Transverse-field activation mechanism in magnetic stimulation of peripheral nerves

A boundary element method of bidomain modeling for predicting cellular responses to electromagnetic fields

Objective.Commonly used cable equation approaches for simulating the effects of electromagnetic fields on excitable cells make several simplifying assumptions that could limit their predictive power. Bidomain or 'whole' finite element methods have been developed to fully couple cells and electric fields for more realistic neuron modeling. Here, we introduce a novel bidomain integral equation designed for determining the full electromagnetic coupling between stimulation devices and the intracellular, membrane, and extracellular regions of neurons.Approach.Our proposed boundary element formulation offers a solution to an integral equation that connects the device, tissue inhomogeneity, and cell membrane-induced E-fields. We solve this integral equation using first-order nodal elements and an unconditionally stable Crank-Nicholson time-stepping scheme. To validate and demonstrate our approach, we simulated cylindrical Hodgkin-Huxley axons and spherical cells in multiple brain stimulation scenarios.Main Results.Comparison studies show that a boundary element approach produces accurate results for both electric and magnetic stimulation. Unlike bidomain finite element methods, the bidomain boundary element method does not require volume meshes containing features at multiple scales. As a result, modeling cells, or tightly packed populations of cells, with microscale features embedded in a macroscale head model, is simplified, and the relative placement of devices and cells can be varied without the need to generate a new mesh.Significance.Device-induced electromagnetic fields are commonly used to modulate brain activity for research and therapeutic applications. Bidomain solvers allow for the full incorporation of realistic cell geometries, device E-fields, and neuron populations. Thus, multi-cell studies of advanced neuronal mechanisms would greatly benefit from the development of fast-bidomain solvers to ensure scalability and the practical execution of neural network simulations with realistic neuron morphologies.

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.

A Boundary Element Method of Bidomain Modeling for Predicting Cellular Responses to Electromagnetic Fields

Objective

Commonly used cable equation-based approaches for determining the effects of electromagnetic fields on excitable cells make several simplifying assumptions that could limit their predictive power. Bidomain or "whole" finite element methods have been developed to fully couple cells and electric fields for more realistic neuron modeling. Here, we introduce a novel bidomain integral equation designed for determining the full electromagnetic coupling between stimulation devices and the intracellular, membrane, and extracellular regions of neurons.

Methods

Our proposed boundary element formulation offers a solution to an integral equation that connects the device, tissue inhomogeneity, and cell membrane-induced E-fields. We solve this integral equation using first-order nodal elements and an unconditionally stable Crank-Nicholson time-stepping scheme. To validate and demonstrate our approach, we simulated cylindrical Hodgkin-Huxley axons and spherical cells in multiple brain stimulation scenarios.

Main Results

Comparison studies show that a boundary element approach produces accurate results for both electric and magnetic stimulation. Unlike bidomain finite element methods, the bidomain boundary element method does not require volume meshes containing features at multiple scales. As a result, modeling cells, or tightly packed populations of cells, with microscale features embedded in a macroscale head model, is made computationally tractable, and the relative placement of devices and cells can be varied without the need to generate a new mesh.

Significance

Device-induced electromagnetic fields are commonly used to modulate brain activity for research and therapeutic applications. Bidomain solvers allow for the full incorporation of realistic cell geometries, device E-fields, and neuron populations. Thus, multi-cell studies of advanced neuronal mechanisms would greatly benefit from the development of fast-bidomain solvers to ensure scalability and the practical execution of neural network simulations with realistic neuron morphologies.

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.

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.

Simulation-Based Optimization of Figure-of-Eight Coil Designs and Orientations for Magnetic Stimulation of Peripheral Nerve

Although magnetic neural stimulation has many advantages over electrical neural stimulation, its main disadvantages are higher energy requirement and poor stimulation selectivity. The orientation and location of the coil with respect to the stimulation site play a critical role in determining the stimulation threshold and stimulation selectivity. Utilizing numerical simulations in this work, we optimized the design parameters, orientation, and positioning of magnetic coils with respect to the peripheral nerve for improved stimulation efficacy. Specifically, we investigated different orientations and positions of the figure-of-eight coils for neural stimulation of the rat sciatic nerve. We also examined the effect of coil design parameters (number of layers and turns) and different coil electrical configurations (opposite vs. same direction of coil currents and series vs. parallel coil connections) on the stimulation threshold. We leveraged the multi-resolution impedance method and a heterogeneous multi-fascicular anatomical model of rat sciatic nerve to explore the possibility of selective stimulation as well. Neural excitation of a nerve fiber was implemented by an equivalent cable model and Frankenhaeuser-Huxley equations using NEURON software. Results suggest that inter-fascicular selectivity could be achieved by properly orienting and positioning the coil with respect to the nerve. Further, by orienting the figure-of-eight coil at an angle of 90° and 6 mm offset, we could switch between primarily activating one fascicle (and barely activating the other) and reversing those roles by merely switching the current direction in the two coils of the figure-of-eight coil.

Modified cable equation incorporating transverse polarization of neuronal membranes for accurate coupling of electric fields

OBJECTIVE

We present a theory and computational methods to incorporate transverse polarization of neuronal membranes into the cable equation to account for the secondary electric field generated by the membrane in response to transverse electric fields. The effect of transverse polarization on nonlinear neuronal activation thresholds is quantified and discussed in the context of previous studies using linear membrane models.

APPROACH

The response of neuronal membranes to applied electric fields is derived under two time scales and a unified solution of transverse polarization is given for spherical and cylindrical cell geometries. The solution is incorporated into the cable equation re-derived using an asymptotic model that separates the longitudinal and transverse dimensions. Two numerical methods are proposed to implement the modified cable equation. Several common neural stimulation scenarios are tested using two nonlinear membrane models to compare thresholds of the conventional and modified cable equations.

MAIN RESULTS

The implementations of the modified cable equation incorporating transverse polarization are validated against previous results in the literature. The test cases show that transverse polarization has limited effect on activation thresholds. The transverse field only affects thresholds of unmyelinated axons for short pulses and in low-gradient field distributions, whereas myelinated axons are mostly unaffected.

