Vol. 12, No. 5, 2019

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Sparse neural codes and convexity

R. Amzi Jeffs, Mohamed Omar, Natchanon Suaysom, Aleina Wachtel and Nora Youngs

Vol. 12 (2019), No. 5, 737–754
Abstract

Determining how the brain stores information is one of the most pressing problems in neuroscience. In many instances, the collection of stimuli for a given neuron can be modeled by a convex set in d . Combinatorial objects known as neural codes can then be used to extract features of the space covered by these convex regions. We apply results from convex geometry to determine which neural codes can be realized by arrangements of open convex sets. We restrict our attention primarily to sparse codes in low dimensions. We find that intersection-completeness characterizes realizable 2-sparse codes, and show that any realizable 2-sparse code has embedding dimension at most 3. Furthermore, we prove that in 2 and 3 , realizations of 2-sparse codes using closed sets are equivalent to those with open sets, and this allows us to provide some preliminary results on distinguishing which 2-sparse codes have embedding dimension at most 2.

Keywords
neural code, sparse, convexity
Mathematical Subject Classification 2010
Primary: 05C62, 52A10, 92B20
Milestones
Received: 26 October 2017
Revised: 24 October 2018
Accepted: 5 December 2018
Published: 22 May 2019

Communicated by Ann N. Trenk
Authors
R. Amzi Jeffs
Department of Mathematics
University of Washington
Seattle, WA
United States
Mohamed Omar
Department of Mathematics
Harvey Mudd College
Claremont, CA
United States
Natchanon Suaysom
Harvey Mudd College
Claremont, CA
United States
Aleina Wachtel
Harvey Mudd College
Claremont, CA
United States
Nora Youngs
Department of Mathematics and Statistics
Colby College
Waterville, ME
United States