Stochastic phase reduction for brain oscillations

From the periodic spiking of single neurons to brain rhythms, oscillations are a ubiquitous neural phenomenon. Phase reduction is a powerful way of analysing deterministic systems exhibiting such oscillatory behavior: the N-dimensional state of the system is encoded into a one dimensional phase variable via projection onto the closed limit cycle attractor. Such a reduction gives rise to one-dimensional models that have been used to model a variety of single cell (e.g. theta-neuron) and network (e.g. Kuramoto oscillator networks) dynamics. However, real-world oscillations are often noisy and better modelled using stochastic differential equations (SDEs). In this case, the deterministic phase reduction approach is ill-defined. Hence, finding an analogous method for stochastic oscillators is an open question, especially in the case where the oscillations are fluctuation-driven.

In this contribution, we will present a novel stochastic phase reduction framework, designed with the deterministic approach in mind [1]. It relies on the spectral decomposition of the Kolmogorov backwards operator (aka the generator of the stochastic Koopman operator). We note that this object is the source of much current interest due to the development of modern data analysis methods aiming to reconstruct it from time series, such as Dynamic Mode Decomposition (DMD) [2].

We first review the phase reduction approach for deterministic systems, and show its limitations when noise is present. We then introduce our phase reduction framework. We then show how it can be applied to systems relevant to modelling in neurobiology. Notably, we will show how a 2-dimensional Wilson-Cowan and a 4-dimensional stochastic Hodgkin-Huxley model can be reduced to a phase description.