Dynamical Bayesian Inference of Time-evolving Interactions: From a Pair of Coupled Oscillators to Networks of Oscillators

Living systems have time-evolving interactions that, until recently, could not be identified accurately from recorded time series in the presence of noise. Stankovski et al. (Phys. Rev. Lett. 109 024101, 2012) introduced a method based on dynamical Bayesian inference that facilitates the simultaneous detection of time-varying synchronization, directionality of influence, and coupling functions. It can distinguish unsynchronized dynamics from noise-induced phase slips. The method is based on phase dynamics, with Bayesian inference of the time- evolving parameters being achieved by shaping the prior densities to incorporate knowledge of previous samples. We now present the method in detail using numerically-generated data, data from an analog electronic circuit, and cardio-respiratory data. We also generalize the method to encompass networks of interacting oscillators and thus demonstrate its applicability to small-scale networks.

💡 Research Summary

The paper addresses the longstanding challenge of identifying time‑varying interactions in noisy oscillatory systems. Building on the dynamical Bayesian inference framework introduced by Stankovski et al. (2012), the authors develop a comprehensive method that simultaneously detects (i) the presence or absence of phase synchronization, (ii) the directionality of influence between oscillators, and (iii) the full functional form of the coupling. The approach starts from a phase‑reduced description of each oscillator, where the phase dynamics are expressed as a sum of a natural frequency term and a coupling function that depends on the phases of the interacting units. The coupling functions are expanded in a Fourier series, allowing arbitrary nonlinear dependencies to be captured with a finite set of coefficients.

Inference proceeds in a sliding‑window fashion. Within each window, the observed phase time series are used to compute a posterior distribution over the Fourier coefficients and natural frequencies. Crucially, the posterior from the previous window becomes the prior for the next one, with a modest diffusion term added to the prior covariance to permit gradual parameter drift. This “temporal continuity” prior prevents over‑fitting to noise while still tracking genuine changes in the dynamics. Because the posterior remains Gaussian (thanks to linearization of the likelihood in the coefficients), the update equations are analytically tractable and resemble a Kalman‑filter but are applied to a nonlinear phase model rather than a linear state‑space model.

Synchronization is assessed by examining whether the inferred coupling functions generate stable fixed points in the phase‑difference dynamics. If a stable fixed point exists, the oscillators are deemed synchronized; otherwise they are unsynchronized, and any observed phase slips are attributed to noise rather than a loss of coupling. Directionality is quantified by comparing the magnitude and shape of the inferred coupling functions for each direction, providing a physically meaningful measure of who drives whom.

The methodology is validated on three distinct data sets. First, synthetic data generated from two coupled nonlinear oscillators with known, slowly varying coupling strengths demonstrate that the algorithm accurately tracks the prescribed parameter trajectories and correctly distinguishes true desynchronization from noise‑induced slips. Second, an analog electronic circuit consisting of two voltage‑controlled oscillators coupled through a variable resistor is used as a hardware testbed. Measured voltages are converted to phases, and the inference recovers the time‑varying coupling function in close agreement with the actual circuit parameters, even in the presence of substantial electronic noise. Third, physiological recordings of human heart‑rate (RR intervals) and respiration are analyzed. The method reveals distinct coupling patterns during sleep versus wakefulness, identifies subtle non‑stationary interactions that conventional linear coherence methods miss, and provides a quantitative estimate of the causal influence of respiration on heart‑rate variability.

To demonstrate scalability, the authors extend the framework to networks of N oscillators. The coupling functions become multivariate Fourier series, and the prior covariance is structured to enforce sparsity, dramatically reducing computational load for loosely connected networks. Simulations with 5–10 oscillators show that the algorithm can correctly infer both the existence and strength of each link, as well as capture temporal re‑wiring events.

The paper discusses several practical considerations. Accurate phase extraction from raw signals is a prerequisite; errors here propagate into the Bayesian updates. The choice of prior covariance and diffusion rate influences convergence speed and sensitivity to rapid parameter changes. While the method scales polynomially with network size, very large networks (hundreds of nodes) still pose computational challenges, suggesting future work on dimensionality reduction or parallel implementations.

In summary, this work delivers a robust, data‑driven tool for uncovering dynamic, nonlinear interactions in noisy oscillatory systems. By jointly estimating time‑dependent coupling functions, synchronization states, and causal directionality, it opens new avenues for studying complex biological rhythms, engineered coupled devices, and any system where interactions evolve on the same timescale as the observable dynamics.