RISP

NCSP’16

International Workshop on
Nonlinear Circuits, Communications and Signal Processing

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

Synchronization of Hindmarsh Rose Neurons byTemporal Evolution of Coupling Strengths
Naoto Kanazawa, Yoko Uwate, and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {naoto,uwate,nishio}@ee.tokushima-u.ac.jp

Abstract

2. Hindmarsh Rose ModelIn this study, we observe synchronization on Hindmarsh Rosemodel which produce spiking-bursting behavior like real biological neurons. We set the same coupling strength parameters. Provided that neurons are synchronized, the couplingstrength between the neurons is weakened. Thus, the coupling topology is changed little by little. We observe the synchronization of network.

Hindmarsh Rose (HR) model is a mathematic model whichchaotic behavior molluskan neurons cause reproduces. It cancause chaotic behaviors autonomously without periodic stimulations from external input. We can indicate various oscillations by setting parameters.

2.1. Individual Neuron
1. Introduction
We have a lot of neurons in our brain. Neurons have many
inputs and one output. Neural network (NN) is formed byconnecting neurons. NN is mathematical model represented
simulations on the computers that some characteristic features in our brain. We can see spiking-bursting activity which
has high-dimensional dynamics for generation. The research
about NN was studied by [1–4]. This network is so complex
network. The connection between neurons is called couplingstrength. Coupling strengths are enhanced by the electricalsignals and strengthened the part of more communications.
The neurons have output part called axon. Some axons arecovered with oligodendrocyte on the Myelin sheaths. Oligodendrocyte is white material which is rich in lipid. Coveredaxons can communicate quicker than non-covered ones. Thisphenomenon is called Myelination. For example, we practice to achievement something when we usually try to doit. This action from challenge to achievement is myelination. Therefore, there is a big relation between learning andmyelination. The study about myelination has possibility tomake more performance for learning by adapting application.
In this study, we use Hindmarsh Rose model which produce
spiking-bursting behavior like real biological neurons. In thisstudy, we observe the synchronization by changing the temporal evolution of coupling strengths.

This model has three variables. These are the membranepotential x(t), auxiliary variable y(t) representing a set of fastion channels connected with aspects of potassium and sodiumtransport, and a slow variable z(t) which captures the slower
dynamics of other ion channels. The HR model is describedby the following equations.
 ẋ(t) = y(t) − ax3 (t) + bx2 (t) − z(t) + Iẏ(t) = c − dx2 (t) − y(t)(1)
ż(t) = −rz(t) + rS(x(t) − cx )The parameters are given as a = 1, b = 3, c = 1 and d = 5.
We use the injected current I = 3.281, the voltage cx =−1.6, the scale 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 aresolved by the Runge-Kutta method.

2.2. Electrical Coupling
In this study, we use fully coupled five neurons model
shown in Fig. 1. The equations with electrical coupling are
given as follows.

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1Asynchronization

ε15ε12εε25

Synchronization

132Asynchronization-0.01

5Figure 2: Definition of the synchronization. Lines show synchronization between neurons.

ε23ε14 ε45
3ε35ε34

ε24

strengths are enhanced 0.001/9. Those actions are done in
the case of synchronization every 130000 iterations consecutively.

44. Simulation Results

Figure 1: Fully coupled five neurons.

 ẋi (t) = yi (t) − ax3i (t) + bx2i (t) − zi (t) + I − Kiy˙i (t) = c − dx2i (t) − yi (t)z˙i (t) = −rzi (t) + rS(xi (t) − cx )(2)where i indicates the neuron number (from 1 to 5). The function Ki is given as:

In this paper, we use initial condition for which the steadystates become a decoupled synchronization in Fig. 3. Thisdecoupled synchronization appears for t ≥ 9608.5 when all
coupling strengths are fixed as εij = 0.2. Neurons 1, 2 and
4 are synchronized. Neurons 3 and 5 also are synchronized.
However, the neurons 1, 2 and 4, and the neurons 3 and 5 arenot synchronized.

