A critical analysis of linear inverse solutions to the neuroelectromagnetic inverse problem

This paper explores the possibilities of using linear inverse solutions to reconstruct arbitrary current distributions within the human brain. We formally prove that due to the underdetermined character of the problem, the only class of measurable current distributions that can be totally retrieved are those of minimal norm. The reconstruction of smooth or averaged versions of the currents is also explored. A solution that explicitly attempts to reconstruct averages of the current is proposed and compared with the minimum norm and the minimum Laplacian solution. In contrast to the majority of previous analysis carried out in the field, in the comparisons, we avoid the use of measures designed for the case of dipolar sources. To allow for the evaluation of distributed solutions in the case of arbitrary current distributions we use the concept of resolution kernels. Two summarizing measures, source identifiability and source visibility, are proposed and applied to the comparison. From this study can be concluded: 1) linear inverse solutions are unable to produce adequate estimates of arbitrary current distributions at many brain sites and 2) averages or smooth solutions are better than the minimum norm solution estimating the position of single point sources. However, they systematically underestimate their amplitude or strength especially for the deeper brain areas. Based on these result, it appears unlikely that a three-dimensional (3-D) tomography of the brain electromagnetic activity can be based on linear reconstruction methods without the use of a significant amount of a priori information.