A Quasi-Static Boundary Element Approach With Fast Multipole Acceleration for High-Resolution Bioelectromagnetic Models

Decoding auditory working memory content from EEG responses to auditory-cortical TMS

Working memory (WM), short term maintenance of information for goal directed behavior, is essential to human cognition. Identifying the neural mechanisms supporting WM is a focal point of neuroscientific research. One prominent theory hypothesizes that WM content is carried in "activity-silent" brain states involving short-term synaptic changes. Information carried in such brain states could be decodable from content-specific changes in responses to unrelated "impulse stimuli". Here, we used single-pulse transcranial magnetic stimulation (spTMS) as the impulse stimulus and then decoded content maintained in WM from EEG using multivariate pattern analysis (MVPA) with robust non-parametric permutation testing. The decoding accuracy of WM content significantly enhanced after spTMS was delivered to the posterior superior temporal cortex during WM maintenance. Our results show that WM maintenance involves brain states, which are activity silent relative to other intrinsic processes visible in the EEG signal.

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

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 method matches TMS experimental thresholds.

CONCLUSIONS

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

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

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.

Fast EEG/MEG BEM-based forward problem solution for high-resolution head models

A fast BEM (boundary element method) based approach is developed to solve an EEG/MEG forward problem for a modern high-resolution head model. The method utilizes a charge-based BEM accelerated by the fast multipole method (BEM-FMM) with an adaptive mesh pre-refinement method (called b-refinement) close to the singular dipole source(s). No costly matrix-filling or direct solution steps typical for the standard BEM are required; the method generates on-skin voltages as well as MEG magnetic fields for high-resolution head models within 90 s after initial model assembly using a regular workstation. The forward method is validated by comparison against an analytical solution on a spherical shell model as well as comparison against a full h-refinement method on realistic 1M facet human head models, both of which yield agreement to within 5 % for the EEG skin potential and MEG magnetic fields. The method is further applied to an EEG source localization (inverse) problem for real human data, and a reasonable source dipole distribution is found.

Decoding auditory working memory content from EEG aftereffects of auditory-cortical TMS

Working memory (WM), short term maintenance of information for goal directed behavior, is essential to human cognition. Identifying the neural mechanisms supporting WM is a focal point of neuroscientific research. One prominent theory hypothesizes that WM content is carried in "activity-silent" brain states involving short-term synaptic changes. Information carried in such brain states could be decodable from content-specific changes in responses to unrelated "impulse stimuli". Here, we used single-pulse transcranial magnetic stimulation (spTMS) as the impulse stimulus and then decoded content maintained in WM from EEG using multivariate pattern analysis (MVPA) with robust non-parametric permutation testing. The decoding accuracy of WM content significantly enhanced after spTMS was delivered to the posterior superior temporal cortex during WM maintenance. Our results show that WM maintenance involves brain states, which are activity silent relative to other intrinsic processes visible in the EEG signal.

Enabling electric field model of microscopically realistic brain

BACKGROUND

Modeling brain stimulation at the microscopic scale may reveal new paradigms for various stimulation modalities.

OBJECTIVE

We present the largest map to date of extracellular electric field distributions within a layer L2/L3 mouse primary visual cortex brain sample. This was enabled by the automated analysis of serial section electron microscopy images with improved handling of image defects, covering a volume of 250 × 140 × 90 μm³.

METHODS

The map was obtained by applying a uniform brain stimulation electric field at three different polarizations and accurately computing microscopic field perturbations using the boundary element fast multipole method. We used the map to identify the effect of microscopic field perturbations on the activation thresholds of individual neurons. Previous relevant studies modeled a macroscopically homogeneous cortical volume.

RESULT

Our result shows that the microscopic field perturbations - an 'electric field spatial noise' with a mean value of zero - only modestly influence the macroscopically predicted stimulation field strengths necessary for neuronal activation. The thresholds do not change by more than 10 % on average.

CONCLUSION

Under the stated limitations and assumptions of our method, this result essentially justifies the conventional theory of "invisible" neurons embedded in a macroscopic brain model for transcranial magnetic and transcranial electrical stimulation. However, our result is solely sample-specific and is only relevant to this relatively small sample with 396 neurons. It largely neglects the effect of the microcapillary network. Furthermore, we only considered the uniform impressed field and a single-pulse stimulation time course.

Highly accurate calculation of electric field for transcranial magnetic stimulation using hybridizable discontinuous galerkin method

The computation of electric field in transcranial magnetic stimulation (TMS) is essentially a problem of gradient calculation for thin layers. This paper introduces a hybrid-order hybridizable discontinuous Galerkin finite element method (HDG-FEM) and systematically demonstrates its superiority in TMS computations. The discrete format of HDG-FEM employing hybrid orders for TMS is derived and, from a fundamental numerical principle perspective, this study provides the elucidation of why HDG-FEM exhibits superior gradient computation capabilities compared to the widely used CG-FEM. Furthermore, the exceptional performance of HDG-FEM in thin layer calculation is demonstrated on both modified head models and realistic head models, focusing on three aspects: calculation errors, utilization of hybrid order, and computational cost. For the calculation of E-field in thin-layer regions with parameter mutation, the L∞ norm error of the first-order HDG-FEM with the same tetrahedral mesh is comparable to the L∞ norm error of the second-order CG-FEM. The L2 norm error of the same-order HDG-FEM is smaller than that of the same-order CG-FEM. By utilizing the hybrid order, HDG-FEM achieves a rapid reduction in errors of thin layers without a significant increase in the computational cost. This study transforms the three-dimensional TMS problem into a special two-dimensional problem for computation, reducing computational complexity from p3 in three dimensions to p2 in two dimensions, while achieving significantly higher accuracy compared to the commonly used CG-FEM. The utilization of hybrid orders in thin layers of the head demonstrates significant flexibility, making HDG-FEM a new alternative choice for TMS computations.

Novel Volume Integral Equation Approach for Low-Frequency E-Field Dosimetry of Transcranial Magnetic Stimulation

A new volume integral equation (VIE) approach is introduced to study transcranial magnetic stimulation (TMS) and high-contrast media at low frequencies. This new integral equation offers a simple solution to the high-contrast breakdown observed in low-frequency electric field (E-field) dosimetry of conductive media. Specifically, we employ appropriate approximations that are valid for low frequencies and stabilize the VIE by introducing a basis expansion set that removes solutions associated with high eigenvalues in the equation. The new equation is devoid of high-contrast breakdown and does not require the use of auxiliary surface variables or projectors, providing a straightforward practical solution for the VIE analysis of TMS. Our results indicate that the novel VIE formulation matches boundary element, finite element, and analytical solutions. This new VIE represents a first step towards including anisotropy in integral equation E-field dosimetry for brain stimulation.

Enabling Electric Field Model of Microscopically Realistic Brain

Modeling brain stimulation at the microscopic scale may reveal new paradigms for various stimulation modalities. We present the largest map to date of extracellular electric field distributions within a layer L2/L3 mouse primary visual cortex brain sample. This was enabled by the automated analysis of serial section electron microscopy images with improved handling of image defects, covering a volume of 250 × 140 × 90 μm 3 . The map was obtained by applying a uniform brain stimulation electric field at three different polarizations and accurately computing microscopic field perturbations using the boundary element fast multipole method. We used the map to identify the effect of microscopic field perturbations on the activation thresholds of individual neurons. Previous relevant studies modeled a macroscopically homogeneous cortical volume. Our result shows that the microscopic field perturbations - an 'electric field spatial noise' with a mean value of zero - only modestly influence the macroscopically predicted stimulation field strengths necessary for neuronal activation. The thresholds do not change by more than 10% on average. Under the stated limitations and assumptions of our method, this result justifies the conventional theory of "invisible" neurons embedded in a macroscopic brain model for transcranial magnetic and transcranial electrical stimulation. However, our result is solely sample-specific and largely neglects the effect of the microcapillary network. Furthermore, we only considered the uniform impressed field and a single- pulse stimulation time course.

