Modeling extracellular electrical stimulation: I. Derivation and interpretation of neurite equations

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.

Numerical simulation of neural excitation during brain tumor ablation by microsecond pulses

Replacing monopolar pulse with bipolar pulses of the same energized time can minimize unnecessary neurological side effects during irreversible electroporation (IRE). An improved neural excitation model that considers dynamic conductivity and thermal effects during brain tumor IRE ablation was proposed for the first time in this study. Nerve fiber excitation during IRE ablation by applying a monopolar pulse (100 μs) and a burst of bipolar pulses (energized time of 100 μs with both the sub-pulse length and interphase delay of 1 μs) was investigated. Our results suggest that both thermal effects and dynamic conductivity change the onset time of action potential (AP), and dynamic conductivity also changes the hyperpolarization amplitude. Considering both thermal effects and dynamic conductivity, the hyperpolarization amplitude in nerve fibers located 2 cm from the tumor center was reduced by approximately 23.8 mV and the onset time of AP was delayed by approximately 17.5 μs when a 500 V monopolar pulse was applied. Moreover, bipolar pulses decreased the excitable volume of brain tissue by approximately 68.8 % compared to monopolar pulse. Finally, bipolar pulses cause local excitation with lesser damage to surrounding healthy tissue in complete tumor ablation, demonstrating the potential benefits of bipolar pulses in brain tissue ablation.

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.

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.

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.

Unifying information theory and machine learning in a model of electrode discrimination in cochlear implants

Despite the development and success of cochlear implants over several decades, wide inter-subject variability in speech perception is reported. This suggests that cochlear implant user-dependent factors limit speech perception at the individual level. Clinical studies have demonstrated the importance of the number, placement, and insertion depths of electrodes on speech recognition abilities. However, these do not account for all inter-subject variability and to what extent these factors affect speech recognition abilities has not been studied. In this paper, an information theoretic method and machine learning technique are unified in a model to investigate the extent to which key factors limit cochlear implant electrode discrimination. The framework uses a neural network classifier to predict which electrode is stimulated for a given simulated activation pattern of the auditory nerve, and mutual information is then estimated between the actual stimulated electrode and predicted ones. We also investigate how and to what extent the choices of parameters affect the performance of the model. The advantages of this framework include i) electrode discrimination ability is quantified using information theory, ii) it provides a flexible framework that may be used to investigate the key factors that limit the performance of cochlear implant users, and iii) it provides insights for future modeling studies of other types of neural prostheses.

Excitation properties of computational models of unmyelinated peripheral axons

Biophysically based computational models of nerve fibers are important tools for designing electrical stimulation therapies, investigating drugs that affect ion channels, and studying diseases that affect neurons. Although peripheral nerves are primarily composed of unmyelinated axons (i.e., C-fibers), most modeling efforts focused on myelinated axons. We implemented the single-compartment model of vagal afferents from Schild et al. (1994) (Schild JH, Clark JW, Hay M, Mendelowitz D, Andresen MC, Kunze DL. J Neurophysiol 71: 2338-2358, 1994) and extended the model into a multicompartment axon, presenting the first cable model of a C-fiber vagal afferent. We also implemented the updated parameters from the Schild and Kunze (1997) model (Schild JH, Kunze DL. J Neurophysiol 78: 3198-3209, 1997). We compared the responses of these novel models with those of three published models of unmyelinated axons (Rattay F, Aberham M. IEEE Trans Biomed Eng 40: 1201-1209, 1993; Sundt D, Gamper N, Jaffe DB. J Neurophysiol 114: 3140-3153, 2015; Tigerholm J, Petersson ME, Obreja O, Lampert A, Carr R, Schmelz M, Fransén E. J Neurophysiol 111: 1721-1735, 2014) and with experimental data from single-fiber recordings. Comparing the two models by Schild et al. (1994, 1997) revealed that differences in rest potential and action potential shape were driven by changes in maximum conductances rather than changes in sodium channel dynamics. Comparing the five model axons, the conduction speeds and strength-duration responses were largely within expected ranges, but none of the models captured the experimental threshold recovery cycle-including a complete absence of late subnormality in the models-and their action potential shapes varied dramatically. The Tigerholm et al. (2014) model best reproduced the experimental data, but these modeling efforts make clear that additional data are needed to parameterize and validate future models of autonomic C-fibers.NEW & NOTEWORTHY Peripheral nerves are primarily composed of unmyelinated axons, and there is growing interest in electrical stimulation of the autonomic nervous system to treat various diseases. We present the first cable model of an unmyelinated vagal nerve fiber and compare its ion channel isoforms and conduction responses with other published models of unmyelinated axons, establishing important tools for advancing modeling of autonomic nerves.

