On open and closed convex codes

Neural codes serve as a language for neurons in the brain. Open (or closed) convex codes, which arise from the pattern of intersections of collections of open (or closed) convex sets in Euclidean space, are of particular relevance to neuroscience. Not every code is open or closed convex, however, and the combinatorial properties of a code that determine its realization by such sets are still poorly understood. Here we find that a code that can be realized by a collection of open convex sets may or may not be realizable by closed convex sets, and vice versa, establishing that open convex and closed convex codes are distinct classes. We establish a non-degeneracy condition that guarantees that the corresponding code is both open convex and closed convex. We also prove that max intersection-complete codes (i.e. codes that contain all intersections of maximal codewords) are both open convex and closed convex, and provide an upper bound for their minimal embedding dimension. Finally, we show that the addition of non-maximal codewords to an open convex code preserves convexity.