Introduction

Transcranial magnetic stimulation (TMS) is a non-invasive brain stimulation technology1 based on electromagnetic induction. It serves not only as a diagnostic and therapeutic tool for nervous system diseases2,3, but also as a means to study brain functions4,5. Although a comprehensive understanding of its mechanism remains lacking6, the electric field (E-field) induced by the pulsed magnetic field of TMS is generally regarded as a crucial physical factor for exciting neurons7,8. The distribution of the E-field is influenced by the head’s anatomical structure and the orientation of the coil. This is due to the tissue conductivity inhomogeneity disrupting the E-field’s continuous distribution. The most crucial factor lies in the complex boundary between cerebrospinal fluid (CSF) and gray matter (GM)9. Compared with the smooth spherical model, the cortical folding of the brain not only generates an enhancement effect on the E-field but also brings about orientation-dependence of the E-field9,10,11,12,13. The folding morphology of the cerebral cortex significantly affects the distribution of the E-field induced by TMS and the stimulation effect.

Precisely for this reason, the placement of the coil needs to be guided by personalized brain imaging results. In studies seeking to determine optimal positioning of the Figure-8 coil through electric-field modeling, researchers typically analyze multiple grid-like points and angular orientations near the target brain region. An extremum of specific E-field parameters (such as the maximum value14, average strength15,16 is then selected as the optimization criterion. Although recent studies have integrated particle swarm algorithm17 and deep learning algorithm18 to accelerate this process, these methods still require extensive pre-training on heterogeneous brain models and provide limited mechanistic interpretability. Integrating the universal E-field patterns governed by neuroanatomical features may offer a more efficient optimization strategy.

Currently, the gyrus morphological characteristic parameter that has received explicit attention is the direction of the gyrus. The hand knob region of M1 has been extensively studied. Some studies show that when the coil orientation is perpendicular to the sulcus direction, a greater E-field strength and penetration depth can be obtained13,19. Similar conclusions have also been drawn for the leg motor area20. In other brain regions, this direction is sometimes effective and sometimes exhibits certain deviations21. However, relying solely on the direction of the gyrus is insufficient to understand the reasons for the significant differences in electric field distribution and the optimal coil orientation among different brain regions14,16,19. There has also been a study that takes into account the curvature of the hand knob region, and the results show that the median curvature of this region is correlated with the different coil orientation preferences of motor evoked potential (MEP) responses20. Nevertheless, how the specific morphology of the gyrus affects the distribution of the electric field remains to be investigated.

Directly assessing the impact of gyral morphology on the E-field distribution within realistic head models proves to be a hard task. The cerebral cortex presents an intricate tapestry of morphological features, leaving us in a state of uncertainty regarding which morphological parameters to consider. Deriving consistent and generalizable conclusions under such circumstances is akin to searching for a needle in a vast expanse of hay. Consequently, a prudent preliminary research approach involves abstracting the gyral morphological characteristics, formulating a series of simplified gyrus models, and meticulously analyzing how diverse morphological features influence the distribution of the E-field induced by TMS. In the past, most studies on simplified models of the cerebral cortex adopted a 3-layer planar tissue model, including CSF, GM, and white matter (WM). These models used idealized gyrus/sulcus shapes where sulcal walls are perfectly perpendicular to the coil plane and gyral curvatures are stretched along straight lines, a configuration that represents the relationship between E-field strength variations and the coil’s orientation relative to gyri12. Alternatively, when combined with neuronal distribution data, this approach was used to identify probable TMS activation sites, including the gyrus apex, transitional margins, and sulcus walls9.

In this study, we investigated how cortical folding morphology influences transcranial magnetic stimulation (TMS)-induced electric fields. We constructed a simplified multi-layered curved cortical fold model to quantitatively analyze the relationships between key morphological parameters (cross-sectional shape and gyral crest curvature) and spatial electric field characteristics.

Methods

Simplified gyrus model based on multi-layer curved surfaces

As depicted in Fig. 1, the simplified gyrus models based on multi-layer curved surfaces consist of five tissue layers. The skin, skull, and CSF are simplified to spheres, forming a concentric-sphere structure with the enveloping spherical surface atop the GM and WM. The cross-sections of the GM and WM are simplified into the trapezoid-like shape and stretched along the cortical surface crest lines. During the adjustment of the gyrus’ morphological characterization parameters, the thicknesses of the skin, bone, CSF above the gyral crown and GM remain constant, set at 5 mm, 7 mm, 3 mm, and 3 mm respectively. The spherical models (Sph.), consisting of only five layers of spheres, serve as the comparison benchmark in the following text.

