Current approaches to model extracellular electrical neural microstimulation

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.

Cellular mechanisms underlying carry-over effects after magnetic stimulation

Magnetic fields are widely used for neuromodulation in clinical settings. The intended effect of magnetic stimulation is that neural activity resumes its pre-stimulation state right after stimulation. Many theoretical and experimental works have focused on the cellular and molecular basis of the acute neural response to magnetic field. However, effects of magnetic stimulation can still last after the termination of the magnetic stimulation (named "carry-over effects"), which could generate profound effects to the outcome of the stimulation. However, the cellular and molecular mechanisms of carry-over effects are largely unknown, which renders the neural modulation practice using magnetic stimulation unpredictable. Here, we investigated carry-over effects at the cellular level, using the combination of micro-magnetic stimulation (µMS), electrophysiology, and computation modeling. We found that high frequency magnetic stimulation could lead to immediate neural inhibition in ganglion neurons from Aplysia californica, as well as persistent, carry-over inhibition after withdrawing the magnetic stimulus. Carry-over effects were found in the neurons that fired action potentials under a variety of conditions. The carry-over effects were also observed in the neurons when the magnetic field was applied across the ganglion sheath. The state of the neuron, specifically synaptic input and membrane potential fluctuation, plays a significant role in generating the carry-over effects after magnetic stimulation. To elucidate the cellular mechanisms of such carry-over effects under magnetic stimulation, we simulated a single neuron under magnetic stimulation with multi-compartment modeling. The model successfully replicated the carry-over effects in the neuron, and revealed that the carry-over effect was due to the dysfunction of the ion channel dynamics that were responsible for the initiation and sustaining of membrane excitability. A virtual voltage-clamp experiment revealed a compromised Na conductance and enhanced K conductance post magnetic stimulation, rendering the neurons incapable of generating action potentials and, therefore, leading to the carry over effects. Finally, both simulation and experimental results demonstrated that the carry-over effects could be controlled by disturbing the membrane potential during the post-stimulus inhibition period. Delineating the cellular and ion channel mechanisms underlying carry-over effects could provide insights to the clinical outcomes in brain stimulation using TMS and other modalities. This research incentivizes the development of novel neural engineering or pharmacological approaches to better control the carry-over effects for optimized clinical outcomes.

Finite-element modeling of neuromodulation via controlled delivery of potassium ions using conductive polymer-coated microelectrodes

Objective. The controlled delivery of potassium is an interesting neuromodulation modality, being potassium ions involved in shaping neuron excitability, synaptic transmission, network synchronization, and playing a key role in pathological conditions like epilepsy and spreading depression. Despite many successful examples of pre-clinical devices able to influence the extracellular potassium concentration, computational frameworks capturing the corresponding impact on neuronal activity are still missing.Approach. We present a finite-element model describing a PEDOT:PSS-coated microelectrode (herein, simplyionic actuator) able to release potassium and thus modulate the activity of a cortical neuron in anin-vitro-like setting. The dynamics of ions in the ionic actuator, the neural membrane, and the cellular fluids are solved self-consistently.Main results. We showcase the capability of the model to describe on a physical basis the modulation of the intrinsic excitability of the cell and of the synaptic transmission following the electro-ionic stimulation produced by the actuator. We consider three case studies for the ionic actuator with different levels of selectivity to potassium: ideal selectivity, no selectivity, and selectivity achieved by embedding ionophores in the polymer.Significance. This work is the first step toward a comprehensive computational framework aimed to investigate novel neuromodulation devices targeting specific ionic species, as well as to optimize their design and performance, in terms of the induced modulation of neural activity.

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.

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.

Effect of electric field direction on neuronal activity: an ex-vivo study. ANNUAL INTERNATIONAL CONFERENCE OF THE IEEE ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY. IEEE ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY. ANNUAL INTERNATIONAL CONFERENCE 2023;2023:1-4. [PMID: 38083738 DOI: 10.1109/embc40787.2023.10340225] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

The effect of electrical stimulation on neurons depends on the spatiotemporal properties of the applied electric field as well as on the biophysical properties of the neural tissue, which includes geometric and electrical characteristics of the cells, and the neural circuit dynamics. In this work, we characterize the effect of electric field direction on neural response in cortical layers. This can, for instance, enable more efficient (e.g., with reduced currents) and/or more selective stimulation. We stimulated mice brain slices using a recently developed brain slice platform to study transcranial currents in an ex-vivo model, where electrodes are separated from the brain slice to inject electric fields at a distance. By rotating the electrode array with respect to the slice, we changed the direction of electric field with respect to the cortical column. Our results demonstrate that in somatosensory cortex, the maximum local field potential (LFP) response is attained when the electric field is oriented parallel to the cortical column. For the same field intensity, when the field is oriented perpendicular to the cortical column, the LFP response is absent. This confirms that electric field direction is an important quantity to determine the effect of neuronal stimulation.

