Stability of Equilibria for Piecewise-linear Models of Genetic Regulatory Networks

A formalism based on piecewise-linear (PL) differential equations has been shown to be well-suited to modelling genetic regulatory networks. The discontinuous vector field inherent in the PL models leads to the approach of Filippov, which extends the vector field to a differential inclusion. We study the stability of equilibria (called singular equilibrium sets) that lie on the surfaces of discontinuity. We prove several theorems that characterize the stability of these singular equilibria directly from the state transition graph, which is a qualitative representation of the dynamics of the system.