A generalized cable equation for magnetic stimulation of axons

During magnetic stimulation, electric fields are induced both on the inside (intracellular region) and the outside (extracellular region) of nerve fibers. The induced electric fields in each region can be expressed as the sum of a primary and a secondary component. The primary component arises due to an applied time varying magnetic field and is the time derivative of a vector potential. The secondary component of the induced field arises due to charge separation in the volume conductor surrounding the nerve fiber and is the gradient of a scalar potential. The question, "What components of intracellular fields and extracellular 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 and 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 components, only the first spatial derivative of the intracellular primary component and the extracellular secondary component of the fields contribute to excitation of a nerve fiber. An earlier form of the cable equation for magnetic stimulation has been shown to result in solutions 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 results in solutions identical to 3-D volume conductor models even for complex 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 bundle such as anisotropic monodomain and bidomain models.