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

Abstract

This paper provides an optimal mechanism for the introduction of temporal constraints into linear imaging formulations of the inverse electroencephalography problem. The method is based on derivation of a "virtual-SVD," an extension of generalized singular value decomposition to the setting of random matrices. Surprisingly, the formalism is superior, in principle, to standard regularization methods even in the absence of known temporal constraints. Investigation of this basic temporally unconstrained setting was undertaken to illustrate the application of the method, and as a necessary first step in its systematic evaluation. Although abstract simulations demonstrate superior accuracy for the virtual-SVD method as compared with standard methods, investigation of a particular realistic simulation involving spatiotemporally distributed temporal lobe interictal spikes indicates that significant improvement in solution estimate quality under temporally unconstrained conditions may be limited to a very narrow range of the signal-to-noise ratio (particularly in the context of a markedly row-deficient transfer matrix). 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 epilepsy associated wave forms, evoked potentials, etc.) within the derived formalism.