2006 International Joint Conference on Neural Networks
Sheraton Vancouver Wall Centre Hotel, Vancouver, BC, Canada
July 16-21, 2006

Complex Phase Synchronization in an Array of Oscillators
Coupled by Time-Varying Resistor
Yoko Uwate, Student Member, IEEE and Yoshifumi Nishio Member, IEEE

Abstract— In recent years, many people have been trying
to develop some applications to information processing by exploiting oscillatory phenomena in neural networks. Bifurcationand stability of equilibrium points in a simple neural oscillator
consisting of two neurons have been analyzed in detail. On
the other hand, oscillatory phenomena in the simple neuraloscillator can be modeled by electrical circuit such as van der
Pol oscillators. In this study, we propose a network model ofsimple oscillators coupled by time-varying resistor, which can
explain some interesting complex phenomena observed in alarge scale network of neurons coupled by both excitability
and inhibitory synapses. By carrying out computer simulations
and circuit experiments, we confirm the generation of various
interesting phenomena which cannot be observed in simple
coupled oscillatory networks.

I. I NTRODUCTION
In recent years, many people have been trying to develop
some applications to information processing by exploiting
oscillatory phenomena in neural networks. Such neural networks can produce some kinds of phase patterns, and they
may be utilized for associative memory or image processing [1]-[4]. Bifurcation and stability of equilibrium points
in a simple neural oscillator consisting of two neurons havebeen analyzed in detail. Further, two neural oscillators coupled by a simple connection have been reported to produce
phase synchronization.
On the other hand, oscillatory phenomena in simple neuraloscillators can be modeled by electrical circuits such as vander Pol oscillator and BVP oscillator. Studies on synchronization phenomena of coupled oscillators are extensively
carried out in various fields, physics [5]-[8], biology [9][10],electrical engineering [11]-[20], and so on. Endo et al. havereported details of theoretical analysis and circuit experiments about some coupled oscillators as a ladder, a ring
and a two-dimensional array [13]-[15]. Yamauchi et al. havediscovered very interesting wave propagation phenomena ofphase states between two adjacent oscillators in an arrayof van der Pol oscillators coupled by inductors [18][19].
Because many researchers suggest that synchronization phenomena of coupled oscillators have some relations to information processing in the brain, we consider that it is very
important to investigate the synchronization phenomena of
coupled oscillators to realize a brain-like computer in futureapplication.
In this study, we propose a network model of simple oscillators coupled by time-varying resistor, which can explainsome interesting complex phenomena observed in a large
scale network of neurons coupled by both excitability andinhibitory synapses. We realize the time-varying resistor byswitching a positive and a negative resistors periodically.
We consider that the positive and the negative effects ofthe coupling resistor can correspond to the excitability andthe inhibitory synapses qualitatively. First, we analyze phasesynchronization observed in two van der Pol oscillators
coupled by the time-varying resistor as a basic oscillators.
Next, we investigate phase synchronization observed in a
one-dimensional array of coupled van der Pol oscillators. By
carrying out computer simulations and circuit experiments,
we confirm the generation of various interesting phenomena
which cannot be observed in simple oscillatory networks
coupled by resistors. We consider that switching between the
excitability and the inhibitory produce the complexity in the
network. Furthermore, we carry out computer simulations aschanging the timing of the switchings and confirm that the
behavior of the whole network becomes more complex.
II. C OUPLED O SCILLATORS M ODEL
Figure 1 shows the circuit model. In this circuit, two
identical van der Pol oscillators are coupled by a TimeVarying Resistor (TVR). We have known the synchronization
phenomena for the case that the coupling resistor is a
simple time-invariant resistor [11][12]. Namely, the in-phasesynchronization is stable for a positive coupling resistor,
while the anti-phase synchronization is stable for a negative
coupling resistor. In this study, we consider the case thatthe coupling resistance R(t) of the TVR varies with time.
We consider that the positive and the negative effects of thecoupling resistor can correspond to the excitability and theinhibitory synapses qualitatively. The characteristics of the
TVR are shown in Fig. 2. In this study, we consider the casethat the duty ratio is fixed as p = 0.5.
Firstly, the vk −iRk characteristics of the nonlinear resistor
are defined as follows,
iRk = −g1 vk + g3 vk 3 .
By changing the variables and the parameters,



√
g1
g1
C
vk =
yk , t = LCτ ,xk , ik =g3
3g3 L

Yoko Uwate and Yoshifumi Nishio are with Dept. of Electrical andElectronic Engineering, Tokushima University, 2-1 Minami-Josanjima,
Tokushima, Japan, (phone: +81-88-656-7470; fax: +81-88-656-7471; email:{uwate, nishio}@ee.tokushima-u.ac.jp).

