International Symposium on Nonlinear Theory and its Applications
Xi’an, PRC, October 7–11, 2002

Classical Wave Propagation Phenomena in Two-Layer Cellular Neural NetworksZonghuang YANG, Yoshifumi NISHIO and Akio USHIDA
Department of Electrical and Electronic Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470, Fax:+81-88-656-7471
Email: yangzh@ee.tokushima-u.ac.jp

1. Introduction

formulated as follows:Studies of dynamic phenomena in arrays composed ofoscillatory and chaotic elements are very important insuch investigations as they provide a universal model forphenomena observed in other domains. From the middle of previous decade, the investigations of the spatiotemporal dynamics in cellular neural networks have been
widely carried out, and many papers have been published. The obtained sound results are pattern formations and various forms of autowaves such as excitability waves, concentration waves and so on, which havebeen observed in many disciplines ranging from biology
to physics and chemistry, etc. Up to the present, theCNN arrays, which are composed of Chua’s oscillators[2]-[4], second nonlinear circuit obtained by the suitably“reducing” Chua’s circuit [5] and etc. [6][7] have been
mainly considered in order to generate these phenomena. A common technique in all these cases is almost
using CNNs to approximate the various types of nonlinear partial diﬀerential equations, especial the well-known
reaction-diﬀusion equations that have shown to generate
Turing pattern and propagation phenomena in various
continuous media.
The purpose of this paper is to investigate anotherclass of spatio-temporal phenomena occurring in a simpler CNN structure - mutually coupled two-layer CNNs.
We found that this type of propagation phenomena has
the properties of reﬂection, permeation, superpositionand etc., which are completely diﬀerent from those nonlinear propagation phenomena. In particular, the conditions for generating the classical wave will be discussed.

2. Two-Layer CNNs

dx1,ijdt

dx2,ijdt=+++=+++

−x
 1,ij + I1A(i,j;k,l)y
11,kl
C(k,l)∈Nr (i,j)
B(i,j;k,l)u11,kl C(k,l)∈N(i,j)
r
C(k,l)∈Nr (i,j) C1 (i, j; k, l)y2,kl−x 2,ij + I2

A(i,j;k,l)y22,kl C(k,l)∈N(i,j)
rB2 (i, j; k, l)u2,kl C(k,l)∈N(i,j)
rC(k,l)∈Nr (i,j) C2 (i, j; k, l)y1,kl

y1,ij = 0.5(|x1,ij + 1| − |x1,ij − 1|)y2,ij = 0.5(|x2,ij + 1| − |x2,ij − 1|)

(1)

(2)

where (i, j) stands for the position of a cell in the array,for 1 ≤ i ≤ M and 1 ≤ j ≤ N , and the subscript 1 and2 stand for ﬁrst layer and second layer of the two-layer
CNN array. The CNNs are eﬃcient for image processing
applications such as center point detection, skeletonization and etc., which have been reported in our previousstudies [7][8]. In the following paragraph, we will investigate the spatio-temporal phenomena in the two-layer
CNNs.
3. Investigation of Classical Wave Propagation
We had shown that the two-layer CNNs could reproduce the pattern formation and active propagation phenomena in reference 9, in which there is a common property that at least one of temporal eigenvalues of two-layer
CNNs has positive real part, even if the property is also
for image processing applications of the CNNs. However,when all of the temporal eigenvalues have zero or negative real parts, what will be taken place in the CNNs?
This work will be discussed in this section.

3.1. Necessary Condition
Firstly, let us recall simply the mutually coupled twoLet us consider an autonomous two-layer CNN withlayer CNNs [7]-[9], which are described based on the wellknown Chua-Yang’s CNNs [1]. The system equations are mutual couple between layers. Its equation is described-981-

as the following form:

Then, the expected solution of the Eq.(5) can be expressed as a weighted sum of M × N orthogonal spaceẋ1,ijdependent eigenfunctions φMN (m, n; i, j) in the followingform:(3)ẋ2,ijM−1N −1
λmn1 tλmn2 tx1,ij(t)=(αmne+βmne)φMN(m,n;i,j)

Where a1 ,a2 ,c1 ,c2 ,d1 and d2 are parameters. By expandm=0 n=0(9)M−1N −1ing the Laplacian operator in discrete form, and comλmn1 tλmn2 t(γmne+δmne)φMN(m,n;i,j)

paring with Eq.(1), the two-layer CNN has the following x2,ij(t)=m=0 n=0template:
where αmn , βmn , γmn , δmn are constants depending on
00d1
theinitial conditions. λmn1 , λmn2 are the roots of theA1 = a1 + 1, C1 =  d1 −4d1 + c1 d1  ,following characteristic Eq.(10), which are inﬂuenced by0
d102
corresponding to spatial eigenspatial eigenvalue kmn

00d2function.
A2 = a2 + 1, C2 =  d2 −4d2 + c2 d2  ,2
c1 −kmn
d1
1 0
a10d2
0
= 0 (10)det λmn−2c2 −kmn
d1
a20 1(4)B1 = 0, B2 = 0, I1 = 0, I2 = 0.

