What makes a neural code convex?

Carina Curto, Elizabeth Gross, Jack Jeffries, Katherine Morrison, Mohamed Omar, Zvi Rosen, Anne Shiu, Nora Youngs

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)222-238
Number of pages17
JournalSIAM Journal on Applied Algebra and Geometry
Volume1
Issue number1
DOIs
StatePublished - 2017

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Geometry and Topology
  • Applied Mathematics

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