The "mirror" estimate: an intuitive predictor of membrane polarization during extracellular stimulation

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.

Galvanic vs. pulsatile effects on decision-making networks: reshaping the neural activation landscape

Objective. Primarily due to safety concerns, biphasic pulsatile stimulation (PS) is the present standard for electrical excitation of neural tissue with a diverse set of applications. While pulses have been shown to be effective to achieve functional outcomes, they have well-known deficits. Due to recent technical advances, galvanic stimulation (GS), delivery of current for extended periods of time (>1 s), has re-emerged as an alternative to PS.Approach. In this paper, we use a winner-take-all decision-making cortical network model to investigate differences between pulsatile and GS in the context of a perceptual decision-making task.Main results. Based on previous work, we hypothesized that GS would produce more spatiotemporally distributed, network-sensitive neural responses, while PS would produce highly synchronized activation of a limited group of neurons. Our results in-silico support these hypotheses for low-amplitude GS but deviate when galvanic amplitudes are large enough to directly activate or block nearby neurons.Significance. We conclude that with careful parametrization, GS could overcome some limitations of PS to deliver more naturalistic firing patterns in the group of targeted neurons.

Thermal effects on neurons during stimulation of the brain

All electric and magnetic stimulation of the brain deposits thermal energy in the brain. This occurs through either Joule heating of the conductors carrying current through electrodes and magnetic coils, or through dissipation of energy in the conductive brain.Objective.Although electrical interaction with brain tissue is inseparable from thermal effects when electrodes are used, magnetic induction enables us to separate Joule heating from induction effects by contrasting AC and DC driving of magnetic coils using the same energy deposition within the conductors. Since mammalian cortical neurons have no known sensitivity to static magnetic fields, and if there is no evidence of effect on spike timing to oscillating magnetic fields, we can presume that the induced electrical currents within the brain are below the molecular shot noise where any interaction with tissue is purely thermal.Approach.In this study, we examined a range of frequencies produced from micromagnetic coils operating below the molecular shot noise threshold for electrical interaction with single neurons.Main results.We found that small temperature increases and decreases of 1∘C caused consistent transient suppression and excitation of neurons during temperature change. Numerical modeling of the biophysics demonstrated that the Na-K pump, and to a lesser extent the Nernst potential, could account for these transient effects. Such effects are dependent upon compartmental ion fluxes and the rate of temperature change.Significance.A new bifurcation is described in the model dynamics that accounts for the transient suppression and excitation; in addition, we note the remarkable similarity of this bifurcation's rate dependency with other thermal rate-dependent tipping points in planetary warming dynamics. These experimental and theoretical findings demonstrate that stimulation of the brain must take into account small thermal effects that are ubiquitously present in electrical and magnetic stimulation. More sophisticated models of electrical current interaction with neurons combined with thermal effects will lead to more accurate modulation of neuronal activity.

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.

Neural Tissue Degeneration in Rosenthal's Canal and Its Impact on Electrical Stimulation of the Auditory Nerve by Cochlear Implants: An Image-Based Modeling Study

Sensorineural deafness is caused by the loss of peripheral neural input to the auditory nerve, which may result from peripheral neural degeneration and/or a loss of inner hair cells. Provided spiral ganglion cells and their central processes are patent, cochlear implants can be used to electrically stimulate the auditory nerve to facilitate hearing in the deaf or severely hard-of-hearing. Neural degeneration is a crucial impediment to the functional success of a cochlear implant. The present, first-of-its-kind two-dimensional finite-element model investigates how the depletion of neural tissues might alter the electrically induced transmembrane potential of spiral ganglion neurons. The study suggests that even as little as 10% of neural tissue degeneration could lead to a disproportionate change in the stimulation profile of the auditory nerve. This result implies that apart from encapsulation layer formation around the cochlear implant electrode, tissue degeneration could also be an essential reason for the apparent inconsistencies in the functionality of cochlear implants.

