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Limitations of perturbative techniques in the analysis of rhythms and oscillations

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

Authors

Kevin K. Lin

Kyle C.A. Wedgwood pmxkw2@nottingham.ac.uk

Lai-Sang Young



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.

Journal Article Type Article
Publication Date Jan 1, 2013
Journal Journal of Mathematical Biology
Print ISSN 0303-6812
Electronic ISSN 0303-6812
Publisher Humana Press
Peer Reviewed Peer Reviewed
Volume 66
Issue 1-2
APA6 Citation Lin, K. K., Wedgwood, K. C., Coombes, S., & Young, L. (2013). Limitations of perturbative techniques in the analysis of rhythms and oscillations. Journal of Mathematical Biology, 66(1-2), doi:10.1007/s00285-012-0506-0
DOI https://doi.org/10.1007/s00285-012-0506-0
Keywords Oscillators; Perturbation theory; Phase response curve; Neuron models; Shear-induced chaos
Publisher URL http://link.springer.com/article/10.1007%2Fs00285-012-0506-0
Copyright Statement Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by/4.0

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by/4.0





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