/
Hopf

Hopf/
fold cycle

subcritical
AndronovHopf

subHopf/
circlesubHopf/
homoclinic

subHopf/
Hopf

subHopf/
fold cycle

Figure 9.23: Classification of planar point-cycle fast-slow bursters based on thecodimension-1 bifurcations of the resting and spiking states of the fast subsystem.
A complete topological classification of bursters based on these two bifurcations isprovided by Izhikevich (2000a), who identified 120 different topological types. Here,we consider only 16 planar point-cycle codimension-1 fast-slow bursters. We say that a
fast-slow burster is planar when its fast subsystem is two-dimensional. We emphasize
planar bursters because they have a greater chance of being encountered in computer
simulations (but not necessarily in nature). We say that a burster is of the pointcycle type when its resting state is a stable equilibrium point and its spiking state is astable limit cycle. All bursters considered so far, including those in Fig.9.1, are of thepoint-cycle type. Other, less common types, such as cycle-cycle and point-point, areconsidered as exercises.
We consider here only bifurcations of codimension 1, that is, those that need only
one parameter and hence are more likely to be encountered in nature. Having a twodimensional fast subsystem imposes severe restrictions on possible codimension-1 bifurcations of the resting and spiking states. In particular, there are only four bifurcationsof equilibria and four bifurcations of limit cycles, which we considered in chapter 6
and summarized in figures 6.46 and 6.47. Any combination of them results in a distinct topological type of fast-slow bursting; hence there are 4 × 4 = 16 such bursters,summarized in Fig.9.23.
We name the bursters according to the types of the bifurcations of the restingand spiking states. To keep the names short, we refer to saddle-node on invariant
circle bifurcation as a “circle” bifurcation because it is the only codimension-1 bifurcation on a circle manifold S1 . We refer to supercritical Andronov-Hopf bifurcation

Bursting349
bifurcation of spiking statesaddlehomoclinic orbit

supercritical
Andronov-Hopf

fold limit cycle"Fold/Circle" Bursting"Fold/Homoclinic" Bursting"Fold/Hopf" Bursting"Fold/Fold Cycle" Bursting"Circle/Circle" Bursting"Circle/Homoclinic" Bursting"Circle/Hopf" Bursting"Circle/Fold Cycle" Bursting"Hopf/Circle" Bursting"Hopf/Homoclinic" Bursting"Hopf/Hopf" Bursting"Hopf/Fold Cycle" Bursting"SubHopf/Circle" Bursting"SubHopf/Homoclinic"
Bursting"SubHopf/Hopf" Bursting"SubHopf/Fold Cycle"
Bursting

saddle-node on
invariant circle
supercritical
Andronov-Hopf
subcritical
Andronov-Hopfbifurcation of resting statesaddle-node (fold)saddle-node on
invariant circle

Figure 9.24: Examples of “2+1” point-cycle fast-slow codimension-1 bursters ofhysteresis-loop type (modified from Izhikevich 2000a). Dashed chains of arrows showtransitions that might involve bifurcations not relevant to the bursting type.

350

Bursting

sp

ikin

g

fold
bifurcationxufold bifurcationrestsaddlehomoclinic
orbit
bifurcation

saddle homoclinic
orbit bifurcation

Figure 9.25: “Fold/homoclinic” bursting. The resting state disappears via saddle-node(fold) bifurcation, and the spiking limit cycle disappears via saddle homoclinic orbit
bifurcation.
as just the “Hopf” bifurcation, the subcritical Andronov-Hopf as the “subHopf”, thefold limit cycle bifurcation as the “fold cycle”, and the saddle homoclinic orbit bifurcation as the “homoclinic” bifurcation. Thus, the bursting pattern exhibited by theINa,p +IK +IK(M) -model in Fig.9.13 is of the “fold/homoclinic” type because the restingstate disappears via “fold” bifurcation, and the spiking limit cycle attractor disappears
via saddle “homoclinic” orbit bifurcation.
In a way similar to Fig.9.13, we depict the geometry of the other bursters in Fig.9.24.
This figure gives only examples, and does not exhaust all possibilities. Let us consider
some of the most common bursting types in detail.

9.3.1

Fold/Homoclinic

When the resting state disappears via a saddle-node (fold) bifurcation and the spiking
limit cycle disappears via a saddle homoclinic orbit bifurcation, the burster is said toBursting

351

Figure 9.26: Putative “fold/homoclinic” bursting in a pancreatic β-cell. (Modified
from Kinard et al. 1999.)

0 mV
20 mV
1 sec

Figure 9.27: Putative ”fold/homoclinic” bursting in a cell located in the pre-Botzinger
complex of rat brain stem. (Data shared by Christopher A. Del Negro and Jack L.
Feldman, Systems Neurobiology Laboratory, Department of Neurobiology, UCLA.)
be of the “fold/homoclinic” type depicted in Fig.9.25. Note the bistability of restingand spiking states, resulting in the hysteresis loop oscillation of the slow subsystem.
“Fold/homoclinic” bursting is quite common in neuronal models, such as in theINa,p +IK +IK(M) -model considered in this chapter; see Fig.9.13. It was first characterized in the context of the insulin-producing pancreatic β-cells in Fig.9.26, with intracellular concentration of Ca2+ ions being the slow resonant variable (Chay and Keizer1983). Neurons located in the pre-Botzinger complex, a region that is associated withgenerating the rhythm for breathing, also exhibit this kind of bursting (Butera et al.
1999), as shown in Fig.9.27. Intrinsic bursting (IB) and chattering (CH) behavior of
the simple model in section 8.2 could be of the “fold/homoclinic” type too, provided
the parameter a is sufficiently small. Because of the distinct square-wave shape ofoscillations of the membrane potential in figures 9.26 and Fig.9.27, this bursting wascalled “square-wave” bursting in earlier studies. Since many types of bursters resemble
square waves, referring to a burster by its shape is misleading and should be avoided.
In Fig.9.25 (bottom) we depict a typical configuration of nullclines of the fast subsystem during “fold/homoclinic” bursting. The resting state of the membrane potential
corresponds to the left stable equilibrium, which is the intersection of the left knee ofthe fast N-shaped nullcline with the slow nullcline. During resting, the N-shape null-Burstingcircle/circle
bursting
circlesaddle-node on
invariant circle bifurcation352

saddle-nodehomoclinic orbit
bifurcation

fold/circle
bursting

r
ifu

ca

oclin
hom

ic
fold/homoclinic
bursting

fold

nic
ocli
hom

it b
orb

saddle-node bifurcation

ti o

n

circle/homoclinic
bursting

Figure 9.28: A neural system near codimension-2 saddle-node homoclinic orbit bifurcation (center dot) can exhibit four different types of fast-slow bursting, depending on
the trajectory of the slow variable u ∈ R2 in the two-dimensional parameter space.
Solid (dotted) lines correspond to spiking (resting) regimes.

cline slowly moves upward, until its knee touches the slow nullcline at a saddle-nodepoint. Right after this moment, the resting state disappears via saddle-node (fold)bifurcation; hence the “fold” in the name of the burster. After the fold bifurcation,
the membrane potential jumps up to the stable limit cycle corresponding to repetitive
spiking. During the spiking state, the N-shaped nullcline slowly moves downward, andthe middle (saddle) equilibrium moves away from the resting state toward the limit
cycle. After a while, the limit cycle becomes a homoclinic trajectory to the saddle,
and then the cycle disappears via saddle homoclinic orbit bifurcation; hence the “homoclinic” in the name of the burster. After this bifurcation, the membrane potential
jumps down to the resting state and closes the hysteresis loop.
Suppose that the hysteresis loop oscillation of the slow variable u has a small amplitude. That is, the saddle-node bifurcation and the saddle homoclinic orbit bifurcation
occur for nearby values of the parameter u. In this case, the fast subsystem of (9.1) isnear codimension-2 saddle-node homoclinic orbit bifurcation, depicted in Fig.9.28 andstudied in section 6.3.6. The figure shows a two-parameter unfolding of the bifurcation,
treating u ∈ R2 as the parameter. A stable equilibrium (resting state) exists in the left
half-plane, and a stable limit cycle (spiking state) exists in the right half-plane of the
figure and in the shaded (bistable) region. “Fold/homoclinic” bursting occurs when
the bifurcation parameter, being a slow variable, oscillates between the resting andspiking states through the shaded region. Due to the bistability, the parameter could
be one-dimensional. Other trajectories of the slow parameter correspond to other typesof bursting.

Bursting353

10

spiking
fast variable, v

86homoclinic

42

thresholdreset
0restingfoldslow variable, u30.5

homoclinic

2
1
0

1

1.5

2

2.5

slow variable, u

fold
0

50

time

100

150

Figure 9.29: “Fold/homoclinic” bursting in the canonical model (9.7) with parametersµ = 0.02, I = 1, and d = 0.2.
In exercise 16 we prove that there is a piecewise continuous change of variables thattransforms any “fold/homoclinic” burster with a fast subsystem near such a bifurcation
into the canonical model (see section 8.1.5)v̇ = I + v 2 − u ,u̇ = −µu ,(9.7)

with an after-spike resetting
if v = +∞, then v ← 1 and u ← u + d.
Here v ∈ R is the re-scaled membrane potential of the neuron; u ∈ R is the re-scaled
net outward (resonant) current that provides a negative feedback to v; and I, d, andµ ≪ 1 are parameters. This model is related to the canonical model considered in
section 8.1.4, and it is simplified further in exercise 15.
The fast subsystem v̇ = (I − u) + v 2 is the normal form for the saddle-node bifurcation, and with the resetting it is known as the quadratic integrate-and-fireneuron√(section 3.3.8). When u > I, there is a stable equilibrium vrest = − u − I corresponding to the resting state. While the parameter u slowly decreases
toward u = 0,√
the stable equilibrium and the saddle equilibrium vthresh = + u − I approach andannihilate each other at u = I via saddle-node (fold) bifurcation. When u < I, themembrane potential v increases and escapes to infinity in a finite time, that is, it firesa spike. (Instead of infinity, any large value can be used in simulations.) The spike
activates fast outward currents and resets v to 1, as in Fig.9.29. It also activates slow
currents and increments u by d. If the reset value 1 is greater than the threshold potential vthresh , the fast subsystem fires another spike, and so on, even when u > I; see354

Bursting
sp

iki
n

g

saddle-node on
invariant circle bifurcation

x

re

st

saddle-node on invariant
circle bifurcationusaddle-node
on invariant
circle bifurcation

saddle-node
on invariant
circle bifurcation

Figure 9.30: “Circle/circle” bursting. The resting state disappears via saddle-node on
invariant circle bifurcation, and so does the spiking limit cycle.
Fig.9.29. Since each spike increases u, the repetitive spiking stops when u = I + 1 via
saddle homoclinic orbit bifurcation. The membrane potential jumps downward to theresting state, the hysteresis loop is closed, and the variable u decreases (recovers) toinitiate another “fold/homoclinic” burst. One can vary I in the canonical model (9.7)
to study transitions from quiescence to bursting to tonic spiking, as in Fig.9.20.

9.3.2

Circle/Circle

When the equilibrium corresponding to the resting state disappears via a saddle-node
on invariant circle bifurcation, and the limit cycle attractor corresponding to the spiking state disappears via another saddle-node on invariant circle bifurcation, the burster
is said to be of the “circle/circle” type shown in Fig.9.30. Since the bifurcation does notproduce a coexistence of attractors, there is usually no hysteresis loop, and the bursting is of the slow-wave type with at least two slow variables. (An unusual example
of “circle/circle” hysteresis loop bursting in a “2+1” system is provided by Izhikevich(2000a)).
“Circle/circle” bursting is a prominent feature of the R15 cells in the abdominal ganglion of the mollusk Aplysia, shown in Fig.9.31 (Plant 1981). It was called “parabolic”
bursting in earlier studies because the interspike period depicted in Fig.9.32 was erroneously thought to be a parabola. In section 6.1.2 we showed that when a system√
undergoes a saddle-node on invariant circle bifurcation, its period scales as 1/ λ,where λ is the distance to the bifurcation. Two pieces of this function, put together

Bursting

355

Figure 9.31: Putative “circle/circle” bursting pacemaker activity of neuron R15 in theabdominal ganglion of the mollusk Aplysia. (Modified from Levitan and Levitan 1988.)
2.5

interspike frequency (Hz)interspike period (sec)

2

1.5

1

0.5

0

0

5

10

spike number

15

2
1.5
1
0.5
0

0

5

10

1

spike number

Figure 9.32: The interspike period in the “circle/circle” bursting in Fig.9.31 resembles
a parabola, which led to the name “parabolic bursting” used in earlier studies.
as in Fig.9.32, do indeed resemble a parabola. But so does the interspike period of a“circle/homoclinic” burster.
To transform the INa,p +IK -model to a “circle/circle” burster, take the parametersin Fig.4.1a so that there is a saddle-node on invariant circle bifurcation when I = 4.51
(see section 6.1.2). Its nullclines and phase portrait look similar to those in Fig.9.30.
Then, add one amplifying and one resonant current with gating variablesṁslow = (m∞,slow (V ) − mslow )/τNa,slow (V ) (slow INa,slow ),ṅslow = (n∞,slow (V ) − nslow )/τK(M) (V )(slow IK(M) ),having parameters as in Fig.9.33. Note that these equations are uncoupled and hencecannot oscillate by themselves without the feedback from variable V .
Let us describe the bursting mechanism in the full INa,p +IK +INa,slow + IK(M) -modelwith I = 5. Since I > 4.51, the resting state of the fast subsystem does not exist,
and the model generates action potentials, depicted in Fig.9.33a. Each spike activates
INa,slow , producing even more inward current and, hence, more spikes. This, however,activates a much slower K+ current (see Fig.9.33b) and produces a net outward current
that moves the fast nullcline downward and eventually terminates spiking. The transition from spiking to resting occurs via saddle-node on invariant circle bifurcation.
While at rest, both currents deactivate and the fast nullcline slowly moves upward.
The net inward current, consisting mostly of the injected DC current I = 5, drives thefast subsystem via the same saddle-node on invariant circle bifurcation and initiates
another burst, as shown in Fig.9.33a.
Using the averaging technique described in section 9.2.3, one can reduce the fourdimensional INa,p + IK + INa,slow + IK(M) -model to a simpler, two-dimensional slow

356

Bursting(c) 0.12

-20

-40

-60

-80

0

100

200
time (ms)

300

0.1
0.08
0.06
re

in
st

g

0.04
0.02
0

400

(b)

ing

K+ activation gate, nslow

membrane potential, V (mV)

0

sp
ik

(a)

0

0.05
0.1
0.15
0.2
Na+ activation gate, mslow

0.25

(d)

0.25

0.1
0.05

200
time (ms)

300

400

-40
-60
-80

0.2

K + a 0.1
ctiva
tion g

g

100

-20

in

0

0

st

0

nslow(t)20

remembrane potential, V (mV)

activation gate

0.15

ng
iki
sp

mslow(t)0.2

0.05
ate, nslow

0 0

Na+ activation
gate, mslow

Figure 9.33: “Circle/circle” bursting in the INa,p +IK +INa,slow +IK(M) -model. Parameters of the fast INa,p +IK -subsystem are the same as in Fig.4.1a with I = 5. Slow Na+
current has V1/2 = −40 mV, k = 5 mV, gNa,slow = 3, τNa,slow (V ) = 20 ms. Slow K+
current has V1/2 = −20 mV, k = 5 mV, gK(M) = 20, τK(M) (V ) = 50 ms.
INa,slow +IK(M) -subsystem of the form (9.4). Bursting of the full model corresponds toa limit cycle attractor of the averaged slow subsystem depicted as a bold curve on the(mslow , nslow ) plane in Fig.9.33c. Superimposed is the projection of the bursting solutionof the full system (thin, wobbly curve). In Fig.9.33d we project a four-dimensional
bursting trajectory onto the three-dimensional subspace (V, mslow , nslow ).
The INa,p +IK +INa,slow +IK(M) -model in Fig.9.33 has a remarkable property: it generates slow-wave bursts even though its slow INa,slow + IK(M) -subsystem consists of
two uncoupled equations, and hence cannot oscillate by itself! Another example ofthis phenomenon is presented in exercise 12. Thus, the slow wave that drives the fastINa,p +IK -subsystem through the two circle bifurcations is not autonomous: it needs
feedback from V . In particular, the oscillation will disappear in a voltage-clamp experiment, that is, when the membrane potential is fixed.
Now consider a “circle/circle” burster with a slow subsystem performing smallamplitude oscillations so that the fast subsystem is always near the saddle-node on
invariant circle bifurcation. If the slow subsystem has an autonomous limit cycle at-Bursting357

fast variable, v

10
8
6
4
2

unstable0

atoll

spiking0

sin ystablerestingphase of slow oscillation, y

2p

Figure 9.34: “Circle/circle” burstingin the Ermentrout-Kopell canonicalmodel (9.8) with r(ψ) = sin ψ andω = 0.1. The fast variable fires spikes
while sin ψ > 0 and is quiescent
while sin ψ < 0. The shaded atoll issurroundedby the equilibria curvesp± | sin ψ|. The fast subsystem undergoes saddle-node on invariant circle
bifurcation when sin ψ = 0.

tractor that exists without feedback from V , then such a burster can be reduced to theErmentrout-Kopell (1986) canonical modelv̇ = v 2 + r(ψ) ,ψ̇ = ω ,if v = +∞, then v = −1,(9.8)

which was originally written in the ϑ-form; see exercise 13. Here, ψ is the phase of
autonomous oscillation of the slow subsystem, ω ≈ 0 is the frequency of the slow
oscillation, and r(ψ) is a periodic function that changes sign and slowly drives the fast
quadratic integrate-and-fire neuron (9.8) back and forth through the bifurcation, as
illustrated in Fig.9.34.
Alternatively, suppose that the slow subsystem cannot have sustained oscillations
without the fast subsystem, that is, the slow subsystem has a stable equilibrium if v isfixed. In exercise 17 we prove that there is a piecewise continuous change of variables
that transforms any such “circle/circle” burster into one of the two canonical models
below, depending on the type of equilibrium. If the equilibrium of the slow subsystem
is a stable node, then the canonical model has the formv̇ = I + v 2 + u1 − u2 ,u̇1 = −µ1 u1 ,u̇2 = −µ2 u2 .

(9.9)

If the equilibrium of the slow subsystem is a stable focus, then the canonical modelhas the formv̇ = I + v 2 + u1 ,u̇1 = −µ1 u2 ,(9.10)
u̇2 = −µ2 (u2 − u1 ) ,with µ2 < 4µ1 . In both cases, there is an after-spike resetting:if v = +∞, then v ← −1, and (u1 , u2 ) ← (u1 , u2 ) + (d1 , d2 ).
Similar to (9.7), the variable v ∈ R is the re-scaled membrane potential of the neuron.
The positive feedback variable u1 ∈ R describes activation of slow amplifying currents

358

Bursting
stable node
10

4

resting

6
4

slow variable, u2

fast variable, v

8

2

slow variables

0
reset

-2
4

=I
u2

2

+u

1

spiking0equilibrium ofslow subsystem

u2

2
0

u1
0

50

100

150

200

-2

250

0

2

4

slow variable, u1

timestable focus
10

4

2

4

slow variable, u2

fast variable, v

6

2
0
reset

slow variables

spiking

resting

8

u2

2

0

-2

u1

0

u1=-I

-2
0

20

40
time

60

80

-4

-2

0

2

4

slow variable, u1

Figure 9.35: “Circle/circle” bursting in the canonical models (9.9) (top, parameters:
I = 1, µ1 = 0.1, µ2 = 0.02, d1 = 1, d2 = 0.5) and (9.10) (bottom, parameters:
I = 1, µ1 = 0.2, µ2 = 0.1, d1 = d2 = 0.5).
or potential at a dendritic compartment, whereas the negative feedback variable u2 ∈ R
describes activation of slow resonant currents. I, d1 , d2 , and µ1 , µ2 ≪ 1 are parameters.
When µ2 > 4µ1 , the equilibrium of the slow subsystem in (9.10) is a stable node,so (9.10) can be transformed into (9.9) by a linear change of slow variables. If d1 = 0,then u1 → 0 and (9.9) is equivalent to (9.7).
Both canonical models above exhibit “circle/circle” slow-wave bursting, as depicted
in Fig.9.35. When I > 0, the equilibrium of the slow subsystem is in the shaded
area corresponding to spiking dynamics of the fast subsystem. When the slow vector(u1 , u2 ) enters the shaded area, the fast subsystem fires spikes, prevents the vector
from converging to the equilibrium, and eventually pushes it out of the area. While
it is outside, the vector follows the curved trajectory of the linear slow subsystem andthen reenters the shaded area. Such a slow wave oscillation corresponds to the thicklimit cycle attractor in Fig.9.35, which looks remarkably similar to the one for theINa,p +IK +INa,slow +IK(M) -model in Fig.9.33.

Bursting359
subcritical
Andronov-Hopfbifurcation

fold limit
cycle bifurcation
subcritical
Andronov-Hopfbifurcation

fold limit cycle
bifurcation

Figure 9.36: “SubHopf/fold cycle” burster: The middle equilibrium corresponding to
the resting state loses stability via subcritical Andronov-Hopf bifurcation, and the outer
limit cycle attractor corresponding to repetitive spiking disappears via fold limit cycle
bifurcation. The two top images are different views of the same 3-D structure.

9.3.3

SubHopf/Fold Cycle

When the resting state loses stability via subcritical Andronov-Hopf bifurcation, andthe spiking state disappears via fold limit cycle bifurcation, the burster is said to be of
the “subHopf/fold cycle” type depicted in Fig.9.36. Because there is a coexistence ofresting and spiking states, such bursting usually occurs via a hysteresis loop with only
one slow variable.
This kind of bursting was one of the three basic types identified by Rinzel (1987). It
was called “elliptic” in earlier studies because the profile of oscillation of the membranepotential resembles an ellipse, or at least a half-ellipse; see Fig.9.37. Rodent trigeminalinterneurons in Fig.9.38, and dorsal root ganglia and mesV neurons in Fig.9.39 are“subHopf/fold cycle” bursters, yet the bursting profiles do not look like ellipses. Manymodels of “subHopf/fold cycle” bursters do not generate elliptic profiles either; hence,referring to this type of bursting by its shape is misleading and should be avoided.
It is quite easy to transform the INa,p +IK -model into a “subHopf/fold cycle” burster.
First, we chose the parameters of the model as in Fig.6.16, so that the phase portrait
depicted in Fig.9.40 is the same as in Fig.9.36 (bottom). The coexistence of the stableequilibrium, an unstable limit cycle, and a stable limit cycle is essential for producing

360

Bursting
r
r(t)

1
-1

0

u

Re z(t)slow passage
effectrr(t)1

Re z(t)-1

0uFigure 9.37: Phase portrait and solution of the canonical model (9.11) for µ = 0.1, ω =3, and a = 0.25 (top) and a = 0.8 (bottom), where r = |z| is the amplitude of oscillation(modified from Izhikevich 2000b).

10 mV-39 mV

Figure 9.38: Putative “subHopf/fold cycle” bursting in rodent trigeminal neurons.
(Modified from Del Negro et al. 1998.)

Figure 9.39: Putative “subHopf/fold cycle” bursting in (a) injured dorsal root ganglion(data modified from Jian et al. 2004) and in (b) rat mesencephalic layer 5 neurons.

Bursting

361

c
cy

le

n-nu

sta
ble

llclin

e

it

e

un
sta

t cycle
imi

lin

el
bl

llc

0.5

nu
V-

K+ activation variable, nlim

1stable equilibrium (rest)

0
-80

-70

-60

-50

-40

-30

-20

membrane voltage, V (mV)

-10

0

10

Figure 9.40: Phase portrait of
the INa,p +IK -model with parameters corresponding to subcritical
Andronov-Hopf bifurcation andfold limit cycle bifurcation.

the hysteresis loop oscillation. Then, we add a slow K+ M-current that activates while
the fast subsystem fires spikes, and deactivates while it is resting. Such a resonant current provides a negative feedback to the fast subsystem, and the full INa,p +IK +IK(M) model exhibits “subHopf/fold cycle” bursting, shown in Fig.9.41.
As in the previous examples, the burster in this figure is conditional: it needs
an injection of a DC current I, so that the equilibrium corresponding to the restingstate of the fast subsystem is unstable. If the subsystem is near such an equilibrium,
it slowly diverges from the equilibrium and jumps to the large-amplitude limit cycle
attractor corresponding to spiking behavior, as one can see in Fig.9.41a. Each spike
activates a slow K+ M-current, see Fig.9.41b, and results in the buildup of a net outward
current that makes the fast subsystem less and less excitable. Geometrically, the largeamplitude limit cycle attractor is approached by a smaller-amplitude unstable limit
cycle; they coalesce, and annihilate each other via fold limit cycle bifurcation at nslow ≈0.14; see Fig.9.41c. The trajectory jumps to the stable equilibrium corresponding to
the resting state. At this moment, the slow K+ current starts to deactivate, and thenet outward current decreases. Since the activation gate nslow moves in the opposite
direction, the fold limit cycle bifurcation gives birth to large-amplitude stable andunstable limit cycles, but the trajectory remains on the steady-state branch. The
unstable limit cycle slowly shrinks, and makes the resting equilibrium lose stability via
subcritical Andronov-Hopf bifurcation. Once the resting state becomes unstable, thetrajectory diverges from it and jumps back to the large-amplitude limit cycle, thereby
closing the hysteresis loop.
A prominent feature of “subHopf/fold cycle” bursting, as well as any other typeof fast-slow bursting involving Andronov-Hopf bifurcation (“subHopf/*” or “Hopf/*”,where the wild card “*” means any bifurcation) is that the transition from resting tospiking does not occur at the moment the resting state becomes unstable. The fastsubsystem remains at the unstable equilibrium for quite some time before it jumps
rather abruptly to a spiking state, as we can clearly see in Fig.9.41. This delayed

362

Bursting(c)fold cycle0membrane potential, V (mV)membrane potential, V (mV)

(a)

-20
subHopf-40-60
delayed transition
-80

0

50

100

150 200
time (ms)

250

300

(b)

limit cycle (max0-20

fold cycle)

delayedtransition

-40

-60

subHopf
limit cycle (min)

-80

0

0.05
0.1
0.15
slow K+ activation gate, nslow(d)fold limit cycle
bifurcation

membrane potential, V (mV)slow K+ activation gate, nslow

0.2

0.15
subcritical
Andronov-Hopfbifurcation

0.1

0.05

0

0

50

100

150 200
time (ms)

250

300

0
-20
-40
-60
-80

0
slo
ga w K +te,
0.1
ns actlow iva

1

tio

n

0.2 0

0.5 ationtiv+ ac , nt K gatesfa

Figure 9.41: “SubHopf/fold cycle” bursting in the INa,p +IK +IK(M) -model. Parametersof the fast INa,p +IK -subsystem are the same as in Fig.6.16 with I = 55. Slow K+
M-current has V1/2 = −20 mV, k = 5 mV, τ (V ) = 60 ms, and gK(M) = 1.5.
transition is due to the slow passage through Andronov-Hopf bifurcation, discussed in
section 6.1.4. Delayed transitions through Andronov-Hopf bifurcation are ubiquitousin neuronal models, but they have never been seen in real neurons. Conductance noise,always present at physiological temperatures, constantly kicks the membrane potential
away from the stable equilibrium, as one can see in the inset in Fig.9.38, so transition
to spiking in real neurons is never delayed. Instead, it can occur even before theequilibrium becomes unstable, as we show in section 6.1.4.
Suppose that the hysteresis loop oscillation of the slow variable has a small amplitude. That is, the subcritical Andronov-Hopf bifurcation and the fold limit cycle
bifurcation of the fast subsystem in (9.1) occur for nearby values of the parameteru. In this case, the fast subsystem is near a codimension-2 Bautin bifurcation, which
was studied in section 6.3.5. Its two-parameter unfolding is depicted in Fig.9.42 (left).
“SubHopf/fold cycle” bursting occurs when the bifurcation parameter, being a slow
variable, oscillates between the resting and spiking regions through the shaded area.

363

uil
ibr
ble
sta

un

unst
a
limit
cycle

eq

eq
ble
sta

ium

lim
sta
b

uil
ibr

le

ium

Bautinbifurcation

it c
yc
le

Bursting|V|usupercritical Andronov-Hopfbifurcation

Hopf

Hopf/Hopf
bursting
subHopf/Hopfbursting
Hopf/fold cycle
bursting

subHopf/fold cycle
bursting
fold limit cycle
bifurcation

subcritical
Andronov-Hopfbifurcation

fold cycle

subHopf

Figure 9.42: A neural system near codimension-2 Bautin bifurcation (central dot) can
exhibit four different types of fast-slow bursting, depending on the trajectory of theslow variable u ∈ R2 in the parameter space. The “subHopf/fold cycle” bursting
occurs via a hysteresis loop and requires only one slow variable. Solid (dotted) lines
correspond to spiking (resting) regimes. (Modified from Izhikevich 2000a.)

Due to the bistability, the parameter could be one-dimensional. Other trajectories ofthe slow parameter correspond to other types of bursting shown in Fig.9.42 (right).
If the slow variable has an equilibrium near the Bautin bifurcation point, then the
fast-slow burster (9.1) can be transformed into the following canonical “2+1” modelby a continuous change of variablesż = (u + iω)z + 2z|z|2 − z|z|4 ,u̇ = µ(a − |z|2 ),(9.11)

where z ∈ C and u ∈ R are the canonical fast and slow variables, respectively, anda, ω, and µ ≪ 1 are parameters. In exercise 14 we show that the model exhibits
the hysteresis loop periodic point-cycle bursting behavior depicted in Fig.9.37 when
0 < a < 1.

364

Bursting

sp
ikin

g

saddlehomoclinic
orbit
bifurcation

foldbifurcationresxfold limit cycle
bifurcation

t

u

fold

saddlehomoclinic orbit

fold limit
cycle

Figure 9.43: “Fold/fold cycle” bursting. The resting state disappears via saddle-node(fold) bifurcation, and the spiking limit cycle disappears via fold limit cycle bifurcation.
(Modified from Izhikevich 2000a.)

9.3.4

Fold/Fold Cycle

When the stable equilibrium corresponding to the resting state disappears via saddlenode (fold) bifurcation and the limit cycle attractor corresponding to the spiking state
disappears via fold limit cycle bifurcation, the burster is said to be of the “fold/foldcycle” type, as in Fig.9.43. This type was first discovered in the Chay-Cook (1988)
model of a pancreatic β-cell by Bertram et al. (1995), who referred to it as being
Type IV bursting (the three bursters we have considered so far are referred to as
being Types I, II, and III). Since both bifurcations result in a coexistence of restingand spiking states, the “fold/fold cycle” bursting can occur via a hysteresis loop in a“2+1” system.
An interesting geometrical feature of the “fold/fold cycle” bursting is that an unstable limit cycle appears in the middle of a burst and participates in the “fold cycle”
bifurcation to terminate the burst. The cycle appears via saddle homoclinic orbit bifurcation in Fig.9.43, but other scenarios are possible. It is a good exercise of one’s
geometrical intuition and understanding of the fast-slow bursting mechanisms to develop alternative scenarios of the “fold/fold cycle” bursting. For example, consider thecase of the unstable limit cycle being inside the stable one.

Bursting

365
sp
iki

ng

supercritical
Andronov-Hopfbifurcation

foldbifurcation

foldbifurcation

x

res

u

tin

g

fold

supercritical
Andronov-Hopf
fold

Figure 9.44: “Fold/Hopf” bursting: The resting state disappears via saddle-node (fold)bifurcation, and the spiking limit cycle shrinks to a point via supercritical AndronovHopf bifurcation. (Modified from Izhikevich 2000a.)

9.3.5

Fold/Hopf

When the stable equilibrium corresponding to the resting state disappears via saddlenode (fold) bifurcation and the limit cycle attractor corresponding to the spiking stateshrinks to a point via supercritical Andronov-Hopf bifurcation, the burster is said to beof the “fold/Hopf” type (see Fig.9.44). This type of bursting, called “tapered” in someearlier studies, was found in models of insulin-producing pancreatic β-cells (Smolen etal. 1993; Pernarowski 1994) and in models of certain enzymatic systems (Holden and
Erneux 1993a, 1993b).
As one can see in the figure, the fast subsystem undergoes two bifurcations whilein the excited state: One corresponds to the termination of repetitive spiking via
supercritical Andronov-Hopf bifurcation, and the other corresponds to the transition
from the excited equilibrium to resting equilibrium via saddle-node (fold) bifurcation.
The first bifurcation (i.e., bifurcation of a spiking limit cycle attractor) determines the
topological type of bursting. The second bifurcation is essential for the “fold/fold”
hysteresis loop, and it determines only the subtype of the “fold/Hopf” bursting. Using
ideas described in exercise 19, one can come up with another subtype of “fold/Hopf”
burster having a “fold/subHopf” hysteresis loop.

366

Bursting
sp

iki

ng

saddle-node on
invariant circle
bifurcation

foldbifurcation

foldbifurcation

x
u

res

tin

g

Saddle-Node on
Invariant Circle

Fold

Fold

Figure 9.45: “Fold/circle” bursting: The resting state disappears via fold bifurcationand the spiking state disappears via saddle-node on invariant circle bifurcation. (Modified from Izhikevich 2000a.)

