Caitlin Lienkaemper, Pennsylvania State University
Authors: Alexander Kunin, Zvi Rosen, and Caitlin Lienkaemper
2022 AWM Research Symposium
Combinatorial and Homological Methods in Commutative Algebra

A combinatorial neural code describes a pattern of neural activity in terms of which neurons fire together. Many types of neurons respond to a set of stimuli, called the neuron's receptive field. The neural ideal is a pseudomonial ideal which encodes the relationships between receptive fields entailed by a neural code. Oriented matroids are combinatorial objects arising from hyperplane arrangements in much the same way that combinatorial codes arise from receptive fields. In this talk, we relate combinatorial neural codes to oriented matroids and neural ideals to oriented matroid ideals. We define a contravariant functor from the category of oriented matroids to the category of neural codes, and show that it commutes with the functors taking oriented matroids and neural codes to their respective monomial ideals. This lets us view the neural ideal as a generalization of the oriented matroid ideal, and lets us see the receptive field relationships of a neural code as a generalization of the circuits of an oriented matroid.

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