Frontiers in Cellular Neuroscience (Aug 2016)
The frequency-dependent neuronal length constant in transcranial magnetic stimulation
Abstract
Background: The behavior of the dendritic or axonal membrane voltage due to transcranial magnetic stimulation (TMS) is often modeled with the one-dimensional cable equation. For the cable equation, a length constant is defined; it describes the axial decay of the membrane voltage in the case of constant applied electric field. In TMS, however, the induced electric field waveform is typically a segment of a sinusoidal wave, with characteristic frequencies of the order of several kHz. Objective: To show that the high frequency content of the stimulation pulse causes deviations in the spatial profile of the membrane voltage as compared to the steady state. Methods: We derive the cable equation in complex form utilizing the complex frequency-dependent representation of the membrane conductivity. In addition, we define an effective length constant $lambda_{mathrm{eff}}$, which governs the spatial decay of the membrane voltage. We model the behavior of a dendrite in an applied electric field oscillating at 3.9 kHz with the complex cable equation and by solving the traditional cable equation numerically. Results: The effective length constant decreases as a function of frequency. For a model dendrite or axon, for which the low-frequency length constant is 1.5 mm, the effective length constant at 3.9 kHz is is decreased by a factor 10 to 0.13 mm. Conclusion: The frequency dependency of the neuronal length constant has to be taken into account when predicting the spatial behavior of the membrane voltage as a response to TMS.
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