Significance statement

This study is arguably the first attempt to model brain stimulation at the microscopic scale, enabled by automated analysis of modern scanning electron microscopy images of the brain. It concentrates on modeling microscopic perturbations of the extracellular electric field caused by the physical cell structure and is applicable to any type of brain stimulation.

Data availability statement

Post-processed cell CAD models (383, stl format), microcapillary CAD models (34, stl format), post-processed neuron morphologies (267, swc format), extracellular electric field and potential distributions at different polarizations (267x3, MATLAB format), *.ses projects files for biophysical modeling with Neuron software (267x2, Neuron format), and computed neuron activating thresholds at different conditions (267x8, Excel tables, without the sample polarization correction from Section 2.8) are made available online through BossDB , a volumetric open-source database for 3D and 4D neuroscience data.

Improving EEG Forward Modeling Using High-Resolution Five-Layer BEM-FMM Head Models: Effect on Source Reconstruction Accuracy

Electroencephalographic (EEG) source localization is a fundamental tool for clinical diagnoses and brain-computer interfaces. We investigate the impact of model complexity on reconstruction accuracy by comparing the widely used three-layer boundary element method (BEM) as an inverse method against a five-layer BEM accelerated by the fast multipole method (BEM-FMM) and coupled with adaptive mesh refinement (AMR) as forward solver. Modern BEM-FMM with AMR can solve high-resolution multi-tissue models efficiently and accurately. We generated noiseless 256-channel EEG data from 15 subjects in the Connectome Young Adult dataset, using four anatomically relevant dipole positions, three conductivity sets, and two head segmentations; we mapped localization errors across the entire grey matter from 4000 dipole positions. The average location error among our four selected dipoles is ∼5mm (±2mm) with an orientation error of ∼12∘ (±7∘). The average source localization error across the entire grey matter is ∼9mm (±4mm), with a tendency for smaller errors on the occipital lobe. Our findings indicate that while three-layer models are robust under noiseless conditions, substantial localization errors (10-20mm) are common. Therefore, models of five or more layers may be needed for accurate source reconstruction in critical applications involving noisy EEG data.

How conductivity boundaries influence the electric field induced by transcranial magnetic stimulation in in vitro experiments

BACKGROUND

Although transcranial magnetic stimulation (TMS) has become a valuable method for non-invasive brain stimulation, the cellular basis of TMS activation of neurons is still not fully understood. In vitro preparations have been used to understand the biophysical mechanisms of TMS, but in many cases these studies have encountered substantial difficulties in activating neurons.

OBJECTIVE/HYPOTHESIS

The hypothesis of this work is that conductivity boundaries can have large effects on the electric field in commonly used in vitro preparations. Our goal was to analyze the resulting difficulties in in vitro TMS using a simulation study, using a charge-based boundary element model.

METHODS

We decomposed the total electric field into the sum of the primary electric field, which only depends on coil geometry and current, and the secondary electric field arising from conductivity boundaries, which strongly depends on tissue and chamber geometry. We investigated the effect of the conductivity boundaries on the electric field strength for a variety of in vitro experimental settings to determine the sources of difficulty.

RESULTS

We showed that conductivity boundaries can have large effects on the electric field in in vitro preparations. Depending on the geometry of the air-saline and the saline-tissue interfaces, the secondary electric field can significantly enhance, or attenuate the primary electric field, resulting in a much stronger or weaker total electric field inside the tissue; we showed this using a realistic preparation. Submerged chambers are generally much more efficient than interface chambers since the secondary field due to the thin film of saline covering the tissue in the interface chamber opposes the primary field and significantly reduces the total field in the tissue placed in the interface chamber. The relative dimensions of the chamber and the TMS coil critically determine the total field; the popular setup with a large coil and a small chamber is particularly sub-optimal because the secondary field due to the air-chamber boundary opposes the primary field, thereby attenuating the total field. The form factor (length vs width) of the tissue in the direction of the induced field can be important since a relatively narrow tissue enhances the total field at the saline-tissue boundary.

CONCLUSIONS

Overall, we found that the total electric field in the tissue is higher in submerged chambers, higher if the chamber size is larger than the coil and if the shorter tissue dimension is in the direction of the electric field. Decomposing the total field into the primary and secondary fields is useful for designing in vitro experiments and interpreting the results.

A survey on integral equations for bioelectric modeling

Bioelectric modeling problems, such as electroencephalography, magnetoencephalography, transcranial electrical stimulation, deep brain stimulation, and transcranial magnetic stimulation, among others, can be approached through the formulation and resolution of integral equations of theboundary element method(BEM). Recently, it has been realized that thecharge-based formulationof the BEM is naturally well-suited for the application of thefast multipole method(FMM). The FMM is a powerful algorithm for the computation of many-body interactions and is widely applied in electromagnetic modeling problems. With the introduction of BEM-FMM in the context of bioelectromagnetism, the BEM can now compete with thefinite element method(FEM) in a number of application cases. This survey has two goals: first, to give a modern account of the main BEM formulations in the literature and their integration with FMM, directed to general researchers involved in development of BEM software for bioelectromagnetic applications. Second, to survey different techniques and available software, and to contrast different BEM and FEM approaches. As a new contribution, we showcase that the charge-based formulation is dual to the more common surface potential formulation.

Quasistatic approximation in neuromodulation

We define and explain the quasistatic approximation (QSA) as applied to field modeling for electrical and magnetic stimulation. Neuromodulation analysis pipelines include discrete stages, and QSA is applied specifically when calculating the electric and magnetic fields generated in tissues by a given stimulation dose. QSA simplifies the modeling equations to support tractable analysis, enhanced understanding, and computational efficiency. The application of QSA in neuromodulation is based on four underlying assumptions: (A1) no wave propagation or self-induction in tissue, (A2) linear tissue properties, (A3) purely resistive tissue, and (A4) non-dispersive tissue. As a consequence of these assumptions, each tissue is assigned a fixed conductivity, and the simplified equations (e.g. Laplace's equation) are solved for the spatial distribution of the field, which is separated from the field's temporal waveform. Recognizing that electrical tissue properties may be more complex, we explain how QSA can be embedded in parallel or iterative pipelines to model frequency dependence or nonlinearity of conductivity. We survey the history and validity of QSA across specific applications, such as microstimulation, deep brain stimulation, spinal cord stimulation, transcranial electrical stimulation, and transcranial magnetic stimulation. The precise definition and explanation of QSA in neuromodulation are essential for rigor when using QSA models or testing their limits.

Fast and Accurate EEG/MEG BEM-Based Forward Problem Solution for High-Resolution Head Models

A BEM (boundary element method) based approach is developed to accurately solve an EEG/MEG forward problem for a modern high-resolution head model in approximately 60 seconds using a common workstation. The method utilizes a charge-based BEM with fast multipole acceleration (BEM-FMM) and a "smart" mesh pre-refinement (called b-refinement) close to the singular source(s). No costly matrix-filling or direct solution steps typical for the standard BEM are required; the method generates on-skin voltages as well as MEG magnetic fields for high-resolution head models in approximately 60 seconds after initial model assembly. The method is verified both theoretically and experimentally.

