Inference in alpha rhythm phase and amplitude modeled on Markov random field using belief propagation from electroencephalograms.

Alpha rhythm is a major component of spontaneous electroencephalographic (EEG) data. We develop a novel method that can be used to estimate the instantaneous phases and amplitudes of the alpha rhythm with high accuracy by modeling the alpha rhythm phase and amplitude as Markov random field (MRF) models. By using a belief propagation technique, we construct an exact-inference algorithm that can be used to estimate instantaneous phases and amplitudes and calculate the marginal likelihood. Maximizing the marginal likelihood enables us to estimate the hyperparameters on the basis of type-II maximum likelihood estimation. We prove that the instantaneous phase and amplitude estimation by our method is consistent with that by the Hilbert transform, which has been commonly used to estimate instantaneous phases and amplitudes, of a signal filtered from observed data in the limited case that the observed data consist of only one frequency signal whose amplitude is constant and a Gaussian noise. Comparison of the performances of observation noise reduction by our method and by a Gaussian MRF model of alpha rhythm signal indicates that our method reduces observation noise more efficiently. Moreover, the instantaneous phase and amplitude estimates obtained using our method are more accurate than those obtained by the Hilbert transform. Application of our method to experimental EEG data also demonstrates that the relationship between the alpha rhythm phase and the reaction time emerges more clearly by using our method than the Hilbert transform. This indicates our method's practical usefulness. Therefore, applying our method to experimental EEG data will enable us to estimate the instantaneous phases and amplitudes of the alpha rhythm more precisely.