Statistical method accounts for microscopic electric field distortions around neurons when simulating activation thresholds

Abstract

Introduction: Notwithstanding advances in computational models of neuromodulation, there are mismatches between simulated and experimental activation thresholds. Transcranial Magnetic Stimulation (TMS) of the primary motor cortex generates motor evoked potentials (MEPs). At the threshold of MEP generation, whole-head models predict macroscopic (at millimeter scale) electric fields (50-70 V/m) which are considerably below conventionally simulated cortical neuron thresholds (175-350 V/m).

Methods: We hypothesize that this apparent contradiction is in part a consequence of electrical field warping by brain microstructure. Classical neuronal models ignore the physical presence of neighboring neurons and microstructure and assume that the macroscopic field directly acts on the neurons. In previous work, we performed advanced numerical calculations considering realistic microscopic compartments (e.g., cells, blood vessels), resulting in locally inhomogeneous (micrometer scale) electric field and altered neuronal activation thresholds. Here we combine detailed neural threshold simulations under homogeneous field assumptions with microscopic field calculations, leveraging a novel statistical approach.

Results: We show that, provided brain-region specific microstructure metrics, a single statistically derived scaling factor between microscopic and macroscopic electric fields can be applied in predicting neuronal thresholds. For the cortical sample considered, the statistical methods match TMS experimental thresholds.

Conclusions: Our approach can be broadly applied to neuromodulation models, where fully coupled microstructure scale simulations may not be computationally tractable.

Keywords: Multiscale brain modeling; TMS; biophysical modeling; brain stimulation.

Fig. 1:

Distribution of the extracellular electric field magnitude (V/m) inside the sample when a uniform impressed electric field of $E_x^{(p)}=100\text{V}/\text{m}$ is applied in the medial-lateral direction (from left to right in the (a)), showing how low conducting barriers (e.g., cell membranes) cause charge accumulation and associated field distortion in (b) and (c). The intracellular space (grey) is not included in the computations. Tissue segmentation and electric field computations are from Qi et al. (2025).

Fig. 2.

The influence of the accumulation of charges on the membranes onto the collinear electric field at the centerlines of neuronal processes for an example neuron. Top: A uniform impressed electric field is applied along the x-axis (from dorsal to ventral) with $E_x^{(p)}=100\text{V}/\text{m}$. The influence of the charges is not considered and the impressed field is directly projected to the centerlines. Hence, the maximum achievable collinear field (for an optimally oriented neuronal segment) at the neurons $E_{max}^{(c)}$ is equal to the impressed field. Bottom: The same uniform impressed field is applied, but realistic neuronal (sub-)compartments are included into the voltage simulation and the impressed field is distorted by the field of the induced charges. In consequence, the maximum achievable collinear field $E_{max}^{(c)}$ can be larger than the impressed field $E_x^{(p)}$.

Fig. 3:

Histogram and probability density $p\left(s_e\right)$ of the electric field scaling factor $\left(s_e=\frac{E_{micro}}{E_{macro}}\right)$ between microscopic and macroscopic electric fields (black dashed line). The median of the scaling factor is ${\overline{s}}_e=1.06$ (red dashed line).

Fig. 4:

Uncorrected and corrected recruitment rates in comparison to experimental MEPs. Dashed black line: original recruitment rates from Weise et al. (2023b) of L2/3 PC for $\theta =0^{°}$ and $\text{Δ}|\widetilde{\mathbf{\text{E}}}|=0\%/\text{mm}$ considering homogeneous macroscopic electric fields and a biphasic TMS pulse, without taking into account microscopic electric field effects. Red line: corrected recruitment rates determined from eqs. (1) and (2) using the probability density $p\left(s_e\right)$ of the electric field scaling factor between microscopic and macroscopic electric fields from Fig. 2, together with the recruitment rate determined using macroscopic electric fields from Weise et al. (2023b) assuming an average number of axon terminals of N=35. Grey lines: Recruitment rate curves after sampling the number of axon terminals N from a normal distribution with the given mean of N=35 and standard deviation of 13.7. Colored dots: MEPs as function of the external macroscopic electric field determined experimentally in Numssen et al. (2021) after motor mapping (different colors represent different subjects).

Fig. 5:

Recruitment rates of L2/3 PC without (a) and with (b) microscopic field corrections stimulated by biphasic TMS pulses for different electric field angles. No e-field gradient $(\text{Δ}|\widetilde{\mathbf{\text{E}}}|=0\%/\text{mm})$ was assumed. (a) The neuronal recruitment rate from Weise et al. (2023b) did not consider microscopic electric field variations, yielding e-field thresholds of above 200 V/m. (b) Recruitment rate of a neuronal population taking electric field variations from Fig. 3 into account yields thresholds of below 50 V/m. (c) and (d) Recruitment rates for field angles of $\theta =0^{°},\theta ={90}^{°}$, and $\theta ={180}^{°}$ without and with microscopic field correction, respectively.