A Kuramoto-von Mises Time Series Model for Probabilistic Modeling of Coupled Oscillators

Abstract

A system of coupled oscillators provides a fundamental framework for modeling a wide range of physical and biological phenomena. In neuroscience, the central nervous system exhibits synchronized oscillatory activity with adjacent brain regions, giving rise to traveling wave dynamics for instance during sleep. Similarly, in the gastrointestinal system, neuromuscular cells coordinate their oscillations to generate propagating waves of slow wave activity. To estimate probability distributions of multivariate phase relationships, existing approaches typically rely on equilibrium thermodynamics, expressing the system in a Boltzmann form through a pairwise exponential family distribution. However, these assumptions are often violated in real-world systems, which are inherently dynamic and frequently transition between equilibrium and non-equilibrium regimes. To address this, we propose an efficient method for estimating the probability distribution of coupled oscillators that does not assume thermodynamic equilibrium. Using a Langevin dynamics-based construction, the approach enables accurate modeling even in non-equilibrium regimes. The maximum likelihood estimation method is shown to have a closed form algebraic solution in the high sampling rate regime, a condition commonly satisfied by modern data acquisition systems, which makes it readily applicable in practice. We demonstrate its robustness on simulated data, where it outperforms existing approaches in non-equilibrium settings, and further illustrate its utility for characterizing dynamic brain traveling waves in response to brain stimulation and in hypothesis testing within the context of electrophysiologic recordings of the human stomach.