2008 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 5-6, 2008

Relationship Between Oscillatory Phenomena andBoundary Conditions in Two-template CNN
Junji Fujii, Yasuteru Hosokawa

Yoshifumi Nishio

Dept. Information Science,
Shikoku University
Email: {i1739082, hosokawa}@keiei.shikoku-u.ac.jp

Dept. Electrical and Electronic Eng.,
Tokushima University
nishio@ee.tokushima-u.ac.jp

II. CNN USING TWO KINDS OF TEMPLATESAbstract— In our previous study, we proposed a two-template
CNN. We investigated oscillatory phenomena in two-template
CNN. As a result, it was conﬁrmed that the boundary condition of CNN affects the oscillatory phenomena. In this study,
we investigate relationship between oscillatory phenomena andboundary conditions. We consider that investigating the relationship between oscillatory phenomena and boundary conditions areimportant to understanding coupled oscillatory systems. Because
we consider that this system is a new class of coupled oscillatory
systems.

I. INTRODUCTION
Cellular Neural Networks (CNNs) [1]-[3] are one kind of
mutually coupled neural networks. The main characteristics
are the local connection and the parallel signal processing.
There have been many studies on CNNs and many kindsof CNNs have been proposed. One of them is two-layer
CNN [4]. Two-layer CNN can generate many interestingphenomena. For instance, self-organizing pattern, active wave
propagation [5] and so on are observed. Like this, some kindsof CNNs generate interesting phenomena.
In our previous study [6], A CNN using two kinds oftemplates (two-template CNN) was proposed. Cells having
one template and the other are placed as checkered. Thesystem generates similar phenomena to Two-layer CNN. For
instance, self-organizing patterns, active wave propagation
phenomena, clustering phenomena and oscillatory phenomena
were observed. Because the structure is simpler than TwoLayer CNN, the implementation is easier than Two-Layer
CNN.
We consider that this system is a new class of coupledoscillatory system. Because oscillatory factors are coupledcheckered by using two templates. Thus, we consider that investigating oscillatory phenomena is very important work. Additionally, realizing this structure by using oscillatory circuit
is not so easy. Before now, a relationship between oscillatory
phenomena and initial values was investigated in detail. Asa result, we conﬁrmed that the boundary condition of CNN
affects the oscillatory phenomena.
In this study, we investigate the relationship between oscillatory phenomena and boundary conditions. In the case of
normal CNN, oscillatory phenomena could not generated by
the case of symmetry and linear templates. From this point ofview, boundary conditions play a role of asymmetrization of
the system are guessed.

Fig. 1.

System model of CNN using two kinds of templates.

Figure 1 shows a system model of proposed system. We
assume that the system has a two-dimensional M by N arraystructure. Each cell in the array is denoted as c(i, j), where(i, j) is the position of the cell, 1 ≤ i ≤ M and 1 ≤ 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 Cellβ. 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.

- 56 -dxij= − xij + Iα
dt∑+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.
dxij= − xij + Iβ
dt∑+Aβ (i, j; k, l)ykl
c(k,l)+∑(2)

initial state values set as 1. The CNN array consists of 15 ×
15. The template is set as−p p −pp −p pAα =  p p p  , Aβ = −p −p −p ,(6)−p p −pp −p p
B α = 0, B β = 0, Iα = q, Iβ = q,Bβ (i, j; k, l)ukl

c(k,l)