ROOTS: An Algorithm to Generate Biologically Realistic Cortical Axons and an Application to Electroceutical Modeling

Advances in computation and neuronal modeling have enabled the study of entire neural tissue systems with an impressive degree of biological realism. These efforts have focused largely on modeling dendrites and somas while largely neglecting axons. The need for biologically realistic explicit axonal models is particularly clear for applications involving clinical and therapeutic electrical stimulation because axons are generally more excitable than other neuroanatomical subunits. While many modeling efforts can rely on existing repositories of reconstructed dendritic/somatic morphologies to study real cells or to estimate parameters for a generative model, such datasets for axons are scarce and incomplete. Those that do exist may still be insufficient to build accurate models because the increased geometric variability of axons demands a proportional increase in data. To address this need, a Ruled-Optimum Ordered Tree System (ROOTS) was developed that extends the capability of neuronal morphology generative methods to include highly branched cortical axon terminal arbors. Further, this study presents and explores a clear use-case for such models in the prediction of cortical tissue response to externally applied electric fields. The results presented herein comprise (i) a quantitative and qualitative analysis of the generative algorithm proposed, (ii) a comparison of generated fibers with those observed in histological studies, (iii) a study of the requisite spatial and morphological complexity of axonal arbors for accurate prediction of neuronal response to extracellular electrical stimulation, and (iv) an extracellular electrical stimulation strength-duration analysis to explore probable thresholds of excitation of the dentate perforant path under controlled conditions. ROOTS demonstrates a superior ability to capture biological realism in model fibers, allowing improved accuracy in predicting the impact that microscale structures and branching patterns have on spatiotemporal patterns of activity in the presence of extracellular electric fields.

Extracellular Stimulation of Neural Tissues: Activating Function and Sub-threshold Potential Perspective

Electric stimulation of neural tissues has been an effective clinical intervention to address a variety of pathological issues such as profound deafness, retinal diseases, and Parkinson's disease. However, the knowledge about the exact mechanism of neural excitation, especially activation sites is still ambiguous. Nevertheless, in silico models utilize two approaches namely activating function and sub-threshold potential to predict the activation sites of neural tissues. This paper studies the applicability of these two approaches to model the electric stimulation of pyramidal neuron and spiral ganglion neurons using finite element models. The simulation results suggest that the activating function could be prone to geometrical irregularities of the neural tissues, yet realistically predicts the activation sites on the myelinated neurons. In contrast, the sub-threshold potential predicts the activation of unmyelinated axons by considering the electrophysiological properties of neural tissues. The present study suggests that it is necessary to choose an appropriate method to estimate the neural activation sites while modeling the extracellular stimulation of neural tissues.

The Quasi-uniform assumption for Spinal Cord Stimulation translational research

BACKGROUND

Quasi-uniform assumption is a general theory that postulates local electric field predicts neuronal activation. Computational current flow model of spinal cord stimulation (SCS) of humans and animal models inform how the quasi-uniform assumption can support scaling neuromodulation dose between humans and translational animal.

NEW METHOD

Here we developed finite element models of cat and rat SCS, and brain slice, alongside SCS models. Boundary conditions related to species specific electrode dimensions applied, and electric fields per unit current (mA) predicted.

RESULTS

Clinically and across animal, electric fields change abruptly over small distance compared to the neuronal morphology, such that each neuron is exposed to multiple electric fields. Per unit current, electric fields generally decrease with body mass, but not necessarily and proportionally across tissues. Peak electric field in dorsal column rat and cat were ∼17x and ∼1x of clinical values, for scaled electrodes and equal current. Within the spinal cord, the electric field for rat, cat, and human decreased to 50% of peak value caudo-rostrally (C5-C6) at 0.48 mm, 3.2 mm, and 8 mm, and mediolaterally at 0.14 mm, 2.3 mm, and 3.1 mm. Because these space constants are different, electric field across species cannot be matched without selecting a region of interest (ROI).

COMPARISON WITH EXISTING METHOD

This is the first computational model to support scaling neuromodulation dose between humans and translational animal.

CONCLUSIONS

Inter-species reproduction of the electric field profile across the entire surface of neuron populations is intractable. Approximating quasi-uniform electric field in a ROI is a rational step to translational scaling.

Network remodeling induced by transcranial brain stimulation: A computational model of tDCS-triggered cell assembly formation

Transcranial direct current stimulation (tDCS) is a variant of noninvasive neuromodulation, which promises treatment for brain diseases like major depressive disorder. In experiments, long-lasting aftereffects were observed, suggesting that persistent plastic changes are induced. The mechanism underlying the emergence of lasting aftereffects, however, remains elusive. Here we propose a model, which assumes that tDCS triggers a homeostatic response of the network involving growth and decay of synapses. The cortical tissue exposed to tDCS is conceived as a recurrent network of excitatory and inhibitory neurons, with synapses subject to homeostatically regulated structural plasticity. We systematically tested various aspects of stimulation, including electrode size and montage, as well as stimulation intensity and duration. Our results suggest that transcranial stimulation perturbs the homeostatic equilibrium and leads to a pronounced growth response of the network. The stimulated population eventually eliminates excitatory synapses with the unstimulated population, and new synapses among stimulated neurons are grown to form a cell assembly. Strong focal stimulation tends to enhance the connectivity within new cell assemblies, and repetitive stimulation with well-chosen duty cycles can increase the impact of stimulation even further. One long-term goal of our work is to help in optimizing the use of tDCS in clinical applications.

