平成20年度電気関係学会四国支部連合大会

379

17-10

Relationship between Oscillatory Phenomena andInitial Values in Two-Template CNN
Junji FUJII(Shikoku University)

Yasuteru HOSOKAWA(Shikoku University)

Yoshifumi NISHIO(Tokushima University)

1. Introduction
In our earlier study, we proposed Two-Template
CNN[1]. Two kinds of templates are used in this CNN.
Cells having one template and the other are placed ascheckered. This system is proposed in order to investigate a new class of coupled oscillatory system. Namely,
by using two kinds of templates, oscillatory factors arecoupled checkered. Realizing this structure using oscillatory circuits is not so easy. In this system, we observed oscillatory phenomena. The phenomena meanthat this system is a new class of coupled oscillatory
system.
In this study, we investigate the relationship between oscillatory phenomena and initial values.

2. System Model

Figure 1: System model of Two-Template CNN.

Figure 1 shows a system model of our proposedsystem. We assume that the system has a twodimensional M by N array structure. Each cell in thearray is denoted as c(i, j), where (i, j) is the position ofthe cell, 1 ≤ i ≤ M and 1 ≤ j ≤ N . The coupling radius is assumed to be one. In this proposed CNN, twokinds 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.
dxij= − xij + Iα
dt∑+Aα (i, j; k, l)ykl
c(k,l)+∑(1)Bα (i, j; k, l)ukl

c(k,l)

2: The case that i + j is an odd number.
dxij= − xij + 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} arecalled as the feedback coeﬃcient, the control coeﬃcient and the bias current, respectively. The output
equation of the cell is given as follows:yij = f (xij ).

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

(4)

The variables u and y are the input and output variables of the cell, respectively. Aα , B α , Aβ and B β are
3 times 3 matrices.

3. Oscillatory Phenomena and Initial Values
We investigate a relationship between oscillatory
phenomena and initial values in this system. The number of cells are set as 15 × 15. We deﬁne two parameters which concern oscillatory phenomena. Some patterns of initial values are investigated. As a result, we
conﬁrmed that there are two cases. One is the case that
oscillatory phenomena are observed in all patterns ofinitial values. Additionally, in this case, we conﬁrmedthat the ratio of two parameters is a constant. The
other is the case that oscillatory phenomena are observed by depending on initial values. In this case, theregion shown as two parameter is changed by initialvalues.

4. Conclusions
In this study, we investigate a relationship between oscillatory phenomena and initial values in TwoTemplate CNN. As a result, we conﬁrmed that there
are two cases.

References[1] Junji Fujii, Yasuteru Hosokawa and Yoshifumi Nishio,“Wave Phenomena in Cellular Neural Networks Using TwoKinds of Template Sets,” Proc. of 2007 International Symposium on Nonlinear Theory and its Applications, pp. 2326, Sep. 2007.