= −x1,ij + (a1 + 1)y1,ij + c1 y2,ij +d1 ∇2 y2,ij= −x2,ij + (a2 + 1)y2,ij + c2 y1,ij +d2 ∇2 y1,ij .

On the other hand, the Laplacian operator in Eq.(3) actson the output variables instead of the state variables,
and the each output cell is a piecewise-linear nonlinear
function of the cell state. If we assume that all initialstates of the system at the beginning are always in thelinear region, then the Eq.(3) can be written as follows:ẋ1,ij = a1 x1,ij + c1 x2,ij + d1 ∇2 x2,ij(5)
ẋ2,ij = a2 x2,ij + c2 x1,ij + d2 ∇2 x1,ij .
where x1;ij , x2;ij are in linear region, and 1 ≤ i ≤ M ,1 ≤ j ≤ N.
Thus, the above linear diﬀerential equation can besolved by decoupling it into M N -decoupled systems of
two ﬁrst-order linear diﬀerential equations, and considering that the M N orthonormal space-dependent eigenfunction φMN (m, n; i, j) of the discrete Laplacian operator can be assumed as follows for most boundary conditions:
2φMN (m, n; i, j)∇2 φMN (m, n; i, j) = −kMN

(6)

and12
λmn [kmn] = [(a1 + a2 )±2
2 d − c )(k 2 d − c ) (11)(a1 − a2 )2 + 4(kmn221mn 1The equation (9) is important because the cell statesof the two-layer CNN are given as the time-dependent
weighted sums of spatial eigenfunctions. From this equation, it can be derived that, if one of the temporal eigenvalues has positive real part, the two-layer CNN becomes
unstable in this linear region, and the cell states do arise
and enter into the nonlinear region (i.e., saturation region or partial saturation region). In this situation, thetwo-layer CNNs can be used for modeling active media for nonlinear phenomena such as pattern formations
and autowaves [7], and for image processing applications.
However, if all of temporal eigenvalues have zero or negative real part, the all of states of the CNN are in linear
region. Thus, the CNNs have the properties of the linear
space, such as superposition etc. Therefore, we can use
this CNN to model passive media with non-loss or loss
in linear space.

where M and N are the CNN dimensions, m and n are
the summation indexes for the current space variables i2and j (i = 0, 1, · · · , M − 1; j = 0, 1, · · · , N − 1), and kmn
are the corresponding spatial eigenvalues. In particular, 3.2. Classical Wave Propagation
for the zero-ﬂux boundary condition, the spatial eigenIn this subsection, the investigations on passive propfunction and eigenvalue can be assumed as the following
agation phenomena with/without loss are carried out inform:
two-layer CNNs, which enable us to ﬁnd out some basic
properties of these phenomena. In the following simu(2j+1)nπ(2i+1)mπcos(7) lation, the CNN array consists of 100 × 100 cells with
∇2φMN (m, n; i, j) = cos
2M2N
zero-ﬂux boundary condition, and the initial state of theandﬁrst layer CNN is set as Fig.1(a), and the second layer
mπnπ2kmn+ sin2)(8) is set as zero initial state. If the parameter set: a1 = 0,= 4(sin22M2N-982-a2 = 0, d1 = −0.5, d2 = 0.5, c1 = −5, and c2 = 5 is selected, from Eq.(11) we have that all of temporal eigenvalues of the CNN are complex numbers with zero realpart and the isolated cell without couple behaves as oscillator with limited cycle. In other words, the two-layer
CNN is a system without loss. This type of two-layer
CNN can be used for modeling passive media without
loss. Fig.1 shows the simulation result. Analysis of thesimulation result shows that circular waves propagatingfrom the initialized position are generated. These circular waves propagate in all directions through the planearray and their amplitudes decrease with propagating.
When the waves collide with each other, they don’t disappear and reﬂect, but permeate through and superposeon each other. When they collide with the boundary,
the waves are reﬂected. Moreover, after the waves arerepeatedly reﬂected and superposed, the initialized positions become invisible. These behaviors behave those theclassical wave does. The wave propagation phenomenon
keeps up forever duo to zero real parts of temporal eigenvalues. This dynamic process can be observed from thegraph of the sum of State-error Square varying with theintegration time shown by Fig.8. The sum of State-error
Square is deﬁned as:MNSCN N (t) =the necessary condition for these phenomena. We canuse the two-layer CNN to approximate the passive media as well. Based on our research results, the two-layer
CNN can be considered not only for modeling active media, but also for modeling passive media with/without
loss. All of these facts show that the two-layer CNNs
have a real potential for expansion.

