IEICE TRANS. FUNDAMENTALS, VOL.E87–A, NO.4 APRIL 2004

864

PAPER

Generation of Various Types of Spatio-Temporal Phenomena in
Two-Layer Cellular Neural Networks
Zonghuang YANG†a) , Student Member, Yoshifumi NISHIO†† , Member, and Akio USHIDA†† , Fellow

SUMMARY
The paper discusses the spatio-temporal phenomena in autonomous two-layer Cellular Neural Networks (CNNs) with mutually coupled templates between two layers. By computer calculations, we show
how pattern formations, autowaves and classical waves can be regenerated
in the networks, and describe the properties of these phenomena in detail.
In particular, we focus our discussion on the necessary conditions for generating these spatio-temporal phenomena. In addition, the influences of
the template parameters and initial state conditions of CNNs on the spatiotemporal phenomena are investigated.
key words: two-layer CNN array, reaction-diﬀusion equation, pattern formation, autowave, classical wave

1.

Introduction

Studies of dynamic phenomena in arrays composed of oscillatory [1]–[7] and chaotic elements [8]–[18] are very important for understanding the phenomena observed in natural
fields. A Cellular Neural Network (CNN) is a locally connected network [19], in which each cell is connected only to
its neighbor, so the CNN allows eﬃcient VLSI implementation of an analogue array-computing machine [20]–[24].
Because of its extraordinary computational power, the CNN
is an ideal tool for the simulation of various spatio-temporal
dynamics in discrete space.
Since the mid-1990s, investigations into the spatiotemporal dynamics in cellular neural networks have been
widely carried out, and many papers have been published[5]–[18]. They have discussed the pattern formations andvarious autowaves such as excitability waves, concentration
waves and so on, as observed in many fields such as biology,
physics and chemistry. To date, these CNN arrays, composed of chaotic circuits [9]–[18] or second-order nonlinear
circuits obtained by appropriately “reducing” Chua’s circuit[8], have been examined for the generation of the above phenomena. In these studies, CNNs are used to approximate
various nonlinear partial diﬀerential equations, particularly
the well-known reaction-diﬀusion equations that show theTuring pattern and propagation phenomena in various continuous media [25]. In [5]–[7], Arena and coworkers andManganaro et al. conducted studies on some other travelling
Manuscript received March 24, 2003.
Manuscript revised August 28, 2003.
Final manuscript received November 28, 2003.
†
The author is with the Graduate School of Engineering, The
University of Tokushima, Tokushima-shi, 770-8506 Japan.
††
The authors are with the Department of Electrical and Electronic Engineering, the University of Tokushima, Tokushima-shi,
770-8506 Japan.
a) E-mail: yangzh@ee.tokushima-u.ac.jp

waves and self-organization phenomena in two-layer CNNs
under the condition that the CNN cell exhibits slow-fast dynamics. This condition is so rigorous that the CNN places
restrictions on other applications.
The objective of this work is to study spatio-temporalphenomena occurring in the mutually coupled two-layer
CNNs, based on the CNN temporal eigenvalues. The twolayer CNNs possess a simpler structure, compared with thepreviously studied CNNs [8], [18]. We found from numerical simulations that similar phenomena can also be observedin our two-layer CNNs under milder necessary conditions.
In addition to some nonlinear spatio-temporal phenomena
such as autowaves and pattern formations, we found that another class of dynamic phenomena – classical wave propagation phenomena observed everywhere – can be simulated
as well in the two-layer CNNs. These types of propagation
phenomena have the properties of reflection, permeation andsuperposition. In particular, the necessary conditions forgenerating these spatio-temporal phenomena are discussed.
This paper is organized as follows: In Sect. 2, we review the two-layer CNNs with mutually coupled templates
between the two layers. In Sects. 3 and 4, we discuss various spatio-temporal phenomena in terms of the temporaleigenvalue of the mutually coupled two-layer CNNs. The
necessary conditions for these phenomena are indicated by
the use of a decoupling technique. Several simulations forinteresting spatio-temporal phenomena are shown and described in detail. In addition, the influences of the parameters in the cloning templates on propagation phenomena arealso investigated by employing numerical calculation.
2.