Accuracy of dipole source reconstruction in the 3-layer BEM model against the 5-layer BEM-FMM model

Objective

To compare cortical dipole fitting spatial accuracy between the widely used yet highly simplified 3-layer and modern more realistic 5-layer BEM-FMM models with and without adaptive mesh refinement (AMR) methods.

Methods

We generate simulated noiseless 256-channel EEG data from 5-layer (7-compartment) meshes of 15 subjects from the Connectome Young Adult dataset. For each subject, we test four dipole positions, three sets of conductivity values, and two types of head segmentation. We use the boundary element method (BEM) with fast multipole method (FMM) acceleration, with or without (AMR), for forward modeling. Dipole fitting is carried out with the FieldTrip MATLAB toolbox.

Results

The average position error (across all tested dipoles, subjects, and models) is ~4 mm, with a standard deviation of ~2 mm. The orientation error is ~20° on average, with a standard deviation of ~15°. Without AMR, the numerical inaccuracies produce a larger disagreement between the 3- and 5-layer models, with an average position error of ~8 mm (6 mm standard deviation), and an orientation error of 28° (28° standard deviation).

Conclusions

The low-resolution 3-layer models provide excellent accuracy in dipole localization. On the other hand, dipole orientation is retrieved less accurately. Therefore, certain applications may require more realistic models for practical source reconstruction. AMR is a critical component for improving the accuracy of forward EEG computations using a high-resolution 5-layer volume conduction model.

Significance

Improving EEG source reconstruction accuracy is important for several clinical applications, including epilepsy and other seizure-inducing conditions.

A review of algorithms and software for real-time electric field modeling techniques for transcranial magnetic stimulation

Transcranial magnetic stimulation (TMS) is a device-based neuromodulation technique increasingly used to treat brain diseases. Electric field (E-field) modeling is an important technique in several TMS clinical applications, including the precision stimulation of brain targets with accurate stimulation density for the treatment of mental disorders and the localization of brain function areas for neurosurgical planning. Classical methods for E-field modeling usually take a long computation time. Fast algorithms are usually developed with significantly lower spatial resolutions that reduce the prediction accuracy and limit their usage in real-time or near real-time TMS applications. This review paper discusses several modern algorithms for real-time or near real-time TMS E-field modeling and their advantages and limitations. The reviewed methods include techniques such as basis representation techniques and deep neural-network-based methods. This paper also provides a review of software tools that can integrate E-field modeling with navigated TMS, including a recent software for real-time navigated E-field mapping based on deep neural-network models.

Quasistatic approximation in neuromodulation

We define and explain the quasistatic approximation (QSA) as applied to field modeling for electrical and magnetic stimulation. Neuromodulation analysis pipelines include discrete stages, and QSA is applied specifically when calculating the electric and magnetic fields generated in tissues by a given stimulation dose. QSA simplifies the modeling equations to support tractable analysis, enhanced understanding, and computational efficiency. The application of QSA in neuro-modulation is based on four underlying assumptions: (A1) no wave propagation or self-induction in tissue, (A2) linear tissue properties, (A3) purely resistive tissue, and (A4) non-dispersive tissue. As a consequence of these assumptions, each tissue is assigned a fixed conductivity, and the simplified equations (e.g., Laplace's equation) are solved for the spatial distribution of the field, which is separated from the field's temporal waveform. Recognizing that electrical tissue properties may be more complex, we explain how QSA can be embedded in parallel or iterative pipelines to model frequency dependence or nonlinearity of conductivity. We survey the history and validity of QSA across specific applications, such as microstimulation, deep brain stimulation, spinal cord stimulation, transcranial electrical stimulation, and transcranial magnetic stimulation. The precise definition and explanation of QSA in neuromodulation are essential for rigor when using QSA models or testing their limits.

Electric Field Modeling in Personalizing Transcranial Magnetic Stimulation Interventions

The modeling of transcranial magnetic stimulation (TMS)-induced electric fields (E-fields) is a versatile technique for evaluating and refining brain targeting and dosing strategies, while also providing insights into dose-response relationships in the brain. This review outlines the methodologies employed to derive E-field estimations, covering TMS physics, modeling assumptions, and aspects of subject-specific head tissue and coil modeling. We also summarize various numerical methods for solving the E-field and their suitability for various applications. Modeling methodologies have been optimized to efficiently execute numerous TMS simulations across diverse scalp coil configurations, facilitating the identification of optimal setups or rapid cortical E-field visualization for specific brain targets. These brain targets are extrapolated from neurophysiological measurements and neuroimaging, enabling precise and individualized E-field dosing in experimental and clinical applications. This necessitates the quantification of E-field estimates using metrics that enable the comparison of brain target engagement, functional localization, and TMS intensity adjustments across subjects. The integration of E-field modeling with empirical data has the potential to uncover pivotal insights into the aspects of E-fields responsible for stimulating and modulating brain function and states, enhancing behavioral task performance, and impacting the clinical outcomes of personalized TMS interventions.

High-resolution EEG source localization in personalized segmentation-free head model with multi-dipole fitting

Objective. Electroencephalograms (EEGs) are often used to monitor brain activity. Several source localization methods have been proposed to estimate the location of brain activity corresponding to EEG readings. However, only a few studies evaluated source localization accuracy from measured EEG using personalized head models in a millimeter resolution. In this study, based on a volume conductor analysis of a high-resolution personalized human head model constructed from magnetic resonance images, a finite difference method was used to solve the forward problem and to reconstruct the field distribution.Approach. We used a personalized segmentation-free head model developed using machine learning techniques, in which the abrupt change of electrical conductivity occurred at the tissue interface is suppressed. Using this model, a smooth field distribution was obtained to address the forward problem. Next, multi-dipole fitting was conducted using EEG measurements for each subject (N= 10 male subjects, age: 22.5 ± 0.5), and the source location and electric field distribution were estimated.Main results.For measured somatosensory evoked potential for electrostimulation to the wrist, a multi-dipole model with lead field matrix computed with the volume conductor model was found to be superior than a single dipole model when using personalized segmentation-free models (6/10). The correlation coefficient between measured and estimated scalp potentials was 0.89 for segmentation-free head models and 0.71 for conventional segmented models. The proposed method is straightforward model development and comparable localization difference of the maximum electric field from the target wrist reported using fMR (i.e. 16.4 ± 5.2 mm) in previous study. For comparison, DUNEuro based on sLORETA was (EEG: 17.0 ± 4.0 mm). In addition, somatosensory evoked magnetic fields obtained by Magnetoencephalography was 25.3 ± 8.5 mm using three-layer sphere and sLORETA.Significance. For measured EEG signals, our procedures using personalized head models demonstrated that effective localization of the somatosensory cortex, which is located in a non-shallower cortex region. This method may be potentially applied for imaging brain activity located in other non-shallow regions.

Estimations of Charge Deposition Onto Convoluted Axon Surfaces Within Extracellular Electric Fields

OBJECTIVE

Biophysical models of neural stimulation are a valuable approach to explaining the mechanisms of neuronal recruitment via applied extracellular electric fields. Typically, the applied electric field is estimated via a macroscopic finite element method solution and then applied to cable models as an extracellular voltage source. However, the field resolution is limited by the finite element size (typically 10's-100's of times greater than average neuronal cross-section). As a result, induced charges deposited onto anatomically realistic curved membrane interfaces are not taken into consideration. However, these details may alter estimates of the applied electric field and predictions of neural tissue activation.

