Global dynamics of neural nets with infinite gain

We consider a model of neural and gene networks where the nonlinearities in the system of differential equations are discontinuous and piecewise const ant. We develop a framework for study of such systems. As a first step, we associate to the system a graph G on a hypercube and show how the collectio n of strongly connected components of G relates to the dynamics of the flow on the set of rays through the origin. In the second step, we discuss the relationship between the invariant sets of the ray how and the invariant se ts of the original flow. We provide a sufficient condition for a one-to-one correspondence between these sets. Finally, we study the class of binary n etworks within this framework. Under certain conditions, we can determine t he structure of an invariant set corresponding to the lowest strongly conne cted component of the hypercube graph. (C) 2000 Elsevier Science B.V. All r ights reserved.