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

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.

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.

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.

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.

Responses of model cortical neurons to temporal interference stimulation and related transcranial alternating current stimulation modalities

Objective.Temporal interference stimulation (TIS) was proposed as a non-invasive, focal, and steerable deep brain stimulation method. However, the mechanisms underlying experimentally-observed suprathreshold TIS effects are unknown, and prior simulation studies had limitations in the representations of the TIS electric field (E-field) and cerebral neurons. We examined the E-field and neural response characteristics for TIS and related transcranial alternating current stimulation modalities.Approach.Using the uniform-field approximation, we simulated a range of stimulation parameters in biophysically realistic model cortical neurons, including different orientations, frequencies, amplitude ratios, amplitude modulation, and phase difference of the E-fields, and obtained thresholds for both activation and conduction block.Main results. For two E-fields with similar amplitudes (representative of E-field distributions at the target region), TIS generated an amplitude-modulated (AM) total E-field. Due to the phase difference of the individual E-fields, the total TIS E-field vector also exhibited rotation where the orientations of the two E-fields were not aligned (generally also at the target region). TIS activation thresholds (75-230 V m-1) were similar to those of high-frequency stimulation with or without modulation and/or rotation. For E-field dominated by the high-frequency carrier and with minimal amplitude modulation and/or rotation (typically outside the target region), TIS was less effective at activation and more effective at block. Unlike AM high-frequency stimulation, TIS generated conduction block with some orientations and amplitude ratios of individual E-fields at very high amplitudes of the total E-field (>1700 V m-1).Significance. The complex 3D properties of the TIS E-fields should be accounted for in computational and experimental studies. The mechanisms of suprathreshold cortical TIS appear to involve neural activity block and periodic activation or onset response, consistent with computational studies of peripheral axons. These phenomena occur at E-field strengths too high to be delivered tolerably through scalp electrodes and may inhibit endogenous activity in off-target regions, suggesting limited significance of suprathreshold TIS.

Finding the Location of Axonal Activation by a Miniature Magnetic Coil

Magnetic stimulation for neural activation is widely used in clinical and lab research. In comparison to electric stimulation using an implanted electrode, stimulation with a large magnetic coil is associated with poor spatial specificity and incapability to stimulate deep brain structures. Recent developments in micromagnetic stimulation (μMS) technology mitigates some of these shortcomings. The sub-millimeter coils can be covered with soft, biocompatible material, and chronically implanted. They can provide highly specific neural stimulation in the deep neural structure. Although the μMS technology is expected to provide a precise location of neural stimulation, the exact site of neural activation is difficult to determine. Furthermore, factors that could cause the shifting of the activation site during μMS have not been fully investigated. To estimate the location of axon activation in μMS, we first derived an analytical expression of the activating function, which predicts the location of membrane depolarization in an unmyelinated axon. Then, we developed a multi-compartment, Hodgkin-Huxley (H-H) type of NEURON model of an unmyelinated axon to test the impact of several important coil parameters on the location of axonal activation. The location of axonal activation was dependent on both the parameters of the stimulus and the biophysics properties of the targeted axon during μMS. The activating function analysis predicted that the location of membrane depolarization and activation could shift due to the reversal of the coil current and the change in the coil-axon distance. The NEURON modeling confirmed these predictions. Interestingly, the NEURON simulation further revealed that the intensity of stimulation played a significant role in the activation location. Moderate or strong coil currents activated the axon at different locations, mediated by two distinct ion channel mechanisms. This study reports several experimental factors that could cause a potential shift in the location of neural activation during μMS, which is essential for further development of this novel technology.

2-ns Electrostimulation of Ca2+ Influx into Chromaffin Cells: Rapid Modulation by Field Reversal

Cellular effects of nanosecond-pulsed electric field exposures can be attenuated by an electric field reversal, a phenomenon called bipolar pulse cancellation. Our investigations of this phenomenon in neuroendocrine adrenal chromaffin cells show that a single 2-ns, 16 MV/m unipolar pulse elicited a rapid, transient rise in intracellular Ca²⁺ levels due to Ca²⁺ influx through voltage-gated calcium channels. The response was eliminated by a 2-ns bipolar pulse with positive and negative phases of equal duration and amplitude and fully restored (unipolar-equivalent response) when the delay between each phase of the bipolar pulse was 30 ns. Longer interphase intervals evoked Ca²⁺ responses that were greater in magnitude than those evoked by a unipolar pulse (stimulation). Cancellation was also observed when the amplitude of the second (negative) phase of the bipolar pulse was half that of the first (positive) phase but progressively lost as the amplitude of the second phase was incrementally increased above that of the first phase. When the amplitude of the second phase was twice that of the first phase, there was stimulation. By comparing the experimental results for each manipulation of the bipolar pulse waveform with analytical calculations of capacitive membrane charging/discharging, also known as accelerated membrane discharge mechanism, we show that the transition from cancellation to unipolar-equivalent stimulation broadly agrees with this model. Taken as a whole, our results demonstrate that electrostimulation of adrenal chromaffin cells with ultrashort pulses can be modulated with interphase intervals of tens of nanoseconds, a prediction of the accelerated membrane discharge mechanism not previously observed in other bipolar pulse cancellation studies. Such modulation of Ca²⁺ responses in a neural-type cell is promising for the potential use of nanosecond bipolar pulse technologies for remote electrostimulation applications for neuromodulation.

