Another bioelectric inverse formulation that appears in the literature poses both the direct and inverse problems in terms of scalar values at the surfaces. For example, both the EEG and ECG problems can be posed in this manner. For the EEG problem, one would take the surface of the brain (cortex) one bounded surface and the surface of the scalp as the other surface. The direct problem would involve making measurements of voltage of the surface of the cortex at discrete locations and then calculating the voltages on the surface of the scalp. Similarly, for the ECG problem, voltages could be measured on the surface of the heart and used to calculate the voltages at the surface of the torso, as well as within the volume conductor of the thorax. The inverse problems are formulated as using measurements on the surface of the skull (torso) and calculating the voltages on the surface of the brain (heart). Here we are solving Laplace's equation instead of Poisson's equation, because we are solving for distributions of voltages on a surface instead of current sources within a volume.
with boundary conditions (5) and (6).