Global analysis of regulatory network dynamics: equilibria and saddle-node bifurcations

In this paper we describe a combined combinatorial/numerical approach to studying equilibria and bifurcations in network models arising in Systems Biology. ODE models of the dynamics suffer from high dimensional parameters which presents a significant obstruction to studying the global dynamics via numerical methods. The main point of this paper is to demonstrate that combining classical techniques with recently developed combinatorial methods provides a richer picture of the global dynamics despite the high parameter dimension. Given a network topology describing state variables which regulate one another via monotone and bounded functions, we first use the Dynamic Signatures Generated by Regulatory Networks (DSGRN) software to obtain a combinatorial summary of the dynamics. This summary is coarse but global and we use this information as a first pass to identify "interesting'' subsets of parameters in which to focus. We construct an associated ODE model with high parameter dimension using our {\em Network Dynamics Modeling and Analysis} (NDMA) Python library. We introduce algorithms for efficiently investigating the dynamics in these ODE models restricted to these parameter subsets. Finally, we perform a statistical validation of the method and several interesting dynamical applications including finding saddle-node bifurcations in a 54 parameter model.