Combinatorial neural codes are 0/1 vectors that are used to model the co-firing patterns of a set of certain neurons in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically represented as a Venn diagram-like figure called an Euler diagram. Significant progress has been made recently by recasting this problem in terms of polynomials and using tools from commutative algebra. In particular, we will describe the toric ideal of a code and a special generating set, called the universal Gröbner basis, which contains an astounding amount of information about the ideal.

We will pay special attention to two infinite classes of combinatorial neural codes. For each code, we explicitly compute the universal Gröbner basis of its toric ideal. These computations allow one to compute the state polytopes of the corresponding toric ideals, which encode all of the distinct initial ideals arising from weight orders. Moreover, we show that the state polytopes are combinatorially equivalent to well-known polytopes: the permutohedron and the stellohedron.