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INVESTIGATION OF PHASE-WAVE PROPAGATION PHENOMENAIN SECOND ORDER CNN ARRAYS
Z. YANG, K. TSURUTA and Y. NISHIO

A. USHIDA

Tokushima University
Department of E. E. Eng.
2-1 Minami-Josanjima, Tokushima, Japan

Tokushima Bunri University
Department of M. E. Eng.
1314-1 Shido, Kagawa, Japan

ABSTRACTThe work is concerned with the phase-wave propagation
phenomena in second order Cellular Neural Network (CNN)
arrays. We investigate the phase relationship between theadjacent cells, observe that how the phase difference is propagated in the second order arrays, and find the regular pattern of the propagation phenomena.

encountered in a simpler circuit array consisted of second
order CNN cells, in which there exists linear couplings between cells and each cell has a well-known piecewise-linear
nonlinear output function at the output stage. Dependingon the application of sevreal difference conditions, we investigate the phase relationship between the adjacent cells,
and observe the regular pattern of the phase difference propagation phenomena, and discuss how to control the phasepropagation phenomena.

1. INTRODUCTION
Dynamic property of networks of oscillatory and chaotic elements is one of the very lively studied topics. Many papers of international conferences and journals are devoted
uniquely to studies of spatially extended systems, active
media, and coupled lattices showing important areas wherestudies of dynamic phenomena in coupled oscillators findpotential applications. Since CNNs were invented in 1988,
the CNN paradigm provides a flexible framework (or universal model) to describe spatio-temporal dynamics in discrete space and - perhaps more importantly from a practicalpoint of view - allows for efficient VLSI implementation ofanalogue, array-computing structure. Many nonlinear phenomena such as pattern formation and autowaves have been
reproduced in the CNNs, and the related literatures havebeen reported [1]–[5].
On the other hand, Investigation on phase relationshipof mutually coupled oscillators or chaotic circuits is also
very important work, which is well known by physicians
but little explited in CNN. Endo and Mori [6]–[7] have studied several modes of synchronization in several van der Pol
oscillators arrays. We have also discovered from pheas difference approach a very interesting phase-wave propagation
phenomena in van der Pol oscillators coupled by inductors
as a ladder [8]. Unfortunately, the basic circuits of the above
models contain inductive and high-order nonlinearly resistive devices resulting in circuit sophistication, so that it is
not suitable to develop the implementation of the large-scale
arrays of these circuits.
In this paper, we investigate these dynamic behaviors

0-7803-8251-X/04/$17.00 ©2004 IEEE

2. TWO-LAYER CNN ARRAY
In this section, we describe the circuit array used for investigating propagation phenomena of phase difference between
adjacent cells. This array is a one-dimensional two-layer
CNN with a set of constant template. Its circuit topology
simpler than those reported in literatures [6]–[8]. The state
equation of each cell is simply described as follows:⎫−x1,ij + I1
ẋ1,ij = ⎪⎪⎪+ C(k,l)∈Nr (i,j) A1 (i, j; k, l)y1,kl ⎪⎪⎪⎪⎪+ C(k,l)∈Nr (i,j) B1 (ij; kl)u1,kl ⎪⎪⎪+ C(k,l)∈Nr (i,j) C1 (ij; kl)y2,kl ⎬(1)ẋ2,ij = −x2,ij + I2⎪⎪⎪
+ C(k,l)∈Nr (i,j) A2 (ij; kl)y2,kl ⎪⎪⎪⎪⎪+ C(k,l)∈Nr (i,j) B2 (ij; kl)u2,kl ⎪⎪⎪⎭+C (ij; kl)yC(k,l)∈Nr (i,j)

21,klwith output equationy1;ij = 0.5(|x1;ij + 1| − |x1;ij − 1|)y2;ij = 0.5(|x2;ij + 1| − |x2;ij − 1|)

(2)i = 1, 2, ..., N.
where A, B, C, and I are respectively the feedback template,control template, coupled template, and bias current, and thesubscripts 1 and 2 stand for the first layer and the second
layer of the two-layer CNN array. In this paper, we restrict

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our discussions to the following template.
⎤⎤⎡⎡
0 d1 00 d2 0
A1 = ⎣ d1 a1 d1 ⎦ , A2 = ⎣ d2 a2 d2 ⎦ ,0 d1 00 d2 0
C1 = c1 , C2 = c2 , B1 = B2 = 0, I1 = I2 = 0.

(3)

It has been proved in references [9] that for a particular
choice of the parameters a1 , a2 , c1 , and c2 , this nonlinear
second-order cell in absence of couple to its neighbor cells
oscillates with a limit cycle centered to the origin and symmetric with respect to the phase plane axis.

(a)3. DEFINITION OF PHASE DIFFERENCE
The phase-waves in an array of oscillators were firstly introduced in [8], which are observed from the changes ofphase difference between adjacent oscillators in a laddercomposed of van der Pol oscillators coupled by inductors.
Assume that yk and yk+1 in Fig. 1 are the outputs of theadjacent cells k and k + 1, then the phase difference in ourcase is defined as follows:
φi,i+1 (n) = 2π ×ti (n) − ti+1 (n)ti (n) − ti (n − 1)

(4)
(c)

where ti (n) is the time when the state x1;i crosses 0[V] atn-th period.
ti (n)

ykcelliO
yk+1

celli+1O

n−1n−1(b)nti+1 (n)tntFig. 1. Definition of phase difference.

