Equilibria and their Stability in Networks with Steep Sigmoidal Nonlinearities

2021/03/31 by Duncan, William, Gedeon, Tomas, Kokubu, Hiroshi +2
#37N25 (Primary) 92C42 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2103.17184

Abstract

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.

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