2006 RISP International Workshop
on Nonlinear Circuits and Signal Processing (NCSP'06)
Waikiki Beach Marriott, Honolulu,Hawaii, USA, March 3-5, 2006

Phase Propagation Phenomena in Two-Layer Cellular Neural Networks
Koji Urabe † and Yoshifumi Nishio ‡†‡Dept. of Electronic Control Engineering,Niihama National College of Technology
7-1 Yagumo-cho, Niihama, Ehime, Japan
Email: coji@ect.niihama-nct.ac.jp

1. Introduction

Dept. of Electrical and Electronics Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: nishio@ee.tokushima-u.ac.jp

The circuit equations governing these cells are as follows:x˙1 = −x1 + a1 y1 + c1 y2(1)
Cellular Neural Networks (CNNs) [1] are constructed bycells connected each other. The cell contains linear and nonx˙2 = −x2 + a2 y2 + c2 y1(2)linear current sources controlled by voltage. It is known thaty` = f (x` ) = 0.5 (|x` + 1| − |x` − 1|)(3)
two-layer CNNs can exhibit oscillation by choosing an ap(` = 1, 2)
propriate set of the parameters.[2]In this work, we present that a simplification of CNN con- where x is state, y is output in cell `.
``structed by two cells can be described by van der Pol equaFor simplicity, Equation (1), (2) are rewritten as follows:tion. Moreover, we confirm the generation of the phase propx˙1 = −x1 + a1 y1 + c1 x2(4)
agation phenomena, which can be observed in coupled vander Pol oscillators [3] , in the two-layer CNNs by computerx˙2 = −x2 + a2 x2 + c2 x1(5)simulations. We discuss the similarity between the two-layer
And we approach piece-wised-linear function f (·) of outCNNs and coupled van der Pol oscillators.
put feedback from its own cell by 3rd order polynomial expression.
2. Oscillation in CNN constructed by two cells−x1 + a1 y1 = −x1 + 0.5a1 (|x1 + 1| − |x1 − 1|) (6)¡¢CNN constructed by two cells used in this study is shown∼ ² x1 − x31 /3(7)
in Fig. 1.
Thus, from (4), (5),x¨1 − ²(1 − x21 )x˙1 − c1 (a2 − 1)x2 − c1 c2 x1 = 0(8)If a2 = 1, Equation (8) is corresponding to van der Pol
equation.
x¨1 − ²(1 − x21 )x˙1 − c1 c2 x1 = 0,(9)Numerical analysis of (9) with ² = 0.30, c1 c2 = 1 shownin Fig 2 is oscillated.
Equation (7) with ² = 0.30 is shown in a dotted line ofFig.3. Determining a1 equation (6) is similar to the dottedline. Equation (6) with a1 = 1.2 is shown in a solid line ofFig.3,We expect that the CNN constructed by 2 cells (1)–(3)given same parameters of oscillated van der Pol equation willbe oscillated. Equation (1)–(3) with a1 = 1.2, a2 = 1.0,
c1 = 1.0, c2 = −1.0 calculated by Runge-Kutta-Gill method
is shown in Fig. 4.
Figure 1: CNN constructed by two cells

3. Phase propagation phenomena in coupled van der Pol
oscillators- 102 -

Figure 2: Oscillation in van der Pol Equation
Figure 3: Approach by piece-wised-linear function
3.1. Circuit model
The van der Pol oscillations coupled by inductor L0 as acomparative object used in this study is shown in Fig. 5.
We assume the v - i characteristics of the nonlinear negative resistors in the each circuit as the following function.

L1
α=, ² = g1
L0r

L1
C(14)

It should be noted that α corresponds to the coupling oftheoscillators and ² corresponds to the nonlinearity of the(g1 , g3 > 0)
(10)
oscillators. In this study, we calculate (2)-(4) by using theThe circuit equations governing the circuit in Fig.5 are Runge-Kutta-Gill method.
written asir (vk ) = −g1 vk + g3 vk3[First Oscillator]
du1= w1dτ¢¡dw1= −u1 + α(u2 − u1 ) + ² w1 − w13 /3dτ[Middle Oscillator]
duk= wkdτdwk= −uk + α(uki+1 − 2uk + uk−1 )dτ¡¢+ ² wk − wk3 /3

3.2. Phase propagation phenomena(11)

In this study, we use N = 8, α = 0.050, ² = 0.30 for
numerical analysis. And the initial conditions are given as
follows:
1. Setting the initial conditions of all oscillators as thesame.

