2014 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'14, Honolulu, Hawaii, USA. February 28 - March 3, 2014

Relationship between a Number of Cells and Phenomena in Cellular
Neural Networks Using Three Kinds of Cloning Templates
Mana Tanaka1 , Yasuteru Hosokawa1 and Yoshifumi Nishio2
1

Shikoku University
Furukawa, Ohjin, Tokushima, Japan
Phone:+81-88-665-1300
FAX:+81-88-637-1318
E-mail: mana.8sk@gmail.com
E-mail: hosokawa@keiei.shikoku-u.ac.jp

2Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
FAX:+81-88-656-7471
E-mail: nishio@ee.tokushima-u.ac.jp

Abstract
Cellular Neural Networks (CNNs) have been investigated bymany researchers. Additionally, many kinds of modified
CNNs have been proposed. In this study, CNN using threekinds of cloning templates which is one of modified CNNs isinvestigated. The cells are coupled as triangle lattices. Threekinds of cloning templates are placed uniformly. The system
was proposed as one of coupled oscillatory systems. Someoscillatory phenomena including chaos are observed.
1. Introduction
There are many studies of coupled oscillatory systems. In
these systems, observed phenomena are influenced by the network structures which are ladder, ring, full-coupled and so on.
On the other hand, some interesting phenomena can be observed in Cellular Neural Networks (CNN)[1][2] using two
kinds of cloning templates in [3]. Oscillatory phenomena
are observed in the system. It mean that the system is oneof novel coupled oscillatory systems. However, remarkablephenomena are not observed.
Cellular neural networks using three kinds of cloning templates are reported in [4]. This system consists of three kindsof cells which have different cloning templates. Chaotic
phenomena in CNN using three cells which have different
cloning templates are reported in [5]. Therefore, the proposedsystem is one of chaotic coupled oscillatory systems and anovel coupled system. The element of the system is coupled
as triangle lattice. By applying symmetric cloning template,the system becomes homogeneous coupled system.
In this study, a relationship between a number of cells andphenomena in cellular neural networks using three kinds ofcloning templates is investigated.

Figure 1: Structure of cellular neural networks using
three kind of cloning templates.

Figure 1 shows a structure of cellular neural networks using three kinds of cloning templates. Cells are coupled as
triangle lattice. The system consists of three kinds of cells
which name is Cell α, Cell β or Cell γ. The difference of
three kinds of cells is only values of cloning templates. By
setting the boundary condition as a periodic condition, threekinds of cells are placed uniformly as shown in Fig. 1.
The state equations are shown as follows.
Cell α:dxij= − xij + Iαdt∑
+Aα (i, j; k, l)ykl
c(k,l)

2. Cellular neural networks using three kinds of cloning
templates- 533 -+∑c(k,l)Bα (i, j; k, l)ukl(1)

Cell β:dxij
= − xij + Iβ
dt∑+Aβ (i, j; k, l)ykl
c(k,l)

+

∑

(1,1)

(1,2)

(1,3)

=One CNN Unit
(2,1)

(2,2)

(2,3)

(2)

Bβ (i, j; k, l)ukl

c(k,l)

Cell γ:dxij= − xij + Iγ
dt∑+Aγ (i, j; k, l)ykl
c(k,l)+∑xa x(3)Bγ (i, j; k, l)ukl

c(k,l)

where A{αβγ} (i, j; k, l)ykl , B{αβγ} (i, j; k, l)ukl , I{αβγ}show feedback value, input value, bias value, respectively.
The output function is shown as follows.
yij = f (xij ).

(4)f (x) = 0.5(|x + 1| − |x − 1|).

(5)

Figure 2: Definition of CNN units in this study.

