Polarization of neural codes

Abstract

The neural rings and ideals as an algebraic tool for analyzing the intrinsic structure of neural codes wereintroduced by C. Curto, V. Itskov, A. Veliz-Cuba, and N. Youngs in 2013. Since then they were investigated in severalpapers, including the 2017 paper by S. Güntürkün, J. Jeffries, and J. Sun, in which the notion of polarization of neuralideals was introduced. In our paper we extend their ideas by introducing the notions of polarization of motifs and neuralcodes. We show that the notions that we introduce have very nice properties which allow the studying of the intrinsicstructure of neural codes of length n via the square-free monomial ideals in 2n variables and interpreting the resultsback 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 partialmotifs, as well as the notions of polarization of these codes and motifs. We use these notions to give a new proof of atheorem from the paper by Güntürkün, Jeffries, and Sun that we mentioned above.