Journal of Signal Processing, Vol.24, No.1, pp.1-9, January 2020

PAPER

Applications and Oscillatory Phenomena of Cellular Neural Network
Using Two Kinds of Cloning Templates
Yasuteru Hosokawa1 and Yoshifumi Nishio2
1

2

Department of Media and Information Systems, Shikoku University, 123-1 Furukawa, Ohjin, Tokushima 771-1192, Japan
Department of Electrical and Electronic Engineering, Tokushima University, 2-1 Minami-Josanjima, Tokushima 770-8506, Japan
E-mail: hosokawa@keiei.shikoku-u.ac.jp, nishio@ee.tokushima-u.ac.jp

Abstract
In this paper, a cellular neural network using two kinds of cloning templates, which is a kind of modified cellularneural networks (CNN), is proposed and investigated. The structure of this system is almost the same as that of the conventional
CNN. Namely, the only diﬀerence between the conventional CNN and the proposed system is the number of signal lines used forcloning templates. Therefore, the diﬃculty of IC implementation is almost the same. This system can carry out more complicatedprocessing than the conventional CNN. Additionally, cloning templates of the conventional CNN can be applied to this system. As
applications, five diﬀerent cloning templates for image processing that can carry out complicated processing and a demonstrationof an image-processing application are shown. In the demonstration, by using two cloning templates for the proposed system andthree cloning templates for the conventional CNN, we demonstrate the extraction of numbers drawn with a hard-to-distinguishcolor in the background of postage stamps. On the other hand, oscillatory phenomena can be observed in the case of a periodic
boundary condition and a symmetry-cloning template. In the case of the conventional CNN with a corresponding condition,
oscillatory phenomena cannot be observed. The proposed system consists of two kinds of cells that cannot oscillate on their own.
Namely, elements of the oscillator are shared among each other. In this point, the proposed system is of interest. Synchronization,
quasi-synchronization and clustering phenomena are observed in the proposed system. The system is analyzed and the oscillationof the system is verified.
Keywords: cellular neural network, nonlinear system, oscillation, coupled system

1.

Introduction

The cellular neural network (CNN) was introduced by
Chua and Yang [1] in 1988. This system is a kind of analog computer and its processing is faster than that of a von
Neumann computer. The system can process many items
of information in real time and its architecture is similar tothat of a retina. Therefore, its main application is image
processing [2] [3]. Many cloning templates, which determine the characteristics of the system, have been proposed.
Combining these cloning templates can realize useful applications [4]-[8]. To realize applications, IC implementation is needed. Fortunately, the system consists of a regulararray of the same cells, and this architecture is suitable for
IC implementation [9]-[11].
On the other hand, many kinds of modified CNNs havebeen proposed. For instance, the delayed cellular neuralnetwork [12], fussy cellular neural network [13], quantum
cellular neural network [14], small-world cellular neuralnetwork [15], shunting inhibitory cellular neural network[16] and two-layer cellular neural network [17] [18] havebeen proposed. Basically, these modified CNNs can carryout more complicated processing than the conventional
CNN. To perform image processing using these CNNs,
many processes are needed. Thus, in recent studies, the development of novel architectures applying novel semiconducting devices [19]-[21] has been addressed. However,these architectures are more complicated than that of theconventional CNN. Therefore, the advantage of the ease of
IC implementation is lost.
In this paper, a cellular neural network using two kindsof cloning templates (two-template CNN) is proposed andanalyzed. The only diﬀerence in the architecture between
the conventional CNN and the proposed CNN is the number of signal lines used for cloning templates. Therefore,
the advantage of IC implementation is not lost. The mainfeature of this system is that it performs more complicatedprocessing in spite of having almost the same architecture
as the conventional CNN. Furthermore, cloning templatesof the conventional CNN can also be applied to this system.
The proposed system can also be utilized as a conventional
CNN. Hence, the advantage of the proposed system is that