[1] L.O. Chua and L. Yang, “Cellular neural networks:
theory and applications,” IEEE Trans. Circuits &Syst., vol.35 pp.1257–1290, Oct. 1988.
[2] L.O. Chua, Martin Hasler, George S. Mochytz, andJacques Neirynck, “Autonomous cellular neural networks: a uniﬁed paradigm for pattern formation
and active wave propagation,” IEEE Trans. Circuits& Syst., vol.42 pp.559-577, 1995.
[3] M.J. Ogorzalek, Z. Galias, A.M. Dabrowski,
and W.R. Dabrowki, “Chaotic waves and spatiotemporal patterns in large arrays of doubly-coupled
Chua’s circuits,” IEEE Trans. Circuits & Syst.,
vol.42 pp.706-714, 1995.
[4] Ladislav Pivka, “Autowave and spatio-temporalchaos in CNNs part 1-2: a tutorial,” IEEE Trans.
Circuits & Syst., vol.42 pp.638-663, 1995((x1,ij (t + ∆t) − x1,ij (t))2i=1 j=1+(x2,ij (t + ∆t) − x2,ij (t))2 )References(12)

From the graph, we can see that the sum of State-error
Square converges on a horizontal line. On the other
hand, if we select the parameter set to satisfy all temporal eigenvalues of the CNN having negative real parts,
then the CNN is a damped system, each cell state will
decay to zero, which can be derived from Eq.(9). Thissituation is suitable for model passive media with loss.
For example, if the parameter set as a1 = −0.04, a2 = 0,d1 = −0.5, d2 = 0.5, c1 = −5, and c2 = 5 is selected,
then all of temporal eigenvalues of the CNN are complex numbers with negative real part and the isolated
cell without couple behaves damped oscillator. The simulation result is shown by Fig.3. From the analysis of thesimulation result, we can see that the propagation phenomenon has the same properties with the above one.
However, the network is calm in the ﬁnals due to thetemporal eigenvalues having negative real part. The dynamic process can be similarly demonstrated from thegraph of sum of State-Error Square shown by Fig.4.
4. Conclusions
In this paper, we have investigated the spatiotemporal phenomena in the two-layer CNNs, and found-983-[5] L. Goras, L.O. Chua, D.M.W. Leenaerts, and L.
Pivka, “Turing patterns in CNNs part 1-3:,” IEEE
Trans. Circuits & Syst., vol.42 pp.602-637, Oct.
1995.
[6] G. Manganaro, P. Arena and L. Fortuna, “Cellularneural networks,” Springer-Verlag, Berlin, Heidelberg, 1999.
[7] Z. Yang, Y. Nishio and A. Ushida, “A model of nonlinear phenomena: a uniﬁed framework two-layer
CNN,” in Proc. of the 2001 International Technical Conference on Circuits/Systems, Computers
and Communications, Tokushima, Japan, pp.12881291.
[8] Z. Yang, Y. Nishio and A. Ushida, “Templates andalgorithms for two-layer cellular neural networks,”
in Proc. of the 2002 International Joint Conference
on Neural Networks, Hawaii, USA, pp.1946-1951.
[9] Z. Yang, Y. Nishio and A. Ushida, “A two-layer
CNN in image processing applications,” in Proc.
of the 2001 International Symposium on Nonlinear
Theory and its Applications, Miyagi, Japan, pp.6770.

(a)

(a)

(b)

(b)

(c)

(c)

(d)

(b)

Figure 1: A simulation for classical wave propagation Figure 3: A simulation for classical wave propagation
without loss. (a) is initial state, (b)-(d) show three snap- with loss. (a) is initial state, (b) (c) and (d) show foursnapshots observed in diﬀerent iterates.
shots observed in diﬀerent iterates.

Figure 2: Graph of sum of state-error square via time Figure 4: Graph of sum of state-error square via time
for classical wave propagation without loss.
for classical wave propagation with loss.

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