  The neural ideal of a binary code $\mathbb{C} \subseteq \mathbb{F}_2^n$ is an
ideal in $\mathbb{F}_2[x_1,\ldots, x_n]$ closely related to the vanishing ideal
of $\mathbb{C}$. The neural ideal, first introduced by Curto et al, provides an
algebraic way to extract geometric properties of realizations of binary codes.
In this paper we investigate homomorphisms between polynomial rings
$\mathbb{F}_2[x_1,\ldots, x_n]$ which preserve all neural ideals. We show that
all such homomorphisms can be decomposed into a composition of three basic
types of maps. Using this decomposition, we can interpret how these
homomorphisms act on the underlying binary codes. We can also determine their
effect on geometric realizations of these codes using sets in $\mathbb{R}^d$.
We also describe how these homomorphisms affect a canonical generating set for
neural ideals, yielding an efficient method for computing these generators in
some cases.