Noninvasive brain stimulation techniques like tDCS have the potential to directly interfere with neural activity, but may also trigger activity-dependent plasticity. We propose a model to study the mechanism of tDCS and persistent aftereffects that may be induced as a consequence of homeostatic structural plasticity. Based on the idea that tDCS perturbs the ongoing activity of neurons, our model predicts that the stimulation also triggers a rearrangement of synapses among stimulated and unstimulated neurons, eventually leading to network remodeling and cell assembly formation. Focal and strong stimulation leads to stronger cell assemblies, and so does repetitive stimulation with optimized stimulation protocols. This is the first original work studying possible long-lasting aftereffects of transcranial stimulation at the mesoscopic neuronal network level using a computational model.

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.

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.

Ih interacts with somato-dendritic structure to determine frequency response to weak alternating electric field stimulation

Transcranial current stimulation (tCS) modulates brain dynamics using weak electric fields. Given the pathological changes in brain network oscillations in neurological and psychiatric illnesses, using alternating electric field waveforms that engage rhythmic activity has been proposed as a targeted, network-level treatment approach. Previous studies have investigated the effects of electric fields at the neuronal level. However, the biophysical basis of the cellular response to electric fields has remained limited. Here, we characterized the frequency-dependent response of different compartments in a layer V pyramidal neuron to exogenous electric fields to dissect the relative contributions of voltage-gated ion channels and neuronal morphology. Hyperpolarization-activated cation current (I_h) in the distal dendrites was the primary ionic mechanism shaping the model's response to electric field stimulation and caused subthreshold resonance in the tuft at 20 ± 4 Hz. In contrast, subthreshold I_h-mediated resonance in response to local sinusoidal current injection was present in all model compartments at 11 ± 2 Hz. The frequencies of both resonance responses were modulated by I_h conductance density. We found that the difference in resonance frequency between the two stimulation types can be explained by the fact that exogenous electric fields simultaneously polarize the membrane potentials at the distal ends of the neuron (relative to field direction) in opposite directions. Our results highlight the role of I_h in shaping the cellular response to electric field stimulation and suggest that the common model of tCS as a weak somatic current injection fails to capture the cellular effects of electric field stimulation. NEW & NOTEWORTHY Modulation of cortical oscillation by brain stimulation serves as a tool to understand the causal role of network oscillations in behavior and is a potential treatment modality that engages impaired network oscillations in disorders of the central nervous system. To develop targeted stimulation paradigms, cellular-level effects must be understood. We demonstrate that hyperpolarization-activated cation current (I_h) and cell morphology cooperatively shape the response to applied alternating electric fields.

Animal models of transcranial direct current stimulation: Methods and mechanisms

The objective of this review is to summarize the contribution of animal research using direct current stimulation (DCS) to our understanding of the physiological effects of transcranial direct current stimulation (tDCS). We comprehensively address experimental methodology in animal studies, broadly classified as: (1) transcranial stimulation; (2) direct cortical stimulation in vivo and (3) in vitro models. In each case advantages and disadvantages for translational research are discussed including dose translation and the overarching "quasi-uniform" assumption, which underpins translational relevance in all animal models of tDCS. Terminology such as anode, cathode, inward current, outward current, current density, electric field, and uniform are defined. Though we put key animal experiments spanning decades in perspective, our goal is not simply an exhaustive cataloging of relevant animal studies, but rather to put them in context of ongoing efforts to improve tDCS. Cellular targets, including excitatory neuronal somas, dendrites, axons, interneurons, glial cells, and endothelial cells are considered. We emphasize neurons are always depolarized and hyperpolarized such that effects of DCS on neuronal excitability can only be evaluated within subcellular regions of the neuron. Findings from animal studies on the effects of DCS on plasticity (LTP/LTD) and network oscillations are reviewed extensively. Any endogenous phenomena dependent on membrane potential changes are, in theory, susceptible to modulation by DCS. The relevance of morphological changes (galvanotropy) to tDCS is also considered, as we suggest microscopic migration of axon terminals or dendritic spines may be relevant during tDCS. A majority of clinical studies using tDCS employ a simplistic dose strategy where excitability is singularly increased or decreased under the anode and cathode, respectively. We discuss how this strategy, itself based on classic animal studies, cannot account for the complexity of normal and pathological brain function, and how recent studies have already indicated more sophisticated approaches are necessary. One tDCS theory regarding "functional targeting" suggests the specificity of tDCS effects are possible by modulating ongoing function (plasticity). Use of animal models of disease are summarized including pain, movement disorders, stroke, and epilepsy.

