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 λ0 is defined; λ0 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 λeff, 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 λ0 = 1.5 mm, the effective length constant at 3.9 kHz 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.
Highlights
In transcranial magnetic stimulation (TMS; Barker et al, 1985), strong, rapidly changing magneticfield pulses are delivered to the brain in order to induce an electric field at the target site
The primary electric field is very homogeneous on the cellular scale; the complicated conductivity structure of the neurons, to a large extent defined by cell membranes, changes the local electric current patterns dramatically
We study the frequency-dependent behavior of a neuron and formulate the effective length constant that governs the spatial decay of the membrane voltage
Summary
In transcranial magnetic stimulation (TMS; Barker et al, 1985), strong, rapidly changing magneticfield pulses are delivered to the brain in order to induce an electric field at the target site. The electric field, in turn, produces electric currents that, when directed through cell membranes, change transmembrane potentials. The primary electric field is very homogeneous on the cellular scale; the complicated conductivity structure of the neurons, to a large extent defined by cell membranes, changes the local electric current patterns dramatically. The behavior of the dendritic or axonal membrane voltage due to transcranial magnetic stimulation (TMS) is often modeled with the one-dimensional cable equation. A length constant λ0 is defined; λ0 describes the axial decay of the membrane voltage in the case of constant applied electric field. In TMS, the induced electric field waveform is typically a segment of a sinusoidal wave, with characteristic frequencies of the order of several kHz
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