RELATIONS BETWEEN SPATIO-TEMPORAL PHENOMENA AND EIGENVALUESIN MUTUALLY COUPLED CNNS
Zonghuang YANG, Masayuki YAMAUCHI, Yoshifumi NISHIO and Akio USHIDA
Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, JAPANABSTRACTlows:ẋ1;ij = −x1;ij +A1 ∗y1;ij +B1 ∗u1;ij +C1 ∗y2;ij +I1
ẋ2;ij = −x2;ij +A2 ∗y2;ij +B2 ∗u2;ij +C2 ∗y1;ij +I2

The work is concerned with investigation of spatio-temporalphenomena on mutually coupled CNN array. Besides pattern formation and active wave propagation, we found that
other interesting spatio-temporal phenomena such as phaseinversion-wave and classical waves can be also observed in
the same network structure. In particular, we focus our discussion on the relations of spatio-temporal phenomena andtemporal eigenvalues.

(1)with output equationy1,ij = 0.5(|x1,ij + 1| − |x1,ij − 1|)y2,ij = 0.5(|x2,ij + 1| − |x2,ij − 1|)

1. INTRODUCTION
Studies of dynamic phenomena in arrays composed of chaotic[1]–[6] and oscillatory [7]–[9] elements are very importantfor understanding various phenomena observed in natural
fields such as biology, physics, chemistry, etc. From the
middle of the last decade, the investigations about the spatiotemporal dynamics in CNNs have been widely carried out,
and many papers have been published, where they have discussed pattern formations and various types of autowaves
such as excitability waves, concentration waves and so on.
Up to the present, these CNN arrays, which are composedof chaotic oscillators [1]–[6], second order nonlinear circuitobtained by the suitably “reducing” Chua’s circuit [7], andetc. have been mainly considered. A common techniquein all these cases usually use the CNNs as the approximations of the various types of nonlinear partial differential
equations, especial the well-known reaction-diffusion equations that show Turing pattern and propagation phenomena
in various continuous media [10]. The purpose of this paper
is to study spatio-temporal phenomena occurred in a simpler
CNN structure - mutually coupled two-layer CNNs, basedon the CNNs’ temporal eigenvalues.
2. MUTUALLY COUPLED TWO-LAYER CNNS
The mutually coupled two-layer CNNs in this paper are described as the extension of the single-layer continuous timeChua-Yang CNN; their state equation is formulated as fol-

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(2)

for 1 ≤ i ≤ M and 1 ≤ j ≤ N , where two mutually coupled templates C1 and C2 are introduced and the subscript
1 and 2 stand for first layer and second layer of the twolayer CNN array. We assume that all cells remain in linear
region, then the solution of the linear system of the CNN ingeneral case has the following form for the simple temporaleigenvalue.
x(t) =

2MNri , x(0)eλi t qi

(3)

i=1

where x(t) is the state vector, qi is the eigenvector of thesystem matrix, ri represents the reciprocal basis, x(0) is
the initial condition, and ., . denotes the scalar product in
R2MN . The solution shows various possibilities for the different ranges of the temporal eigenvalues. In the following section, we investigate the spatio-temporal phenomena
based on the temporal eigenvalues in the two-layer CNNs.
3. SPATIO-TEMPORAL PHENOMENA
3.1. Pattern Formations and Autowaves
Many nonlinear partial differential equations (PDEs) haveshown to generate various types of nonlinear spatio-temporalphenomena. The concept of reaction-diffusion CNNs havebeen formalized to reproduce similar phenomena in [1], because they are mathematically described by a discretizedversion of the following well-known system of nonlinear
partial equations – reaction-diffusion equations:∂u= f (u) + D∇2 u∂t

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(4)

(a)

(b)

(c)

Figure 1: An example of tripe pattern formation with D1 =0.2. (a) is initial state, (b) is transient result, and (c) is steady
output result.

(a)

(b)

(c)

Figure 2: An example of active wave propagation phenomena. (a) is initial state, (b) is transient result, and (c) is
steady output result.
where D is a diagonal matrix, which describes the intensityof diffusive coupling between the components of u. Similarly, if we use the two-layer CNNs to approximate theabove reaction-diffusion equation with u∈ R2 , f ∈ R2 ,then the two-layer CNNs can be written as follows:ẋ1,ij = −x1,ij + a1 y1,ij + c1 y2,ij + d1 ∇2 y1,ij(5)
ẋ2,ij = −x2,ij + a2 y2,ij + c2 y1,ij + d2 ∇2 y2,ij
Thus, We have the following results that, if at least one of
the temporal eigenvalues has positive real part, the CNN
constructed by the connection between diffusing cells will
become unstable in the linear region, and behave as patternformations or active propagation phenomena. The necessary condition is described by (6), which is directly derived
from those conditions obtained Turing patterns in array ofcoupled circuits [7].
(a1 − 1) (a2 − 1)

