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.
Highlights
One of the problems that neuroscience is faced with is to analyze the intrinsic structure of the so-called neural codes resulting from the activity of the certain type of neurons in the brain of an organism
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
In our paper we extend their ideas by introducing the notions of polarization of motifs and neural codes
Summary
One of the problems that neuroscience is faced with is to analyze the intrinsic structure of the so-called neural codes resulting from the activity of the certain type of neurons in the brain of an organism. The definitions of these notions with respect to F22n are the same as the ones with respect to Fn2 , we just need to take into account the notation for the variables This works for other notions as well (for example, minimal pseudomonomials in an ideal, the neural ideal of a code, the canonical form of a pseudomonomial ideal), while some notions (for example, minimal primes of an ideal) can be given in the form that does not depend on the notation for the variables. On we have the following convention: if the length of motifs and codes is denoted by n , the associated rings and ideals will always be in n variables X1, . For lengths given by concrete numbers it will always be clear from the context if the number is n or 2n
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