Paper Info
Title
Grobner Bases Of Neural Ideals
Abstract
The brain processes information about the environment via neural codes. The neural ideal was introduced recently as an algebraic object that can be used to better understand the combinatorial structure of neural codes. Every neural ideal has a particular generating set, called the canonical form, that directly encodes a minimal description of the receptive field structure intrinsic to the neural code. On the other hand, for a given monomial order, any polynomial ideal is also generated by its unique (reduced) Grobner basis with respect to that monomial order. How are these two types of generating sets - canonical forms and Grobner bases - related? Our main result states that if the canonical form of a neural ideal is a Grobner basis, then it is the universal Grobner basis (that is, the union of all reduced Grobner bases). Furthermore, we prove that this situation - when the canonical form is a Grobner basis - occurs precisely when the universal Grobner basis contains only pseudo-monomials (certain generalizations of monomials). Our results motivate two questions: (1) When is the canonical form a Grobner basis? (2) When the universal Grobner basis of a neural ideal is not a canonical form, what can the non-pseudo-monomial elements in the basis tell us about the receptive fields of the code? We give partial answers to both questions. Along the way, we develop a representation of pseudo-monomials as hypercubes in a Boolean lattice.
Year
DOI
Venue
2018
10.1142/S0218196718500261
INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION
Keywords
DocType
Volume
Neural code, receptive field, canonical form, Grobner basis, Boolean lattice
Journal
28
Issue
ISSN
Citations 
4
0218-1967
0
PageRank 
References 
Authors
0.34
3
8
Name
Order
Citations
PageRank
Rebecca Garcia141.17
Luis David García-Puente28110.52
Ryan Kruse300.34
Jessica Liu400.34
Dane Miyata510.73
Ethan Petersen610.73
Kaitlyn Phillipson700.34
Anne Shiu88714.47