Improving focality and consistency in micromagnetic stimulation

The novel micromagnetic stimulation (μMS) technology aims to provide high resolution on neuronal targets. However, consistency of neural activation could be compromised by a lack of surgical accuracy, biological variation, and human errors in operation. We have recently modeled the activation of an unmyelinated axon by a circular micro-coil. Although the coil could activate the axon, its performance sometimes lacked focality and consistency. The site of axonal activation could shift by several experimental factors, including the reversal of the coil current, displacement of the coil, and changes in the intensity of the stimulation. Current clinical practice with transcranial magnetic stimulation (TMS) has suggested that figure-eight coils could provide better performance in magnetic stimulation than circular coils. Here, we estimate the performance of μMS by a figure-eight micro-coil, by exploring the impact of the same experimental factors on its focality and consistency in axonal activation. We derived the analytical expression of the electric field and activating function generated by the figure-eight micro-coil, and estimated the location of axonal activation. Using NEURON modeling of an unmyelinated axon, we found two different types (A and B) of axon activation by the figure-eight micro-coil, mediated by coil currents of reversed direction. Type A activation is triggered by membrane hyperpolarization followed by depolarization; Type B activation is triggered by direct membrane depolarization. Consequently, the two types of stimulation are governed by distinct ion channel mechanisms. In comparison to the circular micro-coil, the figure-eight micro-coil requires significantly less current for axonal activation. Under figure-eight micro-coil stimulation, the site of axonal activation does not change with the reversal of the coil current, displacement of the coil, or changes in the intensity of the stimulation. Ultimately, the figure-eight micro-coil provides a more efficient and consistent site of activation than the circular micro-coil in μMS.

Modeling the Impact of the Variation in Peripheral Nerve Anatomy on Stimulation

Introduction

The peripheral nervous system has a complex anatomical structure. Stimulation of nerve fibers in the peripheral nervous system depends on the fiber diameter and myelination as well as its location within the nerve, packing fraction and fascicle distribution within the nerve bundle. This paper analyzes the impact of the variation in peripheral nervous system anatomy and the distance of the stimulating electrodes on the probability of generating an action potential.

Methods

A mathematical model for effective fascicle conductivity has been developed to capture the variation in the packing fraction and fiber diameter. A linear activating function is utilized to analyze the impact of this effective conductivity and fascicle distribution as an indicator of generating an action potential.

Results

Finite element simulations are performed for the nerve-electrode configuration to evaluate the electric field. The simulation results are used to analyze the activating function for different packing fractions and type of nerve fibers. The effect of electrode distance on activating function and the total current through a nerve bundle has also been studied.

Discussion

The simulation results indicate that the peripheral nerve anatomy and electrode distance have a significant effect on the action potential generation.

Neuron matters: neuromodulation with electromagnetic stimulation must consider neurons as dynamic identities

Neuromodulation with electromagnetic stimulation is widely used for the control of abnormal neural activity, and has been proven to be a valuable alternative to pharmacological tools for the treatment of many neurological diseases. Tremendous efforts have been focused on the design of the stimulation apparatus (i.e., electrodes and magnetic coils) that delivers the electric current to the neural tissue, and the optimization of the stimulation parameters. Less attention has been given to the complicated, dynamic properties of the neurons, and their context-dependent impact on the stimulation effects. This review focuses on the neuronal factors that influence the outcomes of electromagnetic stimulation in neuromodulation. Evidence from multiple levels (tissue, cellular, and single ion channel) are reviewed. Properties of the neural elements and their dynamic changes play a significant role in the outcome of electromagnetic stimulation. This angle of understanding yields a comprehensive perspective of neural activity during electrical neuromodulation, and provides insights in the design and development of novel stimulation technology.

A varying-radius cable equation for the modelling of impulse propagation in excitable fibres

In this study, we present a varying-radius cable equation for nerve fibres taking into account the varying diameter along the neuronal segments. Finite element neuronal models utilising the classical (fixed-radius) and varying-radius cable formulations were compared using simple and realistic morphologies under intra- and extracellular electrical stimulation protocols. We found that the use of the classical cable equation to model intracellular neural electrical stimulation exhibited an error of 17% in a passive resistive cable model with abrupt change in radius from 1 to 2 μm, when compared to the known analytical solution and varying-radius cable formulation. This error was observed to increase substantially using more realistic neuron morphologies and branching structures. In the case of extracellular stimulation however, the difference between the classical and varying-radius formulations was less pronounced, but we expect this difference will increase under more complex stimulation paradigms such as high-frequency stimulation. We conclude that for computational neuroscience applications, it is essential to use the varying-radius cable equation for accurate prediction of neuronal responses under electrical stimulation.