4. AUTOWAVE AND PHASE-WAVE PROPAGATIONPHENOMENASince the middle of the last decade, the investigations into
the spatio-temporal dynamics in CNNs have been widely
carried out. They have discussed various autowaves such as
excitability waves, concentration and so on. We have also
observed these propagation phenomena by using the simple
second order array. Figure 2 shows an example of autowave
with two concentric circular waves under the array parameters: a1 = 1.1, a2 = 1.04, c1 = −1, c2 = 1, d1 = 0.1,
and d2 = −0.01. All of these propagation phenomena areobserved from the output amplitude of cell varying with theFig. 2. A simulation for autowaves with two concentric circular waves. (a) is initial state; (b) and (c) show two snapshots observed in different iterates.
integration time. However, the investigation on phase relationship between adjcent cells is also an important work,
which is well known by physicians but little explited in
CNN so far. Here, we observe the propagation phenomena from the changes of phase difference between adjacentcells, we found a very interesting phase propagation phenomena. For simplicity, we consider one-dimensional second order array. Figure 4 show a simulation example for theone-dimensional array composed of 14 cells under the initial condition shown in Fig. 3, zero-flux boundary condition
and the system parameter: a1 = 1.2, a2 = 1.1, c1 = −2,c2 = 4, d1 = 0.1, and d2 = −0.1. In fig. 4, the verticalaxes are the sum of the first layer outputs of adjacent cells,
and the horizontal axes are the time. Hence, the diagrams
show qualitatively how phase differences between adjacentcells change as time goes, where white regions correspondto the state that two adjacent cells/oscillators are anti-phasesynchronization, and black regions to the in-phase synchronization. Analysis of this simulation indicates that the antiphase between adjacent cells is transferred gradually fromone end to the other end of the one-dimensional CNNs array, and reflected at the end of array. This transfer process

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of phase continuously exists in the CNN array, so called“phase-inversion-wave”.
x1;1x1;ix1;14(a)Fig. 3. A initial condition for one-dimensional array of 14cells.

(b)

(c)
Fig. 4. Phase-inversion-wave in an one-dimensional arrayof 14 cells.

Fig. 5. Influence of couple parameters on phase-inversionwave

5. INVESTIGATIONS OF PHASE-WAVES

5.2. Phase Waves in In-Anti Phase Synchronization ModeIn this section, we investigate the influence of various system parameters on the propagation of phase-waves; findtheir general regular pattern, so that we can control the phasewave propagation as well.

In this one-dimensional array, We also observe another typeof phase wave propagation in in-anti phase synchronization
mode [10]. In this mode, in-phase and anti phase between
adjacent cells exist alternately, morwever the edge oscillators and their adjacent oscillators cannot be synchronized atin-phase. Hence, this Synchronization state can be observed
only when the number of oscillators is even. We observe thephase inversion waves in this mode by using three different initial states. Figure 7 shows a simulation result of thismode, which is carried out in a one-dimensional array of 8cells under the parameter: a1 = 1.2, a2 = 1.1, c1 = −1,c2 = 2.5, d1 = 0.1, and d2 = −0.1. Figure 7(a) shows thein-anti phase synchronization state, where no phase wave
exist; but in Figs. 7(b) and (c), the phase waves are generated, and they are similarly reflected at both ends of thearray.

5.1. Phase Waves in In-Phase Synchronization ModeIn this subsection, the influence of coupling parameters between adjacent cells on phase inversion wave is investigatedin a one-dimensional array composed nine two-layer CNN
cells with zero-flux boundary condition. The parameter set
is the same with the last simulation except parameters d1and d2 . We respectively carry out three simulations by adopting parameters d1 = 0.1 and d2 = −0.1, d1 = 0.15 andd2 = −0.15, and d1 = 0.2 and d2 = −0.2. The simulation
results are respectively shown in Figs. 5(a), (b) and (c). As
we have seen that, the propagation speed of phase-wave isdirectly proportional to the coupling parameters d1 and d2
between adjacent cells.
Figure 6 shows the influence of different initial stateson the phase wave propagation in a one-dimensional arraycomposed of 18 cells. The analysis of the simulation indicates that the anti-phase point is propagates toward bothends of the array, and is reflected at both ends or the position
that two phase waves collide with each other.

6. CONCLUSIONSIn this study, we have proposed a simpler circuit structure– one-dimensional arrays composed of second order CNN
cells for investigating the phase-wave propagation phenomena. We found that the phase wave propagation phenomena
in two kinds of phase synchronization mode can be reproduced in the CNN array composed of almost numbers ofIII - 51➡

➠(a)
(a)

(b)
(b)
Fig. 6. Influence of initial condition on phase-inversionwave
cells. Moreover, we have investigated qualitatively the influences of the coupling parameters and the initial states onthese propagation phenomena. In future works, we will try
to find out the patterns of these dynamics and clarify theirmechanism, so that we can control the phase propagation
for applications.

(c)Fig. 7. Influence of three kinds of initial conditions onphase-inversion-wave

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