Three-way matrix analysis, the MUSIC algorithm and the coupled dipole model

The inverse problem of multi-channel MEG/EEG data is considered as a parameter estimation problem. The stability of the solution of the inverse problem, which decreases with the number of included dipoles, can be improved by either adding constraints to the model parameters, or by adding more data of related data sets. The latter approach was taken by Bijma et al. [Bijma F, de Munck JC, Böcker KBE, Huizenga HM, Heethaar RM. The coupled dipole model: an integrated model for multiple MEG/EEG data sets. NeuroImage 2004;23(3):890-904; Bijma F, de Munck JC, Huizenga HM, Heethaar RM, Nehorai A. Simultaneous estimation and testing in multiple MEG data sets. IEEE Trans SP 2005;53(9):3449-60] by introducing coupling matrices that link dipole parameters and source time functions of different data sets.Here, the theoretical foundations of the coupled dipole model are explored and the MUSIC algorithm is generalised to the analysis of multiple related data sets. Similar to the MUSIC algorithm, the number of sources and the number of constraints are derived from the data by considering the minimum possible residual error as a function of the number of sources and constraints. However, contrary to the MUSIC algorithm, where the minimum residual error can be obtained from an SVD analysis of a two-way data matrix, here we deal with multiple data sets and therefore three-way matrix analysis is used.From a simulation study it appears that the number of sources and constraints can be clearly determined from a generalised SVD analysis. The generalisation of the MUSIC algorithm to three-way data gives reasonable estimates of the dipole parameters. These results can be used in the simultaneous analysis of MEG/EEG data of multiple subjects, multiples data sets of the same subject or models where subsequent trials of data show habituation effects.