### 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 language | English (US) |
---|---|

Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Journal of Mathematical Neuroscience |

Volume | 3 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Neuroscience (miscellaneous)

### Cite this

*Journal of Mathematical Neuroscience*,

*3*(1), 1-22. https://doi.org/10.1186/2190-8567-3-2

**Phase-amplitude descriptions of neural oscillator models.** / Wedgwood, K. C A; Lin, Kevin; Thul, R.; Coombes, S.

Research output: Contribution to journal › Article

*Journal of Mathematical Neuroscience*, vol. 3, no. 1, pp. 1-22. https://doi.org/10.1186/2190-8567-3-2

}

TY - JOUR

T1 - Phase-amplitude descriptions of neural oscillator models

AU - Wedgwood, K. C A

AU - Lin, Kevin

AU - Thul, R.

AU - Coombes, S.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Chaos

KW - Non-weak coupling

KW - Oscillator

KW - Phase-amplitude

UR - http://www.scopus.com/inward/record.url?scp=84881362110&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84881362110&partnerID=8YFLogxK

U2 - 10.1186/2190-8567-3-2

DO - 10.1186/2190-8567-3-2

M3 - Article

C2 - 23347723

AN - SCOPUS:84881362110

VL - 3

SP - 1

EP - 22

JO - Journal of Mathematical Neuroscience

JF - Journal of Mathematical Neuroscience

SN - 2190-8567

IS - 1

ER -