Neural ring homomorphism preserves mandatory sets required for open convexity

TL;DR

This paper investigates how neural code properties related to convexity and local obstructions are affected by elementary code maps, highlighting the preservation of homological mandatory sets and exploring Stanley-Reisner ring relationships.

Contribution

It demonstrates that the homological mandatory set remains invariant under elementary code maps and explores the algebraic structure linking neural codes and simplicial complexes.

Findings

Homological mandatory sets are preserved under elementary code maps.

Relationships between Stanley-Reisner rings and neural codes are established.

Provides an alternative proof of the invariance of homological mandatory sets.

Abstract

It has been studied by Curto et al. (SIAM J. on App. Alg. and Geom., 1(1) : 222 $\unicode{x2013}$ 238, 2017) that a neural code that has an open convex realization does not have any local obstruction relative to the neural code. Further, a neural code $ \mathcal{C} $ has no local obstructions if and only if it contains the set of mandatory codewords, $ \mathcal{C}_{\min}(\Delta),$ which depends only on the simplicial complex $\Delta=\Delta(\mathcal{C})$. Thus if $\mathcal{C} \not \supseteq \mathcal{C}_{\min}(\Delta)$, then $\mathcal{C}$ cannot be open convex. However, the problem of constructing $ \mathcal{C}_{\min}(\Delta) $ for any given code $ \mathcal{C} $ is undecidable. There is yet another way to capture the local obstructions via the homological mandatory set, $ \mathcal{M}_H(\Delta). $ The significance of $ \mathcal{M}_H(\Delta) $ for a given code $ \mathcal{C} $ is that $ \mathcal{M}_H(\Delta) \subseteq \mathcal{C}_{\min}(\Delta)$ and so $ \mathcal{C} $ will have local obstructions if $ \mathcal{C}\not\supseteq\mathcal{M}_H(\Delta). $ In this paper we study the affect on the sets $\mathcal{C}_{\min}(\Delta) $ and $\mathcal{M}_H(\Delta)$ under the action of various surjective elementary code maps. Further, we study the relationship between Stanley-Reisner rings of the simplicial complexes associated with neural codes of the elementary code maps. Moreover, using this relationship, we give an alternative proof to show that $ \mathcal{M}_H(\Delta) $ is preserved under the elementary code maps.