Two-Layer CNN ArchitectureFirst, we review the mutually coupled two-layer CNN [26]–[28] used for investigating spatio-temporal phenomena. The
CNN is described on the basis of the well-known Chua-Yang
CNNs.
The system equations are formulated by introducing
two coupling templates C1 and C2 . In this paper, we also
assume that the two-layer CNNs are composed of twodimensional M by N array structures as shown on the lefthand side in Fig. 1. Each cell in the array is denoted byc(i, j), where (i, j) stands for the position of a cell in thearray, for 1 ≤ i ≤ M and 1 ≤ j ≤ N. The cell local coupling is shown on the right-hand side in Fig. 1. Each cellhas two state variables x1,i j and x2,i j , where the subscripts 1
and 2 denote the first layer and the second layer of the two-YANG et al.: GENERATION OF VARIOUS TYPES OF SPATIO-TEMPORAL PHENOMENA IN TWO-LAYER CNNS

865

u1
B1

First
layer

3.
A1

Pattern Formations and Active Propagation Phenomena

3.1 Common Mechanism
N

C1

C2
A2

Secondlayer u2
B2
A cell c(i, j)
M
Fig. 1 A two-dimensional cellular neural network and a cell linked toneighbor cells with various templates.

layer CNN array. The state equations of the cell are given
by two first-order diﬀerential equations given by Eq. (1),and the output equations are given by Eq. (2), where f (·)
is a piecewise-linear nonlinear function defined by Eq. (3).
The variables u and y refer to the input and the output variables of the cell, respectively. r is the coupling radius [19].
Nr (i, j) denotes the neighbor cells of radius r of a cell c(i, j).
A(i, j; k, l), B(i, j; k, l), C(i, j; k, l) and I denote the feedback
coeﬃcient, control coeﬃcient, coupling coeﬃcient between
layers and bias current, respectively.
dx1,i j=−x1,i j + I1



dt




+ c(k,l)∈Nr (i, j) A1 (i, j; k, l)y1,kl + c(k,l)∈Nr (i, j) B1 (i, j; k, l)u1,kl + c(k,l)∈Nr (i, j) C1 (i, j; k, l)y2,kl 

(1)


dx2,i j=−x2,i j + I2



dt




+ c(k,l)∈Nr (i, j) A2 (i, j; k, l)y2,kl + c(k,l)∈Nr (i, j) B2 (i, j; k, l)u2,kl + c(k,l)∈Nr (i, j) C2 (i, j; k, l)y1,kl y1,i j = f (x1,i j )(2)y2,i j = f (x2,i j )f (x) = 0.5(|x + 1| − |x − 1|).
(3)
When r = 1, the CNN cloning templates [19] A1 , A2 , B1 ,
B1 , C1 and C2 are 3 times 3 matrices, which can be described to have a similar form as that of (4).
 A(i, j; i−1, j−1) A(i, j; i−1, j) A(i, j; i−1, j+1)  A(i, j; i, j−1) (4)A(i, j; i, j)
A(i, j; i, j+1)

A(i, j; i+1, j−1) A(i, j; i+1, j) A(i, j; i+1, j+1)
The CNN is eﬃcient for image processing applications
such as center point detection, skeletonization, and so forth,which have been reported in our previous studies [26]–[28].
In the following sections, we discuss the spatio-temporalphenomena in the two-layer CNN.

Many nonlinear reaction-diﬀusion partial diﬀerential equations (PDEs) [25] have been proven to generate various
types of nonlinear spatio-temporal phenomena, such as pattern formations, autowaves and self-organization phenomena. The concept of reaction-diﬀusion CNNs has been formalized to reproduce similar behaviors in Ref. [14], because
they are mathematically described by a discretized versionof the following well-known system of nonlinear partial differential equations – reaction-diﬀusion equations:∂u= f (u) + D∇2 u,∂t(5)where u∈ Rm represents a vector of continuously changing spatial variables, f (u)∈ Rm their kinetics governing theoscillation in an isolated point of a medium, and the lastterm represents diﬀusion. D is an m × m diagonal matrix,
which describes the intensity of diﬀusive coupling between
the components of u, and∇2 ui =

