Convexity of Neural Codes with Four Maximal Codewords

TL;DR

This work addresses when neural codes realized by convex open sets in $\mathbb{R}^d$ are convex, focusing on codes with at most four maximal codewords. It uses a nerve-based case split of the maximal codewords into 20 types (L9–L28) and analyzes convexity via local obstructions and wheels, employing gluing constructions to assemble convex realizations. The authors prove that for nerves L9–L23, convexity is equivalent to the absence of local obstructions, resolving a large portion of Jeffs' conjecture; for minimal L24 codes, convexity depends on the Path-of-Facets Condition, with sprockets obstructing convexity when this condition holds. The results substantially advance understanding of convexity for 4-maximal codes, provide explicit realizations, and lay groundwork for extending the approach to codes with more maximal codewords and higher dimensions.

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.

Figures (14)

• Figure 1: All simplicial complexes on up to $4$ vertices, up to symmetry. The simplicial complexes are labeled L1 to L28, matching the labels in Curto (in fact, this figure aligns closely with Curto).
• Figure 2: Convex realization of $\mathcal{C}_{22} = \{ \mathbf{134}, \mathbf{1357}, \mathbf{257},\mathbf{356}, 13, 35, 57, \varnothing\}$.
• Figure 4: Convex realization of $\mathcal{C}_{\text{min}}^\ast \cup \mathcal{C}_{\cap F_{4}}$ (Case 1).
• Figure 5: T-shaped convex realization of $\mathcal{C}_{\min} \cup \mathcal{C}_{\cap F_{4}}$ (Case 2).
• Figure 6: Two convex realizations of $\mathcal{C}_{\cap F_4}$.

Theorems & Definitions (73)

• Conjecture 1.1
• Definition 2.1
• Definition 2.2
• Example 2.3: $\mathcal{C}_{22}$
• Definition 2.4
• Remark 2.6
• Remark 2.7: Labels in realizations
• Example 2.8: Example \ref{['ex:first-example-L22']} continued
• Remark 2.9
• Lemma 2.10: Monotonicity of convexity