Cellular mechanisms underlying state-dependent neural inhibition with magnetic stimulation

Novel stimulation protocols for neuromodulation with magnetic fields are explored in clinical and laboratory settings. Recent evidence suggests that the activation state of the nervous system plays a significant role in the outcome of magnetic stimulation, but the underlying cellular and molecular mechanisms of state-dependency have not been completely investigated. We recently reported that high frequency magnetic stimulation could inhibit neural activity when the neuron was in a low active state. In this paper, we investigate state-dependent neural modulation by applying a magnetic field to single neurons, using the novel micro-coil technology. High frequency magnetic stimulation suppressed single neuron activity in a state-dependent manner. It inhibited neurons in slow-firing states, but spared neurons from fast-firing states, when the same magnetic stimuli were applied. Using a multi-compartment NEURON model, we found that dynamics of voltage-dependent sodium and potassium channels were significantly altered by the magnetic stimulation in the slow-firing neurons, but not in the fast-firing neurons. Variability in neural activity should be monitored and explored to optimize the outcome of magnetic stimulation in basic laboratory research and clinical practice. If selective stimulation can be programmed to match the appropriate neural state, prosthetic implants and brain-machine interfaces can be designed based on these concepts to achieve optimal results.

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.

Scalable Three-Dimensional Recording Electrodes for Probing Biological Tissues

Electrophysiological recording technologies can provide critical insight into the function of the nervous system and other biological tissues. Standard silicon-based probes have limitations, including single-sided recording sites and intrinsic instabilities due to the probe stiffness. Here, we demonstrate high-performance neural recording using double-sided three-dimensional (3D) electrodes integrated in an ultraflexible bioinspired open mesh structure, allowing electrodes to sample fully the 3D interconnected tissue of the brain. In vivo electrophysiological recording using 3D electrodes shows statistically significant increases in the number of neurons per electrode, average spike amplitudes, and signal to noise ratios in comparison to standard two-dimensional electrodes, while achieving stable detection of single-neuron activity over months. The capability of these 3D electrodes is further shown for chronic recording from retinal ganglion cells in mice. This approach opens new opportunities for a comprehensive 3D interrogation, stimulation, and understanding of the complex circuitry of the brain and other electrogenic tissues in live animals over extended time periods.

A phenomenological computational model of the evoked action potential fitted to human cochlear implant responses

There is a growing interest in biomedical engineering in developing procedures that provide accurate simulations of the neural response to electrical stimulus produced by implants. Moreover, recent research focuses on models that take into account individual patient characteristics.

We present a phenomenological computational model that is customized with the patient’s data provided by the electrically evoked compound action potential (ECAP) for simulating the neural response to electrical stimulus produced by the electrodes of cochlear implants (CIs). The model links the input currents of the electrodes to the simulated ECAP.

Potentials and currents are calculated by solving the quasi-static approximation of the Maxwell equations with the finite element method (FEM). In ECAPs recording, an active electrode generates a current that elicits action potentials in the surrounding auditory nerve fibers (ANFs). The sum of these action potentials is registered by other nearby electrode. Our computational model emulates this phenomenon introducing a set of line current sources replacing the ANFs by a set of virtual neurons (VNs). To fit the ECAP amplitudes we assign a suitable weight to each VN related with the probability of an ANF to be excited. This probability is expressed by a cumulative beta distribution parameterized by two shape parameters that are calculated by means of a differential evolution algorithm (DE). Being the weights function of the current density, any change in the design of the CI affecting the current density produces changes in the weights and, therefore, in the simulated ECAP, which confers to our model a predictive capacity.

The results of the validation with ECAP data from two patients are presented, achieving a satisfactory fit of the experimental data with those provided by the proposed computational model.

The cochlea, found in the inner ear, is the organ where the sound is transformed into an electrical pulse to be transmitted by the neurons to the auditory cortex. Hearing loss can be caused by damage to the hair cells, in which case neuronal excitation is impaired. CIs are devices that replace the normal function of the impaired/damaged Organ of Corti. Computational models allow a better understanding of the mechanisms involved in the electrical stimulation of the auditory nerve. These models can help biomedical engineers to develop new CIs with improved auditory performance. One important aspect of our model is its customization with the patient’s data provided by the recording of the evoked compound action potential (the synchronous firing of a population of electrically stimulated auditory nerve fibers). This phenomenological model allows us to predict the registers of neural stimulation produced when the auditory nerve is stimulated with the CIs. We have validated the proposed model with real data obtained from two patients with CIs.

Three-dimensional multilayer concentric bipolar electrodes restrict spatial activation in optic nerve stimulation

OBJECTIVE

Intraneural nerve interfaces often operate in a monopolar configuration with a common and distant ground electrode. This configuration leads to a wide spreading of the electric field. Therefore, this approach is suboptimal for intraneural nerve interfaces when selective stimulation is required.

