The frequency-dependent neuronal length constant in transcranial magnetic stimulation

Risto J. Ilmoniemi*, Hanna Mäki, Jukka Saari, Ricardo Salvador, Pedro C. Miranda

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

4 Citations (Scopus)
109 Downloads (Pure)


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.

Original languageEnglish
Article number194
Issue numberAUG
Publication statusPublished - 9 Aug 2016
MoE publication typeA1 Journal article-refereed


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


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