Limitations of perturbative techniques in the analysis of rhythms and oscillations
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,...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Published: |
Springer
2013
|
| Subjects: | |
| Online Access: | https://eprints.nottingham.ac.uk/3202/ |
| _version_ | 1848790976252870656 |
|---|---|
| author | Lin, Kevin K. Wedgwood, Kyle C.A. Coombes, Stephen Young, Lai-Sang |
| author_facet | Lin, Kevin K. Wedgwood, Kyle C.A. Coombes, Stephen Young, Lai-Sang |
| author_sort | Lin, Kevin K. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T18:21:10Z |
| format | Article |
| id | nottingham-3202 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T18:21:10Z |
| publishDate | 2013 |
| publisher | Springer |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-32022020-05-04T20:19:42Z https://eprints.nottingham.ac.uk/3202/ Limitations of perturbative techniques in the analysis of rhythms and oscillations Lin, Kevin K. Wedgwood, Kyle C.A. Coombes, Stephen Young, Lai-Sang 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. Springer 2013-01 Article PeerReviewed Lin, Kevin K., Wedgwood, Kyle C.A., 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 Oscillators; Perturbation theory; Phase response curve; Neuron models; Shear-induced chaos http://link.springer.com/article/10.1007%2Fs00285-012-0506-0 doi:10.1007/s00285-012-0506-0 doi:10.1007/s00285-012-0506-0 |
| spellingShingle | Oscillators; Perturbation theory; Phase response curve; Neuron models; Shear-induced chaos Lin, Kevin K. Wedgwood, Kyle C.A. Coombes, Stephen Young, Lai-Sang Limitations of perturbative techniques in the analysis of rhythms and oscillations |
| title | Limitations of perturbative techniques in the analysis of rhythms and oscillations |
| title_full | Limitations of perturbative techniques in the analysis of rhythms and oscillations |
| title_fullStr | Limitations of perturbative techniques in the analysis of rhythms and oscillations |
| title_full_unstemmed | Limitations of perturbative techniques in the analysis of rhythms and oscillations |
| title_short | Limitations of perturbative techniques in the analysis of rhythms and oscillations |
| title_sort | limitations of perturbative techniques in the analysis of rhythms and oscillations |
| topic | Oscillators; Perturbation theory; Phase response curve; Neuron models; Shear-induced chaos |
| url | https://eprints.nottingham.ac.uk/3202/ https://eprints.nottingham.ac.uk/3202/ https://eprints.nottingham.ac.uk/3202/ |