Convexity of Neural Codes with Four Maximal Codewords

TL;DR

This paper investigates the convexity of neural codes with four maximal codewords, establishing conditions under which such codes are convex, and fully characterizing convexity in most cases based on the nerve complex analysis.

Contribution

It extends the understanding of convex neural codes from up to three maximal codewords to four, identifying new obstructions and providing a complete characterization in most cases.

Findings

Convexity characterized by absence of local obstructions in many four-maximal-codeword cases.

Identification of a second obstruction, called a "wheel," affecting convexity.

Complete characterization of convexity for 15 out of 20 nerve complex cases.

Abstract

Place cells are neurons that act as biological position sensors, associated with and firing in response to regions of an environment to situate an organism in space. These associations are recorded in (combinatorial) neural codes, motivating the following mathematical question: Which neural codes are generated by a collection of convex open sets in Euclidean space? Giusti and Itskov showed that a necessary condition for convexity is the absence of ``local obstructions." This necessary condition is, in fact, sufficient for certain families of codes. One such family consists of all codes with up to three maximal codewords. In this article, we investigate codes with four maximal codewords, showing that for many such codes, convexity is characterized by the absence of local obstructions, whereas for other such codes, convexity is characterized by the absence of local obstructions and a second type of obstruction, a ``wheel". Key to our analysis is a case-by-case investigation based on the nerve complex of the set of maximal codewords of a neural code. Up to symmetry, there are 20 possible nerves; and our results fully characterize convexity in 15 of the 20 cases.