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One of the goals of neuroscience is to determine how the firing of neurons helps the brain understand its environment. Some neurons are observed to fire rapidly in response to particular stimuli; the set of such stimuli is known as the neuron’s [*receptive field*]{}. One aim of the study of such neurons is to determine how their firing patterns encode the relationships among their receptive fields. For example, place cells, discovered in 1971 by O’Keefe and Dostrovsky, fire more rapidly when an animal is in certain regions, allowing the animal to navigate its environment [@Place_cell_discovery]. In this case, the receptive field of a place cell is the spatial region in which it fires, and the firing patterns of the place cells allow the brain to construct a “map” of these regions. We want to understand how the brain does this.

To this end, Curto, Itskov, Veliz-Cuba, and Youngs introduced the neural ideal [@Neural_Rings]. Firing patterns of neurons can be recorded as collections of binary vectors known as neural codes, and the neural ideal $J_C$ of a code $C$ is a polynomial ideal that contains the same information as the code itself. Curto et al. showed that the presence of 