Time/space regularization of the inward continuation problem in EEG using the Boundary Element Method
The inward continuation problem in EEG consists in the computation of the cortical potential distribution from the measured potential distribution on the scalp. Although unique this inverse problem is ill-posed. That is, low-level noise in the scalp potential data or a small error in the geometrical data can lead to unbounded errors in the solution. Regularization techniques have to be used to minimize these effects. The inverse problem is solved in two steps. First Tikhonov regularization is applied yielding a solution of the potential on the inside of the skull surface for every timestep. Than the solution of the first step is used for Twomey regularization. At each moment in time a new solution is found by using as a priori estimate the average of the first solution one timestep prior and one timestep after. This combination of spatial (Tikhonov) and temporal (Twomey) regularization improves the solution and smoothes the solution in space and time. Both simulations and the application to EEG data of a Median Nerve stimulation experiment yield encouraging results. Further comparative studies have to be carried out to evaluate the application of time/space regularization of the inward continuation problem in EEG.