Wheels: A New Criterion for Non-convexity of Neural Codes

TL;DR

This paper introduces new geometric and combinatorial criteria to determine non-convexity in neural codes, providing tools for classification and presenting the first example of a non-convex code without local obstructions.

Contribution

It develops novel criteria for non-convexity, classifies codes on six neurons, and characterizes convexity in codes with low or high-dimensional simplicial complexes.

Findings

First example of a non-convex code with no local obstructions

New criteria for non-convexity of neural codes

Classification of codes on six neurons

Abstract

We introduce new geometric and combinatorial criteria that preclude a neural code from being convex, and use them to tackle the classification problem for codes on six neurons. Along the way, we give the first example of a code that is non-convex, has no local obstructions, and has simplicial complex of dimension two. We also characterize convexity for neural codes for which the simplicial complex is pure of low or high dimension.