SIGNIFICANCE

The modified cable equation captures the membrane's behavior on different time scales and models more accurately the coupling between electric fields and neurons. It addresses the limitations of the conventional cable equation and allows sound theoretical interpretations. The implementation provides simple methods that are compatible with current simulation approaches to study the effect of transverse polarization on nonlinear membranes. The minimal influence by transverse polarization on axonal activation thresholds for the nonlinear membrane models indicates that predictions of stronger effects in linear membrane models with a fixed activation threshold are inaccurate. Thus, the conventional cable equation works well for most neuroengineering applications, and the presented modeling approach is well suited to address the exceptions.

Coupling Magnetically Induced Electric Fields to Neurons: Longitudinal and Transverse Activation

We present a theory and computational models to couple the electric field induced by magnetic stimulation to neuronal membranes. Based on the characteristics of magnetically induced electric fields and the modified cable equation that we developed previously, quasipotentials are derived as a simple and accurate approximation for coupling of the electric fields to neurons. The conventional and modified cable equations are used to simulate magnetic stimulation of long peripheral nerves by circular and figure-8 coils. Activation thresholds are obtained over a range of lateral and vertical coil positions for two nonlinear membrane models representing unmyelinated and myelinated straight axons and also for undulating myelinated axons. For unmyelinated straight axons, the thresholds obtained with the modified cable equation are significantly lower due to transverse polarization, and the spatial distributions of thresholds as a function of coil position differ significantly from predictions by the activating function. However, the activation thresholds of unmyelinated axons obtained with either cable equation are very high and beyond the output capabilities of conventional magnetic stimulators. For myelinated axons, threshold values are similar for both cable equations and within the range of magnetic stimulators. Whereas the transverse field contributes negligibly to the activation thresholds of myelinated fibers, axonal undulation can significantly increase or decrease thresholds depending on coil position. The analysis provides a rigorous theoretical foundation and implementation methods for the use of the cable equation to model neuronal response to magnetically induced electric fields. Experimentally observed stimulation with the electric fields perpendicular to the nerve trunk cannot be explained by transverse polarization and is likely due to nerve fiber undulation and other geometrical inhomogeneities.

Noninvasive Transcranial Magnetic Brain Stimulation in Stroke

It is likely that transcranial magnetic brain stimulation will be used for the clinical treatment of stroke and stroke-related impairments in the future. The anatomic target and stimulation parameters will likely vary for any clinical focus, be it weakness, pain, or cognitive or communicative dysfunction. Biomarkers may also be useful for identifying patients who will respond best, with a goal to enhance clinical decision making. Combination with drugs or specific types of therapeutic exercise may be necessary to achieve maximal response.

Shielding effects of myelin sheath on axolemma depolarization under transverse electric field stimulation

Axonal stimulation with electric currents is an effective method for controlling neural activity. An electric field parallel to the axon is widely accepted as the predominant component in the activation of an axon. However, recent studies indicate that the transverse component to the axolemma is also effective in depolarizing the axon. To quantitatively investigate the amount of axolemma polarization induced by a transverse electric field, we computed the transmembrane potential (Vm) for a conductive body that represents an unmyelinated axon (or the bare axon between the myelin sheath in a myelinated axon). We also computed the transmembrane potential of the sheath-covered axonal segment in a myelinated axon. We then systematically analyzed the biophysical factors that affect axonal polarization under transverse electric stimulation for both the bare and sheath-covered axons. Geometrical patterns of polarization of both axon types were dependent on field properties (magnitude and field orientation to the axon). Polarization of both axons was also dependent on their axolemma radii and electrical conductivities. The myelin provided a significant “shielding effect” against the transverse electric fields, preventing excessive axolemma depolarization. Demyelination could allow for prominent axolemma depolarization in the transverse electric field, via a significant increase in myelin conductivity. This shifts the voltage drop of the myelin sheath to the axolemma. Pathological changes at a cellular level should be considered when electric fields are used for the treatment of demyelination diseases. The calculated term for membrane polarization (Vm) could be used to modify the current cable equation that describes axon excitation by an external electric field to account for the activating effects of both parallel and transverse fields surrounding the target axon.

Mechanic stress generated by a time-varying electromagnetic field on bone surface

Bone cells sense mechanical load, which is essential for bone growth and remodeling. In a fracture, this mechanism is compromised. Electromagnetic stimulation has been widely used to assist in bone healing, but the underlying mechanisms are largely unknown. A recent hypothesis suggests that electromagnetic stimulation could influence tissue biomechanics; however, a detailed quantitative understanding of EM-induced biomechanical changes in the bone is unavailable. This paper used a muscle/bone model to study the biomechanics of the bone under EM exposure. Due to the dielectric properties of the muscle/bone interface, a time-varying magnetic field can generate both compressing and shear stresses on the bone surface, where many mechanical sensing cells are available for cellular mechanotransduction. I calculated these stresses and found that the shear stress is significantly greater than the compressing stress. Detailed parametric analysis suggests that both the compressing and shear stresses are dependent on the geometrical and electrical properties of the muscle and the bone. These stresses are also functions of the orientation of the coil and the frequency of the magnetic field. It is speculated that the EM field could apply biomechanical influence to fractured bone, through the fine-tuning of the controllable field parameters. Graphical abstract Mechanic stress on bone surface in a time-varying magnetic field.

Magnetic stimulation of the upper trapezius muscles in patients with migraine - A pilot study

BACKGROUND

Repetitive peripheral magnetic stimulation (rPMS) has been applied to musculoskeletal pain conditions. Since recent data show that migraine and tension-type headache (TTH) might be closely related to peripheral muscular pain in the neck and shoulder region (supporting the concept of the trigemino-cervical complex (TCC)), this pilot study explores the acceptance of rPMS to the upper trapezius muscles in migraine (partly in combination with TTH).

METHODS

We used rPMS to stimulate active myofascial trigger points (aTrPs) of the upper trapezius muscles in 20 young adults suffering from migraine. Acceptance was assessed by a standardized questionnaire, whereas self-rated effectiveness was evaluated by headache calendars and the Migraine Disability Assessment (MIDAS). Algometry was performed to explore the local effect of rPMS on the muscles.