+
>0
(6)
d1
d2

[(a1 − 1)d2 − (a2 − 1)d1 ]2 > −4d1 d2 c1 c2Figure 1 shows a simulation example for pattern formation, which is carried out under a two-layer CNN array consisted of 100 × 100 cells with zero-flux boundary condition and the following parameter set: a1 = 2.6, a2 = 2.6,
d1 = 0.1, d2 = 0.01, c1 = −1, and c2 = 1. In Fig. 1, (a),(b) and (c) are respectively the initial state, transient result
and steady output of the CNN. The simulation result indicates that, the stripe pattern formation process starts fromthe objects in the image (i.e. non-zero initial conditions),
propagates toward around the objects, generates new stripes

with almost the same geometry with the shape of original
initiated objects. Then, it propagates again until the whole
CNN array reaches a steady state. While the propagation
wave front collides with another wave front or the boundary, the wave fronts annihilate without any interference andreflection. Moreover, the density of stripe pattern can be adjusted by suitably changing the template parameters. From
this example, we have seen that the shape of stripe pattern isdependent on its initial state of the CNN array. Thus, many
interesting stripe patterns such as the coats of animal, fingerprint, and etc. may be regenerated by suitably selecting
initial conditions of the CNN array and the CNN parameters.
Figure 2 shows another simulation example for active
propagation phenomena, which is carried out under the same
conditions with the last example. This simulation indicatesthat two concentric circular waves are generated. Their wave
fronts propagate in all directions through the network fromthe initialized positions. After the waves fronts collide withboundary or collide with each other, they annihilate without any interference and reflection. This propagation phenomenon behaves as autowaves do.
3.2. Phase Wave Propagation Phenomena
A lot of studies on phase relationship of coupled oscillators or chaotic circuits have been widely carried out [6][9].
One of interesting spatio-temporal phenomena – phase inversion wave has been observed in van der Pol oscillators
coupled by inductors as a ladder [9]. In this section, we
will investigate similarly dynamic behaviors encounteredin the simpler two-layer CNN array. Obviously, the CNN
must satisfy the following two conditions that the nonlinear second-order cell in absence of coupling to its neighborcells oscillates with a limit cycle, and the whole CNN isunstable (i.e., at least one of the temporal eigenvalues has
positive real). Next, let us observe how the phase difference
between adjacent cells propagates the CNN array. In Fig.
3(b), the vertical axes are the sum of the first layer outputsof adjacent cells, the horizontal axes are time. Hence, thediagram shows qualitatively how phase differences between
adjacent cells change as time goes, where the white regions
correspond to the state that two adjacent cell/oscillators are
anti-phase synchronization, and the black regions to the inphase synchronization. Figure 3(b) shows an example forthe wave propagation phenomena of the phase difference
that is observed from one-dimensional two-layer CNN array composed of 18 cells with zero-flux boundary condition.
The initial state condition of the first layer of this CNN array is set as Fig. 3(a), where the black squares denote having
value 1 and the white squares value -1, and the initial state
condition of the second layer is zero. The template isA1 = −0.03 1.1 −0.03 , C1 = 0 −3 0 ,III-579

x1;1

x1;i

x1;18

(a)

(b)
Figure 3: Phase wave propagation phenomenon in onedimensional CNN array of 18 cells.
x1;1

x1;i

x1;18

(a)

of temporal eigenvalues of the CNNs has positive real part.
However, when all of the temporal eigenvalues have zero or
negative real parts, what happens in the CNNs? We consider
the phenomena in this subsection.
Let us consider an autonomous two-layer CNN with mutual coupling between layers. Its equation is described as
the following form:
ẋ1,ij = −x1,ij + (a1 + 1)y1,ij + c1 y2,ij +d1 ∇2 y2,ij(8)
ẋ2,ij = −x2,ij + (a2 + 1)y2,ij + c2 y1,ij 2+d2 ∇ y1,ij .
We assume that all cells remain in the linear region (y1,ij =x1,ij , y2,ij = x2,ij ). Thus, the equation (8) can be solved
by decoupling method [9], and its solution can be expressed
as a weighted sum of M × N orthogonal space dependent
eigenfunctions φMN (m, n; i, j) in the following form:

