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 constant. 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 collection 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 flow and the invariant sets 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 networks within this framework. Under certain conditions, we can determine the structure of an invariant set corresponding to the lowest strongly connected component of the hypercube graph.