  A neural code $\mathcal{C}$ is a collection of binary vectors of a given
length n that record the co-firing patterns of a set of neurons. Our focus is
on neural codes arising from place cells, neurons that respond to geographic
stimulus. In this setting, the stimulus space can be visualized as subset of
$\mathbb{R}^2$ covered by a collection $\mathcal{U}$ of convex sets such that
the arrangement $\mathcal{U}$ forms an Euler diagram for $\mathcal{C}$. There
are some methods to determine whether such a convex realization $\mathcal{U}$
exists; however, these methods do not describe how to draw a realization. In
this work, we look at the problem of algorithmically drawing Euler diagrams for
neural codes using two polynomial ideals: the neural ideal, a pseudo-monomial
ideal; and the neural toric ideal, a binomial ideal. In particular, we study
how these objects are related to the theory of piercings in information
visualization, and we show how minimal generating sets of the ideals reveal
whether or not a code is $0$, $1$, or $2$-inductively pierced.
