Impact of Adaptation Currents on Synchronization of Coupled Exponential Integrate-and-Fire Neurons

The ability of spiking neurons to synchronize their activity in a network depends on the response behavior of these neurons as quantified by the phase response curve (PRC) and on coupling properties. The PRC characterizes the effects of transient inputs on spike timing and can be measured experimentally. Here we use the adaptive exponential integrate-and-fire (aEIF) neuron model to determine how subthreshold and spike-triggered slow adaptation currents shape the PRC. Based on that, we predict how synchrony and phase locked states of coupled neurons change in presence of synaptic delays and unequal coupling strengths. We find that increased subthreshold adaptation currents cause a transition of the PRC from only phase advances to phase advances and delays in response to excitatory perturbations. Increased spike-triggered adaptation currents on the other hand predominantly skew the PRC to the right. Both adaptation induced changes of the PRC are modulated by spike frequency, being more prominent at lower frequencies. Applying phase reduction theory, we show that subthreshold adaptation stabilizes synchrony for pairs of coupled excitatory neurons, while spike-triggered adaptation causes locking with a small phase difference, as long as synaptic heterogeneities are negligible. For inhibitory pairs synchrony is stable and robust against conduction delays, and adaptation can mediate bistability of in-phase and anti-phase locking. We further demonstrate that stable synchrony and bistable in/anti-phase locking of pairs carry over to synchronization and clustering of larger networks. The effects of adaptation in aEIF neurons on PRCs and network dynamics qualitatively reflect those of biophysical adaptation currents in detailed Hodgkin-Huxley-based neurons, which underscores the utility of the aEIF model for investigating the dynamical behavior of networks. Our results suggest neuronal spike frequency adaptation as a mechanism synchronizing low frequency oscillations in local excitatory networks, but indicate that inhibition rather than excitation generates coherent rhythms at higher frequencies.

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title: Neurophysiological and computational principles of cortical rhythms in cognition.
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title: Interactions between the Superior Temporal Sulcus and Auditory Cortex Mediate Dynamic Face/Voice Integration in Rhesus Monkeys
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title: Modulation of Oscillatory Neuronal Synchronization by Selective Visual Attention
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title: Cognitive functions of gamma-band activity: memory match and utilization
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title: Matching storage and recall: hippocampal spike timing–dependent plasticity and phase response curves
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title: Pathological synchronization in Parkinson's disease: networks, models and treatments
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title: Abnormal neural oscillations and synchrony in schizophrenia
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title: High-frequency oscillations mirror disease activity in patients with epilepsy
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title: Phase-response curves and synchronized neural networks
abstract: We review the principal assumptions underlying the application of phase-response curves (PRCs) to synchronization in neuronal networks. The PRC measures how much a given synaptic input perturbs spike timing in a neural oscillator. Among other applications, PRCs make explicit predictions about whether a given network of interconnected neurons will synchronize, as is often observed in cortical structures. Regarding the assumptions of the PRC theory, we conclude: (i) The assumption of noise-tolerant cellular oscillations at or near the network frequency holds in some but not all cases. (ii) Reduced models for PRC-based analysis can be formally related to more realistic models. (iii) Spike-rate adaptation limits PRC-based analysis but does not invalidate it. (iv) The dependence of PRCs on synaptic location emphasizes the importance of improving methods of synaptic stimulation. (v) New methods can distinguish between oscillations that derive from mutual connections and those arising from common drive. (vi) It is helpful to assume linear summation of effects of synaptic inputs; experiments with trains of inputs call this assumption into question. (vii) Relatively subtle changes in network structure can invalidate PRC-based predictions. (viii) Heterogeneity in the preferred frequencies of component neurons does not invalidate PRC analysis, but can annihilate synchronous activity.

title: Learning of Phase Lags in Coupled Neural Oscillators
abstract: If an oscillating neural circuit is forced by another such circuit via a composite signal, the phase lag induced by the forcing can be changed by changing the relative strengths of components of the coupling. We consider such circuits, with the forced and forcing oscillators receiving signals with some given phase lag. We show how such signals can be transformed into an algorithm that yields connection strengths needed to produce that lag. The algorithm reduces the problem of producing a given phase lag to one of producing a kind of synchrony with a teaching signal; the algorithm can be interpreted as maximizing the correlation between voltages of a cell and the teaching signal. We apply these ideas to regulation of phase lags in chains of oscillators associated with undulatory locomotion.

title: Synchronization in hybrid neuronal networks of the hippocampal formation.
abstract: Understanding the mechanistic bases of neuronal synchronization is a current challenge in quantitative neuroscience. We studied this problem in two putative cellular pacemakers of the mammalian hippocampal theta rhythm: glutamatergic stellate cells (SCs) of the medial entorhinal cortex and GABAergic oriens-lacunosum-molecular (O-LM) interneurons of hippocampal region CA1. We used two experimental methods. First, we measured changes in spike timing induced by artificial synaptic inputs applied to individual neurons. We then measured responses of free-running hybrid neuronal networks, consisting of biological neurons coupled (via dynamic clamp) to biological or virtual counterparts. Results from the single-cell experiments predicted network behaviors well and are compatible with previous model-based predictions of how specific membrane mechanisms give rise to empirically measured synchronization behavior. Both cell types phase lock stably when connected via homogeneous excitatory-excitatory (E-E) or inhibitory-inhibitory (I-I) connections. Phase-locked firing is consistently synchronous for either cell type with E-E connections and nearly anti-synchronous with I-I connections. With heterogeneous connections (e.g., excitatory-inhibitory, as might be expected if members of a given population had heterogeneous connections involving intermediate interneurons), networks often settled into phase locking that was either stable or unstable, depending on the order of firing of the two cells in the hybrid network. Our results imply that excitatory SCs, but not inhibitory O-LM interneurons, are capable of synchronizing in phase via monosynaptic mutual connections of the biologically appropriate polarity. Results are largely independent of synaptic strength and synaptic kinetics, implying that our conclusions are robust and largely unaffected by synaptic plasticity.

