NOLTA 2006
11-14 september
Bologna - ITALY

Comparing Two-Layer CNN with van der Pol Oscillators Coupled byInductors
Koji Urabe† and Yoshifumi Nishio‡†Department of Electronic Control Engineering, Niihama National College of Tech.
7-1 Yagumo, Niihama, Ehime 792-8580, JAPAN‡Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, JAPAN
Email: coji@ect.niihama-nct.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract—In this work, we present that the 2-layer
CNN is similar to van der Pol oscillators coupled byinductors, which can generate the phase-wave propagation phenomena. However, we cannot observe the phasewave propagation phenomena in the original 2-layer CNN.
Therefore, we introduce a modified 2-layer CNN. We
clearly show the correspondence between the modified 2layer CNN and the van der Pol oscillators coupled by inductors.
1. Introduction
Cellular Neural Networks (CNN) were invented by
L.O.Chua and L.Yang in 1988 [1]. CNN are constructedby cells connected each other. The cell contains linear
and nonlinear current sources controlled by voltage. Already a lot of applications and VLSI implementations of
CNN were reported. Many nonlinear phenomena suchas pattern formation and autowaves could be observed in
CNN. Investigating the nonlinear phenomena is an important work for clarifying dynamics of CNNs. On the other
hand, phase-wave propagation phenomena in van der Pol
oscillators coupled by inductors were reported [2][3]. It
is known that 2-layer CNNs can exhibit phase-wave propagation phenomena by choosing an appropriate set of theparameters [4][5].
In this work, we report the detailed investigation on thesimilarity between the 2-layer CNN and the van der Pol
oscillators coupled by inductors. In particular we makeclear the correspondence of the parameters of the two systems. Further, we confirm that slight difference between
them prevents to generate some of phase-wave propagation
phenomena.
2. van der Pol Oscillators Coupled by Inductors
The van der Pol oscillators coupled by inductors L0 as acomparative object used in this study are shown in Fig. 1.
We assume the v - i characteristics of the nonlinear negative resistors in the circuit as the following function.
ir (vk ) = −g1 vk + g3 v3k(g1 , g3 > 0).

(1)

The circuit equations governing the circuit in Fig. 1 arewritten asduk= wk(2)dτdwk= −uk + α(uk+1 − 2uk + uk−1 )
(3)
dτ

+  wk − w3k /3Figure 1: van der Pol oscillators coupled by inductors.

wheret=ps
Cg1
uk , vk =3L1 g3

L1Cτ, iL1 k =rg1
wk ,3g3r
L1
α = ,  = g1
L0

L1.
C(4)

And, we consider the boundary conditions as follows:
u0 = u1 , u(N+1) = uN .

(5)

It should be noted that α corresponds to the coupling of
the oscillators and  corresponds to the nonlinearity of theoscillators. In this study, we calculate (2) and (3) by using
the Runge-Kutta-Gill method.
In this study, we use N = 8, α = 0.050 and  = 0.30 for
numerical analysis. And the initial conditions are given as
follows:
1. Setting the initial conditions of all oscillators as thesame.
2. Putting the arbitrary phase difference of the voltage andthe current to one oscillator.
Putting the phase difference +180 [deg] to the 1st oscillator, simulation result is shown in Fig. 2. In Fig. 2, the vertical axis is the sum of voltages of adjacent oscillators, andhorizontal axis is time. If the sum of voltages of adjacentoscillators is zero, the phase difference between adjacentoscillators is +180 [deg]. At first, only the 1st oscillator has
phase difference +180[deg], phase difference propagates tothe adjacent oscillator as time goes.

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the output feedbacks y`,k in Eqs. (6) and (7) by the state
feedbacks x`,k , except y2,k , as follows;x1,k˙x2,k˙==−x1,k + a1 x1,k + c1 x2,k+d1 x2,(k−1) + d1 x2,(k+1)−x2,k + a2 y2,k + c2 x1,k+d2 x1,(k−1) + d2 x1,(k+1)

(9)
(10)

By comparing Eq. (9) with Eq. (2), some of the templatevalues are decided as follows;
Figure 2: Phase-wave propagation phenomenon in coupled
van der Pol oscillators.

a1 = −1, c1 = 1, d1 = 0.

