Temporally unconstrained space-time treatment of linear formulations of the inverse problem of electroencephalography

This paper provides an optimal mechanism for the introduction of temporal c onstraints into linear imaging formulations of the inverse electroencephalo graphy problem. The method is based on derivation of a "virtual-SVD," an ex tension of generalized singular value decomposition to the setting of rando m matrices. Surprisingly, the formalism is superior, in principle, to stand ard regularization methods even in the absence of known temporal constraint s. Investigation of this basic temporally unconstrained setting was underta ken to illustrate the application of the method, and as a necessary first s tep in its systematic evaluation. Although abstract simulations demonstrate superior accuracy for the virtual-SVD method as compared with standard met hods, investigation of a particular realistic simulation involving spatiote mporally distributed temporal lobe interictal spikes indicates that signifi cant improvement in solution estimate quality under temporally unconstraine d conditions may be limited to a very narrow range of the signal-to-noise r atio (particularly in the context of a markedly row-deficient transfer matr ix). These results underline the prospective importance of investigation of the efficacy and feasibility of application of temporal constraints (such as those resulting from knowledge of the general time series format of epil epsy associated wave forms, evoked potentials, etc.) within the derived for malism. (C) 2000 Biomedical Engineering Society. [S0090-6964(00)00310-6].