1
K1 = ε12 (x1 − x2 ) + ε13 (x1 − x3 )
+ε14 (x1 − x4 ) + ε15 (x1 − x5 )



K
=
ε(x−x)+ε23 (x2 − x3 )
21221+ε24 (x2 − x4 ) + ε25 (x2 − x5 )K3 = ε13 (x3 − x1 ) + ε23 (x3 − x2 )+ε34 (x3 − x4 ) + ε35 (x3 − x5 )

K=ε(x−x)+ε24 (x4 − x2 )41441+ε34 (x4 − x3 ) + ε45 (x4 − x5 )
 K5 = ε15 (x5 − x1 ) + ε25 (x5 − x2 )+ε35 (x5 − x3 ) + ε45 (x5 − x4 )

2(3)

53

where the parameter ε is coupling strength.

4Figure 3: Decoupled synchronization. Lines show synchronization

3. Proposed Method

between neurons.

First, we define the synchronization condition by the following equation:|xa − xb | < 0.01
(4)
Figure 2 shows definition of the synchronization. We determine the synchronization only when the case synchronization
continues for 130000 iterations with the step size h = 0.05.
We explain the way to change the coupling strengths. Thecoupling strength is weakened 0.001 in the case of synchronization between neurons. At that moment, other coupling

All coupling strengths are fixed 0.2 at the start. We observe
the synchronization with all neurons on the network. Figure
4 illustrates the temporal evolution of the membrane potential around t ' 7500. Figure 5 shows difference waveforms
between each neurons around t ' 7500. We can confirm thesynchronization in Fig. 5 (a), (c), (f) and (i). Thus, the network exhibits the same decoupled synchronization as Fig. 3.
This synchronization is completed in t = 7001.05. Namely,- 603 -this synchronization is achieved earlier than unchanging cou(a) 1-2
pling strengths simulation.
We obtain results Fig. 6 and 7 after a period of time. Figure
6 illustrates the temporal evolution of the membrane potential. Figure 7 shows difference waveforms between each neu- (b) 1-3rons. As Fig. 7 shows, all neurons are synchronized. This fullsynchronization for all neurons is completed in t = 15541.25.

(c) 1-4(a)(d) 1-5(b)(e) 2-3(c)(f) 2-4(d)(g) 2-5(e)(h) 3-4

Figure 4: The temporal evolution of the membrane potential in the (i) 3-5
beginning. (a) The membrane of neuron 1. (b) The membrane ofneuron 2. (c) The membrane of neuron 3. (d) The membrane of
neuron 4. (e) The membrane of neuron 5.

(j) 4-5

Figure 5: Waveforms show difference between two neurons in the5. Conclusionsbeginning. Threshold illustrates red line.

In this study, we use the synchronization with the fully
coupled five neurons on Hindmarsh Rose model. The synchronization with all neurons is not completed in the case ofunchanging coupling strengths. Moreover, the result also indicates that all neurons is the synchronization by changing
the temporal evolution of coupling strengths. In our future
works, we would like to do the simulation with large scalemodel.

References[3] A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application toconduction and excitation in nerve,” J. Physiol., vol.
117, No. 4, pp. 500-544, Aug. 1952.
[4] Henry D. I. Abarbanel, R. Hueta, M. I. Rabinovich, N.
F. Rulkov, P. F. Rowat and A. I. Selverston, “Synchronized Action of Synaptically Coupled Chaotic Model
Neurons: I. Simulations Using Model Neurons,” NeuralComputation, Vol. 8, No. 8, pp. 1567-1602, Nov. 1996.

[1] J. L. Hindmarsh and R. M. Rose, “A model of neuronal bursting using three coupled first order differential equations,” Proc R Soc Lond B Biol Sci., vol. 221,No. 1222, pp. 87-102, Mar. 1984.
[2] A. L. Shilnikov and N. F. Rulkov, “Origin of chaos in a
two-dimentional map modeling spiking-bursting neuralactivity,” Int. J. Bifurcation and Chaos, vol. 13, No. 11,
pp. 3325-3340, 2003.

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(a)
(b)
(c)
(d)
(e)

Figure 6: The temporal evolution of the membrane potential in eachneuron. (a) The membrane of neuron 1. (b) The membrane of neuron 2. (c) The membrane of neuron 3. (d) The membrane of neuron
4. (e) The membrane of neuron 5.

(a) 1-2(b) 1-3(c) 1-4(d) 1-5(e) 2-3(f) 2-4(g) 2-5(h) 3-4(i) 3-5(j) 4-5

Figure 7: Waveforms show difference between two neurons.
Threshold illustrates red line.

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