RISP

NCSP’16

International Workshop on
Nonlinear Circuits, Communications and Signal Processing

Honolulu, Hawaii, USA, March 6-9, 2016

Synchronization Phenomena between Triangle System and Two-Triangles System ofFitzHugh-Nagumo Neurons with Diffusive Coupling
Takato Sei, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {takato, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract
We study synchronization phenomena on FitzHugh-Nagumo(FHN) model. We investigate a three-neuronal system anda four-neuronal system. We focus on the coexistence of theperiodic solutions on the each system by changing the initial value and the coupling strength. Some periodic solutions
depend on the initial value. In addition, we investigate firing patterns on the both systems. The only one firing patternwhich exist on the four-neuronal system is found.
1. Introduction
Many chemical and biological phenomena can be modeled
in reaction-diffusion systems. The reaction-diffusion systems
have various dynamical behavior. Excitability is the most important characteristic and the foundation on numerous physical biological systems.
Creature’s periodic motion is controlled by the neural circuit called Central Pattern Generator (CPG). Neural oscillator
is what has modeled excitation and inhibitory mechanism between the neurons in the neural circuit. FHN model is oneof neural oscillator models. Concretely, for example, we canapply neural oscillators to robot motion control and nervous
disease. The network of neurons in the brain exhibits a subtle balance of dynamic chaos and self-organized order[1]. A
number of neurological disease are characterized by a disturbance of this balance. For instance, it is synchronized firingof electrical pluses of the neurons. Time-delayed feedback
control has recently been applied to suppress this undesired
synchrony.
Periodic neural firing is produced by CPG and has greatsignificance for the control of dynamic functions of thebody[2]. It is worth to investigate the mechanism of neuralnetworks which cause such a variety of periodic activities[3].
These complex firing patterns which include chaotic firings
are modeled through the effect of high dimensional dynamics
of an individual element, interaction of many neurons withspatial degrees of freedom, and time delayed coupling[4].
There are many studies for excitable elements under exter-

nal periodic stimuli, which show various behavior including
phase-locking and chaotic behavior. It has been also studies
for an ensemble of oscillatory or excitatory elements interacting each other in the context of synchronization and chaos[5].
Creatures have the neural circuit in their bodies. It is avery complex structure. For the reason, we would like toinvestigate more complex networks. However, we need toinvestigate fundamental networks. In this study, we investigate synchronization phenomena of a three-neuronal system
and a four-neuronal system on FHN model. We focus on twopoints. The one is the difference of synchronization phenomena by changing the initial value on the each system. The another one is the difference of periodic solutions including firing patterns between the three-neuronal system and the fourneuronal system. We observe the coexistence of the periodicsolutions. In addition to that, we find the only firing patternwhich exist on the four-neuronal system.
2. Fitzhugh-Nagumo model
In the following, we study the system which is modeledby the excitable FHN models. The FHN model is a twodimensional simplification of the Hodgkin-Huxley model of
spike generation in squid giant axons. This model was suggested by FitzHugh and the equivalent circuit was proposed
by Nagumo et al. FHN model is a very simple model for excitable system. The model contains the only two variables,
however, does not describe a specific biochemical reaction. It
can be used to describe neural and cardiac dynamics.
FHN model is given by the following equations:K∑dui= ui (ui − α)(1 − ui ) − vi +(uj − ui )dt
N(1)dvi= τ (ui − γvi )dt(2)i̸=jwhere uij is the activator, vij is the inhibitor, α, τ and γ areparameters, K is the coupling strength and N is the numberof elements. α, τ and γ are fixed at α = 0.01, τ = 0.001 and
γ = 0.0 because these values are used in previous work of apair of excitable FHN elements. The time series are generated- 731 -

by the fourth-order Runge-Kutta method with step size h =0.05.

