Phase oscillators are a common starting point for the reduced description of many single neuron models that exhibit a strongly attracting limit cycle. The framework for analysing such models in response to weak perturbations is now particularly well advanced, and has allowed for the development of a theory of weakly connected neural networks. However, the strong-attraction assumption may well not be the natural one for many neural oscillator models. For example, the popular conductance based Morris-Lecar model is known to respond to periodic pulsatile stimulation in a chaotic fashion that cannot be adequately described with a phase reduction. In this paper, we generalise the phase description that allows one to track the evolution of distance from the cycle as well as phase on cycle. We use a classical technique from the theory of ordinary differential equations that makes use of a moving coordinate system to analyse periodic orbits. The subsequent phase-amplitude description is shown to be very well suited to understanding the response of the oscillator to external stimuli (which are not necessarily weak). We consider a number of examples of neural oscillator models, ranging from planar through to high dimensional models, to illustrate the effectiveness of this approach in providing an improvement over the standard phase-reduction technique. As an explicit application of this phase-amplitude framework, we consider in some detail the response of a generic planar model where the strong-attraction assumption does not hold, and examine the response of the system to periodic pulsatile forcing. In addition, we explore how the presence of dynamical shear can lead to a chaotic response.
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/610255 |
| Date | January 2013 |
| Creators | Wedgwood, Kyle C. A., Lin, Kevin K., Thul, Ruediger, Coombes, Stephen |
| Contributors | School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK, Department of Applied Mathematics, University of Arizona, Tucson, AZ, USA |
| Publisher | BioMed Central |
| Source Sets | University of Arizona |
| Language | English |
| Detected Language | English |
| Type | Article |
| Rights | © 2013 K.C.A. Wedgwood et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0) |
| Relation | http://www.mathematical-neuroscience.com/content/3/1/2 |
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