Volume Conductor Model Research Articles

The Effect of Head Model Simplification on Beamformer Source Localization.

Beamformers are a widely-used tool in brain analysis with magnetoencephalography (MEG) and electroencephalography (EEG). For the construction of the beamformer filters realistic head volume conductor modeling is necessary for accurately computing the EEG and MEG leadfields, i.e., for solving the EEG and MEG forward problem. In this work, we investigate the influence of including realistic head tissue compartments into a finite element method (FEM) model on the beamformer's localization ability. Specifically, we investigate the effect of including cerebrospinal fluid, gray matter, and white matter distinction, as well as segmenting the skull bone into compacta and spongiosa, and modeling white matter anisotropy. We simulate an interictal epileptic measurement with white sensor noise. Beamformer filters are constructed with unit gain, unit array gain, and unit noise gain constraint. Beamformer source positions are determined by evaluating power and excess sample kurtosis (g2) of the source-waveforms at all source space nodes. For both modalities, we see a strong effect of modeling the cerebrospinal fluid and white and gray matter. Depending on the source position, both effects can each be in the magnitude of centimeters, rendering their modeling necessary for successful localization. Precise skull modeling mainly effected the EEG up to a few millimeters, while both modalities could profit from modeling white matter anisotropy to a smaller extent of 5–10 mm. The unit noise gain or neural activity index beamformer behaves similarly to the array gain beamformer when noise strength is sufficiently high. Variance localization seems more robust against modeling errors than kurtosis.

Analysis of a purely conductance-based stochastic nerve fibre model as applied to compound models of populations of human auditory nerve fibres used in cochlear implant simulations.

The study presents the application of a purely conductance-based stochastic nerve fibre model to human auditory nerve fibres within finite element volume conduction models of a semi-generic head and user-specific cochleae. The stochastic, threshold and temporal characteristics of the human model are compared and successfully validated against physiological feline results with the application of a mono-polar, bi-phasic, cathodic first stimulus. Stochastic characteristics validated include: (i) the log(Relative Spread) versus log(fibre diameter) distribution for the discharge probability versus stimulus intensity plots and (ii) the required exponential membrane noise versus transmembrane voltage distribution. Intra-user, and to a lesser degree inter-user, comparisons are made with respect to threshold and dynamic range at short and long pulse widths for full versus degenerate single fibres as well as for populations of degenerate fibres of a single user having distributed and aligned somas with varying and equal diameters. Temporal characteristics validated through application of different stimulus pulse rates and different stimulus intensities include: (i) discharge rate, latency and latency standard deviation versus stimulus intensity, (ii) period histograms and (iii) interspike interval histograms. Although the stochastic population model does not reduce the modelled single deterministic fibre threshold, the simulated stochastic and temporal characteristics show that it could be used in future studies to model user-specific temporally encoded information, which influences the speech perception of CI users.

Estimating cortical column sensory networks in rodents from micro-electrocorticograph (μECoG) recordings

Micro-electrocorticograph (μECoG) arrays offer the flexibility to record local field potentials (LFPs) from the surface of the cortex, using high density electrodes that are sub-mm in diameter. Research to date has not provided conclusive evidence for the underlying signal generation of μECoG recorded LFPs, or if μECoG arrays can capture network activity from the cortex. We studied the pervading view of the LFP signal by exploring the spatial scale at which the LFP can be considered elemental. We investigated the underlying signal generation and ability to capture functional networks by implanting, μECoG arrays to record sensory-evoked potentials in four rats. The organization of the sensory cortex was studied by analyzing the sensory-evoked potentials with two distinct modeling techniques: (1) The volume conduction model, that models the electrode LFPs with an electrostatic representation, generated by a single cortical generator, and (2) the dynamic causal model (DCM), that models the electrode LFPs with a network model, whose activity is generated by multiple interacting cortical sources. The volume conduction approach modeled activity from electrodes separated < 1000 μm, with reasonable accuracy but a network model like DCM was required to accurately capture activity > 1500 μm. The extrinsic network component in DCM was determined to be essential for accurate modeling of observed potentials. These results all point to the presence of a sensory network, and that μECoG arrays are able to capture network activity in the neocortex. The estimated DCM network models the functional organization of the cortex, as signal generators for the μECoG recorded LFPs, and provides hypothesis-testing tools to explore the brain.

