The Frequency-Dependent Neuronal Length Constant in Transcranial Magnetic Stimulation

Area of Science:

• Neuroscience
• Biophysics
• Computational Biology

Background:

• Transcranial magnetic stimulation (TMS) is often modeled using the 1D cable equation, which defines a length constant (λ0) for axial voltage decay under constant electric fields.
• TMS utilizes sinusoidal electric fields with kHz frequencies, differing from the steady-state assumptions of the traditional cable equation.

Purpose of the Study:

• To investigate how the high-frequency content of TMS pulses affects the spatial distribution of membrane voltage.
• To demonstrate deviations from steady-state predictions caused by the frequency-dependent nature of neuronal responses.

Main Methods:

• Derived the cable equation in complex form, incorporating frequency-dependent membrane conductivity.
• Defined an effective length constant (λeff) to characterize spatial voltage decay at specific frequencies.
• Modeled a dendrite's response to a 3.9 kHz electric field using both the complex and traditional numerical cable equations.

Main Results:

• The effective length constant (λeff) was found to decrease as a function of frequency.
• For a model neuron with λ0 = 1.5 mm, λeff dropped by a factor of 10 to 0.13 mm at 3.9 kHz.
• This indicates a significantly reduced spatial spread of membrane voltage at TMS frequencies.

Conclusions:

• The frequency-dependent properties of the neuronal length constant are crucial for accurate modeling of TMS effects.
• Accounting for this frequency dependency is necessary for predicting the spatial spread of membrane voltage during TMS.
• This finding has implications for optimizing TMS protocols and understanding neural responses.