Application of quasi-static magnetic reciprocity to finite element models of the MEG lead-field

The reconstruction of neuronal current sources from magneto- and/or electroencephalography (MEG/EEG) measurements is referred to as an inverse problem. A precursor to most inverse algorithms is a forward transfer, or lead-field, matrix, in which the rows correspond to MEG and/or EEG measurement sites, and each column captures the linear response to a particular unit source. Simple models of the head, such as concentric spheres, result in analytic expressions for the lead-field. More realistic head models, such as those based on medical imagery, require numeric simulations. A straightforward, though inefficient, way to obtain the lead-field is to perform one forward simulation for each source, resulting in one column of the lead-field. For MEG/EEG inverse problems, however, the potential sources (rows) far outnumber the measurement sites (columns). Two approaches have been described for computing the EEG lead-field with a number of forward simulations equal to the number of measurement rows, rather than the number of source columns. One of these approaches is based on the principle of electric reciprocity, and the other approach is based on linear-algebraic manipulations of the forward problem. For the MEG lead-field, only a linear-algebraic approach has been described for numeric approaches such as the finite element method. This paper describes a reciprocal approach for the MEG lead-field and discusses implementation details for both approaches. Index Terms--Finite element methods, inverse problems, lead-field basis, magnetoencephalography, reciprocity.