Quasi-static approximation error of electric field analysis for transcranial current stimulation

Area of Science:

• Neuroscience
• Computational Electromagnetics
• Biophysics

Background:

• Numerical modeling of electric fields is crucial for understanding transcranial alternating current stimulation (tACS).
• The quasi-static approximation (QSA) is commonly used to reduce computational costs in these models.
• Quantifying the validity of QSA across various frequencies is essential for accurate tACS simulations.

Purpose of the Study:

• To analyze and quantify the validity of the quasi-static approximation (QSA) for electric field calculations in tACS over a broad frequency range.
• To compare QSA results with full Maxwell's equations solutions for harmonic and pulsed signals.
• To assess the impact of electrode positioning and tissue dielectric properties on approximation errors.

Main Methods:

• Electromagnetic modeling using an anatomical head model.
• Comparison of purely ohmic (static) and lossy dielectric (QS) formulations against Maxwell's equations.
• Analysis of errors introduced by QSA, including the effect of electrode placement and signal waveform.

Main Results:

• QSA demonstrates validity with <1% relative error up to 1.43 MHz.
• The static formulation (neglecting permittivity) introduces >1% error across the spectrum, reaching 20% in the brain at 10 Hz.
• Capacitive effects are critical for pulsed waveforms to prevent signal distortion; errors up to 22% in electric field at the neuron level can impact firing times.

Conclusions:

• QSA is valid for current tACS frequencies but neglecting permittivity introduces significant errors.
• Accurate low-frequency dielectric data for human tissues are necessary for reliable tACS numerical modeling.
• Understanding these approximations is vital for precise prediction of neural responses to tACS.