Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

We describe generic sliding modes of piecewise-linear systems of differential equations arising in the theory of gene regulatory networks with Boolean interactions. We do not make any a priori assumptions on regulatory functions in the network and try to understand what mathematical consequences this may have in regard to the limit dynamics of the system. Further, we provide a complete classification of such systems in terms of polynomial representations for the cases where the discontinuity set of the right-hand side of the system has a codimension 1 in the phase space. In particular, we prove that the multilinear representation of the underlying Boolean structure of a continuous-time gene regulatory network is only generic in the absence of sliding trajectories. Our results also explain why the Boolean structure of interactions is too coarse and usually gives rise to several non-equivalent models with smooth interactions.