∂2 ui ∂2 ui
+ 2,
∂x2
∂y

i = 1, 2, · · · , m(6)

is the Laplacian operator in R2 .
In order to use the two-layer CNNs to approximate theabove reaction-diﬀusion equation, we consider the equation
with u∈ R2 in finite space, and assume it to be as follows:u̇1 = −u1 + a1 v1 + c1 v2 + D1 ∇2 v1(7)u̇2 = −u2 + a2 v2 + c2 v1 + D2 ∇2 v2
where a1 , a2 , c1 and c2 are parameters, D1 , D2 are diﬀusion
coeﬃcients, and v is a nonlinear function of u defined by (3).
By expanding the Laplacian operator in the discrete space,
Eq. (7) can be written as:u̇1;i j = −u1;i j + a1 v1;i j + c1 v2;i j +D1 (v1;i−1, j + v1;i+1, j+v1;i, j−1 + v1;i, j+1 − 4v1;i j ) (8)u̇2;i j = −u2;i j + a2 v2;i j + c2 v1;i j 






+D2 (v2;i−1, j + v2;i+1, j+v2;i, j−1 + v2;i, j+1 − 4v2;i j )where 1 ≤ i ≤ M and 1 ≤ j ≤ N. Compared with Eq. (1), thestate equation (8) presents a two-layer autonomous reactiondiﬀusion standard CNN. The cloning template of the twolayer CNN array is described as follows:




0 
D1
 0
0 0 0
A1 = D1 −4D1 + a1 D1 , C1 = 0 c1 0 ,
0
D1
0
0 0 0




0 
D2
 0
 0 0 0 A2 =  D2 −4D2 + a2 D2  , C2 =  0 c2 0  ,
0
D2
0
0 0 0

IEICE TRANS. FUNDAMENTALS, VOL.E87–A, NO.4 APRIL 2004

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B1 = 0, B2 = 0, I1 = 0, I2 = 0.

(9)

If at least one of the temporal eigenvalues has a positive
real part, the CNN constructed by the connection between
diﬀusing cells will become unstable and produce the patternformations or active propagation phenomena. The necessary
condition is described by Eq. (10), which is directly derived
from those conditions obtained from the Turing patterns in
an array of coupled circuits [8]:
(a1 − 1) (a2 − 1)



+
>0

(10)
D1
D2



2
[(a1 − 1)D2 − (a2 − 1)D1 ] > −4D1 D2 c1 c2 
The above condition ensures that only the two-layer
CNN array is unstable in the linear region. As time increases, due to the positive real-part temporal eigenvalues,
the cell states enter into nonlinear regions (i.e., saturation region or partial saturation region). In [8], the general principle of pattern formation has been indicated. The key mechanism of pattern formation is based on two types of necessary conditions. In addition to condition (10), the second
condition refers to the requirement for the isolated cell tohave a unique, stable equilibrium point. The two necessary
conditions mean that the above equilibrium is merely a solution to the CNN array, which corresponds to a homogeneous
pattern, but it should become unstable upon introducing thediﬀusion between cells; the pattern will also correspond tosome other equilibrium points that happen to be stable forthe pattern formation. On the other hand, if the isolated cell
has no stable equilibrium point, then it behaves as a nonlinear oscillator with a limit cycle for the active wave propagation. These behaviors concerning the isolated cell have been
discussed in detail in Ref. [29].
3.2 Pattern Formations
In this and the next subsection, we investigate the spatiotemporal phenomena in the two-layer CNNs by applying
the aforementioned necessary conditions. Some significant
findings are shown from the analysis of simulations.
3.2.1 Checkerboard
Experiment 1: This experiment is carried out under a twolayer CNN array consisting of 20 × 20 cells. The initialstates of both layers are initiated by random noise shown in
Fig. 2(a). In this case, the zero-fixed boundary condition isadopted. Under the template:




−0.2
0 
 0
 0 0 0 
A1 =  −0.2 1.8 −0.2  , C1 =  0 −1 0  ,
0
−0.2
0
0 0 0




 0 0.1 0 
 0 0 0 
A2 =  0.1 0.3 0.1  , C2 =  0 1 0  ,0 0.1 00 0 0
(11)B1 = 0, B2 = 0, I1 = 0, I2 = 0,
we found a solution such that a stable output pattern with