0-7803-9490-9/06/$20.00/©2006 IEEE8345


 = g1

L
,
C


γ=r

C
,
L

1
ωt ,
ω= √
LC

(1)

L

I2

I1

L

y1

x2

y2

iR2

iR1
v1

TVR

C

C

v2

x1

x2

(a)

x1
(c)

(b)

x1

OCS-2

OCS-1

x2

(a) Circuit model (TVR is a Time-Varying Resistor).

τ(d)Fig. 3. In-phase synchronization (computer simulation results). (a) 1st
circuit attractor (x1 vs y1 ). (b) 2nd circuit attractor (x2 vs y2 ). (c) Phase
difference (x1 vs x2 ). (d) Time wave form (τ vs x1 and x2 ).
Simple neural oscillator-1

Simple neural oscillator-2y1(b) Network topology.
Fig. 1.

R(t)

x2

y2

Coupled oscillators model.

x1

2π

(a)

r

x1
(c)

x1

ωtt

0

x2
(b)

x2

-r
2πp

(d)
Fig. 2.

τCharacteristics of the TVR.

the normalized circuit equations are given as

dxk


= (1 − xk 2 ) − yk


dτ


2



dyk


= xk − γ(τ )
yj


 dτ
j=1

Fig. 4. Anti-phase synchronization (computer simulation results). (a) 1st
circuit attractor (x1 vs y1 ). (b) 2nd circuit attractor (x2 vs y2 ). (c) Phase
difference (x1 vs x2 ). (d) Time wave form (τ vs x1 and x2 ).

(k = 1, 2) (2)We also confirm that the two coupled oscillators are synchronized at in-phase or at anti-phase in circuit experiments
as shown in Figs. 5 and 6. In the circuit experiments, the TVR
is realized by using an analog switch shown in Fig. 7 [21].
IV. L ARGE S CALE O SCILLATORS

where the sign of the coupling term changes according to
the value of the time-varying resistor.

A. Array of Oscillators

III. S YNCHRONIZATION PHENOMENAWe observed that the two coupled oscillators are synchronized at in-phase or at anti-phase as shown in Figs. 3 and 4.
These two synchronization states can be obtained by giving
different initial conditions. The parameters of the oscillatorsmodel are fixed as  = 2.6, ω = 1.37 and γ = ±0.1.

In this section, we consider a ring of oscillators as shownin Fig. 8. In this circuit adjacent two oscillators are coupledby one TVR. We introduce small resistors rm in computer
simulations to avoid L-loop. By adding the following parameterC,η = rm
L

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+

-

(a)

TVR

(b)

+

-

(c)

(d)

Fig. 7.

Fig. 5.
In-phase synchronization (circuit experimental results). (a) 1st
circuit attractor (v1 vs Ib1 +Ia2 ). (b) 2nd circuit attractor (v2 vs Ia1 +Ib2 ).
(c) Phase difference (v1 vs v2 ). (d) Time wave form (τ vs v1 and v2 ).
L = 10mH, C = 33nF, r = 152Ω.

Circuit realization of the TVR [21].

We can confirm the synchronization phenomena whichhave not been observed in oscillators coupled by timeinvariant resistor.
B. Even Number Coupling: N = 14(a)Figure 9 shows the computer simulated result for the caseof N = 14. N denotes the number of coupled oscillators.
We can see that the array of oscillators coupled by TVR are
synchronized at in-phase or at anti-phase.

(b)

Attractor
Phase
difference

(c)

x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14

(d)

Fig. 6. Anti-phase synchronization (circuit experimental results). (a) 1st
circuit attractor (v1 vs Ib1 +Ia2 ). (b) 2nd circuit attractor (v2 vs Ia1 +Ib2 ).
(c) Phase difference (v1 vs v2 ). (d) Time wave form (τ vs v1 and v2 ).
L = 10mH, C = 33nF, r = 152Ω.