RESULTS

Acceptance of rPMS was shown in all subjects without any adverse events, and rPMS had a statistically significant impact on almost every parameter of the headache calendar and MIDAS. Among others, the number of migraine attacks (p < 0.001) and migraine intensity (p = 0.001) significantly decreased regarding pre- and post-stimulation assessments. Accordingly, 100.0% of subjects would repeat the stimulation, while 90.0% would recommend rPMS as a treatment option for migraine.

CONCLUSIONS

rPMS might represent a promising tool to alleviate migraine symptoms within the context of myofascial pain. This might be due to stimulation-dependent modulation of the peripheral sensory effect within the TCC in migraine. However, sham-controlled studies with larger and more homogeneous cohorts are needed to prove a potential beneficial effect. Ethics Committee Registration Numbers: 356-14 and 447/14.

Vesicle biomechanics in a time-varying magnetic field

Background

Cells exhibit distortion when exposed to a strong electric field, suggesting that the field imposes control over cellular biomechanics. Closed pure lipid bilayer membranes (vesicles) have been widely used for the experimental and theoretical studies of cellular biomechanics under this electrodeformation. An alternative method used to generate an electric field is by electromagnetic induction with a time-varying magnetic field. References reporting the magnetic control of cellular mechanics have recently emerged. However, theoretical analysis of the cellular mechanics under a time-varying magnetic field is inadequate.

We developed an analytical theory to investigate the biomechanics of a modeled vesicle under a time-varying magnetic field. Following previous publications and to simplify the calculation, this model treated the inner and suspending media as lossy dielectrics, the membrane thickness set at zero, and the electric resistance of the membrane assumed to be negligible. This work provided the first analytical solutions for the surface charges, electric field, radial pressure, overall translational forces, and rotational torques introduced on a vesicle by the time-varying magnetic field. Frequency responses of these measures were analyzed, particularly the frequency used clinically by transcranial magnetic stimulation (TMS).

Results

The induced surface charges interacted with the electric field to produce a biomechanical impact upon the vesicle. The distribution of the induced surface charges depended on the orientation of the coil and field frequency. The densities of these charges were trivial at low frequency ranges, but significant at high frequency ranges. The direction of the radial force on the vesicle was dependent on the conductivity ratio between the vesicle and the medium. At relatively low frequencies (<200 KHz), including the frequency used in TMS, the computed radial pressure and translational forces on the vesicle were both negligible.

Conclusions

This work provides an analytical framework and insight into factors affecting cellular biomechanics under a time-varying magnetic field. Biological effects of clinical TMS are not likely to occur via alteration of the biomechanics of brain cells.

Patch-clamp recordings of rat neurons from acute brain slices of the somatosensory cortex during magnetic stimulation

Although transcranial magnetic stimulation (TMS) is a popular tool for both basic research and clinical applications, its actions on nerve cells are only partially understood. We have previously predicted, using compartmental modeling, that magnetic stimulation of central nervous system neurons depolarized the soma followed by initiation of an action potential in the initial segment of the axon. The simulations also predict that neurons with low current threshold are more susceptible to magnetic stimulation. Here we tested these theoretical predictions by combining in vitro patch-clamp recordings from rat brain slices with magnetic stimulation and compartmental modeling. In agreement with the modeling, our recordings demonstrate the dependence of magnetic stimulation-triggered action potentials on the type and state of the neuron and its orientation within the magnetic field. Our results suggest that the observed effects of TMS are deeply rooted in the biophysical properties of single neurons in the central nervous system and provide a framework both for interpreting existing TMS data and developing new simulation-based tools and therapies.

A complete model for the evaluation of the magnetic stimulation of peripheral nerves

In this paper, a numerical procedure for the analysis of peripheral nerve excitation through magnetic stimulation is presented and used to investigate the physical parameters influencing stimulation. The finite difference technique is used to evaluate the electric field distribution induced inside an arm by the current flowing through a coil, and a nonlinear cable model is used to describe the response of the nerve fiber to the induced electric field. The comparison among several forearm structures has evidenced that the heterogeneous non dispersive forearm model is a good reference condition. With this model, the lowest charging voltage on the stimulator capacitance, able to produce the nerve stimulation, is achieved when the coil is shifted, with respect to the nerve, of a quantity slightly lower than the coil radius but it is also possible to excite the nerve fiber by applying a shift equal to zero. The charging voltage increases when the coil radius is increased and when a three-dimensional coil geometry is considered. Moreover, this voltage is strongly dependent on the nerve position inside the forearm and on the kind of tissue surrounding the nerve.

Analysis and optimization of pulse dynamics for magnetic stimulation

Magnetic stimulation is a standard tool in brain research and has found important clinical applications in neurology, psychiatry, and rehabilitation. Whereas coil designs and the spatial field properties have been intensively studied in the literature, the temporal dynamics of the field has received less attention. Typically, the magnetic field waveform is determined by available device circuit topologies rather than by consideration of what is optimal for neural stimulation. This paper analyzes and optimizes the waveform dynamics using a nonlinear model of a mammalian axon. The optimization objective was to minimize the pulse energy loss. The energy loss drives power consumption and heating, which are the dominating limitations of magnetic stimulation. The optimization approach is based on a hybrid global-local method. Different coordinate systems for describing the continuous waveforms in a limited parameter space are defined for numerical stability. The optimization results suggest that there are waveforms with substantially higher efficiency than that of traditional pulse shapes. One class of optimal pulses is analyzed further. Although the coil voltage profile of these waveforms is almost rectangular, the corresponding current shape presents distinctive characteristics, such as a slow low-amplitude first phase which precedes the main pulse and reduces the losses. Representatives of this class of waveforms corresponding to different maximum voltages are linked by a nonlinear transformation. The main phase, however, scales with time only. As with conventional magnetic stimulation pulses, briefer pulses result in lower energy loss but require higher coil voltage than longer pulses.

