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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