whereIn this study, symmetric cloning template is set as follows
for keeping a symmetric property.
kl
1.24k ,Aα = lkl
−mkFigure 3: Chaotic phenomena in the case of three cells.
1.1−m ,Aβ = k−mk(6)3.1 Three cells caselm
As a reference, the case of three cells are shown in Fig. 3.
1.0l ,Aγ = m
This case was reported in [5]. Figure 3 shows the simulationlmresult. A double-scroll type attractor is observed. By changB α = B β = B γ = 0 anding parameter k, some periodic orbits, bifurcation phenomena
Iα = Iβ = Iγ = 0.
and chaos are observed.
where k, l, m show the coupling strengths between Cell α
3.2 One CNN unit case (six cells)and Cell β, Cell α and Cell γ, Cell β and Cell γ, respectively.
B α,β,γ and I α,β,γ are set as 0 because the purpose of thisFigure 4- 6 shows the case of six cells. Clustering phesystem is the investigation as a coupled system. Using system nomena are observed. Six cells are divided into two groups.
equations Eq. (1) - (6), computer simulations are carried out. c(1, 1) and c(1, 2) have very similar waveforms and inverse waveform of c(1, 3) is also similar to c(1, 1). Similarly, c(2, 1), c(2, 2) and c(2, 3) have also similar waveforms.
3. Computer simulations
Namely, each group consists of same row cells in Fig. 2. NoteWe define an unit which consists of two Cell α, two Cell β that these are not synchronized. By changing a parameter k
and two Cell γ as shown in Fig. 2. In this study, one, two and or l, sojourn time of high or low waveforms is changed. Figfour cases are investigated.
ure 5 shows a case of increasing a parameter k. Sojourn time- 534 -Figure 4: Waveforms in the case of six cells. k = −1.06,
l = −1.07 and m = 1.47.

Figure 6: Waveforms in the case of six cells. k = −1.095,
l = −1.07 and m = 1.47.

Figure 5: Waveforms in the case of six cells. k = −1.05,
l = −1.07 and m = 1.47.
of high or low waveforms is increased. On the other hand,decreasing a parameter k decreases sojourn time as shown inFig. 6.
3.3 Two CNN units placed at rows case (Twelve cells)
Figure 7 shows a simulation result in the case of two CNN
units placed in two rows. Four groups which consists of samerow cells are observed. By changing a parameter k or l, sojourn time of high or low waveforms is changed.
3.4 Two CNN units placed at columns case (Twelve cells)

Figure 7: Waveforms in the case of two rows. k = −1.06,
l = −1.07 and m = 1.47.
of Fig. 9 are observed. These clusters are depend on initialstates. Two kinds of oscillatory phenomena ( left side andright side of Fig. 9 ) are observed alternately. These switchingis almost same timing. Note that these waveforms are notsynchronized. By changing a parameter k or l, sojourn timeof high or low waveforms is changed.

Figure 8 shows a simulation result in the case of two CNN
units placed in two columns. However, a synchronization of 4. Considerations
switching phenomena are observed as shown in the center ofIn one CNN unit case and two CNN units placed at rows
Fig. 8. Additionally, the relationship between parameters kcase, same row cells become one cluster. The reason is conor l and sojourn time of waveforms is disappeared.
sidered as follows. Cell α, Cell β and Cell γ are placed at
one row in turn. Therefore, one row becomes one loop which
3.5 Four CNN units caseconsists of Cell α, Cell β and Cell γ.
Figure 9 shows a simulation result in the case of four CNN
In the other case, any cells can not become the loop. Thereunits case. Clustering phenomena as shown in the left side fore, a cluster which consists of same cells can not observed.

- 535 -Figure 8: Waveforms in the case of two columns. k = −1.20,
l = −1.30 and m = 2.50.
In two CNN units placed at columns case, each cell connects to the other one by twice. For instance, c(1, 1) in Fig. 2
is connected to c(2, 1) by upper right and lower right. This is
the reason of why this case differ from other results.
5. Conclusions
In this study, cellular neural networks using three kinds ofcloning templates have been proposed. Some computer simulations were carried out. In cases of one CNN unit case andtwo units case, clustering phenomena are observed. The clusters do not depend on initial values. Additionally, switching
phenomena are observed in all cases.
Difference of results means that clustering phenomena areinfluenced by the number of cells and the structure. In thefuture work, symmetrical case which consists of nine cells
will be investigated.
References

Figure 9: Waveforms in the case of twenty four cells. k =−1.26, l = −1.20 and m = 2.50.
[4] M. Tanaka, Y. Hosokawa and Y. Nishio, “Chaotic Phenomena in Cellular Neural Networks Using Three Kindsof Cloning Templates,” Proc. of NCN 13, pp. 30–33,2013.
[5] F. Zou, and J. A. Nossek, “ Bifurcation and chaos in cellular neural networks,” IEEE Trans. Circuits Syst. I, vol. 40,no. 3, pp. 166–173, 1993.

[1] L. O. Chua and L. Yang, “Cellular Neural Networks:Theory,” IEEE Trans. Circuits Syst., vol. 35, no. 10,pp. 1257–1272, 1988.
[2] L. O. Chua and L. Yang, “ Cellular Neural Networks:Applications,” IEEE Trans. Circuits Syst., vol. 35, no. 10,pp. 1273–1290, 1988.
[3] 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.

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