9.3.6

Fold/Circle

When the stable equilibrium corresponding to the resting state disappears via saddlenode (fold) bifurcation and the limit cycle attractor corresponding to the spiking state
disappears via saddle-node on invariant circle bifurcation, the burster is said to be of
the “fold/circle” type (see Fig.9.45). This type was first discovered in the model of thethalamocortical relay neuron by Rush and Rinzel (1994), and it was called “triangular”
in earlier studies (Wang and Rinzel 1995) because of the shape of the voltage envelope.
As one can see in the figure, the fast subsystem can have five equilibria, two of which
are stable nodes. This is a consequence of the quintic shape of the V -nullcline of thefast subsystem. While the trajectory is at the lower equilibrium, the V -nullcline moves
upward, the equilibrium disappears via fold bifurcation, and the fast subsystem starts
to fire spikes. During this active period, the V -nullcline slowly moves downward, andthe spiking limit cycle disappears via saddle-node on invariant circle bifurcation. The
fast subsystem, however, is at the second stable equilibrium corresponding to a depo-

Bursting367

larized state. The slow V -nullcline continues to move downward, and this equilibrium
disappears via another fold bifurcation, thereby closing the “fold/fold” hysteresis loop.
Alternatively, “fold/circle” bursting can be of the slow-wave type, depicted in Fig.9.28
having only three equilibria. The slow subsystem needs to be at least two-dimensionalin this case, however.

9.4

Neurocomputational Properties

There is more to the topological classification of bursters than just a mathematical
exercise. Indeed, in chapter 7 we showed that the neurocomputational properties ofan excitable system depend on the type of bifurcation of the resting state. The same
is valid for a burster: its neurocomputational properties depend on the kinds of bifurcations of the resting and spiking states, that is, on the burster’s type. Knowing
the topological type of a given bursting neuron, we know what the neuron can do and, more importantly, what it cannot do – regardless of the model that describes its
dynamics.

9.4.1

How to Distinguish?

First, we stress that the topological classification of bursters provided in the previous
section is defined for mathematical models, and not for real neurons. Moreover, thetypes are defined for models of the fast-slow form (9.1) assuming that the ratio of timescales, µ, is sufficiently small. Not all neurons can be described adequately by suchmodels, so extending the classification to those neurons may be worthless. A typicalexample of classification failure is the model of bursting of the sensory processing
neuron in weakly electric fish, known as the “ghostburster” (Doiron et al. 2002), in
which µ > 0.1.
If a bursting neuron can be described accurately by a model having a fast-slow form
(9.1), then there is no problem in determining its topological type – just freeze the slow
subsystem, that is, set µ = 0, and find bifurcations of the fast subsystem treating u
as a parameter. Software packages such as XPPAUT, AUTO, and MATLAB-based
MATCONT, are helpful in bifurcation analyses of such systems.
What if a neuron has an apparent fast-slow dynamics but its model is not knownat present? To determine the types of bifurcations of the fast subsystem, we firstuse noninvasive observations: presence or absence of fast subthreshold oscillations,changes in intraburst (interspike) frequency, changes in spike amplitudes, and so on.
Each piece of information excludes some bifurcations and narrows the set of possible
types of bursting. Then, we can use invasive methods, e.g., small perturbations, totest the coexistence of resting and spiking states, and narrow the choice of bifurcations
further. With some luck, we can exclude enough bifurcations and determine exactly thetype of bursting without knowing the details of the mathematical model that describes
it.

368

Bursting
action potentials cut

2 mV
100 ms

9.4.2

Figure 9.46: The conductance noise destabilizes the focus equilibrium in a mesencephalic V neuron before subcritical Andronov-Hopf bifurcation takes place, thereby
giving an impression of a supercritical Andronov-Hopfbifurcation. (Data modified from Wu et al. 2001.)

Integrators vs. Resonators

A conspicuous feature of neuronal systems near Andronov-Hopf bifurcation, whether
subcritical or supercritical, is the existence of fast subthreshold oscillations of themembrane potential. Quite often, these oscillations are visible in recordings of themembrane potential. If they are not, then they can be evoked by a brief, small pulseof current. Apparently, a bursting neuron exhibiting such oscillations in the quiescentstate is either of the “Hopf/*” type or of the “subHopf/*” type, where the wild card“*” denotes any appropriate bifurcation of the spiking state. All such bursters are in
the lower half of Fig.9.23.
To discern whether the bifurcation is supercritical or subcritical, one needs to study
the amplitude of emerging oscillations, which can be tricky. In models, slow passage
through supercritical Andronov-Hopf bifurcations often results in a delayed transition
to oscillations with an intermediate or large amplitude; hence such a bifurcation may
look like a subcritical one. In recordings like the ones in Fig.9.39a and Fig.9.46, noise
destabilizes the focus equilibrium before the subcritical Andronov-Hopf bifurcation
takes place and gives the impression that the amplitude increases gradually, that is, as
if the bifurcation were supercritical.
The existence of fast subthreshold oscillations indicates that the bursting neuron
acts as a resonator, at least right before the onset of a burst. In section 7.2.2 we showedthat such neurons prefer a certain resonant frequency of stimulation that matches thefrequency of subthreshold oscillations. A resonant input may excite the neuron andinitiate a burst, or it may delay the transition to the burst, depending on its phaserelative to the phase of subthreshold oscillations.
In contrast, all bursters in the upper half of the table in Fig.9.23 (i.e., “fold/*” and“circle/*” types) do not have fast subthreshold oscillations, at least before the onset ofeach burst (see exercise 5). The fast subsystem of such bursters acts as an integrator:
it prefers high-frequency inputs; the higher the frequency, the sooner the transition to
the spiking state. The phase of the input does not play any role here.

9.4.3

Bistability

Suppose the transition from resting to spiking state occurs via saddle-node bifurcation(off an invariant circle) or subcritical Andronov-Hopf bifurcation of the fast subsystem,

Bursting369

bifurcations of equilibria

bifurcations of limit cycles
saddle-node
on invariant
circlesaddlehomoclinic
orbit

supercritical
AndronovHopf

fold
limit
cycle

saddle-node(fold)

fold/
circlefold/homoclinic

fold/
Hopf

fold/
fold cycle

saddle-node
on invariant
circlecircle/circlecircle/homoclinic

circle/Hopfcircle/fold cycle

supercritical
AndronovHopf

Hopf/
circleHopf/homoclinic

Hopf/
Hopf

Hopf/
fold cycle

subcritical
AndronovHopf

subHopf/
circlesubHopf/
homoclinic

subHopf/
Hopf

subHopf/
fold cycle

bistability
beforethe burst

bistability
at the endof the burst

Figure 9.47:
Bistability(i.e., coexistence of restingand spiking states) depends
on the topological type of
bursting.

as in Fig.6.46. In these cases, the trajectory jumps to a preexisting limit cycle attractor
corresponding to the spiking state (not shown in the figure). In contrast, saddle-node
on invariant circle bifurcation or supercritical Andronov-Hopf bifurcation creates sucha limit cycle attractor. Thus, there must be a coexistence of stable resting and stablespiking states in the former case, but not necessarily in the latter case. This simple
observation has far-reaching consequences described below. In particular, it implies
that all “fold/*” and “subHopf/*” bursters exhibit bistability, at least before the onsetof a burst, while “circle/*” and “Hopf/*” bursters may not (see Fig.9.47).
Similarly, if the transition from spiking to resting state of the fast subsystem occurs
via saddle homoclinic orbit bifurcation or fold limit cycle bifurcation, then there is apreexisting stable equilibrium, and hence a coexistence of attractors. Thus, “*/homoclinic” and “*/fold cycle” bursters also exhibit bistability, at least at the end of a burst,
while “*/circle” and “*/Hopf” bursters may not, as we summarize in Fig.9.47.
An obvious consequence of bistability is that an appropriate stimulus can switch thesystem from resting to spiking and back. We illustrate this phenomenon in Fig.9.48,
using the INa,p +IK +IK(M) -model, which exhibits a hysteresis loop “fold/homoclinic”
bursting when I = 5. All three simulations in the figure start with the same initialconditions. In Fig.9.48b we apply a brief pulse of current while the fast subsystem is at
the resting state. This stimulation pushes the membrane potential over the thresholdstate into the attraction domain of the spiking limit cycle of the fast subsystem, thereby
evoking a burst.
Note that the evoked burst is one spike shorter than the control burst in Fig.9.48a.
This is expected, since the K+ M-current did not have enough time to recover fromthe previous burst (not shown in the figure); therefore, there is a residual outward

370

Bursting
spikin

g

V

t
fas n

ho

m

oc
li

ni

c

(a)

fold
thresholdnslow

resting(b)stimulation

stimulation
stimulation(c)stimulation

Figure 9.48: Bistability of resting and spiking states in a “fold/homoclinic” burster. A
brief stimulus can initiate a premature transition to the spiking state (b) or to quiescentstate (c). Shown are simulations of the INa,p +IK +IK(M) -model with parameters as in
Fig.9.4b.
current that shortens the active phase. From the geometrical point of view, this occurs
because the transition to the spiking manifold in Fig.9.48b (right) occurs before theslow variable reaches the fold knee; hence the distance to the homoclinic bifurcation isshorter. An interesting observation is that the first spike in the evoked burst actuallycorresponds to the second spike in the control burst in Fig.9.48a. The earlier the
stimulation acts, the sooner the trajectory jumps to the spiking manifold and thefewer spikes the evoked burst has.
In Fig.9.48c we applied a brief pulse of current in the middle of a burst to switch
the system to the resting state. Note that the quiescent period, that is, the time tothe second burst, is shorter than the control period in Fig.9.48a or 9.48b. This is also
to be expected, since the K+ M- current was not fully activated during the interrupted
burst and therefore does not need much time to deactivate during the resting period.
Geometrically, the short duration of the resting phase is a consequence of the distancethe slow variable needs to travel to get to the fold knee being small.

Bursting

9.4.4

371

Bursts as a Unit of Neuronal Information

There are many hypotheses on the importance of bursting activity in neural computation (Izhikevich 2006).
• Bursts are more reliable than single spikes in evoking responses in postsynaptic
cells. Indeed, excitatory post-synaptic potentials (EPSP) from each spike in aburst add up and may result in a superthreshold EPSP.
• Bursts overcome synaptic transmission failure. Indeed, postsynaptic responses toa single presynaptic spike may fail (release does not occur), however in response
to a bombardment of spikes, i.e., a burst, synaptic release is more likely (Lisman
1997).
• Bursts facilitate transmitter release whereas single spikes do not (Lisman 1997).
Indeed, a synapse with strong short-term facilitation would be insensitive tosingle spikes or even short bursts, but not to longer bursts. Each spike in thelonger burst facilitates the synapse so the effect of the last few spikes may bequite strong.
• Bursts evoke long-term potentiation and hence affect synaptic plasticity muchgreater, or differently than single spikes (Lisman 1997).
• Bursts have higher signal-to-noise ratio than single spikes (Sherman 2001). Indeed, burst threshold is higher than spike threshold, i.e., generation of bursts
requires stronger inputs.
• Bursts can be used for selective communication if the postsynaptic cells havesubthreshold oscillations of membrane potential. Such cells are sensitive to thefrequency content of the input. Some bursts resonate with oscillations and elicita response, others do not, depending on the interburst frequency (Izhikevich etal. 2003).
• Bursts can resonate with short-term synaptic plasticity making a synapse a bandpass filter (Izhikevich et al. 2003). A synapse having short-term facilitation anddepression is most sensitive to a burst having certain resonant interspike frequency. Such a burst evokes just enough facilitation, but not too much depression, so its effect on the postsynaptic target is maximal.
• Bursts encode different features of sensory input than single spikes (Gabbiani etal. 1996, Oswald et al. 2004). For example, neurons in the electrosensory lateralline lobe (ELL) of weakly electric fish fire network induced-bursts in response tocommunication signals and single spikes in response to prey signals (Doiron etal. 2003). In the thalamus of the visual system bursts from pyramidal neurons
encode stimuli that inhibit the neuron for a period of time and then rapidly excite
the neuron (Lesica and Stanely, 2004). Natural scenes are often composed of such
events.

372

Bursting

Figure 9.49: The instantaneous spike frequency of a trigeminal motor neuron (a) anda trigeminal interneuron (b) of a rodent. (Modified from Del Negro et al. 1998.)• Bursts have more informational content than single spikes when analyzed as
unitary events (Reinagel et al. 1999). This information may be encoded into theburst duration or in the fine temporal structure of interspike intervals within aburst.
In summary, burst input is more likely to have a stronger impact on the postsynaptic
cell than single spike input, so some believe that bursts are all-or-none events, whereas
single spikes may be noise.

9.4.5

ChirpsAn important information may be carried in the intraburst frequency. Consider theeffect of a burst on a postsynaptic resonator neuron, that is, a neuron with a restingstate near an Andronov-Hopf bifurcation. Such a neuron is sensitive to the frequency
content of the burst (i.e., whether it is resonant or not, as discussed in section 7.2.2).
Some types of bursters have relatively constant intraburst (instantaneous interspike)frequencies, as in Fig.9.49b, which may be resonant for some postsynaptic neurons
but not for others. In contrast, other topological types of bursters have widely varying instantaneous interspike frequencies, as in Fig.9.49a, that scan or sweep a broadfrequency range going all the way to zero.
When the bifurcation from resting to spiking state is of the saddle-node on invariant
circle type (i.e., the system is Class 1 excitable), the frequency of emerging spiking is
first small, and then increases. Therefore, all “circle/*” bursters generate chirps withinstantaneous interspike frequencies increasing from zero to a relatively large value,at least at the beginning of the burst. Similarly, when the bifurcation of the spiking
state is of the saddle-node on invariant circle or saddle homoclinic orbit type, thefrequency of spiking at the end of the burst decreases to zero, so all “*/circle” and“*/homoclinic” bursters also generate chirps, as in Fig.9.49a. In summary, all shaded
bursters in Fig.9.50 have sweeping interspike frequencies, so that one part of the burst
is resonant for one neuron and another part of the same burst is resonant for anotherneuron.

Bursting373

bifurcations of equilibria

bifurcations of limit cycles
saddle-node
on invariant
circlesaddlehomoclinic
orbit

supercritical
AndronovHopf

fold
limit
cycle

saddle-node(fold)

fold/
circlefold/homoclinic

fold/
Hopf

fold/
fold cycle

saddle-node
on invariant
circlecircle/circlecircle/homoclinic

circle/Hopfcircle/fold cycle

supercritical
AndronovHopf

Hopf/
circleHopf/homoclinic

Hopf/
Hopf

Hopf/
fold cycle

subcritical
AndronovHopf

subHopf/
circlesubHopf/
homoclinic

subHopf/
Hopf

subHopf/
fold cycle

increasing
frequency
at thebeginning

decreasing
frequency
at theend

Figure 9.50: Topological types of bursters in the shaded regions can produce chirpbursts that sweep a frequency range.

9.4.6

Synchronization

Consider two coupled bursting neurons of the fast-slow type. Since each burster has
two time scales, one for rhythmic spiking and one for repetitive bursting, there are two
synchronization regimes:• Spike synchronization, as in Fig.9.51 (left).
• Burst synchronization, as in Fig.9.51 (right).
One of them does not imply the other. Of course, there is an additional regime inwhich spikes and bursts are synchronized. We will study synchronization phenomena

spike synchronization

burst synchronization

Figure 9.51: Various regimes of synchronization of bursters.

374

Bursting
spike
synchronization

spike
de-synchronization

sp
ik

ing

burster B
AB

excitatory coupling

inhibitory
stimulation

excitatory
stimulation
res

tin

g

A
B
inhibitory coupling

Figure 9.52:
Burst synchronization and desynchronization of two coupled“fold/homoclinic” bursters. (Modified from Izhikevich 2000a.)in detail in chapter 10; here we just mention how they depend on the topological typeof bursting.
Let us consider spike synchronization first. Since we are interested in the fast timescale, we neglect the slow variable dynamics for now and treat two bursters as coupledoscillators. A necessary condition for synchronization of two weakly coupled oscillators
is that they have nearly equal frequencies. How near is “near” depends on the strengthof the coupling. Thus, spike synchronization depends crucially on the instantaneousinterspike frequency, which may vary substantially during a burst. Indeed, a small
perturbation of the slow variable may result in large perturbations of the interspikefrequency in any shaded burster in Fig.9.50; hence such a burster would be unlikely to
exhibit spike synchronization unless the coupling is strong.
Studying burst synchronization of weakly coupled neurons involves the same mathematical methods as studying synchronization of strongly coupled relaxation oscillators,
which we consider in detail in chapter 10. The mechanisms of synchronization depend
on whether the bursting is of the hysteresis loop type or of the slow wave type, andwhether the resting state is an integrator or a resonator.
In Fig.9.52 we illustrate the geometry of burst synchronization of two coupled“fold/homoclinic” bursters of the hysteresis loop type. Burster A is slightly aheadof burster B, so that A starts the spiking phase while B is still resting. If the synaptic
connections between the bursters are excitatory, firing of A causes B to jump to thespiking state prematurely, thereby shortening the time difference between the bursts.
In addition, the evoked burst of B is shorter, which also speeds up the synchronization process. In contrast, when the connections are inhibitory, firing of A delays thetransition of B to the spiking state, thereby increasing the time difference between
the bursts and desynchronizing the bursters. Thus, the “fold/homoclinic” burster behaves according to the principle excitation means synchronization, inhibition means

Bursting375

desynchronization. Since the instantaneous interspike frequency of “fold/homoclinic”
bursting decays to zero, small deviations of the slow variable result in large deviationsof the period of oscillation. Typically, the periods of fast oscillations of the two bursters
can diverge slowly from each other. As a result, spikes may start synchronized andthen desynchronize during the burst, as we indicate in the figure.
If the bursting neuron is a resonator, that is, it is of the “Hopf/*” or “subHopf/*”
type, then both excitation and inhibition may evoke premature spiking, as we haveshown in chapter 7, and lead to burst synchronization. An important feature hereis that the interspike frequency of one burster must be resonant to the subthresholdoscillations of the other one. We study these and other issues related to synchronizationin chapter 10.

Review of Important Concepts• A burst of spikes is a train of action potentials followed by a period
of quiescence.
• Bursting activity typically involves two time scales: fast spiking andslow modulation via a resonant current.
• Many mathematical models of bursters have the fast-slow formẋ = f (x, u)u̇ = µg(x, u)(fast spiking),(slow modulation).

• To dissect a burster, one freezes its slow subsystem (i.e., sets µ = 0)
and uses the slow variable u as a bifurcation parameter to study thefast subsystem.
• The fast subsystem undergoes two important bifurcations during aburst: (1) bifurcation of an equilibrium resulting in transition to thespiking state and (2) bifurcation of a limit cycle attractor resulting intransition to the resting state.
• Different types of bifurcations result in different topological types ofbursting.
• There are 16 basic types of bursting, summarized in Fig.9.23.
• Different topological types of bursters have different neurocomputational properties.

376

Bursting

Bifurcations

Saddle-Nodeon Invariant
CircleSaddleHomoclinic
Orbit

Supercritical
AndronovHopf

Fold
Limit
Cycle

triangularsquare-wave
Type I

taperedType V

Type IV

Fold

Saddle-Nodeon Invariant
CircleparabolicType II

Supercritical
AndronovHopf

Subcritical
AndronovHopf

elliptic
Type III

Figure 9.53: Bifurcation mechanisms and classical nomenclature of the six bursters
known in the twentieth century. Compare with Fig.9.23 and Fig.9.24.

Bibliographical Notes
The history of formal classification of bursting starts with the seminal paper by Rinzel(1987), who contrasted the bifurcation mechanisms of the “square-wave”, “parabolic”,
and “elliptic” bursters. Bertram et al. (1995) followed Rinzel’s suggestion and referredto the bursters using Roman numerals, adding a new burster, Type IV. Another,“tapered” type of bursting was studied simultaneously and independently by Holden
and Erneux (1993a, 1993b), Smolen et al. (1993), and Pernarowski (1994). Later,de Vries (1998) suggested referring to it as a Type V burster. A “triangular” type of
bursting was studied by Rush and Rinzel (1994), making the total of identified bursters
six. To honor these pioneers, we described these six classical bursters in the order ofthe numbering of Bertram et al. (1995). Their bifurcation mechanisms are summarized
in Fig.9.53.
The complete classification of bursters, provided by Izhikevich (2000a), was motivated by Guckenheimer et al. (1997). There is a drastic difference between Izhikevich’s
approach and that of the scientists mentioned above. The latter used a bottom-up
approach; that is, they considered biophysically plausible conductance-based models
describing experimentally observable cellular behavior, then determined the types ofbursting these models exhibited. In contrast, Izhikevich (2000a) used the top-down

Bursting377

spiking

resting

Figure 9.54: A hedgehog-like limit cycle attractor results in bursting dynamics even in
two-dimensional systems; see exercise 1. (Modified from Hoppensteadt and Izhikevich
1997.)
approach and considered all possible pairs of codimension-1 bifurcations of rest andspiking states, which resulted in different types of bursting. (It was an easy task toprovide a conductance-based model exhibiting each bursting type.) Thus, many of the
bursters are “theoretical” in the sense that they have yet to be seen in experiments.
A challenging problem was to suggest a naming scheme for the bursters. The
names should be self-explanatory, and easy to remember and understand. Thus, thenumbering scheme suggested by Bertram et al. (1995) could lead to bursters of TypeXXVII, Type LXIII, Type CLXVI, and so on. We cannot use descriptions such as“elliptic”, “parabolic”, “hyperbolic”, “triangular”, “rectangular”, and such because
they are misleading. In this book we follow Izhikevich (2000a) and name the bursters
according to the two bifurcations involved, as in Fig.9.23.
Not all bursters can be represented in the fast-slow form with a clear separation of
the time scales. Those that cannot, are referred to as hedgehog bursters (Izhikevich
2000a), since they have a limit cycle (or a more complicated attractor) with somespiky parts corresponding to repetitive spiking and some smooth parts corresponding
to quiescence, as in Fig.9.54. An interesting example of the hedgehog burster is themodel of the sensory processing neuron of weakly electric fish (Doiron et al. 2002). The
authors refer to the model as “ghostburster” because repetitive spiking corresponds toa slow transition of the full system through the ghost of a fold limit cycle attractor.
As a dynamical system, the ghostburster is near a codimension-2 bifurcation of limit
cycle attractor, and it exhibits chaotic dynamics.
Betram et al. (1995) noticed that bursting often occurs when the fast subsystem
is near a codimension-2 bifurcation. Izhikevich (2000a) suggested that many simple
models of bursters could be obtained by considering unfoldings of various degeneratebifurcations of high codimension (organizing centers) and treating the unfolding parameters as slow variables rotating around the bifurcation point, as in Fig.9.28 or Fig.9.42.
Considering the Bautin bifurcation, Izhikevich (2001) obtained the canonical model forthe “subHopf/fold cycle” (“elliptic”) burster (9.11). Golubitsky et al. (2001) applied
this idea to other local bifurcations (spiking with infinitesimal amplitude). Global

378

Bursting

bifurcations are considered in exercise 26.
Izhikevich and Hoppensteadt (2004) extend the classification of bursters to one- andtwo-dimensional mappings, identifying 3 and 20 different classes, respectively. A recent
book Bursting: The Genesis of Rhythm in the Nervous System edited by Coombes andBressloff (2005), presents recent developments in the field of bursting dynamics.
Studying bursting dynamics is still one of the hardest problems in applied mathematics. The method of dissection of fast-slow bursters of the form (9.1), pioneeredby Rinzel (1987), is part of the asymptotic theory of singularly perturbed dynamicalsystems (Mishchenko et al. 1994). One would expect the theory to suggest other,quantitative methods of analyzing fast-slow bursters. However, the basic assumptionof the theory is that the fast subsystem has only equilibria, e.g., up- and down-states,
as in the point-point hysteresis loops in exercise 19. This assumption is violated when
the neuron fires a burst of spikes, since repetitive spikes correspond to limit cycles.
Thus, the theory is of no help in studying fast-slow point-cycle bursters. An exception is Pontryagin’s problem, which is related to “fold cycle/fold cycle” bursting; see
exercise 21 and section 7 in Mishchenko et al. (1994). Pontryagin and Rodygin (1960)
pioneered the method of averaging of the fast subsystem, which was used in the context
of bursters by Rinzel and Lee (1986), Pernarowski et al. (1992), Smolen et al. (1993),
and Baer et al. (1995). Shilnikov et al. (2005) introduced an average nullcline of theslow subsystem, and showed how the averaging method can be used to study coexistence of spiking and bursting states in a model neuron, and bifurcations in bursters ingeneral. Some of the transitions “resting ↔ bursting ↔ tonic spiking” were consideredby Ermentrout and Kopell (1986a), Terman (1991), Destexhe and Gaspard (1993),
Shilnikov and Cymbalyuk (2004, 2005), and Medvedev (2005).
The averaging method, like many other classical methods of analysis of dynamicalsystems, breaks down when the fast subsystem slowly passes a bifurcation point. The
development of early dynamical system theory was largely motivated by studies of
periodic oscillators. It is reasonable to expect that the next major developments ofthis theory will come from studies of bursters.

Exercises
1. (Planar burster) Invent a planar system of ODEs having a hedgehog limit cycle
attractor (as in Fig.9.54) and capable of exhibiting periodic bursting activity.
2. (Noise-induced bursting) Explain why the INa,p +IK -model with a phase portrait
as in Fig.9.55 bursts even though it has only two dimensions.
3. (Noise-induced bursting) Explore numerically the INa,p +IK -model with phase
portrait as in Fig.6.7 (top) and make it burst as in Fig.9.56, without adding
any new current or gating variable.
4. (Rebound bursting) Explain the mechanism of rebound bursting in the twodimensional FitzHugh-Nagumo oscillator (4.11, 4.12), shown in Fig.9.57.

Bursting379

1

K+ activation gate, n

0.8

0.6

0.4

0.2

0
-80

-60

-40
-20
membrane potential, V (mV)0

Figure 9.55: Bursting in two-dimensional INa,p +IK -model with parameters as inFig.6.16 and I = 43.
0

-20
-30
-40
-50
-60
-70
0

10

20

30
40
time (ms)

50

60

0.2

1

llcline

V-nullcliw

70

Figure 9.56:
Bursting in a twodimensional INa,p +IK -model;
see
exercise 3.

ne

w-nu

membrane potential, V (mV)

-10

0.5
0.1

V(t)

0

0
-0.5

I(t)
0

0.5
V

1

-0.5

0

1000

2000
time

3000

4000

Figure 9.57: Rebound bursting in the FitzHugh-Nagumo oscillator; see exercise 4.

380

Bursting
x1

0.5

t

Figure 9.58: Hopf/Hopf bursting
without coexistence of attractors;
see exercise 6. (Modified from Hoppensteadt and Izhikevich 1997.)

5. Can “circle/*” and “fold/*” bursters have fast subthreshold oscillations of membrane potential? Explain.
6. (Hopf/Hopf bursting) The systemẋ = (y + i)x − x|x|2 ,x = x1 + ix2 ∈ C ,has a unique attractor for any value of the parameter y ∈ R. Ifpy1ẏ = µ(2aS( − a) − |x|) , µ = 0.05 , a = µ/20 , and S(u) =,a
1 + e−uthen the “2+1” system above can burst, as we show in Fig.9.58. Explore thesystem numerically and explain the origin of bursting.
7. (Hopf/Hopf canonical model) Consider the “2+1” fast-slow burster (9.1) andsuppose that x0 is the supercritical Andronov-Hopf bifurcation point of the fastsubsystem when u = u0 . Also suppose that u0 is a stable equilibrium of theslow subsystem when x = x0 is fixed. Show that there is a continuous change ofvariables that transforms (9.1) into the canonical modelz ′ = (u + iω)z − z|z|2 ,u′ = µ(±1 ± u − a|z|2 ) ,where z ∈ C is the new fast variable, u ∈ R is a slow variable, and ω, a, and µ
are parameters.
8. (Bursting in the INa,t +INa,slow -model) Take advantage of the phenomenon of
inhibition-induced spiking described in section 7.2.8 to show that a slow persistent inward current, say INa,slow , can stop spiking and create bursts.
9. Modify the example above to obtain repetitive bursting in a model consisting ofa fast INa,t current, a leak current, and a slow passive dendritic compartment.
10. (Bursting in the INa,p +IK +INa,slow -model) Numerically explore this model with afast subsystem as in Fig.6.16 and a slow Na+ current with parameters gNa,slow =0.5, m∞,slow (V ) with V1/2 = −50 mV and k = 10 mV, and τslow (V ) = 5 +100 exp(−(V + 20)2 /252 ). Explain the origin of bursting oscillations when I = 27
in Fig.9.59.

Bursting381

membrane potential, V (mV)

0
-10
-20
-30
-40
-50
-60
-70

0

50

100

150
200
time (ms)

250

2

300

350

Figure 9.59:
Bursting in theINa,p +IK +INa,slow -model;
see exercise 10.

1y-nullcline

1.5
1

x

0
-1
-2

y 0.5
1

0

0
-1

e

-1
-3

I(t)

x-nullclin

-0.5

-2

-1

0
x

1

2

3

0

200

time

400

600

Figure 9.60: The phase
portrait of the system in
exercise 11 shows that
there is only one stableequilibrium for any valueof I.
Nevertheless, thesystem bursts when I isperiodically modulated.

11. The Bonhoeffer–van der Pol oscillatorẋ = I + x − x3 /3 − y ,ẏ = 0.2(1.05 + x) ,with nullclines as in Fig.9.60, is Class 3 excitable. It has a unique stable equilibrium for any value of I. Periodic modulations of I shift the x-nullcline upwardand downward but do not change the stability of the equilibrium. Why does thesystem burst in Fig.9.60? Explore the phenomenon numerically and explain theexistence of repetitive spikes without a limit cycle.
12. Prove (without simulations) that the fast-slow “2+2” systemż = (1 + u + iω)z − z|z|2 ,u̇ = µ(u − u3 − w) ,ẇ = µ(|z|2 − 1) ,z∈C,is a slow-wave burster, even though the slow subsystem cannot oscillate for any
fixed value of the fast subsystem z.
13. (Ermentrout and Kopell 1986) Consider the systemϑ̇ = 1 − cos ϑ + (1 + cos ϑ)r(ψ) ,ψ̇ = ω ,382

Burstingwith ϑ and ψ being phase variables on the unit circle S1 and r(ψ) being any
continuous function that changes sign. Show that this system exhibits bursting
activity when ω is sufficiently small but positive. What type of bursting is that?

14. Prove that the canonical model for “subHopf/fold cycle” bursting (9.11) exhibits
sustained bursting activity when 0 < a < 1. What happens when a approaches
0 or 1?
15. Show that the canonical model for “fold/homoclinic” bursting (9.7) is equivalent
to a simpler model (equation 27 in Izhikevich 2000a and chapter 8 in this volume),v̇ = v 2 + w ,ẇ = µ ,with after-spike (v = +∞) resetting v ← 1 and w ← w − d, when I is sufficiently
large and µ and d are sufficiently small.
16. Derive the canonical model for “fold/homoclinic” bursting (9.7), assuming thatthe fast subsystem is near a saddle-node homoclinic orbit bifurcation point atsome u = u0 , which is an equilibrium of the slow subsystem.
17. Derive the canonical models (9.9) and (9.10) for “circle/circle” bursting.
18. Show that the averaged slow subsystems of the canonical models for “circle/circle”
bursters (9.9) and (9.10) have the formu̇1 = −µ1 u1 + d1 f (I + u1 − u2 ) ,u̇2 = −µ2 u2 + d2 f (I + u1 − u2 ) ,andu̇1 = −µ1 u2 + d1 f (I + u1 ) ,u̇2 = −µ2 (u2 − u1 ) + d2 f (I + u1 ) ,respectively, where√u√
π/2 + arcot u
is the frequency of spiking of the fast subsystem (in Hz).
f (u) =

19. (Point-point hysteresis loops) Consider (9.1) and suppose that the fast subsystem
has only equilibria for any value of the one-dimensional slow variable u. If there isa coexistence of equilibrium points of the fast subsystem, then (9.1) can exhibit
point-point hysteresis loop oscillation. Classify all codimension-1 point-point
hysteresis loops.
20. (Point-point bursting) In Fig.9.61 we present two geometrical examples of pointpoint bursters that have no limit cycle attractors, yet are capable of exhibiting
spike like dynamics in the active phase. Construct a model for each type ofpoint-point burster in the figure. Use phase portrait snapshots at the bottom of
the figure as hints. What makes such bursting possible?

Bursting

383"Fold/Fold" Bursting"Fold/SubHopf" BurstingUp(R States et)

Up(R Stat
es e
t)
Fold
Bifurcation

Fold
Bifurcation

Fold
Bifurcation

Saddle Homoclinic
Orbit Bifurcation

Subcritical
Andronov-Hopf
Bifurcation

x
x
u

Do
w
(R n-Stes atet)Douw(R n-Stes atet)

Fold

Subcritical
Andronov-Hopf
Bifurcation

Fold
Fold

Saddle Homoclinic
Orbit Bifurcation

Figure 9.61: Two examples of point-point (not fast-slow) bursters. (Modified from
Izhikevich 2000a.)

Figure 9.62: A cycle-cycle bursting:
The resting state is not an equilibrium,
but a small-amplitude limit cycle attractor.
21. (Cycle-cycle bursters) Consider a fast-slow burster (9.1) and suppose that theresting state is not an equilibrium, but a limit cycle attractor, as in Fig.9.62.
Such a bursting is called cycle-cycle. Classify all codimension-1 planar cyclecycle fast-slow bursters. Is bursting in Fig.9.10 of the cycle-cycle type?
22. (Minimal models for bursting) Fill in the blank squares in Fig.9.8.
23. Choose a minimal model from Fig.9.8 and simulate it. Change the parameters
to get as many different bursting types as possible.
24. [M.S.] Determine the bifurcation diagram of the canonical model for “fold/homoclinic”
bursting (9.7).
25. [M.S.] Determine the bifurcation diagrams of the canonical models for “circle/circle” bursters (9.9) and (9.10).

