TY - GEN

T1 - Neural Ring Homomorphisms and Maps Between Neural Codes

AU - Curto, Carina Pamela

AU - Youngs, Nora

PY - 2020

Y1 - 2020

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=85087791248&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85087791248&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-43408-3_7

DO - 10.1007/978-3-030-43408-3_7

M3 - Conference contribution

AN - SCOPUS:85087791248

SN - 9783030434076

T3 - Abel Symposia

SP - 163

EP - 180

BT - Topological Data Analysis - The Abel Symposium, 2018

A2 - Baas, Nils A.

A2 - Quick, Gereon

A2 - Szymik, Markus

A2 - Thaule, Marius

A2 - Carlsson, Gunnar E.

PB - Springer

T2 - Abel Symposium, 2018

Y2 - 4 June 2018 through 8 June 2018

ER -