Minimum variance beamformer weights revisited.

Adaptive minimum variance beamformers are widely used analysis tools in MEG and EEG. When the target brain activity presents in the form of spatially localized responses, the procedure usually involves two steps. First, positions and orientations of the sources of interest are determined. Second, the filter weights are calculated and source time courses reconstructed. This last step is the object of the current study. Despite different approaches utilized at the source localization stage, basic expressions for the weights have the same form, dictated by the minimum variance condition. These classic expressions involve covariance matrix of the measured field, which includes contributions from both the sources of interest and the noise background. We show analytically that the same weights can alternatively be obtained, if the full field covariance is replaced with that of the noise, provided the beamformer points to the true sources precisely. In practice, however, a certain mismatch is always inevitable. We show that such mismatch results in partial suppression of the true sources if the traditional weights are used. To avoid this effect, the "alternative" weights based on properly estimated noise covariance should be applied at the second, source time course reconstruction step. We demonstrate mathematically and using simulated and real data that in many situations the alternative weights provide significantly better time course reconstruction quality than the traditional ones. In particular, they a) improve source-level SNR and yield more accurately reconstructed waveforms; b) provide more accurate estimates of inter-source correlations; and c) reduce the adverse influence of the source correlations on the performance of single-source beamformers, which are used most often. Importantly, the alternative weights come at no additional computational cost, as the structure of the expressions remains the same.