The Effects of Phase Durations on the Spatial Responses of Retinal Ganglion Cells to Epi- and Sub-Retinal Electrical Stimulation

Retinal prostheses have the potential to restore vision to blind patients that have retinitis pigmentosa or similar hereditary degenerative disorders, by electrically stimulating surviving retinal neurons through implanted electrode arrays. Current retinal prostheses provide limited visual acuity and one challenge is to spatially control neural activation following electrical stimulation. Most of the retinal prostheses are either epi-retinal - in front of the retinal ganglion cell layer, or sub-retinal - behind photoreceptor layer. In this study, we performed calcium imaging of ganglion cells from whole mounted retinas and compared the spread of neural activation between epi-retinal stimulation with a fiber electrode and sub-retinal stimulation with a disk electrode. We investigated the effects of phase durations on the spatial resolution of biphasic stimulation. Our results suggest that with fiber electrode epi-retinal stimulation, the axon bundles activation can lead to significant spread of stimulation, and cannot be avoided simply by changing the phase durations. However, sub-retinal stimulation with very short pulses (phase duration 0.033ms) can effectively confine the activation of retinal ganglion cells.

Stimulation Strategies for Improving the Resolution of Retinal Prostheses

Electrical stimulation using implantable devices with arrays of stimulating electrodes is an emerging therapy for neurological diseases. The performance of these devices depends greatly on their ability to activate populations of neurons with high spatiotemporal resolution. To study electrical stimulation of populations of neurons, retina serves as a useful model because the neural network is arranged in a planar array that is easy to access. Moreover, retinal prostheses are under development to restore vision by replacing the function of damaged light sensitive photoreceptors, which makes retinal research directly relevant for curing blindness. Here we provide a progress review on stimulation strategies developed in recent years to improve the resolution of electrical stimulation in retinal prostheses. We focus on studies performed with explanted retinas, in which electrophysiological techniques are the most advanced. We summarize achievements in improving the spatial and temporal resolution of electrical stimulation of the retina and methods to selectively stimulate neurons with different visual functions. Future directions for retinal prostheses development are also discussed, which could provide insights for other types of neuromodulatory devices in which high-resolution electrical stimulation is required.

Determination of the electrical impedance of neural tissue from its microscopic cellular constituents

The electrical properties of neural tissue are important in a range of different applications in biomedical engineering and basic science. These properties are characterized by the electrical admittivity of the tissue, which is the inverse of the specific tissue impedance.

OBJECTIVE

Here we derived analytical expressions for the admittivity of various models of neural tissue from the underlying electrical and morphological properties of the constituent cells.

APPROACH

Three models are considered: parallel bundles of fibers, fibers contained in stacked laminae and fibers crossing each other randomly in all three-dimensional directions.

MAIN RESULTS

An important and novel aspect that emerges from considering the underlying cellular composition of the tissue is that the resulting admittivity has both spatial and temporal frequency dependence, a property not shared with conventional conductivity-based descriptions. The frequency dependence of the admittivity results in non-trivial spatiotemporal filtering of electrical signals in the tissue models. These effects are illustrated by considering the example of pulsatile stimulation with a point source electrode. It is shown how changing temporal parameters of a current pulse, such as pulse duration, alters the spatial profile of the extracellular potential. In a second example, it is shown how the degree of electrical anisotropy can change as a function of the distance from the electrode, despite the underlying structurally homogeneity of the tissue. These effects are discussed in terms of different current pathways through the intra- and extra-cellular spaces, and how these relate to near- and far-field limits for the admittivity (which reduce to descriptions in terms of a simple conductivity).

SIGNIFICANCE

The results highlight the complexity of the electrical properties of neural tissue and provide mathematical methods to model this complexity.

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.

