On the non-uniqueness of the quasi-static magnetic forward problem and its impact on source current reconstruction
We introduce a formulation where individual line segments of a current loop have translationally non-invariant contributions to the electro-quasi-static magnetic scalar potential and magnetic field in source-free regions. While closed current loops composed of these open segments have translationally invariant contributions, our formulation indicates that there are multiple ways to decompose the magnetic field due to a closed current loop into the open current segments. By defining the path-independent magnetic scalar potential using a radial integration path with respect to a given origin, a formula has been derived that shows that only non-radial line current segments have non-zero contributions, whereas radial current segments are magnetically silent. This indicates that magnetic forward models for open current segments are non-unique, since the orientations of the currents are dependent on the choice of the computational origin. This finding affects the conventional physical interpretation of the significance of primary/volume currents in biomagnetic forward and inverse models of brain activity, since their contributions are shown to vary with the choice of origin in our formulation. As an example, we perform a simple magnetoencephalography (MEG) equivalent current dipole (ECD) fit for a spherical head model with various origin choices to localize a primary current source. The primary current can be made to have zero contributions in certain translated origin choices (and the volume currents contribute to the signal entirely). Thus, there exist origins in which the ECD model fits the data with high confidence but incorrect estimated location of the primary current segment. In such cases, the dipole is fit to a segment of the volume current. If one were unaware of any origin translations, such ECD fits may be incorrectly interpreted as the primary current source.