  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 {\it 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 {\it 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 {\it 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.
