2016 International Symposium on Nonlinear Theory and Its Applications,NOLTA2016, Yugawara, Japan, November 27th-30th, 2016

Synchronization in Dynamical Polygonal Oscillatory Networkswith Switching Topology
Yoko Uwate† , Yoshifumi Nishio† and Ruedi Stoop‡†Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp‡Institute of Neuroinformatics, University/ETH Zurich,
Winterthurerstrasse 190, CH-8057 Zurich, Switzerland
Email: ruedi@ini.phys.ethz.ch
Abstract— In this study, we propose dynamical polygonal oscillator networks with switching topology by using
van der Pol oscillators. In the proposed network, the network topology is switching with time. We confirmed thatnew synchronization state can be occurred by stochasticallyswitching the network topology.
1. Introduction
The synchronization phenomena observed from coupled nonlinear oscillators are suitable model to analyze thenatural phenomena [1],[2]. Therefore, many researchers
have proposed different coupled oscillatory networks andhave discovered many interesting synchronization phenomena [3]-[6].
In the human brain, synchronization of neural activity
between different cortical areas may proved several kindsof neural information functions depending on the network
topology (see Fig. 1(a)) [7]. Recently, grid cells which
are the type of neurons in the entorhinal cortex show remarkable hexagonal activity patterns [8], [9]. The polygonal structure of neuronal firing pattern has possibility
important role for emerging unified computational framework [10].
Namely, it is important to study synchronization phenomena observed from the polygonal oscillatory networks
with the several characteristics in the human brain as we described above, for modeling neuro-biological systems andapplying its high-functional information processing to engineering applications.
Neurons in the brain are expressed by stable oscillators.
Neuronal oscillators using mathematical models are often
used for modeling brain networks. However, there are notmany models of neuronal oscillators using electrical circuits. In order to realize the brain network by real physicalsystems, we need to investigate synchronization of neuralactivity in analog electrical circuits. Here, van der Pol oscillator which is simple oscillatory circuit model is used asneural oscillator.
We have investigated the synchronization phenomenain the coupled polygonal oscillatory networks sharing

branches [11], [12]. In this system, odd number of van der
Pol oscillators are connected to every corner of each polygonal network. Namely, frustration is occurred between theadjacent oscillators. By using computer simulations andtheoretical analysis, we confirm that the coupled oscillators
tend to synchronize to minimize the power consumption of
the whole system. The phase difference of the shared oscillators is solved by finding the minimum value of the powerconsumption function.
Furthermore, coupling strength between neural activities
are changed depending on the information processing withtime (see Fig. 1(b)). However, in Ref. [11], we have considered the static network model.

Time(a) Sync. in neural activity.

(b) Time evolution.

Figure 1: Brain network.
In this study, we propose new dynamical polygonal circuit system which is including the switching couplings, inorder to understand the mechanism of high functional neural information processing in the human brain. In the proposed network, external signal is replaced with stochastic
factor and we consider that the coupling strength is expressed by on/off as an extreme example. The synchronization phenomena in the proposed networks are investigated by using computer simulations. First, we investigate
synchronization states in two coupled polygonal networks(triangular and quadrangular networks) with one switching coupling, in order to understand the basic phenomena.
Next, the system model is extended to six coupled polygonal networks with three switching couplings for understanding more complex behavior. By using the computer
simulations, we confirm that new synchronization states- 578 -1st oscillator
ic1
ib1
ia1
rm

3L
rm

3L

5Switching coupling(coupling probability: p)

Figure 2: Conceptual circuit model for 3-4 coupling network.
43R

rmj

i
22v3

3L

Normal coupling(no switching)

ii

1

C

rm

i

j

i

4

rm

1

3L

3L

ib5

ia5

ic5

3L

rm

iR5

iR2

3rd oscillator

Figures 2 and 4 show the conceptual circuit model andcircuit realization using van der Pol oscillators. The triangular and the quadrangular oscillatory networks are coupled by sharing a branch. We call this circuit system “3-4coupling network.” In this circuit model, we consider thecoupling method which two adjacent oscillators are tend tosynchronize at anti-phase state.
In Fig. 2, a chain line denotes the switching coupling(SC). If the coupling edge is selected as the switching coupling, the two oscillators connect or disconnect according
to the coupling probability (p). Namely, the switching coupling is connecting and disconnecting stochastically. If thecoupling probability is p = 0.0, the coupling strength isγ = 0. The state (connecting/disconnecting) of the switching coupling is updated at every switching time (ST) in thesimulation. The switching time denotes the iteration number. One example of operation of the switching coupling isshown in Fig. 3.

ic2

rm

i R3
RR

ib2

ia2

3L

3L
rm

2.1. Circuit Model

3

ib3

v4

R

ia3

RC3L

2. Two Coupled Network- Number of Switching Coupling : 1 -ic3

iR4
R

rm

R

3L

rm

C

v1

4th oscillator
3L
ib4 ic4

3L
rm

3Li R13L

3L

can be produced when the network topology is changed byswitching the connection (on/off) of the edges in the proposed network.

