2009 International Workshop on Nonlinear Circuits and Signal Processing
NCSP'09, Waikiki, Hawaii, March 1-3, 2009

Relationship Between Oscillatory Phenomena andTemplates on a 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
Email: nishio@ee.tokushima-u.ac.jp

Abstract— In our previous studies, we have proposed a twotemplate CNN and have investigated some kinds of boundary
conditions. As a result, it was revealed that oscillatory phenomena
are not observed in the case of periodic boundary conditions.
In this study, we investigate the relationship between oscillatory phenomena and templates on a two-template CNN whose
output function is modiﬁed. By using modiﬁed output function,
oscillatory phenomena are observed in the case of periodic boundary conditions. Additionally, the synchronization phenomena areobserved by changing templates parameters. We consider thatinvestigating this system is important to understanding coupledoscillatory systems. Because this system is a new class of coupledoscillatory systems.

II. CNN USING TWO KINDS OF TEMPLATE
Figure 1 shows a system model of two-template CNN. 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. 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.
dxij= − xij + Iα
dt∑+Aα (i, j; k, l)ykl

I. I NTRODUCTION
Cellular Neural Networks (CNNs) [1]-[3] is 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 interesting
phenomena. For instance, self-organizing pattern [4], active
wave propagation [5] and so on are observed. Like this, somekinds of CNNs generate interesting phenomena. Investigatingthese system contributes to understand complex systems.
In our previous studies [6], we have proposed a twotemplate CNN and have investigated some observed phenomena. Additionally, we investigated some kinds of boundary
conditions in this system. As a result, oscillatory phenomena,clustering phenomena, pattern formation, active wave propagation and so on are observed. We consider that this system
is a novel coupled oscillatory system. The reason of why weconsider is that the cell connects with four the other cells
and the other cell also connects with four neighbor cells.
Namely, a pair of cells which have two kinds of templates
are needed for a simple oscillation. Like this, these cells aresharing a factor of the oscillation. This type of connection
may be difﬁcult to realize by coupling normal oscillators.
However, these phenomena are not observed in the case ofperiodic boundary conditions.
In this study, we investigate the relationship between oscillatory phenomena and templates on a two-template CNN.
By using the modiﬁed output function. We consider thatinvestigating this system is important to understanding coupledoscillatory systems.

c(k,l)+∑(1)Bα (i, j; k, l)ukl

c(k,l)

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

- 253 -dxij= − xij + Iβdt∑
+Aβ (i, j; k, l)ykl
c(k,l)+∑Bβ (i, j; k, l)ukl

c(k,l)

Fig. 1.

Structure of two-template CNN.

(2)A{αβ} (i, j; k, l)ykl , B{αβ} (i, j; k, l)ukl and I{αβ} are called
as the feedback coefﬁcient, the control coefﬁcient and the bias
current, respectively. The output equation of the cell is given
as follows:

Aβ . Thus, the templates are set as follows.
−u v −uu −v
Aα =  v w v  , Aβ = −v −w−u v −uu −v

yij = f (xij ).

(3)

B α = 0, B β = 0, Iα = 0, Iβ = 0.

f (x) = 0.5(|x + 1| − |x − 1|).

(4)

where,The variables u and y are the input and output variables of thecell, respectively. Aα , B α , Aβ and B β are 3 times 3 matrices,which can be described to have a similar form to Eq. (5).
0Aα (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)u−v  ,u(8)Namely, we guess that coupling coefﬁcients and oscillation
factors are corresponding to u and v, respectively. The numberof cells is ﬁxed as 8 × 8. Because that in the case of lower
than 8 × 8, only one kind of synchronization phenomena isobserved and in the case of over 8 × 8, various kinds ofoscillatory phenomena are observed.

1
Aα (i, j; i − 1, j + 1)
Aα (i, j; i, j + 1) A
Aα (i, j; i + 1, j + 1)
(5)

In previous study, a piecewise-linear function shown in Eq. 4
was applied as the output function. We could not observe
oscillatory phenomena in the case of periodic boundary condition. However, the oscillatory phenomena are observed bymodifying a output function. The modiﬁed output function isgiven as a following function.
f (x) = −x + |x + 1| − |x − 1|.

(6)This proposed system is more complex than the normal CNNs.
Although 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. This typeof connection may be difﬁcult to realize by coupling normal
oscillators. Hence, we consider that this system is a new classof coupled oscillatory systems.
III. R ELATIONSHIP BETWEEN OSCILLATORY PHENOMENAAND PARAMETERS

In our previous study, some kinds of boundary conditions
have been investigated. Then, initial state values set as random
values. The parameter of templates are given as follows:−1 1 −1
1 −1 1Aα =  1 1 1  , Aβ = −1 −1 −1 ,(7)−1 1 −11 −1 1
B α = 0, B β = 0, Iα = 0, Iβ = 0.
As a result, it was revealed that oscillatory phenomena areaffected by virtual cells. Furthermore, it was also revealed thatoscillatory phenomena are not observed in the case of periodic
boundary conditions which has no virtual cells.
In this study, we investigate in the case of modiﬁed output
function shown in Eq. 6. We guess that coupling coefﬁcientsand oscillation factors are corresponding to values of Aα and(a)

(b)

(c)

Fig. 2. Simulation results in the case of periodic boundary condition. Thenumber of cells is 8 × 8. u = 1, v = 1 and w = 1. (a) Initial state. (b)Transient state. (c) Stable state.

