On the calculation of magnetic fields based on multipole modeling of focal biological current sources

Spatially restricted biological current distributions, like the primary neuronal response in the human somatosensory cortex evoked by electric nerve stimulation, can be described adequately by a current multipole expansion. Here analytic formulas are derived for computing magnetic fields induced by current multipoles in terms of an nth-order derivative of the dipole field. The required differential operators are given in closed form for arbitrary order. The concept is realized in different forms for an expansion of the scalar as well as the dyadic Green's function, the latter allowing for separation of those multipolar source components that are electrically silent but magnetically detectable. The resulting formulas are generally applicable for current sources embedded in arbitrarily shaped volume conductors. By using neurophysiologically relevant source parameters, examples are provided for a spherical volume conductor with an analytically given dipole field. An analysis of the signal-to-noise ratio for multipole coefficients up to the octapolar term indicates that the lateral extent of cortical current sources can be detected by magnetoencephalographic recordings.