(11)Further, we approximate the piecewise linear functionf (·) of the output feedback from its own cell by the 3rdorder polynomial expression.

3. Two-Layer CNN−x2,k + a2 y2,k= −x2,k + 0.5a2 |x2,k + 1| − |x2,k − 1|3
∼  x2,k − x2,k/3

In this study, we use 1-dimensional 2-layer CNN asshown in Fig. 3.

(12)
(13)

Equation (13) with  = 0.30 is shown by a dotted line inFig. 4. We approximate this curve by Eq. (12) with a2 =1.2, whose curve is shown by a solid line in Fig. 4.

Figure 3: One-dimensional 2-layer CNN.

Figure 4: Approximation of piecewise linear function.

The circuit equations governing the CNN in Fig. 3 arewritten asEquation (10) can be rewritten using the Eqs. (12) and(13) as follows;

3x2,k
˙
=  x2,k − x2,k/3 + c2 x1,kx1,k
˙x2,k˙y`,k= −x1,k + a1 y1,k + c1 y2,k+d1 y2,(k−1) + d1 y2,(k+1)= −x2,k + a2 y2,k + c2 y1,k+d2 y1,(k−1) + d2 y1,(k+1)
= f (x`,k ) = 0.5 |x`,k + 1| − |x`,k − 1|

(6)
(7)
(8)

+d2 x1,(k−1) + d2 x1,(k+1)

By comparing Eq. (14) with Eq. (3), the other templatevalues are decided as follows;
a2 = 1.2, c2 = −(1 + 2α), d2 = α(k = 1, 2, · · · , N, ` = 1, 2)
where x`,k is the state, y`,k is the output of CELL`,k , a` , c` ,and d` are the feedback parameters from the output of its
own cell, from the output of the cell which is at the sameposition in the other layer, and from the output of the neighborhood cell in the other layer, respectively.
In order to investigate the correspondence between the2-layer CNN and the van der Pol oscillators, we replace

(14)

(15)

Using the template values in Eqs. (11) and (15), Eqs. (9)and (10) are rewritten as follows;

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x1,k
˙x2,k˙==x2,k−x1,k + α(x1,(k−1) − 2x1,k + x1,(k+1) )
3+ x2,1 − x2,1/3

(16)
(17)

Now, we consider the boundary conditions as follows:
x1,0x1,(N+1)==x1,1x1,N(18)(19)

Equations (16)-(19) are completely the same as the equations of the van der Pol Oscillators coupled by the inductors(2), (3), and (5).
We expect that the phase-wave propagation phenomena,
which are observed in coupled van der Pol oscillators, canbe generated in the 2-layer CNN (6) and (7) with the template values (11) and (15). Namely,x1,k
˙x2,k˙= y2,k= −y1,k + α(y1,(k−1) − 2y1,k + y1,(k+1) )
3+ x2,1 − x2,1/3(20)Figure 6: Simulated result of modified 2-layer CNN (N =8).

(21)

However, the simulated results of Eqs. (20) and (21) with
α = 0.050 and  = 0.30 did not show the continuously existing phase-wave propagation phenomena. Figure 5 shows
an example of the observed phenomena for N = 8 obtainedby using the Runge-Kutta-Gill Method.

From this result we consider that the cells at the layer 1
correspond to the differentiation of uk . The output feedbacks from their own cells at the layer 2 correspond to thenonlinear resistors, and the output feedbacks from the cells
at the layer 1 correspond to the couplings between the oscillators.
5. Phase-Wave Propagation Phenomena
The simulated result of the coupled van der Pol oscillators with different coupling parameter α = 0.10 is shown in
Fig. 7. The speed of the phase-wave propagation is fasterthan Fig. 2 with α = 0.05. Namely, the propagation speed
can be controlled by changing the coupling inductor L0 in
the coupled oscillators.
We can confirm similar effect in the modified 2-layer
CNN by changing the template values. From α = 0.10,

Figure 5: Simulated result of 2-layer CNN (N = 8).

c2 = −(1 + 2α) = −1.2, d2 = α = 0.10.

