Vol. 14, No. 5, 2021

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Universal Gröbner bases of toric ideals of combinatorial neural codes

Melissa Beer, Robert Davis, Thomas Elgin, Matthew Hertel, Kira Laws, Rajinder Mavi, Paula Mercurio and Alexandra Newlon

Vol. 14 (2021), No. 5, 723–742
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 01-vectors which encode the patterns of cofiring 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 cofiring patterns. This article continues their work by closely focusing on an assortment of classes of combinatorial neural codes. In particular, we identify universal Gröbner bases of the toric ideals for these codes.

Keywords
combinatorial neural codes, place cells
Mathematical Subject Classification 2010
Primary: 13P25, 92B20
Milestones
Received: 22 April 2019
Revised: 15 June 2021
Accepted: 18 June 2021
Published: 9 February 2022

Communicated by Ann N. Trenk
Authors
Melissa Beer
Department of Mathematics and Computing
Franklin College
Franklin, IN
United States
Robert Davis
Department of Mathematics
Colgate University
Hamilton, NY
United States
Thomas Elgin
Department of Mathematics
University of South Carolina
Columbia, SC
United States
Matthew Hertel
Department of Mathematics
Michigan State University
East Lansing, MI
United States
Kira Laws
Department of Mathematical Sciences
Applachian State University
Boone, NC
United States
Rajinder Mavi
Department of Mathematics
University of Cincinnati
Cincinnati, OH
United States
Paula Mercurio
Department of Mathematics
Michigan State University
Lansing, MI
United States
Alexandra Newlon
Department of Mathematics
Colgate University
Hamilton, NY
United States