APPROACH

We designed a multilayer electrode array embedding three-dimensional concentric bipolar electrodes. First, we validated the higher stimulation selectivity of this new electrode array compared to classical monopolar stimulation using simulations. Next, we compared them in-vivo by intraneural stimulation of the rabbit optic nerve and recording evoked potentials in the primary visual cortex.

MAIN RESULTS

Simulations showed that three-dimensional concentric bipolar electrodes provide a high localisation of the electric field in the tissue so that electrodes are electrically independent even for high electrode density. Experiments in-vivo highlighted that this configuration restricts spatial activation in the visual cortex due to the fewer fibres activated by the electric stimulus in the nerve.

SIGNIFICANCE

Highly focused electric stimulation is crucial to achieving high selectivity in fibre activation. The multilayer array embedding three-dimensional concentric bipolar electrodes improves selectivity in optic nerve stimulation. This approach is suitable for other neural applications, including bioelectronic medicine.

A finite element method framework to model extracellular neural stimulation

OBJECTIVE

Increasing complexity in extracellular stimulation experiments and neural implant design also requires realistic computer simulations capable of modeling the neural activity of nerve cells under the influence of an electrical stimulus. Classical model approaches are often based on simplifications, are not able to correctly calculate the electric field generated by complex electrode designs, and do not consider electrical effects of the cell on its surrounding. A more accurate approach is the finite element method (FEM), which provides necessary techniques to solve the Poisson equation for complex geometries under consideration of electrical tissue properties. Especially in situations where neurons experience large and non-symmetric extracellular potential gradients, a FEM solution that implements the cell membrane model can improve the computer simulation results. To investigate the response of neurons in an electric field generated by complex electrode designs, a FEM framework for extracellular stimulation was developed in COMSOL.

APPROACH

Methods to implement morphologically- and biophysically-detailed neurons including active Hodgkin-Huxley (HH) cell membrane dynamics as well as the stimulation setup are described in detail. Covered methods are (i) development of cell and electrode geometries including meshing strategies, (ii) assignment of physics for the conducting spaces and the realization of active electrodes, (iii) implementation of the HH model, and (iv) coupling of the physics to get a fully described model.

MAIN RESULTS

Several implementation examples are briefly presented: (i) a full FEM implementation of a HH model cell stimulated with a honeycomb electrode, (ii) the electric field of a cochlear electrode placed inside the cochlea, and (iii) a proof of concept implementation of a detailed double-cable cell membrane model for myelinated nerve fibers.

SIGNIFICANCE

The presented concepts and methods provide basic and advanced techniques to realize a full FEM framework for innovative studies of neural excitation in response to extracellular stimulation.

Somatic inhibition by microscopic magnetic stimulation

Electric currents can produce quick, reversible control of neural activity. Externally applied electric currents have been used in inhibiting certain ganglion cells in clinical practices. Via electromagnetic induction, a miniature-sized magnetic coil could provide focal stimulation to the ganglion neurons. Here we report that high-frequency stimulation with the miniature coil could reversibly block ganglion cell activity in marine mollusk Aplysia californica, regardless the firing frequency of the neurons, or concentration of potassium ions around the ganglion neurons. Presence of the ganglion sheath has minimal impact on the inhibitory effects of the coil. The inhibitory effect was local to the soma, and was sufficient in blocking the neuron's functional output. Biophysical modeling confirmed that the miniature coil induced a sufficient electric field in the vicinity of the targeted soma. Using a multi-compartment model of Aplysia ganglion neuron, we found that the high-frequency magnetic stimuli altered the ion channel dynamics that were essential for the sustained firing of action potentials in the soma. Results from this study produces several critical insights to further developing the miniature coil technology for neural control by targeting ganglion cells. The miniature coil provides an interesting neural modulation strategy in clinical applications and laboratory research.

Deep learning of material transport in complex neurite networks

Neurons exhibit complex geometry in their branched networks of neurites which is essential to the function of individual neuron but also brings challenges to transport a wide variety of essential materials throughout their neurite networks for their survival and function. While numerical methods like isogeometric analysis (IGA) have been used for modeling the material transport process via solving partial differential equations (PDEs), they require long computation time and huge computation resources to ensure accurate geometry representation and solution, thus limit their biomedical application. Here we present a graph neural network (GNN)-based deep learning model to learn the IGA-based material transport simulation and provide fast material concentration prediction within neurite networks of any topology. Given input boundary conditions and geometry configurations, the well-trained model can predict the dynamical concentration change during the transport process with an average error less than 10% and [Formula: see text] times faster compared to IGA simulations. The effectiveness of the proposed model is demonstrated within several complex neurite networks.