Fig. 1
figure 1

Parametric gyrus simplified modeling. (a) Cross-sectional shape parameters: Bcrown (gyrus crown width), θwall (sulcal wall inclination angle), hsulcus (sulcal depth), and material property distribution (skin, skull, CSF, GM, WM). (b) 3D model with gyral crest curvature parameters: RGMES (radius of the gray matter (GM) envelope sphere ), rCL (radius of curvature of the gyral crest line, limiting case is rCL→∞=straight within the xy-plane). Crest line: ridge trajectories on the folded cortex.

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The lower surface of the coil is placed on a tangent plane 5 mm away from the skin. The projection of the coil center is at the center of the gyrus top. For the Figure-8 coil, the orientation of the central current (as for Coil Orientation) is 0° relative to the tangential direction of the crest lines, and 90° when perpendicular to them. Due to the symmetry of the established models, only the coil orientations ranging from 0° to 90° are considered. The projection of the coil center is at the symmetric center of the cortical surface crest lines. The rotation orientation of the coil is counterclockwise when observed from the positive direction of the z-axis.

Definition of gyral morphological parameters

Two parameter classes characterize gyral morphology:

  1. 1.

    Cross-sectional shape

    Quantifies cortical folding geometry, where the cross-sectional normal aligns with the tangent of cortical surface crest lines. Three parameters define the cross-sectional shape (Fig. 1a), which are Bcrown: gyrus crown width, θwall: sulcal wall inclination angle, hsulcus: Sulcal depth. Gray matter (GM) thickness remains uniform along the gyrus surface normal. A 3 mm-radius chamfer at the gyrus lip ensures smooth crown-to-wall transitioning. Guided by morphometric data from a realistic head model and literature22,23, we set computational ranges as Bcrown: 7–19 mm, hsulcus: 6–18 mm, θwall: -15° to 30° .These ranges are adaptively refined during simulations to account for geometric interference.

  2. 2.

    Gyral crest curvature

    Gyral crest curvature describes the 3D trajectory of cortical surface crest lines (ridge trajectories on folded cortex, corresponding to gyral apices with maximal interface curvature), which governs the overall bending and alignment of gyri. In our simplified model, these crest lines serve as extrusion paths for sweeping cross-sectional profiles. The characteristic parameters of gyral crest curvature (Fig. 1b) are quantified as follows. RGMES: the radius of the gray matter (GM) envelope sphere, reflects the brain’s globular shape. Due to the ellipsoidal geometry of the human brain, RGMES varies across regions and individuals. Based on anatomical measurements from a realistic head model and literature data24, we constrain RGMES to 50–90 mm. rCL: the radius of curvature of the gyral crest line. This parameter quantifies the folding behavior of gyri within a spherical coordinate framework. Based on anatomical measurements from a realistic head model, we constrain rCL to the range of 10–40 mm. In the limit rCL→∞, the crest lines asymptotically approach straight lines within the xy-plane.

Realistic head model and analyzed target areas

The data used to construct the realistic head model in this paper is sourced from the symmetric segmentation data (in the .nii format) of the ICBM-NY model (also known as the New York Head) provided by the Yu Huang research group at the City University of New York, USA25. The head model also consists of five layers of tissues (Fig. 2).

Fig. 2
figure 2

Five-layer tissue segmentation and coil model in the realistic head geometry.

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The dorsal postcentral gyrus (dPoG), the middle precentral gyrus (mPrG; corresponds to hand motor area), and the middle superior temporal gyrus (mSTG) exhibit distinct gyral morphological characteristics. Accordingly, these regions were selected as regions of interest (ROIs) and designated as Regions A–C for analysis (Fig. 6a). Due to the use of a symmetric brain model, the calculations are only performed on the left hemisphere. Project the center of the coil to be positioned at the center of the gyral crest curvature. According to the gyral crest curvatures, define the coil orientation tangent to the crest lines as 0° and the perpendicular direction as 90°.

