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Nonautonomous Dynamical Systems in the Life Sciences pp 89–132Cite as

Canard Theory and Excitability

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Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 2102))

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.

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Acknowledgements

This work was supported in part by NIH grant DC008543-01 (JR) and by the ARC grant FT120100309 (MW). MW would like to thank the organisers of the Inzell workshop for their hospitality and financial support.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sydney, Sydney, Australia

    Martin Wechselberger & John Mitry

  2. Courant Institute, New York University, New York, USA

    John Rinzel

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  1. Martin Wechselberger
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Correspondence to Martin Wechselberger .

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Editors and Affiliations

  1. Institut für Mathematik, Goethe-Universität Frankfurt, Frankfurt am Main, Germany

    Peter E. Kloeden

  2. Institut für Mathematik, Alpen-Adria Universität Klagenfurt, Klagenfurt am Wörthersee, Austria

    Christian Pötzsche

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Wechselberger, M., Mitry, J., Rinzel, J. (2013). Canard Theory and Excitability. In: Kloeden, P., Pötzsche, C. (eds) Nonautonomous Dynamical Systems in the Life Sciences. Lecture Notes in Mathematics(), vol 2102. Springer, Cham. https://doi.org/10.1007/978-3-319-03080-7_3

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