METHODS

To estimate microscopic variations of the electric field, data for intra-axonal space segmented from 3D scanning electron microscopy of the mouse brain genu of corpus callosum were used. The boundary element fast multipole method was applied to accurately compute the extracellular solution. Neuronal recruitment was then estimated via an activating function.

RESULTS

Taking the physical structure of the arbor into account generally predicts higher values of the activating function. The relative integral 2-norm difference is 90% on average when the entire axonal arbor is present. A large fraction of this difference might be due to the axonal body itself. When an isolated physical axon is considered with all other axons removed, the relative integral 2-norm difference between the single-axon solution and the complete solution is 25% on average.

CONCLUSION

Our result may provide an explanation as to why Deep Brain Stimulation experiments typically predict lower activation thresholds than commonly used FEM/Cable model approaches to predicting neuronal responses to extracellular electrical stimulation.

SIGNIFICANCE

These results may change methods for bi-domain neural modeling and neural excitation.

Detailed high quality surface-based mouse CAD model suitable for electromagnetic simulations

Transcranial magnetic stimulation (TMS) studies with small animals can provide useful knowledge of activating regions and mechanisms. Along with this, functional magnetic resonance imaging (fMRI) in mice and rats is increasingly often used to draw important conclusions about brain connectivity and functionality. For cases of both low- and high-frequency TMS studies, a high-quality computational surface-based rodent model may be useful as a tool for performing supporting modeling and optimization tasks. This work presents the development and usage of an accurate CAD model of a mouse that has been optimized for use in computational electromagnetic modeling in any frequency range. It is based on the labeled atlas data of the Digimouse archive. The model includes a relatively accurate four-compartment brain representation (the 'whole brain' according to the original terminology, external cerebrum, cerebellum, and striatum [9]) and contains 21 distinct compartments in total. Four examples of low- and high frequency modeling have been considered to demonstrate the utility and applicability of the model.

A fast direct solver for surface-based whole-head modeling of transcranial magnetic stimulation

When modeling transcranial magnetic stimulation (TMS) in the brain, a fast and accurate electric field solver can support interactive neuronavigation tasks as well as comprehensive biophysical modeling. We formulate, test, and disseminate a direct (i.e., non-iterative) TMS solver that can accurately determine global TMS fields for any coil type everywhere in a high-resolution MRI-based surface model with ~ 200,000 or more arbitrarily selected observation points within approximately 5 s, with the solution time itself of 3 s. The solver is based on the boundary element fast multipole method (BEM-FMM), which incorporates the latest mathematical advancement in the theory of fast multipole methods-an FMM-based LU decomposition. This decomposition is specific to the head model and needs to be computed only once per subject. Moreover, the solver offers unlimited spatial numerical resolution. Despite the fast execution times, the present direct solution is numerically accurate for the default model resolution. In contrast, the widely used brain modeling software SimNIBS employs a first-order finite element method that necessitates additional mesh refinement, resulting in increased computational cost. However, excellent agreement between the two methods is observed for various practical test cases following mesh refinement, including a biophysical modeling task. The method can be readily applied to a wide range of TMS analyses involving multiple coil positions and orientations, including image-guided neuronavigation. It can even accommodate continuous variations in coil geometry, such as flexible H-type TMS coils. The FMM-LU direct solver is freely available to academic users.

Influence of charges deposited on membranes of human hyperdirect pathway axons on depolarization during subthalamic deep brain stimulation

Objective.The motor hyperdirect pathway (HDP) is a key target in the treatment of Parkinson's disease with deep brain stimulation (DBS). Biophysical models of HDP DBS have been used to explore the mechanisms of stimulation. Built upon finite element method volume conductor solutions, such models are limited by a resolution mismatch, where the volume conductor is modeled at the macro scale, while the neural elements are at the micro scale. New techniques are needed to better integrate volume conductor models with neuron models.Approach.We simulated subthalamic DBS of the human HDP using finely meshed axon models to calculate surface charge deposition on insulting membranes of nonmyelinated axons. We converted the corresponding double layer extracellular problem to a single layer problem and applied the well-conditioned charge-based boundary element fast multipole method (BEM-FMM) with unconstrained numerical spatial resolution. Commonly used simplified estimations of membrane depolarization were compared with more realistic solutions.Main result.Neither centerline potential nor estimates of axon recruitment were impacted by the estimation method used except at axon bifurcations and hemispherical terminations. Local estimates of axon polarization were often much higher at bifurcations and terminations than at any other place along the axon and terminal arbor. Local average estimates of terminal electric field are higher by 10%-20%.Significance. Biophysical models of action potential initiation in the HDP suggest that axon terminations are often the lowest threshold elements for activation. The results of this study reinforce that hypothesis and suggest that this phenomenon is even more pronounced than previously realized.

A fast direct solver for surface-based whole-head modeling of transcranial magnetic stimulation

Background

When modeling transcranial magnetic stimulation (TMS) in the brain, a fast and accurate electric field solver can support interactive neuronavigation tasks as well as comprehensive biophysical modeling.

Objective

We formulate, test, and disseminate a direct (i.e., non-iterative) TMS solver that can accurately determine global TMS fields for any coil type everywhere in a high-resolution MRI-based surface model with ~200,000 or more arbitrarily selected observation points within approximately 5 sec, with the solution time itself of 3 sec.

Method

The solver is based on the boundary element fast multipole method (BEM-FMM), which incorporates the latest mathematical advancement in the theory of fast multipole methods - an FMM-based LU decomposition. This decomposition is specific to the head model and needs to be computed only once per subject. Moreover, the solver offers unlimited spatial numerical resolution.

Results

Despite the fast execution times, the present direct solution is numerically accurate for the default model resolution. In contrast, the widely used brain modeling software SimNIBS employs a first-order finite element method that necessitates additional mesh refinement, resulting in increased computational cost. However, excellent agreement between the two methods is observed for various practical test cases following mesh refinement, including a biophysical modeling task.

Conclusion

The method can be readily applied to a wide range of TMS analyses involving multiple coil positions and orientations, including image-guided neuronavigation. It can even accommodate continuous variations in coil geometry, such as flexible H-type TMS coils. The FMM-LU direct solver is freely available to academic users.

Network-specific metabolic and haemodynamic effects elicited by non-invasive brain stimulation

Repetitive transcranial magnetic stimulation (TMS), when applied to the dorsolateral prefrontal cortex (dlPFC), treats depression. Therapeutic effects are hypothesized to arise from propagation of local dlPFC stimulation effects across distributed networks; however, the mechanisms of this remain unresolved. dlPFC contains representations of different networks. As such, dlPFC TMS may exert different effects depending on the network being stimulated. Here, to test this, we applied high-frequency TMS to two nearby dlPFC targets functionally embedded in distinct anti-correlated networks-the default and salience networks- in the same individuals in separate sessions. Local and distributed TMS effects were measured with combined 18fluorodeoxyglucose positron emission tomography and functional magnetic resonance imaging. Identical TMS patterns caused opposing effects on local glucose metabolism: metabolism increased at the salience target following salience TMS but decreased at the default target following default TMS. At the distributed level, both conditions increased functional connectivity between the default and salience networks, with this effect being dramatically larger following default TMS. Metabolic and haemodynamic effects were also linked: across subjects, the magnitude of local metabolic changes correlated with the degree of functional connectivity changes. These results suggest that TMS effects upon dlPFC are network specific. They also invoke putative antidepressant mechanisms of TMS: network de-coupling.

