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A neural code $\mathcal{C}$ is a collection of binary vectors of a givenlength n that record the co-firing patterns of a set of neurons. Our focus ison neural codes arising from place cells, neurons that respond to geographicstimulus. 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 thatthe arrangement $\mathcal{U}$ forms an Euler diagram for $\mathcal{C}$. Thereare some methods to determine whether such a convex realization $\mathcal{U}$exists; however, these methods do not describe how to draw a realization. Inthis work, we look at the problem of algorithmically drawing Euler diagrams forneural codes using two polynomial ideals: the neural ideal, a pseudo-monomialideal; and the neural toric ideal, a binomial ideal. In particular, we studyhow these objects are related to the theory of piercings in informationvisualization, and we show how minimal generating sets of the ideals revealwhether or not a code is $0$, $1$, or $2$-inductively pierced.

Neural ideals and stimulus space visualization