Time-frequency analysis of short-lasting modulation of EEG induced by intracortical and transcallosal paired TMS over motor areas

Dynamic changes in spontaneous electroencephalogram (EEG) rhythms can be seen to occur with a high rate of variability. An innovative method to study brain function is by triggering oscillatory brain activity with transcranial magnetic stimulation (TMS). EEG-TMS coregistration was performed on five healthy subjects during a 1-day experimental session that involved four steps: baseline acquisition, unconditioned single-pulse TMS, intracortical inhibition (ICI, 3 ms) paired-pulse TMS, and transcallosal stimulation over left and right primary motor cortex (M1). A time-frequency analysis based on the wavelet method was used to characterize rapid modifications of oscillatory EEG rhythms induced by TMS. Single, paired, and transcallosal TMS applied on the sensorimotor areas induced rapid desynchronization over the frontal and central-parietal electrodes mainly in the alpha and beta bands, followed by a rebound of synchronization, and rapid synchronization of delta and theta activity. Wavelet analysis after a perturbation approach is a novel way to investigate modulation of oscillatory brain activity. The main findings are consistent with the concept that the human motor system may be based on networklike oscillatory cortical activity and might be modulated by single, paired, and transcallosal magnetic pulses applied to M1, suggesting a phenomenon of fast brain activity resetting and triggering of slow activity.

Asymptotic model of electrical stimulation of nerve fibers

We present a novel theory and computational algorithm for modeling electrical stimulation of nerve fibers in three dimensions. Our approach uses singular perturbation to separate the full 3D boundary value problem into a set of 2D "transverse" problems coupled with a 1D "longitudinal" problem. The resulting asymptotic model contains not one but two activating functions (AF): the longitudinal AF that drives the slow development of the mean transmembrane potential and the transverse AF that drives the rapid polarization of the fiber in the transverse direction. The asymptotic model is implemented for a prototype 3D cylindrical fiber with a passive membrane in an isotropic extracellular region. The validity of this approach is tested by comparing the numerical solution of the asymptotic model to the analytical solutions. The results show that the asymptotic model predicts steady-state transmembrane potential directly under the electrodes with the root mean square error of 0.539 mV, i.e., 1.04% of the maximum transmembrane potential. Thus, this work has created a computationally efficient algorithm that facilitates studies of the complete spatiotemporal dynamics of nerve fibers in three dimensions.

Modeling extracellular electrical neural stimulation: from basic understanding to MEA-based applications

Extracellular electrical stimulation of neural networks has been widely used empirically for decades with individual electrodes. Since recently, microtechnology provides advanced systems with high-density microelectrode arrays (MEAs). Taking the most of these devices for fundamental goals or developing neural prosthesis requires a good knowledge of the mechanisms underlying electrical stimulation. Here, we review modeling approaches used to determine (1) the electric potential field created by a stimulation and (2) the response of an excitable cell to an applied field. Computation of the potential field requires solving the Poisson equation. While this can be performed analytically in simple electrode-neuron configurations, numerical models are required for realistic geometries. In these models, special care must be taken to model the potential drop at the electrode/tissue interface using appropriate boundary conditions. The neural response to the field can then be calculated using compartmentalized cell models, by solving a cable equation, the source term of which (called activating function) is proportional to the second derivative of the extracellular field along the neural arborization. Analytical and numerical solutions to this equation are first presented. Then, we discuss the use of approximated solutions to intuitively predict the neuronal response: Either the "activating function" or the "mirror estimate", depending on the pulse duration and the cell space constant. Finally, we address the design of optimal electrode configurations allowing the selective activation of neurons near each stimulation site. This can be achieved using either multipolar configurations, or the "ground surface" configuration, which can be easily integrated in high-density MEAs. Overall, models highlighting the mechanisms of electrical microstimulation and improving stimulating devices should help understanding the influence of extracellular fields on neural elements and developing optimized neural prostheses for rehabilitation.

Mechanisms of magnetic stimulation of central nervous system neurons

Transcranial magnetic stimulation (TMS) is a stimulation method in which a magnetic coil generates a magnetic field in an area of interest in the brain. This magnetic field induces an electric field that modulates neuronal activity. The spatial distribution of the induced electric field is determined by the geometry and location of the coil relative to the brain. Although TMS has been used for several decades, the biophysical basis underlying the stimulation of neurons in the central nervous system (CNS) is still unknown. To address this problem we developed a numerical scheme enabling us to combine realistic magnetic stimulation (MS) with compartmental modeling of neurons with arbitrary morphology. The induced electric field for each location in space was combined with standard compartmental modeling software to calculate the membrane current generated by the electromagnetic field for each segment of the neuron. In agreement with previous studies, the simulations suggested that peripheral axons were excited by the spatial gradients of the induced electric field. In both peripheral and central neurons, MS amplitude required for action potential generation was inversely proportional to the square of the diameter of the stimulated compartment. Due to the importance of the fiber's diameter, magnetic stimulation of CNS neurons depolarized the soma followed by initiation of an action potential in the initial segment of the axon. Passive dendrites affect this process primarily as current sinks, not sources. The simulations predict that neurons with low current threshold are more susceptible to magnetic stimulation. Moreover, they suggest that MS does not directly trigger dendritic regenerative mechanisms. These insights into the mechanism of MS may be relevant for the design of multi-intensity TMS protocols, may facilitate the construction of magnetic stimulators, and may aid the interpretation of results of TMS of the CNS.

Transcranial magnetic stimulation (TMS) is a widely applied tool for probing cognitive function in humans and is one of the best tools for clinical treatments and interfering with cognitive tasks. Surprisingly, while TMS has been commercially available for decades, the cellular mechanisms underlying magnetic stimulation remain unclear. Here we investigate these mechanisms using compartmental modeling. We generated a numerical scheme allowing simulation of the physiological response to magnetic stimulation of neurons with arbitrary morphologies and active properties. Computational experiments using this scheme suggested that TMS affects neurons in the central nervous system (CNS) primarily by somatic stimulation. Since magnetic stimulation appears to cause somatic depolarization, its effects are highly correlated with the neuron's current threshold. Our simulations therefore predict that subpopulations of CNS neurons with different firing thresholds will respond differently to magnetic stimulation. For example, low-intensity TMS may be used to stimulate low-threshold cortical inhibitory interneurons. At higher intensities we predict that both inhibitory and excitatory neurons are activated. These predictions may be tested at the cellular level and may impact cognitive experiments in humans. Furthermore, our simulations may be used to design TMS coils, devices and protocols.

