2011 International Workshop on Nonlinear Circuits,Communication and Signal Processing
NCSP'11, Tianjin, China, March 1-3, 2011

Relationship between Synchronization and Parameter in Two-Template CNN
with Periodic Boundary Conditions
Junji Fujii† , Yasuteru Hosokawa‡ and Yoshifumi Nishio††Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
Email: {fujii,nishio}@ee.tokushima-u.ac.jp‡Shikoku University
Furukawa, Ohjin, Tokushima, Japan
Phone:+81-88-665-1300
Email: hosokawa@keiei.shikoku-u.ac.jp

In this study, relationship between template parameters andoscillatory phenomena are investigated in the case of periodic
In our previous study, we have proposed Two-Template CNN. boundary condition. Additionally, observed oscillatory pheThis system was proposed in order to investigate a new class nomena are investigated in details.
of coupled oscillatory systems. The system was investigatedin some conditions of templates or boundaries. As a result,some interesting phenomena are observed. In this study, we 2. Two-Template CNN
investigate a relationship between parameters of templates
and oscillatory phenomena in Two-Template CNN with periodic boundary conditions.
Abstract

1. Introduction
Cellular Neural Networks (CNN) [1],[2] is one kinds ofmutually neural networks. The main characteristic is thelocal connection and the parallel signal processing. There
have been many studies on CNN and many kinds of modified
CNNs have been proposed. One of them is Two-layer CNN.
Two-layer CNN [3], [4] can generate many interesting phenomena. For instance, self-organizing pattern, active wave
propagation, clustering and so on are observed.
In our previous studies [5], [6], we have proposed Twotemplate CNN. The architecture of Two-Template CNN isvery simpler than Two-layer CNN. In the case of comparing
a conventional CNN, control signal lines of template values
becomes two times only. The architecture and the number ofcoupling lines of one cell are same as a conventional CNN.
Additionally, this system can generate similar phenomena to
Two-layer CNN and some interesting phenomena can be obtained. In these studied, fixed boundary conditions cases areinvestigated mainly. Since many kinds of complex phenomena are observed. Additionally, some simple oscillatory phenomena are observed only in the case of periodic boundary
conditions. However, it is important to be able to investigate oscillatory phenomena in the case of periodic boundary
conditions. Basically, this condition of our system is corresponding to the case of non-oscillation in conventional CNN.
Therefore, to confirm oscillatory phenomena is very interesting.

Figure 1: Structure of TwoTemplate CNN.

Figure 2: A periodic
boundary conditions.

Figure 1 shows an architecture of Two-Template CNN. We
assume that the system has a two-dimensional M by N arrayarchitecture. 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 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.

- 52 -dxij= − xij + Iαdt∑+Aα (i, j; k, l)yklc(k,l)+∑c(k,l)Bα (i, j; k, l)ukl(1)

2: The case that i + j is an odd number.
dxij= − xij + Iβdt∑
+Aβ (i, j; k, l)ykl
c(k,l)+∑Bβ (i, j; k, l)ukl

c(k,l)(2)

hypotheses. Therefore, the investigation of oscillatory phenomena is very interesting. In order to carry out computer
simulations, following configurations are applied. The initialstate values set as random values. A boundary condition isset as a periodic boundary condition as shown in Fig. 2.
The template is set as follows.
−u v −uu −v
uAα =  v w v  , Aβ = −v −w −v  ,−u v −uu −v
uA{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
B α = 0, B β = 0, Iα = 0, Iβ = 0.
current, respectively. The output equation of the cell is given(7)
as follows:
We consider that coupling coefficients and oscillation factors
are corresponding to u and v, respectively. These templates
yij = f (xij ),(3) are symmetric each other. This setting satisfy the hypothesis 1. The number of cells is fixed as 8 × 8. In the case ofwhere,f (x) = 0.5(|x + 1| − |x − 1|).
(4) lower than 8 × 8, only one kind of synchronization phenomena is observed and in the case of over 8 × 8, various kinds
This system is more complex than a conventional CNN. Al- of oscillatory phenomena are observed. Hence, 8 × 8 whichthough this system has a unique characteristic. Namely, a is lowest number in various kinds of oscillatory phenomena
pair of cell α and cell β are needed for a simple oscillation. is selected.
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.
(a)
(b)
(c)
3. Relationship between oscillatory phenomena and paFigure 3: Simulation result in the case of periodic boundary
rameterscondition. The number of cells is 8 × 8. u = 2, v = 5 andIn our previous study, some kinds of boundary conditions w = 1. (a) Initial state. (b) and (c) Snapshot of a stable state.
have been investigated. As a result, some kinds of interesting
phenomena was observed. Additionally, it was assumed that
Figure 3 shows a computer simulation result in the case ofboundary conditions affect oscillatory phenomena. In fact, u = 2, v = 5 and w = 1. Figure 3 (a) shows an initialthere are complete stability theorem [7] of conventional CNN state. Figures 3 (b) and (c) show snapshots of a stable state.
shown as follows.
The oscillatory phenomena are observed. These oscillations
Any M × N space-invariant CNN of arbitrary neighbor- are divided into four groups is shown in Fig. 4. Cells of eachhood size with constant inputs and constant threshold is com- groups are synchronized. In order to investigate the phenompletely stable if the following three hypotheses are satisfied. ena in detail, some computer simulations are carried out.
Figure 5 shows state variables of cells. Vertical axes show
state variables. Horizontal axes show time. The first row
shows state variables of group α1 . In the same way, the secA(i, j; k, l) = A(k, l; i, j)(5)ond, third and fourth rows show state variables of groups β1 ,2. The nonlinear function yij = f (xij ) is differentiable, β2 and α2 , respectively. Each row includes sixteen waveforms. This result is corresponding to Fig. 3. Synchronizabounded, andtion phenomena of each group are observed. Additionally,f 0 (xij ) > 0, for all − ∞ < xij < ∞(6) groups β1 and β2 are also synchronized. Moreover, a difference of groups α1 and α2 is an equilibrium point only.
3. All equilibrium points are isolated.
Namely, groups α1 and α2 are also synchronized. In thissimulation, the other result is obtained. The difference of two
In this study, Two-Template CNN is set the condition results is that α2 is an inverse value of α1 . On some paramwhich is corresponding to the condition which satisfy these eter sets observed oscillatory phenomena in this study, same
1. The A template is symmetric- 53 -Figure 4: Classification of cells for the investigation of oscillatory phenomena.
Figure 6: Relationship between oscillatory phenomena andparameters u and v. “Osc” means an oscillation. “N” meansno oscillation. −10 ≤ v ≤ 10 and −10 ≤ u ≤ 10. Theinterval is set as 1.0.

