ABSTRACT
In the present chapter we will concern ourselves with artificial recur rent neural networks which can be described by systems of first order ordinary differential equations given by
x = - B x + TS(x) + I (S)
where x is a real n-vector (denoting the neuron variables), x de notes the time derivative of x, B is a real n x n diagonal matrix with positive elements (representing self-feedback), T is a real n x n matrix (representing the interconnections among the neurons), / is a real n-vector (representing bias terms), and the real n-vector val ued function S(x) (representing the neurons) will assume one of the following forms:
A) each component Si(xi) of S(x) — (si(£i), • • •, sn(xn))T is a sig moidal function [i.e., Si maps the real numbers into the real interval (—1,1), it is smooth, it is monotonically increasing, and Si(0) = 0]; or
When the activation functions Si(x{) are sigmoidal functions and the matrix T is symmetric, system (S) constitutes the Hopfield model. When the components of S(x) are saturation functions, system (S) has been used, among other applications, to store bipolar memories and as cellular neural networks.
