  To any inhibition-dominated threshold-linear network (TLN) we can associate a
directed graph that captures the pattern of strong and weak inhibition between
neurons. Robust motifs are graphs for which the structure of fixed points in
the network is independent of the choice of connectivity matrix $W$, and whose
dynamics are thus greatly constrained. This makes them ideal building blocks
for constructing larger networks whose behaviors are robust to changes in
connection strengths. In contrast, flexible motifs correspond to networks with
multiple dynamic regimes. In this work, we give a purely graphical
characterization of both flexible and robust motifs of any size. We find that
all but a few robust motifs fall into two infinite families of graphs with
simple feedforward architectures.