Influence of Neuronal Morphology on the Shape of Extracellular Recordings With Microelectrode Arrays: A Finite Element Analysis

OBJECTIVE

Measuring neuronal cell activity using microelectrode arrays reveals a great variety of derived signal shapes within extracellular recordings. However, possible mechanisms responsible for this variety have not yet been entirely determined, which might hamper any subsequent analysis of the recorded neuronal data.

METHODS

To investigate this issue, we propose a computational model based on the finite element method describing the electrical coupling between an electrically active neuron and an extracellular recording electrode in detail. This allows for a systematic study of possible parameters that may play an essential role in defining or altering the shape of the measured electrode potential.

RESULTS

Our results indicate that neuronal geometry, neurite structure, as well as the actual pathways of input potentials that evoke action potential generation, have a significant impact on the shape of the resulting extracellular electrode recording and explain most of the known variations of signal shapes.

CONCLUSION

The presented models offer a comprehensive insight into the effect of geometrical and morphological factors on the resulting electrode signal.

SIGNIFICANCE

Computational modeling complemented with experimental measurements shows much promise to yield meaningful insights into the electrical activity of a neuronal network.

Toward Closed-Loop Electrical Stimulation of Neuronal Systems: A Review

Biological neuronal cells communicate using neurochemistry and electrical signals. The same phenomena also allow us to probe and manipulate neuronal systems and communicate with them. Neuronal system malfunctions cause a multitude of symptoms and functional deficiencies that can be assessed and sometimes alleviated by electrical stimulation. Our working hypothesis is that real-time closed-loop full-duplex measurement and stimulation paradigms can provide more in-depth insight into neuronal networks and enhance our capability to control diseases of the nervous system. In this study, we review extracellular electrical stimulation methods used in in vivo, in vitro, and in silico neuroscience research and in the clinic (excluding methods mainly aimed at neuronal growth and other similar effects) and highlight the potential of closed-loop measurement and stimulation systems. A multitude of electrical stimulation and measurement-based methods are widely used in research and the clinic. Closed-loop methods have been proposed, and some are used in the clinic. However, closed-loop systems utilizing more complex measurement analysis and adaptive stimulation systems, such as artificial intelligence systems connected to biological neuronal systems, do not yet exist. Our review promotes the research and development of intelligent paradigms aimed at meaningful communications between neuronal and information and communications technology systems, "dialogical paradigms," which have the potential to take neuroscience and clinical methods to a new level.

Axonal blockage with microscopic magnetic stimulation

Numerous neurological dysfunctions are characterized by undesirable nerve activity. By providing reversible nerve blockage, electric stimulation with an implanted electrode holds promise in the treatment of these conditions. However, there are several limitations to its application, including poor bio-compatibility and decreased efficacy during chronic implantation. A magnetic coil of miniature size can mitigate some of these problems, by coating it with biocompatible material for chronic implantation. However, it is unknown if miniature coils could be effective in axonal blockage and, if so, what the underlying mechanisms are. Here we demonstrate that a submillimeter magnetic coil can reversibly block action potentials in the unmyelinated axons from the marine mollusk Aplysia californica. Using a multi-compartment model of the Aplysia axon, we demonstrate that the miniature coil causes a significant local depolarization in the axon, alters activation dynamics of the sodium channels, and prevents the traveling of the invading action potentials. With improved biocompatibility and capability of emitting high-frequency stimuli, micro coils provide an interesting alternative for electric blockage of axonal conductance in clinical settings.

Reduced order models of myelinated axonal compartments

The paper presents a hierarchical series of computational models for myelinated axonal compartments. Three classes of models are considered, either with distributed parameters (2.5D EQS-ElectroQuasi Static, 1D TL-Transmission Lines) or with lumped parameters (0D). They are systematically analyzed with both analytical and numerical approaches, the main goal being to identify the best procedure for order reduction of each case. An appropriate error estimator is proposed in order to assess the accuracy of the models. This is the foundation of a procedure able to find the simplest reduced model having an imposed precision. The most computationally efficient model from the three geometries proved to be the analytical 1D one, which is able to have accuracy less than 0.1%. By order reduction with vector fitting, a finite model is generated with a relative difference of 10^- 4 for order 5. The dynamical models thus extracted allow an efficient simulation of neurons and, consequently, of neuronal circuits. In such situations, the linear models of the myelinated compartments coupled with the dynamical, non-linear models of the Ranvier nodes, neuronal body (soma) and dendritic tree give global reduced models. In order to ease the simulation of large-scale neuronal systems, the sub-models at each level, including those of myelinated compartments should have the lowest possible order. The presented procedure is a first step in achieving simulations of neural systems with accuracy control.

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.

