Limitations of perturbative techniques in the analysis of rhythms and oscillations

Kevin K. Lin, Kyle C.A. Wedgwood, Stephen Coombes, Lai Sang Young

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)139-161
Number of pages23
JournalJournal Of Mathematical Biology
Volume66
Issue number1-2
DOIs
StatePublished - Jan 1 2013

    Fingerprint

Keywords

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

ASJC Scopus subject areas

  • Modeling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics

Cite this