Figure 2 shows a computer simulation result in the caseof u = 1, v = 1 and w = 1. Each square shows a outputof cells. Oscillatory phenomena are observed in Fig. 2 (b)and (c). In Fig 2 (b), each cell oscillates independently. Thisphenomenon is a transient state. After that, synchronizationlike phenomena are observed like as Fig. 2 (c). In this result,
we seem to be able to classify synchronization-like phenomena
into four groups as shown in Fig. 3. In order to investigate thephenomena in detail, some computer simulations are carriedout. The template parameters set as Eq. 8.
Figure 4 shows waveforms of state values. Vertical axes
are state variables. Horizontal axes are time. The ﬁrst row
shows state variables of α1 group. In the same way, the second,
third and fourth row shows state variables of β1 , β2 and α2 ,respectively. Each row includes sixteen waveforms. This result
is corresponding to Fig. 2. This result shows that these are notsynchronized. However, some waveforms of them are similarforms. Like this method, we investigate waveforms in someparameter and initial values. Figures 5–8 show some typicalresults.
Figure 5 shows waveforms of state values in the case ofu = 1.2, v = 1 and w = 1. By changing initial values, twokinds of stable states are observed like as Fig. 5 (a) and (b).
All cells of each row are synchronized. Thus, we can see one
waveform per one row. A difference of Fig. 5 (a) and (b)is that ﬁrst row and fourth row are replaced each other. The
reason of why is symmetrical property of the system. Namely,
by rotating the system 90 degree, Fig. 5 (a) becomes Fig. 5(b).
Figure 6 shows the case that some waves of each groupsynchronize and over three stable states exist. Different initial- 254 -values are applied to Fig. 6 (a), (b) and (c). In this case, wavesof each row are classiﬁed some groups. Waves of each groupare synchronized. Frequencies of these waves are same. Someof waves are same waveforms and different phases. Anti-phasesynchronization phenomena are also observed. The ﬁrst rowof Fig. 6 (b) is similar to the inverted fourth row of Fig. 6(c). Likewise, the second, third and fourth rows of Fig. 6 (b)is similar to the inverted third, second and ﬁrst rows of Fig. 6(c), respectively. Like this, many stable states are observed bychanging initial values.
Figure 7 shows the case that switching phenomena areobserved. In the left side of ﬁgure, all waves exist negativesides of vertical axes and these are not synchronized. In thecenter of ﬁgure, almost waves switch to the positive values
in same time. Like this switching phenomena are observed
randomly. Sojourn time until switching is corresponding toparameter v.
Figure 8 shows the case of increasing a parameter v fromFig. 6. Different initial values are applied to Fig. 8 (a) and(b). Same synchronization phenomena as shown in Fig. 6 areobserved in Fig. 8 (a). On the other hand, similar phenomena
as shown in Fig. 4 are observed in Fig. 8 (b). Like this,two kinds of phenomena are observed by initial values. By
increasing or decreasing a parameter v from Fig. 6, a decreaseof observation rates of synchronization phenomena is caused.

Fig. 3.

As future works, we will investigate these phenomena indetail.
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The CNN array consists of 8 × 8.
Fig. 4.
Waveforms of state values. u = 1.0, w = 1.0 and v = 1.0. This
result is corresponding to Fig. 2.

IV. C ONCLUSIONSIn this study, we have investigated the relationship between
oscillatory phenomena and templates on a two-template CNN
whose output function is modiﬁed. As results, follows phenomena are conﬁrmed.
Coexistence of some kinds of stable states
Various kinds of synchronization phenomena• Switching phenomena• Existence of a relationship between synchronization phenomena and parameters••- 255 -(a)(b)

Fig. 5. Waveforms of state values in the case that some waves of each group synchronize and two stable states exist. u = 1.2, w = 1.0 and v = 1.0.
Different initial values are applied to (a) and (b).

(a)

(b)

(c)

Fig. 6.
Waveforms of state values in the case that waves of each group synchronize and over three stable states exist. u = 1.0, w = 1.0 and v = 0.6.
Different initial values are applied to (a), (b) and (c).

Fig. 7.

Waveforms of state values in the case that switching phenomena are observed. u = 1.0, w = 1.0 and v = 0.75.

(a)

(b)

Fig. 8. Waveforms of state values in the case of changing a parameter from Fig. 6. u = 1.0, w = 1.0 and v = 0.67. Different initial values are applied to(a) and (b).

- 256 -