DYNAMIC PROPERTIES OF CELLULAR NEURAL NETWORKS

Dynam}c behav}or of a new class of information-processing systems called Cellular Neural Networks s investigated. In th}s paper we introduce a small parameter n the state equat}on of a cellular neural network and we seek for period}c phenomena. New approach is used for proving stability of a cellular neural network by constructing Lyapunov’s major}z}ng equat}ons. Th}s algorithm }s helpful for find}ng a map from initial continuous state space of a cellular neural network nto d}screte output. A compar}son between cellular neural networks and cellular automata s made.


INTRODUCTION
Cellular Neural Networks present a new class of information-processing systems. It is a large-scale nonlinear analog circuit which process signals in real time. The basic circuit unit of a cellular neural network is called a cell, an analog processor element which contains linear and nonlinear circuit elements, typically-linear capacitors, linear resistors, linear and nonlinear coatrolled sources, and independent sources [1, fig.3] The architecture of typical cellular neural networks is similar to that found in cellular automata [7,8]: any cell in a cellular neural network is connected only to its neighbor cells. Adjacent cells can interact directly with each other. Cells not directly connected may affect each other indirectly,because of the propagation effects of the continuous-time dynamics of the networks. The main difference between cellular automata and cellular neural networks is in their dynamic behavior. A cellular neural network is a continuous time dynamical system. A cellular automaton is a discrete time dynamical system. 1Received: February, 1993 where the ith row and jth column cell is indicated as C(i, j); by definition the r-neighborhood N r of radius r of a cell, C(i, j), in a cellular neural network is [1] N(i,j)= {C(k,l)lmaz{Ik-il, l/-jl} <r, I <_k<_M, I_</_<N}; vzij, vuij, vuij refer to the state, output and input voltage of a cell C(i,j); C,R x are fixed values of a linear capacitor and a linear resistor in the cell; I is an independent voltage; A(i,j;k,l) is a feedback operator and B(i,j;k,l) control operator, for which Ixu(i j; k, l) = A(i, j;, k, l)Vuk and Ixu(i j; k, l) = B(i, j; k, l)Vuk1(Izu(i, j; k, l) and Iu(i, j; k, l) are linear voltage-controlled sources for all C(i, j) e N(i, j)).
As mentioned above in [1] it is proved that cellular neural networks, described by the equations (1-7) must always converge to a constant steady-state after the transient has decayed to zero.
Moreover, it is obtained that cellular neural networks have binary-valued outputs, which have applications in image-processing [2]. These properties of cellular neural networks imply that the circuit will not oscillate and become chaotic.
Our interests are in obtaining some periodic or chaotic behavior of a cellular neural network. For this purpose we introduce a small parameter u in front of function f. Then the state equation (1) can be rewritten in the following form: where v = vxij(t), A = A(i,j;k,l), f(v) = f(vxij) and we also assume that B(i,j;k,l) = O. I is convenient for our further analysis and comparison of cellular neural networks to cellular automata.
One of the most popular techniques for analyzing the stability of dynamical nonlinear circuits is Lyapunov's method by defining an appropriate Lyapunov function for cellular neural networks [1].
Since we introduce a small parameter/z in equation (8), our approach will differ from the above. We will use the Lyapunov's majorizing equations method to investigate the convergence properties of cellular neural networks, described by state equation (8). Moreover, we will seek for the periodic solutions of (8) and dynamic range of cellular neural network.
Stability of equilibrium points of cellular neural networks will change as an effect of a small parameter / and we can estimate the upper bound of the interval of values of/z, in which periodic solution exists.