Anodal transcranial direct current stimulation boosts synaptic plasticity and memory in mice via epigenetic regulation of Bdnf expression

The effects of transcranial direct current stimulation (tDCS) on brain functions and the underlying molecular mechanisms are yet largely unknown. Here we report that mice subjected to 20-min anodal tDCS exhibited one-week lasting increases in hippocampal LTP, learning and memory. These effects were associated with enhanced: i) acetylation of brain-derived neurotrophic factor (Bdnf) promoter I; ii) expression of Bdnf exons I and IX; iii) Bdnf protein levels. The hippocampi of stimulated mice also exhibited enhanced CREB phosphorylation, pCREB binding to Bdnf promoter I and recruitment of CBP on the same regulatory sequence. Inhibition of acetylation and blockade of TrkB receptors hindered tDCS effects at molecular, electrophysiological and behavioral levels. Collectively, our findings suggest that anodal tDCS increases hippocampal LTP and memory via chromatin remodeling of Bdnf regulatory sequences leading to increased expression of this gene, and support the therapeutic potential of tDCS for brain diseases associated with impaired neuroplasticity.

Multilevel computational models for predicting the cellular effects of noninvasive brain stimulation

Since 2000, there has been rapid acceleration in the use of tDCS in both clinical and cognitive neuroscience research, encouraged by the simplicity of the technique (two electrodes and a battery powered stimulator) and the perception that tDCS protocols can be simply designed by placing the anode over the cortex to "excite," and the cathode over cortex to "inhibit." A specific and predictive understanding of tDCS needs experimental data to be placed into a quantitative framework. Biologically constrained computational models provide a useful framework within which to interpret results from empirical studies and generate novel, testable hypotheses. Although not without caveats, computational models provide a tool for exploring cognitive and brain processes, are amenable to quantitative analysis, and can inspire novel empirical work that might be difficult to intuit simply by examining experimental results. We approach modeling the effects of tDCS on neurons from multiple levels: modeling the electric field distribution, modeling single-compartment effects, and finally with multicompartment neuron models.

Modeling sequence and quasi-uniform assumption in computational neurostimulation

Computational neurostimulation aims to develop mathematical constructs that link the application of neuromodulation with changes in behavior and cognition. This process is critical but daunting for technical challenges and scientific unknowns. The overarching goal of this review is to address how this complex task can be made tractable. We describe a framework of sequential modeling steps to achieve this: (1) current flow models, (2) cell polarization models, (3) network and information processing models, and (4) models of the neuroscientific correlates of behavior. Each step is explained with a specific emphasis on the assumptions underpinning underlying sequential implementation. We explain the further implementation of the quasi-uniform assumption to overcome technical limitations and unknowns. We specifically focus on examples in electrical stimulation, such as transcranial direct current stimulation. Our approach and conclusions are broadly applied to immediate and ongoing efforts to deploy computational neurostimulation.

Current approaches to model extracellular electrical neural microstimulation

Nowadays, high-density microelectrode arrays provide unprecedented possibilities to precisely activate spatially well-controlled central nervous system (CNS) areas. However, this requires optimizing stimulating devices, which in turn requires a good understanding of the effects of microstimulation on cells and tissues. In this context, modeling approaches provide flexible ways to predict the outcome of electrical stimulation in terms of CNS activation. In this paper, we present state-of-the-art modeling methods with sufficient details to allow the reader to rapidly build numerical models of neuronal extracellular microstimulation. These include (1) the computation of the electrical potential field created by the stimulation in the tissue, and (2) the response of a target neuron to this field. Two main approaches are described: First we describe the classical hybrid approach that combines the finite element modeling of the potential field with the calculation of the neuron's response in a cable equation framework (compartmentalized neuron models). Then, we present a “whole finite element” approach allowing the simultaneous calculation of the extracellular and intracellular potentials, by representing the neuronal membrane with a thin-film approximation. This approach was previously introduced in the frame of neural recording, but has never been implemented to determine the effect of extracellular stimulation on the neural response at a sub-compartment level. Here, we show on an example that the latter modeling scheme can reveal important sub-compartment behavior of the neural membrane that cannot be resolved using the hybrid approach. The goal of this paper is also to describe in detail the practical implementation of these methods to allow the reader to easily build new models using standard software packages. These modeling paradigms, depending on the situation, should help build more efficient high-density neural prostheses for CNS rehabilitation.