Fig. 2 Checkerboard pattern formation in a two-layer CNN with random
initial state condition. (a) is initial state, (b) is transient result, and (c) is
steady output result.

the white-black grid alternating between the cells (so-called
checkerboard) shown in Fig. 2(c) finally results in the firstlayer output of the CNN, where the states of all cells settle
into the saturation region (i.e., each cell output is 1 or −1
corresponding to black or white, respectively).
Here, it should be emphasized that, the checkerboard
pattern and stripe pattern formations in the next subsection
can be obtained in a single-layer CNN array only with an infinite CNN structure [30]. Namely, a single-layer CNN withfinite structure can only generate a defective checkerboard
and stripe patterns, because the states of some cells, particularly near the boundary region, do not entirely enter into
the saturation region. Note that the perfect checkerboard
can be obtained in the two-layer CNN array with the finite
structure. Moreover, we found that, irrespective of the initial states and the size of the array, the perfect checkerboard
pattern can be obtained with the template parameter (11).
3.2.2 Stripe
Experiment 2: In this experiment, the procedure of stripe
pattern formation is investigated. The CNN array consistsof 100 × 100 cells with a zero-fixed boundary condition; its
template is as follows: 0 0.1 0 
 0 0 0 
A1 =  0.1 2.2 0.1  , C1 =  0 −1 0  ,0 0.1 0
0 0 0




0.01
0 
 0
 0 0 0 A2 =  0.01 2.56 0.01  , C2 =  0 1 0  ,
0
0.01
0
0 0 0
(12)
B1 = 0, B2 = 0, I1 = 0, I2 = 0.
We observe that a stripe pattern formation occurs in the
CNN, as shown in Fig. 3, where (a), (b) and (c) are respectively the initial state, the transient result and the steady output of the CNN. From the analysis of the simulation results,
it is seen that the pattern formation process starts from theobjects in the image (i.e., nonzero initial conditions), propagates around the objects, and generates new stripes withalmost the same geometry as that of the original initiatedobjects. Then, it propagates again until the entire CNN array reaches the steady state. When the propagation wave
front collides with another wave front or the boundary, thewave fronts eliminate each other without any interference orYANG et al.: GENERATION OF VARIOUS TYPES OF SPATIO-TEMPORAL PHENOMENA IN TWO-LAYER CNNS

867

(a)
Fig. 3 Stripe pattern formation with D1 = 0.1. (a) is initial state, (b) istransient result, and (c) is steady output result.

(b)Fig. 4 Stripe pattern formation with D1 = 0.2. (a) is initial state, (b) istransient result, and (c) is steady output result.
(c)reflection.
Moreover, the density of the stripe pattern can be adjusted by suitably changing the template parameters. The
simulation example is shown in Fig. 4, where the templateis the same as (12) except that the diﬀusion coeﬃcient D1
is set to 0.2 instead of 0.1. Thus, the lower density stripe
pattern formation is obtained.
From the above two simulation examples, we foundthat the shape of the stripe pattern is dependent on the initialstate of the CNN array, and the density of the stripe pattern
can be controlled by the template parameters. Thus, many
interesting stripe patterns such as the coats of animals, fingerprints, and so forth may be regenerated by suitably selecting the initial conditions of the CNN array and the system parameters.
3.3 Active Propagation Phenomena
In this subsection, an investigation into active propagation
phenomena is carried out in the two-layer CNN, which enables us to determine some basic properties of these phenomena. In the following experiment, an array of 100 × 100
cells is considered. Similarly, the selected parameters of thetemplate satisfy the necessary condition (10) to guarantee
the instability of the entire CNN with diﬀusion. Next, let
us observe how the disturbance initiated inside the network
propagates in the plane array.
Experiment 3: In this experiment, the initial conditionis the same as in Experiment 2. The zero-flux boundary
condition is adopted and the template is selected as follows: 0 0.1 0 
 0 0 0 
A1 =  0.1 1.1 0.1  , C1 =  0 −1 0  ,0 0.1 00 0 0

(d)Fig. 5 A simulation for autowaves with two concentric circular waves.
(a) is the initial state, (b), (c) and (d) show three snapshots obtained for
diﬀerent iterates.





