Equilibria and their Stability in Networks with Steep 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.
Cite
@article{arxiv.2103.17184,
title = {Equilibria and their Stability in Networks with Steep Sigmoidal Nonlinearities},
author = {William Duncan and Tomas Gedeon and Hiroshi Kokubu and Konstantin Mischaikow and Hiroe Oka},
journal= {arXiv preprint arXiv:2103.17184},
year = {2021}
}
Comments
29 pages, 3 figures, submitted to SIAM Journal on Applied Dynamical Systems