Modeling, Segmenting and Statistics of Transient Spindles via Two-Dimensional Ornstein-Uhlenbeck Dynamics

TL;DR

This paper introduces a stochastic modeling framework using two-dimensional Ornstein-Uhlenbeck processes to analyze, segment, and statistically characterize transient spindle-like oscillations in EEG signals, capturing their morphological features and temporal dynamics.

Contribution

It presents a novel low-dimensional stochastic dynamical system model for EEG spindles and a segmentation method combining Empirical Mode Decomposition with extremum detection.

Findings

Empirical laws for amplitude, interval, and duration distributions of spindles.

Exponential tail behavior consistent with OU process dynamics.

Extension to coupled OU processes models mixed spindle types.

Abstract

We develop here a stochastic framework for modeling and segmenting transient spindle-like oscillatory bursts in electroencephalogram (EEG) signals. At the modeling level, individual spindles are represented as path realizations of a two-dimensional Ornstein{Uhlenbeck (OU) process with a stable focus, providing a low-dimensional stochastic dynamical system whose trajectories reproduce key morphological features of spindles, including their characteristic rise{decay amplitude envelopes. On the signal processing side, we propose a segmentation procedure based on Empirical Mode Decomposition (EMD) combined with the detection of a central extremum, which isolates single spindle events and yields a collection of oscillatory atoms. This construction enables a systematic statistical analysis of spindle features: we derive empirical laws for the distributions of amplitudes, inter-spindle intervals, and rise/decay durations, and show that these exhibit exponential tails consistent with the underlying OU dynamics. We further extend the model to a pair of weakly coupled OU processes with distinct natural frequencies, generating a stochastic mixture of slow, fast, and mixed spindles in random temporal order. The resulting framework provides a data-driven framework for the analysis of transient oscillations in EEG and, more generally, in nonstationary time series.