From neural codes to homological invariants: regularity and projective dimension of polarized neural ideals

TL;DR

This paper investigates the algebraic properties of polarized neural ideals, specifically their projective dimension and regularity, providing a complete characterization of their possible values and conditions for linear resolutions.

Contribution

It introduces a comprehensive analysis of the projective dimension and regularity of polarized neural ideals, including classification of their possible values and criteria for linear resolutions.

Findings

All possible values for projective dimension and regularity are identified.

Conditions for polarized neural ideals to have linear resolution or linear quotients are characterized.

The study links algebraic invariants to neural code properties in mathematical neuroscience.

Abstract

Neural codes form an algebraic framework to study the nervous system, and understanding neural codes is a key goal of mathematical neuroscience. Neural rings and ideals are the tools connecting neuroscience and commutative algebra. In this article, we study the projective dimension and (Castelnuovo-Mumford) regularity of polarized neural ideals on $n$ neurons. Particularly, we find all the possible values for these two invariants. Moreover, we characterize when these ideals have linear resolution or linear quotients, assuming that they are generated in degree $n$.