2007 International Symposium on Nonlinear Theory and itsApplications
NOLTA'07, Vancouver, Canada, September 16-19, 2007

Wave Phenomena in Cellular Neural Networks
Using Two Kinds of Template Sets
Junji Fujii, † Yasuteru Hosokawa,† and Yoshifumi Nishio‡†Dept. of Information Science, Shikoku University
Furukawa, Ojin-cho, Tokushima, 771-1192 JAPAN‡Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN
Email: i1739082@keiei.shikoku-u.ac.jp, hosokawa@keiei.shikoku-u.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract—In this study, we propose cellular neural networks using two kinds of template sets. The computer simulations show that the proposed system can generate similar phenomena which is observed from the two-layer cellular neural networks. Investigating these phenomena contributes to understand complex systems and to apply themto engineering systems.

1: The case that i + j is an even number.
dxi j= − x i j + Iαdt∑
+Aα (i, j; k, l)ykl
c(k,l)+∑(1)Bα (i, j; k, l)ukl

c(k,l)

1. Introduction

2: The case that i + j is an odd number.

Cellular Neural Network (CNN) [1]-[3] is one kind ofneural networks. The main characteristic is the local connection. There have been many studies on CNN and manykinds of CNN have been proposed. One of them is twolayer CNN. Two-layer CNN can generate many interesting
phenomena. For instance, self-organizing pattern [4], active wave propagation [5] and so on. Investigating these
phenomena contributes to understand complex systems andto apply them to engineering systems.
In this study, we propose CNN using two kinds of template sets. One of the aim of this study is a reproduce ofactive wave propagation phenomena observed from coupled two-layer CNN. Our ﬁnal aim is investigation of a
new class of complex systems using this CNN. The proposed system consists of two types of cells which have different template sets. These cells are places as checkered.
The computer simulations show that the proposed CNN
can generate similar phenomena which is observed fromthe two-layer CNN.

xi j
= − x i j + Iβdt∑
+Aβ (i, j; k, l)ykl
c(k,l)+∑(2)Bβ (i, j; k, l)ukl

c(k,l)

A{αβ} (i, j; k, l)ykl , B{αβ} (i, j; k, l)ukl and I{αβ} are called as thefeedback coeﬃcient, the control coeﬃcient and the bias
current, respectively.

2. CNN Using Two Kinds of Template Sets
Figure 1 shows a system model in this study. We assume that the system has a two-dimensional M by N array structure. Each cell in the array is denoted as c(i, j),where (i, j) is the position of the cell, where 1 ≤ i ≤ M
and 1 ≤ j ≤ N. The coupling radius is assumed to be one
in this study. In this proposed CNN, we use two kinds oftemplate sets. Cells having one template set are called ascell α and the other are called as cell β. These two typesof the cells are placed as checkered. The state equations of
the cells are given as follows:

Figure 1: Structure of CNN using two kinds of templatesets.

- 23 -The output equation of the cell is given as follows:
yi j = f (xi j ).

(3)where,f (x) = 0.5(|x + 1| − |x − 1|).

(4)

Figure 2 shows simulation results using the following
template set.
−0.2 0 −0.20.1 0 0.1
1.8 −1  , Aβ =  1 0.3 0  ,Aα =  0
−0.2 0 −0.2
0.1 0 0.1

The variables u and y are the input and output variables ofthe cell, respectively. Aα , Bα , Aβ and Bβ are 3 times 3matrices, which can be described to have a similar form toEq. (5).
Aα (i, j; i − 1, j − 1) Aα (i, j; i, j − 1)Aα (i, j; i + 1, j − 1)

Aα (i, j; i − 1, j)
Aα (i, j; i, j)
Aα (i, j; i + 1, j1)

Aα (i, j; i − 1, j + 1)

Aα (i, j; i, j + 1) Aα (i, j; i + 1, j + 1)
(5)This proposed system is more complex than the normal
CNN. This system has a peculiar characteristic in order
to investigate a new class of coupled oscillatory systems.
Namely, a pair of cell α and cell β are needed for a simple
oscillation. However, one cell α connects with four neighbor cells β and one cell β also connects with four neighborcells α. Like this, these cells are sharing a factor of oscillation. This type of connection may be diﬃcult to realize bycoupling normal oscillators.

Bα = 0, Bβ = 0, Iα = 0, Iβ = 0,(6)We can observe a stripe line formation. In the case of thetwo-layer CNN, a checkerboard pattern formation is observed using the following template sets.
 0 0 0.1 0 −0.20 
A1 = −0.2 1.8 −0.2 , A2 = 0.1 0.3 0.1 ,
0−0.2
00 0.1 0
0 0 0
0 0 0

C1 = 0 −1 0 , C2 = 0 1 0 ,
0 0 00 0 0
Bα = 0, Bβ = 0, Iα = 0, Iβ = 0,First, we try to reproduce some phenomena observed
from mutually coupled two-layer CNN [5]. The CNN arrayconsists of 50×50 cells with a zero-ﬁxed boundary condition.

(7)
In this template set, all corners of matrices are set to zero.
By utilizing this characteristic, Eq. (6) can be equivalent toEq. (7). As a results, cell β corresponds to the second layercell. Thus, a checkerboard pattern formation in the twolayer CNN becomes a stripe line pattern formation. In thefollowing subsections, the same method is used.

