In the 1970s, James O’Keefe and his team observed that certain neurons in the brain, called place cells, spike in their firing rates when the animal is in a particular physical location within its arena. If a place cell is thought of as either “active” or “silent,” then one may represent the co-firing patterns of place cells by a combinatorial neural code: a set of 0/1 vectors whose coordinates represent that status of distinct place cells. From the code, we can try to reconstruct a geometric picture of the neural activity by sketching a disjoint union of simple closed curves in the plane. Ideally, each curve corresponds to a unique place cell and the interiors of the curves are convex. However, this is not always possible, and identifying criteria which makes this possible is a difficult problem.
In this talk, we will discuss approaches to the problem of representing combinatorial neural codes using convex sets. We will see how turning the codewords into polynomials can reveal hidden information about the code, and how this naturally leads to examining properties of related polyhedra. In particular, we will present progress on using polyhedra to identify representability of a code with circles in the plane.