Fast computational E-field dosimetry for transcranial magnetic stimulation using adaptive cross approximation and auxiliary dipole method (ACA-ADM)

Transcranial Magnetic Stimulation (TMS) is a non-invasive brain stimulation technique that uses a coil to induce an electric field (E-field) in the brain and modulate its activity. Many applications of TMS call for the repeated execution of E-field solvers to determine the E-field induced in the brain for different coil placements. However, the usage of solvers for these applications remains impractical because each coil placement requires the solution of a large linear system of equations. We develop a fast E-field solver that enables the rapid evaluation of the E-field distribution for a brain region of interest (ROI) for a large number of coil placements, which is achieved in two stages. First, during the pre-processing stage, the mapping between coil placement and brain ROI E-field distribution is approximated from E-field results for a few coil placements. Specifically, we discretize the mapping into a matrix with each column having the ROI E-field samples for a fixed coil placement. This matrix is approximated from a few of its rows and columns using adaptive cross approximation (ACA). The accuracy, efficiency, and applicability of the new ACA approach are determined by comparing its E-field predictions with analytical and standard solvers in spherical and MRI-derived head models. During the second stage, the E-field distribution in the brain ROI from a specific coil placement is determined by the obtained rows and columns in milliseconds. For many applications, only the E-field distribution for a comparatively small ROI is required. For example, the solver can complete the pre-processing stage in approximately 4 hours and determine the ROI E-field in approximately 40 ms for a 100 mm diameter ROI with less than 2% error enabling its use for neuro-navigation and other applications. Highlight: We developed a fast solver for TMS computational E-field dosimetry, which can determine the ROI E-field in approximately 40 ms for a 100 mm diameter ROI with less than 2% error.

Devices and Technology in Transcranial Magnetic Stimulation: A Systematic Review

The technology for transcranial magnetic stimulation (TMS) has significantly changed over the years, with important improvements in the signal generators, the coils, the positioning systems, and the software for modeling, optimization, and therapy planning. In this systematic literature review (SLR), the evolution of each component of TMS technology is presented and analyzed to assess the limitations to overcome. This SLR was carried out following the PRISMA 2020 statement. Published articles of TMS were searched for in four databases (Web of Science, PubMed, Scopus, IEEE). Conference papers and other reviews were excluded. Records were filtered using terms about TMS technology with a semi-automatic software; articles that did not present new technology developments were excluded manually. After this screening, 101 records were included, with 19 articles proposing new stimulator designs (18.8%), 46 presenting or adapting coils (45.5%), 18 proposing systems for coil placement (17.8%), and 43 implementing algorithms for coil optimization (42.6%). The articles were blindly classified by the authors to reduce the risk of bias. However, our results could have been influenced by our research interests, which would affect conclusions for applications in psychiatric and neurological diseases. Our analysis indicates that more emphasis should be placed on optimizing the current technology with a special focus on the experimental validation of models. With this review, we expect to establish the base for future TMS technological developments.

[Simulation study of force and temperature field during transcranial magnetic stimulation application working with magnetic resonance imaging simultaneously]

Currently, transcranial magnetic stimulation (TMS) has been widely used in the treatment of depression, Parkinson's disease and other neurological diseases. To be able to monitor the brain's internal activity during TMS in real time and achieve better treatment outcomes, the researchers proposed combining TMS with neuroimaging methods such as magnetic resonance imaging (MRI), both of which use Tesla-level magnetic fields. However, the combination of strong current, large magnetic field and small size is likely to bring physical concerns which can lead to mechanical and thermal instability. In this paper, the MRI static magnetic field, the TMS coil and human head model were built according to the actual situations. Through the coupling of the magnetic field and the heat transfer module in the finite element simulation software COMSOL, the force and temperature of the TMS coil and head were obtained when the TMS was used in combination with MRI (TMS-MRI technology). The results showed that in a 3 T MRI environment, the maximum force density on the coil could reach 2.51 × 109 N/m3. Both the direction of the external magnetic field and the current direction in the coil affected the force distributions. The closer to the boundary of the external magnetic field, the greater the force. The magnetic field generated by the coil during TMS treatment increased the temperature of the brain tissue by about 0.16 °C, and the presence of the MRI static magnetic field did not cause additional thermal effects. The results of this paper can provide a reference for the development of the use guidelines and safety guidelines of TMS-MRI technology.

The effect of meninges on the electric fields in TES and TMS. Numerical modeling with adaptive mesh refinement

BACKGROUND

When modeling transcranial electrical stimulation (TES) and transcranial magnetic stimulation (TMS) in the brain, the meninges - dura, arachnoid, and pia mater - are often neglected due to high computational costs.

OBJECTIVE

We investigate the impact of the meningeal layers on the cortical electric field in TES and TMS while considering the headreco segmentation as the base model.

METHOD

We use T1/T2 MRI data from 16 subjects and apply the boundary element fast multipole method with adaptive mesh refinement, which enables us to accurately solve this problem and establish method convergence at reasonable computational cost. We compare electric fields in the presence and absence of various meninges for two brain areas (M1_HAND and DLPFC) and for several distinct TES and TMS setups.

RESULTS

Maximum electric fields in the cortex for focal TES consistently increase by approximately 30% on average when the meninges are present in the CSF volume. Their effect on the maximum field can be emulated by reducing the CSF conductivity from 1.65 S/m to approximately 0.85 S/m. In stark contrast to that, the TMS electric fields in the cortex are only weakly affected by the meningeal layers and slightly (∼6%) decrease on average when the meninges are included.

CONCLUSION

Our results quantify the influence of the meninges on the cortical TES and TMS electric fields. Both focal TES and TMS results are very consistent. The focal TES results are also in a good agreement with a prior relevant study. The solver and the mesh generator for the meningeal layers (compatible with SimNIBS) are available online.

Optimal Design of Transcranial Magnetic Stimulation Coil with Iron Core

OBJECTIVE

Iron core coils offer a passive way to increase the induced electric field intensity during transcranial magnetic stimulation (TMS), but the influences of core position and dimensions on coil performance have not been elaborately discussed before.

APPROACH

In this study, with the basic figure-of-eight (Fo8) and slinky coil structures, iron core coil optimization is performed with the finite element method considering core position and dimensions. A performance factor combining performance parameters, including the maximum induced electric field, stimulation depth, focus, and heat loss, is utilized to evaluate the comprehensive coil performance.

MAIN RESULTS

According to the performance factor, both iron core coils obtain the best overall performance with a fill factor 0.4 and the two legs of the iron core close to the inner sides of the coil. Finally, three prototypes are constructed-the basic, optimized, and full-size slinky iron core coil-and magnetic field detection demonstrates a good agreement with the simulation results.

SIGNIFICANCE

The proposed systematic optimization approach for iron core coil based on Fo8 and slinky basic structure can be applied to improve TMS coil performance, reduce power requirements, and guide the design of other iron core TMS coils.

Uncertainty quantification of TMS simulations considering MRI segmentation errors

OBJECTIVE

Transcranial Magnetic Stimulation (TMS) is a non-invasive brain stimulation method that is used to study brain function and conduct neuropsychiatric therapy. Computational methods that are commonly used for electric field (E-field) dosimetry of TMS are limited in accuracy and precision because of possible geometric errors introduced in the generation of head models by segmenting medical images into tissue types. This paper studies E-field prediction fidelity as a function of segmentation accuracy.

