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.

Software Package for Transcranial Magnetic Stimulation Coil and Coil Array Analysis and Design.

Objective: This study aims to describe a MATLAB software package for transcranial magnetic stimulation (TMS) coil analysis and design. Approach: Electric and magnetic fields of the coils as well as their self- and mutual (for coil arrays) inductances are computed, with or without a magnetic core. Solid and stranded (Litz wire) conductors are also taken into consideration. The starting point is the centerline of a coil conductor(s), which is a 3D curve defined by the user. Then, a wire mesh and a computer aided design (CAD) mesh for the volume conductor of a given cross-section (circular, elliptical, or rectangular) are automatically generated. Self- and mutual inductances of the coil(s) are computed. Given the conductor current and its time derivative, electric and magnetic fields of the coil(s) are determined anywhere in space.Computations are performed with the fast multipole method (FMM), which is the most efficient way to evaluate the fields of many elementary current elements (current dipoles) comprising the current carrying conductor at a large number of observation points. This is the major underlying mathematical operation behind both inductance and field calculations. Main Results: The wire-based approach enables precise replication of even the most complex physical conductor geometries, while the FMM acceleration quickly evaluates large quantities of elementary current filaments. Agreement to within 0.74% was obtained between the inductances computed by the FMM method and ANSYS Maxwell 3D for the same coil model. Although not provided in this study, it is possible to evaluate non-linear magnetic cores in addition to the linear core exemplified. An experimental comparison was carried out against a physical MagVenture C-B60 coil; the measured and simulated inductances differed by only 1.25%, and nearly perfect correlation was found between the measured and computed E-field values at each observation point. Significance: The developed software package is applicable to any quasistatic inductor design, not necessarily to the TMS coils only.

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.

Modeling transcranial magnetic stimulation coil with magnetic cores.

Objective.Accurate modeling of transcranial magnetic stimulation (TMS) coils with the magnetic core is largely an open problem since commercial (quasi) magnetostatic solvers do not output specific field characteristics (e.g. induced electric field) and have difficulties when incorporating realistic head models. Many open-source TMS softwares do not include magnetic cores into consideration. This present study reports an algorithm for modeling TMS coils with a (nonlinear) magnetic core and validates the algorithm through comparison with finite-element method simulations and experiments.Approach.The algorithm uses the boundary element fast multipole method applied to all facets of a tetrahedral core mesh for a single-state solution and the successive substitution method for nonlinear convergence of the subsequent core states. The algorithm also outputs coil inductances, with or without magnetic cores. The coil-core combination is solved only once i.e. before incorporating the head model. The resulting primary TMS electric field is proportional to the total vector potential in the quasistatic approximation; it therefore also employs the precomputed core magnetization.Main results.The solver demonstrates excellent convergence for typical TMS field strengths and for analyticalB-Happroximations of experimental magnetization curves such as Froelich's equation or an arctangent equation. Typical execution times are 1-3 min on a common multicore workstation. For a simple test case of a cylindrical core within a one-turn coil, our solver computed the small-signal inductance nearly identical to that from ANSYS Maxwell. For a multiturn rodent TMS coil with a core, the modeled inductance matched the experimental measured value to within 5%.Significance.Incorporating magnetic core in TMS coil design has advantages of field shaping and energy efficiency. Our software package can facilitate model-informed design of more efficiency TMS systems and guide selection of core material. These models can also inform dosing with existing clinical TMS systems that use magnetic cores.

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 (M1HAND 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.

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.