ON COUPLED OSCILLATORS NETWORKSFOR THE CELLULAR NEURAL NETWORK~~

Yoshifumi NISHIO t, Shinsaku MORI tand Akio USHIDA $t Dept. of Electrical Engineering, Keio University
3-14-1 Hiyoshi, Kouhoku-ku, Yokohama, 223 Japan
Tel. +81-45-563-1141 Ext. 3319
Fax. +81-45-563-2773
Email nishio@mori.elec.keio.ac.jp$ Dept. of Electrical Engineering, Tokushima University
2-1 Minami- Jousanjima, Tokushima, 770 Japan
ABSTRACT : In this study, we propose two typesof coupled oscillators networks and investigate their steadystates. One network has two-dimensional honeycomb structure. The other network has twedimensional lattice structure. In the Honeycomb circuit, adjacent three oscillators arecoupled by one coupling resistor and it is considered that thevoltages of three oscillators around a coupling resistor makethree-phase synchronization. On the other hand, in the Lattice circuit, four-oscillators are coupled by one coupling resistor and it is considered that the voltages of four oscillators
around a coupling resistor make four-phase synchronization.
As a result, our networks are expected to generate various
synchronization patterns. In our networks each oscillator isconnected to only its adjacent oscillators and various patterns
are generated according to the initial condition. Therefore,
our network is expected to be utilized as a cellular neuralnetwork in the future.

N oscillators with the same natural frequency mutually coupled by one resistor [5], The system is shown in Fig. 1. In
the system various synchronization phenomena can be stably
observed, because the system tends t o minimize the current
through the coupling resistor. Especially, we have confirmed
experimentally that N-phase oscillation (N = 2 N 13) can
be stably excited for the case that each oscillator has strong
nonlinearity. In this case, there exist ( N - l)! phase states.
Because of the coupling structure and extremely large numberof phase states of the system in Fig. 1, this system would be
structural element of the cellular neural network.

INTRODUCTION

os2
Cellular neural network proposed by Chua and Yang [l]have two major features; continuous time feature allowing
real-time signal processing and local interconnection feature
making it tailor made for VLSI implementation. And various
applications for image processing and pattern recognition ofcellular neural network have been reported.
Recently, neural network using a van der Pol oscillator as aneuron has been proposed by Endo et al [2]. This system has
the advantage that no annealing is needed to achieve good
convergence to an optimum solution. Because the static characteristic of the van der Pol oscillator is completely binary andthe dynamic characteristic is continuous and smooth. Moreover, it achieve fast convergence. Therefore, coupled oscillators systems are expected to be utilized for the neural network.
On the other hand, there have been many investigations of
the mutual synchronization of oscillators ([3]-[5] and therein).
We have reported synchronization phenomena observed from0-7803-1254-6/93$03.00
Q 1993 EEE0Fig. 1 Basic circuit model used as a structuralelement of the network.

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Fig. 2 Coupled oscillators network (Honeycomb circuit).

Fig. 3 Coupled oscillators network (Lattice circuit).

In this study, we propose two types of coupled oscillators
networks whose structural element is the circuit in Fig. 1
and investigate their steady states. One network has twodimensional honeycomb structure and it is shown in Fig. 2.
Also the other network has twedimensional lattice structure
and it is shown in Fig. 3. In the Honeycomb circuit, threeoscillators are coupled by one coupling resistor and it is considered that the voltages of three oscillators around a coupling resistor make three-phase synchronization according to
the result in [3]. On the other hand, in the Lattice circuit,
four oscillators are coupled by one coupling resistor and it isconsidered that the voltages of four oscillators around a coupling resistor make four-phase synchronization. Therefore,
our networks can generate various synchronization patterns.
We carry out circuit experiments for two sizes of Honeycomb
circuit and one size of Lattice circuit and confirm the generation of various pat terns.
In our networks each oscillator is connected to only its adjacent oscillators and various patterns are generated according
to the initial condition. Therefore, our network is expected tobe utilized as a cellular neural network in the future.

other has equilateral triangle form with four oscillators on one
side (see Fig. 5(a)). We call the former as Honeycomb1 andthe latter as Honeycomb2. The circuit experimental results
are shown in Figs. 4 and 5 . In both figures, v k is the voltage of the Osck and the notations A , B and C are mean thethree phase states. Though we omit the computer calculation
results, two synchronization patterns corresponding to Figs.
4(b)(c) and 5(b)(c) can be obtained.
In this circuit model, there exist only two patterns. However, it is considererd that various synchronization can bestably excited by inputting some signals. The investigation of
the generation of such new patterns is our future problem.

