Year: 2017
Type: article
Abstract: Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets? In this work, we provide a complete characterization of local obstructions to convexity. This motivates us to define max intersection-complete codes, a family guaranteed to have no local obstructions. We then show how our characterization enables one to use free resolutions of Stanley--Reisner ideals in order to detect violations of convexity. Taken together, these results provide a significant advance in our understanding of the intrinsic combinatorial properties of convex codes.
DOI: 10.1137/16m1073170
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
Awards: RUI: Computational algebraic geometry and combinatorial algorithms for neuroscience and biological networks, Biochemical reaction systems: multistationarity, persistence, and identifiability, Theory of threshold-linear networks and combinatorial neural codes., Mathematics Research Communities, PostDoctoral Research Fellowship, PostDoctoral Research Fellowship, Biochemical reaction systems: multistationarity, persistence, and identifiability, Memory encoding in spatially structured networks: dynamics, discrete geometry & topology, PostDoctoral Research Fellowship, Memory encoding in spatially structured networks: dynamics, discrete geometry & topology
OpenAlex ID: W2240184727