Morphisms, Minors, and Minimal Obstructions to Convexity of Neural Codes

Abstract

We study open and closed convex codes from a geometric and combinatorial point of view. We prove constructive geometric results that establish new upper bounds on the open and closed embedding dimensions of intersection complete codes. We introduce a combinatorial framework of morphisms and minors for the study of convex codes, and show that open and closed embedding dimension are monotone invariants when codes are partially ordered by minors (in particular, open or closed convex codes form a minor-closed family). We establish new discrete geometry theorems and use them to exhibit infinite families of minimally non- convex codes, including new local obstructions to convexity. We also describe families of codes with novel embedding dimension behavior: arbitrary disparity between open and closed embedding dimension, open embedding dimensions that are exponential in the number of neurons in a code, and large increases in closed embedding dimension when adding a new non-maximal codeword. We conclude with an extensive discussion of open questions.