Obstructions to convexity in neural codes

TL;DR

This paper investigates the conditions under which neural codes, specifically convex neural codes related to hippocampal place cells, have local obstructions, providing classifications and counterexamples for codes on up to five neurons.

Contribution

It demonstrates that the previously assumed equivalence between convexity and absence of local obstructions does not hold in general, and classifies all codes on five neurons with no local obstructions.

Findings

Counterexample on five neurons disproves the converse for neural codes.

Complete classification of codes on five neurons with no local obstructions.

New criterion for non-mandatory intersections of maximal codewords.

Abstract

How does the brain encode spatial structure? One way is through hippocampal neurons called place cells, which become associated to convex regions of space known as their receptive fields: each place cell fires at a high rate precisely when the animal is in the receptive field. The firing patterns of multiple place cells form what is known as a convex neural code. How can we tell when a neural code is convex? To address this question, Giusti and Itskov identified a local obstruction, defined via the topology of a code's simplicial complex, and proved that convex neural codes have no local obstructions. Curto et al. proved the converse for all neural codes on at most four neurons. Via a counterexample on five neurons, we show that this converse is false in general. Additionally, we classify all codes on five neurons with no local obstructions. This classification is enabled by our enumeration of connected simplicial complexes on 5 vertices up to isomorphism. Finally, we examine how local obstructions are related to maximal codewords (maximal sets of neurons that co-fire). Curto et al. proved that a code has no local obstructions if and only if it contains certain "mandatory" intersections of maximal codewords. We give a new criterion for an intersection of maximal codewords to be non-mandatory, and prove that it classifies all such non-mandatory codewords for codes on up to 5 neurons.