Stimulation scenarios and finite element calculation of electric field

Based on the geometric dimensions of the Magstim 70 mm Figure-8 coil, a uniform multi-turn coil model is established. The number of turns in a single wing of this coil is 9. The outer radius is 4.4 cm, the inner radius is 2.6 cm, and the thickness is 8 mm26,27. The current of a single-turn coil is set to a unit value of 1. The tissue is assumed to be isotropic, and the conductivity is set to the average values reported in the reference10. Specifically, the conductivity values are assigned as 0.465 S/m for the skin, 0.010 S/m for the skull, 1.654 S/m for CSF, 0.276 S/m for the GM, and 0.126 S/m for the WM. The relative permittivity is assigned according to the 4-Cole-Cole model at 3 kHz28. Calculations are performed in the Magnetic Field Frequency Domain module of COMSOL Multiphysics ® V. 6.1 (by COMSOL AB, www.comsol.com), with the frequency set at 3 kHz.

Results

Electric field analysis in the simplified gyrus model

Impact of gyral cross-sectional shape parameters in the straight crest line gyrus model

The relationship between gyral cross-sectional shape parameters (Bcrown: coronal width; θwall: sulcal wall inclination; hsulcus: sulcal depth) and induced electric fields was tested in straight crest line models (RGMES = 70 mm). In these models, cortical peak electric field strength (Ep, maximum value in scientific notation with three significant digits) increased monotonically with coil orientation across all cross-sectional configurations (Fig. 3b). At 0° coil orientation, Ep exhibited negligible sensitivity to cross-sectional geometry, matching the predictions of homogeneous spherical models while concentrating at the crown of the gyrus (Fig. 3a). However, a substantial 3.91-fold enhancement of Ep occurred at 90° orientation in gyral models compared to spherical counterparts, demonstrating the critical role of structural complexity.

Fig. 3
figure 3

Impact of gyral cross-sectional shape on induced electric fields (E) in straight crest gyrus models. Cross-sectional shape parameter: Bcrown (gyrus crown width), θwall (sulcal wall inclination angle), hsulcus (sulcal depth). (a) E-field distributions in coronal cross-sections under the coil center. (b) Normalized peak E-field (Ep) strength versus coil orientation (relative to spherical model baseline). (c) Ep variation with cross-sectional parameters at 90° (min-normalized). (d) E along the depth of subpial Path (1 mm below GM-WM interface, Coil Orientation at 90°; max-normalized).

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Further parametric analysis revealed that Ep intensified in gyri characterized by narrower coronal widths, flatter sulcal walls (decreased θwall), and deeper sulci. Normalizing Ep to minimum values within the parameter space revealed a 2.38-fold amplitude difference between extreme geometries (Fig. 3c). Notably, higher field strengths in tightly folded configurations extended deeper into sulcal regions, indicating enhanced penetration driven by cortical folding (Fig. 3a, d).

Modulation of gyral crest curvature parameters

To assess gyral crest curvature effects, cross-sectional geometry was fixed (Bcrown = 7 mm; θwall = 0°; hsulcus = 12 mm) while varying radius of the GM envelope sphere (RGMES) and radius of curvature of the gyral crest line (rCL). Ep showed predictable orientation dependence, peaking when the coil axis was perpendicular to the gyrus (90°) and declining to a minimum at parallel alignment (0°) (Fig. 4a, b). Increasing RGMES systematically enhanced Ep, whereas smaller rCL amplified directional sensitivity—models with rCL =10 mm reduced Ep variability by 89% compared to straight gyri.

Fig. 4
figure 4

Effects of gyral crest curvature on induced electric fields. RGMES (radius of the gray matter (GM) envelope sphere ), rCL (radius of curvature of the gyral crest line, limiting case is rCL→∞=straight within the xy-plane). Crest line: ridge trajectories on the folded cortex. (a) Normalized peak electric field (Ep) strength versus coil orientation (normalized according to the sphere model with RGMES = 50 mm). (b) Ep strength (min-normalized) as a function of curvature parameters and coil orientation. (c) Electric field distribution in white matter, with asterisks marking Ep positions and dot-tipped arrows indicating coil orientations. (d) Ep offset distance (Euclidean distance from peak to point beneath coil center) versus coil orientation. (e) Comparison between maximum Ep offset distance (at 0° coil orientation) and chord length of a quarter-circle of the crest line.