title: Spike Frequency Adaptation Affects the Synchronization Properties of Networks of Cortical Oscillators
abstract: Oscillations in many regions of the cortex have common temporal characteristics with dominant frequencies centered around the 40 Hz (gamma) frequency range and the 510 Hz (theta) frequency range. Experimental results also reveal spatially synchronous oscillations, which are stimulus dependent (Gray&Singer, 1987;Gray, Knig, Engel, & Singer, 1989; Engel, Knig, Kreiter, Schillen, & Singer, 1992). This rhythmic activity suggests that the coherence of neural populations is a crucial feature of cortical dynamics (Gray, 1994). Using both simulations and a theoretical coupled oscillator approach, we demonstrate that the spike frequency adaptation seen in many pyramidal cells plays a subtle but important role in the dynamics of cortical networks. Without adaptation, excitatory connections among model pyramidal cells are desynchronizing. However, the slow processes associated with adaptation encourage stable synchronous behavior.

title: Phase Response Curves to Measure Ion Channel Effects on Neurons
abstract: In this chapter, we explore how different common ionic currents change the shape of the PRC using several well-known conductance-based models. By also studying the simplest possible network – two pulse coupled neural oscillators – we can use the shape of the PRC to quantify the degree of synchrony in such a pair. We find a number of qualitative effects that include changing the sign of the PRC during certain phases and altering the skew of the PRC. Using bifurcation theory, we trace these effects back to the underlying dynamics that govern the onset of repetitive firing as the neuron is depolarized by a constant current. The key bifurcation involves the so-called Takens-Bogdanov bifurcation. We show that many of the shape changes of the PRC can be understood by examining the dynamics of the neuron near this bifurcation.

title: Comparative electrophysiology of pyramidal and sparsely spiny stellate neurons of the neocortex.
abstract: Slices of sensorimotor and anterior cingulate cortex from guinea pigs were maintained in vitro and bathed in a normal physiological medium. Electrophysiological properties of neurons were assessed with intracellular recording techniques. Some neurons were identified morphologically by intracellular injection of the fluorescent dye Lucifer yellow CH. Three distinct neuronal classes of electrophysiological behavior were observed; these were termed regular spiking, bursting, and fast spiking. The physiological properties of neurons from sensorimotor and anterior cingulate areas did not differ significantly. Regular-spiking cells were characterized by action potentials with a mean duration of 0.80 ms at one-half amplitude, a ratio of maximum rate of spike rise to maximum rate of fall of 4.12, and a prominent afterhyperpolarization following a train of spikes. The primary slope of initial spike frequency versus injected current intensity was 241 Hz/nA. During prolonged suprathreshold current pulses the frequency of firing adapted strongly. When local synaptic pathways were activated, all cells were transiently excited and then strongly inhibited. Bursting cells were distinguished by their ability to generate endogenous, all-or-none bursts of three to five action potentials. Their properties were otherwise very similar to regular-spiking cells. The ability to generate a burst was eliminated when the membrane was depolarized to near the firing threshold with tonic current. By contrast, hyperpolarization of regular-spiking (i.e., nonbursting) cells did not uncover latent bursting tendencies. The action potentials of fast-spiking cells were much briefer (mean of 0.32 ms) than those of the other cell types.(ABSTRACT TRUNCATED AT 250 WORDS)

title: Intrinsic firing patterns of diverse neocortical neurons
abstract: Neurons of the neocortex differ dramatically in the patterns of action potentials they generate in response to current steps. Regular-spiking cells adapt strongly during maintained stimuli, whereas fast-spiking cells can sustain very high firing frequencies with little or no adaptation. Intrinsically bursting cells generate clusters of spikes (bursts), either singly or repetitively. These physiological distinctions have morphological correlates. RS and IB cells can be either pyramidal neurons or spiny stellate cells, and thus constitute the excitatory cells of the cortex. FS cells are smooth or sparsely spiny non-pyramidal cells, and are likely to be GABAergic inhibitory interneurons. The different firing properties of neurons in neocortex contribute significantly to its network behavior.

title: Multiple time scales of temporal response in pyramidal and fast spiking cortical neurons.
abstract: Neural dynamic processes correlated over several time scales are found in vivo, in stimulus-evoked as well as spontaneous activity, and are thought to affect the way sensory stimulation is processed. Despite their potential computational consequences, a systematic description of the presence of multiple time scales in single cortical neurons is lacking. In this study, we injected fast spiking and pyramidal (PYR) neurons in vitro with long-lasting episodes of step-like and noisy, in-vivo-like current. Several processes shaped the time course of the instantaneous spike frequency, which could be reduced to a small number (1-4) of phenomenological mechanisms, either reducing (adapting) or increasing (facilitating) the neuron's firing rate over time. The different adaptation/facilitation processes cover a wide range of time scales, ranging from initial adaptation (<10 ms, PYR neurons only), to fast adaptation (<300 ms), early facilitation (0.5-1 s, PYR only), and slow (or late) adaptation (order of seconds). These processes are characterized by broad distributions of their magnitudes and time constants across cells, showing that multiple time scales are at play in cortical neurons, even in response to stationary stimuli and in the presence of input fluctuations. These processes might be part of a cascade of processes responsible for the power-law behavior of adaptation observed in several preparations, and may have far-reaching computational consequences that have been recently described.