384

Bursting
bifurcations of limit cycles
saddle-node
on invariant
circlebifurcations of equilibria

saddle-node(fold)

saddle-node
on invariant
circlesupercritical
AndronovHopf
subcritical
AndronovHopf

saddlehomoclinic
orbit

supercritical
AndronovHopf

fold
limit
cycle

saddle-nodehomoclinic orbit
v' = I+v2-u
u' = -µuv' = I+v2+u1
u1' = -µ1u2u2' = -µ2(u2-u1)
Bautinz'=(u+iω)z+2z|z|2-z|z|4
u'=µ(a-|z|2)

Figure 9.63: Some canonical models of fast-slow bursters; see exercise 26.
26. [Ph.D.] Consider fast-slow bursters of the form (9.1) and assume that the fastsubsystem is near a bifurcation of high codimension, as in Fig.9.28 or in Fig.9.42.
Treating the bifurcation point as an organizing center for the fast subsystem(Bertram et al. 1995; Izhikevich 2000a; Golubitsky et al. 2001), use unfolding
theory to derive canonical models for the remaining fast-slow bursters in Fig.9.63.
Do not assume that the slow subsystem has an autonomous oscillation or thatthe fast oscillations have small amplitude.
27. [Ph.D.] Classify all possible mechanisms of emergence of bursting oscillations
from resting or spiking, as in Fig.9.19.
28. [Ph.D.] Develop an asymptotic theory of singularly perturbed systems of theformẋ = f (x, u)(fast subsystem),u̇ = µg(x, u)(slow modulation)that can deal with transitions between equilibria and limit cycle attractors of thefast subsystem.

Chapter 10
Synchronization
In this chapter1 we consider networks of tonically spiking neurons. Like any other kindof physical, chemical, or biological oscillators, such neurons can synchronize and exhibit
collective behavior that is not intrinsic to any individual neuron. For example, partial
synchrony in cortical networks is believed to generate various brain oscillations, such as
the alpha and gamma EEG rhythms. Increased synchrony may result in pathologicaltypes of activity, such as epilepsy. Coordinated synchrony is needed for locomotion andswim pattern generation in fish. There is an ongoing debate on the role of synchrony
in neural computation, see e.g., the special issue of Neuron (September 1999) devoted
to the binding problem.
Depending on the circumstances, synchrony can be good or bad, and it is important
to know what factors contribute to synchrony and how to control it. This is the subjectof the present chapter – the most advanced chapter of the book. It provides a niceapplication of the theory developed earlier and hopefully gives some insight into why
the previous chapters may be worth mastering.
Our goal is to understand how the behavior of two coupled neurons depends on theirintrinsic dynamics. First, we introduce the method of description of an oscillation by
its phase. Then, we describe various methods of reduction of coupled oscillators tosimple phase models. The reduction method and the exact form of the phase model
depend on the type of coupling (i.e., whether it is pulsed, weak, or slow) and on thetype of bifurcation of the limit cycle attractor generating tonic spiking. Finally, weshow how to use phase models to understand the collective dynamics of many coupledoscillators.

10.1

Pulsed Coupling

In this section we consider oscillators of the formẋ = f (x) + Aδ(t − ts ) ,1

x ∈ Rm ,(10.1)

The first two editions of the book had this chapter available only in electronic form on the author’s
website www.izhikevich.com.

385

386

Synchronization
(b)

20

-60
-80
0

2
4
6 T
time (ms)

2T

3T

0.4
T/4

0.2

3T/8
5T/8
T/2
3T/4 7T/8

0-80phase of oscillation

x(q)

x(q)

(c)

x(q)

T/8

cle
l i m it c y

attr

ac

to

-60
-40
-20
0
20
membrane potential, V (mV)

(d)

T/4
3T/8

T/8

q

T

q

T/2

0

x0

0,T
r

-40

cline

-20

K+ activation gate, n

0.6n-nullx(t)0V-nullcline

membrane potential, V (mV)

(a)

7T/8

5T/8

0

T

2T

3T

0,T

3T/4

Figure 10.1: Definition of a phase of oscillation, ϑ, in the INa + IK -model with parameters as in Fig.4.1a and I = 10.
having exponentially stable limit cycles and experiencing pulsed stimulation at times
ts that instantaneously increases the state variable by the constant A. The Diracdelta function δ(t) is a mathematical shorthand notation for resetting x by A. The
strength of pulsed stimulation, A, is not assumed to be small. Most of the results ofthis section can also be applied to the case in which the action of the input pulse is notinstantaneous, but smeared over an interval of time, typically shorter than the periodof oscillation.

10.1.1

Phase of Oscillation

Many types of physical, chemical, and biological oscillators share an astonishing feature:they can be described by a single phase variable ϑ. In the context of tonic spiking, thephase is usually taken to be the time since the last spike, as in Fig.10.1a.
We say that a function x(t) is periodic if there is a constant T > 0 such thatx(t + T ) = x(t) for any t. The minimal value of the constant is the period of x(t).
Periodic functions appear in dynamical systems having limit cycle attractors.
The notion of the phase of oscillation is related to the notion of parametrizationof a limit cycle attractor, as in Fig.10.1b. Take a point x0 on the attractor and plot
the trajectory x(t) with x(0) = x0 . Then the phase of x(t) is ϑ = t. As t increases

Synchronization
V-nullcline

0.6
0.5

hr
on

0.4
0.3

y(t3)

iso
c

K+ activation gate, n

387

y(t2)

0.2

x(t2)

y0

0.1
0

x(t3)

y(t1)
x0

n-nullclinex(t1)K+ activation gate, n

0.6
0.5
0.4
0.3
0.2
0.1
0
-80

-70

-60

-50
-40
-30
-20
-10
membrane potential, V (mV)

0

10

20

Figure 10.2: Top.
An
isochron, or a stable manifold, of a point x0 on thelimit cycle attractor is theset of all initial conditions
y0 such that y(t) → x(t)as t → +∞.
Bottom.
Isochrons of the limit cycle attractor in Fig.10.1
corresponding to 40 evenly
distributed phases nT /40,
n = 1, . . . , 40.

past the period T , then 2T , an so on, the phase variable ϑ wraps around the interval[0, T ], jumping from T to 0; see Fig.10.1c. Gluing together the points 0 and T , as in
Fig.10.1d, we can treat the interval [0, T ] as a circle, denoted as S1 , with circumference
T . The parametrization is the mapping of S1 in Fig.10.1d into the phase space R2 in
Fig.10.1b, given by ϑ 7→ x(ϑ).
We could put the initial point x0 corresponding to the zero phase anywhere else on
the limit cycle, and not necessarily at the peak of the spike. The choice of the initialpoint introduces an ambiguity in parameterizing the phase of oscillation. Different
parametrizations, however, are equivalent up to a constant phase shift (i.e., translationin time). In the rest of the chapter, ϑ always denotes the phase of oscillation, theparameter T denotes the period of oscillation, and ϑ = 0 corresponds to the peak of
the spike unless stated otherwise. If the system has two or more coexisting limit cycle
attractors, then a separate phase variable needs to be defined for each attractor.

10.1.2Isochrons

The phase of oscillation can also be introduced outside the limit cycle. Consider, forexample, point y0 in Fig.10.2 (top). Since the trajectory y(t) is not on a limit cycle,
it is not periodic. However, it approaches the cycle as t → +∞. Hence, there is some388

Synchronization

point x0 on the limit cycle, not necessarily the closest to y0 , such thaty(t) → x(t) as t → +∞ .

(10.2)

Now take the phase of the nonperiodic solution y(t) to be the phase of its periodic
proxy x(t).
Alternatively, we can consider a point on the limit cycle x0 and find all the otherpoints y0 that satisfy (10.2). The set of all such points is called the stable manifoldof x0 . Since any solution starting on the stable manifold has an asymptotic behavior
indistinguishable from that of x(t), its phase is the same as that of x(t). For thisreason, the manifold represents solutions having equal phases, and it is often referred
to as being the isochron of x0 (iso, equal; chronos, time, in Greek), a notion going backto Bernoulli and Leibniz.
Every point on the plane in Fig.10.2, except the unstable equilibrium, gives rise toa trajectory that approaches the limit cycle. Therefore, every point has some phase.
Let ϑ(x) denote the phase of the point x. Then, isochrons are level contours of thefunction ϑ(x), since the function is constant on each isochron.
The entire plane is foliated by isochrons. We depict only 40 representative onesin Fig.10.2. In this chapter we consider neighborhoods of exponentially stable limit
cycles, where the foliation is continuous and invariant (Guckenheimer 1975):• Continuity. The function ϑ(x) is continuous so that nearby points have nearby
phases.
• Invariance. If ϑ(x(0)) = ϑ(y(0)), then ϑ(x(t)) = ϑ(y(t)) for all t. Isochrons are
mapped to isochrons by the flow of the vector field.
Fig.10.3 shows the geometry of isochrons of various oscillators. The Andronov-Hopfoscillator in the figure is often called a radial isochron clock for the obvious reason. It
is simple enough to be solved explicitly (see exercise 1). In general, finding isochrons
is a daunting mathematical task. In exercise 3 we present a MATLAB program thatfinds isochrons numerically.

10.1.3PRC

Consider a periodically spiking neuron (10.1) receiving a single brief pulse of current
that increases the membrane potential by A = 1 mV, as in Fig.10.4 (left). Such a
perturbation may not elicit an immediate spike, but it can change the timing, that is,
the phase, of the following spikes. For example, the perturbed trajectory (solid line in
Fig.10.4, left) fires earlier than the free-running unperturbed trajectory (dashed line).
That is, right after the perturbation, the phase, ϑnew , is greater than the old phase,ϑ. The magnitude of the phase shift of the spike train depends on the exact timing of
the stimulus relative to the phase of oscillation ϑ. Stimulating the neuron at different
phases, we can measure the phase response curve (also called phase-resetting curve
PRC, or spike time response curve STRC)
PRC (ϑ) = {ϑnew − ϑ}(shift = new phase – old phase) ,Synchronization

389

Andronov-Hopf oscillator

van der Pol oscillator

1.5

2
1.5

1

1

Im z

0.5

0.5

y 0

0

-0.5

-0.5

-1
-1

-1.5

-1.5
-1.5

-1

-0.5

0

0.5

1

-2
-1.5

1.5

-1

-0.5

0

0.5

1

Re z

x

INa+IK-model (Class 1)

INa+IK-model (Class 2)

1.5

0.8

0.6

0.7

0.5

0.6
0.4

n

n 0.3

0.5
0.4

0.2

0.3

0.1

0.2
0.1

0

0
-80

-60

-40

-20

V

0

20

-80

-60

-40

-20

0

V

Figure 10.3: Isochrons of various oscillators. Andronov-Hopf oscillator: ż = (1 + i)z −z|z|2 , z ∈ C. van der Pol oscillator: ẋ = x − x3 − y, ẏ = x. The INa + IK -model withparameters as in Fig.4.1a and I = 10 (Class 1) and I = 35 (Class 2). Only isochrons
corresponding to phases nT /20, n = 1, . . . , 20, are shown.

depicted in Fig.10.4, right. Positive (negative) values of the function correspond tophase advances (delays) in the sense that they advance (delay) the timing of the next
spike.
In contrast to the common folklore, the function PRC (ϑ) can be measured for an
arbitrary stimulus, not necessarily weak or brief. The only caveat is that to measure
the new phase of oscillation perturbed by a stimulus, we must wait long enough fortransients to subside. This becomes a limiting factor when PRCs are used to study
synchronization of oscillators to periodic pulses, as we do in section 10.1.5.
There is a simple geometrical relationship between the structure of isochrons of an
oscillator and its PRC, illustrated in Fig.10.5 (see also exercise 6). Let us stimulate
the oscillator at phase ϑ with a pulse, which moves the trajectory from point x lyingSynchronization
phase of stimulation, q
0

20

T/2

PRC=qnew-q
phase resetting, qnew-q

membrane potential, V (mV)

390

T

0
-20
-40
-60
I(t)

-80
0

10

20

30

40
50
time (ms)

60

70

PRC
T/4

T/8

0
0

80

T/2
T
phase of stimulation, q

Figure 10.4: Phase response of the INa + IK -model with parameters as in Fig.4.1a andI = 4.7. The dashed voltage trace is the free-running trajectory.
on the intersection of isochron ϑ and the limit cycle attractor to a point y lying onsome isochron ϑnew . From the definition of PRC, it follows that ϑnew = ϑ+PRC (ϑ).
In general, one uses simulations to determine PRCs, as we do in Fig.10.4. Using
the MATLAB program presented in exercise 5, we can determine PRCs of all four
oscillators in Fig.10.3 and plot them in Fig.10.6. It is a good exercise to explain the
shape of each PRC in the figure, or at least its sign, using the geometry of isochronsof corresponding oscillators. In section 10.2.4 we discuss pitfalls of using the straightforward method in Fig.10.4 to measure PRCs in biological neurons, and we present a
better technique.
Note that the PRC of the INa + IK -model in Fig.10.6 is mainly positive in theClass 1 regime, that is, when the oscillations appear via saddle-node on invariant circle

pulse

K+ activation gate, n

0.6
pulse

0.5

pulse

pulse

0.4
0.3

pulse

0.2

PRC(q)=qnew-q

pulse

0.1

x

q

pulse

y

pulse

0
qnew=q+PRC(q)

-80

-70

-60

-50
-40
-30
-20
-10
membrane potential, V (mV)

0

10

20

Figure 10.5: The geometrical relationship between isochrons and the phase response
curve (PRC) of the INa + IK -oscillator in Fig.10.1.

Synchronization

391

Andronov-Hopf oscillator
Re z(t)

0
PRC2
-0.2

0
PRC2

-0.2

PRC1

0

x(t)

0.2
PRC(q)

PRC(q)

0.2

van der Pol oscillatorT2
4
stimulus phase, q

PRC10

INa+IK-model (Class 1)
V(t)

PRC1

T

PRC2

V(t)

0
PRC2

-0.5

6

INa+IK-model (Class 2)

PRC(q)

PRC(q)

0.5

2
4
stimulus phase, q

0.2
0
-0.2
PRC1

0

2
4
6
stimulus phase, q

T

0

1
2
3
stimulus phase, qTFigure 10.6: Examples of phase response curves (PRC) of the oscillators in Fig.10.3.
PRC1 (ϑ): Horizontal pulses (along the first variable) with amplitudes 0.2, 0.2, 2, 0.2 forAndnronov-Hopf, van der Pol, Class 1 and Class 2 oscillators, respectively. PRC2 (ϑ):
Vertical pulses (along the second variable) with amplitudes 0.2, 0.2, 0.02, 0.002, respectively. An example of oscillation is plotted as a dotted curve in each subplot (not toscale).
bifurcation, but changes sign in the Class 2 regime, corresponding in this case to thesupercritical Andronov-Hopf bifurcation. In section 10.4 we find PRCs analyticallyin the case of weak coupling, and show that the PRC of a Class 1 oscillator has theshape sin2 ϑ (period T = π) or 1 − cos ϑ (period T = 2π), whereas that of a Class
2 oscillator has the shape sin ϑ (period T = 2π). We show in section 10.1.7 how thesynchronization properties of an oscillator depend on the shape of its PRC.

10.1.4

Type 0 and Type 1 Phase Response

Instead of phase-resetting curves, many researchers in the field of circadian rhythms
consider phase transition curves (Winfree 1980)
ϑnew = PTC (ϑold ).
Since
PTC (ϑ) = {ϑ + PRC (ϑ)} mod T,392

Synchronization
Type 1 (weak) resetting

Type 0 (strong) resetting

q=p/2

q=p

q=0,2pq=3p/2pp

PRC(q)
phase resetting

phase resetting

PRC(q)

0

-p

0
2p

2p

0

stimulus phase, q

-p

0

p

phase transition

PTC(q)={q+PRC(q)} mod 2pphase transition0

stimulus phase, q0

2p

stimulus phase, q

2p

PTC(q)={q+PRC(q)} mod 2p

0

-p

0

stimulus phase, q

2p

Figure 10.7: Types of phase-resetting of the Andronov-Hopf oscillator in Fig.10.3.
the two approaches are equivalent. PRCs are convenient when the phase shifts are
small, so that they can be magnified and seen clearly. PTCs are convenient when thephase shifts are large and comparable with the period of oscillation. We present PTCs
in this section solely for the sake of review, and we use PRCs throughout the rest of
the chapter.
In Fig.10.7 (top) we depict phase portraits of the Andronov-Hopf oscillator having
radial isochrons and receiving pulses of magnitude A = 0.5 (left) and A = 1.5 (right).
Note the drastic difference between the corresponding PRCs or PTCs. Winfree (1980)
distinguishes two cases:• Type 1 (weak) resetting results in continuous PRCs and PTCs with mean slope
1.
• Type 0 (strong) resetting results in discontinuous PRCs and PTCs with meanslope 0.

Synchronization

393

0.36

0
1.1

1

0.6
0.4

6

1

4

0.5
stimulus
amplitude, A

02
stimulus phase, θ

0.2
0

1.47
2.20 1.83
2.57
2.94

0.73

0.8

black
hole

2
4
stimulus phase, θ

5.88
5.51

1.5

1

1.2

0.36

0
2

5.5

5.15
4.784.41
4.04
3.67
3.31

stimulus amplitude, A

PTC

2

1.4

3
0.7

4

5.88

1.6

6

0.36

1.8

5.88

2

6

phase of oscillation q

Figure 10.8: Time crystal (left) and its contour plot (right). Shown is the PTC (ϑ, A)of the Andronov-Hopf oscillator (see exercise 4).

qn+1=qn+PRC(qn)+Ts

qn+PRC(qn)

PRC(qn)qn

Tspulse ntime

Figure 10.9: Calculation of
the Poincare phase map.

pulse n+1

(Do not confuse these classes with Class 1, 2, or 3 excitability.) The discontinuityof the Type 0 PRC in Fig.10.7 is a topological property that cannot be removed byreallocating the initial point x0 that corresponds to zero phase. As an exercise, provethat the discontinuity stems from the fact that the shifted image of the limit cycle(dashed circle) goes beyond the central equilibrium at which the phase is not defined.
If we vary not only the phase ϑ of the applied stimulus, but also its amplitude
A, then we obtain parameterized PRC and PTC. In Fig.10.8 we plot PTC (ϑ, A) of
the Andronov-Hopf oscillator (the corresponding PRC is derived in exercise 4). The
surface is called time crystal and it can take quite amazing shapes (Winfree 1980). The
contour plot of PTC (ϑ, A) in the figure contains the singularity point (black hole) that
corresponds to the phaseless equilibrium of the Andronov-Hopf oscillator. Stimulation
at phase ϑ = π with magnitude A = 1 pushes the trajectory into the equilibrium andstalls the oscillation.

394

Synchronization
(a)

(b)
q1

q3

q4

q5

q6

q7

K+ gating variable, bq2

membrane potential, V

?

0.6
0.4

q2

0.2

q1

0

q3
q4-7

-80 -60 -40 -20

Ts

(d)
T

phase, qn

Poincare phase map
q3
q4
q5,6,7

T

q3

mod T

q4

q5

q6

q7

qn+1
stable
fixed
point

10

10

q1

q1
q2
1

20

time, t(c)00membrane potential, V

2

3

4

5

6

7

0

q2
0

pulse number, n

10

qn

T

Figure 10.10: Description of synchronization of INa + IK -oscillator in Fig.10.4, using
Poincare phase map.

10.1.5

Poincare Phase Map

The phase-resetting curve (PRC) describes the response of an oscillator to a single
pulse, but it can also be used to study its response to a periodic pulse train using thefollowing “stroboscopic” approach. Let ϑn denote the phase of oscillation at the time
the nth input pulse arrives. Such a pulse resets the phase by PRC (ϑn ), so that the newphase right after the pulse is ϑn +PRC (ϑn ) (see Fig.10.9). Let Ts denote the period of
pulsed stimulation. Then the phase of oscillation before the next, (n + 1)th, pulse isϑn +PRC (ϑn ) + Ts . Thus, we have a stroboscopic mapping of a circle to itself,
ϑn+1 = (ϑn + PRC (ϑn ) + Ts ) mod T,(10.3)

called the Poincare phase map (two pulse-coupled oscillators are considered in exercise 11). Knowing the initial phase of oscillation ϑ1 at the first pulse, we can determineϑ2 , then ϑ3 , and so on. The sequence {ϑn } with n = 1, 2, . . . , is called the orbit of themap, and it is quite easy to find numerically.
Let us illustrate this concept using the INa + IK -oscillator with PRC shown in
Fig.10.4. Its free-running period is T ≈ 21.37 ms, and the period of stimulationin Fig.10.10a is Ts = 18.37, which results in the Poincare phase map depicted in
Fig.10.10d. The cobweb in the figure is the orbit going from ϑ1 to ϑ2 to ϑ3 , and soSynchronization395

on. Note that the phase ϑ3 cannot be measured directly from the voltage trace in
Fig.10.10a because pulse 2 changes the phase, so it is not the time since the last spike
when pulse 3 arrives. The Poincare phase map (10.3) takes into account such multiple
pulses. The orbit approaches a point (called a fixed point; see below) that corresponds
to a synchronized or phase-locked state.
A word of caution is in order. Recall that PRCs are measured on the limit cycle
attractor. However, each pulse displaces the trajectory away from the attractor, as
in Fig.10.5. To use the PRC formalism to describe the effect of the next pulse, theoscillator must be given enough time to relax back to the limit cycle attractor. Thus,
if the period of stimulation Ts is too small, or the attraction to the limit cycle is too
slow, or the stimulus amplitude is too large, the Poincare phase map may be not anappropriate tool to describe the phase dynamics.

10.1.6

Fixed points

To understand the structure of orbits of the Poincare phase map (10.3), or any othermap
ϑn+1 = f (ϑn ) ,(10.4)
we need to find its fixed points
ϑ = f (ϑ)(ϑ is a fixed point),

which are analogues of equilibria of continuous dynamical systems. Geometrically, a
fixed point is the intersection of the graph of f (ϑ) with the diagonal line ϑn+1 = ϑn(see Fig.10.10d or Fig.10.11). At such a point, the orbit ϑn+1 = f (ϑn ) = ϑn is fixed.
A fixed point ϑ is asymptotically stable if it attracts all nearby orbits, i.e., if ϑ1 is ina sufficiently small neighborhood of ϑ, then ϑn → ϑ as n → ∞, as in Fig.10.11, left.
The fixed point is unstable if any small neighborhood of the point contains an orbit
diverging from it, as in Fig.10.11 (right).
The stability of the fixed point is determined by the slope
m = f ′ (ϑ)
of the graph of f at the point, which is called the Floquet multiplier of the mapping.
It plays the same role as the eigenvalue λ of an equilibrium of a continuous dynamicalsystem. Mnemonically, the relationship between them is µ = eλ , to which the fixed
point is stable when |m| < 1 (λ < 0) and unstable when |m| > 1 (λ > 0). Fixed
points bifurcate when |m| = 1 (λ is zero or purely imaginary). They lose stability
via flip bifurcation (a discrete analogue of Andronov-Hopf bifurcation) when m = −1
and disappear via fold bifurcation (a discrete analogue of saddle-node bifurcation)
when m = 1. The former plays an important role in the period-doubling phenomenon
illustrated in Fig.10.14 (bottom trace). The latter plays an important role in thecycle-slipping phenomenon illustrated in Fig.10.16.

396

Synchronization

stable fixed points

unstable fixed points
f(q)

qn
qn+1

qn+1

f(q)
qn
qn

qn

n
f(q)

qn

f(q)

n

qn

qn+1

qn+1

qn

qn

n

n

Figure 10.11: The stability of fixed points of the mapping (10.4) depends on the slopeof the function f .

10.1.7

Synchronization

We say that two periodic pulse trains are synchronous when the pulses occur at thesame time or with a constant phase shift, as in Fig.10.12a. Each subplot in the figurecontains an input pulse train (bottom) and an output spike train (top), assumingthat spikes are fired at zero crossings of the phase variable, as in Fig.10.1. Sucha synchronized state corresponds to a stable fixed point of the Poincare phase map(10.3). The in-phase, anti-phase, or out-of-phase synchronization corresponds to thephase shift ϑ = 0, ϑ = T /2, or some other value, respectively. Many scientists referto the in-phase synchronization simply as “synchronization”, and use the adjectives
anti-phase and out-of-phase to denote the other types of synchronization.
When the period of stimulation, Ts , is near the free-running period of tonic spiking,T , the fixed point of (10.3) satisfies
PRC (ϑ) = T − Ts ,that is, it is the intersection of the PRC and the horizontal line, as in Fig.10.13. Thus,
synchronization occurs with a phase shift ϑ that compensates for the input period
mismatch T − Ts . The maxima and the minima of the PRC determine the oscillator’s
tolerance of the mismatch. As an exercise, check that stable fixed points lie on the sideof the graph with the slope−2 < PRC ′ (ϑ) < 0(stability region)

Synchronization

397(a) synchronization
in-phaseanti-phase

out-of-phase

2:1

3:2

(b) phase-locking
1:2

Figure 10.12: Examples of fundamental types of synchronization of spiking activity toperiodic pulsed inputs (synchronization is 1:1 phase-locking).
unstable

PRC(q)

T-Ts

0stability
regionstable
phase
shift

0

stability
regionstimulus phase, qTFigure 10.13: Fixed points of the Poincare
phase map (10.3).

marked by the bold curves in Fig.10.13.
Now consider the Class 1 and Class 2 INa + IK -oscillators shown in Fig.10.6. The
PRC in the Class 1 regime is mostly positive, implying that such an oscillator caneasily synchronize with faster inputs (T − Ts > 0) but cannot synchronize with slower
inputs. Indeed, the oscillator can advance its phase to catch up with faster pulse trains,but it cannot delay the phase to wait for the slower input. Synchronization with the
input having Ts ≈ T is only marginal. In contrast, the Class 2 INa + IK -oscillator does
not have this problem because its PRC has well-defined positive and negative regions.

10.1.8Phase-Locking

The phenomenon of p:q-phase-locking occurs when the oscillator fires p spikes for every
q input pulses, such as the 3:2-phase-locking in Fig.10.12b or the 2:2 phase-locking in
Fig.10.14, which typically occurs when pT ≈ qTs . The integers p and q need not be
relatively prime in the case of pulsed-coupled oscillators. Synchronization, that is, 1:1
phase-locking, as well as p:1 phase-locking corresponds to a fixed point of the Poincare
phase map (10.3) with p fired spikes per single input pulse. Indeed, the map tells thephase of the oscillator at each pulse, but not the number of oscillations between thepulses.
Each p:q-locked solution corresponds to a stable periodic orbit of the Poincare

398

Synchronization
Poincare phase map
synchronization (1:1)
qn+1=f(qn)phase-locking (2:2)

phase-locked

sync

0

qn

Figure 10.14: Coexistence of synchronized and phase-locked solutions corresponds tocoexistence of a stable fixed point and a stable periodic orbit of the Poincare phase
map.
phase map with the period q (so that ϑn = ϑn+q for any n). Such orbits in maps (10.4)
correspond to stable equilibria in the iterates ϑk+1 = f q (ϑk ), where f q = f ◦ f ◦ · · · ◦ f
is the composition of f with itself q times. Geometrically, studying such maps is likeconsidering every qth input pulse in Fig.10.12b and ignoring all the intermediate pulses.
Since maps can have coexistence of stable fixed points and periodic orbits, various
synchronized and phase-locking states can coexist in response to the same input pulse
train, as in Fig.10.14. The oscillator converges to one of the states, depending on theinitial phase of oscillation, but can be switched between states by a transient input.

10.1.9Arnold Tongues

To synchronize an oscillator, the input pulse train must have a period Ts sufficiently
near the oscillator’s free-running period T so that the graph of the PRC and the horizontal line in Fig.10.13 intersect. The amplitude of the function |PRC (ϑ, A)| decreases
as the strength of the pulse A decreases, because weaker pulses produce weaker phaseshifts. Hence the region of existence of a synchronized state shrinks as A → 0, and it
looks like a horn or a tongue on the (Ts , A)-plane depicted in Fig.10.15, called Arnold
tongue. Each p:q-phase-locked state has its own region of existence (p:q-tongue in the
figure), which also shrinks to a point pT /q on the Ts -axis. The larger the order oflocking, p + q, the narrower the tongue and the more difficult it is to observe such aphase-locked state numerically, let alone experimentally.
The tongues can overlap, leading to the coexistence of phase-locked states, as in
Fig.10.14. If A is sufficiently large, the Poincare phase map (10.3) becomes noninvertible, that is, it has a region of negative slope, and there is a possibility of chaotic
dynamics (Glass and Mackey 1988).
In Fig.10.16 we illustrate the cycle slipping phenomenon that occurs when the input
period Ts drifts away from the 1:1 Arnold tongue. The fixed point of the Poincare
phase map corresponding to the synchronized state undergoes a fold bifurcation anddisappears. In a way similar to the case of saddle-node on invariant circle bifurcation,

amplitude of stimulation, A

Synchronization399

1:6 1:4
synchronization

1:2
1:3

0

T/4 T/3 T/2

1:1

5:2
3:2

2:1

2:3

3T/4

T
5T/4
3T/2
7T/4
period of stimulation, Ts

2T

9T/4

5T/2

Figure 10.15: Arnold tongues are regions of existence of various phase-locked states on
the “period-strength” plane.
T/2

Poincare phase map

qn+1=f(qn)cycle slipping0?

ghost ofattractor
-T/2
-T/2

0

qn

T/2

Figure 10.16: Cycle slipping phenomenon at the edge of the Arnold tongue corresponding to a synchronized state.
the fold fixed point becomes a ghost attractor that traps orbits and keeps them near
the synchronized state for a long period of time. Eventually the orbit escapes, thesynchronized state is briefly lost, and then the orbit returns to the ghost attractor to
be trapped again. Such an intermittently synchronized orbit typically corresponds toa p:q-phase-locked state with a high order of locking p + q.

10.2

Weak Coupling

In this section we consider dynamical systems of the formẋ = f (x) + εp(t) ,(10.5)

describing periodic oscillators, ẋ = f (x), forced by a time-depended input εp(t), forinstance, from other oscillators in a network. The positive parameter ε measures theoverall strength of the input, and it is assumed to be sufficiently small, denoted asε ≪ 1. We do not assume ε → 0 here. In fact, most of the results in this section can

400

Synchronization

Figure 10.17: Arthur Winfree in 2001.
(Photo provided by Martin Homer,
University of Bristol.)be cast in the form “there is an ε0 such that for all ε < ε0 , the following holds. . .”(Hoppensteadt and Izhikevich 1997), with ε0 depending on the function f (x) in (10.5)
and sometimes taking not so small values, such as, ε0 = 1.
Note that if ε = 0 in (10.5), we can transform ẋ = f (x), to ϑ̇ = 1 using the theory
presented in section 10.1. What happens when we apply the same transformation to(10.5) with ε ̸= 0? In this section we present three different but equivalent approaches
that transform (10.5) into the phase modelϑ̇ = 1 + ε PRC (ϑ)p(t) + o(ε) .
Here, Landau’s “little oh” function o(ε) denotes the error terms smaller than ε so thato(ε)/ε → 0 if ε → 0. For the sake of clarity of notation, we omit o(ε) throughout thebook, and implicitly assume that all equalities are valid up to the terms of order o(ε).
Since we do not impose restrictions on the form of p(t), the three methods arereadily applicable to the caseXp(t) =
gs (x(t), xs (t)) ,s

where the set {xs (t)} denotes oscillators in the network connected to x, and p(t) is thepostsynaptic current.

10.2.1

Winfree’s Approach

A sufficiently small neighborhood of the limit cycle attractor of the unperturbed(ε = 0) oscillator (10.5), magnified in Fig.10.18, has nearly collinear uniformly spaced
isochrons. Collinearity implies that a point x on the limit cycle in Fig.10.18 has thesame phase-resetting as any other point y on the isochron of x near the cycle. Uniform
density of isochrons implies that the phase-resetting scales linearly with the strengthof the pulse, that is, a half-pulse at point z in Fig.10.18 produces a half-resetting of
the phase.
Linear scaling of PRC with respect to the strength of the pulse motivates thesubstitution
PRC (ϑ, A) ≈ Z(ϑ)A ,Synchronization

401

0.6

n

z

0.5

y

0.007

0.4

x

n 0.3
0.2

limit

c y cl

0.005

-62

-61

e at
trac
to

-60

V

0.1

r

0
-80

-59

-60

-40

-20

0

20

V

Figure 10.18: Magnification of isochrons in a small neighborhood of the limit cycle ofthe INa + IK -model in Fig.10.3. Isochron time step: 0.025 ms on the left, 0.35 ms on
the right.
p(t2)h

p(t)
p(t2)
h
t0

t1

t2

t3

t4

t5

time, tFigure 10.19: A continuous function p(t) is replaced by an equivalent train of pulses ofvariable amplitudes.
where Z(ϑ) = ∂ PRC(ϑ, A)/∂A at A = 0 is the linear response or sensitivity function(Winfree 1967) describing the slight alteration of rate, or of instantaneous frequency ofoscillation, accompanying application of a small stimulus. Some call it the infinitesimal
PRC.
Now suppose ε ̸= 0 but is sufficiently small that the trajectory of the weakly perturbed oscillator (10.5) remains near the limit cycle attractor all the time. Let us replace the continuous input function εp(t) with the equivalent train of pulses of strengthA = εp(tn )h, where h is a small interpulse interval (denoted as Ts in section 10.1), and
tn = nh is the timing of the nth pulse, see Fig.10.19. We rewrite the corresponding
Poincare phase map (10.3)
PRC
z
}|
{
ϑ(tn+1 ) = {ϑ(tn ) + Z(ϑ(tn )) εp(tn )h +h} mod T| {z }
A

402

Synchronization

Figure 10.20: Yoshiki Kuramoto in
1988, while he was visiting Jim Murray’s institute at Oxford University.
(Picture provided by Dr.
Y. Kuramoto.)in the form
ϑ(tn + h) − ϑ(tn )= Z(ϑ(tn ))εp(tn ) + 1 ,hwhich is a discrete version ofϑ̇ = 1 + εZ(ϑ) · p(t),(10.6)

in the limit h → 0.
To be consistent with all the examples in section 10.1, we implicitly assume herethat p(t) perturbs only the first, voltage-like variable x1 of the state vector x =(x1 , . . . , xm ) ∈ Rm and that Z(ϑ) is the corresponding sensitivity function. However,the phase model (10.6) is also valid for an arbitrary input p(t) = (p1 (t), . . . , pm (t)).
Indeed, let Zi describe the linear response to perturbations of the ith state variable
xi , and Z(ϑ) = (Z1 (ϑ), . . . , Zm (ϑ)) denote the corresponding linear response vectorfunction. Then the combined phase shift Z1 p1 + · · · + Zm pm is the dot product Z · p in
(10.6).

