A Robust Statistical Framework for Instantaneous Electroencephalogram Phase and Frequency Analysis

In recent decades, instantaneous phase (IP) and instantaneous frequency (IF) of the electroencephalogram (EEG) has received great attention as a notable complement for conventional EEG spectral analysis. The calculation of these parameters commonly includes narrow-bandpass filtering followed by the calculation of the analytical form of the signal using the Hilbert transform. In this research, using this widely accepted phase calculation approach and well-established methods from statistical signal processing, a stochastic model is proposed for the superposition of narrow-band foreground and background EEG activities. Using this model, the probability density functions of the instantaneous envelope (IE) and IP of EEG signals are derived analytically. It is rigorously shown that the IP estimation quality highly depends on the IE and any phase of frequency interpretations in low IE are unreliable for phase analysis. Based on these findings, a Monte Carlo estimation scheme is proposed for the accurate estimation and smoothing of phase related parameters. The impact of this approach on previous studies including time-domain phase synchrony, phase resetting, phase locking value and phase amplitude coupling are studied with examples.