title: The Effects of Spike Frequency Adaptation and Negative Feedback on the Synchronization of Neural Oscillators
abstract: There are several different biophysical mechanisms for spike frequency adaptation observed in recordings from cortical neurons. The two most commonly used in modeling studies are a calcium-dependent potassium current Iahp and a slow voltage-dependent potassium current, Im. We show that both of these have strong effects on the synchronization properties of excitatorily coupled neurons. Furthermore, we show that the reasons for these effects are different. We show through an analysis of some standard models, that the M-current adaptation alters the mechanism for repetitive firing, while the after hyperpolarization adaptation works via shunting the incoming synapses. This latter mechanism applies with a network that has recurrent inhibition. The shunting behavior is captured in a simple two-variable reduced model that arises near certain types of bifurcations. A one-dimensional map is derived from the simplified model.

title: The effects of cholinergic neuromodulation on neuronal phase-response curves of modeled cortical neurons
abstract: The response of an oscillator to perturbations is described by its phase-response curve (PRC), which is related to the type of bifurcation leading from rest to tonic spiking. In a recent experimental study, we have shown that the type of PRC in cortical pyramidal neurons can be switched by cholinergic neuromodulation from type II (biphasic) to type I (monophasic). We explored how intrinsic mechanisms affected by acetylcholine influence the PRC using three different types of neuronal models: a theta neuron, single-compartment neurons and a multi-compartment neuron. In all of these models a decrease in the amount of a spike-frequency adaptation current was a necessary and sufficient condition for the shape of the PRC to change from biphasic (type II) to purely positive (type I).

title: Voltage clamp analysis of cholinergic action in the hippocampus
abstract: A slow muscarinic EPSP, accompanied by an increase in membrane input resistance, can be elicited in hippocampal CA1 pyramidal cells in vitro by electrical stimulation of cholinergic afferents in the slice preparation. Associated with the slow EPSP is a blockade of calcium- activated potassium afterhyperpolarizations (AHPs) (Cole and Nicoll, 1984a). In this study a single-electrode voltage clamp was used to examine the currents affected by activation of muscarinic receptors, using either bath application of carbachol or electrical stimulation of the cholinergic afferents. The 3 main findings of this study are that (1) of the 2 calcium-activated potassium currents (termed IAHP and IC) in hippocampal pyramidal cells, only IAHP is sensitive to carbachol; (2) IAHP is approximately 10-fold more sensitive to carbachol than is another muscarine-sensitive current, IM; and (3) neither blockade of IAHP nor of IM can account for the production of the slow EPSP. Rather, the slow EPSP appears to be generated by the blockade of a nonvoltage- dependent, resting potassium current. We propose that the muscarinic blockade of IAHP, which largely accounts for spike frequency adaptation, is primarily involved in enhancing action potential discharge to depolarizing stimuli, while the slow EPSP acts directly to cause action potential discharge.

title: Cholinergic Neuromodulation Changes Phase Response Curve Shape and Type in Cortical Pyramidal Neurons
abstract: Spike generation in cortical neurons depends on the interplay between diverse intrinsic conductances. The phase response curve (PRC) is a measure of the spike time shift caused by perturbations of the membrane potential as a function of the phase of the spike cycle of a neuron. Near the rheobase, purely positive (type I) phase-response curves are associated with an onset of repetitive firing through a saddle-node bifurcation, whereas biphasic (type II) phase-response curves point towards a transition based on a Hopf-Andronov bifurcation. In recordings from layer 2/3 pyramidal neurons in cortical slices, cholinergic action, consistent with down-regulation of slow voltage-dependent potassium currents such as the M-current, switched the PRC from type II to type I. This is the first report showing that cholinergic neuromodulation may cause a qualitative switch in the PRCs type implying a change in the fundamental dynamical mechanism of spike generation.

title: From subthreshold to firing-rate resonance.
abstract: Many types of neurons exhibit subthreshold resonance. However, little is known about whether this frequency preference influences spike emission. Here, the link between subthreshold resonance and firing rate is examined in the framework of conductance-based models. A classification of the subthreshold properties of a general class of neurons is first provided. In particular, a class of neurons is identified in which the input impedance exhibits a suppression at a nonzero low frequency as well as a peak at higher frequency. The analysis is then extended to the effect of subthreshold resonance on the dynamics of the firing rate. The considered input current comprises a background noise term, mimicking the massive synaptic bombardment in vivo. Of interest is the modulatory effect an additional weak oscillating current has on the instantaneous firing rate. When the noise is weak and firing regular, the frequency most preferentially modulated is the firing rate itself. Conversely, when the noise is strong and firing irregular, the modulation is strongest at the subthreshold resonance frequency. These results are demonstrated for two specific conductance-based models and for a generalization of the integrate-and-fire model that captures subthreshold resonance. They suggest that resonant neurons are able to communicate their frequency preference to postsynaptic targets when the level of noise is comparable to that prevailing in vivo.

title: Adaptive exponential integrate-and-fire model as an effective description of neuronal activity.
abstract: We introduce a two-dimensional integrate-and-fire model that combines an exponential spike mechanism with an adaptation equation, based on recent theoretical findings. We describe a systematic method to estimate its parameters with simple electrophysiological protocols (current-clamp injection of pulses and ramps) and apply it to a detailed conductance-based model of a regular spiking neuron. Our simple model predicts correctly the timing of 96% of the spikes (+/-2 ms) of the detailed model in response to injection of noisy synaptic conductances. The model is especially reliable in high-conductance states, typical of cortical activity in vivo, in which intrinsic conductances were found to have a reduced role in shaping spike trains. These results are promising because this simple model has enough expressive power to reproduce qualitatively several electrophysiological classes described in vitro.

title: Simple model of spiking neurons
abstract: A model is presented that reproduces spiking and bursting behavior of known types of cortical neurons. The model combines the biologically plausibility of Hodgkin-Huxley-type dynamics and the computational efficiency of integrate-and-fire neurons. Using this model, one can simulate tens of thousands of spiking cortical neurons in real time (1 ms resolution) using a desktop PC.

title: Which model to use for cortical spiking neurons?
abstract: We discuss the biological plausibility and computational efficiency of some of the most useful models of spiking and bursting neurons. We compare their applicability to large-scale simulations of cortical neural networks.