APPROACH

The errors in the segmentation of medical images into tissue types are modeled as geometric uncertainty in the shape of the boundary between tissue types. For each tissue boundary realization, we then use an in-house boundary element method to perform a forward propagation analysis and quantify the impact of tissue boundary uncertainties on the induced cortical E-field.

MAIN RESULTS

Our results indicate that predictions of E-field induced in the brain are negligibly sensitive to segmentation errors in scalp, skull and white matter, compartments. In contrast, E-field predictions are highly sensitive to possible CSF segmentation errors. Specifically, the segmentation errors on the CSF and gray matter interface lead to higher E-field uncertainties in the gyral crowns, and the segmentation errors on CSF and white matter interface lead to higher uncertainties in the sulci. Furthermore, the uncertainty of the average cortical E-fields over a region exhibits lower uncertainty relative to point-wise estimates.

SIGNIFICANCE

The accuracy of current cortical E-field simulations is limited by the accuracy of CSF segmentation accuracy. Other quantities of interest like the average of the E-field over a cortical region could provide a dose quantity that is robust to possible segmentation errors.

Uncertainty quantification of TMS simulations considering MRI segmentation errors

OBJECTIVE

Transcranial Magnetic Stimulation (TMS) is a non-invasive brain stimulation method that is used to study brain function and conduct neuropsychiatric therapy. Computational methods that are commonly used for electric field (E-field) dosimetry of TMS are limited in accuracy and precision because of possible geometric errors introduced in the generation of head models by segmenting medical images into tissue types. This paper studies E-field prediction fidelity as a function of segmentation accuracy.

APPROACH

The errors in the segmentation of medical images into tissue types are modeled as geometric uncertainty in the shape of the boundary between tissue types. For each tissue boundary realization, we then use an in-house boundary element method to perform a forward propagation analysis and quantify the impact of tissue boundary uncertainties on the induced cortical E-field.

MAIN RESULTS

Our results indicate that predictions of E-field induced in the brain are negligibly sensitive to segmentation errors in scalp, skull and white matter, compartments. In contrast, E-field predictions are highly sensitive to possible CSF segmentation errors. Specifically, the segmentation errors on the CSF and gray matter interface lead to higher E-field uncertainties in the gyral crowns, and the segmentation errors on CSF and white matter interface lead to higher uncertainties in the sulci. Furthermore, the uncertainty of the average cortical E-fields over a region exhibits lower uncertainty relative to point-wise estimates.

SIGNIFICANCE

The accuracy of current cortical E-field simulations is limited by the accuracy of CSF segmentation accuracy. Other quantities of interest like the average of the E-field over a cortical region could provide a dose quantity that is robust to possible segmentation errors.

Stochastic modeling of geometrical uncertainties on complex domains, with application to additive manufacturing and brain interface geometries

We present a stochastic modeling framework to represent and simulate spatially-dependent geometrical uncertainties on complex geometries. While the consideration of random geometrical perturbations has long been a subject of interest in computational engineering, most studies proposed so far have addressed the case of regular geometries such as cylinders and plates. Here, standard random field representations, such as Kahrunen-Loève expansions, can readily be used owing, in particular, to the relative simplicity to construct covariance operators on regular shapes. On the contrary, applying such techniques on arbitrary, non-convex domains remains difficult in general. In this work, we formulate a new representation for spatially-correlated geometrical uncertainties that allows complex domains to be efficiently handled. Building on previous contributions by the authors, the approach relies on the combination of a stochastic partial differential equation approach, introduced to capture salient features of the underlying geometry such as local curvature and singularities on the fly, and an information-theoretic model, aimed to enforce non-Gaussianity. More specifically, we propose a methodology where the interface of interest is immersed into a fictitious domain, and define algorithmic procedures to directly sample random perturbations on the manifold. A simple strategy based on statistical conditioning is also presented to update realizations and prevent self-intersections in the perturbed finite element mesh. We finally provide challenging examples to demonstrate the robustness of the framework, including the case of a gyroid structure produced by additive manufacturing and brain interfaces in patient-specific geometries. In both applications, we discuss suitable parameterization for the filtering operator and quantify the impact of the uncertainties through forward propagation.

Fast Individualized High-resolution Electric Field Modeling for Computational TMS Neuronavigation

Transcranial Magnetic Stimulation (TMS) is a non-invasive method for safe and painless activation of cortical neurons. On-line visualization of the induced Electric field (E-field) has the potential to improve quantitative targeting and dosing of stimulation, however present commercially available systems are limited by simplified approximations of the anatomy. Here, we developed a near real-time method to accurately approximate the induced E-field of a freely moving TMS coil with an individualized high-resolution head model. We use a set of magnetic dipoles around the head to approximate the total E-field of a moving TMS coil. First, we match the incident field of the dipole basis set with the incident E-field of the moving coil. Then, based on the principle of superposition and uniqueness of the solutions, we apply same basis coefficients to the total E-field of the basis set. The computed E-fields results show high similarity with an established TMS solver both in terms of the amplitude and the spatial distribution patterns. The proposed method enables rapid visualization of the E-field with ~100 ms of computation time enabling interactive planning, targeting, dosing and coil positioning tasks for TMS neuronavigation.

Precise Modulation Strategies for Transcranial Magnetic Stimulation: Advances and Future Directions

Transcranial magnetic stimulation (TMS) is a popular modulatory technique for the noninvasive diagnosis and therapy of neurological and psychiatric diseases. Unfortunately, current modulation strategies are only modestly effective. The literature provides strong evidence that the modulatory effects of TMS vary depending on device components and stimulation protocols. These differential effects are important when designing precise modulatory strategies for clinical or research applications. Developments in TMS have been accompanied by advances in combining TMS with neuroimaging techniques, including electroencephalography, functional near-infrared spectroscopy, functional magnetic resonance imaging, and positron emission tomography. Such studies appear particularly promising as they may not only allow us to probe affected brain areas during TMS but also seem to predict underlying research directions that may enable us to precisely target and remodel impaired cortices or circuits. However, few precise modulation strategies are available, and the long-term safety and efficacy of these strategies need to be confirmed. Here, we review the literature on possible technologies for precise modulation to highlight progress along with limitations with the goal of suggesting future directions for this field.

Boundary element fast multipole method for modeling electrical brain stimulation with voltage and current electrodes

Objective. To formulate, validate, and apply an alternative to the finite element method (FEM) high-resolution modeling technique for electrical brain stimulation-the boundary element fast multipole method (BEM-FMM). To include practical electrode models for both surface and embedded electrodes.Approach. Integral equations of the boundary element method in terms of surface charge density are combined with a general-purpose fast multipole method and are expanded for voltage, shunt, current, and floating electrodes. The solution of coupled and properly weighted/preconditioned integral equations is accompanied by enforcing global conservation laws: charge conservation law and Kirchhoff's current law.Main results.A sub-percent accuracy is reported as compared to the analytical solutions and simple validation geometries. Comparison to FEM considering realistic head models resulted in relative differences of the electric field magnitude in the range of 3%-6% or less. Quantities that contain higher order spatial derivatives, such as the activating function, are determined with a higher accuracy and a faster speed as compared to the FEM. The method can be easily combined with existing head modeling pipelines such as headreco or mri2mesh.Significance.The BEM-FMM does not rely on a volumetric mesh and is therefore particularly suitable for modeling some mesoscale problems with submillimeter (and possibly finer) resolution with high accuracy at moderate computational cost. Utilizing Helmholtz reciprocity principle makes it possible to expand the method to a solution of EEG forward problems with a very large number of cortical dipoles.