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Our field analysis revealed that when the coil orientation deviates from perpendicular alignment with the gyral crest line, Ep shifts away from directly beneath the coil center (Fig. 4c, top row). The offset magnitude exhibits an inverse correlation with Ep intensity, reaching its minimum value at 90° alignment (Fig. 4d). We define the “optimal peak position” at the quarter-circle locus (Fig. 4c; RGMES =50 mm, rCL =10 mm) where the field direction regains perpendicularity to the crest line. As rCL increases, the maximum displacement amplifies incrementally but remains constrained, with Ep positions increasingly diverging from the “optimal peak position” (Fig. 4e). Straight gyral configurations maintain fixed peak positions regardless of orientation. These findings demonstrate the deterministic influence of gyral curvature on both directional sensitivity of peak field intensity and displacement magnitude. Crucially, the complete absence of orientation dependence in spherical models further validates these structural effects.

Combined impact of cross-sectional and curvature parameters

At the maximum field orientation (90°), the study investigated the combined effects of gyral cross-sectional folding parameters (Bcrown, θwall, hsulcus) and crest-line curvature parameters (RGMES, rCL) on induced electric fields (Fig. 5). The influence of gyral crest-line curvature on Ep intensity remained significant. Tightly folded gyri (characterized by reduced coronal width and increased sulcal depth) exhibited stronger curvature-dependent Ep variations (within-group σ = 0.03), whereas loosely folded structures showed lower variation (σ = 0.008). However, the impact of folding severity on field intensity was pronounced at large curvatures but became negligible under minimal curvature conditions, due to geometric constraints imposed by cross-sectional morphology. To avoid geometric interference, simplified models of highly folded gyri required larger minimal rCL values, increasing from 10 mm (Group 1) to 25 mm (Final Group).

Fig. 5
figure 5

Variations in peak electric field (Ep) with cross-sectional shapes (Bcrown, θwall, hsulcus) and curvatures (RGMES, rCL) at 90° coil orientation. Mean values with standard deviations represent within-group averages for gyrus models exhibiting different trends while maintaining identical cross-sectional shapes.

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Collectively, these results demonstrate that cortical folding geometry—encompassing both cross-sectional parameters and curvature metrics—regulates transcranial magnetic stimulation-induced electric field intensity, penetration depth, and directional specificity. The interplay between gyral morphology and coil positioning establishes a preliminary yet rapid optimization framework for stimulation strategies.

Electric field validation in the realistic head model

To validate geometric principles derived from simplified models, we analyzed Ep distributions in three gyral regions of a realistic head model exhibiting distinct morphological signatures. Morphological parameters (Bcrown, θwall, hsulcus, RGMES, rCL) were quantified from cross-sections perpendicular to crest lines beneath the coil center (Fig. 6a; Table 1).

Fig. 6
figure 6

Electric field analysis of ROIs in the realistic head model. (a) Morphological features and Ep distributions in three gyral regions of interest (ROIs). Dashed lines: crest lines; asterisks: Ep peak positions; dot-tipped arrows : coil orientations. Morphological parameters (Bcrown, θwall, hsulcus, RGMES, rCL) and skin-to-cortex distances (SCD) are quantified in Table 1. (b) Inter-regional comparison of peak electric field Ep strengths (green) and peak offset distances (orange) across coil orientations (0°,45°, 90°, 135°). Upper-right inset: SCD variation across ROIs.

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Table 1 The morphological parameter values of the gyri in ROIs.
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Despite comparable skin-to-cortex distances (SCD) between Regions A and C (Δdistance < 2%), orthogonal coil orientation (90°) resulted in a 2.1-fold decrease in Ep amplitude in Region C (RGMES=70 mm, rCL=80 mm) compared to Region A (RGMES =55 mm, rCL =15 mm). This demonstrates that cross-sectional folding degree governs field strength modulation more significantly than global curvature parameters( larger RGMES and rCL correlate with greater Ep).

The coil orientation-dependent spatial modulation of peak electric field (Ep) intensity and location broadly aligns with our simplified model. Maximal Ep amplitudes consistently occurred at 90° coil-to-gyrus alignment (perpendicular orientation), while minimal values were observed at parallel alignment (0°). In Regions A and B, peak field locations shifted toward the gyral crest line perpendicular to the coil axis, with offset distances inversely proportional to Ep magnitude. Region A—characterized by tightly folded geometry (Bcrown=10 mm, θwall = 5°, hsulcus=11 mm)—demonstrated significantly smaller peak offsets (average shift reductions: 77% vs. Region B, 60% vs. Region C; maximum shift reductions: 39% vs. Region B, 34% vs. Region C).

