  In the 1970s, O'Keefe and Dostrovsky discovered that certain neurons, called
place cells, in an animal's brain are tied to its location within its arena. A
combinatorial neural code is a collection of $0/1$-vectors which encode the
patterns of co-firing activity among the place cells. Gross, Obatake, and
Youngs have recently used techniques from toric algebra to study when a neural
code is $0$- $1$-, or $2$-inductively pierced: a property that allows one to
reconstruct a Venn diagram-like planar figure that acts as a geometric
schematic for the neural co-firing patterns. This article examines their work
closely by focusing on a variety of classes of combinatorial neural codes. In
particular, we identify universal Gr\"obner bases of the toric ideal for these
codes.
