Non-monotonicity of closed convexity in neural codes

TL;DR

This paper investigates the relationship between open and closed convex neural codes, revealing that closed convex codes lack certain monotonicity properties of open convex codes and can have significantly higher embedding dimensions.

Contribution

It demonstrates that closed convex codes do not share the monotonicity property of open convex codes and can have arbitrarily larger embedding dimensions, disproving a previous conjecture.

Findings

Closed convex codes lack monotonicity property.

Adding a codeword can greatly increase closed embedding dimension.

Counterexample code is neither open nor closed convex.

Abstract

Neural codes are lists of subsets of neurons that fire together. Of particular interest are neurons called place cells, which fire when an animal is in specific, usually convex regions in space. A fundamental question, therefore, is to determine which neural codes arise from the regions of some collection of open convex sets or closed convex sets in Euclidean space. This work focuses on how these two classes of codes -- open convex and closed convex codes -- are related. As a starting point, open convex codes have a desirable monotonicity property, namely, adding non-maximal codewords preserves open convexity; but here we show that this property fails to hold for closed convex codes. Additionally, while adding non-maximal codewords can only increase the open embedding dimension by 1, here we demonstrate that adding a single such codeword can increase the closed embedding dimension by an arbitrarily large amount. Finally, we disprove a conjecture of Goldrup and Phillipson, and also present an example of a code that is neither open convex nor closed convex.