Selective recruitment of cortical neurons by electrical stimulation

Despite its critical importance in experimental and clinical neuroscience, at present there is no systematic method to predict which neural elements will be activated by a given stimulation regime. Here we develop a novel approach to model the effect of cortical stimulation on spiking probability of neurons in a volume of tissue, by applying an analytical estimate of stimulation-induced activation of different cell types across cortical layers. We utilize the morphology and properties of axonal arborization profiles obtained from publicly available anatomical reconstructions of the twelve main categories of neocortical neurons to derive the dependence of activation probability on cell type, layer and distance from the source. We then propagate this activity through the local network incorporating connectivity, synaptic and cellular properties. Our work predicts that (a) intracranial cortical stimulation induces selective activation across cell types and layers; (b) superficial anodal stimulation is more effective than cathodal at cell activation; (c) cortical surface stimulation focally activates layer I axons, and (d) there is an optimal stimulation intensity capable of eliciting cell activation lasting beyond the end of stimulation. We conclude that selective effects of cortical electrical stimulation across cell types and cortical layers are largely driven by their different axonal arborization and myelination profiles.

Brain stimulation is widely used to probe the neural system to learn about its properties, to normalize dysfunction (e.g., deep brain stimulation for Parkinsonian patients), or to manipulate brain activity, including enhancing memory and learning. Despite its critical importance in experimental and clinical neuroscience, at present there are no systematic methods to predict which neural elements of the brain will be activated by a given stimulation regime. To address this question, we propose a novel theoretical framework that models the effect of cortical stimulation on the spiking probability of a neuron based on its location, type and morphology. Our study predicts that short-lived superficial electrical stimulation has the ability to trigger spiking in layer IV pyramidal cells, and to evoke network activity that could persist for hundreds of milliseconds. It further predicts a much higher spiking response to anodal stimulation compared to cathodal one, as the existence of an optimal stimulation intensity, capable of inducing a maximal response in a population of cortical cells. The results of our study can be directly taken into account in planning future electrical stimulation experiments.

Hydrogel bioelectronics

Hydrogels have emerged as a promising bioelectronic interfacing material. This review discusses the fundamentals and recent advances in hydrogel bioelectronics.

On the parameters used in finite element modeling of compound peripheral nerves

OBJECTIVE

Computational modeling is an important tool for developing and optimizing implantable neural stimulation devices, but requires accurate electrical and geometrical parameter values to improve predictive value. We quantified the effects of perineurial (resistive sheath around each fascicle) and endoneurial (within each fascicle) parameter values for modeling peripheral nerve stimulation.

APPROACH

We implemented 3D finite element models of compound peripheral nerves and cuff electrodes to quantify activation and block thresholds of model axons. We also implemented a 2D finite element model of a bundle of axons to estimate the bulk transverse endoneurial resistivity; we compared numerical estimates to an analytical solution.

MAIN RESULTS

Since the perineurium is highly resistive, potentials were approximately constant over the cross section of a fascicle, and the perineurium resistivity, longitudinal endoneurial resistivity, and fascicle diameter had important effects on thresholds. Activation thresholds increased up to ~130% for higher perineurium resistivity (~400 versus 2200 Ω m) and by ~35%-250% for lower longitudinal endoneurial resistivity (3.5 versus 0.75 Ω m), with larger increases for smaller diameter axons and for axons in larger fascicles. Further, thresholds increased by ~30%-180% for larger fascicle radii, yielding a larger increase with higher perineurium resistivity. Thresholds were largely insensitive to the transverse endoneurial resistivity, but estimates of the bulk resistivity increased with extracellular resistivity and axonal area fraction; the numerical and analytical estimates were in strong agreement except at high axonal area fractions, where structured axon placements that achieved tighter packing produced electric field inhomogeneities.

SIGNIFICANCE

We performed a systematic investigation of the effects of values and methods for modeling the perineurium and endoneurium on thresholds for neural stimulation and block. These results provide guidance for future modeling studies, including parameter selection, data interpretation, and comparison to experimental results.

Biophysical basis of the linear electrical receptive fields of retinal ganglion cells

Responses of retinal ganglion cells to direct electrical stimulation have been shown experimentally to be well described by linear-nonlinear models. These models rely on the simplifying assumption that retinal ganglion cell responses to stimulation with an array of electrodes are driven by a simple linear weighted sum of stimulus current amplitudes from each electrode, known as the 'electrical receptive field'.

