Strengthening Relationships between Neural Ideals and Receptive Fields

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 JC . In turn, the polynomials in JC 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 JC 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 JC 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 JC and I(C). Using the modified relations, we uncover a new relationship involving JC , I(C), and the Type 1-6 relations.