Modeling trans-spinal direct current stimulation for the modulation of the lumbar spinal motor pathways

Objective. Trans-spinal direct current stimulation (tsDCS) is a potential new technique for the treatment of spinal cord injury (SCI). TsDCS aims to facilitate plastic changes in the neural pathways of the spinal cord with a positive effect on SCI recovery. To establish tsDCS as a possible treatment option for SCI, it is essential to gain a better understanding of its cause and effects. We seek to understand the acute effect of tsDCS, including the generated electric field (EF) and its polarization effect on the spinal circuits, to determine a cellular target. We further ask how these findings can be interpreted to explain published experimental results. Approach. We use a realistic full body finite element volume conductor model to calculate the EF of a 2.5 mA direct current for three different electrode configurations. We apply the calculated electric field to realistic motoneuron models to investigate static changes in membrane resting potential. The results are combined with existing knowledge about the theoretical effect on a neuronal level and implemented into an existing lumbar spinal network model to simulate the resulting changes on a network level. Main results. Across electrode configurations, the maximum EF inside the spinal cord ranged from 0.47 V m−1 to 0.82 V m−1. Axon terminal polarization was identified to be the dominant cellular target. Also, differences in electrode placement have a large influence on axon terminal polarization. Comparison between the simulated acute effects and the electrophysiological long-term changes observed in human tsDCS studies suggest an inverse relationship between the two. Significance. We provide methods and knowledge for better understanding the effects of tsDCS and serve as a basis for a more targeted and optimized application of tsDCS.

O203 Neuromodulation of lower limb motor circuitry with transcutaneous lumbar spinal cord direct current stimulation

Objectives The advent of transcranial direct current stimulation (tDCS) introduced growing interest in modulating spinal circuits, altered in many neurologic conditions. Transcutaneous spinal direct current stimulation (tsDCS) of cervical and thoracic spine regions was observed to modulate sensory and motor responses. Combining clinical studies with computational modelling can be a powerful tool to establish tsDCS protocols for specific therapeutic purposes. The aim of this study was to measure the effects of tsDCS delivered on the lumbar region on motor spinal responses and observe if these were consistent with the electric field (E-field) distribution predicted in the spinal cord using a computational model. Methods The exploratory study design was double-blind crossover pseudo-randomized. tsDCS was delivered for 15 min (anodal, cathodal, sham) at L2 vertebra level (2.5 mA, 90 C/cm2) in 15 healthy subjects. The F-wave, H-reflex, cortical silent period, motor evoked potential and sympathetic skin response were analysed. Statistical methods were applied with Bonferroni correction for multiple comparisons (p = 0.05). A human volume conductor model was obtained from available databases. E-field distributions in the spinal grey matter (GM) and white matter (WM) were calculated considering the same protocol and tissue conductivities based on literature review. Results We observed the increase of the F-wave mean latency, however this was independent of the type of intervention. No other differences were observed (p > 0.05, Bonferroni corrected). The E-field magnitude predicted for the lumbosacral spinal GM and WM was 0.15 V/m in protocols with observed modulation of motor responses. Discussion Therefore, the E-field obtained is not sufficient to ensure neuromodulation, which is consistent with the absence of effects observed. The tsDCS protocol applied did not change motor response to delivered stimulus, having no effect on lumbosacral motor circuits. Conclusions Future studies should address diversified protocols sustained in computational models. Significance Computer models of E-field for tsDCS should be optimized to increase chances of a biological effect.

Simulations of high-density surface electromyograms in dynamic muscle contractions.

We describe the extension of pre-existing cylindrical volume conductor model to synthetic high-density surface electromyograms (hdEMG), simulated during dynamic contractions of fusiform skeletal muscles. Its modular structure comprises two main parts. First, dynamic changes of motor unit action potentials (MUAPs) during 36 discrete steps of muscle shortening are simulated. Second, the increase in depth of simulated motor units (MUs) due to shortening and thickening of muscle fibers is simulated. MU firing patterns are generated with the model proposed by Fuglevand et al. and convolved with simulated MUAPs. In this way, the hdEMG simulator can be used to generate dynamic hdEMG of arbitrary muscle shortening, thickening and excitation profiles. In order to demonstrate the value of the aforementioned simulator we independently analyze the impact of muscle shortening and muscle thickening on MU identification by Convolution Kernel Compensation (CKC) technique.