The difference between Eqs. (20) and (21) and Eqs. (2)and (3) is only the feedbacks. Namely, the output feedbacks in the 2-layer CNN and the state feedbacks in the
coupled van der Pol oscillators.
4. Modified 2-Layer CNN
We introduce the modified 2-layer CNN that the output
feedbacks from the cell at the same position in the otherlayer are replaced by the state feedback. The equations
governing the modified 2-layer CNN are written as follows;
x1,k
˙x2,k
˙

= −x1,k + a1 y1,k + c1 x2,k= −x2,k + a2 y2,k + c2 x1,k+d2 y1,(k−1) + d2 y1,(k+1)

The simulated result of the modified 2-layer CNN with thecalculated template values is shown in Fig. 8. We can seethat the speed of the phase-wave propagation is faster thanFig. 6 similar to Fig. 7.
Figure 9 shows some typical examples of the phase-wave
propagation phenomena in the van der Pol oscillators coupled by the inductors.
The corresponding simulated results by the modified 2layer CNN are shown in Fig. 10. We can observe the reflection of the waves, but cannot observe the extinction bythe collision.

(22)
(23)

Equations (22) and (23) are rewritten using the templates(11) and (15);x1,k
˙x2,k˙= x2,k= −x1,k + α(y1,(k−1) − 2x1,k + y1,(k+1) )
3+ x2,1 − x2,1/3(24)(25)

The simulation result of Eqs. (24) and (25) is shown inFig. 6. In Fig. 6, we can observe the phase-wave propagation phenomena similar to Fig.2.

Figure 7: Simulated result of van der Pol oscillators withdifferent α = 0.10.

725Figure 8: Simulated result of modified 2-layer CNN withdifferent template values. a1 = 1, c1 = 1, d1 = 0, a2 = 1.2,
c2 = −1.2, and d2 = 0.10.

(a) Extinction by collision of two waves.

6. Conclusions
In this study, we clearly explained the correspondence
between an array of van der Pol oscillators coupled byinductors and 1-dimensional 2-layer CNN. By investigating the generated phase-wave propagation phenomena, wefound that the difference between the output feedbacks andthe state feedbacks played an important role to make difference in the observed phenomena.
In future works, we will try to clarify the mechanismof the phase-wave propagation phenomena by investigating the modified 2-layer CNN, so that we can control thephase-wave propagation for applications.

(b) Reflection by collision of two waves.
Figure 9: Examples of phase-wave propagation phenomena
in van der Pol oscillators coupled by the inductors. N = 15,
α = 0.10 and  = 0.30.

References[1] L.O. Chua and L. Yang, “Cellular neural network: theory and practice”, IEEE Trans. Circuits and Systems,
vol. 35, no. 10, pp. 1257-1290, 1988.
[2] M. Yamauchi, M. Wada, Y. Nishio and A. Ushida,“Wave propagation phenomena of phase states in oscillators coupled by inductor as a ladder”, IEICE Trans.
Fundamentals, vol. E82-A, no. 11, pp. 2592-2598,
1999.
[3] M. Yamauchi, Y. Nishio and A. Ushida, “Phase-Waves
in a ladder of oscillators”, IEICE Trans. Fundamentals,
vol. E86-A, no. 4, pp. 891-899, 2003.

(a) Extinction by collision can not be observed.

[4] Z. Yang, K. Tsuruta, Y. Nishio and A. Ushida, “Investigation of phase-wave propagation phenomena insecond order CNN arrays”, Proc. of ISCAS’04, vol. 3,
pp. 49-52, 2004.
[5] K. Urabe and Y. Nishio, “Phase propagation phenomena in two-layer cellular neural networks”, Proc. ofNCSP’06, pp. 102-105, 2006.

(b) Reflection by collision of two waves.
Figure 10: Example of phase-wave propagation phenomena in modified 2-layer CNN, N = 15, a1 = 1, c1 = 1, d1 =0, a2 = 1.2, c2 = −1.2, and d2 = 0.10.

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