−0.01
0 
 0
 0 0 0 

A2 =  −0.01 1.04 −0.01  , C2 =  0 1 0  ,
0
−0.01
0
0 0 0
(13)
B1 = 0, B2 = 0, I1 = 0, I2 = 0.
Figure 5 shows four snapshots of this simulation. The simulation result indicates that two concentric circular waves
are generated. Their wave fronts propagate in all directions
through the network from the initialized positions. Behind
the wave fronts, the concentric waves centered at the cells
where the process was initiated are visible. Moreover, they
do not conserve energy but preserve amplitude and waveform, they are not reflected from obstacles but the colliding
wave fronts are annihilated, and they do not produce any interference but diﬀract and pass around obstacles. Namely,
this phenomenon is considered to be an autowave.
Note that we have used a simpler two-layer CNN array to reproduce the same phenomena with those observedin arrays made of chaotic cells [17], [18], and here the cellis merely a general nonlinear oscillator having a limit cycle,
but does not exhibit slow-fast or chaotic dynamics. Simi-IEICE TRANS. FUNDAMENTALS, VOL.E87–A, NO.4 APRIL 2004

868

larly, if we suitably adjust the diﬀusion coeﬃcients D1 or
D2 , we can control the wavelength and speed of the propagation.

eigenvalue can be assumed as follows:

4.

and

Classical Wave Propagation

φ MN (m, n; i, j) = cos

2= 4 sin2
kmn

4.1 Necessary Condition
We have shown in Sect. 3 that the two-layer CNNs can reproduce the pattern formation and the active propagation
phenomena. These CNNs have a common property in thatat least one of the temporal eigenvalues of CNNs has a positive real part. Moreover, this property is also one of thenecessary conditions for the binary image processing applications of the two-layer CNNs [30]. However, when all the
temporal eigenvalues have zero or negative real parts, what
happens in the CNNs? We examine the phenomena in thissection.
Let us consider an autonomous two-layer CNN withmutual coupling between layers. The equation is describedin the following form:
ẋ1,i j = −x1,i j + (a1 + 1)y1,i j + c1 y2,i j 



2

+d1 ∇ y2,i j(14)
ẋ2,i j = −x2,i j + (a2 + 1)y2,i j + c2 y1,i j 




2
+d2 ∇ y1,i j
where a1 , a2 , c1 , c2 , d1 and d2 are parameters. By expanding
the Laplacian operator in discrete form, and in comparison
with Eq. (1), the two-layer CNN has the following template:




0 
0
0 
d1
 0
 0


A1 =  0 a1 + 1 0  , C1 =  d1 −4d1 + c1 d1  ,
0
d1
0
0
0
0




0 
0
0 
d2
 0
 0





A2 =  0 a2 + 1 0  , C2 =  d2 −4d2 + c2 d2  ,
0
d2
0
0
0
0
(15)
B1 = 0, B2 = 0, I1 = 0, I2 = 0.
If we assume that all the cells remain in the linear region, then the state equation (14) can be written as follows:ẋ1,i j = a1 x1,i j + c1 x2,i j + d1 ∇2 x2,i j(16)ẋ2,i j = a2 x2,i j + c2 x1,i j + d2 ∇2 x1,i j .
Thus, the above linear diﬀerential equation can be solved
by decoupling it into MN-decoupled systems of two firstorder linear diﬀerential equations, and considering the MN
orthonormal space-dependent eigenfunction φ MN (m, n; i, j)of the discrete Laplacian operator, it can be assumed for almost all boundary conditions that:2
φ MN (m, n; i, j),∇ φ MN (m, n; i, j) = −kmn2(17)where M and N are the CNN dimensions, m and n are thesummation indexes for the current space variables i and j2(i = 0, 1, · · · , M − 1; j = 0, 1, · · · , N − 1), and kmn
arethe corresponding spatial eigenvalues. In particular, for thezero-flux boundary condition, the spatial eigenfunction and(2i+1)mπ(2 j+1)nπ
cos
2M
2N

mπ
nπ 
+ sin2
.
2M
2N

(18)