title: Dynamics and bifurcations of the adaptive exponential integrate-and-fire model
abstract: Recently, several two-dimensional spiking neuron models have been introduced, with the aim of reproducing the diversity of electrophysiological features displayed by real neurons while keeping a simple model, for simulation and analysis purposes. Among these models, the adaptive integrate-and-fire model is physiologically relevant in that its parameters can be easily related to physiological quantities. The interaction of the differential equations with the reset results in a rich and complex dynamical structure. We relate the subthreshold features of the model to the dynamical properties of the differential system and the spike patterns to the properties of a Poincaré map defined by the sequence of spikes. We find a complex bifurcation structure which has a direct interpretation in terms of spike trains. For some parameter values, spike patterns are chaotic.

title: Firing patterns in the adaptive exponential integrate-and-fire model
abstract: For simulations of large spiking neuron networks, an accurate, simple and versatile single-neuron modeling framework is required. Here we explore the versatility of a simple two-equation model: the adaptive exponential integrate-and-fire neuron. We show that this model generates multiple firing patterns depending on the choice of parameter values, and present a phase diagram describing the transition from one firing type to another. We give an analytical criterion to distinguish between continuous adaption, initial bursting, regular bursting and two types of tonic spiking. Also, we report that the deterministic model is capable of producing irregular spiking when stimulated with constant current, indicating low-dimensional chaos. Lastly, the simple model is fitted to real experiments of cortical neurons under step current stimulation. The results provide support for the suitability of simple models such as the adaptive exponential integrate-and-fire neuron for large network simulations.

title: Large-scale model of mammalian thalamocortical systems
abstract: The understanding of the structural and dynamic complexity of mammalian brains is greatly facilitated by computer simulations. We present here a detailed large-scale thalamocortical model based on experimental measures in several mammalian species. The model spans three anatomical scales. (i) It is based on global (white-matter) thalamocortical anatomy obtained by means of diffusion tensor imaging (DTI) of a human brain. (ii) It includes multiple thalamic nuclei and six-layered cortical microcircuitry based on in vitro labeling and three-dimensional reconstruction of single neurons of cat visual cortex. (iii) It has 22 basic types of neurons with appropriate laminar distribution of their branching dendritic trees. The model simulates one million multicompartmental spiking neurons calibrated to reproduce known types of responses recorded in vitro in rats. It has almost half a billion synapses with appropriate receptor kinetics, short-term plasticity, and long-term dendritic spike-timing-dependent synaptic plasticity (dendritic STDP). The model exhibits behavioral regimes of normal brain activity that were not explicitly built-in but emerged spontaneously as the result of interactions among anatomical and dynamic processes. We describe spontaneous activity, sensitivity to changes in individual neurons, emergence of waves and rhythms, and functional connectivity on different scales.

title: Self-sustained asynchronous irregular states and Up–Down states in thalamic, cortical and thalamocortical networks of nonlinear integrate-and-fire neurons
abstract: Randomly-connected networks of integrate-and-fire (IF) neurons are known to display asynchronous irregular (AI) activity states, which resemble the discharge activity recorded in the cerebral cortex of awake animals. However, it is not clear whether such activity states are specific to simple IF models, or if they also exist in networks where neurons are endowed with complex intrinsic properties similar to electrophysiological measurements. Here, we investigate the occurrence of AI states in networks of nonlinear IF neurons, such as the adaptive exponential IF (Brette-Gerstner-Izhikevich) model. This model can display intrinsic properties such as low-threshold spike (LTS), regular spiking (RS) or fast-spiking (FS). We successively investigate the oscillatory and AI dynamics of thalamic, cortical and thalamocortical networks using such models. AI states can be found in each case, sometimes with surprisingly small network size of the order of a few tens of neurons. We show that the presence of LTS neurons in cortex or in thalamus, explains the robust emergence of AI states for relatively small network sizes. Finally, we investigate the role of spike-frequency adaptation (SFA). In cortical networks with strong SFA in RS cells, the AI state is transient, but when SFA is reduced, AI states can be self-sustained for long times. In thalamocortical networks, AI states are found when the cortex is itself in an AI state, but with strong SFA, the thalamocortical network displays Up and Down state transitions, similar to intracellular recordings during slow-wave sleep or anesthesia. Self-sustained Up and Down states could also be generated by two-layer cortical networks with LTS cells. These models suggest that intrinsic properties such as adaptation and low-threshold bursting activity are crucial for the genesis and control of AI states in thalamocortical networks.

title: A comprehensive workflow for general-purpose neural modeling with highly configurable neuromorphic hardware systems
abstract: In this article, we present a methodological framework that meets novel requirements emerging from upcoming types of accelerated and highly configurable neuromorphic hardware systems. We describe in detail a device with 45 million programmable and dynamic synapses that is currently under development, and we sketch the conceptual challenges that arise from taking this platform into operation. More specifically, we aim at the establishment of this neuromorphic system as a flexible and neuroscientifically valuable modeling tool that can be used by non-hardware experts. We consider various functional aspects to be crucial for this purpose, and we introduce a consistent workflow with detailed descriptions of all involved modules that implement the suggested steps: The integration of the hardware interface into the simulator-independent model description language PyNN; a fully automated translation between the PyNN domain and appropriate hardware configurations; an executable specification of the future neuromorphic system that can be seamlessly integrated into this biology-to-hardware mapping process as a test bench for all software layers and possible hardware design modifications; an evaluation scheme that deploys models from a dedicated benchmark library, compares the results generated by virtual or prototype hardware devices with reference software simulations and analyzes the differences. The integration of these components into one hardware–software workflow provides an ecosystem for ongoing preparative studies that support the hardware design process and represents the basis for the maturity of the model-to-hardware mapping software. The functionality and flexibility of the latter is proven with a variety of experimental results.