τ(a) In-phase synchronization.
Attractor

the normalized circuit equations of the array of oscillators
are given as
 dx
k
2


 dτ = (1 − xk ) − (yak + ybk )







1
dyak


= xk − ηyak − γ(τ )(yak) + yb(k+1) )
dτ
2
(3)



1
dybk


= xk − ηybk − γ(τ )(ya(k−1) + ybk) )



dτ
2




(k = 1, 2, · · · , N )where
ya0 = yaN ,yb(N +1) = yb1 .

Phasedifference
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14

τ

(b) Anti-phase synchronization.
Fig. 9. Computer simulated result for N = 14. ε = 2.6, ω = 1.37, γ =(0.4 or −0.12), η = 0.01. Upper figures: xk vs yak + ybk . Middle
figures: xk vs x(k+1) . Lower figures: τ vs xk . k = 1, 2, 3, . . . , 14.

(4)

8347

2L

Iak

Ibk

2L

2L I a(k+1) Ib(k+1) 2L

rm

rm

iR2

iR1
TVR

rm

rm

v1

C

TVR

C

v2

TVR

(a) Ring of oscillators.
OSC-N
OSC-1

OSC-2
OSC-3

(b) Network topology.
Fig. 8.

Ring of circuits model.

C. Odd Number Coupling: N = 15

V. C OMPLEX P HENOMENA

Figure 10 shows the computer simulated result for the caseof N = 15. We can see that the array of oscillators coupled
by TVR are synchronized with in-phase (Fig. 10(a)). However, the adjacent oscillators are almost synchronized with
anti-phase as shown in Fig. 10(b). Because, the boundary
condition is the ring structure, the phase difference between
the adjacent oscillators in not around π. Namely, in this case15-phase synchronization are observed.

D. Wave Propagation Phenomena
In this section, wave propagation phenomena observed
from the 15-array of oscillators coupled by TVR are investigated. Figure 11 shows one example of wave propagation
phenomena. In this figure, the vertical axis is the sum of
the voltages of adjacent oscillators and the horizontal axis
is time. White regions in the diagram correspond to thestates that the sum of the voltages is close to zero, namely
the adjacent two oscillators are synchronized at anti-phase.
While, black regions correspond to the state the sum of thevoltages has large amplitude. We can see that the adjacent
two oscillators are synchronized at in-phase in the brackregions from Fig. 11.

We carry out computer simulations as changing the timingof the switchings. Some examples of complex phenomena
are shown in Fig. 12. From these figures, various kinds ofcomplex synchronization are confirmed. We consider thatswitching between the excitability and the inhibitory produce
the complex phenomena such as pattern formations.
VI. C ONCLUSIONSIn this study, we proposed a network model of simple oscillators coupled by time-varying resistor, which can explainsome interesting complex phenomena observed in a large
scale network of neurons coupled by both excitability andinhibitory synapses. By carrying out computer simulations
and circuit experiments, we confirmed the generation ofvarious interesting phenomena which cannot be observed insimple oscillatory networks coupled by resistors. We consider
that switching between the excitability and the inhibitory
produce the complexity in the network.
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Attractor
Phase
difference
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15

τ

(a) In-phase synchronization.
Attractor
Phase
difference
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x4
x5
x6
x7
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x9
x10
x11
x12
x13
x14
x15

τ

(b) 15-phase synchronization.
Fig. 10. Computer simulated result for N = 15. ε = 2.6, ω = 1.37, γ = (0.4 or −0.12), η = 0.01. Upper figures: xk vs yak + ybk . Middle
figures: xk vs x(k+1) . Lower figures: τ vs xk . k = 1, 2, 3, . . . , 15.

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x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1τFig. 11.

Wave propagation phenomena. ε = 2.6, ω = 1.37, γ = (0.4 or −0.12), η = 0.01.

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1

τ(a) ε = 2.6, ω = 0.87, γ = (0.2 or −0.2), η = 0.01.
x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1τ(b) ε = 2.6, ω = 1.00, γ = (0.2 or −0.2), η = 0.01.
x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1τ(c) ε = 2.6, ω = 2.52, γ = (0.2 or −0.2), η = 0.01.
Fig. 12.

Complex phenomena.

8350