  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 is determined by a combinatorial
analysis of the limiting switching system with piece-wise constant
non-linearities. 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.