10.2.2

Kuramoto’s Approach

Consider the unperturbed (ε = 0) oscillator (10.5), and let the function ϑ(x) denotethe phases of points near its limit cycle attractor. Obviously, isochrons are the level
contours of ϑ(x) since the function is constant on each isochron. Differentiating thefunction using the chain rule yieldsdxdϑ(x)= grad ϑ ·= grad ϑ · f (x) ,dt
dt
where grad ϑ = (ϑx1 (x), . . . , ϑxm (x)) is the gradient of ϑ(x) with respect to the statevector x = (x1 , . . . , xm ) ∈ Rm . However,
dϑ(x)
=1
dt

Synchronization

grad q(x)

ro n

q

x

403

iso

ch

limit cyf(x)cleFigure 10.21: Geometrical interpretationof the vector grad ϑ.

Figure 10.22: Ioel Gil’evich Malkin (Io l~Gil~eviq Malkin, 1907–1958).
near the limit cycle because isochrons are mapped to isochrons by the flow of the vector
field f (x). Therefore, we obtain a useful equality,
grad ϑ · f (x) = 1 .

(10.7)

Figure 10.21 shows a geometrical interpretation of grad ϑ(x): it is the vector based atpoint x, normal to the isochron of x and with a length equal to the number density ofisochrons at x. Its length can also be found from (10.7).
Kuramoto (1984) applied the chain rule to the perturbed system (10.5),dxdϑ(x)= grad ϑ ·= grad ϑ · {f (x) + εp(t)} = grad ϑ · f (x) + ε grad ϑ · p(t) ,dt
dtand, using (10.7), obtained the phase modelϑ̇ = 1 + ε grad ϑ · p(t) ,(10.8)

which has the same form as (10.6). Subtracting (10.8) from (10.6) yields (Z(ϑ) −grad ϑ) · p(t) = 0. Since this is valid for any p(t), we conclude that Z(ϑ) = grad ϑ(see also exercise 6). Thus, Kuramoto’s phase model (10.8) is indeed equivalent to
Winfree’s model (10.8).

10.2.3

Malkin’s Approach

Yet another equivalent method of reduction of weakly perturbed oscillators to theirphase models follows from Malkin’s theorem (1949, 1956), which we state in the simplest form below. The most abstract form and its proof are provided by Hoppensteadt
and Izhikevich (1997).

404

Synchronization
Andronov-Hopf oscillator
1

1

Q1(q)

Q(q)

Q2

0

-1

-1
0

Q1

Q(q)

1

Q2(q)

0

0

-1
0

2

4

6

phase, q-1-20

INa+IK-model (Class 1)
Q2
0
-10

2

0

2

4

6

phase, q

INa+IK-model (Class 2)
100.01Q2(q)

Q(q)

800

5

600

0

-20

400

-30

-0.2

-40

-0.4

-0.1

Q1

Q2

Q1(q)

0.2

Q(q)

Q1(q)

1

Q2

Q2(q)

0

-1

van der Pol oscillator
1

0

0.1

0.2

Q1

0

200

0.01Q2(q)024phase, q

0
6

-200

Q1(q)

-5
-5

0

Q1

5

0

1

2

phase, q3

Figure 10.23: Solutions Q = (Q1 , Q2 ) to adjoint problem (10.10) for oscillators in
Fig.10.3.
Malkin’s theorem. Suppose the unperturbed (ε = 0) oscillator in (10.5) has an
exponentially stable limit cycle of period T . Then its phase is described by the equationϑ̇ = 1 + εQ(ϑ) · p(t) ,(10.9)

where the T -periodic function Q is the solution to the linear “adjoint” equationQ̇ = −{Df (x(t))}⊤ Q ,with Q(0) · f (x(0)) = 1 ,(10.10)where Df (x(t))⊤ is the transposed Jacobian of f (matrix of partial derivatives) at
the point x(t) on the limit cycle, and the normalization condition can be replaced byQ(t) · f (x(t)) = 1 for any, and hence all, t. Here Q · f is the dot product of two vectors,
which is the same as Q⊤ f .
Though this theorem looks less intuitive than the methods of Winfree and Kuramoto, it is actually more useful because (10.10) can be solved numerically quite easily. Applying the MATLAB procedure in exercise 12 to the four oscillators in Fig.10.3,
we plot their functions Q in Fig.10.23. It is not a coincidence that each component of Q
looks like PRC along the first or second state variable, shown in Fig.10.6. Subtracting(10.9) from (10.8) or from (10.6), we conclude that
Z(ϑ) = grad ϑ(x) = Q(ϑ) ,(see also exercise 7), so that we can determine the linear response function of the phasemodel using any of the three alternative methods: via PRCs, via isochrons, or solving

Synchronization405

the adjoint equation (10.10). This justifies the reason why many refer to the function
simply as PRC, implicitly assuming that it is measured to the infinitesimal stimuli andthen normalized by the stimulus amplitude.

10.2.4

Measuring PRCs Experimentally

In Fig.10.24 we exploit the relationship (10.9) and measure the infinitesimal PRCs of alayer 5 pyramidal neuron of mouse visual cortex. First, we stimulate the neuron with
40 pA DC current to elicit periodic spiking. Initially, the firing period starts at 50 ms,
and then relaxes to the averaged value of 110 ms (Fig.10.24a). The standard method offinding PRCs consists in stimulating the neuron with brief pulses of current at different
phases of the cycle and measuring the induced phase shift, which can be approximatedby the difference between two successive periods of oscillation. The method works wellin models (see exercise 5), but should be used with caution in real neurons because
their firing is too noisy, as we demonstrate in Fig.10.24b. Thus, one needs to apply
hundreds, if not thousands, of pulses and then average the resulting phase deviations(Reyes and Fetz 1993).
Starting with time 10 sec, we inject a relatively weak noisy current εp(t) that
continuously perturbs the membrane potential (Fig.10.24c) and, hence, the phase of
oscillation (the choice of p(t) is important; its Fourier spectrum must span a range of
frequencies that depends on the frequency of firing of the neuron). Knowing εp(t), themoments of firing of the neuron, which are zero crossings ϑ(t) = 0, and the relationshipϑ̇ = 1 + PRC (ϑ)εp(t) ,we solve the inverse problem for the infinitesimal PRC (ϑ) and plot the solution in
Fig.10.24d. As one expects, the PRC is mostly positive, maximal just before the spikeand almost zero during the spike. It would resemble the PRC in Fig.10.23 (Q1 (ϑ) inClass 1) if not for the dip in the middle, for which we have no explanation (probably it
is due to overfitting). The advantage of this method is that it is more immune to noise,because intrinsic fluctuations are spread over the entire p(t) and not concentrated at
the moments of pulses – unless, of course p(t), consists of random pulses, in which casethis method is equivalent to the standard one. The drawback is that we need to solve
the equation above, which we do in exercise 13, using an optimization technique.

10.2.5Phase Model for Coupled Oscillators

Now consider n weakly coupled oscillators of the formpi (t)}|{znX
gij (xi , xj ) ,ẋi = fi (xi ) + ε
j=1

xi ∈ Rm ,(10.11)

Synchronization

T/2

(a)

150
100
50

p(t)

injected current, 40 pA
5 time (sec)

0

10

15

period difference, Ti+1-Ti

interspike periods, Ti (ms)

406

(b)

0

-T/2

0

2

membranepotential

PRC(q), (pA-1)

Ti Ti+1-30 mV

0

10 mV
10 pA

p(t)

(d)

pyramidal neuron8

a v e ra-0.1

Andronov-Hopf

sp
ged

ik e

T

INa+IK-model (Class 1)

(f)

1

10

phase of oscillation, q

0

(e)

8

0.3

(c)

1

4
6
time (sec)

0.2

0

2

1

3

0

2

3

-1
0

2
4
phase of oscillation, q

6

0

2

4
phase of oscillation, q

6

Figure 10.24: Measuring the infinitesimal PRC experimentally in a layer 5 pyramidal
neuron of mouse visual cortex. (a) Interspike periods in response to the injection of DC
current. (b) Differences between successive periods. (c) Spiking 1 second before andafter the noisy current p(t) is injected. (d) Infinitesimal PRC of the neuron (continuous
curve) obtained from 40 cycles and the MATLAB program in exercise 13 (first eight
Fourier terms). Averaged voltage trace during the spike (dotted curve) is plotted forreference. The same procedure is applied to (e) the Andronov-Hopf oscillator and (f)the INa,p +IK -model. Numbers in boxes represent the number of Fourier terms used to
fit the curve; theoretical curves (functions Q1 (ϑ) from Fig.10.23) are dashed.

Synchronization

407
V2

0 V1
-20
-40
-60
-80
T

phase
difference

phasedeviation

phase

membranepotential (mV)

20

q2
0
T

0
T

0

0

q1

j2
j1
c=j2-j1
5

10

15

20

25
time (ms)

30

35

40

45

50

Figure 10.25: The relationship between membrane potential oscillation of two neurons,
V1 (solid) and V2 (dashed), their phases, phase deviations, and phase difference. Shownare simulation of two INa + IK -models with parameters as in Fig.10.3 and coupled
symmetrically via gap junctions 0.1(Vj − Vi ) (see section 2.3.4).
and assume that the oscillators, when uncoupled (ε = 0), have equal free-running
periods T1 = · · · = Tn = T . Applying any of the three methods above to such a weakly
perturbed system, we obtain the corresponding phase modelpi (t)

ϑ̇i = 1 + ε Qi (ϑi ) ·

z
n
X

}|

{

gij (xi (ϑi ), xj (ϑj )) ,(10.12)j=1

where each xi (ϑi ) is the point on the limit cycle having phase ϑi . Note that (10.11) isdefined in Rnm , whereas the phase model (10.12) is defined on the n-torus, denoted asTn .
To study collective properties of the network, such as synchronization, it is convenient to represent each ϑi (t) as
ϑi (t) = t + φi ,(10.13)

with the first term capturing the fast free-running natural oscillation ϑ̇i = 1, and thesecond term capturing the slow network-induced build-up of phase deviation from thenatural oscillation. The relationship between xi (t), ϑi (t) and φi (t) is illustrated in
Fig.10.25.
Substituting (10.13) into (10.12) results in
φ̇i = ε Qi (t + φi ) ·

n
X
j=1

gij (xi (t + φi ), xj (t + φj )) .

(10.14)

408

Synchronization

Note that the right-hand side is of order ε, reflecting the slow dynamics of phase deviations φi seen in Fig.10.25. Thus, it contains two time scales: fast oscillations (variablet) and slow phase modulation (variables φ). The classical method of averaging, reviewed by Hoppensteadt and Izhikevich (1997, Chap. 9), consists in a near-identitychange of variables that transforms the system into the form
φ̇i = εωi + εnX

Hij (φj − φi ) ,(10.15)

j̸=i

where
1
Hij (φj − φi ) =TZ T
Qi (t) · gij (xi (t), xj (t + φj − φi )) dt ,(10.16)0and each ωi = Hii (φi − φi ) = Hii (0) describes a constant frequency deviation fromthe free-running oscillation. Figure 10.26 depicts the functions Hij corresponding to
gap-junction (i.e., electrical; see section 2.3.4) coupling of oscillators in Fig.10.3. Provethat H(χ) = Q(χ) · A/T in the case of pulse-coupling (10.1), so that H(χ) is justre-scaled PRC.
A special case of (10.15) occurs when H is replaced by its first Fourier term, sin.
The resulting system, written in the slow time τ = εt,φ′i = ωi +nX

cij sin(φj − φi + ψij ),j=1

is called the Kuramoto phase model (Kuramoto 1975). Here, the frequency deviations
ωi are interpreted as intrinsic frequencies of oscillators. The strengths of connections
cij are often assumed to be equal to K/n for some constant K, so that the model canbe studied in the limit n → ∞. The phase deviations ψij are often neglected for thesake of simplicity.

Andronov-Hopf oscillator

van der Pol oscillator
Hij(c)

Hij(c)

0.5

0.5

0

0
-0.5

-0.5
G(c)
0

G(c)
2

T

4

INa+IK-model (Class 1)

2

4

6

T

INa+IK-model (Class 2)

4

Hij(c)

20

Hij(c)

0
0

0

G(c)

-4

G(c)

-2

2
4
phase difference, c

6

T

0

1
2
3
phase difference, cTFigure 10.26:
Solid
curves.
Functions
Hij (χ)definedby(10.16) with the input
g(xi , xj ) = (xj1 − xi1 , 0)
corresponding
toelectricalsynapsevia
gap-junction.
Dashedcurves.
Functions G(χ)=
Hji (−χ) − Hij (χ).
Parameters are as in
Fig.10.3.

Synchronization

409
(b)

identify

q2
q2

q1

(c)
identify

(a)

q2

q1

q1

Figure 10.27: Torus knot of type (2, 3) (a) and its representation on the square (b).
The knot produces frequency-locking and phase-locking. (c) Torus knot that does notproduce phase-locking.
To summarize, we transformed the weakly coupled system (10.11) into the phasemodel (10.15) with H given by (10.16) and each Q being the solution to the adjoint
problem (10.10). This constitutes the Malkin theorem for weakly coupled oscillators(Hoppensteadt and Izhikevich 1997, theorem 9.2).

10.3

Synchronization

Consider two coupled phase variables (10.12) in a general formϑ̇1 = h1 (ϑ1 , ϑ2 ) ,ϑ̇2 = h2 (ϑ1 , ϑ2 ) ,with some positive functions h1 and h2 . Since each phase variable is defined on thecircle S1 , the state space of this system is the 2-torus T2 = S1 ×S1 depicted in Fig.10.27,
with ϑ1 and ϑ2 being the longitude and the latitude, respectively. The torus can be
represented as a square with vertical and horizontal sides identified, so that a solution
disappearing at the right side of the square appears at the left side.
The coupled oscillators above are said to be frequency-locked when there is a periodic
trajectory on the 2-torus, which is called a torus knot. It is said to be of type (p, q)
if ϑ1 makes p rotations while ϑ2 makes q rotations, and p and q are relatively prime
integers, that is, they do not have a common divisor greater than 1. Torus knots oftype (p, q) produce p:q frequency-locking, e.g., the 2:3 frequency-locking in Fig.10.27.
A 1:1 frequency-locking is called entrainment. There can be many periodic orbits
on the torus, with stable orbits between unstable ones. Since the orbits on the 2torus cannot intersect, they are all knots of the same type, resulting in the same p:qfrequency-locking.
Let us follow a trajectory on the torus and count the number of rotations of thephase variables. The limit of the ratio of rotations as t → ∞ is independent of the
trajectory we follow, and it is called the rotation number of the torus flow. It is rational
if and only if there is a (p, q) periodic orbit, in which case the rotation number is p/q.
An irrational rotation number implies there are no periodic orbits, and it corresponds to410

Synchronization
frequency locking
in-phaseentrainment

synchronization

phase locking(1:1 frequency locking)anti-phase

Figure 10.28: Various degrees of locking of oscillators.
a quasi-periodic or multifrequency torus flow. Oscillators exhibit phase drifting in thiscase. Denjoy (1932) proved that such coupled oscillators are topologically equivalent
to the uncoupled system ϑ̇1 = r, ϑ̇2 = 1 with irrational r.
Suppose the oscillators are frequency-locked; that is, there is a p:q limit cycle attractor on the torus. We say that the oscillators are p:q phase-locked ifqϑ1 (t) − pϑ2 (t) = const
on the cycle. The value of the constant determines whether the locking is in-phase(const= 0), anti-phase (const= T /2; half-period), or out-of-phase. Frequency-locking
does not necessarily imply phase-locking: the (2, 3) torus knot in Fig.10.27b corresponds to phase-locking, whereas that in Fig.10.27c does not. Frequency-locking without phase-locking is called phase trapping. Finally, synchronization is a 1:1 phaselocking. The phase difference ϑ2 − ϑ1 is also called phase lag or phase lead. The
relationships between all these definitions are shown in Fig.10.28.
Frequency-locking, phase-locking, entrainment, and synchronization of a networkof n > 2 oscillators are the same as pairwise locking, entrainment, and synchronizationof the oscillators comprising the network. In addition, a network can exhibit partial
synchronization when only a subset of oscillators is synchronized.
Synchronization of oscillators with nearly identical frequencies is described by thephase model (10.15). Existence of one equilibrium of (10.15) implies the existence ofthe entire circular family of equilibria, since translation of all φi by a constant phaseshift does not change the phase differences φj − φi , and hence the form of (10.15).
This family corresponds to a limit cycle of (10.11), on which all oscillators, xi (t + φi ),have equal frequencies and constant phase shifts (i.e., they are synchronized, possibly
out-of-phase).

10.3.1

Two Oscillators

Consider (10.11) with n = 2, describing two coupled oscillators, as in Fig.10.29. Let us
introduce the “slow” time τ = εt and rewrite the corresponding phase model (10.15)

Synchronization

411

Figure 10.29: A major part of computational neuroscience concerns coupled oscillators.
in the form
φ′1 = ω1 + H1 (φ2 − φ1 ) ,φ′2 = ω2 + H2 (φ1 − φ2 ) ,where ′ = d/dτ is the derivative with respect to slow time. Let χ = φ2 − φ1 denotethe phase difference between the oscillators. Then the two-dimensional system above
becomes the one-dimensional
χ′ = ω + G(χ) ,(10.17)whereω = ω2 − ω1andG(χ) = H2 (−χ) − H1 (χ)

are the frequency mismatch and the anti-symmetric part of the coupling, respectively(illustrated in Fig.10.26, dashed curves). A stable equilibrium of (10.17) corresponds
to a stable limit cycle of the phase model.
All equilibria of (10.17) are solutions to G(χ) = −ω, and they are intersectionsof the horizontal line −ω with the graph of G, as illustrated in Fig.10.30a. They arestable if the slope of the graph is negative at the intersection. If the oscillators are
identical, then G(χ) = H(−χ) − H(χ) is an odd function (i.e., G(−χ) = −G(χ)), andχ = 0 and χ = T /2 are always equilibria (possibly unstable) corresponding to thein-phase and anti-phase synchronized solutions. The stability condition of the in-phasesynchronized state isG′ (0) = −2H ′ (0) < 0(stability of in-phase synchronization)

412
(a)

Synchronization
(b)

max G

max G

-w
0

0
G(c)

min G
0

G(c)

-w
T

1
2
3
phase difference, c

0

1
2
3
phase difference, cTFigure 10.30: Geometrical interpretation of equilibria of the phase model (10.17) for
gap-junction-coupled Class 2 INa + IK -oscillators (see Fig.10.26).
non-relaxation oscillator

relaxation oscillator

G(c)

G(c)

0

0

-w

-w

0

phase difference, cT0phase difference, cTFigure 10.31: Functions G(χ) for weakly coupled oscillators of non-relaxation (smooth)and relaxation types. The frequency mismatch ω creates a phase difference in thesmooth case, but not in the relaxation case.
The in-phase synchronization of electrically (gap-junction) coupled oscillators in Fig.10.26
is stable because the slope of G (dashed curves) is negative at χ = 0. Simulation of two
coupled INa + IK -oscillators in Fig.10.25 confirms that. Coupled oscillators in the Class
2 regime also have a stable anti-phase solution, since G′ < 0 at χ = T /2 in Fig.10.30a.
The max and min values of the function G determine the tolerance of the network
for the frequency mismatch ω, since there are no equilibria outside this range. Geometrically, as ω increases (the second oscillator speeds up), the horizontal line −ωin Fig.10.30a slides downward, and the phase difference χ = φ2 − φ1 increases, compensating for the frequency mismatch ω. When ω > − min G, the second oscillator
becomes too fast, and the synchronized state is lost via saddle-node on invariant circle
bifurcation (see Fig.10.30b). This bifurcation corresponds to the annihilation of stableand unstable limit cycles of the weakly coupled network, and the resulting activity iscalled drifting, cycle slipping, or phase walk-through. The variable χ slowly passes
the ghost of the saddle-node point, where G(χ) ≈ 0, then increases past T , appears
at 0, and approaches the ghost again, thereby slipping a cycle and √walking throughall the phase values [0, T ]. The frequency of such slipping scales as ω + min G; seesection 6.1.2.
In Fig.10.31 we contrast synchronization properties of weakly coupled oscillators ofSynchronization

413

relaxation and non-relaxation type. The function G(χ) of the former has a negative
discontinuity at χ = 0 (section 10.4.4). An immediate consequence is that the in-phasesynchronization is rapid and persistent in the presence of the frequency mismatchω. Indeed, if G is smooth, then χ slows down while it approaches the equilibriumχ = 0. As a result, complete synchronization is an asymptotic process that requires
an infinite period of time to attain. In contrast, when G is discontinuous at 0, thevariable χ does not slow down, and it takes a finite period of time to lock. Changing
the frequency mismatch ω shifts the root of −ω = G(χ) in the continuous case, but
not in the discontinuous case. Hence, the in-phase synchronized state χ = 0 of coupled
relaxation oscillators exists and is stable in a wide range of ω.

10.3.2

ChainsUnderstanding the synchronization properties of two coupled oscillators helps one instudying the dynamics of chains of n > 2 oscillators
φ′i = ωi + H + (φi+1 − φi ) + H − (φi−1 − φi ) ,(10.18)where the functions H + and H − describe the coupling in the ascending and descending
directions of the chain, as in Fig.10.32. Any phase-locked solution of (10.18) has theform φi (τ ) = ω0 τ + ϕi , where ω0 is the common frequency of oscillation and ϕi areconstants. These satisfy n conditions
ω0 = ω1 + H + (ϕ2 − ϕ1 ) ,ω0 = ωi + H + (ϕi+1 − ϕi ) + H − (ϕi−1 − ϕi ) ,ω0 = ωn + H − (ϕn−1 − ϕn ) .

i = 2, . . . , n − 1 ,A solution with ϕ1 < ϕ2 < · · · < ϕn or with ϕ1 > ϕ2 > · · · > ϕn (as in Fig.10.32)
is called a traveling wave. Indeed, the oscillators oscillate with a common frequency
ω0 but with different phases that increase or decrease monotonically along the chain.
Such a behavior is believed to correspond to central pattern generation (CPG) in
crayfish, undulatory locomotion in lampreys and dogfish, and peristalsis in vascular andintestinal smooth muscles. Below we consider two fundamentally different mechanismsof generation of traveling waves.
Frequency Differences
Suppose the connections in (10.18) look qualitatively similar to those in Fig.10.26, inparticular, H + (0) = H − (0) = 0. If the frequencies are all equal, then the in-phasesynchronized solution φ1 = · · · = φn exists and is stable. A traveling wave exists when
the frequencies are not all equal.
Let us seek the conditions for the existence of a traveling wave with a constantphase shift, say χ = ϕi+1 − ϕi , along the chain. Subtracting each equation from thesecond one, we find that0 = ω2 − ω1 + H − (−χ) ,0 = ω2 − ωi ,0 = ω2 − ωn + H + (χ) ,414

Synchronization
H-

H-

j1

j2
H+

H-

H-

H-

H-

H-

H+

H+

H+

H+

H+

j3
H+

jn

traveling wave

Figure 10.32: Traveling wave solutions in chains of oscillators (10.18) describe undulatory locomotion and central pattern generation.
and ω0 = ω1 + ωn − ω2 . In particular, if ω1 ≤ ω2 = · · · = ωn−1 ≤ ωn , which corresponds
to the first oscillator being tuned up and the last oscillator being tuned down, then
χ < 0 and the traveling wave moves upward, as in Fig.10.32, that is, from the fastest to
the slowest oscillator. Interestingly, such an ascending wave exists even when H − = 0,
that is, even when the coupling is only in the opposite, descending direction.
When there is a linear gradient of frequencies (ω1 > ω2 > · · · > ωn or vice versa),
as in the cases of the smooth muscle of intestines or leech CPG for swimming, one may
still observe a traveling wave, but with a non-constant phase difference along the chain.
When the gradient is large enough, the synchronized solution corresponding to a single
traveling wave disappears, and frequency plateaus may appear (Ermentrout and Kopell
1984). That is, solutions occur in which the first k < n oscillators are phase-locked
and the last n − k oscillators are phase-locked as well, but the two pools, forming two
clusters, oscillate with different frequencies. There may be many frequency plateaus.

Coupling Functions
A traveling wave solution may exist even when all the frequencies are equal, if either
H + (0) ̸= 0 or H − (0) ̸= 0. As an example, consider the case of descending coupling(H − = 0)φ′i = ω + H + (φi+1 − φi ) ,i = 1, . . . , n − 1 .

From φ′n = ω we find that ω0 = ω, that is, the common frequency is the frequencyof the free oscillation of the last, uncoupled oscillator. The phase lag along the chain,χ = φi+1 − φi , satisfies n − 1 identical conditions 0 = H + (χ). Thus, the traveling wave
with a constant phase shift exists when H + has a zero crossing with positive slope, incontrast to Fig.10.26. The sign of χ, and not the direction of coupling, determines the
direction of wave propagation.

Synchronization

10.3.3

415

Networks

Now let us consider weakly connected networks (10.11) with arbitrary, possibly allto-all coupling. To study synchronized states of the network, we need to determine
whether the corresponding phase model (10.15) has equilibria and to examine theirstability properties. A vector ϕ = (ϕ1 , . . . , ϕn ) is an equilibrium of (10.15) when0 = ωi +n
X

Hij (ϕj − ϕi )for all i .

(10.19)j̸=1

It is stable when all eigenvalues of the linearization matrix (Jacobian) at ϕ have negative real parts, except one zero eigenvalue corresponding to the eigenvector along thecircular family of equilibria (ϕ plus a phase shift is a solution of (10.19), too, since thephase differences ϕj − ϕi are not affected).
In general, determining the stability of equilibria is a difficult problem. Ermentrout(1992) found a simple sufficient condition. Namely, if• aij = Hij′ (ϕj − ϕi ) ≥ 0, and• the directed graph defined by the matrix a = (aij ) is connected (i.e., each oscillator is influenced, possibly indirectly, by every other oscillator),then the equilibrium ϕ is neutrally stable, and the corresponding limit cycle x(t + ϕ)of (10.11) is asymptotically stable.
Another sufficient condition was found by Hoppensteadt and Izhikevich (1997). It
states that if system (10.15) satisfies• ω1 = · · · = ωn = ω(identical frequencies) and• Hij (−χ) = −Hji (χ) (pairwise odd coupling)
for all i and j, then the network dynamics converge to a limit cycle. On the cycle, alloscillators have equal frequencies 1 + εω and constant phase deviations.
The proof follows from the observation that (10.15) is a gradient system in therotating coordinates φ = ωτ + ϕ, with the energy function
n

n

1 XX
Rij (ϕj − ϕi ) ,E(ϕ) =
2 i=1 j=1
One can check that dE(ϕ)/dτ = −equality only at equilibria.

Z χ
where

Rij (χ) =Hij (s) ds .
0

P ′ 2
(ϕi ) ≤ 0 along the trajectories of (10.15), with

416

Synchronization

j

i

j

j

r

reiy

y
Figure 10.33: The Kuramoto synchronization index (10.21) describes the degree of coherence inthe network (10.20).

10.3.4Mean-Field Approximations

Synchronization of the phase model (10.15) with randomly distributed frequency deviations ωi can be analyzed in the limit n → ∞, often called the thermodynamic limit
by physicists. We illustrate the theory using the special case H(χ) = sin χ (Kuramoto
1975)
n
KX
sin(φj − φi ) ,φi ∈ [0, 2π] ,(10.20)φ′i = ωi +n j=1
where K > 0 is the coupling strength and the factor 1/n ensures that the model
behaves well as n → ∞. The complex-valued sum of all phases,1reiψ =

n
X

n j=1

eiφj

(Kuramoto synchronization index),(10.21)

describes the degree of synchronization in the network. The parameter r is often
called the order parameter by physicists. Apparently, the in-phase synchronized state
φ1 = · · · = φn corresponds to r = 1, with ψ being the population phase. In contrast,
the incoherent state, with all φi having different values randomly distributed on theunit circle, corresponds to r ≈ 0. (The case r ≈ 0 can also correspond to two or moreclusters of synchronized neurons, oscillating anti-phase or out-of-phase and canceling
each other.) Intermediate values of r correspond to a partially synchronized or coherent
state, depicted in Fig.10.33. Some phases are synchronized in a cluster, while othersroam around the circle.
Multiplying both sides of (10.21) by e−iφi and considering only the imaginary parts,
we can rewrite (10.20) in the equivalent form
φ′i = ωi + Kr sin(ψ − φi ) ,which emphasizes the mean-filed character of interactions between the oscillators: they
are all pulled into the synchronized cluster (φi → ψ) with the effective strength proportional to the cluster size r. This pull is offset by the random frequency deviations
ωi , which pull away from the cluster.

Synchronization

417

Let us assume that the frequencies ωi are distributed randomly around 0 with asymmetric probability density function g(ω) (e.g., Gaussian). Kuramoto (1975) has
shown that in the limit n → ∞, the cluster size r obeys the self-consistency equation
Z +π/2r = rKg(rK sin φ) cos2 φ dφ

(10.22)−π/2derived in exercise 21. Note that r = 0, corresponding to the incoherent state, is always
a solution of this equation. When the coupling strength K is greater than a certain
critical value,
2
,
Kc =
πg(0)
an additional, nontrivial solution r >p 0 appears, which corresponds to a partiallysynchronized state. It scales as r = 16(K − Kc )/(−g ′′ (0)πKc4 ), as the reader can
prove by expanding g in a Taylor series. Thus, the stronger the coupling K relative
to the random distribution of frequencies, the more oscillators synchronize into a coherent cluster. The issue of stability of incoherent and partially synchronized states isdiscussed by Strogatz (2000).

10.4

ExamplesBelow we consider simple examples of oscillators to illustrate the theory developed inthis chapter. Our goal is to understand which details of oscillators are important inshaping the PRC, the form of the function H in the phase deviation model, and, hence,the existence and stability of synchronized states.

10.4.1

Phase Oscillators

Let us consider the simplest possible kind of a nonlinear oscillator, known as the phaseoscillator:ẋ = f (x) + εp(t) ,x ∈ S1 ,(10.23)where f (x) > 0 is a periodic function, for example, f (x) = a + sin x with a > 1.
Note that this kind of oscillator is quite different from the two- or high-dimensional
conductance-based models with limit cycle attractors that we considered in the earlier
chapters. Here, the state variable x is one-dimensional, defined on a circle S1 . It may beinterpreted as a measure of distance along a limit cycle attractor of a multi-dimensional
system.
Consider the unperturbed (ε = 0) phase oscillator ẋ = f (x), and let x(t) be its
solution with some period T > 0. Following Kuramoto’s idea, we substitute x(ϑ) into
(10.23) and use the chain rule,f (x(ϑ)) + εp(t) = {x(ϑ)}′ = x′ (ϑ) ϑ′ = f (x(ϑ))ϑ′ ,418

Synchronization

to get the new phase equationϑ̇ = 1 + εp(t)/f (x(ϑ)) ,(10.24)

which is equivalent to (10.23) for any, not necessarily small, ε.
We can also obtain (10.24) by using any of the three methods of reduction ofoscillators to phase models:• Malkin’s method is the easiest one. We do not even have to solve the onedimensional adjoint equation (10.10) having the form Q̇ = −f ′ (x(t)) Q, because
we can obtain the solution Q(t) = 1/f (x(t)) directly from the normalization
condition Q(t)f (x(t)) = 1.
• Kuramoto’s method relies on the function ϑ(x), which we can find implicitly.
Since the phase at a point x(t) on the limit cycle is t, x(ϑ) is the inverse of ϑ(x).
Using the rule for differentiating of inverse functions, ϑ′ (x) = 1/x′ (ϑ), we findgrad ϑ = 1/f (x(ϑ)).
• Winfree’s method relies on PRC (ϑ), which we find using the following procedure:
A pulsed perturbation at phase ϑ moves the solution from x(ϑ) to x(ϑ) + A,
which is x(ϑ + PRC (ϑ, A)) ≈ x(ϑ) + x′ (ϑ)PRC (ϑ, A) when A is small. Hence,
PRC (ϑ, A) ≈ A/x′ (ϑ) = A/f (x(ϑ)), and the linear response is Z(ϑ) = 1/f (x(ϑ))
when A → 0.
Two coupled identical oscillatorsẋ1 = f (x1 ) + εg(x2 )ẋ2 = f (x2 ) + εg(x1 )can be reduced to the phase model (10.17) with G(χ) = H(−χ) − H(χ), where1
H(χ) =
T

Z T
0

1
Q(t) g(x(t + χ)) dt =
T

Z T
0

g(x(t + χ))dt .
f (x(t))

The condition for exponential stability of the in-phase synchronized state, χ = 0, canbe expressed in the following three equivalent forms
Z TZZg ′ (x)f ′ (x)′g (x(t)) dt > 0
ordx > 0org(x) dx > 0 , (10.25)
20
S1 f (x)S1 f (x)as we prove in exercise 24.