Journal of Signal Processing, Vol.24, No.1, January 20201

it has both unique characteristics and those of a conventional CNN. By demonstrating the unique characteristicsof the proposed system, its eﬀectiveness can be shown.
Five diﬀerent cloning templates for image processing
using the two-template CNN are proposed. These cloning
templates can carry out complicated processing that cannot
be carried out by the single cloning template of the conventional CNN. Four of the proposed cloning templates generate checkered patterns, from which one cloning templateis proposed for the extraction of checkered patterns. Using
these cloning templates, we show that one cloning template
for the two-template CNN can perform more complicatedprocessing than one cloning template for the conventional
CNN.
Additionally, by using two cloning templates for the
proposed system and three cloning templates for the conventional CNN, we demonstrate the extraction of numbers
drawn with a hard-to-distinguish color in the backgroundof postage stamps. The proposed system has the advantages of the ability to perform complicated processing andthe flexibility of using cloning templates of the conventional CNN. This demonstration shows the superiority andthe flexibility of the two-template CNN.
Another feature is that oscillatory phenomena can beobserved when setting symmetry-cloning templates anda periodic boundary condition. In the case of the conventional CNN with symmetry-cloning templates and a
periodic boundary condition, oscillatory phenomena cannot be observed. However, by investigating this condition, synchronization, quasi-synchronization and clustering phenomena are observed in the proposed system. This
result shows that this is a novel coupled oscillatory system.
This system has the feature that elements used for the oscillation are shared among each other in this system. To
verify the existence of oscillatory phenomena under thiscondition, the case of four coupled cells is analyzed.
In Sect. 2, the architecture of the system is introduced.
In Sect. 3, five diﬀerent cloning templates for image processing using two-template CNN are proposed. Additionally, a demonstration of image-processing application isshown. In Sect. 4, synchronization, quasi-synchronizationand clustering phenomena are shown. These phenomena
are observed when setting symmetry-cloning templates
and a periodic boundary condition. Additionally, the caseof four coupled cells is analyzed and the existence of oscillatory phenomena is verified. Finally, conclusions are
given.
2.

Two-Template CNN

2.1 Architecture
Figure 1 shows the system model of the two-template
CNN. 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.
2Fig. 1 System model of two-template CNN
Cells having one cloning template are called cell α and theother cells are called cell β. These two types of cells areplaced as a checkered pattern, as shown in Fig. 1.
The only diﬀerence between the conventional CNN andthe proposed system is the number of signal lines used forcloning templates. Therefore, the proposed system is suitable for IC implementation, similarly to the conventional
CNN.
2.2

Continuous-time version

In the continuous-time version of the two-template
CNN, the state equations of the cells are given as below.
1: For cell α.
∑dxi j= − xi j + Iα +Aα (i, j; k, l)ykl
dt
(k,l)
∑
+
Bα (i, j; k, l)ukl

(1)

(k,l)

2: For cell β.
∑
dxi j= − xi j + Iβ +Aβ (i, j; k, l)ykl
dt
(k,l)
∑
+
Bβ (i, j; k, l)ukl

(2)

(k,l)

A{αβ} (i, j; k, l), B{αβ} (i, j; k, l) and I{αβ} are called the feedback coeﬃcient, control coeﬃcient and bias current, respectively. The variables u, x and y are the input, state andoutput variables of the cell, respectively. Aα , Bα , Aβ andBβ are 3 × 3 matrices, that can be described as, for example,

Journal of Signal Processing, Vol.24, No.1, January 2020

Aα =
 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)Aα (i, j; i − 1, j + 1)

Aα (i, j; i, j + 1) Aα (i, j; i + 1, j + 1)

yi j = 0.5(|xi j + 1| − |xi j − 1|)(4)zero and the initial state is 1, which corresponds to black.
The cloning template is set as

Discrete-time version

In the discrete-time version of the two-template CNN,
the state equations of the cells are given as follows: 0 0.1 0 
0 0 0




