  Nonlinear network dynamics are notoriously difficult to understand. Here we
study a class of recurrent neural networks called combinatorial
threshold-linear networks (CTLNs) whose dynamics are determined by the
structure of a directed graph. They are a special case of TLNs, a popular
framework for modeling neural activity in computational neuroscience. In prior
work, CTLNs were found to be surprisingly tractable mathematically. For small
networks, the fixed points of the network dynamics can often be completely
determined via a series of graph rules that can be applied directly to the
underlying graph. For larger networks, it remains a challenge to understand how
the global structure of the network interacts with local properties. In this
work, we propose a method of covering graphs of CTLNs with a set of smaller
directional graphs that reflect the local flow of activity. While directional
graphs may or may not have a feedforward architecture, their fixed point
structure is indicative of feedforward dynamics. The combinatorial structure of
the graph cover is captured by the nerve of the cover. The nerve is a smaller,
simpler graph that is more amenable to graphical analysis. We present three
nerve theorems that provide strong constraints on the fixed points of the
underlying network from the structure of the nerve. We then illustrate the
power of these theorems with some examples. Remarkably, we find that the nerve
not only constrains the fixed points of CTLNs, but also gives insight into the
transient and asymptotic dynamics. This is because the flow of activity in the
network tends to follow the edges of the nerve.
