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 et al. in 2013. Since then they were investigated in several papers, including the 2017 paper by Güntürkün 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 Güntürkün et al. that we mentioned above.

Neural codes and the factor complex

“…Theorem 3.3 (Codes, factor complexes, and neural ideals). Let C be a code on n neurons, and C ′ its complement code defined in (1). The following two maps are bijections:…”

“…Like previous works, we are motivated by the question of convexity in neural codes [3,6,14,15,16,19,21], with a specific interest in using neural ideals to study convexity [5,7,8,10,11,17]. Also, our factor complexes are motivated by the closely related polar complexes introduced recently by Güntürkün et al [9] (see also [1,11]).…”