10.4.2SNIC Oscillators

Let us go through all the steps of derivation of the phase equation using a neuronmodel exhibiting low-frequency periodic spiking. Such a model is near the saddle-nodeSynchronization
-

419

+

-

+

+

-

spike

spike

spike

1
0
x'=-1+x2

0
x'=0+x2

0
x'=1+x2

-1

x(t)

spike

spike

x(t)=-cot tx'=1+x2

A0xnormalized PRC (PRC(q,A)/A)

Figure 10.34: Phase portraits and typical oscillations of the quadratic integrate-and-fireneuron ẋ = I + x2 with x ∈ R ∪ {±∞}. Parameter: I = −1, 0, +1.
1

A=0.1
A=1
A=2
A=3

0

0

phase of oscillation, qTFigure 10.35: The dependence of the PRC of the quadratic integrate-and-fire modelon the strength of the pulse A.
on invariant circle (SNIC) bifurcation studied in section 6.1.2. Appropriate rescalingof the membrane potential and time converts the model into the normal form
x ′ = 1 + x2 ,x∈R.

Because of the quadratic term, x escapes to infinity in a finite time, producing a spike
depicted in Fig.10.34. If we identify −∞ and +∞, then x exhibits periodic spiking of
infinite amplitude. Such a spiking model is called a quadratic integrate-and-fire (QIF)
neuron (see also section 8.1.3 for some generalizations of the model).
Strong Pulse
The solution of this system, starting at the spike, that is, at x(0) = ±∞, isx(t) = − cot t ,as the reader can check by differentiating. It is a periodic function with T = π;hence, we can introduce the phase of oscillation via the relation x = − cot ϑ. The
corresponding PRC can be found explicitly (see exercise 9) and it has the form
PRC (ϑ, A) = π/2 + atan (A − cot ϑ) − ϑ ,420Synchronization

depicted in Fig.10.35, where A is the magnitude of the pulse. Note that the PRC tilts
to the left as A increases. Indeed, the density of isochrons, denoted by black points on
the x-axis in the figure, is maximal at the ghost of the saddle-node point x = 0, where
the parabola 1 + x2 has the knee. This corresponds to the inflection point of the graphof x(t) in Fig.10.34, where the dynamics of x(t) is the slowest. The effect of a pulse
is maximal just before the ghost because x can jump over the ghost and skip the slow
region. The stronger the pulse, the earlier it should arrive; hence the tilt.
Weak Coupling
The PRC behaves as A sin2 ϑ, with ϑ ∈ [0, π], when A is small, as the reader cansee in Fig.10.35 or prove by differentiating the function PRC (ϑ, A) with respect to A.
Therefore, Z(ϑ) = sin2 ϑ, and we can use Winfree’s approach to transform the weakly
perturbed quadratic integrate-and-fire (QIF) oscillator
x′ = 1 + x2 + εp(t)into its phase modelx′ = 1 + ε(sin2 ϑ)p(t) ,ϑ ∈ [0, π] .

The results of the previous section, Q(ϑ) = 1/f (x(ϑ)) = 1/(1+cot2 ϑ) = sin2 ϑ, confirm
the phase model. In fact, any neuronal model C V̇ = I − I∞ (V ) near saddle-node on
invariant circle bifurcation point (Isn , Vsn ) has infinitesimal PRC:
PRC (ϑ) =C
sin2 ϑ ,I − Isnϑ ∈ [0, π] ,as the reader can prove as an exercise. The function sin2 ϑ has the same form as(1 − cos θ) if we change variables θ = 2ϑ (notice the font difference). The change ofvariables scales the period from π to 2π.
In Fig.10.36a we compare the function with numerically obtained PRCs for theINa + IK -model in the Class 1 regime.
Since the ghost of the saddle-node point,
revealing itself as an inflection of the voltage trace in Fig.10.36b, moves to the right as
I increases away from the bifurcation value I = 4.51, so does the peak of the PRC.
Figure 10.36a emphasizes the common features of all systems undergoing saddlenode on invariant circle bifurcation: they are insensitive to the inputs arriving during
the spike, since PRC ≈ 0 when ϑ ≈ 0, T . The oscillators are most sensitive to the input
when they are just entering the ghost of the resting state, where PRC is maximal.
The location of the maximum tilts to the left as the strength of the input increases,
and may tilt to the right as the distance to the bifurcation increases. Finally, PRCs
are non-negative, so positive (negative) inputs can only advance (delay) the phase of
oscillation.

Synchronization

421

(b)
1

0.8

I=4.52

0.6

sin2q

0.4
0.2

I=4.55
I=4.6
I=4.7
I=5
I=10

0
0

20

membrane potential, (mV)normalized PRC (Q1(q))(a)

phase of oscillation, q

T

0
-20
-40

inflection
points-60-800phase of oscillation, qTFigure 10.36: (a) Numerically found PRCs of the INa + IK -oscillator in Class 1 regime(parameters as in Fig.4.1a) and various I using the MATLAB program in exercise 12.
(b) Corresponding voltage traces show that the inflection point (slowest increase) of V
moves right as I increases.
Gap Junctions
Now consider two oscillators coupled via gap junctions (discussed in section 2.3.4):
x′1 = 1 + x21 + ε(x2 − x1 ) ,x′2 = 1 + x22 + ε(x1 − x2 ) .
Let us determine the stability of the in-phase synchronized state x1 = x2 . The corresponding phase model (10.12) has the form
ϑ′1 = 1 + ε(sin2 ϑ1 )(cot ϑ1 − cot ϑ2 ) ,ϑ′2 = 1 + ε(sin2 ϑ2 )(cot ϑ2 − cot ϑ1 ) .
The function (10.16) can be found analytically:
Z
1
1 π 2
sin t (cot t − cot(t + χ)) dt = sin 2χ ,H(χ) =
π 02
so that the model in the phase deviation coordinates, ϑ(t) = t + φ, has the form
φ′1 = (ε/2) sin{2(φ2 − φ1 )} ,φ′2 = (ε/2) sin{2(φ1 − φ2 )} .
The phase difference, χ = φ2 − φ1 , satisfies the equation (compare with Fig.10.26)
χ′ = −ε sin 2χ ,and, apparently, the in-phase synchronized state, χ = 0, is always stable while theanti-phase state χ = π/2 is not.

422

Synchronization

Weak Pulses
Now consider two weakly pulse-coupled oscillators
x′1 = 1 + x21 + ε1 δ(t − t2 ) ,x′2 = 1 + x22 + ε2 δ(t − t1 ) ,where t1 and t2 are the moments of firing (x(t) = ∞) of the first and the second
oscillator, respectively, and ε1 and ε2 are the strengths of synaptic connections. The
corresponding phase model (10.12) has the form
ϑ′1 = 1 + ε1 (sin2 ϑ1 )δ(t − t2 )
ϑ′2 = 1 + ε2 (sin2 ϑ2 )δ(t − t1 ) .
Since

Z
1 π 2
1
H(χ) =
sin t δ(t + χ) dt = sin2 χ ,π 0
πthe corresponding phase deviation model (10.15) isε1
sin2 (φ2 − φ1 ) ,π
ε2=
sin2 (φ1 − φ2 ) .
π

φ′1 =
φ′2

The phase difference χ = φ2 − φ1 satisfies the equation
χ′ =ε2 − ε1
sin2 χ,π

which becomes χ′ = 0 when the coupling is symmetric. In this case, the oscillators preserve (on average) the initial phase difference. When ε1 ̸= ε2 , the in-phase synchronizedstate χ = 0 is only neutrally stable. Interestingly, it becomes exponentially unstable
in a network of three or more pulse-coupled Class 1 oscillators (see exercise 23).
Weak Pulses with Delays
The synchronization properties of weakly pulse-coupled oscillators can change significantly when explicit axonal conduction delays are introduced. As an example, consider
the system
x′1 = 1 + x21 + εδ(t − t2 − d) ,x′2 = 1 + x22 + εδ(t − t1 − d) ,where d ≥ 0 is the delay. exercise 18 shows that delays introduce simple phase shifts,
so that the phase model has the form
εφ′1 =sin2 (φ2 − φ1 − d) ,
π
ε
′
φ2 =
sin2 (φ1 − φ2 − d) ,π

Synchronization

423
sin2t

sin2t

1

p(t)

0
0

phase of oscillationp

Figure 10.37: Synaptic transmission function p(t) typically has an asymmetric shapewith fast rise and slow decay.
The phase difference χ = φ2 − φ1 satisfies
χ′ = ε sin 2dεsin 2χ .
sin2 (χ + d) − sin2 (χ − d) =ππThe stability of synchronized states is determined by the sign of the function sin 2d.
The in-phase state χ = 0 is unstable when sin 2d > 0, that is, when the delay is shorter
than the half-period π/2, stable when the delay is longer than the half-period but
shorter than one period π, unstable for even longer delays, and so on. The stability of
the anti-phase state χ = π/2 is reversed, that is, it is stable for short delays, unstable
for longer delays, then stable again for even longer delays, and so on. Finally, when thepulses are inhibitory (ε < 0), the (in)stability character is flipped so that the in-phasestate becomes stable for short delays.
Weak Synapses
Now suppose that each pulse is not a delta function, but is smeared in time (i.e., it
has a time course p(t − ti ) with p(0) = p(π) = 0). That is, the synaptic transmission
starts right after the spike of the presynaptic neuron and ends before the onset of
the next spike. The function p has a typical unimodal shape with fast rise and slow
decay, depicted in Fig.10.37. The discussion below is equally applicable to the case ofp(t, x) = g(t)(E − x) with g > 0 being the synaptic conductance with the shape in
the figure and E being the synaptic reverse potential, positive (negative) for excitatory(inhibitory) synapses.
Two weakly synaptically coupled SNIC (Class 1) oscillators
x′1 = 1 + x21 + εp(t − t2 ) ,x′2 = 1 + x22 + εp(t − t1 )can be converted into a general phase model with the connection function (10.16) inthe form
Z
1 π 2
sin t p(t + χ) dt ,H(χ) =π 0424Synchronization
INa+IK-model

n

V(t)

V
quadratic integrate-and-fire model

x

reset

-1 0 +1

reset

xreset

reset

reset

spike

x(t)=-coth2(t-T)

Figure 10.38: Top. Periodic spiking of the INa +IK -neuron near saddle-node homoclinic
orbit bifurcation (parameters as in Fig.4.1a with τ (V ) = 0.167 and I = 4.49). Bottom.
Spiking in the corresponding quadratic integrate-and-fire model.
and it can be computed explicitly for some simple p(t).
The in-phase synchronized solution, χ = 0, is stable whenZ
1 π 2 ′′
sin t p (t) dt > 0 .
H (0) =
π 0
Since the function sin2 t depicted in Fig.10.37 is small at the ends of the interval andlarge in the middle, the integral is dominated by the sign of p′ in the middle. Fast-risingand slowly decaying excitatory (p > 0) synaptic transmission has p′ < 0 in the middle(as in the figure), so the integral is negative and the in-phase solution is unstable. In
contrast, fast-rising slowly decaying inhibitory (p < 0) synaptic transmission has p′ > 0
in the middle, so the integral is positive and the in-phase solution is stable.
AnotherR
way to see this is to integrate the equation by parts, reduce it to − p(t) sin 2t dt,
and note that p(t) is concentrated in the first (left) half of the period, where sin 2t
is positive. Hence, positive (excitatory) p results in H ′ (0) < 0, and vice versa. Bothapproaches confirm the theoretical results independently obtained by van Vreeswijk
et al. (1994) and Hansel et al. (1995) that inhibition, not excitation, synchronizes
Class 1 (SNIC) oscillators. The relationship is inverted for the anti-phase solutionχ = π/2 (the reader should prove this), and no relationships are known for other typesof oscillators.

10.4.3

Homoclinic Oscillators

Besides the SNIC bifurcation considered above, low-frequency oscillations may also
indicate the proximity of the system to a saddle homoclinic orbit bifurcation, as inSynchronization

425
(b)

normalized PRC (Q1(q))(a)

nsinh2(q-T)-n'(q)spike
downstrokeAsinh2(q-T)

0B

numerical
0

0.1T

0
phase advancephase delay
0

0.1T

phase of oscillation, qV

TFigure 10.39: (a) Numerically found PRCs of the INa + IK -oscillator near saddle-nodehomoclinic orbit bifurcation (as in Fig.10.38), using the MATLAB program in exercise 12. Magnification shows the divergence from the theoretical curve sinh2 (ϑ − T )during the spike. (b) A pulsed input during the downstroke of the spike can produce asignificant phase delay (pulse A) or advance (pulse B) not captured by the quadratic
integrate-and-fire model.
Fig.10.38 (top). The spiking trajectory in the figure quickly approaches a small shaded
neighborhood of the saddle along the stable direction, and then slowly diverges fromthe saddle along the unstable direction, thereby resulting in a large-period oscillation.
As often the case in neuronal models, the saddle equilibrium is near a stable nodeequilibrium corresponding to the resting state, and the system is near the codimension2 saddle-node homoclinic orbit bifurcation studied in section 6.3.6. As a result, there
is a drastic difference between the attraction and divergence rates to the saddle, sothat the dynamics in the shaded neighborhood of the saddle-node in the figure can be
reduced to the one-dimensional V -equation, which in turn can be transformed into the“quadratic integrate-and-fire” form
x′ = −1 + x2 ,if x = +∞, then x ← xreset ,with solutions depicted in Fig.10.38 (bottom). The saddle and the node correspondto x = +1 and x = −1, respectively. One can check by differentiating that thesolution of the model with x(0) = xreset > 1 is x(t) = − coth(t − T ), where coth(s) =(es + e−s )/(es − e−s ) is the hyperbolic cotangent and T = acoth (xreset ) is the period of
oscillation, which becomes infinite as xreset → 1.
Using the results of section 10.4.1, we find the function
Q(ϑ) = 1/(−1 + coth2 (ϑ − T )) = sinh2 (ϑ − T )whose graph is shown in Fig.10.39a. For comparison, we plotted the numerically found
PRC for the INa + IK -oscillator to illustrate the disagreement between the theoreticaland numerical curves in the region ϑ < 0.1T corresponding to the downstroke ofthe spike. Such a disagreement is somewhat expected, since the quadratic integrateand-fire model ignores spike downstroke. If a pulse arriving during the downstroke

426

Synchronization

displaces the trajectory to the exterior of the limit cycle (as in Fig.10.39b, pulse A),
then the trajectory becomes closer to the saddle equilibrium when it reenters a smallneighborhood of the saddle, thereby leading to a significant phase delay. Displacements
to the interior of the cycle (as in Fig.10.39b, pulse B) push away from the saddle
and lead to phase advances. The direction and the magnitude of displacements are
determined by the derivative of the slow variable n′ along the limit cycle.
The region of disagreement between theoretical and numerical PRCs becomes infinitesimal relative to T → ∞ near the bifurcation. Theoretical PRC can be used to
study anti-phase and out-of-phase synchronization of pulse-coupled oscillators, but notof in-phase synchronization, because the region of breakdown is the only importantregion in this case. Finally, note that as T → ∞, the spiking limit cycle fails to be exponentially stable, and the theory of weakly coupled oscillators is no longer applicable
to it.
Though the PRC in Fig.10.39 is quite different from the one corresponding to
SNIC oscillators in Fig.10.36, there is an interesting similarity between these two cases:
both can be reduced to quadratic integrate-and-fire neurons, and both have cotangentshaped periodic spiking solutions and sine-squared-shape PRCs, except that they are“regular” in the SNIC case and hyperbolic in the homoclinic case (see also exercise 26).

10.4.4

Relaxation Oscillators and FTMConsider two relaxation oscillators having weak fast → fast connectionsµẋi = f (xi , yi ) + εpi (xi , xk ) ,ẏi = g(xi , yi ) ,(10.26)

where i = 1, 2 and k = 2, 1. This system can be converted to a phase model in therelaxation limit ε ≪ µ → 0 (Izhikevich 2000b). The connection functions Hi (χ) have apositive discontinuity at χ = 0, which occurs because the x-coordinate of the relaxation
limit cycle is discontinuous at the jump points. Hence, the phase difference function
G(χ) = H2 (−χ) − H1 (χ) has a negative discontinuity at χ = 0 (depicted in Fig.10.31).
This reflects the profound difference between behaviors of weakly coupled oscillatorsof the relaxation and non-relaxation types, discussed in section 10.3.1: The in-phasesynchronized solution, χ = 0, in the relaxation limit µ → 0 is stable and persistentin the presence of the frequency mismatch ω, and it has a rapid rate of convergence.
The reduction to a phase model breaks down when ε ≫ µ → 0, that is, when theconnections are relatively strong. One can still analyze such oscillators in the special
case considered below.
Fast Threshold Modulation
Consider (10.26) and suppose that p1 = p2 = p is a piecewise constant function: p = 1
when the presynaptic oscillator, xk , is on the right branch of the cubic x-nullcline
corresponding to an active state, and p = 0 otherwise (see Fig.10.40a). Somers andSynchronization

427

(a)

(c)

p(x)=1

y1(t)

y2(t)

p(x)=0

t0 t1

(b)

(d)
f(x
, y)=0
f(x

d
a

b'

t0

=0
)+e
,y

=0
)+e
,y

, y)=0
f(x

f(x

d'
a

t1
b'

Figure 10.40: Fast threshold modulation of relaxation oscillation. (a) The Heavisideor sigmoidal coupling function p(x) is constant while x is on the left or right branch of
the x-nullcline. (b) In the relaxation limit µ = 0, the synchronized limit cycle consistsof the left branch of the nullcline f (x, y) = 0 and the right branch of the nullclinef (x, y) + ε = 0. When oscillator 1 is ahead of oscillator 2 (c), the phase difference
between them decreases after the jump (d).
Kopell (1993, 1995) referred to such a coupling in the relaxation limit µ → 0 as fast
threshold modulation (FTM), and found a simple criterion of stability of synchronizedstate that works even for strong coupling.
Since the oscillators are identical, the in-phase synchronized state exists, during
which the variables x1 and x2 follow the left branch of the x-nullcline defined byf (x, y) = 0 (see Fig.10.40b) until they reach the jumping point a. During the instantaneous jump, they turn on the mutual coupling ε, and land at some point b′ on
the perturbed x-nullcline defined by f (x, y) + ε = 0. They follow the new nullcline to
the right (upper) knee, and then jump back to the left branch.
To determine the stability of the in-phase synchronization, we consider the casewhen oscillator 1 is slightly ahead of oscillator 2, as in Fig.10.40c. We assume that thephase difference between the oscillators is so small – or alternatively, the strength ofcoupling is so large – that when oscillator 1 jumps and turns on its input to oscillator
2, the latter, being at point d in Fig.10.40d, is below the left knee of the perturbed
x-nullcline f (x, y) + ε = 0 and therefore jumps, too. As a result, both oscillators jump
to the perturbed x-nullcline and reverse their order. Although the apparent distance
between the oscillators, measured by the difference of their y-coordinates, is preserved
during such a jump, the phase difference between them usually is not.
The phase difference between two points on a limit cycle is the time needed totravel from one point to the other. Let τ0 (d) be the time needed to slide from pointd to point a along the x-nullcline in Fig.10.40d (i.e., the phase difference just before

428

Synchronization

the jump). Let τ1 (d) be the time needed to slide from point b′ to point d′ (i.e., thephase difference after the jump). The phase difference between the oscillators during
the jump changes by the factor C(d) = τ1 (d)/τ0 (d), called the compression function.
The difference decreases when the compression function C(d) < 1 uniformly for all dnear the left knee a. This condition has a simple geometrical meaning: the rate ofchange of y(t) is slower before the jump than after it, so that y(t) has a “scalloped”
shape, as in Fig.10.40c. As an exercise, prove that C(d) → |g(a)|/|g(b′ )| as d → a.
If the compression function at the right (upper) knee is also less than 1, then thein-phase synchronization is stable. Indeed, the phase difference does not change while
the oscillators slide along the nullclines, and it decreases geometrically with each jump.
In fact, it suffices to require that the product of compression factors at the two knees
be less than 1, so that any expansion at one knee is compensated for by even stronger
compression at the other knee.

10.4.5Bursting Oscillators

Let us consider bursting neurons coupled weakly through their fast variables:ẋi = f (xi , yi ) + εp(xi , xj ) ,ẏi = µg(xi , yi ) ,(10.27)(10.28)i = 1, 2 and j = 2, 1. Since bursting involves two time scales, fast spiking and slow
transition between spiking and resting, there are two synchronization regimes: spikesynchronization and burst synchronization, illustrated in Fig.9.51 and discussed in
section 9.4. Below, we outline some useful ideas and methods of studying both regimes.
Our exposition is not complete, but it lays the foundation for a more detailed research
program.
Spike Synchronization
To study synchronization of individual spikes within the burst, we let µ = 0 in order tofreeze the slow subsystem (10.28), and consider the fast subsystem (10.27) describing
weakly coupled oscillators. When yi ≈ yj , the fast variables oscillate with approximately equal periods, so (10.27) can be reduced to the phase modelφ̇i = εH(φj − φi , yi ) ,where yi = const parameterize the form of the connection function. For example,
during the “circle/Hopf” burst, the function is transformed from H(χ) = sin2 χ or
1 − cos χ at the beginning of the burst (saddle-node on invariant circle bifurcation)to H(χ) = sin χ at the end of the burst (supercritical Andronov-Hopf bifurcation).
Changing yi slowly, one can study when spike synchronization appears and when it
disappears during the burst. When the slow variables yi have different values, fastvariables typically oscillate with different frequencies, so one needs to look at low-orderresonances to study the possibility of spike synchronization.

429

a2'

b2

y(t)

w

a1

spikingl c li n e, g=0

Xequiv(y, e)n ulslow variable, yresting

Synchronization

sl o

b1'

C2=

|g(a2')||g(b2)|

C1=|g(a1)||g(b1')|

Xequiv(y, 0)

fast variable, x

Figure 10.41: Reduction of the INa,p +IK +IK(M) -burster to a relaxation oscillator. The
slow variable exhibits “scalloped” oscillations needed for stability of in-phase burst
synchronization. C1 and C2 are compression functions at the two jumps.
Burst synchronization
In chapter 9 we presented two methods, averaging and equivalent voltage, that remove
fast oscillations and reduce bursters to slow relaxation oscillators. Burst synchronization is then reduced to synchronization of such oscillators, and it can be studied using
phase reduction or fast threshold modulation (FTM) approaches.
To apply FTM, we assume that the coupling in (10.27) is piecewise constant, thatis, p(xi , xj ) = 0 when the presynaptic burster xj is resting, and p(xi , xj ) = 1 (or any
function of xi ) when the presynaptic burster is spiking. We also assume that the slow
subsystem (10.28) is one-dimensional, so that we can use the equivalent voltage method(section 9.2.4) and reduce the coupled system to
0 = Xequiv (yi , εp) − xi ,yi′ = g(xi , yi ) .
When the burster is of the hysteresis loop type (i.e., the resting and spiking states
coexist), the function x = Xequiv (y, εp) often, but not always, has a Z-shape on theslow/fast plane, as in Fig.9.16, so that the system corresponds to a relaxation oscillator with nullclines as in Fig.10.41. Fast threshold modulation occurs via the constantεp, which shifts the fast nullcline up or down. The compression criterion for stabilityof the in-phase burst synchronization, presented in the previous section, has a simple
geometrical illustration in the figure. The slow nullcline has to be sufficiently close
to the jumping points that y(t) slows before each jump and produces the “scalloped”
curve. Many hysteresis loop fast/slow bursters do generate such a shape. In particular, “fold/*” bursters exhibit robust in-phase burst synchronization when they arenear the bifurcation from quiescence to bursting, since the slow nullcline is so close to
the left knee that the compression during the resting → spiking jump (C1 in Fig.10.41)
dominates the expansion, if any, during the spiking → resting jump.

430

Synchronization

Review of Important Concepts• Oscillations are described by their phase variables ϑ rotating on a circle
S1 . We define ϑ as the time since the last spike.
• The phase response curve, PRC (ϑ), describes the magnitude of the phaseshift of an oscillator caused by a strong pulsed input arriving at phase ϑ.
• PRC depends on the bifurcations of the spiking limit cycle, and it defines
synchronization properties of an oscillator.
• Two oscillators are synchronized in-phase, anti-phase, or out-of-phasewhen their phase difference, ϑ2 − ϑ1 , equals 0, half-period, or some othervalue, respectively; see Fig.10.42.
• Synchronized states of pulse-coupled oscillators are fixed points of thecorresponding Poincare phase map.
• Weakly coupled oscillatorsẋi = f (xi ) + εXgij (xj )can be reduced to phase modelsϑ̇i = 1 + ε Q(ϑi )X

gij (xj (ϑj )) ,where Q(ϑ) is the infinitesimal PRC defined by (10.10).
• Weak coupling induces a slow phase deviation of the natural oscillation,
ϑi (t) = t + φi , described by the averaged modelXφ̇i = ε ωi +Hij (φj − φi ) ,where ωi denote the frequency deviations, andZ
1 T
Q(t) gij (xj (t + φj − φi )) dt
Hij (φj − φi ) =T 0
describes the interactions between the phases.
• Synchronization of two coupled oscillators corresponds to equilibria of
the one-dimensional systemχ̇ = ε(ω + G(χ)) ,χ = φ2 − φ1 ,where G(χ) = H21 (−χ) − H12 (χ) describes how the phase difference χ
compensates for the frequency mismatch ω = ω2 − ω1 .

Synchronization431

in-phaseanti-phase

out-of-phase

Figure 10.42: Different types of synchronization.

Bibliographical Notes
Surprisingly, this chapter turned out to be quite different from chapter 9 (“Weakly
Connected Oscillators”) of the book Weakly Connected Neural Networks by Hoppensteadt and Izhikevich (1997) and from the book Synchronization: A Universal Concept
in Nonlinear Sciences by Pikovsky, Rosenblum, and Kurths (2001). All three texts,
though devoted to the same subject, do not repeat, but rather complement, each other.
The last provides an excellent historical overview of synchronization, starting with thework of the famous Dutch mathematician, astronomer, and physicist Christiaan Huygens (1629–1695), who was the first to describe synchronization of two pendulum clocks
hanging from a common support (which was, incidentally, anti-phase). While providing
many examples of synchronization of biological, chemical, and physical systems, thebook by Pikovsky et al. also discusses the definition of a phase and synchronization of
nonperiodic, e.g., chaotic, oscillators, a topic not covered here. A major part of Spiking
Neuron Models by Gerstner and Kistler (2002) is devoted to synchronization of spiking neurons, with the emphasis on the integrate-and-fire model and the spike-response
method.
The formalism of the phase response curve (PRC) was introduced by Hastingsand Sweeney (1958), and it has been used extensively in the context of resetting thecircadian rhythms. “Forty Years of PRC – What Have We Learned?” by Johnson(1999) gives a historical overview of this idea and some recent developments. John
Guckenheimer (1975) used the theory of normally hyperbolic invariant manifolds toprovide a mathematical foundation for the existence of isochrons, and their geometricalproperties. An encyclopedic exposition on isochrons and phase-resettings in nature, as
well as numerous anecdotes, can be found in Arthur Winfree’s remarkable book The
Geometry of Biological Time (1980, 2nd ed., 2001). In particular, Winfree describes
the work of George R. Mines (1914), who was doing phase-resetting experiments byshocking rabbits at various phases of their heartbeat. He found the phase and shockthat could stop a rabbit’s heart (black hole in Fig.10.8), and then applied it to himself.
He died.
Glass and Mackey (1988) provide a detailed exposition of circle phase maps. Although the structure of phase-locking regions in Fig.10.15 was discovered by Cartwrightand Littlewood (1945), it is better known today as Arnold tongues (Arnold 1965). Gue-

432

Synchronization

Figure 10.43: Frank Hoppensteadt,
the author’s adviser and mentor, circa
1989.
vara and Glass (1982) found this structure analytically for the Andronov-Hopf oscillator
in Fig.10.3 (radial isochron clock). Hoppensteadt (1997, 2000) provides many examplesof oscillatory systems arising in biology and neuroscience (see also Hoppensteadt andPeskin 2002).
Malkin’s method of reduction of coupled oscillators to phase equations has been
known, at least to Russian scientists, since the early 1950s (Malkin 1949, 1956; Blechman 1971). For example, Melnikov (1963) applied Malkin’s theorem to a homoclinic
orbit of infinite period to obtain the transversality condition for the saddle homoclinic
orbit bifurcation (Kuznetsov 1995).
Malkin’s method was rediscovered in the West by Neu (1979), and hoorayed by
Winfree (1980), who finally saw a mathematical justification for his usage of phasevariables. Since then, the field of phase equations has been largely dominated by Bard
Ermentrout and Nancy Kopell (Ermentrout 1981, 1986, 1992; Ermentrout and Kopell
1986a,b, 1990, 1991, 1994; Kopell and Ermentrout 1990; Kopell 1986; Kopell et al.
1991). In particular, they developed the theory of traveling wave solutions in chainsof oscillators, building on the seminal paper by Cohen et al. (1982). Incidentally, theone-page proof of Malkin’s theorem provided by Hoppensteadt and Izhikevich (1997,
Sect. 9.6) is based on Ermentrout and Kopell’s idea of using the Fredholm alternative;
Malkin’s and Neu’s proofs are quite long, mostly because they reprove the alternative.
There are only a handful of examples in which the Malkin adjoint problem can besolved analytically (i.e., without resort to simulations). The SNIC, homoclinic andAndronov-Hopf cases are the most important, and have been considered in detail inthis chapter. Brown et al. (2004) also derive PRCs for oscillators with homoclinic
orbits to pure saddles (see exercise 25) and for Bautin oscillators.
Throughout this chapter we define the phase ϑ or φ on the interval [0, T ], where T
is the period of free oscillation, and do not normalize it to be on the interval [0, 2π].

Synchronization

433

Figure 10.44: Nancy Kopell in her Boston University office in 1990 (photograph provided by Dr. Kopell).
As a result, we have avoided the annoying terms 2π/T and 2π/Ω in the formulas. The
only drawback is that some of the results may have an “unfamiliar look”, such as sin2 ϑwith ϑ ∈ [0, π] for the PRC of Class 1 neurons, as opposed to 1 − cos ϑ with ϑ ∈ [0, 2π]
used previously by many authors before.
Hansel, Mato, and Meunier (1995) were the first to note that the shape of the PRC
determines the synchronization properties of synaptically coupled oscillators. Ermentrout (1996) related this result to the classification of oscillators and proved that PRCsof all Class 1 oscillators have the form of 1 − cos ϑ, though the proof can be foundin his earlier papers with Kopell (Ermentrout and Kopell 1986a, 1986b). Reyes andFetz (1993) measured the PRC of a cat neocortical neuron and largely confirmed thetheoretical predictions. The experimental method in section 10.2.4 is related to thatof Rosenblum and Pikovsky (2001). It needs to be developed further, for instance, byincorporating the measurement uncertainty (error bars). In fact, most experimentallyobtained PRCs, including the one in Fig.10.24, are so noisy that nothing useful can bederived from them. This issue is the subject of active research.
Our treatment of the FTM theory in this volume closely follows that of Somers andKopell (1993, 1995). Anti-phase synchronization of relaxation oscillators is analyzed
using phase models by Izhikevich (2000b), and FTM theory, by Kopell and Somers(1995). Ermentrout (1994) and Izhikevich (1998) considered weakly coupled oscillators with axonal conduction delays and showed that delays result in mere phase shifts(see exercise 18). Frankel and Kiemel (1993) observed that slow coupling can be reduced to weak coupling. Izhikevich and Hoppensteadt (2003) used Malkin’s theorem
to extend the results to slowly coupled networks, and to derive useful formulas for the
coupling functions and coefficients. Ermentrout (2003) showed that the result could
be generalized to synapses having fast-rising and slow-decaying conductances. Goel

434

Synchronizationand Ermentrout (2002) and Katriel (2005) obtained interesting results on in-phasesynchronization of identical phase oscillators.
Interactions between resonant oscillators were considered by Ermentrout (1981),
Hoppensteadt and Izhikevich (1997), and Izhikevich (1999) in the context of quasiperiodic (multi-frequency) oscillations. Baesens et al. (1991) undertook the heroictask of studying resonances and toroidal chaos in a system of three coupled phaseoscillators. Mean-field approaches to the Kuramoto model are reviewed by Strogatz(2000) and Acebron et al. (2005). Daido (1996) extended the theory to the general
coupling function H(χ). van Hemmen and Wreszinski (1993) were the first to find the
Lyapunov function for the Kuramoto model, which was generalized (independently) by
Hoppensteadt and Izhikevich (1997) to the arbitrary coupling function H(χ).
Ermentrout (1986), Aronson et al. (1990), and Hoppensteadt and Izhikevich (1996,1997) studied weakly coupled Andronov-Hopf oscillators, and discussed the phenomenaof self-ignition (coupling-induced oscillations) and oscillator death (coupling-induced
cessation of oscillation). Collins and Stewart (1993, 1994) and Golubitsky and Stewart(2002) applied group theory to the study of synchronization of coupled oscillators in
networks with symmetries.
In this chapter we have considered either strong pulsed coupling or weak continuous
coupling. These limitations are severe, but they allow us to derive model-independent
results. Studying synchronization in networks of strongly coupled neurons is an active
area of research, though, most such studies fall into two categories: (1) simulationsand (2) integrate-and-fire networks. In both cases, the results are model-dependent. If
the reader wants to pursue this line of research, he or she will definitely need to read
Mirollo and Strogatz (1990), van Vreeswijk et al. (1994), Chow and Kopell (2000),
Rubin and Terman (2000, 2002), Bressloff and Coombes (2000), van Vreeswijk (2000),
van Vreeswijk and Hansel (2001), Pfeuty et al. (2003), and Hansel and Mato (2003).

Exercises
1. Find the isochrons of the Andronov-Hopf oscillatorż = (1 + i)z − z|z|2 ,z ∈ C,

in Fig.10.3.
2. Prove that the isochrons of the Andronov-Hopf oscillator in the formż = (1 + i)z + (−1 + di)z|z|2 ,z ∈ C,

are the curvesz(s) = s(−1+di) eχi ,s>0,where χ is the phase of the isochron (see Fig.10.45).