Protuberant Electrode Structures for Subretinal Electrical Stimulation: Modeling, Fabrication and in vivo Evaluation

Many neural interfaces used for therapeutic applications are based on extracellular electrical stimulation to control cell polarization and thus functional activity. Amongst them, retinal implants have been designed to restore visual perception in blind patients affected by photoreceptor degeneration diseases, such as age-related macular degeneration (AMD) or retinitis pigmentosa (RP). While designing such a neural interface, several aspects must be taken into account, like the stimulation efficiency related to the current distribution within the tissue, the bio-interface optimization to improve resolution and tissue integration, and the material biocompatibility associated with long-term aging. In this study, we investigate the use of original microelectrode geometries for subretinal stimulation. The proposed structures combine the use of 3D wells with protuberant mushroom shaped electrode structures in the bottom, implemented on a flexible substrate that allows the in vivo implantation of the devices. These 3D microelectrode structures were first modeled using finite element analysis. Then, a specific microfabrication process compatible with flexible implants was developed to create the 3D microelectrode structures. These structures were tested in vivo to check the adaptation of the retinal tissue to them. Finally, preliminary in vivo stimulation experiments were performed.

ELFENN: A Generalized Platform for Modeling Ephaptic Coupling in Spiking Neuron Models

The transmembrane ionic currents that underlie changes in a cell's membrane potential give rise to electric fields in the extracellular space. In the context of brain activity, these electric fields form the basis for extracellularly recorded signals, such as multiunit activity, local field potentials and electroencephalograms. Understanding the underlying neuronal dynamics and localizing current sources using these signals is often challenging, and therefore effective computational modeling approaches are critical. Typically, the electric fields from neural activity are modeled in a post-hoc form, i.e., a traditional neuronal model is used to first generate the membrane currents, which in turn are then used to calculate the electric fields. When the conductivity of the extracellular space is high, the electric fields are weak, and therefore treating membrane currents and electric fields separately is justified. However, in brain regions of lower conductivity, extracellular fields can feed back and significantly influence the underlying transmembrane currents and dynamics of nearby neurons—this is often referred to as ephaptic coupling. The closed-loop nature of ephaptic coupling cannot be modeled using the post-hoc approaches implemented by existing software tools; instead, electric fields and neuronal dynamics must be solved simultaneously. To this end, we have developed a generalized modeling toolbox for studying ephaptic coupling in compartmental neuron models: E_LFENN (E_Lectric Field Effects in Neural Networks). In open loop conditions, we validate the separate components of E_LFENN for modeling membrane dynamics and associated field potentials against standard approaches (NEURON and LFPy). Unlike standard approaches however, E_LFENN enables the closed-loop condition to be modeled as well, in that the field potentials can feed back and influence membrane dynamics. As an example closed-loop case, we use E_LFENN to study phase-locking of action potentials generated by a population of axons running parallel in a bundle. Being able to efficiently explore ephaptic coupling from a computational perspective using tools, such as E_LFENN will allow us to better understand the physical basis of electric fields in the brain, as well as the conditions in which these fields may influence neuronal dynamics in general.

Implantable Direct Current Neural Modulation: Theory, Feasibility, and Efficacy

Implantable neuroprostheses such as cochlear implants, deep brain stimulators, spinal cord stimulators, and retinal implants use charge-balanced alternating current (AC) pulses to recover delivered charge and thus mitigate toxicity from electrochemical reactions occurring at the metal-tissue interface. At low pulse rates, these short duration pulses have the effect of evoking spikes in neural tissue in a phase-locked fashion. When the therapeutic goal is to suppress neural activity, implants typically work indirectly by delivering excitation to populations of neurons that then inhibit the target neurons, or by delivering very high pulse rates that suffer from a number of undesirable side effects. Direct current (DC) neural modulation is an alternative methodology that can directly modulate extracellular membrane potential. This neuromodulation paradigm can excite or inhibit neurons in a graded fashion while maintaining their stochastic firing patterns. DC can also sensitize or desensitize neurons to input. When applied to a population of neurons, DC can modulate synaptic connectivity. Because DC delivered to metal electrodes inherently violates safe charge injection criteria, its use has not been explored for practical applicability of DC-based neural implants. Recently, several new technologies and strategies have been proposed that address this safety criteria and deliver ionic-based direct current (iDC). This, along with the increased understanding of the mechanisms behind the transcutaneous DC-based modulation of neural targets, has caused a resurgence of interest in the interaction between iDC and neural tissue both in the central and the peripheral nervous system. In this review we assess the feasibility of in-vivo iDC delivery as a form of neural modulation. We present the current understanding of DC/neural interaction. We explore the different design methodologies and technologies that attempt to safely deliver iDC to neural tissue and assess the scope of application for direct current modulation as a form of neuroprosthetic treatment in disease. Finally, we examine the safety implications of long duration iDC delivery. We conclude that DC-based neural implants are a promising new modulation technology that could benefit from further chronic safety assessments and a better understanding of the basic biological and biophysical mechanisms that underpin DC-mediated neural modulation.

