Polarization of neural codes

Abstract

The neural rings and ideals as an algebraic tool for analyzing the intrinsic structure of neural codes were introduced by C. Curto, V. Itskov, A. Veliz-Cuba, and N. Youngs in 2013. Since then they were investigated in several papers, including the 2017 paper by S. Güntürkün, J. Jeffries, and J. Sun, in which the notion of polarization of neural ideals was introduced. In our paper we extend their ideas by introducing the notions of polarization of motifs and neural codes. We show that the notions that we introduce have very nice properties which 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 Güntürkün, Jeffries, and Sun that we mentioned above.

Algebraic properties of neural codes.

“…In Chapter 6, we give our conclusions and recommendations for future work. Note that much of Chapters 3 and 4 appear in our recent paper [4], while much of Chapter 5 appears in our recent paper [5], although we have extended several notions and included several additional proofs.…”