Synchronization

435
d=-2

d=+2
s(-1+di)

1

1

0

0

-1

-1

s(-1+di)
-1

0

1

-1

0

1

Figure 10.45: Isochrons of the Andronov-Hopf oscillator; see exercise 2.
(

cos q, sin q)(A+

cos q, sin q)

A
y

q

z(t)

Figure 10.46: Pulsed stimulation of theAndronov-Hopf oscillator in Fig.10.3;
see exercise 4.
3. [MATLAB] To determine isochrons of an oscillator ẋ = F (x), one can start withmany initial points near the limit cycle and integrate the system backwards, thatis, ẋ = −F (x). The images of the points at any time t lie on the same isochron.
Write a MATLAB program that implements this algorithm.
4. Prove that the phase response curve of the Andronov-Hopf oscillator in Fig.10.3
is−ψ when 0 ≤ ϑ ≤ π,
PRC (ϑ, A) =(10.29)+ψ when π ≤ ϑ ≤ 2π,
whereψ = arcos √1 + A cos ϑ
1 + 2A cos ϑ + A2

and A is the magnitude of the horizontal displacement of z(t); see Fig.10.46.
5. [MATLAB] Write a program that stimulates an oscillator at different phases anddetermines its phase response curve (PRC).
6. Show that Z(ϑ) = grad ϑ, so that Winfree’s phase model (10.6) is equivalent toKuramoto’s phase model (10.8).
7. Show that Z(ϑ) = Q(ϑ), so that Winfree’s phase model (10.6) is equivalent toMalkin’s phase model (10.9).

436

Synchronization

8. Show that the PRC of the leaky integrate-and-fire neuron (section 8.1.1)v̇ = b − v ,if v ≥ 1 (threshold), then v ← 0 (reset)with b > 1 has the form
PRC (ϑ) = min {ln(b/(b exp(−ϑ) − A)), T } − ϑ ,where T = ln(b/(b − 1)) is the period of free oscillations and A is the amplitudeof the pulse.
9. Prove that the quadratic integrate-and-fire neuronv̇ = 1 + v 2 ,if v = +∞ (peak of spike), then v ← −∞ (reset)

has PTC (ϑ) = π/2 + atan (A − cot ϑ).
10. Find the PRC of the quadratic integrate-and-fire neuron (section 8.1.3)v̇ = b + v 2 ,if v ≥ 1 (peak of spike), then v ← vreset (reset)with b > 0.
11. Consider two mutually pulsed coupled oscillators with periods T1 ≈ T2 and type1 phase transition curves PTC1 and PTC2 , respectively. Show that the locking
behavior of the system can be described by the Poincare phase map
χn+1 = T1 − PTC1 (T2 − PTC2 (χn )) ,where χn is the phase difference between the oscillators, that is, the phase of
oscillator 2 when oscillator 1 fires a spike.
12. [MATLAB] Write a program that solves the adjoint equation (10.10) numerically.
(Hint: Integrate the equation backward to achieve stability.)
13. [MATLAB] Write a program that finds the infinitesimal PRC using the relationshipϑ̇ = 1 + PRC (ϑ) εp(t) ,the moments of firings of a neuron (zero crossings of ϑ(t)), and the injectedcurrent εp(t); see section 10.2.4 and Fig.10.24.
14. Use the approaches of Winfree, Kuramoto, and Malkin to transform the integrateand-fire neuron v̇ = b − v + εp(t) in exercise 8 into its phase modelϑ̇ = 1 + ε eϑ /b p(t) ,with T = ln(b/(b − 1)).

Synchronization

437

15. Use the approaches of Winfree, Kuramoto, and Malkin to transform the quadratic
integrate-and-fire neuron v̇ = 1 + v 2 + εp(t) in exercise 9 into its phase modelϑ̇ = 1 + ε (sin2 ϑ) p(t) ,with T = π.
16. Use the approaches of Winfree, Kuramoto, and Malkin to transform the AndronovHopf oscillator ż = (1 + i)z − z|z|2 + εp(t) with real p(t) into its phase modelϑ̇ = 1 + ε (− sin ϑ)p(t) ,with T = 2π.
17. (PRC for Andronov-Hopf) Consider a weakly perturbed system near supercritical
Andronov-Hopf bifurcation (see section 6.1.3)ż = (b + i)z + (−1 + di)z|z|2 + ϵp(t) ,z∈C.
√with b > 0. Let ε = b b/ϵ be small. Prove that the corresponding phase modelisθ̇ = 1 + d + ε Im {(1 + di)p(t)e−iθ } .
When the forcing p(t) is one-dimensional (i.e., p(t) = cq(t) with c ∈ C and scalarfunction q(t)), the phase model has the sinusoidal formθ̇ = 1 + d + εs sin(θ − ψ)q(t) ,with the strength s and the phase shift ψ depending only on d and c.
18. (Delayed coupling) Show that weakly coupled oscillatorsẋi = f (xi ) + εnXgij (xi (t), xj (t − dij ))j=1

with explicit axonal conduction delays dij ≥ 0 have the phase model
φ′i = ωi +n
X

Hij (φj − dij − φi )j̸=iwhere ′ = d/dτ , τ = εt is the slow time, and H(χ) is defined by (10.16). Thus,
explicit delays result in explicit phase shifts.
19. Determine the existence and stability of synchronized states in the systemφ̇1 = ω1 + c1 sin(φ2 − φ1 )φ̇2 = ω2 + c2 sin(φ1 − φ2 )as a function of the parameters ω = ω2 − ω1 and c = c2 − c1 .

438

Synchronization

20. Consider the Kuramoto model
φi = ω +nX

cij sin(φj + ψij − φi ) ,j=1

where cij and ψij are parameters. What can you say about its synchronization
properties?
21. Derive the self-consistency equation (10.22) for the Kuramoto model (10.20).
22. Consider the phase deviation model
φ′1 = ω + c1 H(φ2 − φ1 )φ′2 = ω + c2 H(φ1 − φ2 )with an even function H(−χ) = H(χ). Prove that the in-phase synchronizedstate, φ1 = φ2 , if it exists, cannot be exponentially stable. What can you say
about the anti-phase state φ1 = φ2 + T /2?
23. Prove that the in-phase synchronized state in a network of three or more pulse
coupled quadratic integrate-and-fire neurons is unstable.
24. Prove (10.25).
25. (Brown et al. 2004) Show that the PRC for an oscillator near saddle homoclinic
orbit bifurcation scales as PRC (ϑ) ∼ eλ(T −ϑ) , where λ is the positive eigenvalueof the saddle and T is the period of oscillation.
26. Consider the quadratic integrate-and-fire neuron ẋ = ±1 + x2 with the resetting“ if x = +∞, then x ← xreset ”. Prove that
regimemodel

SNIC
x = +1 + x2

homoclinic
x = −1 + x2 , (xreset > 1)′

xreset′xx -1

π/2 − atan xreset− cot(t − T )sin2 (ϑ − T )period T
solution x(t)PRC Q(ϑ)

1xresetacoth xreset− coth(t − T )sinh2 (ϑ − T )15xreset=+1.1

xreset=-1.1
0
0

T

0
0

T

Synchronization

439
g

b2

a1
g(x, y)=0

a2b1
f(x, y)=0

a1

a1
m <<

|g(a1)||g(a1)| << mFigure 10.47: Left: Relaxation oscillator in the limit µ = 0 near the onset of oscillation.
Middle and right: A magnification of a neighborhood of the jump point a1 for variousg(a1 ) and µ. Canard solutions can appear when g(a1 ) ≪ µ.
where coth, acoth, and sinh are hyperbolic cotangent, hyperbolic inverse cotangent and hyperbolic sine, respectively.
27. [M.S.] Derive the PRC for an oscillator near saddle homoclinic orbit bifurcationthat is valid during the spike downstroke. Take advantage of the observation in
Fig.10.39 that the homoclinic orbit consists of two qualitatively different parts.
28. [M.S.] Derive the PRC for a generic oscillator near fold limit cycle bifurcation(beware of the problems of defining the phase near such a bifurcation).
29. [M.S.] Simplify the connection function H for coupled relaxation oscillators(Izhikevich 2000b) when the slow nullcline approaches the left knee, as in Fig.10.47.
Explore the range of parameters ε, µ, and |g(a1 )| where the analysis is valid.
30. [Ph.D.] Use ideas outlined in section 10.4.5 to develop the theory of reduction of
weakly coupled bursters to phase models. Do not assume that bursting trajectoryis periodic.

440

Synchronization

Solutions to Exercises
Solutions for chapter 2
1. T = 20◦ C ≈ 293◦ F.
[Ion]out[Ion]out
RT
8315 · 293 · ln 10[Ion]out
ln
log10
EIon === ± 58 log10
zF[Ion]inz · 96480[Ion]in[Ion]inwhen z = ±1. Therefore,
EK

=

ENa

=

ECl

=

58 log(20/430) = −77 mV
58 log(440/50) =55 mV−58 log(560/65) = −54 mV

2.
I

=
=

ḡNa p (V − ENa ) + ḡK p (V − EK ) = p{(ḡNa + ḡK ) V − ḡNa ENa − ḡK EK }ḡNa ENa + ḡK EK(ḡNa + ḡK ) p V −ḡNa + ḡK| {z }|{z}ḡ
E

3. The answer follows from the equationbL ) ,
I − gL (V − EL ) = −gL (V − E

wherebL = EL + I/gL .
E

4. See Fig.S.1.
Functionn∞ (V )
m∞ (V )h∞ (V )

V1/2
12
25
3

k
15
9
−7

Functionτn (V )
τm (V )
τn (V )

Vmax σ Camp−14 504.7
27 30 0.46
−2 20
7.4

Cbase
1.1
0.04
1.2

Hodgkin and Huxley shifted V1/2 and Vmax by 65 mV so that the resting potential is at V = 0mV.
5. (Willms et al. 1999)
Ṽ1/2

=

k̃

=

V1/2 − k ln(21/p − 1) ,k.
2p(1 − 2−1/p )The first equation is obtained from the condition mp∞ (Ṽ1/2 ) = 1/2. The second equation is
obtained from the condition that the two functions have the same slope at V = Ṽ1/2 .
6. See author’s Web page, www.izhikevich.com.
7. See author’s Web page, www.izhikevich.com.

441

442

Solutions to Exercises, chapter 3
9

1

0.9

h (V)

τ h (V)

8

0.8

7

0.7
6
0.6
5
0.5

m (V)

4

τ n (V)

0.4
3
0.3

n (V)
2

0.2

0
-40

τm(V)

1

0.1

-20

0

20

40

60

0
-40

80

-20

0

20

40

60

80

100

V (mV)V (mV)

Figure S.1: Open dots: The steady-state (in)activation functions and voltage-sensitive time constants in the Hodgkin-Huxley model. Filled dots: steady-state Na+ activation function m∞ (V ) in thesquid giant axon (experimental results by Hodgkin and Huxley 1952, figure 8). Continuous curves:
Approximations by Boltzmann and Gaussian functions. See exercise 4.

Solutions for chapter 3
1. Consider the limit case: (1) activation of the Na+ current is instantaneous, and (2) conductancekinetics of the other currents are frozen. Then, the Na+ current will result in the nonlinear
term gNa m∞ (V ) (V − ENa ) with the parameter h∞ (Vrest ) incorporated into gNa , and all theother currents will result in the linear leak term.
In Fig.3.15, the activation of the Na+ current is not instantaneous; hence the sag right after thepulses. In addition, its inactivation, and the kinetics of the other currents are not slow enough;hence the membrane potential quickly reaches the excited state and then slowly repolarizes tothe resting state.
2. See Fig.S.2. The eigenvalues are negative at each equilibrium marked as a filled circle (stable),and positive at each equilibrium marked as an open circle (unstable). The eigenvalue at thebifurcation point (left equilibrium in Fig.S.2b) is zero.

F(V)

F(V)

F(V)
V

V
a

b

V
c

Figure S.2: Phase portraits of the system V̇ = F (V ) with given F (V ).
3. Phase portraits are shown in Fig.S.3.
(a) The equation 0 = −1 + x2 has two solutions: x = −1 and x = +1; hence there are
two equilibria in the system (a). The eigenvalues are the derivatives at each equilibrium,
λ = (−1 + x2 )′ = 2x, where x = ±1. Equilibrium x = −1 is stable because λ = −2 < 0.
Equilibrium x = +1 is unstable because λ = +2 > 0. The same fact follows from thegeometrical analysis in Fig.S.3.

Solutions to Exercises, chapter 3443(b) The equation 0 = x − x3 has three solutions: x = ±1 and x = 0; hence there are threeequilibria in system (b). The eigenvalues are the derivatives at each equilibrium, λ =(x − x3 )′ = 1 − 3x2 . The equilibria x = ±1 are stable because λ = 1 − 3(±1)2 = −2 < 0.
The equilibrium x = 0 is unstable because λ = 1 > 0. The same conclusions follow fromthe geometrical analysis in Fig.S.3.
1.5

1

x

x

x2 -1

1

x-x3

0.5
0.5

x

0

x
0

-0.5
-0.5
-1
-2

-1
-1

0

1

2

-2

a

-1

0

1

2

b

Figure S.3: Phase portraits of the systems (a) ẋ = −1 + x2 , and (b) ẋ = x − x3 .
4. The equilibrium x = 0 is stable in all three cases.
5. See Fig.S.4. Topologically equivalent systems are in (a), (b), and (c). In (d) there are different
numbers of equilibria; no stretching or shrinking of the rubber phase line can produce new
equilibria. In (e) the right equilibrium is unstable in V̇ = F1 (V ) but stable in V̇ = F2 (V ); no
stretching or shrinking can change the stability of an equilibrium. In (f) the flow between thetwo equilibria is directed rightward in V̇ = F1 (V ) and leftward in V̇ = F2 (V ); no stretching or
shrinking can change the direction of the flow.
6. (Saddle-node [fold] bifurcation in ẋ = ap
+ x2 ) The equation 0 = a + x2 has no real solutions
when a > 0, and two solutions x = ± |a| when a ≤ 0. Hence there are two branches of
equilibria, depicted in Fig.S.5. The eigenvalues arep
λ = (a + x2 )′ = 2x = ±2 |a| .
pp
The lower branch − |a| is stable (λ < 0), and the upper branch + |a| is unstable (λ > 0).
They meet at the saddle-node (fold) bifurcation point a = 0.
7.
(a) x = −1 √at a = 1√(d) x = ±1/ 3 at a = ±2/(3 3)(b) x = −1/2 at a = 1/4(e) x = ±1 at a = ∓2(c) x = 1/2 at a = 1/4(f) x = −1 at a = 1

8. (Pitchfork bifurcation in ẋ = bx − x3 ) The√equation 0 = bx − x3 has one solution x = 0 whenb ≤ 0, and three solutions x = 0, x = ± b when b > 0. Hence there is only one branch ofequilibria for b < 0 and three branches for b > 0 of the pitchfork curve depicted in Fig.S.6. The
eigenvalues are
λ = (bx − x3 )′ = b − 3x2 .
The branch x = 0 exists for any b, and its eigenvalueis λ = b. Thus, it is stable for b < 0 and√unstable for b > 0. The√ 2two branches x = ± b exist only for b > 0, but they are always stablebecause λ = b−3(± b) = −2b < 0. We see that the branch x = 0 loses stability when b passes
the pitchfork bifurcation value b = 0, at which point a pair of new stable branches bifurcates(hence the name bifurcation). In other words, the stable branch x = 0 divides (bifurcates) into
two stable branches when b passes 0.

444

Solutions to Exercises, chapter 3
Topologically
Equivalent

Topologically
Equivalent
F1 (V)V

Topologically
EquivalentVV
F1 (V)F1 (V)F2 (V)F2 (V)F2 (V)

V

V

V

a

b

c

Topologically NOT
Equivalent
F1 (V)

Topologically NOT
Equivalent
F1 (V)
Left

Topologically NOT
F (V) Equivalent?

F2 (V)V

1

V

Unstable

Right

?

?

?

F2 (V)
V

Left
Right

StableedVf

Figure S.4: Answer to chapter 3, exercise 5.
9. Recall that the current IKir is turned off by depolarization and turned on by hyperpolarization.
The dynamics of the IKir -model is similar to that of the INa,p -model in many respects. In particular, this system can also have coexistence of two stable equilibria separated by an unstableequilibrium, which follows from the N-shaped I-V relation. Indeed, when V is hyperpolarized,
the current IKir is turned on (deinactivated), and it pulls V toward EK . In contrast, when V is
depolarized, the current is turned off (inactivated), and does not obstruct further depolarizationof V .
Use (3.11) to find the curve
I = gL (V − EL ) + gKir h∞ (V )(V − EK ) ,in Fig.S.8. (The curve may not be S-shaped if a different bifurcation parameter is used, as inexercise 12a below).
The bifurcation diagram of the IKir -model (3.11) in Fig.S.8 has three branches corresponding
to the three equilibria. When the parameter I is relatively small, the outward IKir current
dominates and the system has only one equilibrium in the low voltage range – the downstate. When the parameter I is relatively large, the injected inward current I dominates, andthe system has one equilibrium in the intermediate voltage range – the up-state. When theparameter I is in neighborhood of I = 6, the system exhibits bistability of the up-state andthe down-state. The states appear and disappear via saddle-node bifurcations. The behaviorof the IKir -model is conceptually (and qualitatively) similar to the behavior of the INa,p -model(3.5) even though the models have completely different ionic mechanisms for bistability.
10. The equilibrium satisfies the one-dimensional equation
0 = I − gK n4∞ (V )(V − EK ) − gNa m3∞ (V )h∞ (V )(V − ENa ) − gL (V − EL ) ,Solutions to Exercises, chapter 3

445
Bifurcation Diagram

Unstableequilibriax1a
Saddle-node(fold)
bifurcation

0

a

-1

Stableequilibriua

a
-2

-1

0.5

0

1

Representative Phase Portraits
1.5

1.5

x

a+x2

1

1.5

a+x2

1

x

0

-1

0

1

2

-0.5

-1

0

a=0

a=-1

x

0

-0.5
-1
-2

a+x2

0.5

x

0

-0.5

x

1

0.5

0.5

-1
-2

x

1

2

-1
-2

-1

0

1

2

a=0.5

Figure S.5: Saddle-node (fold) bifurcation diagram and representative phase portraits of the systemẋ = a + x2 (see chapter 3, exercise 6).
where all gating variables assume their asymptotic values. The solution
I = gK n4∞ (V )(V − EK ) + gNa m3∞ (V )h∞ (V )(V − ENa ) + gL (V − EL )is depicted in Fig.S.9. Since the curve in this figure does not have folds, there are no saddle-nodebifurcations in the Hodgkin-Huxley model (with the original values of parameters).
11. The curves(a)gL (V ) = −gNa m∞ (V )(V − ENa )/(V − EL )

and
(b)

EL (V ) = V + gNa m∞ (V )(V − ENa )(V − EL )/gLare depicted in Fig.S.7.
12. The curves(a)gL (V ) = {I − gKir h∞ (V )(V − EK )}/(V − EL )(b)gKir (V ) = {I − gL (V − EL )}/{h∞ (V )(V − EK )}andare depicted in Fig.S.10. Note that the curve in Fig.S.10a does not have the S shape.
13.
F ′ (V ) = −gL − gK m4∞ (V ) − gK 4m3∞ (V )m′∞ (V )(V − EK ) < 0
because gL > 0, m∞ (V ) > 0, m′∞ (V ) > 0, and V − EK > 0 for all V > EK .
14.
F ′ (V ) = −gL − gh h∞ (V ) − gh h′∞ (V )(V − Eh ) < 0
because gL > 0, h∞ (V ) > 0, but h′∞ (V ) < 0 and V − Eh < 0 for all V < Eh .

446

Solutions to Exercises, chapter 3

2

x
1.5

Bifurcation
Diagramstab

stableunstable

le

b

1

0.5

0

Pitchfork
bifurcation

-0.5

b

stab
le

-1

b

-1.5

-2
-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Representative Phase Portraits
4

2

x

bx-x3

2

x

1

bx-x3

1

x

0

x

0

-2

-1

-4
-2

0

bx-x3
x

0

-0.5

-2
-2

2

x

0.5

0

2

-1
-2

0

b=0

b=-1

2

b=+1

Figure S.6: Pitchfork bifurcation diagram and representative phase portraits of the system ẋ =bx − x3 (see chapter 3, exercise 8).

40

dexcited00Voltage V

Voltage V

20

50

saddle-node(fold)
bifurcations

exc
ite

-20
-40

e
thr

-60
-80
0.5

old

sh

thre

-100

rest
1

1.5

2

2.5

sho

-50

-150
-150

ld
saddle-node(fold)
bifurcations

t
res
-100

-50

leak conductance gLleak reverse potential E L

a

b

0

Figure S.7: Bifurcation diagrams of the INa,p -model (3.5) with bifurcation parameters (a) gL and(b) EL (see chapter 3, exercise 11).

Solutions to Exercises, chapter 3
Monostability

447

Bistability

Monostability

-10

p")

Re

"u
st (

te

sta

-30

Saddle-node(fold) bifurcation

-40

ol

sh
re

Th

-50

d

Membrane Voltage, V (mV)-20

Saddle-node(fold) bifurcation-60

Rest ("d
-70

4

tate
own") s

4.5

5

5.5

Figure S.8: Bifurcation diagram of
6

6.5

7

7.5

the IKir -model (3.11).

8

Injected Curent, I
120
20
membrane voltage, V

membrane voltage, V

100
80
60
40
20
0
-20

0

1000 2000 3000
injected dc-current, I

4000

5000

10

0

magnification

-10

-20

0

50

100
150
injected dc-current, I

20

Figure S.9: Dependence of the position of equilibrium in the Hodgkin-Huxley model on the injected
DC current; see exercise 10.
15. When V is sufficiently large, V̇ ≈ V 2 . The solution of V̇ = V 2 is V (t) = 1/(c − t) (check bydifferentiating), where c = 1/V (0). Another way to show this is to solve (3.9) for V and findthe asymptote of the solution.
16. Each equilibrium of the system ẋ = a + bx − x3 is a solution to the equation 0 = a + bx − x3 .
Treating x and b as free parameters, the set of all equilibria is given by a = −bx + x3 , and it
looks like the cusp surface in Fig.6.34. Each point where the cusp surface folds, corresponds toa saddle-node (fold) bifurcation. The derivative with respect to x at each such point is zero;alternatively, the vector tangent to the cusp surface at each such point is parallel to the x-axis.
The set of all bifurcation points is projected to the (a, b)-plane at the bottom of the figure, andit looks like a curve having two branches. To find the equation for the bifurcation curves, one
needs to remember that each bifurcation point satisfies two conditions:• It is an equilibrium; that is, a + bx − x3 = 0.
• The derivative of a + bx − x3 with respect to x is zero; that is, b − 3x2 = 0.
pSolving the second equation for x and using the solution x = ± b/3 in the first equation yields
a = ∓2(b/3)3/2 . The point a = b = 0 is called a cusp bifurcation point.

448

Solutions to Exercises, chapter 4400Voltage V

saddle-node(fold)
bifurcationteVoltage V

sta
up-

20

-20
-40 threshold-60

down-stat

e

0.1

0.2

EL

-80
0

up-state

-30

0.3

0.4

saddle-node(fold)
bifurcation

-40

ld

thr

-50

o
esh

saddle-node(fold)-60 bifurcation

down-s

tate

-70
1.7

1.8

1.9

2

2.1

2.2

leak conductance gL

Kir conductance gKirab

Figure S.10: Bifurcation diagrams of the IKir -model (3.11), I = 6, with bifurcation parameters (a)gL and (b) gKir (see chapter 3, exercise 12).

17. (Gradient systems) For V̇ = F (V ) take
Z V
E(V ) = −F (v) dv ,cwhere c is any constant.
a. E(V ) = 1
d. E(V ) = V − V 3 /3

b. E(V ) = −Ve. E(V ) = −V 2 /2 + V 4 /4c. E(V ) = V 2 /2
f. E(V ) = − cos V

18. (c) implies (b) because |x(t) − y| < exp(−at) implies that x(t) → y as t → ∞. (b) implies (a)according to the definition.
(a) does not imply (b) because x(t) may not approach y. For example, y = 0 is an equilibrium inthe system ẋ = 0 (any other point is also an equilibrium). It is stable, since |x(t) − 0| < ε for all|x0 −0| < ε and all t ≥ 0. However, it is not asymptotically stable because limt→∞ x(t) = x0 ̸= 0
regardless of how close x0 is to 0 (unless x0 = 0).
(b) does not imply (c). For example, the equilibrium y = 0 in the system ẋ = −x3 is asymp−1/2
totically stable (check by differentiating that x(t) = (2t + x−2→ 0 is a solution with
0 )x(0) = x0 ); however, x(t) approaches 0 with a slower than exponential rate, exp(−at), for any
constant a > 0.

Solutions for chapter 4
1. See figures S.11–S.15.
2. See Fig.S.16.
3. See figures S.17–S.21.
4. The diagram follows from the form of the eigenvalues√τ ± τ 2 − 4∆.
λ=2Solutions to Exercises, chapter 4

Figure S.11: Nullclines of the vector field; see also Fig.S.17.

Figure S.12: Nullclines of the vector field; see also Fig.S.18.

Figure S.13: Nullclines of the vector field; see also Fig.S.19.

449

450

Solutions to Exercises, chapter 4

Figure S.14: Nullclines of the vector field; see also Fig.S.20.

Figure S.15: Nullclines of the vector field; see also Fig.S.21.
If ∆ < 0 (left half-plane in Fig.4.15), then the eigenvalues have opposite signs. Indeed,p√τ 2 − 4∆ > τ 2 = |τ | ,whenceτ+pτ 2 − 4∆ > 0

and

τ−

p

τ 2 − 4∆ < 0 .

The equilibrium is a saddle in this case. Now consider the case ∆ > 0. When τ 2 < 4∆ (inside
the parabola in Fig.4.15), the eigenvalues are complex-conjugate; hence the equilibrium is afocus. It is stable (unstable) when τ < 0 (τ > 0). When τ 2 > 4∆ (outside the parabola inFig.4.15), the eigenvalues are real. Both are negative (positive) when τ < 0 (τ > 0).
5. (van der Pol oscillator) The nullclines of the van der Pol oscillator,y = x − x3 /3(x-nullcline) ,x=0(y-nullcline) ,are depicted in Fig.S.22. There is a unique equilibrium (0, 0). The Jacobian matrix at theequilibrium has the form


1 −1
L=
.
b 0
Since tr L = 1 > 0 and det L = b > 0, the equilibrium is always an unstable focus.

Solutions to Exercises, chapter 4

451

a

b

c

d

Figure S.16: Approximate directions of the vector in each region between the nullclines.
6. (Bonhoeffer–van der Pol oscillator) The nullclines of the Bonhoeffer–van der Pol oscillator withc = 0 have the formy = x − x3 /3(x-nullcline) ,x=a(y-nullcline) ,shown in Fig.S.23. They intersect at the point x = a, y = a − a3 /3. The Jacobian matrix at
the equilibrium (a, a − a3 /3) has the form

1 − a2 −1L=.
b0
Since tr L = 1 − a2 and det L = b > 0, the equilibrium is a stable (unstable) focus when |a| > 1(|a| < 1), as we illustrate in Fig.S.23.
7. (Hindmarsh-Rose spiking neuron) The Jacobian matrix at the equilibrium (x̄, ȳ) is
 ′

f −1
L=
,
g ′ −1
thereforetr L = f ′ − 1anddet L = −f ′ + g ′ .

The equilibrium is a saddle (det L < 0) when g ′ < f ′ , that is, in the region below the diagonalin Fig.S.24. When g ′ > f ′ , the equilibrium is stable (tr L < 0) when f ′ < 1, which is the left
half-plane in Fig.S.24. Using the classification in Fig.4.15, we conclude that it is a focus when(f ′ − 1)2 − 4(g ′ − f ′ ) < 0, that is, when
g′ >

1 ′
(f + 1)2 ,4

452

Solutions to Exercises, chapter 4

Figure S.17: Left: No equilibria. Right: Saddle equilibrium.

Figure S.18: Left: Stable node. Right: Stable focus.

Figure S.19: Left: Excitable system having one stable equilibrium. Right: Two stable nodes
separated by a saddle equilibrium.

Solutions to Exercises, chapter 4453

Figure S.20: Left: Unstable focus inside a stable limit cycle. Right: Stable focus inside an unstable
limit cycle.

Figure S.21: Left: Saddle-node equilibrium. Right: Stable node and saddle equilibria connected by
two heteroclinic trajectories, which form an invariant circle with an unstable focus inside.

1
y
0

-1
-2

-1

0

x

1

2

Figure S.22: Nullclines and phase portrait of the van der Pol oscillator (b = 0.1).

454

Solutions to Exercises, chapter 4
a=-1.2a=-0.8
1
y
0.5
0
-0.5
-1

-2

-1

0

1

x

2

-2

-1

0

x

1

2

Figure S.23: Nullclines and phase portrait of the Bonhoeffer–van der Pol oscillator (b = 0.05 andc = 0).
6

de
no

le

un

st

ab

f'=1

4

g '=

( f' +

1) 2

unstable focus

5stable focus

3

f'=

g'

g'
2

1

Figure S.24:

saddle0stable node
-1
-2

-1

0

1

f'

2

3

4

Stability diagram of
the Hindmarsh-Rose spiking neuronmodel; see exercise 7.

which is the upper part of the parabola in Fig.S.24.
8. (IK -model) The steady-state I-V relation of the IK -model is monotone; hence it has a uniqueequilibrium, which we denoted here as (V̄ , m̄) ∈ R2 , where V̄ > EK and m̄ = m∞ (V̄ ). The
Jacobian at the equilibrium has the form−(gL + ḡK m̄4 )/C −4ḡK m̄3 (V̄ − EK )/CL=,
m′∞ (V̄ )/τ (V̄ )−1/τ (V̄ )with the signs

L=

−
+

−
−


.

Obviously, det L > 0 and tr L < 0; hence the equilibrium (focus or node) is always stable.
9. (Ih -model) The steady-state I-V relation of the Ih -model is monotone; hence it has a uniqueequilibrium denote here as (V̄ , h̄) ∈ R2 , where V̄ < Eh and h̄ = h∞ (V̄ ). The Jacobian at theequilibrium has the form−(gL + ḡh h̄)/C −ḡh (V̄ − Eh )/CL=,
h′∞ (V̄ )/τ (V̄ )−1/τ (V̄ )with the signs

L=

−
−

+
−


.

Solutions to Exercises, chapter 5

455

y

x2

y=

bx

/c

y= a +

unstable (saddle)

stable if x<c/2x
c/2

Figure S.25: The left equilibrium is stable when x < c/2; see exercise 11.
Obviously, det L > 0 and tr L < 0; hence the equilibrium is always stable.
10. (Bendixson’s criterion) The divergence of the vector field of the IK -model∂f (x,y)/∂x∂g(x,y)/∂yz}|{z }| {(−gL − ḡK m4 )/C + −1/τ (V )is always negative; hence the model cannot have a periodic orbit. Therefore, it cannot havesustained oscillations.
11. The x-nullcline is y = a + x2 and the y-nullcline is y = bx/c, as in Fig.S.25. The equilibria(intersections of the nullclines) arep
b/c ± (b/c)2 − 4ax̄ =,ȳ = bx̄/c ,2provided that a < 14 (b/c)2 . The Jacobian matrix at (x̄, ȳ) has the form
2x̄ −1L=b −cwith tr L = 2x̄ − c andp
det L = −2x̄c + b = ∓ b2 − 4ac2 .
pThus, the right equilibrium(i.e., (b/c + (b/c)2 − 4a)/2) is always a saddle and the left equip
librium (i.e., (b/c − (b/c)2 − 4a)/2) is always a focus or a node. It is always stable when it
lies on the left branch of the parabola y = a + x2 (i.e., when x̄ < 0), and also can be stable on
the right branch if it is not too far from the parabola knee (i.e., if x̄ < c/2); see Fig.S.25.

Solutions for chapter 5
1. The IA -model with instantaneous activation has the form

C V̇

=

ḣ =

leak IL
IA
z
}|
{
z
}|
{
I − gL (V − EL ) − ḡA m∞ (V )h(V − EK )(h∞ (V ) − h)/τ (V ) .

456

Solutions to Exercises, chapter 5

K+ activation, n

n-n
ullc
line

K+ activation, n

n-n
ullc
line

e
V-nullclineV-nullclin

membrane voltage, V

membrane voltage, V

Figure S.26: Answer to exercise 2.

-40

membrane voltage, V (mV)membrane voltage, V (mV)-30
INa,t-model(-24, -41)

-50
-60

(0.45, -63)

-70
-80
-30

-20

-10
injected dc-current, I

0

10

0

IA-model(10.75, -19)

-20

-40

(12.7, -42)

-60
0

5

10

15

injected dc-current, I

Figure S.27: The saddle-node bifurcation diagrams of the INa,t - and IA -minimal models.
To apply the Bendixson criterion (chapter 4, exercise 10), we first determine the divergence of
this vector field
∂ V̇
∂ ḣ
+
= −{gL + ḡA m′∞ (V )h(V − EK ) + ḡA m∞ (V )h}/C − 1/τ (V ) < 0 .
∂V∂hSince it is always negative, the IA -model cannot have limit cycle attractors (or any other closed
loop orbit).
2. See Fig.S.26.
3. The curves
I = gL (V − EL ) + ḡNa m3∞ (V )h∞ (V )(V − ENa )and
I = gL (V − EL ) + ḡA m∞ (V ) h∞ (V )(V − EK )are depicted in Fig.S.27.
4. g is not an absolute conductance, but is taken relative to the conductance at the resting state.
Negative values occur because the initial holding voltage value in the voltage-clamp experimentdescribed in Fig.5.22a corresponds to the resting potential, at which the K+ conductance ispartially activated. Indeed, in the INa,p +IK -model the K+ gating variable n ≈ 0.04; hence
the K+ conductance is approximately 0.4 (because ḡK = 10). According to the procedure,
this value corresponds to g = 0. Any small decrease in conductance would result in negativevalues of g. If the initial holding voltage were very negative, say below −100 mV, then the slow
conductance g would have nonnegative values in the relevant voltage range (above −100 mV).