Limited Sensitivity of Hippocampal Synaptic Function or Network Oscillations to Unmodulated Kilohertz Electric Fields

Understanding the cellular mechanisms of kilohertz (kHz) electrical stimulation is of broad interest in neuromodulation including forms of transcranial electrical stimulation, interferential stimulation, and high-rate spinal cord stimulation (SCS). Yet, the well-established low-pass filtering by neuronal membranes suggests minimal neuronal polarization in respond to charge-balanced kHz stimulation. The hippocampal brain slice model is among the most studied systems in neuroscience and exhaustively characterized in screening the effects of electrical stimulation. High-frequency electric fields of varied amplitudes (1–150 V/m), waveforms (sinusoidal, symmetrical pule, asymmetrical pulse) and frequencies (1 and10 kHz) were tested. Changes in single or paired-pulse field EPSPs (fEPSP) in CA1 were measured in response to radial-directed and tangential-directed electric fields, with brief (30 s) or long (30 min) application times. The effects of kHz stimulation on ongoing endogenous network activity were tested in carbachol-induced γ oscillation of CA3a and CA3c. Across 23 conditions evaluated, no significant changes in fEPSP were resolved, while responses were detected for within-slice control direct current (DC) fields; 1-kHz sinusoidal and pulse stimulation (≥60 V/m), but not 10 kHz, induced changes in oscillating neuronal network. We thus report no responses to low-amplitude 1-kHz or any 10-kHz fields, suggesting that any brain sensitivity to these fields is via yet to be-determined mechanism(s) of action which were not identified in our experimental preparation.

Electro-physiology Models of Cells with Spherical Geometry with Non-conducting Center

We study the flow of electrical currents in spherical cells with a non-conducting core, so that current flow is restricted to a thin shell below the cell's membrane. Examples of such cells are fat storing cells (adipocytes). We derive the relation between current and voltage in the passive regime and examine the conditions under which the cell is electro-tonically compact. We compare our results to the well-studied case of electrical current flow in cylinder structures, such as neurons, described by the cable equation. In contrast to the cable, we find that for the sphere geometry (1) the voltage profile across the cell depends critically on the electrode geometry, and (2) the charging and discharging can be much faster than the membrane time constant; however, (3) voltage clamp experiments will incur similar distortion as in the cable case. We discuss the relevance for adipocyte function and experimental electro-physiology.

Temporal interference stimulation targets deep brain regions by modulating neural oscillations

BACKGROUND

Temporal interference (TI) stimulation of the brain generates amplitude-modulated electric fields oscillating in the kHz range with the goal of non-invasive targeted deep brain stimulation. Yet, the current intensities required in human (sensitivity) to modulate deep brain activity and if superficial brain region are spared (selectivity) at these intensities remains unclear.

OBJECTIVE

We developed an experimentally constrained theory for TI sensitivity to kHz electric field given the attenuation by membrane low-pass filtering property, and for TI selectivity to deep structures given the distribution of modulated and unmodulated electric fields in brain.

METHODS

The electric field threshold to modulate carbachol-induced gamma oscillations in rat hippocampal slices was determined for unmodulated 0.05-2 kHz sine waveforms, and 5 Hz amplitude-modulated waveforms with 0.1-2 kHz carrier frequencies. The neuronal effects are replicated with a computational network model to explore the underlying mechanisms, and then coupled to a validated current-flow model of the human head.

RESULTS

Amplitude-modulated electric fields are stronger in deep brain regions, while unmodulated electric fields are maximal at the cortical regions. Both experiment and model confirmed the hypothesis that spatial selectivity of temporal interference stimulation depends on the phasic modulation of neural oscillations only in deep brain regions. Adaptation mechanism (e.g. GABA_b) enhanced sensitivity to amplitude modulated waveform in contrast to unmodulated kHz and produced selectivity in modulating gamma oscillation (i.e. Higher gamma modulation in amplitude modulated vs unmodulated kHz stimulation). Selection of carrier frequency strongly affected sensitivity to amplitude modulation stimulation. Amplitude modulated stimulation with 100 Hz carrier frequency required ∼5 V/m (corresponding to ∼13 mA at the scalp surface), whereas, 1 kHz carrier frequency ∼60 V/m (∼160 mA) and 2 kHz carrier frequency ∼80 V/m (∼220 mA) to significantly modulate gamma oscillation. Sensitivity is increased (scalp current required decreased) for theoretical neuronal membranes with faster time constants.