Notably, deviations from the simplified model predictions emerged in two critical cases. First, Region A exhibited its smallest offset distance at a 45° coil orientation rather than the predicted 90°. Second, despite its near-straight gyrus morphology, Region C demonstrated unexpected offset distances at both 0° (parallel) and 45° orientations. Morphological analysis of reconstructed models revealed that localized increases in cortical folding density or reduced scalp-to-cortex distances near coil projection areas drove these displacements, redirecting peak electric fields toward such regions.

Discussion

Our findings elucidated how gyral cross-sectional shape and crest curvature collectively influence transcranial magnetic stimulation (TMS)-induced electric fields, thereby bridging cortical morphology with physical optimization metrics for stimulation efficacy.

The relationships between morphology and electric fields provide implications for two key clinical considerations. First, cortical atrophy patterns observed in diseases29 — through altered folding geometry — may systematically reduce electric field penetration depth. By tracking patient-specific reductions in gyral crown width, clinicians could calibrate stimulation intensity to counteract morphology-driven field attenuation, offering a targeted adjunct to symptom-guided dosing protocols. Second, the dynamic interaction between gyral crown curvature and coil orientation dictates precision thresholds across regions. While low-curvature targets (e.g., paracentral lobule) demand strict perpendicular coil alignment — exemplified by the directional dependence of motor-evoked potentials (MEPs) in hand motor areas20,30 — highly curved, geometrically complex regions like the dorsolateral prefrontal cortex (DLPFC) tolerate wider orientation ranges31,32.

The folding extent of the cortex extends more significant implications at the micro level. Theoretically, the WM fiber bundles within a highly folded cortex have a greater degree of curvature. This leads to a stronger E-field gradient along the fiber bundles33 and a higher “activation function” value8, further reducing the activation threshold required for this part of the cortical region.

Our simplified model offers initial guidance for balancing placement precision and operational complexity, particularly when anatomical variability obscures standard positioning landmarks. Notably, these advances stem not from overhauling existing electric field modeling frameworks but from strategically integrating patient-specific folding metrics. For atypical anatomies, high-resolution finite element analysis remains irreplaceable. However, our parametric relationships establish a classification mechanism: morphological metrics derived from routine magnetic resonance imaging scans can identify scenarios where rapid biomarker approximations suffice, thereby reserving computational resources for cases demanding highest precision. When high-resolution finite element computations of electric fields are unavailable, our findings guide rapid (albeit approximate) adjustments of stimulation parameters in resource-limited scenarios where personalized realistic brain models are absent.

Future research should refine these insights by incorporating whole-brain morphometric data and validating them against patient-specific simulations. Comprehensive verification would require expanding analyses to more cortical regions and anatomically realistic brain models. While our idealized model oversimplifies the complex distribution of cortical gyri and other physical factors, it establishes a foundational approach to reconcile neuroanatomical complexity with clinical practicality.

Debates persist regarding TMS activation mechanisms. Some studies propose that the induced electric field parallel to pyramidal neuron axes (aligned with cortical columns) induces maximal neuronal excitation34 and serves as a key predictor of TMS effects35,36. Others argue that axonal terminals integrate all field components during depolarization37,38,39. While we employed electric field magnitude as the primary efficacy metric, correlations between the orientation of fields relative to the cortical surface and morphological parameters warrant systematic investigation — a priority for future work.

Finally, our study adopted uniform tissue conductivity values. Differences in inter-tissue conductivity may alter cortical electric field strength, direction, and penetration depth. To enhance generalizability, we will evaluate how reported conductivity extremes influence our key findings.

Conclusion

This study investigated the effects of cortical fold morphological features on transcranial magnetic stimulation (TMS)-induced electric fields. To establish quantitative relationships between morphological characteristics and electric field parameters, we developed a simplified multi-layered curved cortical fold model and examined the associations between two classes of morphological parameters (fold complexity and curvature) with peak electric field intensity and spatial distribution. Results showed that more deeply folded structures induced stronger electric fields penetrating into deeper brain regions, while fold curvature governed the directional dependency of field intensity and the maximum displacement of peak locations. Validation using realistic head models confirmed the significant influence of cross-sectional fold geometry on electric field strength.

In summary, this work establishes a clearer connection between cortical morphology parameters and electric field metrics. It provides practical guidance for adjusting stimulation strategies in scenarios where precise field simulations are unfeasible. Furthermore, it offers direct anatomical parameters to expedite coil placement optimization in real-brain models, enhancing the efficiency of protocol design for targeted neuromodulation.