Equilibria and their Stability in Networks with Steep Sigmoidal Nonlinearities*

In this paper we investigate equilibria of continuous differential equation models of network dynamics. The motivation comes from gene regulatory networks where each directed edge represents either down- or up-regulation, and is modeled by a sigmoidal nonlinear function. We show that the existence and stability of equilibria of a sigmoidal system are determined by a combinatorial analysis of the limiting switching system with piecewise constant nonlinearities. In addition, we describe a local decomposition of a switching system into a product of simpler cyclic feedback systems, where the cycles in each decomposition correspond to a particular subset of network loops.