N1
3. A pair of excitable neurons
As a previous study, there is a study which focuses on aminimal model that consists of repulsively coupled two excitable neurons. This model produces and sustains various
firing patterns. There are many types of sequence of firing.
The diffusive coupling with a negative coefficient has been
considered as the effect of phase-repulsive coupling on two
dimensional coupled FHN arrays. The periodic firings arefound in two neurons with excitatory and inhibitory synapticcouplings.
Five firing patterns have found on two neurons by changing the coupling strength K from −1 to 0. As an example, weshow time waveforms of two neurons in case of K = −0.012
in Fig. 1. We explain two-phase firing pattern in detail. AfterN1 excites, N2 excites soon. After these successive excitations, both neurons stay quiescent state for a while. On thenext successive excitations, N2 excites first. After that, N1excites soon. These states are repeated. Accordingly, we cansymbolize this firing pattern as N1 N2 − N2 N1 −. In the sameway, we can symbolize four other firing patterns as N1 N2 −,N1 N2 N1 − N2 N1 N2 −, N1 N2 − N1 N2 − N2 N1 − N2 N1 −and N1 N2 − N1 N2 − N1 N2 − N2 N1 − N2 N1 − N2 N1 −.
Here, we focus on the three-neuronal system and the fourneuronal system. In this study, we consider analysis of thesesystems by changing the initial value. We find that these systems show a various of firing patterns including chaotic firing.

t(a)N2t(b)

N1
N2t(c)

Figure 1: Time evolution of two neurons on FHN model for thecoupling strength K = −0.012. The solid lines of (a) and (b) correspond to time waveforms of N1 and N2 , respectively. The solidlines of (c) correspond to both neuronal time waveforms.

N1

4. Network Systems
We introduce the network systems used in this study. We
investigate two network systems. The three-neuronal system
is shown in Fig. 2. This system is a very simple system thata neuron is just connected to a system of two neuron as aring topology. The each neuron connects with the other neurons. We name this three-neuronal system triangle system.
The four-neuronal system is shown in Fig. 3. This system isa system that a neuron is connected to a triangle system. We
put the neuron on the place that N2 and the neuron contrast
with a side between N1 and N3 on the triangle system as asymmetry axis. This system has axial symmetry and include
two triangle. Therefore, we name this four-neuronal system
two-triangle system. The each neuron connects with the otherneurons, except that N2 does not connect with N4 .

N2

N3Figure 2: The triangle system.

N4

N1

N2N3Figure 3: The two-triangle system.

- 732 -

35. Numerical Experiments

Triangle system

We investigate synchronization phenomena on triangle
system and two-triangle system by changing the couplingstrength K from −1 to 0. We also focus on the coexistence ofthe periodic solutions by changing the initial value and various firing patterns. In this section, we consider the coexistence of the periodic solutions. We decide the conditions of
the initial value. We define some rules. We use the same absolute value on all neurons and investigate the systems by meansof five types of the absolute values. The absolute values are
0.25, 0.5, 1, 2 and 4. The negative sign is set to odd-numberedneurons. Likewise, the positive sign is set to even-numberedneurons.
In the beginning, we consider the triangle system. Figure 4 shows the number of solutions by changing the couplingstrength K from −1 to 0 on the triangle system. In this figure,“1” of solutions mean that the system has the same solution at
all five types of the initial value. “0” of solutions correspondto chaotic firing patterns. These patterns are observed in the−0.7 to −0.61 range. “2” of solutions are observed, whichshow the coexistence of the periodic solutions.
Secondly, we investigate the two-triangle system. Figure 5 shows the number of solutions by changing the coupling strength K from −1 to 0 on the two-triangle system.
“0” of solutions correspond to chaotic firing patterns likewise. These patterns are observed in the −0.98 to −0.65
range. By connecting a neuron to the triangle system, therange of chaotic firing patterns on the triangle system is larger
than that on the two-triangle system. On the two-triangle system, “2” of solutions are also observed, which show the coexistence of the periodic solutions. There are more cases whichhave the coexistence of the periodic solutions than on the triangle system.
5.2 Comparison of Firing Patterns

The number of solutions

5.1 Coexistence of Periodic Solutions
210-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Coupling strength

Figure 4: The number of solutions by changing the couplingstrength K from −1 to 0 on the triangle system.

