One Definition to Join Them All: The N-Spherical Solution for the EEG Lead Field

Albeit its simplicity, the concentric spheres head model is widely used in EEG. The reason behind this is its simple mathematical definition, which allows for the calculation of lead fields with negligible computational cost, for example, for iterative approaches. Nevertheless, the literature shows contradictory formulations for the electrical solution of this head model. In this work, we study several different definitions for the electrical lead field of a four concentric spheres conduction model, finding that their results are contradictory. A thorough exploration of the mathematics used to build these formulations, provided in the original works, allowed for the identification of errors in some of the formulae, which proved to be the reason for the discrepancies. Moreover, this mathematical review revealed the iterative nature of some of these formulations, which allowed us to develop a formulation to solve the lead field in a head model built from an arbitrary number of concentric, homogeneous, and isotropic spheres.