2008 International Symposium on Nonlinear Theory and its Applications
NOLTA'08, Budapest, Hungary, September 7-10, 2008

Oscillatory Phenomena in Cellular Neural Network
Using Two Kinds of Templates
Junji Fujii,† Yasuteru Hosokawa† and Yoshifumi Nishio‡†Faculty of Information Science, Shikoku University
Furukawa, Ojin-cho, Tokushima, Japan‡Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email:{i1739082,hosokawa}@keiei.shikoku-u.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract—In this study, oscillatory phenomena in a cellular neural network using two kinds of templates are investigated. The number of cells is limited to 3 × 3. The
relationship among oscillatory phenomena and two parameters which are corresponding to coupling factors and biases. We consider that investigating this system is important to understanding coupled oscillatory systems. Because
this system is a new class of coupled oscillatory systems.

2. CNN Using Two Kinds of Template

1. Introduction
Cellular Neural Networks (CNNs) [1]-[3] is one kindof mutually coupled neural networks. The main characteristics are the local connection and the parallel signal
processing. There have been many studies on CNNs andmany kinds of CNNs have been proposed. One of themis two-layer CNN [4]. Two-layer CNN can generate many
interesting phenomena. For instance, self-organizing pattern [4], active wave propagation [5] and so on are observed. Like this, some kinds of CNNs generate interesting
phenomena. Investigating these system contributes to understand complex systems and to apply them to engineeringsystems.
In our earlier study [6], we proposed a CNN using twokinds of templates. This system was proposed in order
to investigate a new class of coupled oscillatory system.
Namely, by using two kinds of templates, oscillatory factors are coupled checkered. It is not so easy to implement
this structure by using oscillatory circuit. On the proposedsystem, self-organizing pattern, active wave propagationand clustering phenomena were observed. Especially, clustering phenomena were investigated in detail. It was conﬁrmed that clusters were classiﬁed into four kinds of patterns. Relationship among four patterns were also investigated.
In this system, we observed oscillatory phenomena bychanging parameters. This phenomena means that this system is a new class of coupled oscillatory system. In thisstudy, we investigate this phenomena. Especially, we pay
attention to the relationship among values of templates andobserved phenomena.

Figure 1: System model of CNN using two kinds of templates.
Figure 1 shows a system model of our proposed system.
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, 1 ≤ i ≤ M and1 ≤ j ≤ N. The coupling radius is assumed to be one. In
this proposed CNN, two kinds of templates are used. Cells
having one template are called as Cell α and the other are
called as Cell β. These two types of the cells are placed
as checkered. The state equations of the cells are given as
follows:
1: The case that i + j is an even number.

- 688 -dxi j= − x i j + Iαdt∑
Aα (i, j; k, l)ykl+c(k,l)+∑c(k,l)Bα (i, j; k, l)ukl(1)2: The case that i + j is an odd number.
dxi j= − xi j + Iβ
dt∑+Aβ (i, j; k, l)ykl
c(k,l)+∑(2)Bβ (i, j; k, l)ukl

c(k,l)(a) Initial state(b) transient state 1(c) transient state 2(d) transient state 3A{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. The output equation of the cell isgiven as follows:
yi j = f (xi j ).

(3)f (x) = 0.5(|x + 1| − |x − 1|).

(4)

where,The variables u and y are the input and output variables ofthe cell, respectively. Aα , Bα , Aβ and Bβ are 3 times 3
matrices, 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, j)

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 normalCNNs. This system has a unique characteristic. Namely, apair of cell α and cell β are needed for a simple oscillation.
Additionally, one cell α connects with four neighbor cells
β and one cell β also connects with four neighbor cells α.
Like this, these cells are sharing a factor of oscillation. Thistype of connection may be diﬃcult to realize by couplingnormal oscillators. Hence, we consider that this system is
a new class of coupled oscillatory systems.
3. Relationship Between Oscillatory Phenomena andParameters
In our earlier study, clustering phenomena observed in
the system have been investigated. The initial conditions
and parameters are set as follows. The values of boundary
cells are set zero. The template is set as 1 −1 1 −1 1 −1



