Fixed points of competitive threshold-linear networks

Threshold-linear networks (TLNs) are models of neural networks that consist of simple, perceptron-like neurons and exhibit nonlinear dynamics that are determined by the network’s connectivity. The set of fixed points of a TLN, including both stable and unstable equilibria, plays a critical role in shaping its emergent dynamics. In this work, we provide two novel characterizations for the set of fixed points of competitive TLNs: the first is in terms of a simple sign condition corresponding to each fixed point support; while the second introduces the concept of domination. We then apply these results to a special family of TLNs, called combinatorial threshold-linear networks (CTLNs), whose connectivity matrix is defined from just two continuous parameters and a directed graph. This leads us to a set of graph rules that enable us to determine fixed points for a CTLN that are independent of the parameters, and can be inferred directly from the underlying graph. Our results thus provide the foundation for a kind of “graphical calculus” to infer features of the dynamics from a network’s connectivity.