## 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 λ_{0} is defined; λ_{0} 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 λ_{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 λ_{0} = 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.

Original language | English |
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Article number | 194 |

Journal | FRONTIERS IN CELLULAR NEUROSCIENCE |

Volume | 10 |

Issue number | AUG |

DOIs | |

Publication status | Published - 9 Aug 2016 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Cable equation
- Length constant
- Membrane potential
- TMS
- Transcranial magnetic stimulation