Aα = 0.1 0 0.1 , Bα = 0 4 0 , Iα = −3.7

0 0.1 0
0 0 0




1 0 10 0 0





Aβ = 0 2 0 , Bβ = 0 4 0 , Iβ = −0.5




1 0 1
0 0 0

1: For cell α.
xi j (t + 1) =∑Aα (i, j; k, l)ykl (t)(k,l)∈Nr (i, j)∑

Bα (i, j; k, l)ukl + Iα∑

Aβ (i, j; k, l)ykl (t)

(5)

(8)

(k,l)∈Nr (i, j)

2: For cell β.
xi j (t + 1) =(k,l)∈Nr (i, j)∑

Bβ (i, j; k, l)ukl + Iβ(b)Fig. 2 Computer simulation of hole filling and noise reduction: The image size is 256 × 256, (a) Input image, (b)ResultThe output equation of the cell is given as follows

2.3

(a)

(3)

(6)

Figure 2 shows a simulation result of hole filling andnoise reduction. Some black dots are removed. Center andright objects become gray areas where white and black dots
are placed as a checkered pattern.

(k,l)∈Nr (i, j)

yi j (t) =

{

1
−1

(xi j (t) ≥ 0)
(xi j (t) < 0)

3.2 Dithering and noise reduction(7)

3. Image Processing
In this section, five diﬀerent cloning templates for image processing are shown. The first three cloning templates
are for the discrete-time version and last two are for thecontinuous-time version. Each processing result for eachcloning template set has a stable state. Namely, transientstates are not used in image processing in this paper. Generally, some cloning template sets utilize transient states orrepeat the image processing of a CNN. The characteristicof the proposed system contributes to reducing the numberof combinations of cloning template sets. Additionally, theproposed system can also be operated as a conventional
CNN. This means that many of investigations results of
conventional CNNs can be utilized in the proposed system.
The potential of the proposed system for image processing
is shown in this section.
3.1 Hole filling and noise reduction
This cloning template can process hole filling and noise
reduction at the same time. The discrete-time version described in Sec. 2.3 is applied. The boundary condition is
Journal of Signal Processing, Vol.24, No.1, January 2020

This cloning template can process dithering and noise
reduction at the same time. The boundary condition andinitial state are the same as described in Sect. 3.1. The
cloning template is set as 0 0.1 0 
0 0 0




Aα = 0.1 0 0.1 , Bα = 0 4 0 , Iα = −3.7

0 0.1 0
0 0 0




−1 −1 −10 0 0




Aβ = 0 0 0 , Bβ = −1 9 −1 , Iβ = −0.5
−1 −1 −1
0 0 0

(9)

Figure 3 shows the simulation result of dithering andnoise reduction. Some black dots are removed and the details become clear, such as the hands, camera and legs.
3.3

Edge detection

This cloning template can perform edge detection. The
boundary condition and initial state are those in Sect. 3.1.
The cloning template is set as

3

(a)

(b)

(a)

Fig. 3 Computer simulation of dithering and noise reduction: The image size is 256 × 256, (a) Input image, (b)Result(a)(b)

Fig. 5 Computer simulation of extraction of parts withintermediate brightness: The image size is 256 × 256, (a)Input image, (b) Result

(a)

(b)

(c)

(d)

(e)

(f)

(b)

Fig. 4 Computer simulation of edge detection: The image
size is 256 × 256, (a) Input image, (b) Result−1 −1 −1Aα = 0, Bα = −1 9 −1 , Iα = 0−1 −1 −1−1 −1 −1Aβ = 0, Bβ = −1 9 −1 , Iβ = 2−1 −1 −1

(10)

Figure 4 shows the simulation result. Edges are detected as white. Additionally, original objects can be recognized. Gray areas are checkered patterns.
3.4 Extraction of parts with intermediate brightness
This cloning template can extract parts with intermediate brightness. The continuous-time version described in
Sect. 2.2 is applied. The boundary condition is 1 and theinitial state is the input image. The cloning template is set
as