OBJECTIVE

This paper aims to demonstrate the biophysical basis of the linear-nonlinear model and the electrical receptive field to facilitate the development of improved stimulation strategies for retinal implants.

APPROACH

We compare the linear-nonlinear model of subretinal electrical stimulation with a multi-layered, biophysical, volume conductor model of retinal stimulation.

MAIN RESULTS

Our results show that the linear electrical receptive field of the linear-nonlinear model matches the transmembrane currents induced by electrodes (the activating function) at the site of the high-density sodium channel band with only minor discrepancies. The discrepancies are mostly eliminated by including axial current flow originating from adjacent cell compartments. Furthermore, for cells where a single linear electrical receptive field is insufficient, we show that cell responses are likely driven by multiple sites of action potential initiation with multiple distinct receptive fields, each of which can be accurately described by the activating function.

SIGNIFICANCE

This result establishes that the biophysical basis of the electrical receptive field of the linear-nonlinear model is the superposition of transmembrane currents induced by different electrodes at and near the site of action potential initiation. Together with existing experimental support for linear-nonlinear models of electrical stimulation, this provides a firm basis for using this much simplified model to generate more optimal stimulation patterns for retinal implants.

Upper stimulation threshold for retinal ganglion cell activation

OBJECTIVE

The existence of an upper threshold in electrically stimulated retinal ganglion cells (RGCs) is of interest because of its relevance to the development of visual prosthetic devices, which are designed to restore partial sight to blind patients. The upper threshold is defined as the stimulation level above which no action potentials (direct spikes) can be elicited in electrically stimulated retina.

APPROACH

We collected and analyzed in vitro recordings from rat RGCs in response to extracellular biphasic (anodic-cathodic) pulse stimulation of varying amplitudes and pulse durations. Such responses were also simulated using a multicompartment model.

MAIN RESULTS

We identified the individual cell variability in response to stimulation and the phenomenon known as upper threshold in all but one of the recorded cells (n  =  20/21). We found that the latencies of spike responses relative to stimulus amplitude had a characteristic U-shape. In silico, we showed that the upper threshold phenomenon was observed only in the soma. For all tested biphasic pulse durations, electrode positions, and pulse amplitudes above lower threshold, a propagating action potential was observed in the distal axon. For amplitudes above the somatic upper threshold, the axonal action potential back-propagated in the direction of the soma, but the soma's low level of hyperpolarization prevented action potential generation in the soma itself.

SIGNIFICANCE

An upper threshold observed in the soma does not prevent spike conductance in the axon.

Minimizing activation of overlying axons with epiretinal stimulation: The role of fiber orientation and electrode configuration

Currently, a challenge in electrical stimulation of the retina with a visual prosthesis (bionic eye) is to excite only the cells lying directly under the electrode in the ganglion cell layer, while avoiding excitation of axon bundles that pass over the surface of the retina in the nerve fiber layer. Stimulation of overlying axons results in irregular visual percepts, limiting perceptual efficacy. This research explores how differences in fiber orientation between the nerve fiber layer and ganglion cell layer leads to differences in the electrical activation of the axon initial segment and axons of passage. Approach. Axons of passage of retinal ganglion cells in the nerve fiber layer are characterized by a narrow distribution of fiber orientations, causing highly anisotropic spread of applied current. In contrast, proximal axons in the ganglion cell layer have a wider distribution of orientations. A four-layer computational model of epiretinal extracellular stimulation that captures the effect of neurite orientation in anisotropic tissue has been developed using a volume conductor model known as the cellular composite model. Simulations are conducted to investigate the interaction of neural tissue orientation, stimulating electrode configuration, and stimulation pulse duration and amplitude. Main results. Our model shows that simultaneous stimulation with multiple electrodes aligned with the nerve fiber layer can be used to achieve selective activation of axon initial segments rather than passing fibers. This result can be achieved while reducing required stimulus charge density and with only modest increases in the spread of activation in the ganglion cell layer, and is shown to extend to the general case of arbitrary electrode array positioning and arbitrary target volume. Significance. These results elucidate a strategy for more targeted stimulation of retinal ganglion cells with experimentally-relevant multi-electrode geometries and achievable stimulation requirements.

