State polytopes related to two classes of combinatorial neural codes

Combinatorial neural codes are collections of 0/1 vectors that used to model the co-firing patterns a set place cells in brain. One wide-open problem this area is determine when given code can be algorithmically drawn on plane as Venn diagram-like figure. A sufficient condition do so for have property called 2-inductively pierced. Gross, Obatake, and Youngs recently toric algebra show three neurons 1-inductively pierced if only ideal trivial or generated by quadratics. No result known additional same generality, part difficulty coming from large number codewords possible used. In article, we study two infinite classes combinatorial detail. For each code, explicitly compute its universal Gröbner basis. This done first class recognizing form Lawrence-type matrix. With second class, showing matrix totally unimodular. These computations allow one state polytopes corresponding ideals, which all distinct initial ideals may computed efficiently. Moreover, combinatorially equivalent well-known polytopes: permutohedron stellohedron.