Evaluation of a minimum-norm based beamforming technique, sLORETA, for reducing tonic muscle contamination of EEG at sensor level

BackgroundCranial and cervical muscle activity (electromyogram, EMG) contaminates the surface electroencephalogram (EEG) from frequencies below 20 through to frequencies above 100Hz. It is not possible to have a reliable measure of cognitive tasks expressed in EEG at gamma-band frequencies until the muscle contamination is removed. New methodIn the present work, we introduce a new approach of using a minimum-norm based beamforming technique (sLORETA) to reduce tonic muscle contamination at sensor level. Using a generic volume conduction model of the head, which includes three layers (brain, skull, and scalp), and sLORETA, we estimated time-series of sources distributed within the brain and scalp. The sources within the scalp were considered to be muscle and discarded in forward modelling. Result(1) The method reduced EMG contamination, more strongly at peripheral channels; (2) task-induced cortical activity was retained or revealed after removing putative muscle activity. Comparison with existing methodsThis approach can decrease tonic muscle contamination in scalp measurements without relying on time-consuming processing of expensive MRI data. In addition, it is competitive to ICA in muscle reduction and can be reliably applied on any length of recorded data that captures the dynamics of the signals of interest. ConclusionThis study suggests that sLORETA can be used as a method to quantitate cranial muscle activity and reduce its contamination at sensor level.

Sub-voxel refinement method for tissue boundary conductivities in volume conductor models

The resolution and element type of the mesh used in the Finite-Element Method modeling of transcranial direct-current stimulation (tDCS) greatly affect both the accuracy of the solution and the computational time. Tetrahedral meshing is usually used in these models as it well approximates curvature, but the models are slow to solve. Using a voxel grid as the mesh significantly reduces the computational time, but the cubical elements are not the most suitable option for curved surfaces. Tissue boundaries can be modeled as a layer of voxels with an average conductivity of the surrounding tissues. However, as the boundary being modeled only rarely divides a voxel into two equally sized portions, this approach is often erroneous, and in particular, with low resolutions. In this paper, we propose a novel method for improving the accuracy of anatomically correct Finite-Element Method simulations by enhancing the tissue boundaries in voxel models. In our method, a voxel model is created from a set of polygonal surfaces segmented from magnetic-resonance imaging (MRI) data. This is done by first voxelizing with a fine resolution, and then increasing the voxel size to the target resolution, and calculating the ratio of fine voxels in and outside the surface within each coarse voxel. More-accurate proportions for the volume of a coarse voxel inside and outside the tissue boundary are thus achieved, and the tissue boundary's conductivity can be better approximated. To test the performance of this method, a series of simulations of motor cortical tDCS were performed using resolutions from 0.2 mm to 2 mm, scaled to zero, two, or four times finer resolution. Based on the results, the voxel size could be doubled with a cost of 3% in relative error by using our method. The model's degrees of freedom (DOF) could thus be decreased by 87%, and the simulation times could be decreased by 82%.

MEG and EEG dipole clusters from extended cortical sources.

Data from magnetoencephalography (MEG) and electroencephalography (EEG) suffer from a rather limited signal-to-noise-ratio (SNR) due to cortical background activities and other artifacts. In order to study the effect of the SNR on the size and distribution of dipole clusters reconstructed from interictal epileptic spikes, we performed simulations using realistically shaped volume conductor models and extended cortical sources with different sensor configurations. Head models and cortical surfaces were derived from an averaged magnetic resonance image dataset (Montreal Neurological Institute). Extended sources were simulated by spherical patches with Gaussian current distributions on the folded cortical surface. Different patch sizes were used to investigate cancellation effects from opposing walls of sulcal foldings and to estimate corresponding changes in MEG and EEG sensitivity distributions. Finally, white noise was added to the simulated fields and equivalent current dipole reconstructions were performed to determine size and shape of the resulting dipole clusters. Neuronal currents are oriented perpendicular to the local cortical surface and show cancellation effects of source components on opposing sulcal walls. Since these mostly tangential aspects from large cortical patches cancel out, large extended sources exhibit more radial components in the head geometry. This effect has a larger impact on MEG data as compared to EEG, because in a spherical head model radial currents do not yield any magnetic field. Confidence volumes of single reconstructed dipoles from simulated data at different SNRs show a good correlation with the extension of clusters from repeated dipole reconstructions. Size and shape of dipole clusters reconstructed from extended cortical sources do not only depend on spike and timepoint selection, but also strongly on the SNR of the measured interictal MEG or EEG data. In a linear approximation the size of the clusters is proportional to the inverse SNR.

