State Polytopes Related to Two Classes of Combinatorial Neural Codes

TL;DR

This paper investigates the algebraic and geometric properties of two classes of combinatorial neural codes, computing their state polytopes and relating them to known polytopes, advancing understanding of their realizability.

Contribution

It explicitly computes universal Gröbner bases and state polytopes for two classes of neural codes, linking them to well-known polytopes like the permutohedron and stellohedron.

Findings

State polytopes are combinatorially equivalent to permutohedron and stellohedron.

Universal Gröbner bases are explicitly computed for the classes.

Codes are shown to have properties related to Lawrence matrices and total unimodularity.

Abstract

Combinatorial neural codes are $0/1$ vectors that are used to model the co-firing patterns of a set of place cells in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically drawn in the plane as a Venn diagram-like figure. A sufficient condition to do so is for the code to have a property called $k$-inductively pierced. Gross, Obatake, and Youngs recently used toric algebra to show that a code on three neurons is $1$-inductively pierced if and only if the toric ideal is trivial or generated by quadratics. No result is known for additional neurons in the same generality, part of the difficulty coming from the large number of codewords possible when additional neurons are used. In this article, we study two infinite classes of combinatorial neural codes in detail. For each code, we explicitly compute its universal Gr\"obner basis. This is done for the first class by recognizing that the codewords form a Lawrence-type matrix. With the second class, this is done by showing that the matrix is totally unimodular. These computations allow one to compute the state polytopes of the corresponding toric ideals, from which all distinct initial ideals may be computed efficiently. Moreover, we show that the state polytopes are combinatorially equivalent to well-known polytopes: the permutohedron and the stellohedron.