Neural Codes With Three Maximal Codewords: Convexity and Minimal Embedding Dimension

TL;DR

This paper investigates convex neural codes with up to three maximal codewords, proving that such codes are convex if they lack local obstructions and establishing that their minimal embedding dimension is at most two.

Contribution

It proves the converse for convexity in neural codes with up to three maximal codewords and determines their minimal embedding dimension.

Findings

Convex neural codes with up to three maximal codewords have no local obstructions.

Such codes are convex if they lack local obstructions.

The minimal embedding dimension for these codes is at most two.

Abstract

Neural codes, represented as collections of binary strings called codewords, are used to encode neural activity. A code is called convex if its codewords are represented as an arrangement of convex open sets in Euclidean space. Previous work has focused on addressing the question: how can we tell when a neural code is convex? Giusti and Itskov identified a local obstruction and proved that convex neural codes have no local obstructions. The converse is true for codes on up to four neurons, but false in general. Nevertheless, we prove this converse holds for codes with up to three maximal codewords, and moreover the minimal embedding dimension of such codes is at most two.