2

Aα = 0

2

2

Aβ = 1

2

4

0
2
0
1
2
1



2
0


0 , Bα = 1


0
2


2
−1


1 Bβ = −1


−1
2


0

1 , Iα = 20−1 −1

2 −1 , Iβ = −2

−1 −1

Fig. 6 Computer simulation of extraction of parts with intermediate brightness from a gradation image: The image
size is 256 × 256, (a) Input image, (b) Iα = −5 and Iβ = −6,(c) Iα = 9 and Iβ = −1, (d) Iα = 2 and Iβ = −2, (e) Iα = 3
and Iβ = −3, (f) Iα = 4 and Iβ = −4
Figure 5 shows a simulation result of extraction of parts
with intermediate brightness. By using this cloning template, parts with intermediate brightness are extracted as
checkered patterns.
Changing the parameters Iα and Iβ corresponds tochanging the brightness and sensitivity of the extracted
area. Namely, by increasing the values of Iα and Iβ , thebrightness of the extracted area becomes high, as shownin Figs. 6(b) and (c). By increasing the value of Iα − Iβ ,the range of the extracted area is expanded, as shown in
Figs. 6(d)-(f).
3.5

1
0
1

Extraction of checkered pattern

To use the proposed two-template CNN, whose output often becomes a checkered pattern, the extraction ofa checkered pattern is necessary. The following cloning
template can extract a checkered pattern.
(11)

Journal of Signal Processing, Vol.24, No.1, January 2020

(a)

(b)

(a)

(b)

(c)

(d)

Fig. 7 Computer simulation of extraction of a checkered
pattern: The image size is 256 × 256, (a) Input image, (b)Result 1 −1 1 Aα = 0, Bα = −1 1 −1 , Iα = −6

1 −1 1−1 1 −1Aβ = 0, Bβ =  1 −1 1  , Iβ = −6
−1 1 −1(12)

Figure 7 shows a simulation result of extraction of acheckered pattern. The input image is the same as that inFig. 5(b). The checkered pattern is extracted.
3.6 Demonstration of image-processing application
The extraction of parts with intermediate brightness
can be realized by combining the cloning templates described in Sects. 3.4 and 3.5. Here, as an application,
the extraction of numbers drawn with a hard-to-distinguishcolor in the background of postage stamps is demonstrated.
Numbers in postage stamps are an important feature ofstamps. However, sometimes the numbers are drawn witha hard-to-distinguish color in the background because thedesign of the postage stamps is emphasized. An algorithm
based on the conventional CNN for the extraction of numbers drawn with a hard-to-distinguish color has been proposed by Kishida et al. [22]. In their algorithm, six ormore cloning templates are needed. Additionally, many iterations, the adjustment of some parameters or additional
processing is needed for the processing of some cloning
templates, and the procedure is complicated. By using thetwo-template CNN, a similar result can be obtained withmuch a simpler algorithm.
Figure 8 shows the extraction of numbers from thebackground of a postage stamp. Figure 8(a) shows an input image. Figure 8(b) shows an image with intermediatebrightness parts (gray areas) extracted as checkered patterns. Here, the cloning template in Eq. (11) with values changed to Iα = −1 and Iβ = −2.5 is applied. Figure 8(c) shows an image with the intermediate-brightness
parts from Fig. 8(b) in black, where the cloning template in
Eq. (12) was applied. By applying peel, grow and subtract

Journal of Signal Processing, Vol.24, No.1, January 2020

Fig. 8 Extraction of numbers drawn with a hard-todistinguish color from the background of postage stamps:
The image size is 166 × 200, (a) Input image, (b) Intermediate brightness parts extracted as checkered patterns, (c)Extracted intermediate-brightness parts in black, (d) Extracted numbers after removing large black areas

templates, which are the cloning templates of the conventional CNN, to Fig. 8(c), large black areas are removed.
Figure 8(d) shows that the numbers are extracted.
4.

