Georgia Institute of Technology College of Sciences

A combinatorial neural code is convex if it arises as the intersection pattern of convex open subsets of Euclidean space. We relate the emerging theory of convex neural codes to the established theory of oriented matroids, both categorically and with respect to feasibility and complexity. By way of this connection, we prove that all convex codes are related to some representable oriented matroid, and we show that deciding whether a neural code is convex is NP-hard.