Canard theory and excitability

Martin Wechselberger, John Mitry, John Rinzel

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

An important feature of many physiological systems is that they evolve on multiple scales. From a mathematical point of view, these systems are modeled as singular perturbation problems. It is the interplay of the dynamics on different temporal and spatial scales that creates complicated patterns and rhythms. Many important physiological functions are linked to time-dependent changes in the forcing which leads to nonautonomous behaviour of the cells under consideration. Transient dynamics observed in models of excitability are a prime example.Recent developments in canard theory have provided a new direction for understanding these transient dynamics. The key observation is that canards are still well defined in nonautonomous multiple scales dynamical systems, while equilibria of an autonomous system do, in general, not persist in the corresponding driven, nonautonomous system. Thus canards have the potential to significantly shape the nature of solutions in nonautonomous multiple scales systems. In the context of neuronal excitability, we identify canards of folded saddle type as firing threshold manifolds. It is remarkable that dynamic information such as the temporal evolution of an external drive is encoded in the location of an invariant manifold - the canard.

Original languageEnglish (US)
Title of host publicationNonautonomous Dynamical Systems in the Life Sciences
PublisherSpringer Verlag
Pages89-132
Number of pages44
ISBN (Print)9783319030791
DOIs
StatePublished - 2013

Publication series

NameLecture Notes in Mathematics
Volume2102
ISSN (Print)0075-8434

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Keywords

  • Canards
  • Excitability
  • Firing threshold manifold
  • Geometric singular perturbation theory
  • Neural dynamics
  • Separatrix
  • Transient attractor

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Wechselberger, M., Mitry, J., & Rinzel, J. (2013). Canard theory and excitability. In Nonautonomous Dynamical Systems in the Life Sciences (pp. 89-132). (Lecture Notes in Mathematics; Vol. 2102). Springer Verlag. https://doi.org/10.1007/978-3-319-03080-7-3