Solutions to Exercises, chapter 6457

5. The curve Islow (V ) defines slow changes of the membrane voltage. The curve I −Ifast (V ) defines
fast changes. Its middle part, which has a positive slope, is unstable. If the I-V curves intersect
in the middle part, the equilibrium is unstable, and the system exhibits periodic spiking: The
voltage slowly slides down the left branch of the fast I-V curve toward the slow I-V curve
until it reaches the left knee, and then jumps quickly to the right branch. After the jump, thevoltage slowly slides up the right branch until it reaches the right knee, and then quickly jumps
to the left branch along the straight line that connects the knee and the point (EK , 0) (seealso previous exercise). Note that the direction of the jump is not horizontal, as in relaxation
oscillators, but along a sloped line. On that line the slow conductance g is constant, but theslow current Islow (V ) = g(V − EK ) changes quickly because the driving force V − EK changes
quickly. When the I-V curves intersect at the stable point (negative slope of I − Ifast (V )), thevoltage variable may produce a single action potential, then slide slowly toward the intersection,
which is a stable equilibrium.

Solutions for chapter 6
1. There are two equilibria: x = 0 and x = b. The stability is determined by the sign of thederivative
λ = (x(b − x))′x = b − 2xat the equilibrium. Since λ = b when x = 0, this equilibrium is stable (unstable) when b < 0(b > 0). Since λ = −b when x = b, this equilibrium is unstable (stable) when b < 0 (b > 0).
2.

(a) The systemẋ = bx2 ,b ̸= 0cannot exhibit saddle-node bifurcation: It has one equilibrium for any nonzero b, or an
infinite number of equilibria when b = 0. The equilibrium x = 0 is non-hyperbolic,and the non-degeneracy condition is satisfied (a = b ̸= 0). However, the transversality
condition is not satisfied at the equilibrium x = 0. Another example is ẋ = b2 + x2 .
(b) The systemẋ = b − x3
has a single stable equilibrium for any b. However, the point x = 0 is non-hyperbolic whenb = 0 and the transversality condition is also satisfied. The non-degeneracy condition isviolated, however.
3. It is easy to check (by differentiating) thatppc(b − bsn )√tan( ac(b − bsn )t)
V (t) =ais a solution to the system. Since tan(−π/2) = −∞ and tan(+π/2) = +∞, it takes
T =pπac(b − bsn )for the solution to go from −∞ to +∞.
4. The first system can be transformed into the second if we use complex coordinates z = u + iv.
To obtain the third system, we use polar coordinatesreiφ = z = u + iv ∈ C ,458

Solutions to Exercises, chapter 6
so that(c(b)+iω(b))z+(a+id)z|z|2

ż

z
}|
{ z
}|
{
i
φ
i
φ
i
φ
3 iφ
ṙe + re iφ̇ = (c(b) + iω(b))re + (a + id)r e .
Next, we divide both sides of this equation by eiφ and separate the real and imaginary parts
to obtain{ṙ − c(b)r − ar3 } + ir{φ̇ − ω(b) − dr2 } = 0 ,which we can write in the polar coordinates form.
5.

(a) The equilibrium r = 0 of the system= br3 ,

ṙ

φ̇ =

1,

has a pair of complex-conjugate eigenvalues ±i for any b, and the non-degeneracy condition is satisfied for any b ̸= 0. However, the transversality condition is violated, andthe system does not exhibit Andronov-Hopf bifurcation (no limit cycle exists near theequilibrium).
(b) The equilibrium r = 0 for b = 0ṙ = br ,φ̇ = 1 ,has a pair of complex-conjugate eigenvalues ±i and the transversality condition is satisfied. However, the bifurcation is not of the Andronov-Hopf type because no limit cycle
exists near the equilibrium for any b.
6. The Jacobian matrix at the equilibrium (u, v) = (0, 0) has the form

L=

b
1

−1
b


.

It has eigenvalues b± i. Therefore, the loss of stability occurs at b = 0, and the non-hyperbolicity
and transversality conditions are satisfied. Since the model can be reduced to the polarcoordinate system (see exercise 4) and a ̸= 0, the non-degeneracy condition is also satisfied,
and the system undergoes an Andronov-Hopf bifurcation.
7. Since(cr + ar3 )′r = c + 3ar2 = c + 3a|c/a| =c + 3|c| when a > 0,c − 3|c| when a < 0,

the limit cycle is stable when a < 0.
8. The sequence of bifurcations is similar to that of the RS neuron in Fig.8.15. The restingstate is a globally asymptotically stable equilibrium for I < 5.64. At this value a stable(spiking) limit cycle appears via a big saddle homoclinic orbit bifurcation. At I = 5.8 asmall-amplitude unstable limit cycle is born via another saddle homoclinic orbit bifurcation.
This cycle shrinks to the equilibrium and makes it lose stability via subcritical Andronov-Hopfbifurcation at I = 6.5. This unstable focus becomes an unstable node when I increases, andthen it coalesces with the saddle (at I = 7.3) and disappears. Note that there is a saddle-nodebifurcation according to the I-V relation, but it corresponds to the disappearance of an unstableequilibrium.

Solutions to Exercises, chapter 6459

9. The Jacobian matrix of partial derivatives has the form−IV′ (V, x)−Ix′ (V, x)L=,x′∞ (V )/τ (V ) −1/τ (V )so thattr L = −{IV′ (V, x) + 1/τ (V )}and′det L = {IV′ (V, x) + Ix′ (V, x)x′∞ (V )}/τ (V ) = I∞
(V )/τ (V ) .

The characteristic equation
λ2 − λ tr L + det L = 0
has two solutions

ω

c
}|
{
z }| { zp(tr L)/2 ± {(tr L)/2}2 − det L

which might be complex-conjugate.
10. Let z = reφi ; then
r′
φ′

=
=

ar + r3 − r5 ,ω.

Any limit cycle is an equilibrium of the amplitude equation, that is,
a + r2 − r4 = 0 .
The system undergoes fold limit cycle bifurcation when the amplitude equation undergoes asaddle-node bifurcation, that is, whena + 3r2 − 5r4 = 0(check the non-degeneracy and transversality
conditions). The two equations have the non√trivial solution (a, r) = (−1/4, 1/ 2).
11. The projection onto the v1 -axis is described by the equationẋ = λ1 x ,x(0) = a .

The trajectory leaves the square when x(t) = aeλ1 t = 1; that is, whent=−11
ln a = − ln τ (I − Ib ) .
λ1
λ1

12. Equation (6.13) has two bifurcation parameters, b and v, and the saddle-node homoclinic
bifurcation occurs when b = bsn and v = Vsn . The saddle-node bifurcation curve is the straightline b = bsn (any v). This bifurcation is on an invariant circle when v < Vsn and off otherwise.
When b > bsn , there are no equilibria and the normal form exhibits periodic spiking. Whenb < bsn , the normal form has two equilibria,
node
z
{
p }|
Vsn − c|b − bsn |/aandsaddlez{p }|Vsn + c|b − bsn |/a .

The saddle homoclinic orbit bifurcation occurs when the voltage is reset to the saddle, that is,whenpv = Vsn + c|b − bsn |/a .

460

Solutions to Exercises, chapter 7

13. The Jacobian matrix at an equilibrium is

L=

2v
ab

−1
−a


.

Saddle-node condition det L = −2va + ab = 0 results in v = b/2. Since v is an equilibrium,
it satisfies v 2 − bv + I = 0; hence b2 = 4I. The Andronov-Hopf condition tr L = 2v − a = 0
results in v = a/2; hence a2 /4 − ab/2 + I = 0. The bifurcation occurs when det L > 0,
resulting in a < b. Combining the saddle-node and Andronov-Hopf conditions results in the
Bogdanov-Takens conditions.
√
14. Change of variables (6.5), v = x and u = µy, transforms the relaxation oscillator into the
form
 ′
√ 
√
f (b) − µẋ = f (x) − µy√√with the Jacobian L =µ(x − b)µ0ẏ =√at the equilibrium v = x = b, u = µy = f (b). The Andronov-Hopf bifurcation occurs whentr L = f ′ (b) = 0 and det L = µ > 0. Using (6.7), we find that it is supercritical when f ′′′ (b) < 0
and subcritical when f ′′′ (b) > 0.
15. The Jacobian matrix at the equilibrium, which satisfies F (v) − bv = 0, is
 ′

F −1
L=
.
µb −µ
The Andronov-Hopf bifurcation occurs when tr L = F ′ − µ = 0 (hence F ′ = µ) and det L =ω 2 = µb − µ2 > 0 (hence b > µ). The change of variables (6.5), v = x and u = µx + ωy,transforms the system into the form
ẋ =
ẏ =

−ωy + f (x)ωx+ g(x),

where f (x) = F (x) − µx and g(x) = µ[bx − F (x)]/ω. The result follows from (6.7).
16. The change of variables (6.5) converts the system into the form
ẋ =
ẏ =

F (x) + linear termsµ(G(x) − F (x))/ω + linear terms.

The result follows from (6.7).
17. The change of variables (6.5), v = x and u = µx + ωy, converts the system intoẋẏ

= F (x) − x(µx + ωy) + linear terms= µ[G(x) − F (x) + x(µx + ωy)]/ω.

The result follows from (6.7).
18. The system undergoes Andronov-Hopf bifurcation when Fv = −µGu and Fu Gv < −µG2u . We
perform all the steps from (6.4) to (6.7), disregarding
linear terms (they do not influence a)√and the terms of the order o(µ). Let ω = −µFu Gv + O(µ), then u = (µGu x − ωy)/Fu =−ωy/Fu + O(µ), andf (x, y) = F (x, −ωy/Fu + O(µ)) = F (x, 0) − Fu (x, 0)ωy/Fu + O(µ)
andg(x, y) = (µ/ω)[Gu F (x, 0) − Fu G(x, 0)] + O(µ) .
The result follows from (6.7).

Solutions to Exercises, chapter 8461

slow K+
activation
homoclinic orbit

fast K+
activation

V

Andronov-Hopf

Figure S.28: Codimension-2 Shilnikov-Hopf bifurcation.

Solutions for chapter 7
1. Take c < 0 so that the slow w-nullcline has a negative slope.
2. The quasi-threshold contains the union of canard solutions.
3. The change of variables z = eiωt u transforms the system into the formu̇ = ε{−u + e−iωt I(t)} ,which can be averaged, yieldingu̇ = ε{−u + I ∗ (ω)} .
Apparently, the stable equilibrium u = I ∗ (ω) corresponds to the sustained oscillation z =eiωt I ∗ (ω).
4. The existence of damped oscillations with frequency ω implies that the system has a focus
equilibrium with eigenvalues −ε ± iω, where ε > 0. The local dynamics near the focus can be
represented in the form (7.3). The rest of the proof is the same as the one for exercise 3.
5. Even though the slow and the fast nullclines in Fig.5.21 intersect in only one point, they
continue to be close and parallel to each other in the voltage range 10 mV to 30 mV. Such aproximity creates a tunneling effect (Rush and Rinzel 1996) that prolongs the time spent nearthose nullclines.
6. (Shilnikov-Hopf bifurcation) The model is near a codimension-2 bifurcation having a homoclinic
orbit to an equilibrium undergoing subcritical Andronov-Hopf bifurcation, as we illustrate in
Fig.S.28. Many weird phenomena can happen near bifurcations of codimension-2 or higher.

Solutions for chapter 8
1. Consider two coupled neurons firing together.
2. The equationV̇ = c(b − bsn ) + a(V − Vsn )2
can be written in the formV̇ = a(V − Vrest )(V − Vthresh )withVrest = Vsn −provided that b < bsn .

pc(bsn − b)/aandVthresh = Vsn +pc(bsn − b)/a,462

Solutions to Exercises, chapter 8

3. The system v̇ = b + v 2 with b > 0 and the initial condition v(0) = vreset has the solution (checkby differentiating)√√v(t) = b tan( b(t + t0 )),where

1vresett0 = √ atan √ .
bbFrom the condition v(T ) = vpeak = 1, we find111
T = √ atan √ − t0 = √
b
b
b



1
vresetatan √ − atan √
b
b


,

which alternatively can be written as
1
T = √ atanb√ vreset − 1b.
vreset + b

4. The system v̇ = −|b| + v 2 with the initial condition v(0) = vreset >by differentiating)√p 1 + e2 b(t+t0 )√v(t) = |b|,
1 − e2 b(t+t0 )p|b| has the solution (checkwherepvreset − |b|1p .
t0 = p ln
2 |b| vreset + |b|
From the condition v(T ) = 1, we find
1
T = p
2 |b|

p
p !
|b|
vreset − |b|p − lnpln.
1 + |b|vreset + |b|1−

5. The saddle-node bifurcation occurs when b = 0, regardless of the value of vreset , which is astraight vertical line in Fig.8.3. If vreset < 0, then the saddle-node bifurcationis on an invariantpcircle. When b < 0, the unstable node (saddle) equilibrium
is
at
v
=
|b|.
Hence,
the saddlephomoclinic orbit bifurcation occurs when vreset = |b|.
6. The change of variables v = g/2 + V , b = g 2 /4 + B transforms v̇√= b − gv + v 2 to V̇ = B + V 2
with Vreset = −∞ and Vpeak = +∞. It has the threshold V = B, the rheobase B = 0, andthe same F-I
p curve as in the original model√ with g = 0. In v-coordinates, the threshold is
v = g/2 + b − g 2 /4, which is greater than b, the new rheobase is b = g 2 /4, which is greaterthan b = 0, and the new F-I curve is the same as the old one, just shifted to the right by g 2 /4.
7. Let b = εr with ε ≪ 1 be a small parameter. The change of variables
v=

√

ε tan

ϑ
2

uniformly transforms (8.2) into the theta-neuron formϑ̇ =√ε{(1 − cos ϑ) + (1 + cos ϑ)r} .

√on the unit circle except for the small interval |ϑ − π| < 2 4 ε corresponding to the action
potential (v > 1); see Hoppensteadt and Izhikevich (1997) for more details.

Solutions to Exercises, chapter 9463

8. Use the change of variables

√

v=

εϑ
.
1 − |ϑ|To obtain other theta neuron models, use the change of variables√v = εh(ϑ) ,where the monotone function h maps (−π, π) to (−∞, ∞) and scales like 1/(ϑ ± π) when
ϑ → ±π. The corresponding model has the form
ϑ′ = h2 (ϑ)/h′ (ϑ) + r/h′ (ϑ) .
In particular, h2 (ϑ)/h′ (ϑ) exists, and is bounded and 1/h′ (ϑ) = 0 when ϑ → ±π. These imply
a uniform velocity independent from the input r when ϑ passes the value ±π corresponding tofiring a spike.
9. The equilibrium v = I/(b + 1), u = bI/(b + 1) has the Jacobian matrix


−1 −1
L=
ab −a
with trL = −(a + 1) and detL = a(b + 1). It is a stable node (integrator) when b < (a +1)2 /(4a) − 1 and a stable focus (resonator) otherwise.
10. The quadratic integrate-and-fire neuron with a dendritic compartment
V̇=

B + V 2 + g1 (Vd − V )V̇d=gleak (Eleak − Vd ) + g2 (V − Vd )can be written in the form (8.3, 8.4), with v = V − g1 /2, u = −g1 Vd , I = B − g12 /4 − (g12 g2 +gleak Eleak )/(gleak + g2 ), a = gleak + g2 , and b = −g1 g2 /a.
11. A MATLAB program generating the figure is provided on the author’s Web page (www.izhikevich.com).
12. An example is in Fig.S.29.
13. An example is in Fig.S.30.

Solutions for chapter 9
1. (Planar burster) Izhikevich (2000a) suggested the systemẋ = x − x3 /3 − u + 4S(x) cos 40u,u̇ = µx ,with S(x) = 1/(1 + e5(1−x) ) and µ = 0.01. It has a hedgehog limit cycle depicted in Fig.S.31.
2. (Noise-induced bursting) Noise can induce bursting in a two-dimensional system with coexistence of resting and spiking states. Indeed, noisy perturbations can randomly push the stateof the system into the attraction domain of the resting state or into the attraction domain of
the limit cycle attractor, as in Fig.S.32. The solution meanders between the states, exhibitinga random bursting pattern as in Fig.9.55 (right). Neocortical neurons of the RS and FS types,
as well as stellate neurons of the entorhinal cortex, exhibit such bursting; see chapter 8.

464

Solutions to Exercises, chapter 9
FS neuron (in vitro)

simple model2

3

I=400 pA

1

I=230 pA
2
1

I=125 pA

-55 mV

reset AHP

200

ullc
line
li n e,
I= 0

reset

v-n

300

AHP

ne I=100 pA

cli

ull

u-n

ullc

25 ms

400

v -n

25 mV

recovery variable, u

2
1

100
0

spike
rest
-60

-40
-20
0
membrane potential, v (mV)

20

Figure S.29: Comparison of in vitro recordings of a fast spiking (FS) interneuron oflayer 5 rat visual cortex with simulations of the simple model with linear slow nullcline
20v̇ = (v + 55)(v + 40) − u + I, u̇ = 0.15{8(v + 55) − u}, if v ≥ 25, then v ← −55,u ← u + 200.
3. (Noise-induced bursting) Bursting occurs because noisy perturbations push the trajectory intoand out of the attraction domain of the limit cycle attractor, which coexists with the resting
equilibrium; see the phase portrait in Fig.S.33.
4. (Rebound bursting in the FitzHugh-Nagumo oscillator) The oscillator is near the fold limit cycle
bifurcation. The solution makes a few rotations along the ghost of the cycle before returning
to rest; see Fig.S.34.
5. Yes, they can, at the end of a burst. Think of a “fold/Hopf” or “circle/Hopf” burster. The
resting equilibrium is a stable focus immediately after the termination of a burst, and then it istransformed into a stable node to be ready for the circle or fold bifurcation. Even “circle/circle”
bursters can exhibit such oscillations, if the resting equilibrium turns into a focus for a shortperiod of time somewhere in the middle of a quiescent phase. In any case, the oscillations
should disappear just before the transition to the spiking state.
6. (Hopf/Hopf bursting) Even though there is no coexistence of attractors, there is a hysteresis
loop due to the slow passage effect through the supercritical Andronov-Hopf bifurcation; see
Fig.S.35. The delayed transition to spiking creates the hysteresis loop and enables bursting.
7. (Hopf/Hopf canonical model) First, we restrict the fast subsystem to its center manifold andtransform it into the normal form for supercritical Andronov-Hopf bifurcation, which afterSolutions to Exercises, chapter 9

layer 5 neuron

465

simple model

I=80 pA

0

I=50 pA

0

I=45 pA
0

recovery, u

100
I=40 pA

u=I-5(v+60)

u=I-5(v+60)+3(v+50)2

50
0
-50
-60

u=-2(v+60)

-55
-50
-45
-40
membrane potential, v (mV)

-3

Figure S.30: Comparison of in vitro recordings of a regular spiking (RS) neuron withsimulations of the simple model 100v̇ = I − 5(v + 60) + 3(v + 50)2+ − u, u̇ = 0.02{−2(v +60) − u}, if v ≥ 35, then v ← −50, u ← u + 70.

0.4

u
0.2

0

0.2

0.4

0.6

0.8

x
1
2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

3

3

2.5

2

1.5

Figure S.31: Solution to exercise 1.

1

0.5

0

0.5

1

x(t)

1.5

2
0

50

100

150

200

250

300

350

400

450

Nullclines, a hedgehog limit cycle, anda bursting solution of a planar system.
(Modified from Izhikevich 2000a.)

466

Solutions to Exercises, chapter 9

1

K+ activation gate, n

0.8

0.6

0.4

0.2

Figure S.32: Noise-induced bursting-40-20
membrane potential, V (mV)in two-dimensional system; See exercise 2.

0

0.8

0

0.7

-10
membrane potential, V (mV)

-60

K+ gating variable, n

0
-80

0.6
0.5
0.4
0.3
0.2

-30
-40
-50
-60

0.1
0
-80

-20

-60

-40
-20
membrane potential, V (mV)

0

-70
0

10

20

30
40
time (ms)

50

60

70

Figure S.33: Noise-induced bursting in a two-dimensional system with coexistence of an equilibrium
and a limit cycle attractor; see exercise 3.

0.2

1

w
ghost offold limit cycle
0.5
0.1

V(t)

0

0
-0.5

I(t)
0

0.5
V

1

-0.5

0

1000

2000
time

3000

4000

Figure S.34: Rebound bursting in the FitzHugh-Nagumo oscillator; see exercise 4.

Solutions to Exercises, chapter 9

467
x1

|x|
y

x2

0.5

y
S
y

0.5

x1

Supercritical Andronov-Hopf Bifurcationt
Slow Passage
EffectFigure S.35: Hopf/Hopf bursting without coexistence of attractors; see exercise 6. (Modified from
Hoppensteadt and Izhikevich 1997.)
appropriate re-scaling, has the formż = (u + iω)z − z|z|2 .
Here, u is the deviation from the slow equilibrium u0 . The slow subsystemu̇ = µg(zeiωt + complex-conjugate, u)
can be averaged and transformed into the canonical form.
8. (Bursting in the INa,t +INa,slow -model) First, determine the parameters of the INa,t -model corresponding to the subcritical Andronov-Hopf bifurcation, and hence the coexistence of the restingand spiking states. Then, add a slow high-threshold persistent Na+ current that activates during spiking, depolarizes the membrane potential, and stops the spiking. During resting, thecurrent deactivates, the membrane potential hyperpolarizes, and the neuron starts to fire again.
9. Replace the slow Na+ current in the exercise above with a slow dendritic compartment with dendritic resting potential far below the somatic resting potential. As the dendritic compartment
hyperpolarizes the somatic compartment, the soma starts to fire (due to the inhibition-induced
firing described in section 7.2.8). As the somatic compartment fires, the dendritic compartment
slowly depolarizes, removes the hyperpolarization and stops firing.
10. (Bursting in the INa,p +IK +INa,slow -model) The time constant τslow (V ) is relatively small in thevoltage range corresponding to the spike after-hyperpolarization (AHP). Deactivation of theNa+ current during each AHP is much stronger than its activation during the spike peak. As
a result, the Na+ current deactivates (turns off) during the burst, and then slowly reactivatesto its baseline level during the resting period, as one can see in Fig.S.36.

Solutions to Exercises, chapter 9
membrane potential,
V (mV)

468
0
-20
-40
-60

Na+ activation
gate, mslow

0

50

100

150

200

250

300

350

400

450

300

350

400

450

deactivation

0.5
reactivation

0.4
0.3

0

50

100

150

200
250
time (ms)Figure S.36: Bursting in the INa,p +IK +INa,slow -model. See exercise 10.
slow increase of I

fast increase of I

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

Rest

0.5

1
2.5

2

1.5

1

0.5

0

0.5

Spiking

0.5

1

1.5

2

2.5

1
2.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

Figure S.37: The system has a unique attractor, equilibrium, yet it can exhibit repetitive spiking
activity when the N-shaped nullcline is moved upward not very slowly.
11. The mechanism of spiking, illustrated in Fig.S.37, is closely related to the phenomenon ofaccommodation and anodal break excitation. The key feature is that this bursting is not
fast-slow.
The system has a unique attractor – a stable equilibrium – and the solution always converges
to it. The slow variable I controls the vertical position of the N-shaped nullcline. If I increases,
the nullcline slowly moves upward, and so does the solution, because it tracks the equilibrium.
However, if the rate of change of I is not small enough, the solution cannot catch up with theequilibrium and starts to oscillate with a large amplitude. Thus, the system exhibits spiking
behavior even though it does not have a limit cycle attractor for any fixed I.
12. From the first equation, we find the equivalent voltage
1 + u if 1 + u > 0 ,|z|2 = |1 + u|+ =0
if 1 + u ≤ 0 ,so that the reduced slow subsystem has the formu̇ = µ[u − u3 − w] ,ẇ= µ[|1 + u|+ − 1] ,and it has essentially the same dynamics as the van der Pol oscillator.

Solutions to Exercises, chapter 9469

Saddle-Node on
Invariant Circle
Bifurcation

Spiking

Saddle-Node on
Invariant Circle
Bifurcation

Rest

θ
ψ

Figure S.38: Answer to exercise 13.
13. The fast equationϑ̇ = 1 − cos ϑ + (1 + cos ϑ)ris the Ermentrout-Kopell canonical model for Class 1 excitability, also known as the theta
neuron (Ermentrout 1996). It is quiescent when r < 0 and fires spikes when r > 0. As ψ
oscillates with frequency ω, the function r = r(ψ) changes sign. The fast equation undergoes asaddle-node on invariant circle bifurcation; hence the system is a “circle/circle” burster of theslow-wave type; see Fig.S.38.
14. To understand the bursting dynamics of the canonical model, rewrite it in polar coordinatesz = reiφ :ṙ = ur + 2r3 − r5 ,u̇ = µ(a − r2 ) ,φ̇ = ω .
Apparently, it is enough to consider the first two equations, which determine the oscillation
profile. Nontrivial (r ̸= 0) equilibria of this system correspond to limit cycles of the canonicalmodel, which may look like periodic (tonic) spiking with frequency ω. Limit cycles of thissystem correspond to quasi-periodic solutions of the canonical model, which look like bursting;see Fig.9.37.
The first two equation above have a unique equilibrium,

  √

r
a
=
u
a2 − 2afor all µ and a > 0, which is stable when a > 1. When a decreases and passes an µ neighborhoodof a = 1, the equilibrium loses stability via Andronov-Hopf bifurcation. When 0 < a < 1, thesystem has a limit cycle attractor. Therefore, the canonical model exhibits bursting behavior.
The smaller the value of a, the longer the interburst period. When a → 0, the interburst period
becomes infinite.
15. Take w = I − u. Then (9.7) becomesv̇
ẇ

= v2 + w ,= µ(I − w) ≈ µI .

16. Let us sketch the derivation. Since the fast subsystem is near saddle-node homoclinic orbit
bifurcation for some u = u0 , a small neighborhood of the saddle-node point v0 is invariantly
foliated by stable submanifolds, as in Fig.S.39. Because the contraction along the stable submanifolds is much stronger than the dynamics along the center manifold, the fast subsystem
can be mapped into the normal form v̇ = q(u) + p(v − v0 )2 by a continuous change of variables.

470

Solutions to Exercises, chapter 9
Saddle-NodeSeparatrix Loop
Saddle-Nodeon Invariant
CircleSpike

Invariant
Foliation

Figure S.39: A small neighborhood of the saddle-node point can be invariantly foliated by stablesubmanifolds.

slow variable u2

42

averaged
full

0

-2

-4
-4

-2

0
2
slow variable u1

4

Figure S.40: exercise 18.
When v escapes the small neighborhood of v0 , the neuron is said to fire a spike, and v is resetto v ← v0 + c(u). Such a stereotypical spike also resets u by a constant d. If g(v0 , u0 ) ≈ 0, then
all functions are small, and linearization and appropriate re-scaling yield the canonical model.
If g(v0 , u0 ) ̸= 0, then the canonical model has the same form as in the previous exercise.
17. The derivation proceeds as in the previous exercise, yieldingv̇ = I + v 2 + (a, u) ,u̇ = µAu .
where (a, u) is the scalar (dot) product of vectors a, u ∈ R2 , and A is the Jacobian matrix
at the equilibrium of the slow subsystem. If the equilibrium is a node, it generically has two
distinct eigenvalues and two real eigenvectors. In this case, the slow subsystem uncouples into
two equations, each along the corresponding eigenvector. Appropriate re-scaling gives the firstcanonical model. If the equilibrium is a focus, the linear part can be made triangular in orderto get the second canonical model.
18. The solution of the fast subsystemv̇ = u + v 2 ,with fixed u > 0 isv(t) =

√

u tan


√

v(0) = −1 ,1ut − atan √u.

The interspike period, T , is defined by v(T ) = +∞, given by the formula


1
π
1
T (u) = √+ atan √.
u 2uSolutions to Exercises, chapter 10

Fold
Bifurcation

471

Fold
Bifurcation

Fold
Bifurcation"Fold/Fold"
Hysteresis Loop

Subcritical
Androno-Hopf
Bifurcation

Subcritical
Andronov-Hopf
Bifurcation

Saddle Homoclinic
Orbit Bifurcation"Fold/SubHopf"
Hysteresis Loop

Saddle Homoclinic
Orbit Bifurcation

Subcritical
Androno-Hopf
Bifurcation

Fold
Bifurcation"SubHopf/Fold"
Hysteresis Loop

SaddleHomoclinic
Orbit
Bifurcations

Subcritical
Andronov-Hopf
Bifurcation"SubHopf/SubHopf"
Hysteresis Loop

Figure S.41: Classification of point-point codimension-1 hysteresis loops.
The result follows from the integral
Z T (u)1d1
di δ(t − T (u)) dt =T (u) 0T (u)and the relationships
f (u) =

1
T (u)

and

√
1
atan √ = arcot u .
uPeriodic solutions of the averaged system (focus case) and the full system are depicted in
Fig.S.40. The deviation is due to the finite size of the parameters µ1 and µ2 in Fig.9.35.
19. There are only two codimension-1 bifurcations of an equilibrium that result in transitions toanother equilibrium: saddle-node off limit cycle and subcritical Andronov-Hopf bifurcation.
Hence, there are four point-point hysteresis loops, depicted in Fig.S.41. More details are provided in Izhikevich (2000a).
20. These figures are modified from Izhikevich (2000a), where one can find two models exhibiting
this phenomenon. The key feature is that the slow subsystem is not too slow, and the rateof attraction to the upper equilibrium is relatively weak. The spikes are actually damped
oscillations that are generated by the fast subsystem while it converges to the equilibrium.
Periodic bursting is generated via the point-point hysteresis loop.
21. There are only two codimension-1 bifurcations of a small limit cycle attractor (subthresholdoscillation) on a plane that result in sharp transitions to a large-amplitude limit cycle attractor(spiking): fold limit cycle bifurcation and saddle-homoclinic orbit bifurcation; see Fig.S.42.
These two bifurcations paired with any of the four bifurcations of the large-amplitude limit
cycle attractor result in eight planar codimension-1 cycle-cycle bursters; see Fig.S.43. Moredetails are provided by Izhikevich (2000a).

472

Solutions to Exercises, chapter 10

Fold Limit Cycle Bifurcation

Saddle Homoclinic Orbit Bifurcation

Figure S.42: Codimension-1 bifurcations of a stable limit cycle in planar systems that result in
sharp loss of stability and transition to a large-amplitude (spiking) limit cycle attractor, not shownin the figure. Fold limit cycle: A stable limit cycle is approached by an unstable one, they coalesce,
and then they disappear. Saddle homoclinic orbit: A limit cycle grows into a saddle. The unstablemanifold of the saddle makes a loop and returns via the stable manifold (separatrix).

Saddle-Nodeon Invariant
CircleSaddleHomoclinic
Orbit

Supercritical
AndronovHopf

Fold
Limit
Cycle

Fold
Limit
Cycle

fold cycle/
circlefold cycle/
homoclinic

fold cycle/
Hopffold cycle/
fold cycle

SaddleHomoclinic
Orbit

homoclinic/
circlehomoclinic/
homoclinic

homoclinic/
Hopfhomoclinic/
fold cycle

Bifurcations

Figure S.43: Classification of codimension-1 cycle-cycle planar bursters.

Solutions to Exercises, chapter 10

473

Solutions to Chapter 10
1. In polar coordinates, z = reiϑ , the system has the formϑ̇ = 1 ,ṙ = r − r3 .

Since the phase of oscillation does not depend on the amplitude, the isochrons have the radial
structure depicted in Fig.10.3.
2. In polar coordinates, the oscillator has the formϑ̇ = 1 + dr2 ,ṙ = r − r3 .

The second equation has an explicit solution r(t), such thatr(t)2 =

1.
1 − (1 − 1/r(0)2 )e−2t

The phase difference between ϑ˙lc = 1 + d(1)2 and ϑ̇ = 1 + dr(t)2 grows as χ̇ = d(r(t)2 − 1), andits asymptotic value isZ∞d(r(t)2 − 1) = d log r(0) .

