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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


Kevin K. Lin

Kyle C.A. Wedgwood

Lai-Sang Young


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.


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),

Journal Article Type Article
Publication Date Jan 1, 2013
Deposit Date May 28, 2014
Publicly Available Date May 28, 2014
Journal Journal of Mathematical Biology
Print ISSN 0303-6812
Electronic ISSN 0303-6812
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 66
Issue 1-2
Keywords Oscillators; Perturbation theory; Phase response curve; Neuron models; Shear-induced chaos
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