Pattern completion in symmetric threshold-linear networks

TL;DR

This paper analyzes symmetric threshold-linear neural networks, proving they support pattern completion by ensuring stable fixed points correspond to maximal cliques, and applies this to designing effective place field code decoders.

Contribution

It establishes the antichain property for symmetric networks and links stable fixed points to maximal cliques, advancing understanding of pattern completion mechanisms.

Findings

Symmetric networks support stable fixed points only for maximal cliques.

Networks can be constructed to encode any graph's maximal cliques as stable fixed points.

Demonstrated effective error correction and pattern completion in place field code decoders.

Abstract

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.