Oscillatory Phenomena

In this section, some oscillatory phenomena are shown.
On a conventional CNN with applied symmetrical-cloning
template values, oscillatory phenomena cannot be observed. There are some modified CNNs that can generate
oscillatory phenomena [23] - [24]. The intentionally modification for oscillatory phenomena is applied to these system. The proposed system is modified utilizing only two
cloning template sets. The values applied to the cloning
template are symmetric values. The structure of the proposed system is interesting as a coupled oscillatory system. Normally, a coupled oscillatory system consists ofoscillators and coupling elements. However, the proposedsystem cannot be divided into oscillators and coupling elements because the elements of oscillator are shared among
each other. There are no similar systems in the field ofelectronic oscillators. Couplied systems are classified into
continuous or discrete. For instance, a system of coupledelectronic oscillators is classified as a continuous-time,
continuous-valued and discrete-space system. As anotherexample, a coupled map lattice is classified as a discretetime, continuous-valued and discrete-space system. The
proposed system is not easy to classify in this way, thespace cannot be classified as continuous or discrete. The
5

interesting point of the proposed system is this structure.
Some observed phenomena in the proposed system and abasic analysis of the oscillation are shown below.
4.1 Simulation settings
In this section, synchronization, quasi-synchronizationand clustering phenomena are shown using the following conditions. Boundary conditions are set as periodic.
Cloning templates are set as symmetrical as follows.
−p q −prq  , Bα = 0, Iα = 0
Aα =  q−p q −p p −q p Aβ = −q −r −q , Bβ = 0 Iβ = 0p −q p(13)

The network size is 8 × 8. The initial state is set at random. The input values do not aﬀect the result because
Bα = Bβ = 0.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)
(1)

(i)

(j)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)
(2)

(i)

(j)

4.2 Synchronization phenomena
Figures 9(1) and (2) show synchronization phenomena. The parameters of the cloning templates are set as
p = 2, q = 4 and r = 0. These two results are obtained from two diﬀerent random initial states, as shownin Figs. 9(1)(a) and (2)(a). The two results show periodic oscillations. The phenomena shown in Figs. 9(1)(b)and (2)(b) to Figs. 9(1)(j) and (2)(j), respectively, are repeated via transient states from the initial state, as shownin Figs. 9(1)(a) and (2)(a). All cells are divided into
three groups. For instance, the value of cell c(1, 1) in
Fig. 9(1) (e) is synchronized with the value of cells c(3, 1),c(1, 3), c(3, 3) and so on. This group is called Cell α1 inthis study. In the same way, the cell c(2, 2) group is called
Cell α2 . All cells β are synchronized. Therefore, this groupis called Cell β. Figures 10(1) and (2) show the waveformsof Figs. 9(1) and (2), respectively. Each group is synchronized via a transient state.
A diﬀerence between the two figures is that Cell α1 andCell α2 are placed on opposite sides from each other. Thesewaves synchronize in phase. Additionally, Cells β are alsosynchronized. Note that all waves synchronize in phaseand the oﬀsets have diﬀerent values.
4.3 Quasi-synchronization phenomena
Figure 11 shows quasi-synchronization phenomena.
The parameters of the cloning templates are set as p = 2,
q = 3 and r = 0. The waveforms are blurred. Additionally, burst parts are observed in the center part. The snapshots when this burst occurred are similar to those in Fig. 9.
However, some cells are delayed temporarily. This delay
is observed as the burst.

6

Fig. 9 Oscillatory phenomena starting from two diﬀerent
initial states: p = 2, q = 4, r = 0, (a) Initial state, (b) (j) Snapshots, The iteration numbers of (1) are (b) 10010,(c) 10040, (d) 10070, (e) 10100, (f) 10130, (g) 10160, (h)10190, (i) 10210 and (j) 10240, The iteration numbers of(2) are (b) 10080, (c) 10110, (d) 10140, (e) 10170, (f)10200, (g) 10260, (h) 10290, (i) 10210 and (j) 10320
4.4