Axon terminal polarization induced by weak uniform DC electric fields: a modeling study

Uniform steady state (DC) electric fields, like those generated during transcranial direct current stimulation (tDCS), can affect neuronal excitability depending on field direction and neuronal morphology. In addition to somatic polarization, subthreshold membrane polarization of axon compartments can play a significant role in modulating synaptic efficacy. The aim of this study is to provide an estimation of axon terminal polarization in a weak uniform subthreshold electric field. Simulations based on 3D morphology reconstructions and simplified models indicate that for axons having long final branches compared to the local space constant (L>4λ) the terminal polarization converges to Eλ for electric fields oriented in the same direction as the branch. In particular we determined how and when analytical approximations could be extended to real cases when considering maximal potential polarization during weak DC stimulation.

Computationally efficient simulation of electrical activity at cell membranes interacting with self-generated and externally imposed electric fields

OBJECTIVE

We present a computational method that implements a reduced set of Maxwell's equations to allow simulation of cells under realistic conditions: sub-micron cell morphology, a conductive non-homogeneous space and various ion channel properties and distributions.

APPROACH

While a reduced set of Maxwell's equations can be used to couple membrane currents to extra- and intracellular potentials, this approach is rarely taken, most likely because adequate computational tools are missing. By using these equations, and introducing an implicit solver, numerical stability is attained even with large time steps. The time steps are limited only by the time development of the membrane potentials.

MAIN RESULTS

This method allows simulation times of tens of minutes instead of weeks, even for complex problems. The extracellular fields are accurately represented, including secondary fields, which originate at inhomogeneities of the extracellular space and can reach several millivolts. We present a set of instructive examples that show how this method can be used to obtain reference solutions for problems, which might not be accurately captured by the traditional approaches. This includes the simulation of realistic magnitudes of extracellular action potential signals in restricted extracellular space.

SIGNIFICANCE

The electric activity of neurons creates extracellular potentials. Recent findings show that these endogenous fields act back onto the neurons, contributing to the synchronization of population activity. The influence of endogenous fields is also relevant for understanding therapeutic approaches such as transcranial direct current, transcranial magnetic and deep brain stimulation. The mutual interaction between fields and membrane currents is not captured by today's concepts of cellular electrophysiology, including the commonly used activation function, as those concepts are based on isolated membranes in an infinite, isopotential extracellular space. The presented tool makes simulations with detailed morphology and implicit interactions of currents and fields available to the electrophysiology community.

Cellular effects of acute direct current stimulation: somatic and synaptic terminal effects

Transcranial direct current stimulation (tDCS) is a non-invasive brain stimulation technique to modulate cortical excitability. Although increased/decreased excitability under the anode/cathode electrode is nominally associated with membrane depolarization/hyperpolarization, which cellular compartments (somas, dendrites, axons and their terminals) mediate changes in cortical excitability remains unaddressed. Here we consider the acute effects of DCS on excitatory synaptic efficacy. Using multi-scale computational models and rat cortical brain slices, we show the following. (1) Typical tDCS montages produce predominantly tangential (relative to the cortical surface) direction currents (4-12 times radial direction currents), even directly under electrodes. (2) Radial current flow (parallel to the somatodendritic axis) modulates synaptic efficacy consistent with somatic polarization, with depolarization facilitating synaptic efficacy. (3) Tangential current flow (perpendicular to the somatodendritic axis) modulates synaptic efficacy acutely (during stimulation) in an afferent pathway-specific manner that is consistent with terminal polarization, with hyperpolarization facilitating synaptic efficacy. (4) Maximal polarization during uniform DCS is expected at distal (the branch length is more than three times the membrane length constant) synaptic terminals, independent of and two-three times more susceptible than pyramidal neuron somas. We conclude that during acute DCS the cellular targets responsible for modulation of synaptic efficacy are concurrently somata and axon terminals, with the direction of cortical current flow determining the relative influence.