Degree of improving TMS focality through a geometrically stable solution of an inverse TMS problem

The Transcranial Magnetic Stimulation (TMS) inverse problem (TMS-IP) investigated in this study aims to focus the TMS induced electric field close to a specified target point defined on the gray matter interface in the M1_HAND area while otherwise minimizing it. The goal of the study is to numerically evaluate the degree of improvement of the TMS-IP solutions relative to the well-known sulcus-aligned mapping (a projection approach with the 90^∘ local sulcal angle). In total, 1536 individual TMS-IP solutions have been analyzed for multiple target points and multiple subjects using the boundary element fast multipole method (BEM-FMM) as the forward solver. Our results show that the optimal TMS inverse-problem solutions improve the focality - reduce the size of the field "hot spot" and its deviation from the target - by approximately 21-33% on average for all considered subjects, all observation points, two distinct coil types, two segmentation types, two intracortical observation surfaces under study, and three tested values of the field threshold. The inverse-problem solutions with the maximized focality simultaneously improve the TMS mapping resolution (differentiation between neighbor targets separated by approximately 10 mm) although this improvement is quite modest. Coil position/orientation and conductivity uncertainties have been included into consideration as the corresponding de-focalization factors. The present results will change when the levels of uncertainties change. Our results also indicate that the accuracy of the head segmentation critically influences the expected TMS-IP performance.

High-Resolution EEG Source Localization in Segmentation-Free Head Models Based on Finite-Difference Method and Matching Pursuit Algorithm

Electroencephalogram (EEG) is a method to monitor electrophysiological activity on the scalp, which represents the macroscopic activity of the brain. However, it is challenging to identify EEG source regions inside the brain based on data measured by a scalp-attached network of electrodes. The accuracy of EEG source localization significantly depends on the type of head modeling and inverse problem solver. In this study, we adopted different models with a resolution of 0.5 mm to account for thin tissues/fluids, such as the cerebrospinal fluid (CSF) and dura. In particular, a spatially dependent conductivity (segmentation-free) model created using deep learning was developed and used for more realist representation of electrical conductivity. We then adopted a multi-grid-based finite-difference method (FDM) for forward problem analysis and a sparse-based algorithm to solve the inverse problem. This enabled us to perform efficient source localization using high-resolution model with a reasonable computational cost. Results indicated that the abrupt spatial change in conductivity, inherent in conventional segmentation-based head models, may trigger source localization error accumulation. The accurate modeling of the CSF, whose conductivity is the highest in the head, was an important factor affecting localization accuracy. Moreover, computational experiments with different noise levels and electrode setups demonstrate the robustness of the proposed method with segmentation-free head model.

Rapid computation of TMS-induced E-fields using a dipole-based magnetic stimulation profile approach

BACKGROUND

TMS neuronavigation with on-line display of the induced electric field (E-field) has the potential to improve quantitative targeting and dosing of stimulation, but present commercially available solutions are limited by simplified approximations.

OBJECTIVE

Developing a near real-time method for accurate approximation of TMS induced E-fields with subject-specific high-resolution surface-based head models that can be utilized for TMS navigation.

METHODS

Magnetic dipoles are placed on a closed surface enclosing an MRI-based head model of the subject to define a set of basis functions for the incident and total E-fields that define the subject's Magnetic Stimulation Profile (MSP). The near real-time speed is achieved by recognizing that the total E-field of the coil only depends on the incident E-field and the conductivity boundary geometry. The total E-field for any coil position can be obtained by matching the incident field of the stationary dipole basis set with the incident E-field of the moving coil and applying the same basis coefficients to the total E-field basis functions.

RESULTS

Comparison of the MSP-based approximation with an established TMS solver shows great agreement in the E-field amplitude (relative maximum error around 5%) and the spatial distribution patterns (correlation >98%). Computation of the E-field took ~100 ms on a cortical surface mesh with 120k facets.

CONCLUSION

The numerical accuracy and speed of the MSP approximation method make it well suited for a wide range of computational tasks including interactive planning, targeting, dosing, and visualization of the intracranial E-fields for near real-time guidance of coil positioning.

Boundary Element Fast Multipole Method for Enhanced Modeling of Neurophysiological Recordings

OBJECTIVE

A new numerical modeling approach is proposed which provides forward-problem solutions for both noninvasive recordings (EEG/MEG) and higher-resolution intracranial recordings (iEEG).

METHODS

The algorithm is our recently developed boundary element fast multipole method or BEM-FMM. It is based on the integration of the boundary element formulation in terms of surface charge density and the fast multipole method originating from its inventors. The algorithm still possesses the major advantage of the conventional BEM - high speed - but is simultaneously capable of processing a very large number of surface-based unknowns. As a result, an unprecedented spatial resolution could be achieved, which enables multiscale modeling.

RESULTS

For non-invasive EEG/MEG, we are able to accurately solve the forward problem with approximately 1 mm anatomical resolution in the cortex within 1-2 min given several thousand cortical dipoles. Targeting high-resolution iEEG, we are able to compute, for the first time, an integrated electromagnetic response for an ensemble (2,450) of tightly packed realistic pyramidal neocortical neurons in a full-head model with 0.6 mm anatomical cortical resolution. The neuronal arbor is comprised of 5.9 M elementary 1.2 μm long dipoles. On a standard server, the computations require about 5 min.

CONCLUSION

Our results indicate that the BEM-FMM approach may be well suited to support numerical multiscale modeling pertinent to modern high-resolution and submillimeter iEEG.

SIGNIFICANCE

Based on the speed and ease of implementation, this new algorithm represents a method that will greatly facilitate simulations at multi-scale across a variety of applications.

A 3-axis coil design for multichannel TMS arrays

Purpose:

Multichannel Transcranial Magnetic Stimulation (mTMS) arrays enable multiple sites to be stimulated simultaneously or sequentially under electronic control without moving the system’s stimulation coils. Here, we build and characterize the performance of a novel modular 3-axis TMS coil that can be utilized as a unit element in large-scale multichannel TMS arrays.

Methods:

We determined the basic physical characteristics of the 3-axis TMS coil x-, y- and z-elements using a custom 2-channel programmable stimulator prototype. We mapped the temporal rate-of-change of the induced magnetic field (dB/dt) on a 2D plane parallel to the coil surface (including an extended line for full spatial coverage) and compared those values with predictions from magnetic field simulations. Temperature measurements were carried out to assess the incorporated air-cooling method. We measured the mutual and self-inductances of the x/y/z-elements to assess coupling between them. Additionally, we measured and calculated the coupling between z-elements in the array configuration. Finally, we performed electric field simulations to evaluate the stimulation intensity and focality of the coil and compared the results to conventional TMS coils as well as demonstrated suitability of the 3-axis coil for a multichannel array configuration.

Results:

The experimentally obtained dB/dt values validated the computational model of the 3-axis coil and therefore confirmed that both the coil and stimulator system are operating as intended. The air-cooling system was effective for brief high-frequency pulse trains and extended single- and paired-pulse TMS protocols. The electromagnetic simulations suggested that an array of the 3-axis coils would have comparable stimulation intensity to conventional TMS coils, therefore enabling clearly suprathreshold stimulation of the human brain. The recorded coil coupling between the x/y/z-elements was < 1% and the maximal coupling between z-elements in the array configuration was 1.8% and 3.4% for the measured and calculated values, respectively.