Analysis of upper threshold mechanisms of spherical neurons during extracellular stimulation

Exceeding a certain stimulation strength can prevent the generation of somatic action potentials, as has been demonstrated in vitro with extracellularly stimulated dorsal root ganglion cells as well as retinal ganglion cells. This phenomenon, termed upper threshold, is currently thought to be a consequence of sodium current reversal in strongly depolarized regions. Here we analyze the contribution of membrane kinetics, using spherical model neurons that are stimulated externally with a microelectrode, in more detail. During extracellular pulse application, the electric field depolarizes one part and hyperpolarizes the other part of the cell. Strong transmembrane currents are generated only in the active depolarized region, changing the overall polarization level. The asymmetric membrane voltage distribution caused by the stimulus strongly influences the cell's behavior during and even after the stimulus. Effects on membrane voltage and transmembrane currents during and after the stimulus are shown and discussed in detail. Aside from the sodium current reversal, two more key mechanisms were identified in causing the upper threshold: strong potassium currents and inactivation of sodium channels. The contributions of the mechanisms involved strongly depend on cell properties, stimulus parameters, and other factors such as temperature. The conclusions presented here are based on several retinal ganglion cell models of the Fohlmeister group, a model with original Hodgkin-Huxley membrane, and a pyramidal cell model. NEW & NOTEWORTHY The upper threshold phenomenon in extracellular stimulation is analyzed in detail for spherical cells. Three main mechanisms were identified that prevent the generation of action potentials at high stimulation strengths: 1) strong potassium currents, 2) inactivating sodium ion channels, and 3) sodium current reversal. Ion channel kinetics in retinal ganglion cells, pyramidal cells, and the original Hodgkin-Huxley model were investigated under the influence of an extracellular stimulus.

Modeling Current Sources for Neural Stimulation in COMSOL

Background: Computational modeling provides an important toolset for designing and analyzing neural stimulation devices to treat neurological disorders and diseases. Modeling enables efficient exploration of large parameter spaces, where preclinical and clinical studies would be infeasible. Current commercial finite element method software packages enable straightforward calculation of the potential distributions, but it is not always clear how to implement boundary conditions to appropriately represent metal stimulating electrodes. By quantifying the effects of different electrode representations on activation thresholds for model axons, we provide recommendations for accurate and efficient modeling of neural stimulating electrodes. Methods: We quantified the effects of different representations of current sources for neural stimulation in COMSOL Multiphysics for monopolar, bipolar, and multipolar electrode designs. Results: We recommend modeling each electrode contact as a thin platinum domain, modeling the electrode substrate with the conductivity of silicone, and either using a point current source in the center of each electrode contact or using a boundary current source. Alternatively, to avoid possible numerical instabilities associated with a large range of conductivity values (i.e., platinum and silicone) and to eliminate the small mesh elements required for thin electrode contacts, the electrode substrate can be assigned the conductivity of platinum by using insulating boundaries between the substrate and surrounding medium, and within the substrate to isolate the contacts from each other. When modeling more than one contact, we recommend using superposition by solving the model once for each contact, leaving inactive contacts floating, and superposing the resulting potentials. We computed comparable errors in activation thresholds across the different implementations in a simplified model (electrode in a homogeneous, isotropic medium), and in realistic models of rat spinal cord stimulation (SCS) and human deep brain stimulation, indicating that the recommended approaches are applicable to different stimulation targets.

A multiobjective optimization procedure for the electrode design of cochlear implants

This paper presents a new procedure to design optimal electrodes for cochlear implants. The main objective of this study is to find a set of electrode designs that maximize the focalization and minimize the power consumption simultaneously. To achieve that, a criterion to measure the ability of focalization of an electrode is proposed. It is presented a procedure to determine (1) the electrical potential induced by an electrode by solving the Laplace equation through the finite element method; (2) the response of a neuron to an applied field using NEURON, a compartmentalized cell model; (3) the optimization to find the best electrode designs according to power consumption and focalization by 2 evolutionary multiobjective methods based on the non-dominated sorting genetic algorithm II: a straight multiobjective approach and a seeded multiobjective approach. An electrode design formed by 2 conductive rings with a possible difference of potential between them is proposed. It is analyzed that the response of the neuron is determined by the shape and the difference of the potential between the electrode rings. Our procedure successfully achieves a nondominated set of optimum electrode designs improving a standard electrode in both objectives, as designs with better focalization allow to include extra electrodes in the cochlear implant, and designs with lower power consumption extend the length of the battery.

