  Sequences of neural activity arise in many brain areas, including cortex,
hippocampus, and central pattern generator circuits that underlie rhythmic
behaviors like locomotion. While network architectures supporting sequence
generation vary considerably, a common feature is an abundance of inhibition.
In this work, we focus on architectures that support sequential activity in
recurrently connected networks with inhibition-dominated dynamics.
Specifically, we study emergent sequences in a special family of
threshold-linear networks, called combinatorial threshold-linear networks
(CTLNs), whose connectivity matrices are defined from directed graphs. Such
networks naturally give rise to an abundance of sequences whose dynamics are
tightly connected to the underlying graph. We find that architectures based on
generalizations of cycle graphs produce limit cycle attractors that can be
activated to generate transient or persistent (repeating) sequences. Each
architecture type gives rise to an infinite family of graphs that can be built
from arbitrary component subgraphs. Moreover, we prove a number of graph rules
for the corresponding CTLNs in each family. The graph rules allow us to
strongly constrain, and in some cases fully determine, the fixed points of the
network in terms of the fixed points of the component subnetworks. Finally, we
also show how the structure of certain architectures gives insight into the
sequential dynamics of the corresponding attractor.
