平成 27 年度電気関係学会四国支部連合大会 講演論文集 (2015 高知工科大学)
2015 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (KOCHI)

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Synchronization Changing Local Coupling Strengthwith Hindmarsh Rose Model
Naoto KANAZAWA

Ryota OSHIMA
Yoko UWATE( Tokushima University)

1. Introduction
In recent years, there has been considerable interested
in building connections of neurons in a brain. The coupling strengths between neurons become strong if the neurons communicate each other. The neurons have the axon.
Some axons are covered with the oligodendrocyte. The covered axons communicate quicker than the non-covered ones.
This phenomenon is called myelination. The myelination
has big relation to learning. Therefore, the study about
myelination has possibility to make more performance forlearning by adapting application. In this study, we investigate synchronization by changing local coupling strengths
between neurons.

1ε1Yoshifumi NISHIO

2ε2

3

ε3

4

Figure 1: The ladder of four neurons on HR model.

(a)
(b)
(c)

2. System model(d)In this study, we use Hindmarsh-Rose (HR) model. Thismodel has three variables. These are the membrane potential x(t), auxiliary variable called y(t) representing a set offast ion channels connected with aspects of potassium andsodium transport, and a ‘slow’ variable z(t) which captures
the slower dynamics of yet other ion channels. The HRmodel is written the following equations.

Figure 2: The temporal evolution of the membrane potential in each neuron. (a) The membrane of neuron 1. (b)The membrane of neuron 2. (c) The membrane of neuron
3. (d) The membrane of neuron 4.

 x˙i (t) = yi (t) − ax3i (t) + bx2i (t) − zi (t) + I − Kiy˙i (t) = c − dx2i (t) − yi (t)z˙i (t) = −rzi (t) + rS(xi (t) − cx )

(1)

X3

X2

where the function K is given as:
K1 = ε1 (x1 − x2 )


K2 = ε1 (x2 − x1 ) + ε2 (x2 − x3 )K3 = ε2 (x3 − x2 ) + ε3 (x3 − x4 )
K4 = ε3 (x4 − x3 )

(2)

X1
(a)

where the parameter i is chosen by 1, 2, 3 and 4. Theparameters are given a = 1, b = 3, c = 1 and d = 5. We use
the injected current I = 3.281, the voltage cx = −1.6, thescale of the influence of the membrane voltage on the slow
dynamics S = 4.0 and the time scale for the slow adaptation
current r = 0.0021. The differential equations are solved by
the Runge-Kutta method.

X2(b)Figure 3: The lissajous figure. (a) In phase. (b) Anti
phase.

4. Conclusions
In this study, we have investigated the synchronization
with the ladder of four neurons in HR model. The couplingstrengths are fixed with different values. The simulation
results indicate in phase and anti phase synchronization.
In our future works, we observe synchronization by othercoupling strengths. Furthermore, we would like to do thesimulation with large scale ladder model.

3. Simulation results
The research of two neurons using HR model has alreadyobtained results[1]. In this simulation, we use the ladderof four neurons which are coupled by electrical coupling.
Figure 1 illustrates in the ladder model. Figure 2 shows thetemporal evolution of the membrane potential in neuron 1,2, 3 and 4. Figure 2 (a), (b), (c) and (d) indicates themembrane potential in the neuron1, 2, 3 and 4. When thecoupling strengths are fixed as ε1 = 0.58, ε2 = 0.28 andε3 = 0.58, we obtain results. The neuron 1 is in phase withneuron 2, however it is the anti phase with the neuron 3and 4. Figure 3 shows lissajous figures about in phase andanti phase. The neuron 1, 2 and 3, 4 indicate Fig. 3 (a)and the others indicate Fig. 3 (b).

Reference[1] Henry D. I. Abarbanel, R. Hueta, M. I. Rabinovich,
N. F. Rulkov, P. F. Rowat, A. I. Selverston, “Synchronized Action of Synaptically Coupled Chaotic Model
Neurons: I. Simulations Using Model Neurons”, Neural Computation Vol. 8, No. 8: 15671602 (November15, 1996)

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