A GENERALIZED CABLE EQUATION FOR MAGNETIC STIMULATION OF AXONS

During magnetic stimulation, electric fields are induced both on the i nside (intracellular region) and the outside (extracellular region) of nerve fibers. The induced electric fields in each region can be expre ssed as the sum of a primary and a secondary component. The primary co mponent arises due to an applied time varying magnetic field and is th e time derivative of a vector potential. The secondary component of th e induced field arises due to charge separation in the volume conducto r surrounding the nerve fiber and is the gradient of a scalar potentia l. The question, ''What components of intracellular fields and extrace llular induced electric fields contribute to excitation?'' has, so far , not been clearly addressed. In this paper, we address this question while deriving a generalized cable equation for magnetic stimulation a nd explicitly identify the different components of applied fields that contribute to excitation. In the course of this derivation, we review several assumptions of the core-conductor cable model in the context of magnetic stimulation. It is shown that out of the possible four com ponents, only the first spatial derivative of the intracellular primar y component and the extracellular secondary component of the fields co ntribute to excitation of a nerve fiber. An earlier form of the cable equation for magnetic stimulation has been shown to result in solution s identical to three-dimensional (3-D) volume-conductor model for the specific configuration of an isolated axon in a located in an infinite homogenous conducting medium. In this paper, we extend and generalize this result by demonstrating that our generalized cable equation resu lts in solutions identical to 3-D volume conductor models even for com plex geometries of volume conductors surrounding axons such as a nerve bundle of different conductivity surrounding axons. This equivalence in the solutions is valid for several representations of a nerve bundl e such as anisotropic monodomain and bidomain models.