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.
Neural code, neural ideal, canonical form, minimal prime ideal, motifs, polarization, monomial ideal, pseudomonomial ideal, square-free monomial ideal, inactive neurons, partial word, partial motif, partial neural code
CHRISTENSEN, KATIE and KULOSMAN, HAMID
"Polarization of neural codes,"
Turkish Journal of Mathematics: Vol. 44:
1, Article 1.
Available at: https://journals.tubitak.gov.tr/math/vol44/iss1/1