Polarization of neural codes

TL;DR

This paper extends the algebraic framework for analyzing neural codes by introducing polarization of motifs and codes, enabling a deeper understanding of their intrinsic structure through square-free monomial ideals.

Contribution

It introduces the polarization of neural motifs and codes, providing new tools to study neural code structures via algebraic methods and offering a novel proof of existing theorems.

Findings

Polarization of neural motifs and codes has desirable algebraic properties.

The approach allows studying neural codes in a higher-dimensional algebraic space.

A new proof of a key theorem on neural codes is provided using these concepts.

Abstract

The neural rings and ideals as an algebraic tool for analyzing the intrinsic structure of neural codes were introduced by C.~Curto et al. in 2013. Since then they were investigated in several papers, including the 2017 paper by G\"unt\"urk\"un et al., in which the notion of polarization of neural ideals was introduced. In this paper we extend their ideas by introducing the notions of polarization of motifs and neural codes. We show that the notions that we introduced have very nice properties which could allow the studying of the intrinsic structure of neural codes of length $n$ via the square free monomial ideals in $2n$ variables and interpreting the results back in the original neural code ambient space. In the last section of the paper we introduce the notions of inactive neurons, partial neural codes, and partial motifs, as well as the notions of polarization of these codes and motifs. We use these notions to give a new proof of a theorem from the paper by G\"unt\"urk\"un et al. that we mentioned above.