title: Synchrony of Neuronal Oscillations Controlled by GABAergic Reversal Potentials
abstract: GABAergic synapse reversal potential is controlled by the concentration of chloride. This concentration can change significantly during development and as a function of neuronal activity. Thus, GABA inhibition can be hyperpolarizing, shunting, or partially depolarizing. Previous results pinpointed the conditions under which hyperpolarizing inhibition (or depolarizing excitation) can lead to synchrony of neural oscillators. Here we examine the role of the GABAergic reversal potential in generation of synchronous oscillations in circuits of neural oscillators. Using weakly coupled oscillator analysis, we show when shunting and partially depolarizing inhibition can produce synchrony, asynchrony, and coexistence of the two. In particular, we show that this depends critically on such factors as the firing rate, the speed of the synapse, spike frequency adaptation, and, most important, the dynamics of spike generation (type I versus type II). We back up our analysis with simulations of small circuits of conductance-based neurons, as well as large-scale networks of neural oscillators. The simulation results are compatible with the analysis: for example, when bistability is predicted analytically, the large-scale network shows clustered states.

title: A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances.
abstract: 1. We have developed a 19-compartment cable model of a guinea pig CA3 pyramidal neuron. Each compartment is allowed to contain six active ionic conductances: gNa, gCa, gK(DR) (where DR stands for delayed rectifier), gK(A), gK(AHP), and gK(C). THe conductance gCa is of the high-voltage activated type. The model kinetics for the first five of these conductances incorporate voltage-clamp data obtained from isolated hippocampal pyramidal neurons. The kinetics of gK(C) are based on data from bullfrog sympathetic neurons. The time constant for decay of submembrane calcium derives from optical imaging of Ca signals in Purkinje cell dendrites. 2. To construct the model from available voltage-clamp data, we first reproduced current-clamp records from a model isolated neuron (soma plus proximal dendrites). We next assumed that ionic channel kinetics in the dendrites were the same as in the soma. In accord with dendritic recordings and calcium-imaging data, we also assumed that significant gCa occurs in dendrites. We then attached sections of basilar and apical dendritic cable. By trial and error, we found a distribution (not necessarily unique) of ionic conductance densities that was consistent with current-clamp records from the soma and dendrites of whole neurons and from isolated apical dendrites. 3. The resulting model reproduces the Ca(2+)-dependent spike depolarizing afterpotential (DAP) recorded after a stimulus subthreshold for burst elicitation. 4. The model also reproduces the behavior of CA3 pyramidal neurons injected with increasing somatic depolarizing currents: low-frequency (0.3-1.0 Hz) rhythmic bursting for small currents, with burst frequency increasing with current magnitude; then more irregular bursts followed by afterhyperpolarizations (AHPs) interspersed with brief bursts without AHPs; and finally, rhythmic action potentials without bursts. 5. The model predicts the existence of still another firing pattern during tonic depolarizing dendritic stimulation: brief bursts at less than 1 to approximately 12 Hz, a pattern not observed during somatic stimulation. These bursts correspond to rhythmic dendritic calcium spikes. 6. The model CA3 pyramidal neuron can be made to resemble functionally a CA1 pyramidal neuron by increasing gK(DR) and decreasing dendritic gCa and gK(C). Specifically, after these alterations, tonic depolarization of the soma leads to adapting repetitive firing, whereas stimulation of the distal dendrites leads to bursting. 7. A critical set of parameters concerns the regulation of the pool of intracellular [Ca2+] that interacts with membrane channels (gK(C) and gK(AHP)), particularly in the dendrites.(ABSTRACT TRUNCATED AT 400

title: Brian: a simulator for spiking neural networks in Python
abstract: “Brian” is a new simulator for spiking neural networks, written in Python (http://brian. di.ens.fr). It is an intuitive and highly fl exible tool for rapidly developing new models, especially networks of single-compartment neurons. In addition to using standard types of neuron models, users can defi ne models by writing arbitrary differential equations in ordinary mathematical notation. Python scientifi c libraries can also be used for defi ning models and analysing data. Vectorisation techniques allow effi cient simulations despite the overheads of an interpreted language. Brian will be especially valuable for working on non-standard neuron models not easily covered by existing software, and as an alternative to using Matlab or C for simulations. With its easy and intuitive syntax, Brian is also very well suited for teaching computational neuroscience.

title: Gamma Oscillation by Synaptic Inhibition in a Hippocampal Interneuronal Network Model
abstract: Fast neuronal oscillations (gamma, 20–80 Hz) have been observed in the neocortex and hippocampus during behavioral arousal. Using computer simulations, we investigated the hypothesis that such rhythmic activity can emerge in a random network of interconnected GABAergic fast-spiking interneurons. Specific conditions for the population synchronization, on properties of single cells and the circuit, were identified. These include the following: (1) that the amplitude of spike afterhyperpolarization be above the GABAA synaptic reversal potential; (2) that the ratio between the synaptic decay time constant and the oscillation period be sufficiently large; (3) that the effects of heterogeneities be modest because of a steep frequency–current relationship of fast-spiking neurons. Furthermore, using a population coherence measure, based on coincident firings of neural pairs, it is demonstrated that large-scale network synchronization requires a critical (minimal) average number of synaptic contacts per cell, which is not sensitive to the network size. By changing the GABAA synaptic maximal conductance, synaptic decay time constant, or the mean external excitatory drive to the network, the neuronal firing frequencies were gradually and monotonically varied. By contrast, the network synchronization was found to be high only within a frequency band coinciding with the gamma (20–80 Hz) range. We conclude that the GABAA synaptic transmission provides a suitable mechanism for synchronized gamma oscillations in a sparsely connected network of fast-spiking interneurons. In turn, the interneuronal network can presumably maintain subthreshold oscillations in principal cell populations and serve to synchronize discharges of spatially distributed neurons.