Clustering phenomena

Figure 12 shows clustering phenomenon. The parameters of the cloning templates are set as p = 3, q = 4
and r = 1. Depending on initial states, similar phenomena to those shown in Figs. 10 or 11 can also be observed
for these parameters. The waveforms are divided into 16
clusters as shown in Fig. 12. Namely, the waveforms are
divided into four, which are named Cell α1 , Cell α2 , Cell
β1 and Cell β2 , and each Cell is divided into four clusters
by phases. The diﬀerence in the phases is 90 degrees per
cluster. Each cluster consists of four Cells. These clusters are divided regularly. For instance, Cell α1 − 1 includes c(1, 1), c(3, 1), c(5, 1) and c(7, 1), Cell α1 − 2 includes c(1, 3), c(3, 3), c(5, 3) and c(7, 3), Cell α1 − 3 includes c(1, 5), c(3, 5), c(5, 5) and c(7, 5) and Cell α1 − 4includes c(1, 7), c(3, 7), c(5, 7) and c(7, 7). In the four clusters of Cell α2 , Cell β1 and Cell β2 , the same relationships
are observed.
4.5

Analysis of oscillation

Normally, coupled oscillatory systems consist of oscillators and coupling elements. Namely, an oscillator canoscillate on its own. However, the proposed system consists of two kinds of cells that cannot oscillate on their
Journal of Signal Processing, Vol.24, No.1, January 2020

10

Cell α 1

10Cell α 1
10Cell α 2

10Cell α 2
10Cell β

10Cell β(1)Fig. 11 Quasi-synchronization including burst parts
10

Cell α 1

10Cell α 2

10Cell β(2)Fig. 10 Time series of oscillatory phenomena shown in
Figs. 9(1) and (2)own. The elements of the oscillator are shared among eachother. Regarding this point, it is considered that the proposed system is a novel oscillatory system. Additionally,
the waveforms, phases and so on of these oscillations are
influenced by the positions of the cells. For instance, theresults in Fig. 10 show that Cell α plays two roles in spiteof it being the same kind of cell in the system. Therefore,
three kinds of waveforms are observed. These phenomena
can only be observed in the proposed system. Oscillatory
phenomena cannot be observed in the conventional CNN
with the symmetry-cloning template. The observation ofoscillatory phenomena in the proposed system is also aninteresting point. In this section, the mechanism of thescillation of the proposed system is verified.
In the proposed system, oscillatory phenomena can beobserved easily with fixed boundary conditions. However,
it is not so easy to observe oscillatory phenomena with periodic boundary conditions. The reason why it is diﬃcult
is that the states of the proposed system basically correspond to the stable state of the conventional CNN, whichis mentioned in [1]. The only diﬀerence is that two kinds oftemplates are applied in the proposed system. Therefore,
the observation of oscillatory phenomena is very interesting and it is considered that using two kinds of templatescauses the oscillatory phenomena.
According to the results of many computer simulations,the minimum number of cells required to observe oscillatory phenomena is four. Namely, the simplest architecture
Journal of Signal Processing, Vol.24, No.1, January 2020

required for an oscillation consists of four cells. In thissection, the simplest architecture shown in Fig. 13 is investigated and the mechanism of the oscillation is revealed.
The output function yi j is defined as the piecewise linear function in Eq. (4). Each region is defined as follows.
D+
D0
D−

≡ {x > 1}≡ {1 ≤ x ≤ −1}≡ {x < −1}(14)

There are four cells, each with three regions. Consequently, the number of combinations is 34 = 81, whichis too many to analyze. Hence, the regions that the orbit
pass through are investigated by computer simulation. Table 1 shows wandering of the orbit. The parameters are set
to be the same as those in Fig. 9. The orbit passes throughregions in numerical order. When the orbit reaches the endof No. 12, it returns to No. 1. Sometimes, Nos. 3 and 9
are skipped. The eigenvalues of these regions are derived
as shown in Table 2. In region Nos. 2, 3, 8 and 9, the orbit does not converge because of the eigenvalues include
a positive value. In the other regions, equilibrium points
exist outside the regions, as shown in Table 3. Therefore,
the orbit does not converge in these regions. These results
show that the system does not oscillate in one region, and a
combination of attraction to the equilibrium points of eachregion generates the oscillation.
According to the result of this analysis, this system canoscillate in the case of symmetry-cloning templates. In theconventional CNN, symmetry-cloning templates could notoscillate. This point is one of the important features of the
two-template CNN.
5. Conclusions
In this paper, a cellular neural network using two kindsof cloning templates, which is a kind of modified cellularneural networks, has been proposed and investigated. Thissystem can carry out more complicated processing than theconventional CNN. Additionally, cloning templates of theconventional CNN can be applied to this system. As applications, five diﬀerent cloning templates for image processing that can carry out complicated processing and a demon7