(19)

Then, the expected solution of Eq. (16) can be expressed asa weighted sum of the M × N orthogonal space dependent
eigenfunctions φ MN (m, n; i, j) in the following form:

N−1
M−1



λmn1 tλmn2 t
x1,i j (t) =(αmn e+ βmn e)φ MN (m, n; i, j)





m=0 n=0

N−1
M−1





λmn1 tλmn2 t(γmn e+ δmn e)φ MN (m, n; i, j) x2,i j (t) =
m=0 n=0(20)
where αmn , βmn , γmn , δmn are constants depending on theinitial conditions. λmn1 and λmn2 are the roots of the following characteristic equation (21), which are influenced by2
corresponding to the spatial eigenthe spatial eigenvalue kmn
function φ MN (m, n; i, j).


 

2

1 0
a1 c1 −kmn
d1 
(21)
λmn 0 1 − c −k2 d = 0
a2
2
mn 1
and

1
2
] = (a1 + a2 )λmn [kmn2
2 d − c )(k2 d − c )± (a1 − a2 )2 + 4(kmn
2
2
1
mn 1

(22)

Equation (20) is important because the cell states of the
two-layer CNN are given as the time-dependent weighted
sums of spatial eigenfunctions. We have discussed in Sect. 3
that, if at least one of the temporal eigenvalues has a positive real part, the two-layer CNNs can be used for modeling the active media for nonlinear phenomena such as pattern formation and autowaves. However, if all the temporaleigenvalues have a zero or negative real part, all the statesof the CNN are in the linear region, which agrees with ourassumption above. Thus, the CNNs have the properties oflinear space, such as superposition. Therefore, we can also
use this CNN to model the passive media with/without loss
in linear space.
4.2 Classical Waves
In this subsection, we investigate the passive propagation
phenomena with/without loss in two-layer CNNs. In order
to compare these with the propagation phenomena (Experiments 2, 3) in active media, we adopt the same-sized arraywith the same initial states and boundary condition (zeroflux) in Experiment 4.
Experiment 4: In this experiment, we observe the propagation phenomena in a two-layer CNN with/without loss.

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869

(a)

Fig. 7 Graph of sum of State-Error Square vs. time for the passive wave
propagation phenomenon without loss.

(b)

(a)

(c)

(b)

(d)
Fig. 6 A simulation for the passive wave propagation phenomenon without loss. (a) is the initial state, (b), (c) and (d) show three snapshots observed in diﬀerent iterates.

If we select the parameter set to satisfy that all the temporaleigenvalues of Eq. (20) have a zero real part, then the twolayer CNN will become a system without loss. This typeof two-layer CNN can be used for modeling passive media
without loss. For example, for the selected parameter set of
a1 = 0, a2 = 0, d1 = −0.5, d2 = 0.5, c1 = −5, and c2 = 5,
all the temporal eigenvalues of the CNN are complex numbers with a zero real part. The simulation result is shown in
Fig. 6, where (a) is the initial state of the CNN and (b)–(d)are three snapshots obtained as time progresses.
The simulation result shows that the circular waves
propagating from the initialized position are generated.
These circular waves propagate in all directions through theplane array and their amplitudes decrease through propagation. When the waves collide with each other, they do not

(c)

(d)
Fig. 8 A simulation for the passive wave propagation phenomenon withloss. (a) is the initial state, (b), (c) and (d) show snapshots observed in
diﬀerent iterates.

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870

Fig. 9 Graph of sum of State-Error Square vs. time for the passive wave
propagation phenomenon with loss.

annihilate and reflect, but permeate through and superposeon each other. When they collide with the boundary, thewaves are reflected. Moreover, after the waves are repeatedly reflected and superposed, the initialized positions become invisible. These behaviors are similar to those of classical waves. The wave propagation continues forever due tothe zero real part temporal eigenvalue. This dynamic process can be observed from the graph of the sum of StateError Square versus the integration time shown in Fig. 7.
The sum of State-Error Square is defined as:
S CNN (t) =

N
M 


((x1,i j (t + ∆t) − x1,i j (t))2i=1 j=1+ (x2,i j (t + ∆t) − x2,i j (t))2 ).

