Phase-amplitude descriptions of neural oscillator models

K. C A Wedgwood, Kevin Lin, R. Thul, S. Coombes

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29 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1-22
Number of pages22
JournalJournal of Mathematical Neuroscience
Volume3
Issue number1
DOIs
Publication statusPublished - 2013

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Keywords

  • Chaos
  • Non-weak coupling
  • Oscillator
  • Phase-amplitude

ASJC Scopus subject areas

  • Neuroscience (miscellaneous)

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