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

在研究期间(1993-1994)得到的结果如下：1.考虑了n个神经元排列在一个环上的神经网络。我们假设每个神经元只与前面的k(*n-1)个神经元相连，并且联接系数是逐渐变小的，即系数的大小随着距离的增加而减小。该模型来源于真实的生物学知识，是一种特殊的细胞神经网络(CNN)。我们考察了上述神经网络的平衡点个数，证明了(1)当k=1或2时，平衡点个数仅为1，当k=3时，平衡点个数为3，当k=4.2时，平衡点个数为4。我们可以推测，对于锥形神经网络，平衡点的数量非常少。因此，我们研究了另一种极限情况，其中k=n-1，并且每个系数的大小为1。在这种情况下，我们证明了：(1)存在一个至少有2^n/2个平衡点的网络；(2)存在一个至少有2^n/4个稳定平衡点的网络。因此，我们证明了平衡点的数量随着K.3的增加而指数增加。关于CNN的稳定性，我们推广了Gilli最近给出的定理。对于给定的矩阵A，Gilli定理要求存在正对角矩阵D使得DA+A^Td是对称矩阵，而我们的定理要求Gamma A+A^T Gamma是对称矩阵，其中Gamma是正定矩阵。给出了利用神经网络实现一组平衡点的充要条件，该神经网络的所有对角线元素具有相同的值，所有非对角线元素也是相同的。在一般情况下，平衡点对应于非线性阻性电路的解。因此，我们调查了这些电路的解的数量，并进行了许多演示，如下面的参考文献所示。

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.