Strengthening Relationships between Neural Ideals and Receptive Fields

Abstract

Neural codes are collections of binary vectors that represent the firing patterns of neurons.The information given by a neural code C can be represented by its neural ideal J C . In turn, the polynomials in J C can be used to determine the relationships among the receptive fields of the neurons. In a paper by Curto et al., three such relationships, known as the Type 1-3 relations, were linked to the neural ideal by three if-and-only-if statements. Later, Garcia et al.discovered the Type 4-6 relations. These new relations differed from the first three in that they were related to J C by one-way implications. In this paper, we first show that the converses of these new implications are false at the level of both the neural ideal J C and the larger ideal I(C) of a code. We then present modified statements of these relations that, like the first three, can be related by if-and-only-if statements to both J C and I(C). Using the modified relations, we uncover a new relationship involving J C , I(C), and the Type 1-6 relations.

Neural codes, decidability, and a new local obstruction to convexity

“…An additional problem suggested by our work pertains to detecting our new local obstructions (those of the second kind). Neural codes have been studied from an algebraic standpoint via neural ideals, which are closely related to the Stanley-Reisner ideal of a code's simplicial complex [4,5,7,11,13,16,18]. Using these algebraic tools, it is possible to find local obstructions that can be detected by homology, which suffices to determine contractibility for small simplicial complexes.…”