Vol. 15, No. 2, 2022

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Neural codes with three maximal codewords: convexity and minimal embedding dimension

Katherine Johnston, Anne Shiu and Clare Spinner

Vol. 15 (2022), No. 2, 333–343
Abstract

Neural codes, represented as collections of binary strings called codewords, are used to encode neural activity. A code is called convex if its codewords are represented as an arrangement of convex open sets in Euclidean space. Previous work has focused on addressing the question: how can we tell when a neural code is convex? Giusti and Itskov (Neural Comput. 26:11 (2014), 2527–2540) identified a local obstruction and proved that convex neural codes have no local obstructions. The converse is true for codes on up to four neurons, but false in general. Nevertheless, we prove that this converse holds for codes with up to three maximal codewords, and, moreover, the minimal embedding dimension of such codes is at most 2.

Keywords
neural codes, convex, simplicial complex, link, contractible
Mathematical Subject Classification
Primary: 05E45, 52A20
Secondary: 92C20
Milestones
Received: 4 June 2021
Revised: 15 September 2021
Accepted: 22 September 2021
Published: 29 July 2022

Communicated by Anant Godbole
Authors
Katherine Johnston
Lafayette College
Easton, PA
United States
Anne Shiu
Department of Mathematics
Texas A&M University
College Station, TX
United States
Clare Spinner
University of Portland
Portland, OR
United States