Stimulation Efficiency With Decaying Exponential Waveforms in a Wirelessly Powered Switched-Capacitor Discharge Stimulation System.

The purpose of this study was to test the feasibility of using a switched-capacitor discharge stimulation (SCDS) system for electrical stimulation, and, subsequently, determine the overall energy saved compared to a conventional stimulator. We have constructed a computational model by pairing an image-based volume conductor model of the cat head with cable models of corticospinal tract (CST) axons and quantified the theoretical stimulation efficiency of rectangular and decaying exponential waveforms, produced by conventional and SCDS systems, respectively. Subsequently, the model predictions were tested in vivo by activating axons in the posterior internal capsule and recording evoked electromyography (EMG) in the contralateral upper arm muscles. Compared to rectangular waveforms, decaying exponential waveforms with time constants >500 μs were predicted to require 2%-4% less stimulus energy to activate directly models of CST axons and 0.4%-2% less stimulus energy to evoke EMG activity in vivo. Using the calculated wireless input energy of the stimulation system and the measured stimulus energies required to evoke EMG activity, we predict that an SCDS implantable pulse generator (IPG) will require 40% less input energy than a conventional IPG to activate target neural elements. A wireless SCDS IPG that is more energy efficient than a conventional IPG will reduce the size of an implant, require that less wireless energy be transmitted through the skin, and extend the lifetime of the battery in the external power transmitter.

Epidural Stimulation of Rat Spinal Cord at Lumbosacral Segment Using a Surface Electrode: A Computer Simulation Study.

Clinical research indicates that the epidural spinal cord stimulation (ESCS) at lumbosacral segment has shown potential for promoting locomotor recovery in patients with incomplete spinal cord injury. However, the underlying neural mechanism needs to be determined by animal experiments. In order to refine experimental protocols, we used a finite element simulation to investigate the activation of nerve fibers in a rat spinal cord model. Our model is composed of a volume conductor model from L1 to S2 spinal segments and the McIntyre-Richard-Grill axon model, which is used to investigate the threshold of selected spinal fibers with different diameters at varied locations and predict the neural responses of any target fibers with bipolar electrode configuration. Mathematical modeling suggests that the electrode-fiber distance may play an important role in the recruitment of nerve fibers, whereas longer pulse width predicted greater activity of spinal root fibers and dorsal column fibers, as well as may exert an effective influence on the motor system by the ability to increase and even "steer" spatial selectivity with deeper penetration into the dorsal columns. The spikes were initiated at sites along the nerve fibers depending on which component was closest to the cathode among the longitudinal part of the fiber, its entrance into spinal cord, or strong bending at the entry. Our simulation results show good agreement with the previous findings from animal studies. It is concluded that the computational ESCS model is a valuable tool to obtain a better insight into the immediately evoked electrophysiological phenomena in animal models, and provides further guidelines for conducting animal experiments to enhance the exploration of basic neural mechanisms.

Integral equations and boundary-element solution for static potential in a general piece-wise homogeneous volume conductor

Boundary element methods (BEM) are used for forward computation of bioelectromagnetic fields in multi-compartment volume conductor models. Most BEM approaches assume that each compartment is in contact with at most one external compartment. In this work, I present a general surface integral equation and BEM discretization that remove this limitation and allow BEM modeling of general piecewise-homogeneous medium. The new integral equation allows positioning of field points at junctioned boundary of more than two compartments, enabling the use of linear collocation BEM in such a complex geometry. A modular BEM implementation is presented for linear collocation and Galerkin approaches, starting from the standard formulation. The approach and resulting solver are verified in four ways, including comparisons of volume and surface potentials to those obtained using the finite element method (FEM), and the effect of a hole in skull on electroencephalographic scalp potentials is demonstrated.

In-vivo measurements of human brain tissue conductivity using focal electrical current injection through intracerebral multicontact electrodes.

