A compact theory of magnetic nerve stimulation: predicting how to aim

A compact theory that predicts quantitatively when and where magnetic neurostimulation will occur is needed as a guide to therapy, ideally providing a single equation that defines the target volume of tissue excited by single or dual coils.

A first-principles analysis of magnetic stimulation incorporating a simplified description of electromagnetic fields and a simplified cable theory of the axon yields a mathematical synthesis predicting how to aim.

Nerve stimulation produced by a single circular coil having one or more closely packed turns occurs in donut shaped volume of tissue beneath the coil. Axons spanning several millimeters are the sites of magnetic stimulation. The sites of maximal transmembrane depolarization in nerve fibers correspond to points where the axons enter or exit this volume of magnetically induced voltage and current. The axonal membrane at one end is depolarized locally during the rising phase of current in the coil. The axonal membrane at the opposite end is depolarized locally during the falling phase of current in the coil. Penetration depths of several centimeters from the skin surface or approximately one to two coil radii are practical. With two coils placed in a figure-of-eight configuration the separate clockwise and counterclockwise currents generate magnetic fields that add, producing maximal stimulation of a spindle shaped volume, centered at a depth of one-third to one-half coil radius from the body surface.

This condensed synthesis of electromagnetic theory and cable theories of axon physiology provides a partial solution to the targeting problem in peripheral and in transcranial magnetic stimulation.