where p and q are corresponding to the coupling factors andA{αβ} (i, j; k, l)ykl , B{αβ} (i, j; k, l)ukl and I{αβ} are called biases Iα and Iβ , respectively. Template parameter p set as
as the feedback coefﬁcient, the control coefﬁcient and the bias 0.5, 1, 1.5, . . . , 6, and bias q set as 0, 0.5, 1, . . . , 13. Bycurrent, respectively. The output equation of the cell is given using these conditions and parameters, computer simulations
are carried out. Some of these conditions were carried out inas follows:
our previous study. As a result of previous study, it is clearthat two kind of oscillatory phenomena are observed. One isyij = f (xij ).
(3)
the case that phenomena are observed in all cells. Anotheris the case that phenomena are observed in around boundary
where,f (x) = 0.5(|x + 1| − |x − 1|).
(4) cells only.
Figures 3 – 8 show simulation results. Horizontal axis is theThe variables u and y are the input and output variables of the value of p. Vertical axis is the value of q. ”A” means that thecell, respectively. Aα , B α , Aβ and B β are 3 times 3 matrices, oscillatory phenomena are observed in all cells. ”R” means thatwhich can be described to have a similar form to Eq. (5).
01 the oscillatory phenomena are observed in boundary cells only.
”N” means that the oscillatory phenomena are not observed.
Aα (i, j; i − 1, j − 1) Aα (i, j; i − 1, j) Aα (i, j; i − 1, j + 1)
@ Aα (i, j; i, j − 1)
Aα (i, j; i, j)
Aα (i, j; i, j + 1) A Figure 3 shows the case that boundary conditions are set as
Aα (i, j; i + 1, j − 1) Aα (i, j; i + 1, j) Aα (i, j; i + 1, j + 1)
0. Two kinds of oscillatory phenomena are observed and the(5)area is divided two areas. One is observed in 1.5 ≤ p ≤ 6This proposed system is more complex than the normal CNNs.
and 0 ≤ q ≤ 0.5. In this area, oscillatory phenomena ”R” areThis system has a unique characteristic. Namely, a pair of cellobserved. Another is observed in the area which satisﬁes a
α and cell β are needed for a simple oscillation. Additionally,ﬁxed ratio of p and q. In this area, oscillatory phenomena ”A”
one cell α connects with four neighbor cells β and one cell β
are observed. Figure 4 shows the case that boundary conditions
also connects with four neighbor cells α. Like this, these cellsare set as 1. Two kinds of oscillatory phenomena are observed.
are sharing a factor of oscillation. This type of connection
The area of ”R” is larger than the case of Fig. 3. Figure 5
may be difﬁcult to realize by coupling normal oscillators.
shows the case that boundary conditions are set as -1. TwoHence, we consider that this system is a new class of coupled
regions of oscillatory phenomena are observed. However, theoscillatory systems.
region of ”R” is very smaller than the cases of Fig. 3 andFig. 4. Figure 6 shows the case that boundary conditions ofIII. RELATIONSHIP BETWEEN OSCILLATORYvirtualcells α are set as 1 and boundary conditions of virtual
PHENOMENA AND BOUNDARY CONDITIONcells β are set as -1. In this case, phenomena ”R” are observed
In our previous study, we observed oscillatory phenomena in large area compared with Figs. 3 – 5. Figure 7 shows theshown in Fig. 2. Figure 2 (a) shows initial state which is case that boundary conditions of virtual cells α are set as -1
set as 1. Figure 2 (b) and (c) show snapshots. We can and boundary conditions of virtual cells β are set as 1. In thisobserve asymmetry oscillation phenomena like as Fig. 2 (b) case, phenomena ”R” are observed in p = 1.5 and q = 0 only.
via symmetry oscillation like as Fig. 2 (c).
Figure 8 shows the case of the periodic boundary condition.
Namely, the shape of CNN becomes torus and there are novirtual cells. In this case, the CNN array consists of 14 × 14
because of keeping symmetrical property of the system. In thiscase, oscillatory phenomena are not observed.
From these results, it is considered that boundary conditions
affects oscillatory phenomena. Especially, conditions of virtual
cells α are very strong. Additionally, it is considered that45
boundary cells are needed for oscillatory phenomena. Because(a) Initial state(b) 10 iteration(c) 10 iteration
oscillatory phenomena are not observed in the case of theFig. 2.
Oscillatory phenomena for the CNN array consists of 49 × 49.
periodic boundary condition.
p = 1 and q = 1.

In this study, we investigate about ﬁve kinds of ﬁxed
boundary conditions and a periodic boundary condition. TheIV. CONCLUSIONSIn this study, the relationship between oscillatory phenomena and boundary conditions of two-template CNN was- 57 -investigated. Computer simulations were carried out about six
kinds of boundary conditions.
As a result, we conﬁrmed that oscillatory phenomena of
the system strongly depends on these boundary conditions.
Especially, conditions of virtual cells α are very strong andboundary cells are needed for oscillatory phenomena. However, we did not investigate the case of the Zero-ﬂux boundary conditions and the other initial conditions. Additionally,we limited values of templates. These conditions should beinvestigated.
In our future works, we will investigate these conditions andcompare this system with other coupled systems.
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Fig. 3.
Relationship between oscillatory phenomena and parameters.
Horizontal axis is p and vertical axis is q. The boundary condition is set
as 0.

Fig. 4.
Relationship between oscillatory phenomena and parameters.
Horizontal axis is p and vertical axis is q. The boundary condition is set
as 1.

- 58 -Fig. 5.
Relationship between oscillatory phenomena and parameters.
Horizontal axis is p and vertical axis is q. The boundary condition is set
as -1.

Fig. 7.
Relationship between oscillatory phenomena and parameters.
Horizontal axis is p and vertical axis is q. Virtual cells α are set as -1 andvirtual cells β are set as 1.

Fig. 6.
Relationship between oscillatory phenomena and parameters.
Horizontal axis is p and vertical axis is q. Virtual cells α are set as 1 andvirtual cells β are set as -1.

Fig. 8.
Relationship between oscillatory phenomena and parameters.
Horizontal axis is p and vertical axis is q. The boundary condition is set
as a periodic boundary condition. CNN array is set as 14 × 14.

- 59 -