Solution of the Forward and Inverse Problems Associated with the Potential Field of a Single Active Fiber in a Volume Conductor
Clark, Jr., J.W.
Laplace; linear; transmembrane
A solution of Laplace's equation relating the transmembrane potential distribution of an active fiber in a volume conductor to its extracellular field distribution utilizing a Fourier transform method  has been reformulated as a one-dimensional linear filtering problem. Formulation of the solution in this manner allows the application of well-known techniques in linear system theory and optimal linear filtering thereby facilitating the solution for both the forward (from transmembrane to field potential distribution) and inverse (from field to transmembrane potential distribution) problems. The forward problem is shown to be a simple two-stage filtering process composed of a membrane and medium filter. In the inverse case the field potential distribution is considered in the presence of additive measurement noise and the best estimate in the least-mean-square sense will be obtained for the transmembrane potential distribution.
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