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

## TL;DR

This paper investigates the algebraic structure of combinatorial neural codes using toric algebra, specifically identifying universal Gröbner bases of their associated toric ideals to understand neural co-firing patterns.

## Contribution

It provides the first detailed analysis of universal Gröbner bases for toric ideals of various classes of neural codes, linking algebraic properties to neural coding features.

## Key findings

- Universal Gröbner bases are identified for classes of neural codes.
- The algebraic structure reveals insights into neural co-firing patterns.
- The work connects combinatorial neural codes with toric algebra techniques.

## 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.