IEEE Workshop on Nonlinear Circuit Networks
December 11-12, 2015

Synchronization Phenomena between Triangle
System and Two-Triangle System ofFitzHugh-Nagumo Neurons
Takato Sei, Yoko Uwate and Yoshifumi Nishio
Dept. of Electricaland 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 FitzHughNagumo (FHN) model. We investigate a three-neuronal system
and a four-neuronal system. We focus on the coexistence of theperiodic solution on each system by changing the initial value
and the coupling strength. Some periodic solution depend on theinitial value. In addition, we investigate firing patterns on bothsystems. The only one firing pattern which exist on the fourneuronal system is found.

I. I NTRODUCTION
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 physicalbiological systems.
Creature’s periodic motion is controlled by the neural circuit
called Central Pattern Generator (CPG). Neural oscillator
is what have 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 instence, it is synchronized firingof electrical pluses of the neurons. Time-delayed feedbackcontrol have recently been applied to suppress this undesiredsynchrony.
Periodic neural firing is produced by CPG and have 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 aremodeled through the effect of high dimensional dynamics ofan individual element, interaction of many neurons with spatial
degrees of freedom, and time delayed coupling[4]. There aremany studies for excitable elements under external periodic
stimuli, which show various behavior including phase-locking

and chaotic behavior. It has been also studies for an ensembleof oscillatory or excitatory elements interacting each other inthe context of synchronization and chaos[5].
Creatures have the neural circuit in their bodies. It is a very
complex structure. For the reason, we’d like to investigate
more complex networks. However, we need to investigate
fundamental networks. In this study, we investigate synchronization phenomena of a three-neuronal system and a fourneuronal system on FHN model. We focus on two points.
The one is the difference of synchronization phenomena bychanging the initial value on each system. The another one
is the difference of periodic solution including firing patterns
between the three-neuronal system and the four-neuronal system We observe the coexistence of the periodic solution. In
addition to that, we find the only firing pattern which exist on
the four-neuronal system.
II. F ITZHUGH -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 medel 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:duiK∑= ui (ui − α)(1 − ui ) − vi +(uj − ui )(1)
dtNi̸=jdvi= τ (ui − γvi )(2)dt
where 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 a- 59 -pair of excitable FHN elements. The time series are generated
by the fourth-order Runge-Kutta method with step size h =0.05.

N1

III. A PAIR OF EXCITABLE NEURONSAs a previous study, there are a study which focuses ona minimal 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 Figure 1. We explain two-phase firing pattern in detail.
After N1 excite, N2 excite soon. After these successiveexcitations, both neurons stay quiescent state for a while. On
the next successive excitations, N2 excite first. After that, N1excite 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 − andN1 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 thesesystems show a various of firing patterns including chaoticfiring.

t(a)N2t(b)

N1
N2t(c)

Fig. 1. Time evolution of two neurons on FHN model for the couplingstrength K = −0.012. The solid lines of (a) and (b) correspond totime waveforms of N1 and N2 , respectively. The solid lines of (c)correspond to both neuronal time waveforms.

N1

IV. N ETWORK S YSTEMS
We introduce the network systems we use in this study. We
investigate two network systems. The three-neuronal system
is shown in Figure 2. This system is a very simple systemthat a neuron is just connected to a system of two neuron as aring topology. Each neuron connects with the other neurons.
We name this three-neuronal system triangle system. The fourneuronal system is shown in Figure 3. This system is a systemthat a neuron is connected to a triangle system. We put theneuron on the place that N2 and the neuron contrast with aside between N1 and N3 on the triangle system as a symmetryaxis. This system has axial symmetry and include two triangle.
Therefore, we name this four-neuronal system two-triangle
system. Each neuron connects with the other neurons, exceptthat N2 does not connect with N4 .
V. N UMERICAL E XPERIMENTS
A. Coexistence of Periodic Solution
We investigate synchronization phenomena on triangle system and two-triangle system by changing the coupling strength
K from −1 to 0 by one hundredth, respectively. We also focus
on the coexistence of the periodic solution by changing theinitial value and various firing patterns. In this section, we
consider the coexistence of the periodic solution. We decide

N2

N3Fig. 2. The triangle system.

the conditions of the initial value. We define some rules. We
use the same absolute value on all neurons and investigate thesystems by means of three types of the absolute values. The
absolute values are 0.25, 0.5 and 1. The negative sign is set
to odd-numbered neurons. Likewise, the positive sign is set toeven-numbered neurons.
In the beginning, we consider the triangle system. Figure 4
shows the number of solution by changing the couplingstrength K from −1 to 0 by one hundredth on the triangle
system. In this figure, ”1” of solution means that the system
has the same solution at all three types of the initial value. ”0”of solution correspond to chaotic firing patterns. These patterns
are observed in the −0.7 to −0.61 range. ”2” of solution areobserved, which show the coexistence of the periodic solution.

