Georgia Institute of Technology College of Sciences

Neural codes are inspired by John O'Keefe's discovery of the place cell, a neuron in the mammalian brain which fires if and only if its owner is in a particular region of physical space. Mathematically, a neural code $C$ on n neurons is a collection of subsets of $\{1,...,n\}$, with the subsets called codewords. The implication is that $C$ encodes how the members of some collection $\{U_i\}_{i=1}^n$ of subsets of $\mathbb{R}^d$ intersect one another. 

The principal question driving the study of neural codes is that of convexity. Given just the codewords of $C$, can we determine if there is a collection of open convex subsets $ \{U_i\}_{i=1}^n$ of some $\mathbb{R}^d$ for which $C$ is the code? A convex code is a code for which there is such a realization of open convex sets. While the question of determining which codes are convex remains open, there has been significant progress as many large families of codes can now be ruled as convex or nonconvex. In this talk, I will give an overview of some of the results from this work. In particular, I will focus on a phenomenon called a local obstruction, which if found in a code forbids convexity.