The algorithm for canonical forms of neural ideals

Licui Zheng; Yiyao Zhang; Jinwang Liu; Licui Zheng; Yiyao Zhang; Jinwang Liu

doi:10.3934/era.2024145

Electronic Research Archive

2024, Volume 32, Issue 5: 3162-3170. doi: 10.3934/era.2024145

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The algorithm for canonical forms of neural ideals

School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, China

Received: 20 December 2023 Revised: 28 March 2024 Accepted: 28 April 2024 Published: 06 May 2024

To elucidate the combinatorial architecture of neural codes, the neural ideal $ J_C $, an algebraic object, was introduced. Represented in its canonical form, $ J_C $ provides a succinct characterization of the inherent receptive field architecture within the code. The polynomials in $ J_C $ are also instrumental in determining the relationships among the neurons' receptive fields. Consequently, the computation of the collection of canonical forms is pivotal. In this paper, based on the study of relations between pseudo-monomials, the authors present a computationally efficient iterative algorithm for the canonical forms of the neural ideal. Additionally, we introduce a new relationship among the neurons' receptive fields, which can be characterized by if-and-only-if statements, relating both to $ J_C $ and to a larger ideal of a code $ I(C) $.
- neural code,
- neural ideals,
- canonical forms
Citation: Licui Zheng, Yiyao Zhang, Jinwang Liu. The algorithm for canonical forms of neural ideals[J]. Electronic Research Archive, 2024, 32(5): 3162-3170. doi: 10.3934/era.2024145

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Abstract

To elucidate the combinatorial architecture of neural codes, the neural ideal $ J_C $, an algebraic object, was introduced. Represented in its canonical form, $ J_C $ provides a succinct characterization of the inherent receptive field architecture within the code. The polynomials in $ J_C $ are also instrumental in determining the relationships among the neurons' receptive fields. Consequently, the computation of the collection of canonical forms is pivotal. In this paper, based on the study of relations between pseudo-monomials, the authors present a computationally efficient iterative algorithm for the canonical forms of the neural ideal. Additionally, we introduce a new relationship among the neurons' receptive fields, which can be characterized by if-and-only-if statements, relating both to $ J_C $ and to a larger ideal of a code $ I(C) $.

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