3.1. Stripe Line (Checkerboard)

3.2. Stripe

3. Computer Simulations

(a)

(b)

(a)

(c)

(b)

(c)

Figure 2: Stripe line pattern formation. (a) Initial state.
(b) Transient state. (c) Stable state.

Figure 3: Stripe pattern formation.
(b) Transient state. (c) Stable state.

- 24 -(a) Initial state.

Figure 3 shows simulation results using the following
template set.
0.1 0 0.1
0.0100.01


2.56
0  ,Aα =  0 2.2 −1  , Aβ =  1

0.1 0 0.1
0.01
00.01
Bα = 0, Bβ = 0, Iα = 0, Iβ = 0,(8)We can observe a stripe pattern formation. In the case of
the two-layer CNN, the boundary between white zone andblack zone is very clear. However, the boundary betweenwhite zone and black zone is not clear for cells β. The
corresponding template set in the two-layer CNN is shownas follows: 0 0.1 0 
 00.010 




A1 = 0.1 2.2 0.1 , A2 = 0.01 2.56 0.01 ,
0 0.1 000.010
0 0 0
0 0 0




C1 = 0 −1 0 , C2 = 0 1 0 ,




0 0 0
0 0 0

Bα = 0, Bβ = 0, Iα = 0, Iβ = 0.
(11)
4. Conclusions
We have proposed CNN using two kinds of templatesets. In the proposed CNN, we could observe similar phenomena to the two-layer CNN. Investigating more complex
phenomena which cannot be seen from the two-layer CNN
is our future research.
References[1] L. O. Chua and L. Yang, “Cellular Neural Networks:Theory,” IEEE Trans. Circuits Syst., vol. 35, no. 10,
pp. 1257–1272, 1988.

Bα = 0, Bβ = 0, Iα = 0, Iβ = 0.
(9)[2] L. O. Chua and L. Yang, “Cellular Neural Networks:Applications,” IEEE Trans. Circuits Syst., vol. 35,
no. 10, pp. 1273–1290, 1988.
[3] L. O. Chua, M. Hasler, G. S. Moschytz andJ. Neirynck, “Autonomous Cellular Neural Networks:
a uniﬁed paradigm for pattern formation and active wave propagation,” IEEE Trans. Circuits Syst. I,
vol. 42, no. 10, pp. 559–577, 1995.

3.3. Active Propagation Phenomena

Figure 4 shows simulation results using the following
template set.
−0.01
0.1 0 0.1
0

1.04
Aα =  0 1.1 −1  , Aβ =  1
−0.01
00.1 0 0.1shown as follows: 0 0.1 0 
 0−0.010 

A1 = 0.1 1.1 0.1 , A2 = −0.01 1.04 −0.01 ,
0 0.1 00−0.010
0 0 0
0 0 0

C1 = 0 −1 0 , C2 = 0 1 0 ,
0 0 00 0 0
−0.01

0  ,
−0.01[4] P. Arena, S. Baglio, L. Fortuna and G. Manganaro,“Self-Organization in a Two-Layer CNN,” IEEE Trans.
Circuits Syst. I, vol. 45, no. 2, pp. 157–162, 1998.
[5] Z. Yang, Y. Nishio and A. Ushida, “Generationof Various Types of Spatio-Temporal Phenomena in
Two-Layer Cellular Neural Networks,” IEICE Trans.
Funds., vol. E87-A, no. 4, pp. 864–871, 2004.

Bα = 0, Bβ = 0, Iα = 0, Iβ = 0,
(10)
Two concentric circular waves are generated. Their wave
fronts propagate in all directions through the network fromtheir initialized positions shown as Figs. 4(a)-(g). When
the waves hit the end of CNN, the waves reﬂect shown asFigs. 4(g)-(l). After that the pattern becomes complex pattern as Figs. 4(m)-(t). In the case of the two-layer CNN,
waves do not reﬂect at the end of CNN. We consider thatthe diﬀerence of the wave reﬂection comes from the factthat the cell α and the cell β on the edge of the proposed
CNN can not be the same as the pairs of non-edge cells.
Detailed investigation of the reason is our future research.
The corresponding template set in the two layer CNN is[6] Z. Yang, Y. Nishio and A. Ushida, “Image Processingof Two-Layer CNNs — Applications and their Stability,” IEICE Trans. Funds., vol. E85-A, no. 9, pp. 473–490, 2002.
[7] Z. Yang, K. Tsuruta, Y. Nishio and A. Ushida, “Investigation of Phase-Wave Propagation Phenomena in Second Order CNN Arrays,” Proc. of ISCAS’04, vol. 3,pp. 49–52, 2004.

- 25 -(a) initial state.

(b) 1000 iteration.

(c) 2000 iteration.

(d) 3000 iteration.

(e) 4000 iteration.

(f) 5000 iteration.

(g) 6000 iteration.

(h) 7000 iteration.

(i) 8000 iteration.

(j) 9000 iteration.

(k) 10000 iteration.

(l) 15000 iteration.

(m) 20000 iteration.

(n) 25000 iteration.

(o) 30000 iteration.

(p) 35000 iteration.

(q) 40000 iteration.

(r) 45000 iteration.

(s) 50000 iteration.

(t) 55000 iteration.

Figure 4: Active Propagation Phenomena. (a) Initial state. (b)-(t) Snapshots obtained at diﬀerent iterations.

- 26 -