implications of our results and suggest topics for future research.

Background
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In this section we introduce neural codes, pseudo-monomials, and neural ideals, as well as the prior results on which our work is based. Our notation matches that in [@Neural_Rings; @Grobner_bases]. We begin with neural codes.

A [**neural code**]{} (or binary code) $C$ is a set of vectors in $\mathbb{F}_2^n$. A vector $c \in C$ is called a [**codeword**]{}.

Each codeword $c$ in a neural code represents a firing pattern of $n$ neurons: the $i$th component of $c$ is 1 if neuron $i$ is firing and $0$ if it is not. For example, the neural code $C = \{ [0,1,1], [1,0,1], [0,0,1], [0,0,0]\}$ consists of the codewords $c_1 = [0,1,1]$, $c_2 = [1,0,1]$, $c_3 = [0, 0, 1]$, and $c_4 = [0,0,0]$. The codeword $c_1$ tells us that neurons 2 and 3 fire together while neuron 1 does not, $c_2$ tells us that neurons 1 and 3 fire together while neuron 2 does not, $c_3$ tells us that neuron 3 fires alone, and $c_4$ tells us that none of the neurons fire.

Alternatively, each codeword $c \in C$ can also be represented by the set $$\text{supp(}c\text{)} = \{i \in 