What makes a neural code convex?

TL;DR

This paper characterizes the combinatorial structures that determine when a neural code can be represented by convex regions, advancing understanding of neural encoding and receptive fields.

Contribution

It provides a complete characterization of local obstructions to convexity in neural codes and introduces max intersection-complete codes as a class with guaranteed convexity.

Findings

Complete characterization of local obstructions to convexity.

Introduction of max intersection-complete codes as convex.

Use of Stanley-Reisner ideals to detect convexity violations.

Abstract

Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets? In this work, we provide a complete characterization of local obstructions to convexity. This motivates us to define max intersection-complete codes, a family guaranteed to have no local obstructions. We then show how our characterization enables one to use free resolutions of Stanley-Reisner ideals in order to detect violations of convexity. Taken together, these results provide a significant advance in understanding the intrinsic combinatorial properties of convex codes.