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.

翻译：耦合振荡器系统为模拟多种物理及生物现象提供了基础框架。在神经科学中，中枢神经系统与相邻脑区表现出同步振荡活动，例如在睡眠期间产生行波动力学。类似地，在胃肠系统中，神经肌肉细胞协调其振荡以产生慢波活动的传播波。为估计多变量相位关系的概率分布，现有方法通常依赖于平衡热力学，通过成对指数族分布将系统表达为玻尔兹曼形式。然而，这些假设在现实系统中常被违背——这些系统本质上是动态的，并频繁在平衡与非平衡态之间转换。针对此问题，我们提出一种高效方法用于估计耦合振荡器的概率分布，该方法无需假设热力学平衡。通过基于朗之万动力学的构建方式，该方法即使在非平衡态下也能实现精确建模。在高采样率条件下（现代数据采集系统通常满足此条件），最大似然估计方法被证明具有闭合形式的代数解，这使得该方法易于实际应用。我们在模拟数据上验证了其鲁棒性——在非平衡场景中表现优于现有方法，并进一步展示了其在表征脑刺激响应中的动态脑行波、以及人体胃部电生理记录假设检验场景中的实用价值。