Kilohertz Frequency Deep Brain Stimulation Is Ineffective at Regularizing the Firing of Model Thalamic Neurons

Deep brain stimulation (DBS) is an established therapy for movement disorders, including tremor, dystonia, and Parkinson's disease, but the mechanisms of action are not well understood. Symptom suppression by DBS typically requires stimulation frequencies ≥100 Hz, but when the frequency is increased above ~2 kHz, the effectiveness in tremor suppression declines (Benabid et al., 1991). We sought to test the hypothesis that the decline in efficacy at high frequencies is associated with desynchronization of the activity generated within a population of stimulated neurons. Regularization of neuronal firing is strongly correlated with tremor suppression by DBS, and desynchronization would disrupt the regularization of neuronal activity. We implemented computational models of CNS axons with either deterministic or stochastic membrane dynamics, and quantified the response of populations of model nerve fibers to extracellular stimulation at different frequencies and amplitudes. As stimulation frequency was increased from 2 to 80 Hz the regularity of neuronal firing increased (as assessed with direct estimates of entropy), in accord with the clinical effects on tremor of increasing stimulation frequency (Kuncel et al., 2006). Further, at frequencies between 80 and 500 Hz, increasing the stimulation amplitude (i.e., the proportion of neurons activated by the stimulus) increased the regularity of neuronal activity across the population, in accord with the clinical effects on tremor of stimulation amplitude (Kuncel et al., 2007). However, at stimulation frequencies above 1 kHz the regularity of neuronal firing declined due to irregular patterns of action potential generation and conduction block. The reductions in neuronal regularity that occurred at high frequencies paralleled the previously reported decline in tremor reduction and may be responsible for the loss of efficacy of DBS at very high frequencies. This analysis provides further support for the hypothesis that effective DBS masks the intrinsic patterns of activity in the stimulated neurons and replaces it with regularized firing.

Computational modeling to improve treatments for essential tremor

Essential tremor (ET) is a neurological disorder of unknown etiology that is typically characterized by an involuntary periodic movement of the upper limbs. No longer considered monosymptomatic, ET patients often have additional motor and even cognitive impairments. Although there are several pharmacological treatments, no drugs have been developed specifically for ET [1], and 30-70% of patients are medication-refractory [2]. A subset of medication-refractory patients may benefit from electrical deep brain stimulation (DBS) of the ventral intermediate nucleus of the thalamus (VIM), which receives cerebellar inputs. Abnormal cerebellar input to VIM is presumed to be a major contributor to tremor symptoms, which is alleviated by DBS. Computational modeling of the effects of DBS in VIM has been a powerful tool to design DBS protocols to reduce tremor activity. However, far less is known about how these therapies affect non-tremor symptoms, and more experimental and computational modeling work is required to address these growing considerations. Models capable of addressing multiple facets of ET will lead to novel, more efficient treatment.

Numerical characterization of intraoperative and chronic electrodes in deep brain stimulation

An intraoperative electrode (microelectrode) is used in the deep brain stimulation (DBS) technique to pinpoint the brain target and to choose the best parameters for the electrical stimulus. However, when the intraoperative electrode is replaced with the chronic one (macroelectrode), the observed effects do not always coincide with predictions. To investigate the causes of such discrepancies, a 3D model of the basal ganglia has been considered and realistic models of both intraoperative and chronic electrodes have been developed and numerically solved. Results of simulations of the electric potential (V) and the activating function (AF) along neuronal fibers show that the different geometries and sizes of the two electrodes do not change the distributions and polarities of these functions, but rather the amplitudes. This effect is similar to the one produced by the presence of different tissue layers (edema or glial tissue) in the peri-electrode space. Conversely, an inaccurate positioning of the chronic electrode with respect to the intraoperative one (electric centers not coincident) may induce a completely different electric stimulation in some groups of fibers.

Modelling extracellular electrical stimulation: part 4. Effect of the cellular composition of neural tissue on its spatio-temporal filtering properties

OBJECTIVE

The objective of this paper is to present a concrete application of the cellular composite model for calculating the membrane potential, described in an accompanying paper.

APPROACH

A composite model that is used to determine the membrane potential for both longitudinal and transverse modes of stimulation is demonstrated.

MAIN RESULTS

Two extreme limits of the model, near-field and far-field for an electrode close to or distant from a neuron, respectively, are derived in this paper. Results for typical neural tissue are compared using the composite, near-field and far-field models as well as the standard isotropic volume conductor model. The self-consistency of the composite model, its spatial profile response and the extracellular potential time behaviour are presented. The magnitudes of the longitudinal and transverse components for different values of electrode-neurite separations are compared.

SIGNIFICANCE

The unique features of the composite model and its simplified versions can be used to accurately estimate the spatio-temporal response of neural tissue to extracellular electrical stimulation.