In-vivo measurements of human brain tissue conductivity at body temperature were conducted using focal electrical currents injected through intracerebral multicontact electrodes. A total of 1,421 measurements in 15 epileptic patients (age: 28 ± 10) using a radiofrequency generator (50 kHz current injection) were analyzed. Each contact pair was classified as being from healthy (gray matter, n = 696; white matter, n = 530) or pathological (epileptogenic zone, n = 195) tissue using neuroimaging analysis of the local tissue environment and intracerebral EEG recordings. Brain tissue conductivities were obtained using numerical simulations based on conductivity estimates that accounted for the current flow in the local brain volume around the contact pairs (a cube with a side length of 13 mm). Conductivity values were 0.26 S/m for gray matter and 0.17 S/m for white matter. Healthy gray and white matter had statistically different median impedances (P < 0.0001). White matter conductivity was found to be homogeneous as normality tests did not find evidence of multiple subgroups. Gray matter had lower conductivity in healthy tissue than in the epileptogenic zone (0.26 vs. 0.29 S/m; P = 0.012), even when the epileptogenic zone was not visible in the magnetic resonance image (MRI) (P = 0.005). The present in-vivo conductivity values could serve to create more accurate volume conduction models and could help to refine the identification of relevant intracerebral contacts, especially when located within the epileptogenic zone of an MRI-invisible lesion. Hum Brain Mapp 38:974-986, 2017. © 2016 Wiley Periodicals, Inc.

Epicardial potentials computed from the body surface potential map using inverse electrocardiography and an individualised torso model improve sensitivity for acute myocardial infarction diagnosis.

Epicardial potentials (EPs) derived from the body surface potential map (BSPM) improve acute myocardial infarction (AMI) diagnosis. In this study, we compared EPs derived from the 80-lead BSPM using a standard thoracic volume conductor model (TVCM) with those derived using a patient-specific torso model (PSTM) based on body mass index (BMI). Consecutive patients presenting to both the emergency department and pre-hospital coronary care unit between August 2009 and August 2011 with acute ischaemic-type chest pain at rest were enrolled. At first medical contact, 12-lead electrocardiograms and BSPMs were recorded. The BMI for each patient was calculated. Cardiac troponin T (cTnT) was sampled 12 hours after symptom onset. Patients were excluded from analysis if they had any ECG confounders to interpretation of the ST-segment. A cardiologist assessed the 12-lead ECG for ST-segment elevation myocardial infarction by Minnesota criteria and the BSPM. BSPM ST-elevation (STE) was ⩾0.2 mV in anterior, ⩾0.1 mV in lateral, inferior, right ventricular or high right anterior and ⩾0.05 mV in posterior territories. To derive EPs, the BSPM data were interpolated to yield values at 352 nodes of a Dalhousie torso. Using an inverse solution based on the boundary element method, EPs at 98 cardiac nodes positioned within a standard TVCM were derived. The TVCM was then scaled to produce a PSTM using a model developed from computed tomography in 48 patients of varying BMIs, and EPs were recalculated. EPs >0.3 mV defined STE. A cardiologist blinded to both the 12-lead ECG and BSPM interpreted the EP map. AMI was defined as cTnT ⩾0.1 µg/L. Enrolled were 400 patients (age 62 ± 13 years; 57% male); 80 patients had exclusion criteria. Of the remaining 320 patients, the BMI was an average of 27.8 ± 5.6 kg/m2. Of these, 180 (56%) had AMI. Overall, 132 had Minnesota STE on ECG (sensitivity 65%, specificity 89%) and 160 had BSPM STE (sensitivity 81%, specificity 90%). EP STE occurred in 165 patients using TVCM (sensitivity 88%, specificity 95%; p < 0.001) and in 206 patients using PSTM (sensitivity 98%, specificity 79%; p < 0.001). Of those with AMI by cTnT and EPs ⩽0.3 mV using TVCM ( n = 22), 18 (82%) patients had EPs >0.3 mV when an individualised PSTM was used. Among patients presenting with ischaemic-type chest pain at rest, EPs derived from BSPM using a novel PSTM significantly improve sensitivity for AMI diagnosis.

Uncertainty Quantification of Oscillation Suppression During DBS in a Coupled Finite Element and Network Model.

Models of the cortico-basal ganglia network and volume conductor models of the brain can provide insight into the mechanisms of action of deep brain stimulation (DBS). In this study, the coupling of a network model, under parkinsonian conditions, to the extracellular field distribution obtained from a three dimensional finite element model of a rodent's brain during DBS is presented. This coupled model is used to investigate the influence of uncertainty in the electrical properties of brain tissue and encapsulation tissue, formed around the electrode after implantation, on the suppression of oscillatory neural activity during DBS. The resulting uncertainty in this effect of DBS on the network activity is quantified using a computationally efficient and non-intrusive stochastic approach based on the generalized Polynomial Chaos. The results suggest that variations in the electrical properties of brain tissue may have a substantial influence on the level of suppression of oscillatory activity during DBS. Applying a global sensitivity analysis on the suppression of the simulated oscillatory activity showed that the influence of uncertainty in the electrical properties of the encapsulation tissue had only a minor influence, in agreement with previous experimental and computational studies investigating the mechanisms of current-controlled DBS in the literature.

