Return current in encephalography. Variational principles.

Abstract

The encephalographic problem of finding the electric potential V and the return current associated with any assumed primary current, Jp, is put in the form of a variational principle. With Jp and the conductivity specified, the correct V is one which makes an integral quantity P[V] a maximum. The terms in P[V] are related to the rates at which work is done by the electric field on the primary and return currents. It is shown that there is a unique solution for the electric field, and it satisfies the conservation of energy; this condition can serve as a check on any numerical solution. With the conductivity a different constant in different regions, the variational principle is recast in terms of the charge density on the surfaces of discontinuity. An iteration-variation method for finding the solution is outlined, and possible computational advantages over other approaches are discussed.