Electro-mechanical response of a 3D nerve bundle model to mechanical loads leading to axonal injury

Traumatic brain injuries and damage are major causes of death and disability. We propose a 3D fully coupled electro-mechanical model of a nerve bundle to investigate the electrophysiological impairments due to trauma at the cellular level. The coupling is based on a thermal analogy of the neural electrical activity by using the finite element software Abaqus CAE 6.13-3. The model includes a real-time coupling, modulated threshold for spiking activation, and independent alteration of the electrical properties for each 3-layer fibre within a nerve bundle as a function of strain. Results of the coupled electro-mechanical model are validated with previously published experimental results of damaged axons. Here, the cases of compression and tension are simulated to induce (mild, moderate, and severe) damage at the nerve membrane of a nerve bundle, made of 4 fibres. Changes in strain, stress distribution, and neural activity are investigated for myelinated and unmyelinated nerve fibres, by considering the cases of an intact and of a traumatised nerve membrane. A fully coupled electro-mechanical modelling approach is established to provide insights into crucial aspects of neural activity at the cellular level due to traumatic brain injury. One of the key findings is the 3D distribution of residual stresses and strains at the membrane of each fibre due to mechanically induced electrophysiological impairments, and its impact on signal transmission.

A Mixed Integer Linear Programming Approach to Electrical Stimulation Optimization Problems

Electrical stimulation optimization is a challenging problem. Even when a single region is targeted for excitation, the problem remains a constrained multi-objective optimization problem. The constrained nature of the problem results from safety concerns while its multi-objectives originate from the requirement that non-targeted regions should remain unaffected. In this paper, we propose a mixed integer linear programming formulation that can successfully address the challenges facing this problem. Moreover, the proposed framework can conclusively check the feasibility of the stimulation goals. This helps researchers to avoid wasting time trying to achieve goals that are impossible under a chosen stimulation setup. The superiority of the proposed framework over alternative methods is demonstrated through simulation examples.

Avoiding nerve stimulation in irreversible electroporation: a numerical modeling study

Electroporation based treatments consist in applying one or multiple high voltage pulses to the tissues to be treated. As an undesired side effect, these pulses cause electrical stimulation of excitable tissues such as nerves and muscles. This increases the complexity of the treatments and may pose a risk to the patient. To minimize electrical stimulation during electroporation based treatments, it has been proposed to replace the commonly used monopolar pulses by bursts of short bipolar pulses. In the present study, we have numerically analyzed the rationale for such approach. We have compared different pulsing protocols in terms of their electroporation efficacy and their capability of triggering action potentials in nerves. For that, we have developed a modeling framework that combines numerical models of nerve fibers and experimental data on irreversible electroporation. Our results indicate that, by replacing the conventional relatively long monopolar pulses by bursts of short bipolar pulses, it is possible to ablate a large tissue region without triggering action potentials in a nearby nerve. Our models indicate that this is possible because, as the pulse length of these bipolar pulses is reduced, the stimulation thresholds raise faster than the irreversible electroporation thresholds. We propose that this different dependence on the pulse length is due to the fact that transmembrane charging for nerve fibers is much slower than that of cells treated by electroporation because of their geometrical differences.

Electrothermal Equivalent Three-Dimensional Finite-Element Model of a Single Neuron

OBJECTIVE

We propose a novel approach for modelling the interdependence of electrical and mechanical phenomena in nervous cells, by using electrothermal equivalences in finite element (FE) analysis so that existing thermomechanical tools can be applied.

METHODS

First, the equivalence between electrical and thermal properties of the nerve materials is established, and results of a pure heat conduction analysis performed in Abaqus CAE Software 6.13-3 are validated with analytical solutions for a range of steady and transient conditions. This validation includes the definition of equivalent active membrane properties that enable prediction of the action potential. Then, as a step toward fully coupled models, electromechanical coupling is implemented through the definition of equivalent piezoelectric properties of the nerve membrane using the thermal expansion coefficient, enabling prediction of the mechanical response of the nerve to the action potential.

RESULTS

Results of the coupled electromechanical model are validated with previously published experimental results of deformation for squid giant axon, crab nerve fibre, and garfish olfactory nerve fibre.

CONCLUSION

A simplified coupled electromechanical modelling approach is established through an electrothermal equivalent FE model of a nervous cell for biomedical applications.