(12)

2. Putting the arbitrary phase difference of the voltage andthe current to one oscillator.

Putting the phase difference +180 [deg] to 1 oscillator, result of simulation is shown in Fig 6. In Fig. 6, the vertical axis
[Last Oscillator]
is the sum of voltages of adjacent oscillators, and horizontal
duN= wN(13) axis is time. If the sum of voltages of adjacent oscillators isdτzero, the phase difference between adjacent oscillators have¡¢dwN
3phase difference +180 [deg]. At first, only 1 oscillator has= −uN + α(uN −1 − u1 ) + ² wN − wN /3dτphase difference +180[deg], phase difference propagating toadjacent oscillator is shown though a time.
wheresrpCg1
g1
uk , vk =wk ,t = L1 Cτ, iL1 k =4. Phase propagation phenomena in 2 layer CNN
3L1 g3
3g3- 103 -Figure 5: coupled van der Pol oscillators

Figure 4: Oscillation in CNN constructed by 2 cells

4.1. Circuit model
In this study, we use 1-dimension 2-layer CNN , shown inFig. 7, that coupled CNN oscillators shown in Fig.1.
The circuit equations governing the CNN in Fig.7 are written asx1,k˙= −x1,k + a1 y1,k + c1 y2,k(15)x2,k˙= −x2,k + a2 y2,k + c2 y1,k+d2 y1,(k−1) + d2 y1,(k+1)

(16)

y`,k

= f (x`,k ) = 0.5 (|x`,k + 1| − |x`,k − 1|) (17)(` = 1, 2)
Figure 6: Phase propagation phenomena in coupled van derPol oscillators

4.2. Comparison with coupled van der Pol Oscillators
For simplicity, Equation (15), (16) are rewritten as follows:x1,k˙x2,k˙= −x1,k + a1 x1,k + c1 x2,k= −x2,k + a2 y2,k + c2 x1,k+d2 x1,(k−1) + d2 x1,(k+1)

(18)
(19)

And we approach piece-wised-linear function f (·) of output feedback from its own cell by 3rd order polynomial expression like (6), (7).

The parameters are determined by comparison with circuit
equations governing coupled van der Pol Oscillotors (12), as
follow:
a1 = 1, c1 = 1, c2 = −(1 + 2α), d2 = αNow, we consider boundary condition as follows:
x1,0−x2,k + a2 y2,k=∼−x2,k + 0.5a2 (|x2,k + 1| − |x2,k − 1|)¡¢² x2,k − x32,k /3x2,k˙= (a1 − 1)x1,k + c1 x2,k¡¢= +c2 x1,k + ² x2,k − x32,k /3+d2 x1,(k−1) + d2 x1,(k+1)

x1,1x1,N

(23)
(24)

4.3. Result of simulation

Thus, from (18), (19),x1,k˙x1,(N +1)

(20)
(21)

==(22)

To calculate the same conditions as 3.2 the parameters in
2-layer CNN are given as follow. Since α = 0.050,
a1 = 1, c1 = 1, c2 = −1.1, d2 = 0.050- 104 -Figure 8: Result of simulation in changed 2-layer CNN

5. Conclusions
In this study, we investigated the relation between CNN
constructed by two cells and van der Pol oscillator and foundthat an array of van der Pol oscillators coupled by inductors
can be modeled by two-layer CNN. We expect that phasepropagation phenomena will be generated in 2-layer CNN
coupled the CNN Oscillators. Though, 2-layer CNN given
same parameter of coupled van der Pol oscillators couldn’t
generate phase propagation phenomena. We observed that 2layer CNN changed into the state feedback in the output feedback from other layer is generated phase propagation phenomena.
In future works, we will try to clarify phase propagation
phenomena by comparing two CNNs, so that we can control
the phase propagation for applications.

Figure 7: Two-layer CNN

Since ² = 0.30 and Fig. 3,
a2 = 1.2

References

The state initial values are given as follow:
1. Setting the state initial conditions of all cell as the same.
2. Putting the arbitrary phase difference of the state value
to one cell.
Putting the phase difference +180 [deg] to state of Cell1,1
result of simulation is not observed phase propagation phenomena.
We consider 2-layer CNN changed into the state feedbackin the output feedback from other layer. The circuit equations
governing the changed 2-layer CNN are following.
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)(25)[1] L.O.Chua and L.Yang, ”Cellular neural network:Theoryand practice”, IEEE Trans. Circuits & System, vol 35,no 10, pp.1257-1290, 1988.
[2] Z.Yang, K.Tsuruta, Y.Nishio and A.Ushida, ”Investigation of phase-wave propagation phenomena in second
order CNN arrays”, Proceedings of 2004 IEEE International Symposium on Circuits and Systems (ISCAS2004), vol. III, pp. 49-52, 2004.
[3] 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.

(26)

Result of simulation in changed 2-layer CNN is shown Fig.
8. In Fig. 8, phase propagation phenomena similar to Fig.6 isobserved.

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