Table 1: Relationship between parameters and oscillation frequencies or amplitude
Quadrant By increasing v
By increasing uFirst
Freq. HighFreq. Low, Amp. HighSecond
Freq. Low
Freq. Low, Amp. HighThird
Freq. Low
Freq. High, Amp. Low
FourthFreq. HighFreq. High, Amp. Low
Figure 5: Waveforms of state values. u = 1.0, w = 1.0 andv = 5.0.
results are obtained.
First of all, in order to investigate the oscillation region indetail,some computer simulation are carried out.
Figure 6 shows the relationships between oscillatory phenomena and parameters u and v. Horizontal axis shows v.
Vertical axis shows u. “Osc” means that the oscillatory phenomena are observed. “N” means that the oscillatory phenomena are not observed. Intervals of u and v are set as 1in Fig. 6. Line-symmetric oscillatory regions are confirmed.
Figures 7 and 8 show waveforms of state variables of cells.
In Figure 7, parameters are set as u = 2, v = 7 and w = 1.
In Figure 8, parameters are set as u = 3, v = 5 and w = 1.
In an oscillatory region of first quadrant of Fig. 6. Following
relatioinships are confirmed by comparing with Fig. 5, 7, 8and 9.
• By increasing parameter u, the amplitude is increased
and the frequency is decreased.
• By increasing parameter u, distances between horizontal axes and equilibrium points of groups α1 and α2 areincrased.
• By increasing parameter w, the frequency is decreased
and the amplitude is decreased.

Quadrant
FirstSecondThird
FourthHigh

By increasing w
Freq. Low
Freq. Low
Freq. HighFreq. HighIn the other quadrant, there are similar relationships. These
relatioinships are shown in Table. 1 and 2. Like this, relationships are line-symmetries. In almost oscillatory regions ofFig. 6, oscillations of cells are divided four groups as shownin Fig. 5, 7, 8 and 9. However, in some areas which are near
non-oscillatory regions, oscillations of cells are divided fiveor more groups as shown in Fig 10.
4. Conclusions
In this study, we have investigated relationship between oscillatory phenomena and parameters in Two-Template CNN
with periodic boundary conditions. Additionally, we have investigated the oscillation region. Furthermore, we could observed coexistences of some kinds of stable states in the caseof nearly convergents of the template parameters u and v.
As future works, we will investigate these phenomena indetail.

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Table 2: Relationship between parameter u and distances between equilibrium points and horizontal axes.
Quadrant
By increasing uFirstDistances of group α1 and α2 are increased.
SecondDistances of group α1 and α2 are increased.
Third
Distances of group β1 and β2 are increased.
FourthDistances of group β1 and β2 are increased.

Figure 7: Waveforms of state values. u = 2.0, w = 1.0 andv = 7.0.

Figure 10: Waveforms of state values. u = 3.0, w = 1.0 andv = 4.0 in the case of the initial values set as random values.
References[1] L. O. Chua and L. Yang, “Cellular Neural Networks:Theory,” IEEE Trans. Circuits Syst., vol. 35, no. 10,pp. 1257–1272, 1988.
Figure 8: Waveforms of state values. u = 3.0, w = 1.0 andv = 5.0.

[2] L. O. Chua and L. Yang, “Cellular Neural Networks: Applications,” IEEE Trans. Circuits Syst., vol. 35, no. 10,
pp. 1273–1290, 1988.
[3] Z. Yang, Y. Nishio and A. Ushida, “Generation of Various
Types of Spatio-Temporal Phenomena in Two-Layer Cellular Neural Networks,” IEICE Trans. Funds., vol. E87A, no. 4, pp. 864–871, 2004.
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[5] J. Fujii, Y. Hosokawa and Y. Nishio, “Wave Phenomena
in Cellular Neural Networks Using Two Kinds of Template Sets,” Proc. of NOTLA’07, pp. 23–26, 2007.
[6] J. Fujii, Y. Hosokawa and Y. Nishio, “Oscillatory Phenomena in Cellular Neural Networks Using Two Kindsof Templates,” Proc. of NOLTA’08, pp. 688–691, 2008.

Figure 9: Waveforms of state values. u = 2.0, w = 2.0 andv = 5.0.

[7] L. O. Chua and L. Tamas Roska, “Cellular Neural
Networks and visual computing,” Cambidge University
Press, 2002.

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