SIGNIFICANCE

One of the key findings is the mechanical characterization of the neural activity in a coupled electromechanical domain, which provides insights into the electromechanical behaviour of nervous cells, such as thinning of the membrane. This is a first step toward modelling three-dimensional electromechanical alteration induced by trauma at nerve bundle, tissue, and organ levels.

A computational model of orientation-dependent activation of retinal ganglion cells

Currently, a challenge in electrical stimulation for epiretinal prostheses is the avoidance of stimulation of axons of passage in the nerve fiber layer that originate from distant regions of the ganglion cell layer. A computational model of extracellular stimulation that captures the effect of neurite orientation in anisotropic tissue is developed using a modified version of the standard volume conductor model, known as the cellular composite model, embedded in a four layer model of the retina. Simulations are conducted to investigate the interaction of neural tissue orientation, electrode placement, and stimulation pulse duration and amplitude. Using appropriate multiple electrode configurations and higher frequency stimulation, preferential activation of the axon initial segment is shown to be possible for a range of realistic electrode-retina separation distances. These results establish a quantitative relationship between the time-course of stimulation and physical properties of the tissue, such as fiber orientation.

Impedance Spectrum in Cortical Tissue: Implications for Propagation of LFP Signals on the Microscopic Level

Brain research investigating electrical activity within neural tissue is producing an increasing amount of physiological data including local field potentials (LFPs) obtained via extracellular in vivo and in vitro recordings. In order to correctly interpret such electrophysiological data, it is vital to adequately understand the electrical properties of neural tissue itself. An ongoing controversy in the field of neuroscience is whether such frequency-dependent effects bias LFP recordings and affect the proper interpretation of the signal. On macroscopic scales and with large injected currents, previous studies have found various grades of frequency dependence of cortical tissue, ranging from negligible to strong, within the frequency band typically considered relevant for neuroscience (less than a few thousand hertz). Here, we performed a detailed investigation of the frequency dependence of the conductivity within cortical tissue at microscopic distances using small current amplitudes within the typical (neuro)physiological micrometer and sub-nanoampere range. We investigated the propagation of LFPs, induced by extracellular electrical current injections via patch-pipettes, in acute rat brain slice preparations containing the somatosensory cortex in vitro using multielectrode arrays. Based on our data, we determined the cortical tissue conductivity over a 100-fold increase in signal frequency (5-500 Hz). Our results imply at most very weak frequency-dependent effects within the frequency range of physiological LFPs. Using biophysical modeling, we estimated the impact of different putative impedance spectra. Our results indicate that frequency dependencies of the order measured here and in most other studies have negligible impact on the typical analysis and modeling of LFP signals from extracellular brain recordings.

A Simple and Accurate Model to Predict Responses to Multi-electrode Stimulation in the Retina

Implantable electrode arrays are widely used in therapeutic stimulation of the nervous system (e.g. cochlear, retinal, and cortical implants). Currently, most neural prostheses use serial stimulation (i.e. one electrode at a time) despite this severely limiting the repertoire of stimuli that can be applied. Methods to reliably predict the outcome of multi-electrode stimulation have not been available. Here, we demonstrate that a linear-nonlinear model accurately predicts neural responses to arbitrary patterns of stimulation using in vitro recordings from single retinal ganglion cells (RGCs) stimulated with a subretinal multi-electrode array. In the model, the stimulus is projected onto a low-dimensional subspace and then undergoes a nonlinear transformation to produce an estimate of spiking probability. The low-dimensional subspace is estimated using principal components analysis, which gives the neuron’s electrical receptive field (ERF), i.e. the electrodes to which the neuron is most sensitive. Our model suggests that stimulation proportional to the ERF yields a higher efficacy given a fixed amount of power when compared to equal amplitude stimulation on up to three electrodes. We find that the model captures the responses of all the cells recorded in the study, suggesting that it will generalize to most cell types in the retina. The model is computationally efficient to evaluate and, therefore, appropriate for future real-time applications including stimulation strategies that make use of recorded neural activity to improve the stimulation strategy.

