TY - GEN
T1 - Neural Ring Homomorphisms and Maps Between Neural Codes
AU - Curto, Carina Pamela
AU - Youngs, Nora
N1 - Publisher Copyright:
© 2020, Springer Nature Switzerland AG.
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
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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 -