Coupling magnetically induced electric fields to neurons: longitudinal and transverse activation
We present a theory and computational models to couple the electric field induced by magnetic stimulation to neuronal membranes. The response of neuronal membranes to induced electric fields is examined under different time scales, and the characteristics of the primary and secondary electric fields from electromagnetic induction and charge accumulation on conductivity boundaries, respectively, are analyzed. Based on the field characteristics and decoupling of the longitudinal and transverse field components along the neural cable, quasi-potentials are a simple and accurate approximation for coupling of magnetically induced electric fields to neurons and a modified cable equation provides theoretical consistency for magnetic stimulation. The conventional and modified cable equations are used to simulate magnetic stimulation of long peripheral nerves by circular and figure-8 coils. Activation thresholds are obtained over a range of lateral and vertical coil positions for two nonlinear membrane models representing unmyelinated and myelinated axons and also for undulating myelinated axons. For unmyelinated straight axons, the thresholds obtained with the modified cable equation are significantly lower due to transverse polarization, and the spatial distributions of thresholds as a function of coil position differ significantly from predictions by the activating function. For myelinated axons, the transverse field contributes negligibly to activation thresholds, whereas axonal undulation can increase or decrease thresholds depending on coil position. The analysis provides a rigorous theoretical foundation and implementation methods for the use of the cable equation to model neuronal response to magnetically induced electric fields. Experimentally observed stimulation with the electric fields perpendicular to the nerve trunk cannot be explained by transverse polarization alone and is likely due to nerve fiber undulation and other geometrical inhomogeneities.