- 60 -3Two-triangle system

N4N2

The number of solution

N1N321Fig. 3. The two-triangle system.
0Secondly, we investigate the two-triangle system. Figure. 5
shows the number of solution by changing the couplingstrength K from −1 to 0 by one hundredth on the two-triangle
system. ”0” of solution 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, the rangeof chaotic firing patterns on the triangle system is larger thanthat on the two-triangle system. On the two-triangle system,”2” of solution are also observed, which show the coexistenceof the periodic solution. There are more cases which have thecoexistence of the periodic solution than on the triangle system.
3The number of solution

Triangle system

2

10-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0Coupling strength

Fig. 4. The number of solution by changing the coupling strength K
from −1 to 0 by one hundredth on the triangle system.

B. Comparison of Firing Patterns
In this section, we focus on firing patterns. First of all, weinvestigate the periodic solution on triangle system and twotriangle system by changing the coupling strength K from −1to 0 by one hundredth, respectively. In almost every cases,
the periodic solution does not depend on the initial value on

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0Coupling strengthFig. 5. The number of solution by changing the coupling strength K

from −1 to 0 by one hundredth on the two-triangle system.

the both systems. That is why we consider the only case thatthe absolute value of the initial value is 0.25 as an example.
Figures 6(a) and 6(b) show the periodic solution by changing
the coupling strength K from −1 to 0 by one hundredth on thetriangle system and on the two-triangle system, respectively.
There are the existence of three periodic solution in Figure 6(b)compared with Figure 6(a). That is clear at a glance. The
triangle system is different from the two-triangle system inthat it has the existence of three periodic solution.
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 thetwo-triangle 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 )− isobserved for the first time by connecting a neuron to thetriangle system. For example, this firing pattern is observedin case of K = −0.990. The observed time waveforms
are shown in Figure 7. Time waveforms N1 and N3 aresynchronized at in-phase with N2 and N4 . On the firstsuccessive excitations, the order of firing is N1 N2 N1 −.
On the next successive excitations, the order of firing isN2 N1 N2 −. These states are repeated. A whole new firing
pattern which is not found on the triangle system is observedby connecting a neuron to the triangle system. Three- 61 -

successive firings on one successive excitations were notobserved until on the two-triangle system.

N1t(d)t(e)t(f)

N2N1N(a) The triangle system.

Fig. 7. Time evolution of the two-triangle system on FHN model forthe coupling strength K = −0.990. The solid lines of (d) and (e)correspond to time waveforms of N1 (N3 ) and N2 (N4 ), respectively.
The solid lines of (f) correspond to all neuronal time waveforms.

[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 coupledslow inhibitory neurons”, Physica D, 241, 1998.

(b) The two-triangle system.
Fig. 6. Periodic solution by changing the coupling strength K from−1 to 0 by one hundredth.

VI. C ONCLUSION
We investigated synchronization phenomena of the triangle
system and the two-triangle system on FHN model by changing the initial value and the coupling strength. Some solution
depended on the initial value, and others did not. There weremore cases which have the coexistence of the periodic solution
as with chaotic firing pattern on the two-triangle system thanthose on the triangle system. As well as on a pair of excitableneurons, some firing patterns were observed on the triangle
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 solution.
R EFERENCES[1] Philipp Hovel, Markus A. Dahlem and Eckehard Scholl, ”Control ofSynchronization in Coupled Neural Systems by Time-Delayed Feedback”, International Journal of Bifurcation and Chaos, Vol. 20, No. 3,2010, 813-825.
[2] T. Yanagita, T. Ichinomiya and Y. Oyama, ”Pair of Excitable FitzHughNagumo Elements: Synchronization, Multistability, and Chaos”,
Phys. Rev. E 72, 056218, 2005.
[3] H. Fujii and I. Tsuda, ”Itinerant Dynamics of Class I* Neurons Coupledby Gap Junctions”, Lect. Notes Comput. Sci. 3146, 140, 2004.

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