BIO

Osiano Isekenegbe is a senior undergraduate mathematics student. His current research interest concerns algebraic, combinatorial, and geometric approaches to mathematical neuroscience. Outside of mathematics, he enjoys learning French, playing pool, and reading science fiction and fantasy novels. He will pursue Ph.D. studies in mathematics and looks forward to a career as a researcher and educator.

Convexity of Four-Maximal Neural Codes

Abstract

A central challenge facing neuroscientists is deciphering neural code and understanding the relationship between stimuli and neural activity. A motivating example comes from hippocampal place cells observed in rats, humans, and other animals, which fire when the animal is in specific regions (place fields) of its environment. This firing helps the brain determine the animal's position. Interestingly, place fields are experimentally observed to be almost convex. To mathematically represent the stimuli-neural activity relationship, we use combinatorial neural codes, which record neural activity as a family of sets called codewords. Convex neural codes are those realizable by a family of convex open sets in Euclidean space. As such, convex codes represent a model of place cells and place fields. We desire simple convexity criteria because detecting convexity in arbitrary codes is otherwise NP-hard. Giusti and Itskov identified local obstructions to convexity and Johnston, Shiu, and Spinner showed that these obstructions characterize convexity for codes with at most 3 maximal codewords. Lienkaemper, Shiu, and Woodstock gave an example showing that this is not the case for codes with 4 maximal codewords. In this work, we further examine 4-maximal neural codes. Using combinatorial, geometric, and topological methods, we show that 4-maximal codes corresponding to certain simplicial complexes are convex if and only if they do not have local obstructions. We also give a hypothesis for characterizing the remaining 4-maximal codes and describe interesting results supporting this hypothesis.

Keywords: mathematics, neuroscience, biology, animal, environment