Cv2

2nd oscillator

RCv5

R

5th oscillator

Figure 4: Circuit model for 3-4 coupling network.
and disconnecting. We consider the four patterns (patternA, B, C, and D) depending on the switching coupling.
“state-1” and “state-2” denote the synchronization state obtained from the two types of the network topologies.
Table 1: Switching Coupling (SC) Patterns.
Patternstate-1 (SC: on)state-2 (SC: off)
pattern1
14433A
2525(SC:γ12 )φ12 ≈138.6◦
φ12 ≈72◦
pattern1
14433
B
2525(SC:γ13 )φ12 ≈138.6◦
φ12 ≈180◦
pattern1
14433C2525(SC:γ14 )φ12 ≈138.6◦
φ12 ≈120◦
pattern1
14433D2525(SC:γ45 )φ12 ≈138.6◦
φ12 ≈120◦
Next, we develop the expression for the circuit equationsof 3-4 coupling oscillatory networks as shown in Fig. 4.
The vk − iRk characteristics of the nonlinear resistor are approximated by the following third order polynomial equation,

51425iRk = −g1 vk + g3 vk 3 (g1 , g3 > 0), (k = 1, 2, 3, 4).

31425(1)

31

42

5Time

31

42

53

Figure 3: Example of operation of switching coupling for
3-4 coupling network.
We consider only one coupling edge can be the switching coupling in the 3-4 coupling network. Table I is summarized the network typologies and synchronization statesof four patterns when the switching coupling is connecting

The normalized circuit equations governing the circuit
are expressed as[kth oscillator]
 dx
1 2k
=ε1−xk xk − (yak + ybk + yck )dτ3
dyak 1=x−ηy−γ(y+y)
kak
kn
akn

dτ3

 dy
1(2)
bk

xk − ηybk − γkn (ybk + yn )
=

dτ3


dyck 1 x−ηy−γ(y+y)
=
kck
kn
ckn

dτ3





(k = 1, 2, 3, 4, 5).

- 579 -

LCτ, vk =rg1
xk , iak =
3g3

r

g1
3g3

C
yak ,L

180

180

160

160

140

140

Phase differencet=√rPhase difference

where120
100806040g1CLibk =ybk , ε = g1,3g3 L
Crr
CC, η = rm,
γ=RL
L0100
80
60
4020r

2001000

2000

τp

3000

4000

05000

0(a) ST=100,000.

In these equations, γkn is the coupling strength, ε denotes
the nonlinearity of the oscillators and yn denotes the currentof neighbor oscillator on the coupling resistor. For the computer simulations, we calculate Eq. (2) using the fourthorder Runge-Kutta method with the step size h = 0.005.
The parameters of this circuit model are fixed as ε = 0.1and η = 0.0001.

180160160

140140120
100
8060402000

τp

3000

4000

5000

4000

5000

120
100
80
60
40

2001000(b) ST=10,000.

180

Phase difference(k = 1, 2, 3, 4, 5).

Phase difference

rr

120

2001000

2000

τp

3000

4000

(c) ST=1,000.

5000

001000

2000

τp

3000

(d) ST=100.

Figure 6: Example of the phase difference for Pattern C(p=0.5).

2.2. Synchronization Phenomena

180

180

160

160

140

140

Phase difference

Phase differenceFirst, we investigate the synchronization state of the 3-4
coupling network with the switching topology. The switching time (ST) is changed from 100 to 10000. Figures 5 and6 show the examples of the phase difference between 1stand 2nd oscillators for Patterns A and C at Table 1 when
the coupling probability is set to p=0.5. The phase difference is calculated by using the Poincaré section : xk < 0 ,yk = 0.

120
100
80
60
40

10080
60
40

2001202001000

2000

τp

3000

4000

0

5000

0

180

160

160

140

140

120
100
80
60
40

τp

3000

4000

5000

3. Conclusions

120
100
80
60
40

2002000(b) ST=10,000.

180

Phase difference

Phase difference(a) ST=100,000.

1000

20010002000τp3000

4000(c) ST=1,000.

5000

0010002000

τp3000Figs. 5(d) and 6(d)). And the converging phase state shows
value between the state-1 and the state-2
Figure 7 shows the simulation results of observing newphase states by changing the coupling probability p for fourpatterns. The phase difference shows the average value of1000 iterations in the simulation.
In the cases of Patterns-A, C and D, the new phase states
can be observed between state-1 and state-2 in the range of0.1 ≤ p ≤ 0.9. While, in the case of Pattern-B, the phasestates show almost similar value with state-2 when the coupling probability is smaller than 0.6. By increasing p, thenew phase states can be obtained as shown in Fig. 7(b).
From these results, we can see that new synchronization
states can be occurred by changing the network topology.

4000

5000(d) ST=100.

Figure 5: Example of the phase difference for Pattern A(p=0.5).
In the both cases of Patterns-A and B, the phase difference changes between two states (state-1 and state-2),
when the switching time (ST) is fixed with 100, 000 asshown in Figs. 5(a) and 6(a). By decreasing the value of ST,
the oscillation range of the phase difference becomes small
and the phase difference converges to the certain value (seeIn this study, we have proposed new dynamical polygonal circuit system which is including the switching couplings, in order to understand the mechanism of high functional neural information processing in the human brain.
In the proposed network, external signal is replaced withstochastic factor and we consider that the coupling strengthis expressed by on/off as an extreme example. By using thecomputer simulations, we confirmed that new synchronization state can be occurred by switching the network topology.
For the future works, we investigate the synchronization state in detail when the three switching couplings ischanged with different iteration in the extended dynamicalnetwork. And, we would like to apply the proposed system
to more large scale networks to model the existing biological complex networks.

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132130128126124
122
120

00.2

0.4

0.6

0.8

1

Coupling probability: p(d) Pattern-D.

Figure 7: Observing new phase states by changing p.

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