CONCLUSION

The TI sensitivity (current required at the scalp) depends on the neuronal membrane time-constant (e.g. axons) approaching the kHz carrier frequency. TI selectivity is governed by network adaption (e.g. GABA_b) that is faster than the amplitude-modulation frequency. Thus, we show neuronal and network oscillations time-constants determine the scalp current required and the selectivity achievable with TI in humans.

Modelling the effects of ephaptic coupling on selectivity and response patterns during artificial stimulation of peripheral nerves

Artificial electrical stimulation of peripheral nerves for sensory feedback restoration can greatly benefit from computational models for simulation-based neural implant design in order to reduce the trial-and-error approach usually taken, thus potentially significantly reducing research and development costs and time. To this end, we built a computational model of a peripheral nerve trunk in which the interstitial space between the fibers and the tissues was modelled using a resistor network, thus enabling distance-dependent ephaptic coupling between myelinated axons and between fascicles as well. We used the model to simulate a) the stimulation of a nerve trunk model with a cuff electrode, and b) the propagation of action potentials along the axons. Results were used to investigate the effect of ephaptic interactions on recruitment and selectivity stemming from artificial (i.e., neural implant) stimulation and on the relative timing between action potentials during propagation. Ephaptic coupling was found to increase the number of fibers that are activated by artificial stimulation, thus reducing the artificial currents required for axonal recruitment, and it was found to reduce and shift the range of optimal stimulation amplitudes for maximum inter-fascicular selectivity. During propagation, while fibers of similar diameters tended to lock their action potentials and reduce their conduction velocities, as expected from previous knowledge on bundles of identical axons, the presence of many other fibers of different diameters was found to make their interactions weaker and unstable.

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.

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.

Biophysically realistic neuron models for simulation of cortical stimulation

OBJECTIVE

We implemented computational models of human and rat cortical neurons for simulating the neural response to cortical stimulation with electromagnetic fields.

APPROACH

We adapted model neurons from the library of Blue Brain models to reflect biophysical and geometric properties of both adult rat and human cortical neurons and coupled the model neurons to exogenous electric fields (E-fields). The models included 3D reconstructed axonal and dendritic arbors, experimentally-validated electrophysiological behaviors, and multiple, morphological variants within cell types. Using these models, we characterized the single-cell responses to intracortical microstimulation (ICMS) and uniform E-field with dc as well as pulsed currents.

MAIN RESULTS

The strength-duration and current-distance characteristics of the model neurons to ICMS agreed with published experimental results, as did the subthreshold polarization of cell bodies and axon terminals by uniform dc E-fields. For all forms of stimulation, the lowest threshold elements were terminals of the axon collaterals, and the dependence of threshold and polarization on spatial and temporal stimulation parameters was strongly affected by morphological features of the axonal arbor, including myelination, diameter, and branching.

SIGNIFICANCE

These results provide key insights into the mechanisms of cortical stimulation. The presented models can be used to study various cortical stimulation modalities while incorporating detailed spatial and temporal features of the applied E-field.

Physiological effects of low-magnitude electric fields on brain activity: advances from in vitro, in vivo and in silico models

While electrical stimulation of brain tissue has been thoroughly investigated over the last decades, ongoing questions remain regarding the neurophysiological effects of low-level electric fields (on the order of 1 V/m) on brain activity. Electric fields at such levels are, for example, induced by transcranial direct/alternating current stimulation (tDCS/tACS). Action potentials can be indeed elicited when applied (supra-threshold) electric fields are in the 10-100 V/m range, while lower (subthreshold) electric fields result in more limited and subtler membrane polarization effects. In this review, we address the question of the mechanisms underlying the immediate effects (also referred to as acute, concurrent or short-term) and the lasting effects (also referred to as long-term or aftereffects) of low-level electric fields on brain tissue. We review recent evidence at the in vitro and in vivo (animal and human) level, and also present mechanistic insights gained from in silico models, which are still few but have received increased attention over the recent past years. We highlight the convergent evidence towards potential mechanisms, and also discuss discrepancies between in vitro studies and human tDCS/tACS studies that require further investigation to bridge the gap between the single-cell and large-scale network level. Possible novel avenues of research are discussed.