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Limitations of perturbative techniques in the analysis of rhythms and oscillations

Abstract

Perturbation theory is an important tool in the analysis of oscillators and their response to external stimuli. It is predicated on the assumption that the perturbations in question are “sufficiently weak”, an assumption that is not always valid when perturbative methods are applied. In this paper, we identify a number of concrete dynamical scenarios in which a standard perturbative technique, based on the infinitesimal phase response curve (PRC), is shown to give different predictions than the full model. Shear-induced chaos, i.e., chaotic behavior that results from the amplification of small perturbations by underlying shear, is missed entirely by the PRC. We show also that the presence of “sticky” phase–space structures tend to cause perturbative techniques to overestimate the frequencies and regularity of the oscillations. The phenomena we describe can all be observed in a simple 2D neuron model, which we choose for illustration as the PRC is widely used in mathematical neuroscience.

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Acknowledgments

We thank one of the anonymous referees for pointing out Şuvak and Demir (2011). KKL is supported in part by the US National Science Foundation (NSF) through grant DMS-0907927. KCAW and SC acknowledge support from the CMMB/MBI partnership for multiscale mathematical modelling in systems biology-United States Partnering Award; BB/G530484/1 Biotechnology and Biological Sciences Research Council (BBSRC). LSY is supported in part by NSF grant DMS-1101594.

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Correspondence to Kevin K. Lin.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Lin, K.K., Wedgwood, K.C.A., Coombes, S. et al. Limitations of perturbative techniques in the analysis of rhythms and oscillations. J. Math. Biol. 66, 139–161 (2013). https://doi.org/10.1007/s00285-012-0506-0

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Keywords

  • Oscillators
  • Perturbation theory
  • Phase response curve
  • Neuron models
  • Shear-induced chaos

Mathematics Subject Classification (2000)

  • 92B25
  • 37N25