2008 International Workshop on Nonlinear Circuits and Signal Processing
NCSP'08, Gold Coast, Australia, March 6-8, 2008

Clustering Phenomena on Cellular Neural Networks Using Two kinds of Templates
Junji Fujii† , Yasuteru Hosokawa† and Yoshifumi Nishio‡†Shikoku University
Furukawa, Ojin-cho, Tokushima, Japan
Email: {i1739082,hosokawa}@keiei.shikoku-u.ac.jp‡Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: nishio@ee.tokushima-u.ac.jp

Abstract
In this study, we investigate clustering phenomena on cellularneural networks using two kinds of templates. It was conﬁrmed that these clusters are classiﬁed into four kinds of patterns.
1. Introduction
Cellular Neural Network (CNN) [1]-[3] is one kind of neural networks. The main characteristic is the local connection.
There have been many studies on CNN and many kinds ofCNN have been proposed. One of them is two-layer CNN.
Two-layer CNN can generate many interesting phenomena.
For instance, self-organizing pattern [4], active wave propagation [5] and so on. Investigating these phenomena contributes to understand complex systems and to apply them toengineering systems.
In our earlier study[6], we proposed CNN using two kindsof templates. In this system, clustering phenomena, patternformation, active wave propagation and so on are observed.
This system has a peculiar characteristic in order to investigate a new class of coupled oscillatory systems. Namely, apair of cells which have two kinds of templates are needed
for a simple oscillation. The cell connects with four the othercells and the other cell also connects with four neighbor cells.
Like this, these cells are sharing a factor of the oscillation.
This type of connection may be diﬃcult to realize by coupling normal oscillators.
In this study, we investigate clustering phenomena observed in this system. In computer simulations, it is conﬁrmed that these clusters are classiﬁed into four kinds of patterns. The relationship of patterns is also investigated.

Figure 1: Structure of CNN using two kinds of templates.
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.
dxi j= − xi j + Iαdt∑+
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.
dxi j= − xi j + Iβdt∑
Aβ (i, j; k, l)ykl+

2. CNN Using Two Kinds of Templates
Figure 1 shows a system model of our proposed system.
We assume that the system has a two-dimensional M by N
array structure. 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∑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} are called as the- 431 -feedback coeﬃcient, the control coeﬃcient and the bias current, respectively.
The output equation of the cell is given as follows:
yi j = f (xi j ).

(3)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 , Bα , Aβ and Bβ are 3 times 3 matrices,which can be described to have a similar form to Eq. (5).
Aα (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)(b) Iteration 8000(c) Iteration 20000(d) Iteration 100000

Aα (i, j; i − 1, j + 1)

Aα (i, j; i, j + 1)  (5)Aα (i, j; i + 1, j + 1)

This proposed system is more complex than the normal
CNN. This system has a peculiar characteristic in order to investigate a new class of coupled oscillatory systems. Namely,a pair 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.

3. Computer Simulations
In all simulation results of this study, we deﬁne -1 and 1 as
white and black, respectively. Initial state values of dummycells which is placed at outside of border cells are set as 0.
Figures 2 show one of the computer simulation results using
following templates.
−0.01
0.1 0 0.10
1.04
Aα =  0 1.1 −1  , Aβ =  1
−0.01
00.1 0 0.1

−0.01
0  ,−0.01Bα = 0, Bβ = 0, Iα = 0, Iβ = 0,(6)Two concentric circular waves are generated. Their wave
fronts propagate in all directions through the network fromtheir initialized positions shown in Fig. 2 (a) (b). When thewaves hit the end of CNN, the waves reﬂect shown in Fig. 2(c). After that the pattern becomes complex pattern shownin Fig. 2 (d). Like this, the proposed system generates someinteresting phenomena.
In this study, the following template case is investigated.
−1 1
Aα =  1 1

−1 1(a) Initial state 1−1
1  , Aβ = −1
1−1Bα = 0, Bβ = 0, Iα = 0, Iβ = 0,−1−1−1
1 
−1 ,1(7)Figure 2: Active Propagation Phenomena.