Conclusion:

We presented a 3-axis coil intended for multichannel TMS arrays. The electromagnetic measurements and simulations verified that the coil fabrication met the desired specifications and that the inductive coupling between the elements was negligible. The air-cooled 3-axis TMS coil appears suitable to be used as an element in multichannel TMS arrays.

A software toolkit for TMS electric-field modeling with boundary element fast multipole method: an efficient MATLAB implementation

Objective

To present and disseminate our transcranial magnetic stimulation (TMS) modeling software toolkit, including several new algorithmic developments, and to apply this software to realistic TMS modeling scenarios given a high-resolution model of the human head including cortical geometry and an accurate coil model.

Approach

The recently developed charge-based boundary element fast multipole method (BEM-FMM) is employed as an alternative to the 1st order finite element method (FEM) most commonly used today. The BEM-FMM approach provides high accuracy and unconstrained numerical field resolution close to and across cortical interfaces. Here, the previously proposed BEM-FMM algorithm has been improved in several novel ways.

Main results

The improvements resulted in a threefold increase in computational speed while maintaining the same solution accuracy. The computational code based on the MATLAB® platform is made available to all interested researchers, along with a coil model repository and examples to create custom coils, head model repository, and supporting documentation. The presented software toolkit may be useful for post-hoc analyses of navigated TMS data using high-resolution subject-specific head models as well as accurate and fast modeling for the purposes of TMS coil/hardware development.

Significance

TMS is currently the only non-invasive neurostimulation modality that enables painless and safe supra-threshold stimulation by employing electromagnetic induction to efficiently penetrate the skull. Accurate, fast, and high resolution modeling of the electric fields may significantly improve individualized targeting and dosing of TMS and therefore enhance the efficiency of existing clinical protocols as well as help establish new application domains.

Accurate TMS Head Modeling: Interfacing SimNIBS and BEM-FMM in a MATLAB-Based Module

We interface the head modelling, coil models, Graphical User Interface (GUI), and post-processing capabilities of the SimNIBS package with the boundary element fast multipole method (BEM-FMM), implemented in a MATLAB-based module. The resulting pipeline combines the best of both worlds: the individualized head modelling and ease-of-use of SimNIBS with the numerical accuracy of BEM-FMM. The corresponding TMS (transcranial magnetic stimulation) modeling package is developed and made available online. It imports a SimNIBS surface segmentation and a coil field, and then exports electric-field values in selected surfaces or volumes. Additional information is also made available, such as discontinuous compartment surface electric fields and associated surface electric charge distributions.

Electric field simulations for transcranial brain stimulation using FEM: an efficient implementation and error analysis

OBJECTIVE

Transcranial magnetic stimulation (TMS) and transcranial electric stimulation (TES) modulate brain activity non-invasively by generating electric fields either by electromagnetic induction or by injecting currents via skin electrodes. Numerical simulations based on anatomically detailed head models of the TMS and TES electric fields can help us to understand and optimize the spatial stimulation pattern in the brain. However, most realistic simulations are still slow, and the role of anatomical fidelity on simulation accuracy has not been evaluated in detail so far.

APPROACH

We present and validate a new implementation of the finite element method (FEM) for TMS and TES that is based on modern algorithms and libraries. We also evaluate the convergence of the simulations and estimate errors stemming from numerical and modelling aspects.

MAIN RESULTS

Comparisons with analytical solutions for spherical phantoms validate our new FEM implementation, which is three to six times faster than previous implementations. The convergence results suggest that accurately capturing the tissue geometry in addition to choosing a sufficiently accurate numerical method is of fundamental importance for accurate simulations.

SIGNIFICANCE

The new implementation allows for a substantial increase in computational efficiency of FEM TMS and TES simulations. This is especially relevant for applications such as the systematic assessment of model uncertainty and the optimization of multi-electrode TES montages. The results of our systematic error analysis allow the user to select the best tradeoff between model resolution and simulation speed for a specific application. The new FEM code is openly available as a part of our open-source software SimNIBS 3.0.

Conditions for numerically accurate TMS electric field simulation

BACKGROUND

Computational simulations of the E-field induced by transcranial magnetic stimulation (TMS) are increasingly used to understand its mechanisms and to inform its administration. However, characterization of the accuracy of the simulation methods and the factors that affect it is lacking.

OBJECTIVE

To ensure the accuracy of TMS E-field simulations, we systematically quantify their numerical error and provide guidelines for their setup.

METHOD

We benchmark the accuracy of computational approaches that are commonly used for TMS E-field simulations, including the finite element method (FEM) with and without superconvergent patch recovery (SPR), boundary element method (BEM), finite difference method (FDM), and coil modeling methods.

RESULTS

To achieve cortical E-field error levels below 2%, the commonly used FDM and 1st order FEM require meshes with an average edge length below 0.4 mm, 1st order SPR-FEM requires edge lengths below 0.8 mm, and BEM and 2nd (or higher) order FEM require edge lengths below 2.9 mm. Coil models employing magnetic and current dipoles require at least 200 and 3000 dipoles, respectively. For thick solid-conductor coils and frequencies above 3 kHz, winding eddy currents may have to be modeled.

CONCLUSION

BEM, FDM, and FEM all converge to the same solution. Compared to the common FDM and 1st order FEM approaches, BEM and 2nd (or higher) order FEM require significantly lower mesh densities to achieve the same error level. In some cases, coil winding eddy-currents must be modeled. Both electric current dipole and magnetic dipole models of the coil current can be accurate with sufficiently fine discretization.

Real-time computation of the TMS-induced electric field in a realistic head model

Transcranial magnetic stimulation (TMS) is often targeted using a model of TMS-induced electric field (E). In such navigated TMS, the E-field models have been based on spherical approximation of the head. Such models omit the effects of cerebrospinal fluid (CSF) and gyral folding, leading to potentially large errors in the computed E-field. So far, realistic models have been too slow for interactive TMS navigation. We present computational methods that enable real-time solving of the E-field in a realistic five-compartment (5-C) head model that contains isotropic white matter, gray matter, CSF, skull and scalp. Using reciprocity and Geselowitz integral equation, we separate the computations to coil-dependent and -independent parts. For the Geselowitz integrals, we present a fast numerical quadrature. Further, we present a moment-matching approach for optimizing dipole-based coil models. We verified and benchmarked the new methods using simulations with over 100 coil locations. The new quadrature introduced a relative error (RE) of 0.3-0.6%. For a coil model with 42 dipoles, the total RE of the quadrature and coil model was 0.44-0.72%. Taking also other model errors into account, the contribution of the new approximations to the RE was 0.1%. For comparison, the RE due to omitting the separation of white and gray matter was >11%, and the RE due to omitting also the CSF was >23%. After the coil-independent part of the model has been built, E-fields can be computed very quickly: Using a standard PC and basic GPU, our solver computed the full E-field in a 5-C model in 9000 points on the cortex in 27 coil positions per second (cps). When the separation of white and gray matter was omitted, the speed was 43-65 cps. Solving only one component of the E-field tripled the speed. The presented methods enable real-time solving of the TMS-induced E-field in a realistic head model that contains the CSF and gyral folding. The new methodology allows more accurate targeting and precise adjustment of stimulation intensity during experimental or clinical TMS mapping.