Coupling Magnetically Induced Electric Fields to Neurons: Longitudinal and Transverse Activation

Abstract

We present a theory and computational models to couple the electric field induced by magnetic stimulation to neuronal membranes. Based on the characteristics of magnetically induced electric fields and the modified cable equation that we developed previously, quasipotentials are derived as a simple and accurate approximation for coupling of the electric fields to neurons. 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 straight 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. However, the activation thresholds of unmyelinated axons obtained with either cable equation are very high and beyond the output capabilities of conventional magnetic stimulators. For myelinated axons, threshold values are similar for both cable equations and within the range of magnetic stimulators. Whereas the transverse field contributes negligibly to the activation thresholds of myelinated fibers, axonal undulation can significantly 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 and is likely due to nerve fiber undulation and other geometrical inhomogeneities.

Figure 1

Simulation setup. (A) Three configurations of magnetic stimulation coils (black) with a nerve (gray) underneath are shown. The coordinate system shows the nerve’s distance to the coil center in the vertical (y) and lateral (x) directions. (B) E-field waveforms of monophasic and half-sine magnetic stimulation pulses, with peak amplitude normalized to unity at pulse onset, are shown. (C) Four types of axon placement within the nerve trunk are shown. The wavelengths and amplitudes of the undulation are exaggerated for visualization.

Figure 2

Distributions of E-field components contributing to neural activation in a plane 1 cm below a circular coil (top row) and a figure-8 coil aligned with or perpendicular to the nerve (middle and bottom rows, respectively). The coil current has a rate of change of 100 A/μs. The gradient of the longitudinal field (left column) and strength of the transverse field (right column) are shown along the nerve (z) for different lateral locations (x) relative to the coils (black outlines). The white contour lines, spaced at 3 mV/mm² and 100 mV/mm intervals, respectively, show positive and negative values with solid and dashed lines, respectively. To see this figure in color, go online.

Figure 3

Activation thresholds of straight unmyelinated HH-model axon for a range of vertical and lateral axon-coil distances. Rows show results for the monophasic and half-sine stimulation waveforms grouped by the three coil configurations. For the aligned figure-8 coil, the left region (negative x) is not fully shown because of the symmetry of the threshold distributions. The left and center columns show threshold values obtained with the conventional and modified CEs, respectively. The right column shows the threshold difference comparing modified and conventional CEs; any unmarked contour lines are spaced 10 and 5% apart for monophasic and half-sine waveforms, respectively. The outlines of the coils are illustrated as gray boxes, with the idealized windings located in the horizontal plane of y = 0. Color scales are the same within each column and shown at the bottom. To see this figure in color, go online.

Figure 4

Activation thresholds for straight myelinated RMG-model axon. These are presented in a similar format as Fig. 3, with a different color scale range for thresholds. To see this figure in color, go online.

Figure 5

Activation thresholds of undulating myelinated RMG-model axon for the three coil configurations (columns) using monophasic waveform. The first, second, and third rows show the threshold values for axons having only axon undulation, only fascicle undulation, and both undulation components. The fourth, fifth, and sixth rows show the threshold difference compared to straight axons (versus left column of Fig. 4) for the three undulating axon models in the first, second, and third rows; contour lines for positive and zero difference are shown with dashed and thick solid lines, respectively. Color scales are the same within each row and shown on the right; the same colors as in Fig. 4 are used for the thresholds and negative threshold differences. To see this figure in color, go online.