Universal Gr\"obner Bases of Toric Ideals of Combinatorial Neural Codes

Abstract

In the 1970s, O'Keefe and Dostrovsky discovered that certain neurons, called place cells, in an animal's brain are tied to its location within its arena. A combinatorial neural code is a collection of $0/1$-vectors which encode the patterns of co-firing activity among the place cells. Gross, Obatake, and Youngs have recently used techniques from toric algebra to study when a neural code is $0$- $1$-, or $2$-inductively pierced: a property that allows one to reconstruct a Venn diagram-like planar figure that acts as a geometric schematic for the neural co-firing patterns. This article examines their work closely by focusing on a variety of classes of combinatorial neural codes. In particular, we identify universal Gr\"obner bases of the toric ideal for these codes.