χ(∞) =
0Thus, on the χ-isochron, we have ϑ + d log r = χ.
3. An example is the file isochrons.mfunction isochrons(F,phases,x0)% plot isochrons of a planar dynamical system x’=F(t,x)% at points given by the vector ’phases’.
% ’x0’ is a point on the limit cycle (2x1-vector)
T= phases(end); % is the period of the cycle
tau = T/600;% time step of integration
m=200;% spatial grid
k=5;
% the number of skipped cycles[t,lc] = ode23s(F,0:tau:T,x0);dx=(max(lc)-min(lc))’/m;
center = (max(lc)+min(lc))’/2;
iso=[x0-m^0.5*dx, x0+m^0.5*dx];% forward integration% spatial resolution% center of the limit cycle% isochron’s initial segment

for t=0:-tau:-(k+1)*T% backward integration
for i=1:size(iso,2)
iso(:,i)=iso(:,i)-tau*feval(F,t,iso(:,i)); % move one step
end;
i=1;
while i<=size(iso,2)% remove infinite solutions
if any(abs(iso(:,i)-center)>1.5*m*dx)% check boundaries
iso = [iso(:,1:i-1), iso(:,i+1:end)]; % remove
else
i=i+1;
end;
end;
i=1;
while i<=size(iso,2)-1

474

Solutions to Exercises, chapter 10
d=sqrt(sum(((iso(:,i)-iso(:,i+1))./dx).^2)); % normalized distanceif d > 2% add a point in the middle
iso = [iso(:,1:i), (iso(:,i)+iso(:,i+1))/2 ,iso(:,i+1:end)];end;
if d < 0.5% remove the point
iso = [iso(:,1:i), iso(:,i+2:end)];else
i=i+1;end;end;
if (mod(-t,T)<=tau/2) & (-t<k*T+tau)% refresh the screen
cla;plot(lc(:,1),lc(:,2),’r’); hold on; % plot the limit cycleend;
if min(abs(mod(-t,T)-phases))<tau/2% plot the isochrons
plot(iso(1,:),iso(2,:),’k-’); drawnow;
end;
end;
hold off;
The call of the function is isochrons(’F’,0:0.1:2*pi,[1;0]); withfunction dx = F(t,x);z=x(1)+1i*x(2);dz=(1+1i)*z-z*z*conj(z);dx=[real(dz); imag(dz)];

4. (Hoppensteadt and Keener 1982) From calculus, B · C = |B| |C| cos(ψ). Since |B| = 1 and
2
2
C = (A + cos ϑ, sin ϑ)√ – see Fig.10.46 – we have B · C = A cos ϑ + cos ϑ + sin ϑ. Hence,2
cos ψ = (1 + A cos ϑ)/ 1 + 2A cos ϑ + A . When ϑ is in the upper (lower) half-plane, the phaseis delayed (advanced).
5. An example is the file prc.mfunction PRC=prc(F,phases,x0,A)% plot phase-resetting curve (PRC) of system x’=F(t,x) + A delta(t)% at points given by the vector ’phases’.
% ’x0’ is a point on the limit cycle with zero phase% A is the strength of stimulation (row-vector)% use peaks of spikes to find the phase differences
T= phases(end); % is the period of the cycle
tau = T/6000;% time step of integration
k=3;% the number of cycles needed to determine the new phase
PRC=[];[tc,lc] = ode23s(F,0:tau:k*T,x0); % find limit cycle
peak=1+find(lc(2:end-1,1)>lc(1:end-2,1)&lc(2:end-1,1)>=lc(3:end,1));
peak0 = tc(peak(end)); % the last peak is used for reference
for i=1:length(phases)[m,j]=min(abs(phases(i)-tc));[t,x] = ode23s(F,phases(i):tau:k*T,lc(j,:)+A); % stimulate
peaks=1+find(x(2:end-1,1)>x(1:end-2,1)&x(2:end-1,1)>=x(3:end,1));
PRC=[PRC, mod(T/2+peak0-t(peaks(end)),T)-T/2];
subplot(2,1,2);drawnow;
plot(phases(1:length(PRC)),PRC);
xlabel(’phase of stimulation’);ylabel(’induced phase difference’);Solutions to Exercises, chapter 10475

subplot(2,1,1);plot(tc,lc(:,1),’r’,t,x(:,1),t(peaks(end)),x(peaks(end),1),’ro’);xlabel(’time’);ylabel(’membrane potential’);end;
An example of a call of the function is PRC=prc(’F’,0:0.1:2*pi,[-1 0],[0.1 0]); withfunction dx = F(t,x);z=x(1)+1i*x(2);dz=(1+1i)*z-z*z*conj(z);dx=[real(dz); imag(dz)];6.
ϑ(x + hm ) − ϑ(x)
ϑ(x + h1 ) − ϑ(x),...,grad ϑ(x) =
h1
hm


PRC1 (ϑ(x), h1 )PRCm (ϑ(x), hm )
=
,...,
h1
hm
(in the limit h → 0)
Z1 (ϑ(x))h1
Zm (ϑ(x))hm
=
,...,
h1
hm
= (Z1 (ϑ(x)), . . . , Zm (ϑ(x))) = Z(ϑ(x)) .



7. (Brown et al. 2004, appendix A) Let x be a point on the limit cycle and z be an arbitrary
nearby point. Let x(t) and z(t) be the trajectories starting from the two points, and y(t) =z(t) − x(t) be the difference. All equations below are valid up to O(y 2 ). The phase shift∆ϑ = ϑ(z(t)) − ϑ(x(t)) = grad ϑ(x(t)) · y(t) does not depend on time. Differentiating withrespect to time and taking grad ϑ(x(t)) = Z(ϑ(t)) (see previous exercise), results in
0=(d/dt) (Z(ϑ(t)) · y(t)) = Z ′ (ϑ(t)) · y(t) + Z(ϑ(t)) · Df (x(t))y(t)⊤= Z ′ (ϑ(t)) · y(t) + Df (x(t)) Z(ϑ(t)) · y(t)⊤=Z ′ (ϑ(t)) + Df (x(t)) Z(ϑ(t)) · y(t) .
⊤Since y is arbitrary, Z satisfies Z ′ (ϑ) + Df (x(ϑ)) Z(ϑ) = 0, that is, the adjoint equation(10.10). The normalization follows from (10.7).
8. The solution to v̇ = b − v with v(0) = 0 is v(t) = b(1 − e−t ) with the period T = ln(b/(b − 1))determined from the threshold crossing v(T ) = 1. From v = b(1−e−ϑ ) we find ϑ = ln(b/(b−v)),
hence
PRC (ϑ) = ϑnew − ϑ = min {ln(b/(b exp(−ϑ) − A), T } − ϑ .
9. The system v̇ = 1 + v 2 with v(0) = −∞ has the solution (the reader should check this bydifferentiating) v(t) = tan(t − π/2) with the period T = π. Since t = π/2 + atan v, we find
PTC (ϑ) = π/2 + atan [A + tan(ϑ − π/2)]
and
PRC (ϑ) = PTC (ϑ) − ϑ = atan [A + tan(ϑ − π/2)] − (ϑ − π/2) .
10. The system v̇ = b + v 2 with b > 0 and the initial condition v(0) = vreset has the solution (thereader should check this by differentiating)√√v(t) = b tan( b(t + t0 ))476

Solutions to Exercises, chapter 10
where1vresett0 = √ atan √ .
bb

Equivalently,
1v
t = √ atan √ − t0 .
bb
From the condition v = 1 (peak of the spike), we find


1
1
1
1
vresetT = √ atan √ − t0 = √atan √ − atan √
,
b
b
b
b
b
Hence
√
1
A
PRC (ϑ) = ϑnew − ϑ = min { √ atan [ √ + tan( b(ϑ + t0 ))] − t0 , T } − ϑ .
bb
11. Let ϑ denote the phase of oscillator 1. Let χn denote the phase of oscillator 2 just before
oscillator 1 fires a spike (i.e., when ϑ = 0). This spike resets χn to PTC2 (χn ). Oscillator 2 firesa spike when ϑ = T2 −PTC2 (χn ), and it resets ϑ to PTC1 (T2 −PTC2 (χn )). Finally, oscillator
1 fires its spike when oscillator 2 has the phase χn+1 = T1 −PTC1 (T2 −PTC2 (χn )).
12. [MATLAB] An example is the file adjoint.mfunction Q=adjoint(F,t,x0)% finds solution to the Malkin’s adjoint equation Q’ = -DF^t Q% at time-points t with t(end) being the period% ’x0’ is a point on the limit cycle with zero phase
tran=3;% the number of skipped cycles
dx = 0.000001; dy = 0.000001;% for evaluation of Jacobian
Q(1,:)=feval(F,0,x0)’;[t,x] = ode23s(F,t,x0);% initial point;% find limit cyclefor k=1:tran
Q(length(t),:)=Q(1,:);% initial point;
for i=length(t):-1:2% backward integration
L = [(feval(F,t(i),x(i,:)+[dx 0])-feval(F,t(i),x(i,:)))/dx,...
(feval(F,t(i),x(i,:)+[0 dy])-feval(F,t(i),x(i,:)))/dy];Q(i-1,:) = Q(i,:) + (t(i)-t(i-1))*(Q(i,:)*L);
end;
end;
Q = Q/(Q(1,:)*feval(F,0,x0));% normalization
An example of a call of the function is Q=adjoint(’F’,0:0.01:2*pi,[1 0]); withfunction dx = F(t,x);z=x(1)+1i*x(2);dz=(1+1i)*z-z*z*conj(z);dx=[real(dz); imag(dz)];
13. [MATLAB] We assume that PRC (ϑ) is given by its truncated Fourier series with unknown
Fourier coefficients. Then, we find the coefficients that minimize the difference between predicted and actual interspike intervals. The MATLAB file findprc.m takes the row vector of
spike moments, not counting the spike at time zero, and the input function p(t), determines the
sampling frequency and the averaged period of oscillation; and then calls the file prcerror.m
to find PRC.

Solutions to Exercises, chapter 10function PRC=findprc(sp,pp)
global spikes p tau n% finds PRC of an oscillator theta’= 1 + PRC(theta)pp(t)% using the row-vector of spikes ’sp’ (when theta(t)=0)
spikes = [0 sp];p=pp;
tau = spikes(end)/length(p) % time step (sampling period)
n=8;% The number of Fourier terms approximating PRC
coeff=zeros(1,2*n+1); % initial approximation
coeff(2*n+2) = spikes(end)/length(spikes); % initial period
coeff=fminsearch(’prcerror’,coeff);a = coeff(1:n)% Fourier coefficients for sin
b = coeff(n+1:2*n) % Fourier coefficients for cos
b0= coeff(2*n+1)% dc term
T = coeff(2*n+2)% period of oscillation
PRC=b0+sum((ones(floor(T/tau),1)*a).*sin((tau:tau:T)’*(1:n)*2*pi/T),2)+...
sum((ones(floor(T/tau),1)*b).*cos((tau:tau:T)’*(1:n)*2*pi/T),2);
The following program must be in the file prcerror.m.

function err=prcerror(coeff)
global spikes p tau n
a = coeff(1:n);% Fourier coefficients for sin
b = coeff(n+1:2*n); % Fourier coefficients for cos
b0= coeff(2*n+1);% dc term
T = coeff(2*n+2);% period of oscillation
err=0;
i=1;
clf;
for s=2:length(spikes)
theta=0;
while i*tau<=spikes(s)PRC=b0+sum(a.*sin((1:n)*2*pi*theta/T))+...
sum(b.*cos((1:n)*2*pi*theta/T));theta = theta + tau*(1+PRC*p(i));
i=i+1;
end;
err = err + (theta-T)^2;
subplot(2,1,1);plot(spikes(s),T,’r.’,spikes(s),theta,’b.’);hold on;end;axis([0 spikes(end) 0.75*T 1.25*T])
subplot(2,1,2);
PRC=b0+sum((ones(floor(T/tau),1)*a).*sin((tau:tau:T)’*(1:n)*2*pi/T),2)+...
sum((ones(floor(T/tau),1)*b).*cos((tau:tau:T)’*(1:n)*2*pi/T),2);plot(PRC);
err = (err/(length(spikes)-1))^0.5; % normalizationtext(0,mean(PRC),[’err=’ num2str(err)]);drawnow;

477

478

Solutions to Exercises, chapter 10

14. Winfree approach: Using results of exercise 8,
∂
b
1
ln
= −ϑ
∂A be−ϑ − A
be − Aand setting A = 0, we obtain Z(ϑ) = eϑ /b.
Kuramoto approach: The solution is v(ϑ) = b(1 − e−ϑ ) with T = ln(b/(b − 1)) and f (v(ϑ)) =be−ϑ . From the condition (10.7), grad (ϑ) = 1/f (v(ϑ)) = eϑ /b.
Malkin approach: Df = −1; hence Q̇ = 1 · Q has the solution Q(t) = Cet . The free constantC = 1/b is found from the normalization condition Q(0) · (b − 0) = 1.
15. Winfree approach: Using results of exercise 10 and the relation PRC=PTC−ϑ, we obtain,1∂(π/2 + atan (A − cot ϑ) − ϑ) =2
∂A
1 + (A − cot ϑ)and, setting A = 0, Z(ϑ) = sin2 ϑ.
Kuramoto approach: The solution is v(ϑ) = − cot ϑ with T = π and f (v(ϑ)) = 1/ sin2 ϑ. From
the normalization condition (10.7), grad (ϑ) = 1/f (v(ϑ)) = sin2 ϑ.
Malkin approach: Df = 2v, hence Q̇ = 2 cot(t) · Q has the solution Q(t) = C sin2 t. The freeconstant C = 1 is found from the normalization condition Q(π/2) · (1 + 02 ) = 1.
16. Winfree approach: Using results of exercise 4,
Z(ϑ) =

1 + A cos ϑ∂
acos √= − sin ϑ∂A
1 + 2A cos ϑ + A2

at A = 0.
Kuramoto approach: Since grad ϑ(x) is orthogonal to the contour line of the function ϑ(x) at
x (i.e., the isochron of x) and the results of exercise 1 that isochrons are radial, we obtain
grad (ϑ) = (− sin ϑ, cos ϑ), using purely geometrical considerations. Since p(t) is real, we needto keep only the first component.
Malkin approach: Let us work in the complex domain. On the circle z(t) = eit we obtain
Df = i. Since Df ⊤ is equivalent to complex-conjugation in the complex domain, we obtain
Q̇ = i · Q, which has the solution Q(t) = Ceit . The free constant C = i is found from thenormalization condition Q(0)∗ i = 1, where ∗ means complex-conjugate.
Alternatively, on the circle z(t) = eit , we have f (z(t)) = f (eit ) = ieit . From the normalization
condition Q(t)∗ f (z(t)) = 1 we find Q(t) = ieit = − sin ϑ + i cos ϑ.
√
17. Rescaling the state variable z = bu and the time, τ = ϵt, we obtain the reduced systemu′ = (1 + i)u + (−1 + di)u|u|2 + εp(t) .
We can apply the theory√ only when ε is small. That is, the theory is guaranteed to work ina very weak limit ϵ ≪ b b ≪ 1. As is often the case, numerical simulations suggest that thetheory works well outside the guaranteed interval. Substituting u = reiθ into this equation,r′ eiθ + reiθ iθ′ = (1 + i)reiθ + (−1 + di)r3 eiθ + εp(t) ,dividing by eiθ , and separating real and imaginary terms, we represent the oscillator in polar
coordinatesr′= r − r3θ′=+ ε Re p(t)e−iθ

1 + dr2 + ε Im r−1 p(t)e−iθ .

Solutions to Exercises, chapter 10

479

When ε = 0, this system has a limit cycle attractor r(t) = 1 and θ(t) = (1 + d)t, provided thatd ̸= −1. On the attractor, the solution to Malkin’s adjoint equation (10.10),
′

Q =−



−2
2d

0
0

⊤


Q with Q(t) ·

0
1+d


=1,

is Q(t) = (d, 1)/(1+d). Indeed, the normalization condition results in Q2 (t) = 1/(1+d). Hence,unique periodic solution of the first equation, Q′1 = 2Q1 − 2d/(1 + d), is Q1 (t) = d/(1 + d).
One can also use Kuramoto’s approach and the results of exercise 2. The corresponding phasemodel,
ϑ′ = 1 + ε{d Re p(t)e−i(1+d)ϑ + Im p(t)e−i(1+d)ϑ }/(d + 1) ,can be simplified via θ = (1 + d)ϑ (notice the font difference) to get the result.
18. (Delayed coupling) Let ϑ(t) = t + φ(τ ), where τ = εt is the slow time. Since ϑ(t − d) =t − d + φ(τ − εd) = t − d + φ(τ ) + O(ε), we have xj (ϑi (t − dij )) = xj (t − dij + φ(τ )), so that
we can proceed as in section 10.2.5, except that there is an extra term, −dij , in (10.16). See
also Izhikevich (1998).
19. Let χ = φ2 − φ1 ; then we haveχ̇ = ω − c sin χ .
If |ω/c| ≤ 1, then there are two synchronized states, χ = arcsin (ω/c) and χ = π − arcsin (ω/c),
one stable and the other unstable.
20. From the theorem of Hoppensteadt and Izhikevich (1997), presented in section 10.3.3, it follows
that Kuramoto’s model is a gradient system when cij = cji and ψij = −ψji . From Ermentrout’s
theorem presented in the same section, it follows that the synchronized state φi = φj is stableif, for example, all ψij = 0 and cij > 0.
21. Since the probability density function g(ω) is symmetric, the averaged frequency deviation of
the network is zero, and, rotating the coordinate system, we can make the cluster phase ψ = 0.
The network is split into two populations: One oscillating with the cluster (|ω| < Kr), thereby
forming the cluster, and one drifting in and out of the cluster. The latter does not contributeto the Kuramoto synchronization index, because contributions from different oscillators cancel
each other on average. In the limit n → ∞, the sum (10.21) becomes the integral
Z
Z
i
φ(ω)
r= e
g(ω)d ω ≈eiφ(ω) g(ω)d ω .
|ω|<KrNext, since there are as many oscillators with positive φ as with negative, the imaginary partsof eiφ(ω) cancel each other, so thatZr=
cos φ(ω)g(ω)d ω .
|ω|<KrUsing the condition for locking with the cluster, ω = Kr sin φ, we change the variables in the
integral and obtain (10.22).
22. Let χ = φ2 − φ1 ; then χ′ = (c2 − c1 )H(χ). The in-phase state χ = 0 exists when either c1 = c2
or H(0) = 0. Since H(χ) is even, H ′ (0) = 0, and hence it is neutrally stable in either case.
The anti-phase state χ = T /2 exists when H(T /2) = 0, and it can be exponentially stable orunstable, depending on the sign of H ′ (T /2).
23. See Izhikevich (1999), Sect IVB.

480

Solutions to Exercises, chapter 10

24. The exponential stability requirement is G′ (0) = −2H ′ (0) < 0. Since x′ (t) = f (x(t)), we have
T H ′ (0) =Z T
0g ′ (x(t))f (x(t))
dt =f (x(t))Z T

g ′ (x(t)) dt =g ′ (x)dx > 0S1 f (x)Z

0Integrating the latter equation by parts, or differentiating
H(χ) =

1
TZ T
0g(x(t))dtf (x(t − χ))

f ′ (x)g(x) dx > 0 .
2
S1 f (x)Z
at χ = 0, we obtain

25. (Brown et al. 2004) The solution of x′ = λx with x(0) = x0 is x(t) = x0 eλt . The period T =log(∆/x0 )/λ is found from the condition x(T ) = ∆. Hence, Q(ϑ) = 1/(λx(ϑ)) = 1/(λx0 eλϑ ) =eλ(T −ϑ) /(∆λ).
26. Let us first consider the SNIC case ẋ = 1 + x2 . The solution starting with x(0) = xreset has theform (the reader should check by differentiating) x(t) = tan(t + t0 ), where t0 = atan xreset . The
period should be found from the condition tan(T + t0 ) = +∞, and it is T = π/2 − t0 . Hence,x(t) = tan(t + π/2 − T ) = − cot(t − T ). Now, Q(ϑ) = 1/(1 + x(ϑ)2 ) = 1/(1 + cot2 (ϑ − T )) =sin2 (ϑ − T ).
The homoclinic case ẋ = −1 + x2 is quite similar. The solution starting with x(0) = xreset > 1
has the form (the reader should check by differentiating) x(t) = − coth(t + t0 ), where t0 =acoth(−xreset ) = − acoth(xreset ). The period is found from the condition − coth(T + t0 ) = +∞resulting in T = −t0 . Hence, x(t) = − coth(t − T ). Finally, Q(ϑ) = 1/(−1 + x(ϑ)2 ) =1/(1 + coth2 (ϑ − T )) = sinh2 (ϑ − T ).

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Indexdiagram, 77
equilibrium, 159
flip, 190, 454
accommodation, 222
fold, 454
action potential, see spike
fold limit cycle, 181
activation, 33, 34
fold limit cycle on homoclinic torus, 192
adaptation variable, 8fold-Hopf, 194
adapting interspike frequency, 236
homoclinic, see saddle homoclinic
adjoint equation, 462
limit cycle, 178
afterdepolarization (ADP), 260
Neimark-Sacker, 192
afterhyperpolarization, 41, 260, 296
pitchfork, 194
AHP, see afterhyperpolarization
saddle homoclinic orbit, 279, 482, 496
amplifying gate, 129
saddle-focus homoclinic, 190
Andronov-Hopf bifurcation, see bifurcation
saddle-node, 11, 74, 78, 113, 162, 271
anodal break excitation, see postinhibitory
saddle-node homoclinic orbit, 201, 483
spike, see postinhibitory
saddle-node on invariant circle, 13, 164,
Arnold tongue, 456
180, 272, 279, 284, 306, 477
attraction domain, 16, 62, 108
subcritical, 209
attractor, 9, 60
subHopf, 348
coexistence, 13, 66
supercritical, 209
ghost, 75, 478
to bursting, 344
global, 63transcritical, 209
limit cycle, 10, 97
bistability, 14, 66, 72, 82, 108, 226, 248, 286,
autonomous dynamical system, 58
299, 316, 328, 368
averaging, 339
black hole, 451
blue-sky catastrophe, 192
basal ganglia, 311
Boltzmann function, 38, 45
basin of attraction, see attraction domain
Bonhoeffer–van der Pol, see model
Bendixson’s criterion, 126
brainstem, 313
bifurcation, 11, 70, 216
Andronov-Hopf, 13, 116, 168, 181, 199, bursting, 288, 296, 325
m+k type, 336
286
autonomous, 328
Bautin, 200, 362
circle/circle, 354
big saddle homoclinic, 189
classification, 347
blue-sky, 192
conditional, 328
Bogdanov-Takens, 194, 251, 284
dissection, 336
circle, 348
excitability, 328, 343
codimension, 75, 163, 169, 192
fast-slow, 335
cusp, 192o(ε), 458
p:q-phase-locking, 456

497

498

Index
fold/circle, 366
fold/fold cycle, 364
fold/homoclinic, 350
fold/Hopf, 365
forced, 327
hedgehog, 377
Hopf/Hopf, 380
hysteresis loop, 343, 352, 359, 363
intrinsic, 328
minimal model, 332
oscillation, 486
planar, 348
point-cycle, 348
point-point, 382
slow-wave, 344, 356
subHopf/fold cycle, 299, 359
synchronization, 373, 487

cable equation, 42
canard, 199, 241, 497
central pattern generator (CPG), 334, 472
CH (chattering), see neuron
chain of oscillators, 471
channels, 25
cobweb diagram, 452
coherent state, 474
coincidence detection, 233
complex spike, 343
compression function, 486
conductance, 27, 32
conductance-based, see model
cortex, 281
coupled bursters, 486
coupled oscillators, 465
coupled relaxation oscillators, 470, 484
couplingdelayed, 480
gap-junction, 479
pulsed, 444, 477
synaptic, 481
weak, 480
current, 27
K+ , 46
Na+ , 45
amplifying, 55, 129, 147
cation, 47

hyperpolarization-activated, 47
Ohmic, 28, 53
persistent, 33, 45
ramp, 221
resonant, 55, 130, 147, 270, 330
rheobase, 155, 242
transient, 33, 35, 45
zap, 232
current threshold, see rheobase
current-voltage relation, see I-V
cycle slipping, 457, 470
DAP, see afterdepolarization
deactivation, 33
deinactivation, 33
delay, 480
delay loss of stability, see stability
dendrite, 43, 292
dendritic compartment, 43, 292
dendritic-somatic ping pong, 290
depolarization, 29, 41
desynchronization, 374
determinant, 103
Dirac delta function, 444
direction field, see vector field
dissection of bursting, 336
down-state, 316
drifting, 470
dynamic clamp, 288
dynamical system, 8, 57
eigenvalue, 61, 102
eigenvector, 102
elliptic bursting, see bursting, subHopf/foldcycleenergy function, 474
entorhinal cortex, 314
entrainment, 467
equilibrium, 60, 99
classification, 103
focus, 104
hyperbolic, 69, 103
node, 103
saddle, 104
stable, 60, 100, 161
unstable, 61
equivalent circuit, 28

Index499

equivalent voltage, 151, 341
Euler method, 58
excitability, 9, 11, 81, 215
Class 1/2, 221, 228
Class 3, 222
class of, 218, 449
Hodgkin’s classification, 14, 218
excitation block, 118
excitation variable, 8
exponential integrate-and-fire, see model

Hodgkin-Frankenhaeuser layer, 331
Hodgkin-Huxley, see model
homoclinic trajectory, see trajectory
Hopf bifurcation, see bifurcation, AndronovHopf
hyperbolic equilibrium, see equilibrium, 103
hyperpolarization, 29
hyperpolarization-activated channels, 36, 131,
136
hysteresis, 13, 67, 259, 342, 382

F-I curve, 15, 188, 218, 227, 255, 321
fast threshold modulation (FTM), 484
fast-slow dynamics, 329, 335
firing threshold, 3
FitzHugh-Nagumo model, see model
fixed point, 453
Floquet multiplier, 454
focus, see equilibrium
FRB (fast rhythmic bursting), see neuron,CH
French duck, see canard
frequency
acceleration, 255
adaptation, 255
mismatch, 470
plateaus, 472
preference, 232, 237, 265
frequency-current curve, see F-I
frequency-locking, 467
FS (fast spiking), see neuron

I-V relation, 30, 54, 77, 151, 155, 161, 256,316
instantaneous, 31, 152
multiple scales, 257
steady-state, 31, 34, 59, 99, 152, 162
IB (intrinsically bursting), see neuron
impedance, 233
in vivo, 287
inactivation, 33, 35
incoherent state, 474
infinitesimal PRC, 459
inhibition-induced spiking, 244
input conductance, 29
input resistance, 29, 155
instantaneous voltage threshold, 282
integrate-and-fire, see model
integrator, 13, 55, 81, 119, 229, 240, 269, 272,
284, 316, 368
interneuron, see neuron
intra-burst, see interspike
ions, 25
isochron, 445

gap-junction, 44, 467, 479
Gaussian function, 38
GENESIS, 6, 24, 44
geometrical analysis, 59
ghost
seeattractor, 478
gradient system, 474
half-center oscillator, 334
hard loss, see stability
Hartman-Grobman, see theorem
hedgehog burster, 377
heteroclinic trajectory, see trajectory
Hindmarsh-Rose, see model
hippocampus, 308

Jacobian matrix, 102, 473
Kirchhoff’s law, 28
Kuramoto phase model, see model
Kuramoto synchronization index, 474
Landau o(ε), 458
latency, see spike
Liapunov coefficient, 200
limit cycle, 10, 96
Bendixson’s criterion, 126
linear analysis, 101
low-threshold, see spike

500

Index

quadratic integrate-and-fire, 80, 203, 270,
279, 353, 477, 483, 494
reduction, 147
resonate-and-fire, 269
manifold
simple, 153, 272
stable, 109, 445
theta, 320, 322
threshold, 240
van der Pol, 123
unstable, 109
MATLAB, 6, 24, 51, 58, 274, 322, 367, 446, modulation
slow, 252
448, 462, 494, 498, 501
monostable
dynamics, 14
mean-field approximation, 474
Morris-Lecar, see model
membrane potentia, see potential
multistability, see bistability
membrane voltage, see potential
mesencephalic V, see neuron
neocortex, 281
minimal model, see model
neostriatum, 311
mitral, see neuron
Nernst, see potential
model
neurocomputational property, 367
IA , 142
NEURON, 6, 24, 44
ICa +IK , 6
neuron, 1
ICl +IK , 158
basal ganglia, 311
BSNP, 297
IK +IKir , 140
CH, 281, 294, 351
INa,p +ENa ([Na+ ]in/out ), 158
FS, 281, 298
INa,p +IK , 6, 9, 89, 128, 132, 163, 172,
hippocampal, 308, 328
182, 201, 225, 257, 327
IB, 281, 288, 351
INa,p +IK +IK(M) , 253, 327
inhibitory, 301
INa,p +Ih , 136
LS, 282, 300
INa,t , 129, 133
LTS, 281, 296
INa +IK , 452
mesencephalic V, 313
Ih +IKir , 138
mitral, 248, 316
Bonhoeffer–van der Pol, 123, 381
neostriatal, 311
canonical, 278, 353, 357, 363
Purkinje, 319
conductance-based, 43
RS, 281, 282
Emrentrout-Kopell, 357
RSNP, 296
exponential integrate-and-fire, 81
RTN, 306
FitzHugh-Nagumo, 21, 106, 223
stellate, 314
Hindmarsh-Rose, 123
TC, 305
Hodgkin-Huxley, 37, 128, 147, 334
theta, 320
integrate-and-fire, 268, 275, 493
node, see equilibrium
irreducible, see minimal
noise, 177
Kuramoto, 467, 474
normal form, 75, 170, 271
minimal, 127
nullcline, 92
Ca2+ -gated, 147
minimal for bursting, 332
olfactory bulb, 316
Morris-Lecar, 6, 89, 132
orbit, see trajectory
phase, 279
order parameter, 474
oscillation, 177planar, 89
LS (late spiking), see neuron
LTS (low-threshold spiking), see neuron

Indexhomoclinic, 482
interburst, 329
intraburst, 329
multifrequency, 468
phase, 444
quasi-periodic, 468
slow, 232
SNIC, 477
subthreshold, 13, 177, 230, 286, 298, 316
slow, 258
oscillator, 385
Andronov-Hopf, 451, 492
half-center, 334
relaxation, 98, 107, 198, 470
oscillator death, 492
pacemaker, 9parabolic bursting, see bursting, circle/circle
partial synchronization, 474
period, 97, 445
periodic orbit, 10
persistent current, see current
phase, see oscillation
phase deviation, 466
phase drifting, 468
phase lag, 468
phase lead, 468
phase line, 58
phase model, see model
coupled oscillators, 465
Kuramoto reduction, 460, 476
linear response, 459
Malkin reduction, 461, 476
Winfree reduction, 459, 476
phase oscillator, 475
phase portrait, 9, 67, 108
geometrical analysis, 59
local equivalence, 69
topological equivalence, 68
phase space, 58
phase transition curve, see PTC
phase trapping, 468
phase walk-through, 470
phase-locking, 456
phase-resetting curve, see PRC
phaseless set, 451

501
ping-pong, 262, 290
Poincare phase map, 452
postinhibitory
depression, 260
facilitation, 243, 260
spike, 5, 242, 252, 259, 314
postsynaptic potential, 2potential
equivalent, 151, 341
Nernst, 26, 32
resting, 29
reverse, 32
PRC, 446, 459, 462
PSP, see postsynaptic potential
PTC, 450
Purkinje neuron, 248
quadratic integrate-and-fire, see model
quasi-threshold, 241
radial isochron clock, 446
Rall’s branching law, 43
ramp input, 224
rebound, see postinhibitory
recovery variable, 8
refractory period, 41, 269
regular point, 73
relaxation oscillator, see oscillator, 484
repeller, 62, 97
repolarization, 41
resonance, 5, 232
resonant gate, 130
resonator, 13, 55, 119, 229, 241, 313, 316, 368,
372
rest point, see equilibrium
resting potential, see potential
reverse potential, see potential
rheobase, 4, 155, 242
rotation number, 467
RS (regular spiking), see neuron
RTN (reticular thalamic nucleus), see neuron
saddle, 18, see equilibrium
saddle quantity, 185
saddle-node bifurcation, see bifurcation
saddle-node equilibrium, 104

502

Index

saddle-node of periodics, see bifurcation, fold stable manifold, see manifold
limit cycle
state line, see phase line
saddle-node on invariant circle bifurcation, see state space, see phase spacebifurcation
stellate cell, see neuronsag, see voltage
step input, 224
self-ignition, 492
striatum, 311
separatrix, 18, 109, 240
stroboscopic map, 452
shock input, 224
stutter, 227, 301, 316
simple model, see model
subcritical Andronov-Hopf, see bifurcation
slow modulation, 252
subthreshold, 63
slow passage effect, 175, 361
subthreshold oscillation, see oscillation
slow subthreshold oscillation, 258
supercritical Andronov-Hopf, see bifurcation
slow transition, 75
superthreshold, 63
slow-wave bursting, see burstingsuprathreshold, see superthreshold
SNIC, see bifurcation, saddle-node on invari- synapse, 2
ant circle
synaptic coupling, 481
SNLC, see bifurcation, saddle-node on invari- synchronization, 385, 443, 454, 467
ant circle
anti-phase, 454
soft loss, see stability
in-phase, 454
somatic-dendritic ping-pong, 262
of bursts, 373, 487
spike, 2, 41, 63
of spikes, 486
all-or-none, 4, 95, 268
out-of-phase, 454
complex, 343
TC (thalamocortical), see neuron
dendritic, 43, 261, 292
thalamic
doublet, 236
relay neuron, 305
frequency modulation, 255
thalamic burst mode, 306
inhibition-induced, 244
thalamic interneuron, 308
latency, 4, 18, 75, 242, 246, 284, 312
thalamic relay mode, 305
low-threshold, 5, 306
thalamus, 304
postinhibitory, 298
theorempotassium, 140
averaging, 340
propagation, 42
Ermentrout, 473
rebound, see postinhibitory
Hartman-Grobman, 69, 103
synchronization, 374, 486
Malkin, 462
upside-down, 145
Pontryagin–Rodygin, 341
upstroke, 41
theta-neuron, see model
spike time response curve, see PRC
square-wave bursting, see bursting, fold/homo- threshold, 3, 63, 95, 111, 238, 268
current threshold, see rheobase
clinicfiring, 3
squid axon, 14
manifold, 240
stability, 60
quasi-, 241
asymptotic, 60, 97, 100, 453
time crystal, 451
delay loss, 175, 361
topological equivalence, 68
exponential, 100, 103
topological normal form, see normal form
loss, hard/soft, 204
neutral, 100
torus knot, 467

Index
trace, 103
trajectory, 94
canard, 199
heteroclinic, 111
homoclinic, 111
periodic, 96transient current, see current
transmembrane potential, see potential
traveling wave, 471
type of excitability, see excitability
unstable equilibrium, 62
unstable manifold, see manifold
up-state, 316
van der Pol, see model
vector fieldplanar, 89
velocity field, see vector field
voltage sag, 259, 284, 314
voltage-clamp, 30
voltage-gated channels, 33
wave, 471
weak coupling, 458
XPP, 6, 24

503