Equilibria and their Stability in Networks with Steep Sigmoidal Nonlinearities

TL;DR

This paper analyzes the existence and stability of equilibria in network models with steep sigmoidal nonlinearities, relevant for gene regulatory networks, using combinatorial and decomposition methods.

Contribution

It introduces a novel combinatorial approach to determine equilibria and stability in sigmoidal network models, and provides a local decomposition technique into cyclic feedback systems.

Findings

Equilibria are characterized by a combinatorial analysis of the switching system.

Stability depends on the structure of network loops and their decomposition.

The methods apply to gene regulatory network models with sigmoidal nonlinearities.

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.