Nondegenerate Neural Codes and Obstructions to Closed-Convexity

TL;DR

This paper explores the conditions under which neural codes are convex or non-convex, introduces criteria to identify obstructions to closed-convexity, and clarifies the relationship between open- and closed-convex realizations.

Contribution

It extends the concept of degeneracy to characterize when closures or interiors preserve neural codes and provides the first general criteria to rule out closed-convexity.

Findings

Nondegeneracy precisely characterizes closure and interior preservation.

New criteria based on rigid structures and neural ideals preclude closed-convexity.

Some codes previously thought non-convex are actually closed-convex in two dimensions.

Abstract

Previous work on convexity of neural codes has produced codes that are open-convex but not closed-convex -- or vice-versa. However, why a code is one but not the other, and how to detect such discrepancies are open questions. We tackle these questions in two ways. First, we investigate the concept of degeneracy introduced by Cruz et al., and extend their results to show that nondegeneracy precisely captures the situation when taking closures or interiors of open or closed realizations, respectively, yields another realization of the code. Second, we give the first general criteria for precluding a code from being closed-convex (without ruling out open-convexity), unifying ad-hoc geometric arguments in prior works. One criterion is built on a phenomenon we call a rigid structure, while the other can be stated algebraically, in terms of the neural ideal of the code. These results complement existing criteria having the opposite purpose: precluding open-convexity but not closed-convexity. Finally, we show that a family of codes shown by Jeffs to be not open-convex is in fact closed-convex and realizable in dimension two.