title: Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients
abstract: We apply the concept of phase synchronization of chaotic and/or noisy systems and the statistical distribution of the relative instantaneous phases to electroencephalograms (EEGs) recorded from patients with temporal lobe epilepsy. Using the mean phase coherence as a statistical measure for phase synchronization, we observe characteristic spatial and temporal shifts in synchronization that appear to be strongly related to pathological activity. In particular, we observe distinct differences in the degree of synchronization between recordings from seizure-free intervals and those before an impending seizure, indicating an altered state of brain dynamics prior to seizure activity.

title: Cellularly-Driven Differences in Network Synchronization Propensity Are Differentially Modulated by Firing Frequency
abstract: Spatiotemporal pattern formation in neuronal networks depends on the interplay between cellular and network synchronization properties. The neuronal phase response curve (PRC) is an experimentally obtainable measure that characterizes the cellular response to small perturbations, and can serve as an indicator of cellular propensity for synchronization. Two broad classes of PRCs have been identified for neurons: Type I, in which small excitatory perturbations induce only advances in firing, and Type II, in which small excitatory perturbations can induce both advances and delays in firing. Interestingly, neuronal PRCs are usually attenuated with increased spiking frequency, and Type II PRCs typically exhibit a greater attenuation of the phase delay region than of the phase advance region. We found that this phenomenon arises from an interplay between the time constants of active ionic currents and the interspike interval. As a result, excitatory networks consisting of neurons with Type I PRCs responded very differently to frequency modulation compared to excitatory networks composed of neurons with Type II PRCs. Specifically, increased frequency induced a sharp decrease in synchrony of networks of Type II neurons, while frequency increases only minimally affected synchrony in networks of Type I neurons. These results are demonstrated in networks in which both types of neurons were modeled generically with the Morris-Lecar model, as well as in networks consisting of Hodgkin-Huxley-based model cortical pyramidal cells in which simulated effects of acetylcholine changed PRC type. These results are robust to different network structures, synaptic strengths and modes of driving neuronal activity, and they indicate that Type I and Type II excitatory networks may display two distinct modes of processing information.

title: On the Phase Reduction and Response Dynamics of Neural Oscillator Populations
abstract: We undertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli. Recalling and extending results from the literature, we compute phase response curves (PRCs) valid near bifurcations to periodic firing for Hindmarsh-Rose, Hodgkin-Huxley, Fitz Hugh-Nagumo, and Morris-Lecar models, encompassing the four generic (codimension one) bifurcations. Phase density equations are then used to analyze the role of the bifurcation, and the resulting PRC, in responses to stimuli. In particular, we explore the interplay among stimulus duration, baseline firing frequency, and population-level response patterns. We interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions.

title: Mathematical foundations of neuroscience
abstract: The Hodgkin-Huxley Equations.- Dendrites.- Dynamics.- The Variety of Channels.- Bursting Oscillations.- Propagating Action Potentials.- Synaptic Channels.- Neural Oscillators: Weak  Fast/Slow Analysis.- Noise.- Firing Rate Models.- Spatially Distributed Networks.

title: The Theory of Weakly Coupled Oscillators
abstract: This chapter focuses on the application of phase response curves (PRCs) in predicting the phase locking behavior in networks of periodically oscillating neurons using the theory of weakly coupled oscillators. The theory of weakly coupled oscillators can be used to predict phase-locking in neuronal networks with any form of coupling. As the name suggests, the coupling between cells must be sufficiently weak for these predictions to be quantitatively accurate. This implies that the coupling can only have small effects on neuronal dynamics over any given cycle. However, these small effects can accumulate over many cycles and lead to phase locking in the neuronal network. The theory of weak coupling allows one to reduce the dynamics of each neuron, which could be of very high dimension, to a single differential equation describing the phase of the neuron.

title: Weakly connected neural networks
abstract: 1 Introduction.- 2 Bifurcations in Neuron Dynamics.- 3 Neural Networks.- 4 Introduction to Canonical Models.- 5 Local Analysis of WCNNs.- 6 Local Analysis of Singularly Perturbed WCNNs.- 7 Local Analysis of Weakly Connected Maps.- 8 Saddle-Node on a Limit Cycle.- 9 Weakly Connected Oscillators.- 10 Multiple Andronov-Hopf Bifurcation.- 11 Multiple Cusp Bifurcation.- 12 Quasi-Static Bifurcations.- 13 Synaptic Organizations of the Brain.- References.

title: Calculation of Lyapunov exponents for dynamic systems with discontinuities
abstract: The model based algorithm for the calculation of the spectrum of Lyapunov exponents is generalized for nonlinear dynamical systems with discontinuities. As the main result, the required linearized equations have to be supplemented by certain transition conditions at the instants of discontinuities.

title: Impulsive differential equations
abstract: General description of impulsive differential systems linear systems stability of solutions periodic and almost periodic impulsive systems integral sets of impulsive systems optimal control in impulsive systems asymptotic study of oscillations in impulsive systems a periodic and almost periodic impulsive system.

title: Principles of Discontinuous Dynamical Systems
abstract: Description of the System with Fixed Moments of Impulses and its Solutions.- Stability and Periodic Solutions of Systems with Fixed Moments of Impulses.- Basics of Linear Systems.- Non-Autonomous Systems with Variable Moments of Impulses.- Differentiability Properties of Non-Autonomous Systems.- Periodic Solutions of Nonlinear Systems.- Discontinuous Dynamical Systems.- Perturbations and Hopf Bifurcation of a Discontinuous Limit Cycle.- Chaos and Shadowing.- Bibliography

title: MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs
abstract: MATCONT is a graphical MATLAB software package for the interactive numerical study of dynamical systems. It allows one to compute curves of equilibria, limit points, Hopf points, limit cycles, period doubling bifurcation points of limit cycles, and fold bifurcation points of limit cycles. All curves are computed by the same function that implements a prediction-correction continuation algorithm based on the Moore-Penrose matrix pseudo-inverse. The continuation of bifurcation points of equilibria and limit cycles is based on bordering methods and minimally extended systems. Hence no additional unknowns such as singular vectors and eigenvectors are used and no artificial sparsity in the systems is created. The sparsity of the discretized systems for the computation of limit cycles and their bifurcation points is exploited by using the standard Matlab sparse matrix methods. The MATLAB environment makes the standard MATLAB Ordinary Differential Equations (ODE) Suite interactively available and provides computational and visualization tools; it also eliminates the compilation stage and so makes installation straightforward. Compared to other packages such as AUTO and CONTENT, adding a new type of curves is easy in the MATLAB environment. We illustrate this by a detailed description of the limit point curve type.

