Algebraic signatures of convex and non-convex codes

TL;DR

This paper develops algebraic methods using neural ideals to determine whether binary codes generated by convex sets are convex or non-convex, with applications to neuroscience.

Contribution

It introduces algebraic signatures based on neural ideals to distinguish convex from non-convex codes, advancing the algebraic understanding of convexity in neural codes.

Findings

Algebraic signatures identify non-convex codes.

Signatures for convex, intersection-complete codes.

Connections between algebraic conditions and set arrangements.

Abstract

A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we use algebraic methods to determine if a code is convex. Specifically, we use the neural ideal of a code, which is a generalization of the Stanley-Reisner ideal. Using the neural ideal together with its standard generating set, the canonical form, we provide algebraic signatures of certain families of codes that are non-convex. We connect these signatures to the precise conditions on the arrangement of sets that prevent the codes from being convex. Finally, we also provide algebraic signatures for some families of codes that are convex, including the class of intersection-complete codes. These results allow us to detect convexity and non-convexity in a variety of situations, and point to some interesting open questions.