Conference abstracts

Session A2 - Computational Algebraic Geometry

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Gr\"obner bases of neural ideals

Rebecca E. Garcia

Sam Houston State University, USA   -   rgarcia@shsu.edu

A major area in neuroscience is the study of how the brain processes spatial information. Neurons in the brain represent external stimuli via neural codes. These codes often arise from regions of space called receptive fields: each neuron fires at a high rate precisely when the animal is in the corresponding receptive field. Much research in this area has focused on understanding what features of receptive fields can be extracted directly from a neural code. In particular, Curto, Itskov, Veliz-Cuba, and Youngs recently introduced the concept of neural ideal, which is an algebraic object that encodes the full combinatorial data of a neural code. 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) Gr\"obner basis with respect to that monomial order. How are these two types of generating sets, canonical forms and Gr\"obner bases, related? Our main result states that if the canonical form of a neural ideal is a Gr\"obner basis, then it is the universal Gr\"obner basis (that is, the union of all reduced Gr\"obner bases). Furthermore, we prove that this situation --- when the canonical form is a Gr\"obner basis --- occurs precisely when the universal Gr\"obner basis contains only pseudo-monomials (certain generalizations of monomials). Our results motivate two questions: (1) When is the canonical form a Gr\"obner basis? (2) When the universal Gr\"obner 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.

Joint work with Luis David Garcia Puente (Sam Houston State University, USA), Ryan Kruse (Central College, USA), Jessica Liu (Bard College, USA), Dane Miyata (Willamette University, USA), Ethan Petersen (Rose-Hulman Institute of Technology, USA), Kaitlyn Phillipson (St. Edward’s University, USA) and Anne Shiu (Texas A\&M University, USA).

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