Transmembrane potential generated by a magnetically induced transverse electric field in a cylindrical axonal model

During the electrical stimulation of a uniform, long, and straight nerve axon, the electric field oriented parallel to the axon has been widely accepted as the major field component that activates the axon. Recent experimental evidence has shown that the electric field oriented transverse to the axon is also sufficient to activate the axon, by inducing a transmembrane potential within the axon. The transverse field can be generated by a time-varying magnetic field via electromagnetic induction. The aim of this study was to investigate the factors that influence the transmembrane potential induced by a transverse field during magnetic stimulation. Using an unmyelinated axon model, we have provided an analytic expression for the transmembrane potential under spatially uniform, time-varying magnetic stimulation. Polarization of the axon was dependent on the properties of the magnetic field (i.e., orientation to the axon, magnitude, and frequency). Polarization of the axon was also dependent on its own geometrical (i.e., radius of the axon and thickness of the membrane) and electrical properties (i.e., conductivities and dielectric permittivities). Therefore, this article provides evidence that aside from optimal coil design, tissue properties may also play an important role in determining the efficacy of axonal activation under magnetic stimulation. The mathematical basis of this conclusion was discussed. The analytic solution can potentially be used to modify the activation function in current cable equations describing magnetic stimulation.

Transmembrane potential induced on the internal organelle by a time-varying magnetic field: a model study

BACKGROUND

When a cell is exposed to a time-varying magnetic field, this leads to an induced voltage on the cytoplasmic membrane, as well as on the membranes of the internal organelles, such as mitochondria. These potential changes in the organelles could have a significant impact on their functionality. However, a quantitative analysis on the magnetically-induced membrane potential on the internal organelles has not been performed.

METHODS

Using a two-shell model, we provided the first analytical solution for the transmembrane potential in the organelle membrane induced by a time-varying magnetic field. We then analyzed factors that impact on the polarization of the organelle, including the frequency of the magnetic field, the presence of the outer cytoplasmic membrane, and electrical and geometrical parameters of the cytoplasmic membrane and the organelle membrane.

RESULTS

The amount of polarization in the organelle was less than its counterpart in the cytoplasmic membrane. This was largely due to the presence of the cell membrane, which "shielded" the internal organelle from excessive polarization by the field. Organelle polarization was largely dependent on the frequency of the magnetic field, and its polarization was not significant under the low frequency band used for transcranial magnetic stimulation (TMS). Both the properties of the cytoplasmic and the organelle membranes affect the polarization of the internal organelle in a frequency-dependent manner.

CONCLUSIONS

The work provided a theoretical framework and insights into factors affecting mitochondrial function under time-varying magnetic stimulation, and provided evidence that TMS does not affect normal mitochondrial functionality by altering its membrane potential.

Influence of coil current configuration in magnetic stimulation of a nerve fiber in inhomogeneous and anisotropic conducting media

In this study, we used a computer simulation to investigate the effects of the coil current waveform and direction on the excitation processes of the nerve axon in inhomogeneous and anisotropic conducting media in magnetic stimulation. We assumed that the nerve axon was located in the media with 2 regions having different conductivities or electrical anisotropy that simulate different tissue types. The distribution of induced electric fields was calculated with the finite element method (FEM). The nerve fiber was modeled after equivalent electrical circuits having active nodes of Ranvier. The direction of the coil current at the intersection of a figure-eight coil was assumed to flow perpendicular to the nerve axon. We observed the excitation threshold when the coil current waveform and direction are changed with varying the electrical properties such as tissue electrical conductivity and anisotropy. The simulation results show that the threshold decreases with the increase of conductivity ratio between 2 regions and it also depends on the coil current waveform and direction. Biphasic coil current has lower threshold than monophasic one when the current direction is the same in both waveforms. The results also suggest that the tissue anisotropy strongly affects the excitation threshold. The threshold increases with the increase of tissue anisotropic ratio of longitudinal direction to the transverse one respect to the nerve axon. The results in this study give useful information to explain the experimental results of the magnetic stimulation of human peripheral nervous systems and the theoretical model is applicable to understand the characteristics in magnetic stimulation of both peripheral and central nervous systems.

Interaction of MRI field gradients with the human body

In this review, the effects of low-frequency electromagnetic fields encountered specifically during magnetic resonance imaging (MRI) are examined. The primary biological effect at frequencies of between 100 and 5000 Hz (typical of MRI magnetic field gradient switching) is peripheral nerve stimulation, the result of which can be a mild tingling and muscle twitching to a sensation of pain. The models for nerve stimulation and how they are related to the rate of change of magnetic field are examined. The experimental measurements, and analytic and computational modelling work in this area are reviewed. The review concludes with a discussion of current regulation in this area and current practice as both are applied to MRI.

Motor threshold in transcranial magnetic stimulation: the impact of white matter fiber orientation and skull-to-cortex distance

The electrophysiology of transcranial magnetic stimulation (TMS) of motor cortex is not well understood. In this study, we investigate several structural parameters of the corticospinal tract and their relation to the TMS motor threshold (MT) in 17 subjects, with and without schizophrenia. We obtained structural and diffusion tensor MRI scans and measured the fractional anisotropy and principal diffusion direction for regions of interest in the corticospinal tract. We also measured the skull-to-cortex distance over the left motor region. The anterior-posterior trajectory of principle diffusion direction of the corticospinal tract and skull-to-cortex distance were both found to be highly correlated with MT, while fractional anisotropy, age and schizophrenia status were not. Two parameters-skull-to-cortex distance and the anterior component of the principle diffusion direction of the corticospinal tract as it passes the internal capsule-are highly predictive of MT in a linear regression model, and account for 82% of the variance observed (R2 = 0.82, F = 20.27, P < 0.0001) in measurements of MT. The corticospinal tract's anterior-posterior direction alone contributes 13% of the variance explained.

