EXACT SYNTHESIS OF ASSOCIATIVE MEMORY CIRCUITS BY MEANS OF RECURSIVE NEURAL NETWORKS

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The results obtained in the research term (1993-1994) are as follows :1. We consider a neural network in which n neurons are arranged on a ring. We assume that each neuron is connected only with the preceding k (*n-1) neurons and the coefficients of connections are tapered, that is, the magnitude of coefficients decrease along with distance. This model is derived from the real biological knowledge and is a special kind of CNNs (Cellular Neural Network). We examined the number of equilibrium points of the above neural networks and proved that (1) the number is only one for k=1 or 2, three for k=3, and four for k=4.2. We may conjecture that the number of equilibrium points are extremely few for tapered neural networks. We therefore examined another limiting case where k=n-1 and the magnitude of each coefficients is unity. We showed in this case that (1) there exist a network which can possess are at least 2^n/2 equilibrium points, and (2) there exists a network which can possess at least 2^n/4 stable equilibrium points. Thus we showed that the number of equilibrium points increases exponentially with k.3. Concerning the stability of the CNN,we generalized the theorem recently given by Gilli. Gilli's theorem requires for a prescribed matrix A the condition that there exists a positive diagonal matrix D such that DA+A^TD is a symmetric matrix, while our theorem requires that GAMMA A+A^T GAMMA is a symmetric matrix where GAMMA is a positive definite matrix.4. We gave the necessary and sufficient condition for a set of equilibrium points to be realizable by means of a neural network whose all diagonal elements have identical value and all nondiagonal elements are also identical.5. In general equilibrium points correspond to solutions of a nonlinear resistive circuits. So we investigated about the number of solutions of these circuits and had many presentations as shown in REFERENCES below.