Numerical Twin with Two Dimensional Ornstein--Uhlenbeck Processes of Transient Oscillations in EEG signal

TL;DR

The paper addresses the problem of modeling transient, burst-like EEG oscillations by introducing a numerical twin based on a two-dimensional Ornstein–Uhlenbeck process with parameters $\lambda$, $\omega$, and $\sigma$ that jointly shape decay, resonance, and noise. It develops two complementary estimators: a global method leveraging the PSD, amplitude distribution, and autocorrelation, and an event-wise approach matching segmented spindle statistics to OU simulations (with a discrete-time likelihood underpinning the fit). The framework extends to multiple frequency bands and piecewise-stationary dynamics, enabling real-time tracking of slow parameter drifts. Applied to EEG under general anesthesia, the OU-based decomposition reproduces alpha-spindle morphology and band-limited spectra with low residual error, yielding interpretable metrics (spindle counts, durations, amplitudes, and parameter trajectories) for brain-state monitoring and potential control.

Abstract

Stochastic burst-like oscillations are common in physiological signals, yet there are few compact generative models that capture their transient structure. We propose a numerical-twin framework that represents transient narrowband activity as a two-dimensional Ornstein-Uhlenbeck (OU) process with three interpretable parameters: decay rate, mean frequency, and noise amplitude. We develop two complementary estimation strategies. The first fits the power spectral density, amplitude distribution, and autocorrelation to recover OU-parameters. The second segments burst events and performs a statistical match between empirical spindle statistics (duration, amplitude, inter-event interval) and simulated OU output via grid search, resolving parameter degeneracies by including event counts. We extend the framework to multiple frequency bands and piecewise-stationary dynamics to track slow parameter drifts. Applied to electroencephalography (EEG) recorded during general anesthesia, the method identifies OU models that reproduce alpha-spindle (8-12 Hz) morphology and band-limited spectra with low residual error, enabling real-time tracking of state changes that are not apparent from band power alone. This decomposition yields a sparse, interpretable representation of transient oscillations and provides interpretable metrics for brain monitoring.

Figures (6)

• Figure 1: $\alpha$ spindles dynamics during anesthesia, sleep and generated by simulations of two-dimensional Ornstein-Uhlenbeck processes. (A) General Anesthesia: from top to bottom: EEG, Spectrogram, Power Spectral Density, Filtered $\alpha$ wave, Spectrogram of the filtered EEG, showing the time-frequency spindle localization. Time-frequency representation (spectrogram) uses a 1 second window and 90% overlap. (B) Same as in (A) for a sleep EEG. (C) Same as in (A) for a signal generated by an OU-process with parameters $r=1$, $\sigma=8$, $\omega=60$ that mimics the $\alpha$ wave signal of general anesthesia.
• Figure 2: Parameter estimation methods from a realization of a OU-process. (A) Example of a signal generated by the Ornstein–Uhlenbeck (OU) model with parameters $\lambda$, $\omega$, and $\sigma$. (B) Estimator $\hat{\lambda}$ of the noise-to-decay ratio $\frac{\sigma}{\lambda}$ obtained via Gaussian fitting of the amplitude distribution. (C) Estimators of the decay coefficient $\lambda$ and noise amplitude $\sigma$ using the autocorrelation function, based on the result from (B). (D) Estimator of the frequency parameter $\omega$ using the mean instantaneous frequency computed via the Hilbert transform and averaged over all segmented spindles.
• Figure 3: Spectral decomposition of EEG in $\alpha-$ and $\delta-$ and residual componentsA. EEG and spectrogram over several seconds (scal bar 5s), showing $\alpha-$ and $\delta-$ spindles. B.$\alpha-$ and $\delta-$ filtered components. C. Power spectral decomposition (PSD). We fitted the PSD in intermediate frequency bands $[1.8,4]$ and $[6.5,8.1]$ Hz with $\log PSD= a \log (f)+b$. D. Statistical result of (a,b) coefficients for the two bands and Lorentzian parameters $\frac{A \sigma}{\pi}\frac{1}{\sigma^2+(f-f_c)^2}$, computed over n=18 EEGs.
• Figure 4: Parameter estimation of the stochastic OU- model to EEG spindles via grid search.(A) Parameter estimation pipeline. A grid search is performed over the parameter space of the two-dimensional Ornstein–Uhlenbeck process ($\lambda$, $\sigma$), while fixing the resonance frequency $f_c = \omega/2\pi$. (B) Grid search minimizing a loss function based on spindle statistics with (duration, amplitude) computed from segmented EEG recordings, this leads to a line of minimum with slope $\lambda/\sigma = 0.082$. However, when considering the spindle statistics with (number, duration, and amplitude), the optimal parameters are $\lambda^* = 1.84$, $\sigma^* = 35.59$, and $f_c^* = 9.63$ Hz. (C) EEG signal (top) and stochastic simulations of the OU process (bottom) using the optimal parameters. (D) Spindle segmentation applied to both real EEG and simulated signals. Events are identified and classified based on amplitude and duration thresholds. (E) Comparison of spindle statistics (number, amplitude, and duration) between EEG and simulations.
• Figure 5: Non-stationary Ornstein–Uhlenbeck model.(A) Dynamics parameters $\lambda$ (decay rate), $\omega$ (rotation frequency), and $\sigma$ (noise amplitude) of the two-dimensional Ornstein–Uhlenbeck (OU) model over time. The parameters are estimated sequentially from EEG recordings filtered in the $\alpha$-band. (B) Piecewise approximation of the parameter trajectories using constant segments, separated by detected transition points (black vertical lines). This segmentation captures long-range trends and identifies parameter regimes with relatively stable dynamical behavior. (C) Time-frequency representation (spectrogram) of the EEG signal, showing spindle activity evolving over the course of the recording. (D) Raw signal (gray) and smoothed EEG (blue) used for parameter extraction. Together, these panels illustrate the feasibility of modeling EEG dynamics with a time-varying stochastic focus model (or equivalently linear-Gaussian continuous-time oscillator), providing a numerical twin of the patient’s brain state during anesthesia or sleep.