Singular Dynamics in Gene Network Models

Gene regulatory networks are commonly modeled by steep sigmoid interactions or by their limiting step functions. This leads to difficulties in dealing with singular dynamics, i.e., when some of the gene expressions are close to their thresholds. Two methods have been proposed to analyze this situation: the steep sigmoid framework based on singular perturbation techniques and the Filippov theory of differential inclusions. However, these lead to different concepts of solutions. Here we revisit these approaches and show their relationship. For the Filippov framework, we emphasize the use of two different solution concepts, namely, with a convex and a not necessarily convex right-hand side of the differential inclusion. We show that the second case is identical, under certain assumptions, to the solution concept used in the steep sigmoid framework. Even without these assumptions we obtain an existence result in the 2-dimensional case. We present 3-dimensional examples that do not fit the classical singular perturbation theory and for which even Filippov solutions may not be meaningful. We therefore suggest a generalization of the classical singular perturbation techniques that is based on the Artstein theory of dynamic (nonstationary) limits of the fast flow.