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CNNA'92
PROCEEDINGS

Second International Workshop on Cellular Neural Networks and their Applications

Technical University Munich Munich, Germany October 14-16,1992

IEEE 92TH0498-6


CNN'92

INHOMOGENEOUS CELLULAR NEURAL NETWORKS: POSSIBILITY OF FUNCTIONAL DEVICE DESIGN.

Yuri ANDREYEV, Yuri BELSKY, Alexander DMITRIEV, Dmitriy KUMINOV Institute of Radio Engineering and Electronics of the Academy of Sciences Marx Avenue 18, 103907 Moscow, Russia Tel.*(095)-203-48-17; Fax*095)-203-84-14; Email dmitr@ire.msk.su

Abstract. A possibility of designing functional devices for various applications in terms of unified technology on the basis of CNNs is discussed on examples of an associative memory device, complex oscillations generator, devices for chaotic memory scanning and pattern recognition.

1. INTRODUCTION. One of the advantages of analog (CNN)cellular neural networks (1] possibility of their realization as chips. But CNNs with relatively small number of elements may also be promising. Even if all the elements are coupled with each other, the total connections is not excessive here. Then an interesting possibility to design chips of devices for various applications in terms of unified basis of CNNs with inhomogeneous couplings, appears. For example, a possibility to use neural networks as A/D converter and integral functional on the number of is a

technology

circuits for solving linear programming problems were discussed in [2]. This approach can also be used for designing various classifiers [3]. We will discuss in this report an application of this approach to the

design of devices for chaotic memory scanning and pattern recognition.

0-7803-0875-1/92 $3.00 ©1992IEEE

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2. ASSOCIATIVE MEMORY, CHAOTIC SCANNING AND PACEMAKER. Two devices are necessary to organize chaotic memory scanning: a one memorized pattern to another. 2.1. Associative memory on the basis of an inhomogeneous CNN. To store images into a neural network, we have to find an forming the matrix of couplings or templates, providing algorithm of memory

[4] itself and some external generator or pacemaker to push the system from

correspondence

between equilibrium states of a CNN circuit and memorized patterns. ADALINE algorithm [5] was chosen as a learning rule (an algorithm next of

coupling matrix forming). This algorithm was chosen for the

reasons:

it is simple in realization and its matrix of couplings is not symmetric. In general, the matrix of couplings is not local. But demands on

the locality of the matrix of couplings can be lowered for the case of a little number of cells and images. This case of a little number of cells and pat terns is important not only as an example, but for classifier design also. 2.2. Generator of complex oscillations with amplitude modulation. The dynamics of a non-autonomous harmonically driven CNN composed of two cells was studied in details in [6]. A chaotic attractor was found in non-autonomous system. Such complex dynamics can obviously be obtained in an autonomous system, replacing an external harmonic signal by the signal from a generator composed of two cells. The dynamics of the four cell-CNN is described by: dY/dt + Y = T * F(Y), Y = { y(1), y(2), y(3), y(4) } F(Y) = { f(y(1)), f(y2)), f(y(3)), f(y(3)) }, f(x) = (|x + 1| - |x - 1|) / 2 To design a pacemaker that could be used to control neural generator in be constructed. We modify then the system (1) to a neural network of Fig. 1. Two which a large amplitude chaotic memory scanning a (1) this

oscillations

are

interchanged by oscillations with a small amplitude near zero value, should

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additional controlling cells 5 and 6 are introduced here. The the cells 5 and 6 are described by equations (t) dx/dt +x = f(u), (t) = 0 +
1

dynamics

of

sin(2 t/T) the cells 5

(2) or 6

where u is an output signal from the cells 1 or 3 for

respectively, T is a period of controlling signal for parameter (t). With an increase of the parameter (t) chaotic oscillations disappear in the system. We can obtain the needed control signal ( Fig.2) for the memory device

with the help of high-pass filter composed of neural cells. Applying the signal from the pacemaker to the associative memory device

we obtain the device for chaotic scanning of memory. Fig. 3 shows the regime of memory scanning for CNN composed of nine cells with tree stored patterns V1, V2 and V3. It should be noted that chaotic memory scanning may also be organized in some other way [7]. 3. PATTERN RECOGNITION. By pattern recognition we mean the following. Let an external is applied to a neural system, in which chaotic scanning to one of of system is organized. If the external signal patterns, stored in the neural network it corresponds (memory signal memory the must

device),the pattern.

stabilize itself in the state, associated with that should continue chaotic memory scanning.

Otherwise,

For the organization of pattern recognition we introduce a

local

feed -

back between the memory and pacemaker. This feedback turns on and

destroys

the complex oscillations of the pacemaker only if a given equilibrium state of the memory device corresponds to the pattern being recognized. The feedback can be described by the relation (|| - Z|| = k/( || - Z||+ ), Y ) Y where k and are some fixed parameters and << 1; vector Z stands for external signal or external pattern; vector Y describes an (section 2.2). internal (3) an

state

of associative memory; || * || is a vector norm. (|| - Z|| is added to (t) Y )

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We organize pattern recognition only on one component of Z. (|| - Z|| = k/(|y(2)-z(2)| + ), Y ) (4)

where y(2) is the second component of Y, z(2) is the second component of Z. Fig. 4 shows the trajectory of the state pattern recognition regime. 4. CONCLUSION. So we discussed the possibility of applying CNNs and their combinations chaotic is variable y(2) for a

to the design of various functional devices. This approach was discussed on the examples of associative memory, complex oscillation generator, in the use of universal CNN chips for different purposes. REFERENCES 1. L.O.Chua, L.Yang, "Cellular Neural Networks", IEEE Transactions Networks: on An A/D memory scanning and pattern recognition. The advantage of this approach

Circuits and Systems, 1988, V.35, pp.1257-1272, pp.1273-1290. 2. D.W.Tank, J.J.Hopfield,"Simple "Neural" Optimization Converter, 541. 3. D.E.Rumelhart, G.E.Hinton, R.G.Wiliams, and "Learning Representation with by Back-Propagating Errors", Nature, 1986, V.323, pp.533-536. 4. J.J.Hopfield, " Neural Networks Physical Systems Emergent of Collective Computational Abilities ", Proceedings Sciences USA, 1982, V.79, pp.2554-2558. 5. J.S.Dencer ," Neural Networks: Modeling and Adaptation ", Physica D, on 1986, V.22, pp.216-232. 6. F.Zou, J.A.Nossek,"A Chaotic Attractor with CNNs", IEEE Transactions Circuits and Systems, 1991, V.38, pp.811-812. 7. A.Dmitriev, L.Jdanova, in Physical Systems and Japan, pp. 501-504. D.Kuminov, 1/f "The Simplest Neural-Like Conference on Systems Noise Kyoto, with Chaos", Proceedings of the International National Academy Signal Decision Circuit, and a Linear Programming

Circuit", IEEE Transactions on Circuits and Systems, 1986, V.33, pp.533-

Fluctuations , November , 1991,

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Fig. 1 The four-cell CNN with two control cells 5 and 6. Couplings between cells are T T
52 11

=T

22

=
64

2.0,

T

33

=

T

44

=1.2,
31

T

15

=

T

36

=

1,

=

-T

21

=1.2, T

=-T

43

= 2.1, T

= 3.5.

Fig. 2. Control signal for the "memory" device.

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Fig. 3. Chaotic memory

scanning.

Memory

device

falls

onto

memorized

pattern V1 for 0 < t < 45, 200 < t < 270, onto V2 for 90
310
Fig.4. Pattern recognition:

k = 0.02,

e = 0.002.

Z

is

an

arbitrary

external pattern for 0 140.

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