Time-frequency characterization of interdependencies in nonstationary signals: application to epileptic EEG

Abstract

For the past decades, numerous works have been dedicated to the development of signal processing methods aimed at measuring the degree of association between electroencephalographic (EEG) signals. This interdependency parameter, which may be defined in various ways, is often used to characterize a functional coupling between different brain structures or regions during either normal or pathological processes. In this paper, we focus on the time-frequency characterization of the interdependency between signals. Particularly, we propose a novel estimator of the linear relationship between nonstationary signals based on the cross correlation of narrow band filtered signals. This estimator is compared to a more classical estimator based on the coherence function. In a simulation framework, results show that it may exhibit better statistical performances (bias and variance or mean square error) when a priori knowledge about time delay between signals is available. On real data (intracerebral EEG signals), results show that this estimator may also enhance the readability of the time-frequency representation of relationship and, thus, can improve the interpretation of nonstationary interdependencies in EEG signals. Finally, we illustrate the importance of characterizing the relationship in both time and frequency domains by comparing with frequency-independent methods (linear and nonlinear).

Fig. 1

Bias and MSE of both estimators for model M1 (stationary situation, constant α[t]) for different duration N₀ of the sliding window a) Bias of |ρ̂[t, f]|². b) Bias of R̂²[t, f] with τ_M = −τ_m = 5. c) Bias of R̂²[t, f] with τ_m = τ_M = 0. d) MSE of |ρ̂[t, f]|². e) MSE of R̂²[t, f] with τ_M = −τ_m = 5. f) MSE of R̂²[t, f] with τ_m = τ_M = 0. These curves are obtained by Monte Carlo simulation for long duration signals and by averaging over time-frequency plane.

Fig. 2

Performances of both estimators for model M1 (nonstationary situation, time-varying α[t] for N₀ =768. a) Time course of parameter α[t] (dashed line) and target relationship curve (solid line) averaged over frequency axis. Standard boxplots obtained by Monte-Carlo simulation for different estimators: b) |ρ̂[t, f]|², c) R̂²[t, f] with τ_M = −τ_m = 5, d) R̂²[t, f] with τ_m = τ_M = 0.

Fig. 3

Model M2, nonstationary multi-component situation, a) Target time-frequency relationship obtained by Monte-Carlo simulation. Different estimations of target: b) |ρ̂[t, f]|², c) R̂²[t, f] with τ_m = τ_M = 0, d) R̂²[t, f] with τ_M = −τ_m = 5.

Fig. 4

Results obtained on real data, a) Two SEEG signals recorded from hippocampus (top) and amygdala (bottom) in an epileptic patient (TLE) and b) corresponding spectrograms, c) Estimated relationship in the time-frequency plane for both methods. (*) Time-frequency representations of R̂²[t, f] maximized for time delay τ (middle, range −20 to 20 ms) and for fixed τ (bottom, τ_m = τ_M = 4 ms). d) Estimated relationship by two frequency-independent methods, one linear (r²[t], top) and the other nonlinear (h²[t], bottom).