Chaotic Circuits Networkwith Scale-Free Coupling Distribution
Yoko Uwate

Yoshifumi Nishio

Dept. of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
uwate@ee.tokushima-u.ac.jp

Dept. of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japannishio@ee.tokushima-u.ac.jp

Abstract—In this study, we investigate synchronization
observed in chaotic circuits network with scale-free coupling
distribution. The network topology is inspired from real brain
network. Here, we consider the network which consists of twomodules including connector and provincial hubs. We confirm
several clustering patterns from the proposed system.

module consists 24 chaotic circuits. The node is expressed by
Nishio-Inaba chaotic circuits as shown in Fig. 2. The chaotic
circuits are coupled by a resistor.

Keywords—brain network; chaotic circuits network; clustering

I.

INTRODUCTION

We have focused on synchronization phenomena observed
from nonlinear oscillatory circuits networks [1], [2]. This isbecause the results of synchronization in complex networks areuseful deeper understanding of control methods in power
networks, communication systems and so on. Also, they can beused as an alternative approach, apart from existing ones, fordescribing mode-locking phenomena in biological networks.
Therefore, we consider the schematic brain networks as one ofcomplex biological networks. Because, we would like topropose modeling of synchronization in brain by using coupledelectrical oscillatory circuits, in order to make clear themechanism of functional operation in brain. Structural andfunctional brain networks are explored using graph theory andthe brain network structures have been made clear [3].
Furthermore, several research groups have reported that thesynaptic connectivity in local cortical circuits has scale-free
distribution [4]. We also apply this scale-free characteristics ofcoupling distribution for the proposed system in this study.
First, a symmetric diagram of a brain network is used as asimple network model. Then, we extend the proposed network
to asymmetric circuits network topologies. By using computer
simulations, we confirm that the chaotic circuits network withscale-free coupling distribution can produce several kinds
clustering patterns.
II.

Fig. 1. Network model.

Fig. 2. Nishio-Inaba chaotic circuit.
Next, we consider link probability in the network. The linkprobability is expressed by the following equation.

PROPOSED SYSTEM

A. Network Model
A network model composed of 51 nodes is shown in Fig. 1.
There are two important hubs in this network, “Connector hub”
and “Provincial hub”. The both hubs are high-degree nodes.
“Connector hub” shows a diverse connectivity by connecting
two sub-networks. “Provincial hub” primarily connects nodes
in the same sub-network. There are two modules and each

978-1-5386-2285-8/17/$31.00 ©2017 IEEE(1)By using Eq. (1), the network coupling is shown in Fig. 3.
Here, the link probability of the connector and provincial hubs
have two times. For the computer simulations, we calculate
circuit equations using the fourth-order Runge- Kutta method.

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Fig. 3. Coupling distribution in symmetric network.

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Fig. 5. Coupling distribution in asymmetirc networks.

B. Synchronization Phenomena
In order to investigate synchronization, we calculate thefive types of the phase differences as follows.

180

180

phase_12
phase_13
phase_23
phase_2gphase_3g160phase difference1401) phase_12: connector hub to provincial hub-12) phase_13: connector hub to provincial hub-23) phase_23: provincial hub-1 to provincial hub-24) phase_2g: circuits in module-15) phase_3g: circuits in module-2

14012010080100
8060

4040
2020

Figure 4 shows the simulation result of phase difference bychanging the coupling strength g. By increasing the couplingstrength, all chaotic circuits are synchronized. Namely, in caseof symmetric network topology, we observe one clustering
phenomena.

120

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00.001phase_12
phase_13
phase_23
phase_2gphase_3g160phase difference

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gg

(a) pattern-1.

(b) pattern-2.

0.008

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Fig. 6. Phase difference in asymmetric networks.

180

phase_12
phase_13
phase_23
phase_2gphase_3g160phase difference

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(a) pattern-1.

(b) pattern-2.

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Fig. 7. Clustering in asymmetric networks.

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40

IV. CONCLUSIONS

20
0
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0.009

In this study, we have investigated synchronization
observed in chaotic circuits network with scale-free coupling
distribution. The network topology is inspired from real brain
network. Here, we consider the network which consists of twomodules including connector and provincial hubs. We haveconfirmed several clustering patterns from the proposed system.

0.01

gFig. 4. Phase difference in symmetric network.
III. ASYMMETRIC TOPOLOGIES
Next, we consider asymmetric networks as shown in Fig. 5.
In Fig. 5(a), the radius of module-2 is smaller than module-1.
While, in Fig. 5(b), the radius of module-2 is larger thanmodule-1. The link probability is applied by using Eq. (1).

REFERENCES

The simulation results of the phase differences are shown inFig. 6. By increasing the coupling strength, we observe
clustering pattern. Figure 7 shows the clustering pattern byusing phase differences. In Fig. 7 (a), there are two clusters andtwo modules do not synchronize with the connector hub. In Fig.
7 (b), there is one cluster and module-1 synchronize withconnector hub. From these results, we can see that the network
symmetrically has important role for synchronization andclustering pattern in complex networks.

978-1-5386-2285-8/17/$31.00 ©2017 IEEE143[1] Y. Uwate and Y. Nishio, “Synchronization in Several Types of CoupledPolygonal Oscillatory Networks,” IEEE Trans. Circuits Syst. I, vol. 59,no. 5, pp. 1042-1050, May 2012.
[2] Y. Uwate and Y. Nishio, ”Frustrated Synchronization in Two Coupled
Polygonal Oscillatory Networks,” Proc. of ISCAS’13, pp. 1781-1784,
May 2013.
[3] E. Bullmore and O. Sporns, “Complex Brain Networks: Graph
Theoretical Analysis of Structural and Functional Systems,” Nature,Reviews, Neuro, vol.10, pp. 186-198, Mar. 2009.
[4] R. Stoop, V. Saase, C. Wagner, B. Stoop and R. Stoop, “Beyond ScaleFree Small-World Networks: Cortical Columns for Quick Brains”
Physical Review Letter, 110, 108105, 2013.

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