2007 IEEE Workshop on Nonlinear Circuit Networks
NCN'07, Tokushima, Japan, Dec 7--8, 2007

Phasewave Propagation Phenomena
in 2-Dimensional Cellular Neural Networks
Koji URABE

Yoshifumi Nishio

Dept. Electronic Control Eng.,
Niihama N.C.Tech.
coji@ect.niihama-nct.ac.jp

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: nishio@ee.tokushima-u.ac.jpiAbstract— Investigating the nonlinear phenomena is an important work for clarifying dynamics of CNNs. One of thenonlinear phenomena observed in 1-Dimensional CNNs is phasewave propagation phenomena. For applying to image processingby the phase-wave propagation phenomena, in this work, wereport nonlinear phenomena could be observed in 2-dimensionalCNNs.

123M

12I. I NTRODUCTION
Cellular Neural Networks (CNN) are constructed by manycells connected each other. The cell contains linear andnonlinear current sources controlled by voltage. Investigating
the nonlinear phenomena is an important work for clarifying
dynamics of CNNs.
One of the nonlinear phenomena observed in CNNs isphase-wave propagation phenomena (1) . The CNN constructedby 2 cells can oscillated by choosing appropriate parameters.
Coupling oscillating CNNs, 1-dimensional 2-layer CNN canbe construc ted. Putting initial phase difference to any cells
in 1-dimensional 2-layer CNN, phase-wave propagation phenomena that the phase difference is propagate to other cells
could be observed.
In this work, we report nonlinear phenomena could beobserved in 2-dimensional CNNs.

3jN
CNN Oscillator
constructed by 2 cells
Fig.1: 2-Dimension Array constructed by CNN Oscillators(N × M )II. 2-D IMENSIONAL CNN
In this study, we consider what kind of phenomena can beobserved in the CNN oscillators coupled in 2-dimension.
We use 2-layer 2-Dimension modified CNN as shown inFig. 1.
The circuit equation governing the CNN in Fig. 1 are written
asx1,i,j˙ = −x1,i,j + a1 y1,i,j + c1 x2,i,j(1)¡x2,i,j˙ = −x2,i,j + a2 y2,i,j + c2 x1,i,j+d2 y1,i,(j−1) + y1,i,(j+1) + y1,(i−1),j + y1,(i+1),jy`,i,j = 0.5 (|x`,i,j + 1| − |x`,i,j − 1|)

the output its own cell, from state of the cell which is at thesame position in the other layer, and from the output of theneighborhood cell in the other layer, respectively.
For numerical analysis we use this set of parameters which
are same values at 1-Dimension CNN, as shown follows,(2)
a1 = 1, c1 = 1, a2 = 1.2, c2 = −1.1, d2 = 0.05¢
(3)

III. S IMULATION R ESULTS(i = 1, 2, · · · , M, j = 1, 2, · · · N, ` = 1, 2)
where x`,i,j is the state, y`,i,j is the output of CELL`,i,j . This
modified CNN is different from the original CNN in state
feedback from the cell which is at the same position in theother layer. a` , c` and d` are the feedback parameters form

For numerical simulation, we consider 2-dimensional CNN
constructed by 9 cells, M = 3, N = 3. And the initialconditions are given as follows:- 12 -

Table 1: Initial State xh,i,j (0)(a) x1,i,j (0)(b) x2,i,j (0)

ij1
2
3

1
1.0
-1.0
-1.0

2-1.0
1.0
1.0

3
-1.0
1.0
1.0

j(a)10.00.00.0

123i20.00.00.0

30.00.00.0(b)Fig.2: Simulation Result
Simulation Results are shown in Fig. 2, The horizontal axis
is time, the vertical axis in Fig. 2 (a) is the sum of voltagesof vertical adjacent outputs and the vertical axis in Fig. 2 (b)is the sum of voltages of horizontal adjacent outputs,
If the sum of voltages of adjacent outputs is zero, thephase difference between adjacent outputs is +180 [deg]. Aftertransition, the phase difference +180[deg] is observed in cells
shown as follows:(A) Between the vertical positions 1 and 2(B) Between the vertical positions 2 and 3(C) Between the horizontal positions 1 and 2(D) Between the horizontal positions 2 and 3(A) and (B) are observed alternately and phase wave propagation phenomana is grenerated to vertical direction. (C)and (D) are observed alternately. and phase wave propagation
phenomana is grenerated to horizontal direction, too.
From Fig. 2 (a) and (b), (A) and (C) are observed in thesame time. (B) and (D) are observed in the same time, too.
IV. C ONCLUSIONSIn this works, we could be observed phase-wave propagation phenomena in 2-dimensional CNNs. We can observe
phase-wave propagation phenomena to horizontal and verticaldirections.
R EFERENCES[1] Z. Yang, K. Tsuruta, Y. Nishio and A. Ushida, “Investigation of phasewave propagation phenomena in second order CNN arrays”, Proc. ofISCAS’04, vol. 3, pp. 49-52, 2004.
[2] K. Urabe and Y. Nishio, “Comparing Two-Layer CNN with van der Pol
Oscillators Coupled by Inductors”, Proc. of NOLTA’06, pp. 723-726, Sep.
2006.

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