This template is used only three values which are 1, 0 and -1.
Investigating simple templates like this, peculiar characteristics of this system can be revealed. Figure 3 shows one of thecomputer simulation results. All initial state values are set as
random numbers. In this case, clustering phenomena are observed. Border lines keep oscillating and these positions don’tmove. These clusters are classiﬁed into four kinds of patterns
as shown in Fig. 4. Pattern A as shown in Fig. 4 shows thecase that cells α which are placed at rows and columns of odd
numbers become black and the others become white. In thesame way, Pattern B shows the case that cells α which areplaced at rows and columns of even numbers become black.
Pattern C shows the case that cells α which are placed at rowsand columns of odd numbers become white. Pattern D shows
the case that cells α which are placed at rows and columns ofeven numbers become white.
This result shows that this system has four stable states.
Therefore, it is supposed that patterns are placed at arbitrary
positions by these patterns are set as initial values.
In order to conﬁrm this supposition, some investigations
are carried out by computer simulations. At ﬁrst, the casethat one of four patterns are set as initial values are investigated. In this case, we can conﬁrm that one pattern ﬁlls
the system. Figure 5 shows the computer simulation resultin the case of pattern A. In all cases, border cells don’t become white or black. However, border cells don’t inﬂuence
inner cells. Figures 6 show left upper sides of the computer
simulation results in cases that one of four patterns are set as
initial values. Next, we investigate cases that one pattern isplaced at the center of the other pattern. The results of this- 432 -(a) Pattern A.

(b) Pattern B.

Figure 3: Clustering phenomena.

investigation are shown in Table 1. Two kinds of results are(c) Pattern C.
(d) Pattern D.
obtained. One is that the center pattern doesn’t move and theborder line keeps oscillating. The other is that the center pattern disappears. Figure 7 shows one of the results. In this Figure 4: Four kinds of patterns observed in clustering phecase, two patterns don’t move and the border line keeps oscil- nomena.
lating. These results show that the center pattern disappears
in the case of combinations of ”A and D” or ”B and C.” Next,
we investigate cases that two patterns are placed at right andleft sides. The results of this investigation are shown in Table 2. Figure 8 shows one of the results. In this case, two patterns don’t move and the border line keeps oscillating. Theseresults show that patterns A and C disappear in the case of
combinations of ”A and D” or ”B and C.”
Finally, we try to set four patterns as the initial state. Figures 9 show the one of the computer simulation results in thecase of setting four patterns. Pattern A, B, C and D are set
at left upper side, right upper side, left lower side and rightlower side, respectively. In this case, these four patterns exist
stable states and border lines keep oscillating. However, fourpatterns don’t exist stable states in some cases. By placements of patterns, the result changes. Namely, it depends onrelationship between two patterns.

Figure 5: Computer simulation result in the case that initialvalues are set as Pattern A.

(a) Pattern A(b) Pattern B(c) Pattern C(d) Pattern D

4. Conclusions
In this study, we investigated clustering phenomena on cellular neural networks using two kinds of templates. It wasconﬁrmed that these clusters are classiﬁed into four kinds ofpatterns. Relationships among four patterns also investigated.
As a result of these investigations, we could conﬁrm that patterns are placed at arbitrary positions with conditions. In future, we will analyze this phenomena theoretically.

Figure 6: left upper sides of the computer simulation resultsin cases that one of four patters are set as initial values.

- 433 -(a) Initial state.

(a) Initial state.

(b) Steady state.

(b) Steady state.

Figure 7: Computer simulation result in the case that pattern
B is placed at center of pattern A.

Figure 9: Computer simulation result in the case of Pattern
B is placed at center of Pattern A.

Table 1: Relationship between two patterns 1References

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

Center pattern
A B C D- O O D
O - D OO D - OD O O -[2] L. O. Chua and L. Yang, “Cellular Neural Networks: Applications,” IEEE Trans. Circuits Syst., vol. 35, no. 10,
pp. 1273–1290, 1988.

O: The center pattern doesn’t move and the border line keeps oscillating.

[3] L. O. Chua, M. Hasler, G. S. Moschytz and J. Neirynck,“Autonomous Cellular Neural Networks: a uniﬁed
paradigm for pattern formation and active wave propagation,” IEEE Trans. Circuits Syst. I, vol. 42, no. 10,pp. 559–577, 1995.

D: The center pattern disappears.

[4] P. Arena, S. Baglio, L. Fortuna and G. Manganaro, “SelfOrganization in a Two-Layer CNN,” IEEE Trans. Circuits Syst. I, vol. 45, no. 2, pp. 157–162, 1998.

(a) Initial state.

[5] 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.

(b) Steady state.

Figure 8: Computer simulation result in the case that pattern
A and B are placed at left and right side respectively.
Table 2: Relationship between two patterns 2

Left pattern
ABCD

Right pattern
A B C D
- O O DO - B OO B - OD O O -O: Two patterns don’t move and the border line keeps oscillating.
B: The system is ﬁlled by pattern B.
D: The system is ﬁlled by pattern D.

[6] J. Fujii, Y. Hosokawa and Y. Nishio, “Wave Phenomena
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[8] Z. Yang, Y. Nishio and A. Ushida, “Image Processingof Two-Layer CNNs — Applications and their Stability,”
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