Neural Ring Homomorphisms and Maps Between Neural Codes

Carina Pamela Curto, Nora Youngs

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Neural codes are binary codes that are used for information processing and representation in the brain. In previous work, we have shown how an algebraic structure, called the neural ring, can be used to efficiently encode geometric and combinatorial properties of a neural code (Curto et al., Bull Math Biol 75(9), 2013). In this work, we consider maps between neural codes and the associated homomorphisms of their neural rings. In order to ensure that these maps are meaningful and preserve relevant structure, we find that we need additional constraints on the ring homomorphisms. This motivates us to define neural ring homomorphisms. Our main results characterize all code maps corresponding to neural ring homomorphisms as compositions of five elementary code maps. As an application, we find that neural ring homomorphisms behave nicely with respect to convexity. In particular, if C and D are convex codes, the existence of a surjective code map C→ D with a corresponding neural ring homomorphism implies that the minimal embedding dimensions satisfy d(D) ≤ d(C).

Original languageEnglish (US)
Title of host publicationTopological Data Analysis - The Abel Symposium, 2018
EditorsNils A. Baas, Gereon Quick, Markus Szymik, Marius Thaule, Gunnar E. Carlsson
PublisherSpringer
Pages163-180
Number of pages18
ISBN (Print)9783030434076
DOIs
StatePublished - 2020
EventAbel Symposium, 2018 - Geiranger, Norway
Duration: Jun 4 2018Jun 8 2018

Publication series

NameAbel Symposia
Volume15
ISSN (Print)2193-2808
ISSN (Electronic)2197-8549

Conference

ConferenceAbel Symposium, 2018
CountryNorway
CityGeiranger
Period6/4/186/8/18

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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