From neural codes to homological invariants: regularity and projective dimension of polarized neural ideals

Abstract

Neural codes form an algebraic framework to study the nervous system, and understanding neural codes is a key goal of mathematical neuroscience. Neural rings and ideals are the tools connecting neuroscience and commutative algebra. In this article, we study the projective dimension and (Castelnuovo-Mumford) regularity of polarized neural ideals on $n$ neurons. Particularly, we find all the possible values for these two invariants. Moreover, we characterize when these ideals have linear resolution or linear quotients, assuming that they are generated in degree $n$.