Global Dynamics of Neural Nets With Infinite Gain.

Abstract

We consider a model of neural and gene networks where the nonlinearities in the system of differential equations are discontinuous and piecewise constant. We associate to the system a graph on a hypercube, which can be used to define a Morse decomposition of a related flow on the set of rays through the origin. We then discuss relationship between the invariant sets in the ray flow and the invariant sets in regular flow. We provide a sufficient conditions when there is a oneto -one correspondence between these sets. Finally, we consider a subclass of gene networks, where under certain conditions we can determine the structure of the lowest (attracting) Morse set directly from the hypercube graph. Key words. Neural and gene networks, Morse decomposition, global dynamics. AMS subject classification 34C25, 68T10, 92B20,92D15. 1 Introduction In this paper we study a set of differential equations, introduced by L. Glass [1] x i = \Gammafl i x i + i (x 1 ; : : : ; x n ); i = 1; : : : ; n..