Characterizations and constructions of reverberating networks

Consider the family $\text{G}$_n of all n neuron networks whose dynamical behaviors are described by Caianiello's neuronic equations, and also its subfamily $\text{R}$_n of all reverberating networks each of whose neuronic equations have only periodic solution (states), i.e., without having any transient states. This paper is specifically concerned with characterizations of the subfamily $\text{R}$_n. First, we show that $\text{R}$_n is contained in a subfamily $\text{S}$_n of $\text{G}$_n consisting of all self-dual networks. We introduce Chow's matrix corresponding to each network of $\text{S}$_n, using Chow parameters and some algebraic operations {$\text{2}$_α} applied to the coefficient matrix of the network in $\text{S}$_n, such as interchanges of coefficients between two neurons or changes of their signs. Then we give some necessary conditions on Chow's matrix under which any network in $\text{S}$_n belong to $\text{R}$_n, and a necessary and sufficient condition on the coefficient matrix. We also discuss relations between Chow's matrix and the maximum period of reverberations. In particular, it is shown that Chow's matrix of a network in $\text{R}$_n is symmetric if and only if the maximum period of reverberations is less than three. By virtue of these results, we propose two methods of construction of networks in $\text{R}$_n. The first method is an inductive construction. The second is based on the algebraic operations.