On graphical domination for threshold-linear networks with recurrent excitation and global inhibition

Graphical domination was first introduced in [1] in the context of combinatorial threshold-linear networks (CTLNs). There it was shown that when a domination relationship exists between a pair of vertices in a graph, certain fixed points in the corresponding CTLN can be ruled out. Here we prove two new theorems about graphical domination, and show that they apply to a significantly more general class of recurrent networks called generalized CTLNs (gCTLNs). Theorem 1 establishes that if a dominated node is removed from a network, the reduced network has exactly the same fixed points. Theorem 2 tells us that by iteratively removing dominated nodes from an initial graph $G$, the final (irreducible) graph $\widetilde{G}$ is unique. We also introduce another new family of TLNs, called E-I TLNs, consisting of $n$ excitatory nodes and a single inhibitory node providing global inhibition. We provide a concrete mapping between the parameters of gCTLNs and E-I TLNs built from the same graph such that corresponding networks have the same fixed points. We also show that Theorems 1 and 2 apply equally well to E-I TLNs, and that the dynamics of gCTLNs and E-I TLNs with the same underlying graph $G$ exhibit similar behavior that is well predicted by the fixed points of the reduced graph $\widetilde{G}$.