M−1
−1
N



t
λmn2 t(αmneλmn1+β)φMN(m,n;i,j)
x1,ij(t)=mne
m=0 n=0
(9)
M−1
N
−1



λmn1 tλmn2 t

(γmne
+δmne
)φMN(m,n;i,j)
x2,ij(t)=
m=0 n=0

where αmn , βmn , γmn , δmn are constants depending on theinitial conditions. λmn1 , λmn2 are temporal eigenvalues,
2which are influenced by spatial eigenvalue kmn
corresponding to spatial eigenfunction.
(b)
Figure 4: Anothor phase wave propagation phenomenon in
one-dimensional CNN array with different initial condition.
A2 =

0.03 1.1 0.03

, C2 =



0 5

0



1
[(a1 + a2 )±2

2 d − c )(k 2 d − c ) (10)(a1 − a2 )2 + 4(kmn
2
2
1
mn 1

For the zero-flux boundary condition, the spatial eigenfunction and eigenvalue can be assumed as the following form:,

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

2λmn [kmn]=(7)

Figure 4(b) shows another example for phase wave propagation with the different initial state conditions shown by Fig.
4(a). We observe that the anti-phase between adjacent cells
is transferred gradually from one end to the other end of the
CNN array, and is reflected when the states of anti-phasecollide with the end of array, or collide with each other.
This propagation process of phase exists continuously in the
two-layer CNN array, so called “phase-inversion-wave” [9].
The two simulations indicate that the phase wave propagation phenomena can be reproduced in the two-layer CNN
with simpler structure.
3.3. Classical Wave Propagation
We have shown in the above subsections that the two-layer
CNNs can reproduce many interesting nonlinear phenomena, in which there is a common property that at least one∇2φMN (m, n; i, j) = cos
and(2j +1)nπ(2i+1)mπ
cos
(11)
2M
2N

mπ
nπ
+ sin2)(12)
2M
2N
The equation (9) is important because the cell states of the
two-layer CNN are given as the time-dependent weighted
sums of spatial eigenfunctions. From this equation, it canbe derived that, if all the temporal eigenvalues have zero or
negative real part, all the states of the CNN are in the linear
region. Thus, the CNNs have the properties of the linear
space, such as superposition etc. Therefore, we can use thisCNN to model passive media with/without loss in the linear
space.
Figure 5 shows a simulation example for the wave propagation phenomenon in a two-layer CNN array composed of
100 × 100 cells, which is executed by selecting the following parameter set: a1 = 0, a2 = 0, d1 = −0.5, d2 = 0.5,

III-5802kmn= 4(sin2

4. CONCLUSIONS

(a)

(b)

(c)

Figure 5: An example for the passive wave propagation phenomena without loss. (a) is initial state, (b) and (c) show two
snapshots observed in different times.

In this paper, we have investigated the spatio-temporal phenomena in the two-layer CNNs in term of the temporal eigenvalues, and found many interesting spatio-temporal phenomena. The necessary conditions for these phenomena havebeen indicated. 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.
Moreover, we found that an interesting phase wave propagation phenomena can be produced in the same two-layer
CNN structure. All of these facts show that the two-layer
CNNs have a real potential for applications.
5. REFERENCES

(a)

(b)

(c)

Figure 6: An example for the passive wave propagation phenomena with loss. (a) is initial state, (b) and (c) show two
snapshots observed in different times.
c1 = −5, and c2 = 5 (i.e., all the temporal eigenvalues of
the CNN are complex numbers with zero real part and theisolated cell behaves as oscillator with limited cycle). The
simulation result shows that the circular waves propagatingfrom the initialized position are generated. These circular
waves propagate in all directions through the plane arrayand their amplitudes decrease with propagating. When the
waves collide with each other, they don’t annihilate and reflect, but permeate through and superpose on each other.
When they collide with the boundary, the waves are reflected. Moreover, after the waves are repeatedly reflected
and superposed, the initialized positions become invisible.
These behaviors are completely different form those of nonlinear propagation phenomena, but behave as the classicalwave does. In this case, the CNN can be used for modeling passive media without loss. On the other hand, if weselect the parameter set to satisfy all temporal eigenvaluesof the CNN having negative real parts, then each cell state
will decay to zero, which can be derived from Eq. (9). Thissituation is suitable for modeling passive media with loss.
For example, if the parameter set: a1 = −0.04, a2 = 0,d1 = −0.5, d2 = 0.5, c1 = −5, and c2 = 5 is selected,
then all the temporal eigenvalues of the CNN are complex
numbers with negative real part and the isolated cell without couple behaves as damped oscillator. Figure 6 shows
the simulation result. From the analysis of the simulation
result, we can see that the propagation phenomenon has thesame properties with the above one. However, the network
is calm at the end due to the temporal eigenvalues having
negative real part.

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