Limitations of perturbative techniques in the analysis of rhythms and oscillations

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

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.

Citation

Lin, K. K., Wedgwood, K. C., Coombes, S., & Young, L.-S. (2013). Limitations of perturbative techniques in the analysis of rhythms and oscillations. Journal of Mathematical Biology, 66(1-2), https://doi.org/10.1007/s00285-012-0506-0

• 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: 1432-1416
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 66
• Issue: 1-2
• DOI: https://doi.org/10.1007/s00285-012-0506-0
• Keywords: Oscillators; Perturbation theory; Phase response curve; Neuron models; Shear-induced chaos
• Public URL: https://nottingham-repository.worktribe.com/output/1003043
• Publisher URL: http://link.springer.com/article/10.1007%2Fs00285-012-0506-0