title: Neurons to Transient Inputs Phase-Response Curves Give the Responses of
abstract: , March 15, 2006; 90 (6): 1925-1938.  Morita, K. Tsumoto and K. Aihara Bidirectional Modulation of Neuronal Responses by Depolarizing GABAergic Inputs [PDF]  [ , November 1, 2006; 18 (11): 2617-2650. Neural Comput. Whittington and N. J. Kopell D. D. Pervouchine, T. I. Netoff, H. G. Rotstein, J. A. White, M. O. Cunningham, M. A. Low-Dimensional Maps Encoding Dynamics in Entorhinal Cortex and Hippocampus

title: Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators
abstract: Gamma oscillations can synchronize with near zero phase lag over multiple cortical regions and between hemispheres, and between two distal sites in hippocampal slices. How synchronization can take place over long distances in a stable manner is considered an open question. The phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike, depending upon where in the cycle it is received. We use PRCs under the assumption of pulsatile coupling to derive existence and stability criteria for 1:1 phase-locking that arises via bidirectional pulse coupling of two limit cycle oscillators with a conduction delay of any duration for any 1:1 firing pattern. The coupling can be strong as long as the effect of one input dissipates before the next input is received. We show the form that the generic synchronous and anti-phase solutions take in a system of two identical, identically pulse-coupled oscillators with identical delays. The stability criterion has a simple form that depends only on the slopes of the PRCs at the phases at which inputs are received and on the number of cycles required to complete the delayed feedback loop. The number of cycles required to complete the delayed feedback loop depends upon both the value of the delay and the firing pattern. We successfully tested the predictions of our methods on networks of model neurons. The criteria can easily be extended to include the effect of an input on the cycle after the one in which it is received.

title: Effects of heterogeneity in synaptic conductance between weakly coupled identical neurons
abstract: A significant degree of heterogeneity in synaptic conductance is present in neuron to neuron connections. We study the dynamics of weakly coupled pairs of neurons with heterogeneities in synaptic conductance using Wang–Buzsaki and Hodgkin–Huxley model neurons which have Types I and II excitability, respectively. This type of heterogeneity breaks a symmetry in the bifurcation diagrams of equilibrium phase difference versus the synaptic rate constant when compared to the identical case. For weakly coupled neurons coupled with identical values of synaptic conductance a phase locked solution exists for all values of the synaptic rate constant, α. In particular, in-phase and anti-phase solutions are guaranteed to exist for all α. Heterogeneity in synaptic conductance results in regions where no phase locked solution exists and the general loss of the ubiquitous in-phase and anti-phase solutions of the identically coupled case. We explain these results through examination of interaction functions using the weak coupling approximation and an in-depth analysis of the underlying multiple cusp bifurcation structure of the systems of coupled neurons.

title: Synchronization induced by temporal delays in pulse-coupled oscillators.
abstract: We derive return maps and phase diagrams to identify mechanisms of synchronization of pulse-coupled oscillators and emphasize the importance of temporal delays and inhibitory coupling. Optimum synchronization between two oscillators is obtained for inhibitory coupling, where delays give rise to stable in-phase synchronization; those with excitatory coupling only synchronize with a phase lag. In large ensembles of globally coupled oscillators the delayed interaction leads to new collective phenomena like synchronization in multistable clusters of common phases for inhibitory coupling; for excitatory coupling a mechanism of emerging and decaying synchronized clusters prevails.

title: Delays and weakly coupled neuronal oscillators
abstract: We use weakly coupled oscillator theory to study the effects of delays on coupled systems of neuronal oscillators. We explore, first, simple pairs with constant delays and then examine the role of distributed delays as would occur in systems with dendritic branches or in networks where there is a distance-dependent conductance delay. In the latter, we use mean field theory to show the emergence of travelling waves and the loss of synchronization. Next, we consider phase models with stronger coupling and delays in the state variables. We show that they have a richer dynamics but one that is still similar to the weakly coupled case.

title: Synaptic mechanisms of synchronized gamma oscillations in inhibitory interneuron networks
abstract: Gamma frequency oscillations are thought to provide a temporal structure for information processing in the brain. They contribute to cognitive functions, such as memory formation and sensory processing, and are disturbed in some psychiatric disorders. Fast-spiking, parvalbumin-expressing, soma-inhibiting interneurons have a key role in the generation of these oscillations. Experimental analysis in the hippocampus and the neocortex reveals that synapses among these interneurons are highly specialized. Computational analysis further suggests that synaptic specialization turns interneuron networks into robust gamma frequency oscillators.

title: When inhibition not excitation synchronizes neural firing
abstract: Excitatory and inhibitory synaptic coupling can have counter-intuitive effects on the synchronization of neuronal firing. While it might appear that excitatory coupling would lead to synchronization, we show that frequently inhibition rather than excitation synchronizes firing. We study two identical neurons described by integrate-and-fire models, general phase-coupled models or the Hodgkin-Huxley model with mutual, non-instantaneous excitatory or inhibitory synapses between them. We find that if the rise time of the synapse is longer than the duration of an action potential, inhibition not excitation leads to synchronized firing.

