%A Ilmoniemi,Risto J.
%A Mäki,Hanna
%A Saari,Jukka
%A Salvador,Ricardo
%A Miranda,Pedro C.
%D 2016
%J Frontiers in Cellular Neuroscience
%C
%F
%G English
%K Transcranial Magnetic Stimulation,TMS,Length Constant,cable equation,membrane potential
%Q
%R 10.3389/fncel.2016.00194
%W
%L
%N 194
%M
%P
%7
%8 2016-August-09
%9 Original Research
%+ Prof Risto J. Ilmoniemi,Department of Neuroscience and Biomedical Engineering, Aalto University School of Science,Espoo, Finland,risto.ilmoniemi@aalto.fi
%#
%! Frequency-dependent length constant in TMS
%*
%<
%T The Frequency-Dependent Neuronal Length Constant in Transcranial Magnetic Stimulation
%U https://www.frontiersin.org/article/10.3389/fncel.2016.00194
%V 10
%0 JOURNAL ARTICLE
%@ 1662-5102
%X 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.