The cortical site of visual suppression by transcranial magnetic stimulation

In visual suppression paradigms, transcranial magnetic stimulation (TMS) applied approximately 90 ms after visual stimulus presentation over occipital visual areas can robustly interfere with visual perception, thereby most likely affecting feedback activity from higher areas (Amassian VE, Cracco RQ, Maccabee PJ, Cracco JB, Rudell A, Eberle L. 1989. Suppression of visual perception by magnetic coil stimulation of human occipital cortex. Electroencephalogr Clin Neurophysiol 74:458-462.). It is speculated that the observed effects might stem primarily from the disruption of V1 activity. This hypothesis, although under debate, argues in favor of a special role of V1 in visual awareness. In this study, we combine TMS, functional magnetic resonance imaging, and calculation of the induced electric field to study the neural correlates of visual suppression. For parafoveal visual stimulation in the lower right half of the visual field, area V2d is shown to be the likely TMS target based on its anatomical location close to the skull surface. Furthermore, isolated stimulation of area V3 also results in robust visual suppression. Notably, V3 stimulation does not directly affect the feedback from higher visual areas that is relayed mainly via V2 to V1. These findings support the view that intact activity patterns in several early visual areas (rather than merely in V1) are likewise important for the perception of the stimulus.

Transcranial magnetic stimulation in heterogeneous brain tissue: clinical impact on focality, reproducibility and true sham stimulation

BACKGROUND

Transcranial magnetic stimulation (TMS) is an attractive research and possibly therapeutic tool for non-invasive central nervous system stimulation. However, relatively little is known about the direction, magnitude and distribution of induced electric field and current flows in tissue, and optimal setup characteristics as well as appropriate sham stimulation conditions remain largely undetermined, hampering reproducibility.

METHODS

We reconstruct the conductive phenomena induced by TMS by implementing digitized coil geometry and realistic stimulator parameters and solving the electromagnetic problem over an MRI-based, realistic head model of 1mm resolution. Findings are validated by recording motor evoked potentials from the right abductor pollicis brevis muscle from healthy subjects stimulated in a stereotaxic framework.

RESULTS

Several commonly used sham stimulation configurations elicit conductive patterns which achieve up to 40% of the strength of real stimulation. Also, variations in coil position of the order of a 7 degrees tilt, which are expected to occur in non-stereotaxic stimulation, can alter the stimulation intensity by up to 25%.

CONCLUSIONS

In accordance with our findings, several clinical studies observe measurable effects during sham stimulation or no significant difference between sham and real stimulation, and the sensitivity of stimulation intensity to tiny coil rotations affords a partial explanation for the poor reproducibility and partial disagreements observed across clinical TMS studies. Knowledge of coil and stimulator specifications alone is hence not sufficient to control stimulation conditions, and a stereotaxic setup coupled with individually adjusted field solvers appear essential in performing reliable TMS studies.

Modulation of cortical oscillatory activity during transcranial magnetic stimulation

Transcranial magnetic stimulation (TMS) can transiently modulate cortical excitability, with a net effect depending on the stimulation frequency (< or =1 Hz inhibition vs. > or =5 Hz facilitation, at least for the motor cortex). This possibility has generated interest in experiments aiming to improve deficits in clinical settings, as well as deficits in the cognitive domain. The aim of the present study was to investigate the on-line effects of low frequency (1 Hz) TMS on the EEG oscillatory activity in the healthy human brain, focusing particularly on the outcome of these modulatory effects in relation to the duration of the TMS stimulation. To this end, we used the event-related desynchronization/synchronization (ERD/ERS) approach to determine the patterns of oscillatory activity during two consecutive trains of sham and real TMS. Each train of stimulation was delivered to the left primary motor cortex (MI) of healthy subjects over a period of 10 min, while EEG rhythms were simultaneously recorded. Results indicated that TMS induced an increase in the power of brain rhythms that was related to the period of the stimulation, i.e. the synchronization of the alpha band increased with the duration of the stimulation, and this increase was inversely correlated with motor-evoked potentials (MEPs) amplitude. In conclusion, low frequency TMS over primary motor cortex induces a synchronization of the background oscillatory activity on the stimulated region. This induced modulation in brain oscillations seems to increase coherently with the duration of stimulation, suggesting that TMS effects may involve short-term modification of the neural circuitry sustaining MEPs characteristics.

Electric fields induced in the human body by time-varying magnetic field gradients in MRI: numerical calculations and correlation analysis

The spatial distributions of the electric fields induced in the human body by switched magnetic field gradients in MRI have been calculated numerically using the commercial software package, MAFIA, and the three-dimensional, HUGO body model that comprises 31 different tissue types. The variation of |J|, |E| and |B| resulting from exposure of the body model to magnetic fields generated by typical whole-body x-, y- and z-gradient coils has been analysed for three different body positions (head-, heart- and hips-centred). The magnetic field varied at 1 kHz, so as to produce a rate of change of gradient of 100 T m(-1) s(-1) at the centre of each coil. A highly heterogeneous pattern of induced electric field and current density was found to result from the smoothly varying magnetic field in all cases, with the largest induced electric fields resulting from application of the y-gradient, in agreement with previous studies. By applying simple statistical analysis to electromagnetic quantities within axial planes of the body model, it is shown that the induced electric field is strongly correlated to the local value of resistivity, and the induced current density exhibits even stronger correlation with the local conductivity. The local values of the switched magnetic field are however shown to bear little relation to the local values of the induced electric field or current density.

Excitation properties of peripheral nerves stimulated by an induced transverse electric field during magnetic stimulation

Interest in magnetic stimulation (MS) of peripheral nerves has grown rapidly, but difficulties in determining excitation sites and computing the stimulation efficiency have prevented magnetic stimulation from becoming a routing clinical tool. The classical cable function has demonstrated that the activating function of nerves is the first spatial derivative of the electric field component parallel to the nerves. Therefore, excellent analyses have been presented in the literatures which predict the excitation properties, for peripheral nerves stimulated by the induced electric field component parallel to the nerves during MS. There is not, however, an adequate analysis for nerves excitation which considers the induced field component perpendicular to the fibers. Based on the improved cable function, which introduces a transverse field component, we have analyzed the excitation sites and the stimulation efficiencies during MS using round coils and 8-shaped coils, respectively. That is helpful to the further development of the functional magnetic nerve stimulation technique and to the clinical application of MS.

