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Polarization of neural codes. (English) Zbl 1443.13007

Summary: The neural rings and ideals as an algebraic tool for analyzing the intrinsic structure of neural codes were introduced by C. Curto et al. [Bull. Math. Biol. 75, No. 9, 1571–1611 (2013; Zbl 1311.92043)] in 2013. Since then they were investigated in several papers, including the 2017 paper by S. Güntürkün et al. [“Polarization of neural rings”, Preprint, arXiv:1706.08559], 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 et al. [loc.cit.] that we mentioned above.

MSC:

13B25 Polynomials over commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
92B20 Neural networks for/in biological studies, artificial life and related topics
94B60 Other types of codes

Citations:

Zbl 1311.92043
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References:

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