  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 fixed points of a TLN, including
both stable and unstable equilibria, play a critical role in shaping its
emergent dynamics. In this work, we provide two novel characterizations for the
set of fixed points of a competitive TLN: the first is in terms of a simple
sign condition, while the second relies on the concept of domination. We apply
these results to a special family of TLNs, called combinatorial
threshold-linear networks (CTLNs), whose connectivity matrices are defined from
directed graphs. This leads us to prove a series of graph rules that enable one
to determine fixed points of a CTLN by analyzing the underlying graph.
Additionally, we study larger networks composed of smaller "building block"
subnetworks, and prove several theorems relating the fixed points of the full
network to those of its components. Our results provide the foundation for a
kind of "graphical calculus" to infer features of the dynamics from a network's
connectivity.