Aα =  1 1 1  , Aβ = −1 −1 −1 ,(6)1 −1 1−1 1 −1Bα = 0, Bβ = 0, Iα = 0, Iβ = 0,The number of cells are set as 49 × 49. In this clustering phenomena, the cells between clusters oscillate and theinner cell of clusters don’t oscillate. Namely, most of thecells don’t oscillate in spite of our purpose which is proposing a new class of coupling oscillatory systems. However,by changing only Iα and Iβ of Eq. (6), oscillatory phenomena are observed shown in Fig. 2. In this case, parameters

Figure 2: Oscillatory phenomena for the CNN array consists of 49 × 49.
Iα and Iβ are set as 1. Initial states of cells are set as 1shown in Fig. 2 (a). By starting the computer simulation,
oscillatory phenomena are observed as shown in Figs. 2(b)-(d) in series. At ﬁrst, the phenomena look like symmetry shown in Figs. 2 (b) and (c). Then, symmetric phenomena is broken shown in Fig. 2 (d), suddenly. After that allcells keep oscillating. In order to investigate this phenomena, the number of cells is limited to 3×3 in this study. The
initial conditions and parameters are set as follows. Initialstate values of boundary cells are set as 0.
 p −p p −p p −p
Aα =  p p p  , Aβ = −p −p −p ,(7)p −p p−p p −pBα = 0, Bβ = 0, Iα = q, Iβ = q,
where p and q are corresponding to the coupling factors andbiases Iα and Iβ , respectively. By using this conditions andparameters, computer calculations are carried out. Here,state variables of each cells are deﬁned as shown in Fig. 3.
Figure 4 shows the computer simulation results in the caseof cell array 3 × 3. Vertical axis is the value of state variables. Horizontal axis is time. Cell A, C, G and I are thesame values each other. In the same way, cell B, D, F and H
are also the same values each other. Therefore, only three
waveforms are observed. The case of p = 1 and q = 1
is corresponding to Fig. 2. In this case, oscillatory phenomena are not observed as shown in Fig. 4 (a). However,oscillatory phenomena are observed in the cases of somecombinations of p and q. Some of these results are shownin Figs. 4 (b)-(d). Parameters of Fig. 4 (b) diﬀer from parameters of Fig. 2 in Iα and Iβ . As a result of this investigation, we can conﬁrm that the waveform frequency and- 689 -amplitude are inﬂuenced by changing variables p and q.
Next, we investigate the relationship among oscillatory
phenomena, variables p and q. Figure 5 shows the relationship among oscillatory phenomena, variables p and q.
Horizontal axis is the value of p. Vertical axis is the valueof q. ”O” means that the oscillatory phenomena are observed. ”N” means that the oscillatory phenomena aren’tobserved. Oscillatory phenomena are observed in two regions. In lower side region, similar waveforms to Fig.4(b) are observed. Increasing the value p increases the amplitude and decreases the frequency. In upper side region,similar phenomena to Fig.4 (c) are observed. Increasing thevalues p and q increases the amplitude and the frequency.
Please note that the oscillatory phenomena are not observedin the case of q = 1.0. This line separates two region. Additionally, in the case of 49 × 49, oscillatory phenomena areobserved. It can be considered that border cells inﬂuence
the oscillatory phenomena.

(a) p = 1 and q = 1.

4. Conclusions
In this study, oscillatory phenomena in cellular neuralnetworks using two kinds of templates were investigated.
Especially, the case that the number of cells are limited to 3
× 3 were investigated. As a result, the relationship among
oscillatory phenomena, values p and q were shown. Additionally, it could be conﬁrmed that the border of CNNs
inﬂuences the oscillatory phenomena.
In our future works, we will investigate the case of thelarge numbers of cells.

(b) p = 1 and q = 0.5.

(c) p = 2 and q = 0.5.

(d) p = 3 and q = 0.5.
Figure 4: Waveforms of state values of each cells. The
number of cells is 3 × 3.

Figure 3: The CNN array consists of 3×3.

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Figure 5: Relationship among the oscillatory phenomena,value of p and q. ”O” means that oscillatory phenomena
are observed. ”N” means that oscillatory phenomena arenot observed.

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