TY - JOUR
T1 - What makes a neural code convex?
AU - Curto, Carina
AU - Gross, Elizabeth
AU - Jeffries, Jack
AU - Morrison, Katherine
AU - Omar, Mohamed
AU - Rosen, Zvi
AU - Shiu, Anne
AU - Youngs, Nora
N1 - Funding Information:
∗Received by the editors May 3, 2016; accepted for publication (in revised form) December 21, 2016; published electronically March 28, 2017. http://www.siam.org/journals/siaga/1/M107317.html Funding: This work began at a 2014 AMS Mathematics Research Community, “Algebraic and Geometric Methods in Applied Discrete Mathematics,” which was supported by NSF DMS-1321794. CC was supported by NSF DMS-1225666/1537228, NSF DMS-1516881, and an Alfred P. Sloan Research Fellowship; EG was supported by NSF DMS-1304167 and NSF DMS-1620109; JJ was supported by NSF DMS-1606353; and AS was supported by NSF DMS-1004380 and NSF DMS-1312473/1513364. †Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 (ccurto@psu.edu). ‡Department of Mathematics, San JoséState University, San José, CA 95192 (elizabeth.gross@sjsu.edu). §Department of Mathematics, University of Utah, Salt Lake City, UT 84112. Current address: Department of Mathematics, University of Michigan, Ann Arbor, MI 48109 (jeffries45@gmail.com). ¶Department of Mathematics, The Pennsylvania State University, University Park, PA 16802. Current address: School of Mathematical Sciences, University of Northern Colorado, Greeley, CO 80639 (kmorris2@gmail.com). ‖Department of Mathematics, Harvey Mudd College, Claremont, CA 91711 (omar@g.hmc.edu). #Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, and Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720. Current address: Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104 (zvihr@sas.upenn.edu). ††Department of Mathematics, Texas A&M University, College Station, TX 77843 (annejls@math.tamu.edu). ‡‡Department of Mathematics, Harvey Mudd College, Claremont, CA 91711. Current address: Department of Mathematics, Colby College, Waterville, ME 04901 (nyoungs@g.hmc.edu).
Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics.
PY - 2017
Y1 - 2017
N2 - 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.
AB - 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.
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U2 - 10.1137/16M1073170
DO - 10.1137/16M1073170
M3 - Article
AN - SCOPUS:85034627464
VL - 1
SP - 222
EP - 238
JO - SIAM Journal on Applied Algebra and Geometry
JF - SIAM Journal on Applied Algebra and Geometry
SN - 2470-6566
IS - 1
ER -