Multichannel electroencephalography (EEG) offers a non-invasive tool to explore spatio-temporal dynamics of brain activity. With EEG recordings consisting of multiple trials, traditional signal processing approaches that ignore inter-trial variability in the data may fail to accurately estimate the underlying spatio-temporal brain patterns. Moreover, precise characterization of such inter-trial variability per se can be of high scientific value in establishing the relationship between brain activity and behavior. In this paper, a statistical modeling framework is introduced for learning spatio-temporal decompositions of multiple-trial EEG data recorded under two contrasting experimental conditions. By modeling the variance of source signals as random variables varying across trials, the proposed two-stage hierarchical Bayesian model is able to capture intertrial amplitude variability in the data in a sparse way where a parsimonious representation of the data can be obtained. A variational Bayesian (VB) algorithm is developed for statistical inference of the hierarchical model. The efficacy of the proposed modeling framework is validated with the analysis of both synthetic and real EEG data. In the simulation study we show that even at low signal-to-noise ratios our approach is able to recover with high precision the underlying spatio-temporal patterns and the dynamics of source amplitude across trials; on two brain-computer interface (BCI) data sets we show that our VB algorithm can extract physiologically meaningful spatio-temporal patterns and make more accurate predictions than other two widely used algorithms: the common spatial patterns (CSP) algorithm and the Infomax algorithm for independent component analysis (ICA). The results demonstrate that our statistical modeling framework can serve as a powerful tool for extracting brain patterns, characterizing trial-to-trial brain dynamics, and decoding brain states by exploiting useful structures in the data. © 2011 Elsevier Inc. All rights reserved. Thanks to advances in data recording technologies, the past decades have witnessed widespread applications of multichannel electroencephalographs (EEG) in neuroscience studies to probe into the working mechanisms of the human brain (Makeig et al., 2002; Miltner et al., 1999; Ray and Cole, 1985) , as well as in clinical applications to monitor brain states (Rampil, 1998) and assist the diagnosis of neurological abnormalities (Cichocki et al., 2005) . In recent years, EEG has also been widely used in emerging fields such as neural engineering (Wolpaw et al., 2002) and neuromarketing (McClure et al., 2004) for decoding brain activity. Compared with microscopic recordings that measure the activities of only single neuron or a group of nearby neurons with a spatial scale of at most millimeters (Wilson and McNaughton, 1993) , multichannel EEG has the advantage of being able to map the macroscopic dynamics across the whole brain, albeit indirectly from the scalp, with a high temporal resolution of milliseconds. On the other hand, in order to gain insights into brain function for answering relevant scientific questions or for practical purposes, it is crucial to link the spatio-temporal dynamics of EEG to the underlying neurophysiological processes or behavioral changes. However, due to volume conduction, scalp EEG hardly preserves the fidelity of the original brain dynamics, which often renders its interpretation difficult (Baillet et al., 2001) . In particular, functionally distinct brain activities that may well be separated in the brain are mixed up in EEG in a simultaneous and linear manner, 1 leading to substantially distorted signals with high correlations between spatially adjacent data channels. The situation is made even worse by the contamination from various artifacts such as electrocardiogram (ECG), electromyogram (EMG), and electrooculogram (EOG). E-mail address: weiwu@neurostat.mit.edu (W. Wu). 1 The validity of simultaneity and linearity is guaranteed by the quasistatic approximation to the Maxwell equations, since the effective frequency range of EEG lies below 1 kHz.
Wu et al. REF presented a hierarchical Bayesian method to model the inter-trial variability of the EEG signals.
A hierarchical Bayesian approach for learning sparse spatio-temporal decompositions of multichannel EEG.
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