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.
Highlights
One only has to look at the plethora of papers and books on the topic of phase oscillators in mathematical neuroscience to see the enormous impact that this tool from dynamical systems theory has had on the way we think about describing neurons and neural networks
It is more common to side-step the need for constructing global isochrons by restricting attention to a small neighbourhood of the limit cycle, where dynamics can be recast in the reduced form θ = 1, where θ is the phase around a cycle
Focusing on pulsatile forcing, we explore the conditions for neural oscillator models to exhibit shear induced chaos [16]
Summary
One only has to look at the plethora of papers and books on the topic of phase oscillators in mathematical neuroscience to see the enormous impact that this tool from dynamical systems theory has had on the way we think about describing neurons and neural networks. The iPRC at a point on cycle is equal to the gradient of the (isochronal) phase at that point This approach forms the basis for constructing models of weakly interacting oscillators, where the external forcing is pictured as a function of the phase of a firing neuron. This has led to a great deal of work on phase-locking and central pattern generation in neural circuitry (see, for example [13]). 3, we consider how inputs to the neuron are transformed under these coordinate transformations and derive the evolution equations for the forced phase-amplitude system This reduces to the standard phase description in the appropriate limit.
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