If we look at types of firing patterns, there are fourtypes of firing patterns on the triangle system. We cansymbolize the four firing patterns on the triangle system
as N1 (N3 )N2 −, N2 N1 (N3 )−, N2 N1 (N3 ) − N2 N1 (N3 )−and N1 (N3 )N2 − N2 N1 (N3 )−. Here, N1 (N3 ) means that
N1 is perfectly synchronized with N3 . On the other hand,
there are six types of firing patterns on the two-triangle system. We can symbolize the six firing patterns on the twotriangle system as N1 (N3 )N2 (N4 )−, N2 (N4 )N1 (N3 )−,N1 (N3 )N2 (N4 ) − N1 (N3 )N2 (N4 )−, N2 (N4 )N1 (N3 ) −N2 (N4 )N1 (N3 )−, N1 (N3 )N2 (N4 ) − N2 (N4 )N1 (N3 )−and N1 (N3 )N2 (N4 )N1 (N3 ) − N2 (N4 )N1 (N3 )N2 (N4 )−.
N1 (N3 )N2 (N4 ) − N1 (N3 )N2 (N4 )− is the same with
N2 (N4 )N1 (N3 ) − N2 (N4 )N1 (N3 )− as a type. However,N1 (N3 )N2 (N4 )N1 (N3 ) − N2 (N4 )N1 (N3 )N2 (N4 )− is observed for the first time by connecting a neuron to the triangle
system. For example, this firing pattern is observed in case ofK = −0.990. The observed time waveforms are shown inFig. 7. Time waveforms N1 and N3 are synchronized at inphase with N2 and N4 . On the first successive excitations,the order of firing is N1 N2 N1 −. On the next successive excitations, the order of firing is N2 N1 N2 −. These states arerepeated. A whole new firing pattern which is not found on
the triangle system is observed by connecting a neuron to thetriangle system. Three successive firings on one successive
excitations were not observed until on the two-triangle system.

In this section, we focus on firing patterns. First of all, weinvestigate the periodic solutions on triangle system and twotriangle system by changing the coupling strength K from
−1 to 0. In almost every cases, the periodic solutions do notdepend on the initial value on the both systems. That is why
we consider the only case that the absolute value of the initialvalue is 0.25 as an example. Figures 6 (a) and 6 (b) show theperiodic solutions by changing the coupling strength K from
−1 to 0 by one hundredth on the triangle system and on thetwo-triangle system, respectively. There are the existence of 6. Conclusions
three periodic solutions in Fig. 6 (b) compared with Fig. 6 (a).
We investigated synchronization phenomena of the triangle
That is clear at a glance. The triangle system is different fromthe two-triangle system in that it has the existence of three system and the two-triangle system on FHN model by changing the initial value and the coupling strength. Some solutions
periodic solutions.

- 733 -3Two-triangles system

The number of solutions

N12t(a)

t(b)

t(c)

N21N1
N0-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0Coupling strength

Figure 5: The number of solutions by changing the couplingstrength K from −1 to 0 on the two-triangle system.

Figure 7: Time evolution of the two-triangle system on FHN modelfor the coupling strength K = −0.990. The solid lines of (a) and(b) correspond to time waveforms of N1 (N3 ) and N2 (N4 ), respectively. The solid lines of (c) correspond to all neuronal time waveforms.

tions as with chaotic firing pattern on the two-triangle systemthan those on the triangle system. As well as on a pair of
excitable neurons, some firing patterns were observed on thetriangle system and the two-triangle system. A whole new
firing pattern was observed by connecting a neuron to the triangle system. The firing pattern has three periodic solutions.
(a) The triangle system.

References[1] P. Hovel, M. A. Dahlem and E. Scholl, “Control ofSynchronization in Coupled Neural Systems by TimeDelayed Feedback”, International Journal of Bifurcation and Chaos, Vol. 20, No. 3, 813-825, 2010.
[2] T. Yanagita, T. Ichinomiya and Y. Oyama, “Pair of Excitable FitzHugh-Nagumo Elements: Synchronization,Multistability, and Chaos”, Phys. Rev. E 72, 056218,2005.
[3] H. Fujii and I. Tsuda, “Itinerant Dynamics of Class I*
Neurons Coupled by Gap Junctions”, Lect. Notes Comput. Sci. 3146, 140, 2004.

(b) The two-triangle system.

Figure 6: Periodic solutions by changing the coupling strength K
from −1 to 0.

depended on the initial value, and others did not. There weremore cases which have the coexistence of the periodic solu-[4] K. Tateno, H. Tomonari, H. Hayashi and S. Ishizuka,“Phase Dependent Transition between Multistable
States in A Neural Network with Reciprocal Inhibition”,
Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 1559, 2004.
[5] D. Terman, N. Kopell and A. Bose, “Dynamics of two
mutually coupled slow inhibitory neurons”, Physica D,241, 1998.

- 734 -