Differential Inclusions and Filippov Solutions in the Analysis of Piecewise Linear Models Describing Gene Regulatory Networks

We study some properties of piecewise-linear differential systems describing gene regulatory networks, where the dynamics is governed by sigmoid-type nonlinearities which are close to or coincide with the step functions. To overcome the difficulty of describing the dynamics of the system near singular stationary points (i.e. belonging to the discontinuity set of the system) we use the concept of Filippov solutions. It consists in replacing differential equations with discontinuous right-hand sides with differential inclusions with multi-valued functions. The global existence and some basic properties of Filippov solutions such as continuous dependence on parameters are studied. We also study uniqueness and non-uniqueness of the Fillipov solutions in singular domains. The concept of Filippov stationary point is extensively exploited in the paper. We compare two approaches in defining the singular stationary points: based on Fillipov theory and based on replacing step functions by steep sigmoids and investigating the smooth systems thus obtained.