Fig. 13 Simplest architecture of two template CNN

Table 1 Wandering of the orbit

Fig. 12 Computer simulation result in which 16 clusters
were observed
stration of an image-processing application were shown.
On the other hand, clustering phenomena of oscillations
could be observed in the case of periodic boundary condition and a symmetry-cloning template. The system wasanalyzed and the oscillation of the system was verified. It
is considered that these results show the potencial of the
two-template CNN.
Developing various new cloning templates, investigations of oscillatory phenomena and so forth, are our future
works.
Acknowledgment
This work was partly supported by JSPS Grant-in-Aid
for Scientific Research 16K06357.

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8

No.
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9
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12

Cell α1
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Table 2 Eigenvalues of each region
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Eigenvalues−1, −1, −1, −1−1, 6, −10, −1−1, −10, 3 ± j7.937−1, −1, −1, −1−1, −1, −1, −1−1, −1, −1, −1−1, −1, −1, −1−1, 6, −10, −1−1, −10, 3 ± j7.937−1, −1, −1, −1−1, −1, −1, −1−1, −1, −1, −1Table 3 Equilibrium points of each region
No.
1
4
5
6
7
10
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12

Equilibrium points (x(1,1) , x(1,2) ,x(2,1) , x(2,2) )(−8, 8, 8, −8)(8, 64, 64, 80)(8, 8, 8, 24)(8, −64, −64, −48)
(8, −8, −8, 8)(8, −48, −64, −64)
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Yasuteru Hosokawa received his
B.E., M.E. and Ph.D. degrees from
Tokushima University, Tokushima
Japan, in 1997, 1999 and 2002, respectively. In 2001, he joined the Faculty ofManagement and Information Scienceat Shikoku University, Tokushima,
Japan, where he is currently an
Associate Professor.
From August
2016, he spent four months at Saginaw
Valley State University (SVSU) as a
Visiting Professor. His research interest
is in chaotic circuit systems. He iscurrently the Vise Chair of the IEEE
CAS Society Shikoku Chapter. He is a member of the IEICE, IEEE andRISP.
Yoshifumi Nishio received his B.E.,
M.E. and Ph.D. degrees in electrical engineering from Keio University, Yokohama, Japan, in 1988, 1990 and 1993,
respectively. In 1993, he joined theDepartment of Electrical and Electronic
Engineering at Tokushima University,
Tokushima, Japan, where he is currently a Professor. From May 2000, he
spent a year in the Laboratory of Nonlinear Systems (LANOS) at the Swiss
Federal Institute of Technology Lausanne (EPFL) as a Visiting Professor.
His research interests include analysis
and application of chaos in electrical circuits, analysis of synchronization in coupled oscillatory circuits, development of methods for analyzing nonlinear circuits, theory and application of cellular neural networks,
and neural network architecture. He was the Chair of the IEEE CAS Society Technical Committee on Nonlinear Circuits and Systems (NCAS)
during 2004–2005, the Steering Committee Secretary of the IEICE Research Society of Nonlinear Theory and its Applications (NOLTA) during 2004–2007, and is currently the Secretary/Treasurer of the IEEE CAS
Society Shikoku Chapter. He was an Associate Editor of IEEE Transactions on Circuits and Systems–I: Regular Papers during 2004–2005 andIEEE CAS Magazine during 2008–2009, and is currently serving as an
Associate Editor for IEEE Transactions on Circuits and Systems–II: Express Briefs, the IEEE CAS Society Newsletter. He is also the Editor of
NOLTA, IEICE and IEICE Fundamentals Review and a member of the
Editorial Board of International Journal of Circuit Theory and Applications. He is a senior member of the IEEE and a member of IEICE and
RISP.
(Received March 16, 2019; revised August 9, 2019)9