  Threshold-linear networks are a common class of firing rate models that
describe recurrent interactions among neurons. Unlike their linear
counterparts, these networks generically possess multiple stable fixed points
(steady states), making them viable candidates for memory encoding and
retrieval. In this work, we characterize stable fixed points of general
threshold-linear networks with constant external drive, and discover
constraints on the co-existence of fixed points involving different subsets of
active neurons. In the case of symmetric networks, we prove the following
antichain property: if a set of neurons $\tau$ is the support of a stable fixed
point, then no proper subset or superset of $\tau$ can support a stable fixed
point. Symmetric threshold-linear networks thus appear to be well suited for
pattern completion, since the dynamics are guaranteed not to get "stuck" in a
subset or superset of a stored pattern. We also show that for any graph G, we
can construct a network whose stable fixed points correspond precisely to the
maximal cliques of G. As an application, we design network decoders for place
field codes, and demonstrate their efficacy for error correction and pattern
completion. The proofs of our main results build on the theory of permitted
sets in threshold-linear networks, including recently-developed connections to
classical distance geometry.