Implantable multi-electrode arrays (MEAs) are used to record neurological signals and stimulate the nervous system to restore lost function (e.g. cochlear implants). MEAs that can combine both sensing and stimulation will revolutionize the development of the next generation of devices. Simple models that can accurately characterize neural responses to electrical stimulation are necessary for the development of future neuroprostheses controlled by neural feedback. We demonstrate a model that accurately predicts neural responses to concurrent stimulation across multiple electrodes. The model is simple to evaluate, making it an appropriate model for use with neural feedback. The methods described are applicable to a wide range of neural prostheses, thus greatly assisting future device development.

Thermo-electrical equivalents for simulating the electro-mechanical behavior of biological tissue

Equivalence is one of most popular techniques to simulate the behavior of systems governed by the same type of differential equation. In this case, a thermo-electrical equivalence is considered as a method for modelling the inter-dependence of electrical and mechanical phenomena in biological tissue. We seek to assess this approach for multi-scale models (from micro-structure to tissue scale) of biological media, such as nerve cells and cardiac tissue, in which the electrical charge distribution is modelled as a heat distribution in an equivalent thermal system. This procedure allows for the reduction in problem complexity and it facilitates the coupling of electrical and mechanical phenomena in an efficient and practical way. Although the findings of this analysis are mainly addressed towards the electro-mechanics of tissue within the biomedical domain, the same approach could be used in other studies in which a coupled finite element analysis is required.

Electrical stimulation of neural tissue modeled as a cellular composite: point source electrode in an isotropic tissue

Standard volume conductor models of neural electrical stimulation assume that the electrical properties of the tissue are well described by a conductivity that is smooth and homogeneous at a microscopic scale. However, neural tissue is composed of tightly packed cells whose membranes have markedly different electrical properties to either the intra- or extracellular space. Consequently, the electrical properties of tissue are highly heterogeneous at the microscopic scale: a fact not accounted for in standard volume conductor models. Here we apply a recently developed framework for volume conductor models that accounts for the cellular composition of tissue. We consider the case of a point source electrode in tissue comprised of neural fibers crossing each other equally in all directions. We derive the tissue admittivity (that replaces the standard tissue conductivity) from single cell properties, and then calculate the extracellular potential. Our findings indicate that the cellular composition of tissue affects the spatiotemporal profile of the extracellular potential. In particular, the full solution asymptotically approaches a near-field limit close to the electrode and a far-field limit far from the electrode. The near-field and far-field approximations are solutions to standard volume conductor models, but differ from each other by nearly an order or magnitude. Consequently the full solution is expected to provide a more accurate estimate of electrical potentials over the full range of electrode-neurite separations.

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.

Internal inconsistencies in models of electrical stimulation in neural tissue

Calculating the membrane potential of a neurite under extracellular electrical stimulation is important in the design of some recent stimulation strategies for neuroprosthetic devices including retinal implants, cochlear implants, deep brain stimulation. A common approach, widely used in the electrical stimulation literature uses a volume conductor model to calculate the electrical potential in the tissue and then extracts the voltage or current density on the surface of a neuron, which is used as input to the cable equation to calculate the neuron's response. However this approach ignores the effect of the neuron itself as well as surrounding neurons on the extracellular potential. Here we highlight that this leads to an internal inconsistency in the overall model because the result depends on whether the voltage or current density is used to calculate the neural response. The magnitude of this discrepancy is calculated for the example of a point source electrode in a homogeneous medium and is shown to be up to several hundred percent under some stimulus conditions. The inconsistency can be resolved by ensuring that the voltage is related to the current density by the transimpedance of the neurite. Deriving a volume conductor model that satisfies this relationship requires further work.

Modeling extracellular electrical stimulation: II. Computational validation and numerical results

The validity of approximate equations describing the membrane potential under extracellular electrical stimulation (Meffin et al 2012 J. Neural Eng. 9 065005) is investigated through finite element analysis in this paper. To this end, the finite element method is used to simulate a cylindrical neurite under extracellular stimulation. Laplace's equations with appropriate boundary conditions are solved numerically in three dimensions and the results are compared to the approximate analytic solutions. Simulation results are in agreement with the approximate analytic expressions for longitudinal and transverse modes of stimulation. The range of validity of the equations describing the membrane potential for different values of stimulation and neurite parameters are presented as well. The results indicate that the analytic approach can be used to model extracellular electrical stimulation for realistic physiological parameters with a high level of accuracy.