Modulation of cortical oscillatory activities induced by varying single-pulse transcranial magnetic stimulation intensity over the left primary motor area: A combined EEG and TMS study

Combined transcranial magnetic stimulation/electroencephalography (TMS/EEG) was used to study the activation and interaction of cortical regions to a variety of focused sub- and suprathreshold magnetic pulses over the left primary motor cortex (M1) in ten healthy subjects. Five single-pulse TMS conditions were performed based on the individual resting motor threshold (RMT): (1) 80%; (2) 100%; (3) 120%; (4) 130%; and (5) sham. Simple self-paced movements of the right first finger were also executed. We evaluated the reactions to magnetic stimulation and movement conditions using event-related power and event-related coherence transformations of alpha and beta rhythms. Event-related power reflected regional oscillatory activity of neural assemblies, while event-related coherence reflected the inter-regional functional coupling of oscillatory neural activity. The event-related power transformation revealed that the magnetic pulse modulated cortical oscillations within the first half second for both frequency ranges. For the alpha rhythm, threshold TMS induced a small decrease in the amplitude of EEG oscillations over the stimulation site, while for both rhythms, a progressive synchronization was observed as the intensity of TMS was increased in both hemispheres. Movement onset produced a greater bilateral decrease of power compared with the effects of a magnetic pulse. The event-related coherence revealed that TMS enhanced the electrode connectivity of both hemispheres. Additionally, it was more enhanced within the first 500 ms following stimulation and was seen only for the alpha frequency rhythm. The increase of functional connectivity between cortical areas was minor for magnetic stimulation conditions compared with that for finger movements. The single-pulse TMS over M1 partially modulated the motor cortex generators of oscillatory activity, while a simple active self-paced movement of the right first finger induced greater cortex activation and coupling between cortical regions. We propose that finger movements impose higher functional demands on the motor system compared to artificial magnetic stimulation. These findings are consistent with the possibility that the human motor system may be based on network-like oscillatory cortical activity and might be modulated by brief electromagnetic sub- and suprathreshold pulses applied to M1, suggesting a phenomenon of resetting.

Therapeutic effects of peripheral repetitive magnetic stimulation on myofascial pain syndrome

OBJECTIVE

To evaluate short- and medium-term effects of peripheral repetitive magnetic stimulation (rMS) on myofascial pain.

METHODS

Eighteen patients who presented with myofascial trigger points (TPs) at the level of the superior trapezius were separated into two groups according to a restricted randomization scheme. Group 1 (n=9) underwent treatment with rMS that consisted of a total of 10 sessions, each lasting 20min, in which 4000 magnetic stimuli were administered in 5s trains at 20Hz at the TP. Group 2 (n=9) received a placebo treatment that consisted of the application of a non-functioning ultrasound therapy device to the TP. Patients were evaluated before treatment, at the end of treatment, and again 1 week and 1 month after the conclusion of the treatment. Clinical evaluation included parameters for measuring pain levels (VAS, NPDVAS and algometry), the myofascial TP characteristics and the range of cervical movement (ROM).

RESULTS

The rMS group showed a significant improvement in VAS, NPDVAS, algometry, as well as in the characteristics of the TP after conclusion of treatment. Improvements in the ROM were also present in rotation and controlateral bending. This improvement persisted after 1 month. On the other hand, the placebo group did not show any significant improvement in the tests considered.

CONCLUSIONS

The results of this study show that peripheral rMS may have positive short- and medium-term therapeutic effects on myofascial pain.

Rigorous Green's function formulation for transmembrane potential induced along a 3-D infinite cylindrical cell

The quasi-static electromagnetic field interaction with three-dimensional infinite-cylindrical cell is investigated for both intracellular (IPS) and extracellular (EPS) current point-source excitation. The induced transmembrane potential (TMP), expressed conventionally via Green's function, may alternatively be expanded into a faster-converging representation using a complex contour integration, consisting of an infinite-discrete set of exponentially decaying oscillating modes (corresponding to complex eigenvalues) and a continuous source-mode convolution integral. The dominant contributions for both the IPS and EPS problems are obtained in simple closed-form expressions, including well documented special mathematical functions. In the IPS case, the dominant modal contribution (of order zero)--an exact solution of the well-known cable equation--is explicitly and analytically corrected by the imaginary part of its eigenvalue and the source-mode convolution contribution. However, the TMP along a fiber was shown to decay at infinity algebraically and not exponentially, as predicted by the classic cable equation solution. In the EPS case, the dominant contribution is expressed as a source-mode convolution integral. However, for a long EPS distance (e.g., >10 cable length constant) the order-one-modes involved in the convolution is not a solution of the cable equation. Only for shorter EPS distance should the cable equation solution (i.e., the order zero dominant mode) be included in addition to the modes of order one. For on-membrane EPS location, additional modes should be included as well. In view of our EPS result, we suggest that the cable equation modeling existing in the literature and related to functional electrical stimulation for EPS problems, should be critically reviewed and corrected.

Ipsi- and contralateral EEG reactions to transcranial magnetic stimulation

OBJECTIVES

Transcranial magnetic stimulation (TMS) and high-resolution electroencephalography (EEG) were used to study the spreading of cortical activation in 6 healthy volunteers.

METHODS

Five locations in the left sensorimotor cortex (within 3cm(2)) were stimulated magnetically, while EEG was recorded with 60 scalp electrodes. A frameless stereotactic method was applied to determine the anatomic locus of stimulation and to superimpose the results on magnetic resonance images. Scalp potential and cortical current-density distributions were derived from averaged electroencephalographic (EEG) data.

RESULTS

The maxima of the ipsilateral activation were detected at the gyrus precentralis, gyrus supramarginalis, and lobulus parietalis superior, depending on the subject. Activation over the contralateral cortex was observed in all subjects, appearing at 22plus minus2ms (range 17--28); the maxima were located at the gyrus precentralis, gyrus frontalis superior, and the lobulus parietalis inferior. Contralateral EEG waveforms showed consistent changes when different sites were stimulated: stimulation of the two most medial points evoked the smallest responses fronto-parietally.

CONCLUSIONS

With the combination of TMS, EEG, and magnetic resonance imaging, an adequate spatiotemporal resolution may be achieved for tracing the intra- and interhemispheric spread of activation in the cortex caused by a magnetic pulse.