(23)

We can see from the graph that the sum of State-Error
Square converges on a horizontal line with a value.
On the other hand, if we select the parameter set tosatisfy all the temporal eigenvalues of the CNN having negative real parts, then each cell state will decay to zero, which
can be derived from Eq. (20). This situation is suitable forthe model of the passive media with loss. For example, if
the parameters are set as a1 = −0.04, a2 = 0, d1 = −0.5,
d2 = 0.5, c1 = −5, and c2 = 5, then all the temporal eigenvalues of the CNN will be complex numbers with a negative
real part. Therefore, the CNN behaves as a damped system.
The simulation result is shown in Fig. 8, where (a) shows theinitial state of the CNN, and (b)–(d) show snapshots taken
at three diﬀerent times. From the simulation results, we cansee that the propagation phenomenon has the same properties as the above loss-less system. However, the network
is calm in the final due to the temporal eigenvalues having
a negative real part. The dynamic process can be similarly
demonstrated from the graph of sum of State-Error Squareas shown in Fig. 9.
5.

Conclusions

In this paper, we have shown that many types of spatiotemporal phenomena, such as pattern formation, autowaves
and classical waves, can be reproduced in the two-layer
CNNs with a simple structure. The conditions necessary

for obtaining these phenomena have been indicated. The effects on propagation phenomena due to changes of systemparameters have also been investigated. On the basis of ourresearch results, we found that the two-layer CNN can beconsidered not only as a model of active media, but also
as a model of passive media with/without loss. Moreover,
it is worth mentioning that the type of two-layer CNN has
a simpler structure compared with those arrays that consistof chaotic circuits or second-order nonlinear circuits. Although the pattern formations can be theoretically obtainedin an infinite single-layer CNN array, applying the two-layer
CNNs with an arbitrary number of cells to these problems
enables us to obtain perfect pattern formations.
In addition, the two-layer CNNs described in this paper have zero B-templates. Thus, we can consider the Ctemplate as a B-template in every layer, and mutually link
one layer output with the other layer input. Therefore, it
is possible to realize the above two-layer CNN Universal
Machine (CNNUM) as a hardware simulator for simulating spatio-temporal phenomena, based on the current singlelayer CNNUM chip. All these findings demonstrate that the
two-layer CNNs have a real potential for a range of applications.
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Zonghuang Yang
received the B.E. degree in Radio Engineering from South East University, Nanjing China in 1985, and the M.E.
degree in Electrical and Electronic Engineeringfrom Tokushima University, Tokushima, Japan,
in 2001. He is currently working toward the
Ph.D. degree in the same department of Tokushima University. His research interests include
the theory and application of cellular neural networks.

Yoshifumi Nishio
received the B.E., M.E.,
and Ph.D. degrees in Electrical Engineeringfrom Keio University, Yokohama, Japan, in
1988, 1990, and 1993, respectively. In 1993,
he joined the Department of Electrical and Electronic Engineering at Tokushima University,
Tokushima, Japan, where he is currently an Associate 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 inelectrical circuits, analysis of synchronization in coupled oscillators circuits, development of analyzing methods for nonlinear circuits and theory
and application of cellular neural networks. Dr. Nishio is a member of theIEEE.

Akio Ushidareceived the B.E. and M.E. degrees in Electrical Engineering from Tokushima
University in 1961 and 1966, respectively, and
the Ph.D. degree in Electrical Engineering from
University of Osaka Prefecture in 1974. He
was an Associate Professor from 1973 to 1980
at Tokushima University. From 1980 to 2003,
he was a Professor in the Department of Electrical and Electronic Engineering at Tokushima
University. Since 2003, he has been with the
Department of Mechanical-Electronic Engineering, Tokushima Bunri University. From 1974 to 1975 he spent one year as avisiting scholar at the Department of Electrical Engineering and Computer
Sciences at the University of California, Berkeley. His current researchinterests include numerical methods and computer-aided analysis of nonlinear systems. Dr. Ushida is a member of the IEEE.