Limitations of perturbative techniques in the analysis of rhythms and oscillations
Lin, Kevin K. and Wedgwood, Kyle C.A. and Coombes, Stephen and Young, Lai-Sang (2013) Limitations of perturbative techniques in the analysis of rhythms and oscillations. Journal of Mathematical Biology, 66 (1-2). pp. 139-161. ISSN 0303-6812
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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