Evaluation of the cable model for electrical stimulation of unmyelinated nerve fibers

The cable model, used to calculate the membrane potential of an unmyelinated nerve fiber due to electrical stimulation, is reexamined under passive steady-state conditions. The validity of two of the assumptions of the cable model are evaluated, namely that the membrane potential be a function of the axial coordinate only and that the extracellular potential due to the presence of the nerve fiber be negligible. The membrane potential calculated from the passive steady-state cable model is compared with the membrane potential obtained from an analytical three-dimensional (3-D) volume conductor model of a nerve fiber. It is shown that for very small electrode-fiber distances (of only a few fiber radii), both assumptions are violated and the two models give quite different results. Over a wide range of the electrode-fiber distance (about 0.1 mm to 1 cm), both assumptions are fulfilled and the two models give approximately the same results. For very large distances (more than 10 cm, independent of fiber diameter) only the second assumption is satisfied, but a modification of the activating function of the cable model allows to calculate the membrane potential in agreement with the 3-D model.

Calculation of electric fields in a multiple cylindrical volume conductor induced by magnetic coils

A method is presented for calculating the electric field, that is induced in a cylindrical volume conductor by an alternating electrical current through a magnetic coil of arbitrary shape and position. The volume conductor is modeled as a set of concentric, infinitely long, homogeneous cylinders embedded in an outer space that extends to infinity. An analytic expression of the primary electric field induced by the magnetic coil, assuming quasi-static conditions, is combined with the analytic solution of the induced electric scalar potential due to the inhomogeneities of the volume conductor at the cylindrical interfaces. The latter is obtained by the method of separation of variables based on expansion with modified Bessel functions. Numerical results are presented for the case of two cylinders representing a nerve bundle with perineurium. An active cable model of a myelinated nerve fiber is included, and the effect of the nerve fiber's undulation is shown.

A comparative study of the effects of magnetic stimulation and electric stimulation on peripheral nerve injury in rat

The influence of pulsed magnetic stimulation (MS) on the sciatic nerve injury was investigated. Thirty rats were divided into three groups equally: MS group (A), electric stimulation (ES) group (B) and the control group (C). The MS and ES were applied immediately after the first 10 min of the sciatic nerve crush. Sciatic function index (SFI), toe spreading reflex (TSR), muscular weight and volume were measured after the experiment. The TSR of in the groups A and B occurred at 4th day while in the control group it occurs at 10th day. There was statistically significant difference in SFI between groups A and B (P < 0.01). The weight and volume of the gastrocnemius muscle were statistically greater in the groups A and B than in the control group (P < 0.01). The effect of MS was similar to that of ES. It was suggested that the application of MS immediately after the nerve injury might have an important clinical value as it can accelerate functional recovery and prevent or minimize muscle atrophy. The technique is easily to operate, non-invasion, painless and permits tolerance of high intensity output to be used.

Magnetic and electrical stimulation of undulating nerve fibres: a simulation study

Mathematical models of myelinated nerve fibres are highly stylized abstractions of real nerve fibres. For example, nerve fibres are usually assumed to be perfectly straight. Such idealizations can cause discrepancies between theoretical predictions and experimental results. One well-known discrepancy is that the currently used models predict (contradictory to experimental findings) that an activation of nerve fibres is not possible with a pure transverse electric field. This situation occurs when a magnetic coil is placed symmetrically above a straight nerve fibre for magnetic nerve stimulation, or when an anode and a cathode are placed equidistantly on a line perpendicular to the fibre in the case of electrical stimulation. It is shown that this discrepancy does not occur if the physiological undulation of peripheral nerve fibres is included in the models. Even for small undulation amplitudes (e.g. 0.02 mm), it is possible to activate the fibre in these positions. For physiological undulations, as found in the literature, and favourable (off-centre) positions, the typical reduction of the thresholds is in a range between one and five, compared with perfectly straight fibres.

Theory of multichannel magnetic stimulation: toward functional neuromuscular rehabilitation

Human excitable cells can be stimulated noninvasively with externally applied time-varying electromagnetic fields. The stimulation can be achieved either by directly driving current into the tissue (electrical stimulation) or by means of electro-magnetic induction (magnetic stimulation). While the electrical stimulation of the peripheral neuromuscular system has many beneficial applications, peripheral magnetic stimulation has so far only a few. This paper analyzes theoretically the use of multiple magnetic stimulation coils to better control the excitation and also to eventually mimic electrical stimulation. Multiple coils allow electronic spatial adjustment of the shape and location of the stimulus without moving the coils. The new properties may enable unforeseen uses for peripheral magnetic stimulation, e.g., in rehabilitation of patients with neuromuscular impairment.

Transcranial magnetic stimulation of the oculomotor and abducens nerves: determining the site of excitation in the cat

The authors have attempted to stimulate the feline oculomotor and abducens nerves using a magnetic coil (MC) and to determine the optimal reliable MC position for eliciting motor evoked potentials. The authors have also determined the site of excitation to analyze the validity and potential advantages of this technique as a means of neurophysiologically studying the oculomotor and abducens nerves. The potentials of both of these muscles are evoked by MC placed to be symmetrical to the line connecting the vertex and the center of the eyeball on the side being examined, as the coil center corresponds to the midpoint of this line. The latencies of the motor responses of the superior rectus and lateral rectus were 1.08 +/- 0.22 and 1.02 +/- 0.21 msec, respectively, suggesting that magnetic stimulation excites both the oculomotor and the abducens nerve at the superior orbital fissure.

The effect of magnetic coil orientation on the excitation of the median nerve

OBJECTIVE

We investigated the effect of magnetic coil orientation on the excitation of the median nerve in healthy subjects.

METHODS

An 8-shaped coil, 3.2 cm in outer diameter, was used. The median nerve was stimulated at the elbow while the compound muscle action potentials (CMAPs) of abductor pollicis brevis muscle were recorded at 4 different directions of the induced current: orthodromic, antidromic, medio-lateral and latero-medial.

RESULTS

We found that the amplitude of the CMAP was the greatest in a medio-lateral (ML) direction. We also measured the induced electric field in the saline tank that mimicked the forearm. The induced electric field and its spatial gradient were the greatest in the ML direction among 4 directions.

CONCLUSION

The fact that the forearm is a restrictive volume conductor may result in the different effects of coil orientation on the excitation of the median nerve at the elbow.