HONEYCOMB CIRCUIT
Fig. 2 shows the Honeycomb circuit. In this circuit adjacent three oscillators (e.g. 0 x 1 , Osc2 and 0 x 3 in Fig. 2) arecoupled by one coupling resistor (R1 in Fig. 2). Therefore, it
is considered that the voltages of these three oscillators makethree-phase synchronization according to the result in [3]. In
the Honeycomb circuit, only 2 steady states can be exist inspite of the size of the network. Because if the relation of
the phases of three oscillators around a coupling resistor isdecided, the phase of the next oscillator is decided uniquely.
This is confirmed by circuit experiment and computer calculation for two sizes of the network; one has equilateral triangle
form with three oscillators on one side (see Fig. 4(a)) and theLATTICE CIRCUIT
Fig. 3 shows the Lattice circuit, In this circuit adjacent
four oscillators (e.g. Oscl, Osc2, 0 x 4 , and 0 x 5 in Fig. 3)
are coupled by one coupling resistor (R.1 in Fig. 3) Therefore,
it is considered that the voltage of these four oscillators makefour-phase synchronization. In this circuit, it is proved that
there exist 3!(2N1-3 2N2-3 - 1) patterns ( N I x N2 is the sizeof the network). Two typical examples of such patterns forthe size 3 x 3 obtained from circuit experiments are shownin Fig. 6. In Fig. 6 the notations A N D mean four phasestates.
Although we have not analyzed the control of the output
patterns by external inputs, we consider that several patterns
can be obtained by inputting some signals to some oscillators.
Therefore, we believe that we can control the output patterns
by external inputs and that some applicatioris using our network as a cellular neural network will be proposed in nearfuture.

+CONCLUSIONSIn this study, we have proposed two types of coupled oscillators networks and have investigated their steady states. One2328

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ox:3

osc2

Oscl

YY0x4oscsYOsc 6A B C
Y Y*AB-,tFig. 4 Circuit experimental results for the Honeycomb1(L= 9.9mH, C = 0.01pF, R = 1.5LR, r = 3000).
(horizontal axis : 10 pec/div., vertical axis : 1 V/div.).
(a) Arrangement of the oscillators. (b) Pattern 1. (c) Pattern 2.

0scl 0 . ~ ~osc3
2 oS,4Y Y Y

0 ~ c S O S C ~ Osc7Y

OSC8Yo s 4Y0% 10*YBYCYABY(b)

CA

(c)

C BY/+Fig. 5 Circuit experimental results for the Honeycomb-2(Parameter values are the same as those in Fig. 4).
(horizontal axLs : 10 psec/div., vertical axis : 1 V/div.).
(a) Arrangement of the oscillators. (b) Pattern 1. (c) Pattern 2 .

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OSC~ 0x2 Osc3x xOs&
Osc5 Osc6x x
Os?

Os&

osc9

*XCXB

A B A
% %x xx BxA

B D A-t

D C D(b) A C B(‘) AFig. 6 Circuit experimental results for the Lattice circuit(L= 9.9mH, C = O.OlpP, R = 1.5kR, r = 1000).
(horizontal axis : 20 ,usec/div., vertical axis : 1 V/div.)(a) Arrangement of the oscillators. (b) Pattern 1. (c) Pattern 2.

network has two-dimensional honeycomb structure. Also theother network has two-dimensional lattice structure. In OUI
networks each oscillator is connected to only its adjacent oscillators and various patterns are generated according to theinitial condition. Therefore, our network is expected to be
utilized as a cellular neural network in the future.

[5] Y. Nishio and S. Mori, ”Mutually Coupled Oscillatorswith an Extremely Large Number of Steady States,”
Proc. of ISCAS’92, (1992), 819.

REFERENCES
L. 0. Chua and L. Yang, ”Cellular Neural Networks: Theory,” IEEE Trans. Circuits Syst., Vol. CAS-35, No. 10,(1988), 1257.

T. Endo, L. 0. Chua and K. Takeyama, ” A Neural Network Using Oscillators,” Proc. of ISCAS’91, (1991), 782.
H. Kimura and K. Mano, ”Some Properties of Mutually
Synchronized Oscillators Coupled by Resistances,” Trans.
IECE, Vol. 48, No. 10, (1965), 1647(in Japanese).
T. Endo and S. Mori, ”Mode Analysis of a Ring of a Large
Number of Mutually Coupled van der Pol Oscillators,”IEEE Trans. Circuits Syst., Vol. CAS-25, No. 1, (1978),
7.

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