PRELIMINARIES
Lyapunov's majorizing equations method can be applied for the operator system of quite general kind with bounded linear operators: where the function F belongs to the class C 1 with respect to x(t, #) and to the class C with respect to t, p. L is a linear and bounded operator. Suppose also that From the boundedness of the operator L it follows the existence of a finite constant p > 0, such that the following inequality is satisfied for any function E C: We introduce the class f of functions (a, p) such that: i) When vxij(t ) < 0, then vuij(t = 0 for all t > 0, no matter if vxij(t = const, or vxij(t # const., which corresponds to the first case. When vxij > 1, then vuij(t = 1 for all t>0, so we have the fourth case. If 0<vxij<l, then vuij(t)=vxi(t ). In the third case vxij(t 5k const, and we can expect some periodic behavior or oscillation. Now let us consider equation (8), which can be rewritten in the following form: The operator L is linear and bounded. Therefore there exists a constant p, such that inequality (11) will be satisfied. If we go back to equation (12), the following can be obtained: v(t,#) = Lg(v,t, lz), which is an operator equation of type (9), equivalent to equation (13). 1. From the properties of Lyapunov's majorizing equations [3] and since function ff(c,p) is linear, it follows that solution c, =a(/z) will exist on the half-open interval 0 _</z </.. The solution c = c(#) may be unbounded for p---,/.. If we consider the plot of curves y-ff(c,#) for # </z., /z-/z., /z >/z. in the plane (c,y) and the plot of the straight line y = c, we can see that curve ff(c,/z) for/z >/. diverges.
2. Small parameter gt affects dynamic range of a cell, in other words if # >/z., state voltage vxij(t will be unbounded and this implies in different unpredictable boundary effects between neighbor cells. If/z E [0,/z.), state voltage vxij(t is bounded (from Theorem 2) and periodic, therefore output vuij(t = vxij, 0 < vxij < 1 will have periodic behavior.

=
gives the dynamic rules of a cellular neural network. Therefore, we can use cellular neural networks to obtain a dynamic transform of an initial state at any time t. In the special cases when t---oo and state variable vij is a constant, the output vui j tends to either 0 or i, (see equation (2)), which are limit points for a cellular neural network. Therefore in 0 _< vxij(t ) <_ 1, the equilibrium points of a typical cell of a cellular neural network C(i, j) are defined as" IRx v;ij (t) = 1"--'#A'R 1 <_ <_ M, 1 <_ j <_ N.
(24) Then we can define stable system equilibrium points of a cellular neural network, which describe its global dynamic behavior.
Definition 2: Stable system equilibrium point of a cellular neural network is I and a state vector with components vxij(t), 1 <_ <_ M, 1 <_ j <_ N, for which limt...oovuij(t = 0 or 1. From the above definition we can see that if we change the parameter, it will change the equilibrium points. Since any stable system equilibrium point is a limit point of a set of trajectories of the corresponding differential equation (8), such an attracting limit point is said to have a basin of attraction [5]. Therefore parameter # can affect the stability of a cellular neural network, in sense that different limit points and basins of attraction will be obtained for different values of small parameter #. So, if we consider the initial state space as [0,1] Mx N and the output space as {0,1) Mx N, then cellular neural network can be used to map an initial state of a system into one of many distinct stable equilibrium points and this map, defined by (22), will depend on #. In other words we have: F," [0,11M x N__{0, 1}M x N, which gives us the dynamic behavior of cellular neural networks.

SIMILARITY BETWEEN CELLULAR NEURAL NETWORKS AND CELLULAR AUTOMATA
In Section 3 we reduced state equation (8) As we said before the structure of the cellular neural networks is similar to that of cellular automata. A typical equation of a two-dimensional cellular automaton is ([7]) aij(n + 1) = ff[akt(n for all C(k,1) Nr(i,j)]. (28) Comparing equation (27) and equation (28) we can see a similarity between them. Therefore we can use cellular automata theory to study dynamic behavior of the cellular neural networks.
It is well known [7], that cellular automata may be considered as discrete dynamical system. In almost all cases, cellular automata evolution is irreversible. Trajectories in the configuration space for cellular automata therefore merge with time, and after many time steps, trajectories starting from almost all initial states become concentrated onto attractors. These attractors typically contain only a very small fraction of possible states. There are four classes of cellular automata, which characterize the attractors in cellular automaton evolution. The attractors in classes 1, 2 and 3 are roughly analogous respectively to the limit points, limit cycles and chaotic attractors found in continuous dynamical systems [8].