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Given an intersection pattern of arbitrary sets in Euclidean space, is therean arrangement of convex open sets in Euclidean space that exhibits the sameintersections? This question is combinatorial and topological in nature, but ismotivated by neuroscience. Specifically, we are interested in a type of neuroncalled a place cell, which fires precisely when an organism is in a certainregion, usually convex, called a place field. The earlier question, therefore,can be rephrased as follows: Which neural codes, that is, patterns of neuralactivity, can arise from a collection of convex open sets? To address thisquestion, Giusti and Itskov proved that convex neural codes have no "localobstructions," which are defined via the topology of a code's simplicialcomplex. Codes without local obstructions are called locally good, because theobstruction precludes the code from encoding the intersections of open setsthat form a good cover. In other words, every good-cover code is locally good.Here we prove the converse: Every locally good code is a good-cover code. Wealso prove that the good-cover decision problem is undecidable. Finally, wereveal a stronger type of local obstruction that prevents a code from beingconvex, and prove that the corresponding decision problem is NP-hard. Ourproofs use combinatorial and topological methods.

Neural codes, decidability, and a new local obstruction to convexity