title: Short Conduction Delays Cause Inhibition Rather than Excitation to Favor Synchrony in Hybrid Neuronal Networks of the Entorhinal Cortex
abstract: How stable synchrony in neuronal networks is sustained in the presence of conduction delays is an open question. The Dynamic Clamp was used to measure phase resetting curves (PRCs) for entorhinal cortical cells, and then to construct networks of two such neurons. PRCs were in general Type I (all advances or all delays) or weakly type II with a small region at early phases with the opposite type of resetting. We used previously developed theoretical methods based on PRCs under the assumption of pulsatile coupling to predict the delays that synchronize these hybrid circuits. For excitatory coupling, synchrony was predicted and observed only with no delay and for delays greater than half a network period that cause each neuron to receive an input late in its firing cycle and almost immediately fire an action potential. Synchronization for these long delays was surprisingly tight and robust to the noise and heterogeneity inherent in a biological system. In contrast to excitatory coupling, inhibitory coupling led to antiphase for no delay, very short delays and delays close to a network period, but to near-synchrony for a wide range of relatively short delays. PRC-based methods show that conduction delays can stabilize synchrony in several ways, including neutralizing a discontinuity introduced by strong inhibition, favoring synchrony in the case of noisy bistability, and avoiding an initial destabilizing region of a weakly type II PRC. PRCs can identify optimal conduction delays favoring synchronization at a given frequency, and also predict robustness to noise and heterogeneity.

title: Stability of two cluster solutions in pulse coupled networks of neural oscillators
abstract: Phase response curves (PRCs) have been widely used to study synchronization in neural circuits comprised of pacemaking neurons. They describe how the timing of the next spike in a given spontaneously firing neuron is affected by the phase at which an input from another neuron is received. Here we study two reciprocally coupled clusters of pulse coupled oscillatory neurons. The neurons within each cluster are presumed to be identical and identically pulse coupled, but not necessarily identical to those in the other cluster. We investigate a two cluster solution in which all oscillators are synchronized within each cluster, but in which the two clusters are phase locked at nonzero phase with each other. Intuitively, one might expect this solution to be stable only when synchrony within each isolated cluster is stable, but this is not the case. We prove rigorously the stability of the two cluster solution and show how reciprocal coupling can stabilize synchrony within clusters that cannot synchronize in isolation. These stability results for the two cluster solution suggest a mechanism by which reciprocal coupling between brain regions can induce local synchronization via the network feedback loop.

title: Low-Dimensional Maps Encoding Dynamics in Entorhinal Cortex and Hippocampus
abstract: Cells that produce intrinsic theta oscillations often contain the hyperpolarization-activated current Ih. In this article, we use models and dynamic clamp experiments to investigate the synchronization properties of two such cells (stellate cells of the entorhinal cortex and O-LM cells of the hippocampus) in networks with fast-spiking (FS) interneurons. The model we use for stellate cells and O-LM cells is the same, but the stellate cells are excitatory and the O-LM cells are inhibitory, with inhibitory postsynaptic potential considerably longer than those from FS interneurons. We use spike time response curve methods (STRC), expanding that technique to three-cell networks and giving two different ways in which the analysis of the three-cell network reduces to that of a two-cell network. We show that adding FS cells to a network of stellate cells can desynchronize the stellate cells, while adding them to a network of O-LM cells can synchronize the O-LM cells. These synchronization and desynchronization properties critically depend on Ih. The analysis of the deterministic system allows us to understand some effects of noise on the phase relationships in the stellate networks. The dynamic clamp experiments use biophysical stellate cells and in silico FS cells, with connections that mimic excitation or inhibition, the latter with decay times associated with FS cells or O-LM cells. The results obtained in the dynamic clamp experiments are in a good agreement with the analytical framework.

title: Spike frequency adaptation and neocortical rhythms.
abstract: Spike-frequency adaptation in neocortical pyramidal neurons was examined using the whole cell patch-clamp technique and a phenomenological model of neuronal activity. Noisy current was injected to reproduce the irregular firing typically observed under in vivo conditions. The response was quantified by computing the poststimulus histogram (PSTH). To simulate the spiking activity of a pyramidal neuron, we considered an integrate-and-fire model to which an adaptation current was added. A simplified model for the mean firing rate of an adapting neuron under noisy conditions is also presented. The mean firing rate model provides a good fit to both experimental and simulation PSTHs and may therefore be used to study the response characteristics of adapting neurons to various input currents. The models enable identification of the relevant parameters of adaptation that determine the shape of the PSTH and allow the computation of the response to any change in injected current. The results suggest that spike frequency adaptation determines a preferred frequency of stimulation for which the phase delay of a neuron's activity relative to an oscillatory input is zero. Simulations show that the preferred frequency of single neurons dictates the frequency of emergent population rhythms in large networks of adapting neurons. Adaptation could therefore be one of the crucial factors in setting the frequency of population rhythms in the neocortex.

title: Beyond Two-Cell Networks: Experimental Measurement of Neuronal Responses to Multiple Synaptic Inputs
abstract: Oscillations of large populations of neurons are thought to be important in the normal functioning of the brain. We have used phase response curve (PRC) methods to characterize the dynamics of single neurons and predict population dynamics. Our past experimental work was limited to special circumstances (e.g., 2-cell networks of periodically firing neurons). Here, we explore the feasibility of extending our methods to predict the synchronization properties of stellate cells (SCs) in the rat entorhinal cortex under broader conditions. In particular, we test the hypothesis that PRCs in SCs scale linearly with changes in synaptic amplitude, and measure how well responses to Poisson process-driven inputs can be predicted in terms of PRCs. Although we see nonlinear responses to excitatory and inhibitory inputs, we find that models based on weak coupling account for scaling and Poisson process-driven inputs reasonably accurately.