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

1993-1994年的研究结果如下：1.我们考虑一个神经网络，其中n个神经元排列在一个环上。我们假设每个神经元只与前面的k（*n-1）个神经元连接，并且连接的系数是锥形的，即系数的大小沿着距离减小。该模型来源于真实的生物知识，是一种特殊的细胞神经网络。我们研究了上述神经网络的平衡点的数量，证明了（1）k=1或2时平衡点的数量只有一个，k=3时只有三个，k=4.2时只有四个。我们可以推测，锥形神经网络的平衡点的数量是非常少的。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）存在一个网络，它至少可以拥有2^n/2个平衡点，（2）存在一个网络，它至少可以拥有2^n/4个稳定的平衡点。因此，我们表明，平衡点的数量随k呈指数增长。关于CNN的稳定性，我们推广了Gilli最近给出的定理。Gilli定理要求给定的矩阵A存在一个正对角矩阵D使得DA+A^TD是对称矩阵，而我们的定理要求<$MA +A^T <$MA是对称矩阵，其中<$MA是正定矩阵。4.我们给出了通过所有对角元素具有相同值且所有非对角元素也相同的神经网络实现一组平衡点的充分必要条件。5.在一般的平衡点对应的非线性电阻电路的解决方案。因此，我们研究了这些电路的解决方案的数量，并在下面的参考文献中显示了许多演示文稿。

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.