Physiological and modeling evidence for focal transcranial electrical brain stimulation in humans: a basis for high-definition tDCS

Transcranial Direct Current Stimulation (tDCS) is a non-invasive, low-cost, well-tolerated technique producing lasting modulation of cortical excitability. Behavioral and therapeutic outcomes of tDCS are linked to the targeted brain regions, but there is little evidence that current reaches the brain as intended. We aimed to: (1) validate a computational model for estimating cortical electric fields in human transcranial stimulation, and (2) assess the magnitude and spread of cortical electric field with a novel High-Definition tDCS (HD-tDCS) scalp montage using a 4 × 1-Ring electrode configuration. In three healthy adults, Transcranial Electrical Stimulation (TES) over primary motor cortex (M1) was delivered using the 4 × 1 montage (4 × cathode, surrounding a single central anode; montage radius ~3 cm) with sufficient intensity to elicit a discrete muscle twitch in the hand. The estimated current distribution in M1 was calculated using the individualized MRI-based model, and compared with the observed motor response across subjects. The response magnitude was quantified with stimulation over motor cortex as well as anterior and posterior to motor cortex. In each case the model data were consistent with the motor response across subjects. The estimated cortical electric fields with the 4 × 1 montage were compared (area, magnitude, direction) for TES and tDCS in each subject. We provide direct evidence in humans that TES with a 4 × 1-Ring configuration can activate motor cortex and that current does not substantially spread outside the stimulation area. Computational models predict that both TES and tDCS waveforms using the 4 × 1-Ring configuration generate electric fields in cortex with comparable gross current distribution, and preferentially directed normal (inward) currents. The agreement of modeling and experimental data for both current delivery and focality support the use of the HD-tDCS 4 × 1-Ring montage for cortically targeted neuromodulation.

High-resolution modeling assisted design of customized and individualized transcranial direct current stimulation protocols

OBJECTIVES

Transcranial direct current stimulation (tDCS) is a neuromodulatory technique that delivers low-intensity currents facilitating or inhibiting spontaneous neuronal activity. tDCS is attractive since dose is readily adjustable by simply changing electrode number, position, size, shape, and current. In the recent past, computational models have been developed with increased precision with the goal to help customize tDCS dose. The aim of this review is to discuss the incorporation of high-resolution patient-specific computer modeling to guide and optimize tDCS.

METHODS

In this review, we discuss the following topics: 1) The clinical motivation and rationale for models of transcranial stimulation is considered pivotal in order to leverage the flexibility of neuromodulation; 2) the protocols and the workflow for developing high-resolution models; 3) the technical challenges and limitations of interpreting modeling predictions; and 4) real cases merging modeling and clinical data illustrating the impact of computational models on the rational design of rehabilitative electrotherapy.

CONCLUSIONS

Though modeling for noninvasive brain stimulation is still in its development phase, it is predicted that with increased validation, dissemination, simplification, and democratization of modeling tools, computational forward models of neuromodulation will become useful tools to guide the optimization of clinical electrotherapy.

Asymptotic model of electrical stimulation of nerve fibers

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

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

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

Improved focalization of electrical microstimulation using microelectrode arrays: a modeling study

Extracellular electrical stimulation (EES) of the central nervous system (CNS) has been used empirically for decades, with both fundamental and clinical goals. Currently, microelectrode arrays (MEAs) offer new possibilities for CNS microstimulation. However, although focal CNS activation is of critical importance to achieve efficient stimulation strategies, the precise spatial extent of EES remains poorly understood. The aim of the present work is twofold. First, we validate a finite element model to compute accurately the electrical potential field generated throughout the extracellular medium by an EES delivered with MEAs. This model uses Robin boundary conditions that take into account the surface conductance of electrode/medium interfaces. Using this model, we determine how the potential field is influenced by the stimulation and ground electrode impedances, and by the electrical conductivity of the neural tissue. We confirm that current-controlled stimulations should be preferred to voltage-controlled stimulations in order to control the amplitude of the potential field. Second, we evaluate the focality of the potential field and threshold-distance curves for different electrode configurations. We propose a new configuration to improve the focality, using a ground surface surrounding all the electrodes of the array. We show that the lower the impedance of this surface, the more focal the stimulation. In conclusion, this study proposes new boundary conditions for the design of precise computational models of extracellular stimulation, and a new electrode configuration that can be easily incorporated into future MEA devices, either in vitro or in vivo, for a better spatial control of CNS microstimulation.