Role of Soft-Tissue Heterogeneity in Computational Models of Deep Brain Stimulation.

Bioelectric field models of deep brain stimulation (DBS) are commonly utilized in research and industrial applications. However, the wide range of different representations used for the human head in these models may be responsible for substantial variance in the stimulation predictions. Determine the relative error of ignoring cerebral vasculature and soft-tissue heterogeneity outside of the brain in computational models of DBS. We used a detailed atlas of the human head, coupled to magnetic resonance imaging data, to construct a range of subthalamic DBS volume conductor models. We incrementally simplified the most detailed base model and quantified changes in the stimulation thresholds for direct activation of corticofugal axons. Ignoring cerebral vasculature altered predictions of stimulation thresholds by <10%, whereas ignoring soft-tissue heterogeneity outside of the brain altered predictions between -44 % and 174%. Heterogeneity in the soft tissues of the head, if unaccounted for, introduces a degree of uncertainty in predicting electrical stimulation of neural elements that is not negligible and thereby warrants consideration in future modeling studies.

A fast, stochastic, and adaptive model of auditory nerve responses to cochlear implant stimulation

Cochlear implants (CIs) rehabilitate hearing impairment through direct electrical stimulation of the auditory nerve. New stimulation strategies can be evaluated using computational models. In this study, a computationally efficient model that accurately predicts auditory nerve responses to CI pulse train input was developed. A three-dimensional volume conduction and active nerve model developed at Leiden University Medical Center was extended with stochasticity, adaptation, and accommodation. This complete model includes spatial and temporal characteristics of both the cochlea and the auditory nerve. The model was validated by comparison with experimentally measured single fiber action potential responses to pulse trains published in the literature. The effects of pulse rate and pulse amplitude on spiking patterns were investigated. The modeled neural responses to CI stimulation were very similar to the single fiber action potential measurements in animal experiments. The model's responses to pulse train stimulation with respect to spatial location were also investigated. Adaptation was stronger at the borders of the stimulated area than in the center. By combining spatial details with long-term temporal components and a broad implementation of stochasticity a comprehensive model was developed that was validated for long duration electric stimulation of a wide range of pulse rates and amplitudes. The model can be used to evaluate auditory nerve responses to cochlear implant sound coding strategies.

EP 29. Field distribution for deep brain stimulation using an anatomical partially anisotropic volume conductor model of the rat brain

Introduction Deep brain stimulation (DBS) is a common therapy for the late-stage treatment of Parkinson’s disease. In order to explore clinically relevant questions, DBS is studied both in vivo in animal models and in silico in computational models. Objectives Creating an anatomically realistic volume conductor model (VCM) of the rat brain allows for a close interrelationship of in vivo and in silico models. Thus, an optimization of the stimulation field distribution and an improvement of DBS in animal models is facilitated. Various quantities affect the resulting spatial distribution of the stimulation field in a VCM, namely the specific topology of the compartments and the electric properties of brain tissue (e.g., its anisotropy). Methods Based on the quasi-static modeling approach Laplace equation div( σ )grad(Φ) = 0 is solved, with σ as electric conductivity and Φ as electric potential. The potential distribution evolved by DBS is computed with COMSOL Multiphysics® (Comsol). With the digital rat brain atlas in Papp et al., 2014 , the target region for DBS is assembled as three-dimensional structures. Modeled grey matter nuclei are the subthalamic nucleus (STN) and the entopeduncular nucleus (EPN). In the cubic region of interest (3 mm edge length) all white matter structures and cerebrospinal fluid cavities are included. Whilst the electrical conductivities of grey matter may be assumed as isotropic, for white matter an anisotropic conductivity should be assigned. By assuming that white matter has a ten times higher longitudinal conductivity along the axonal fibers than in transverse direction ( Tuch et al., 2001 ), the conductivity tensor is defined. The orientation of the anisotropic properties follows curvilinear coordinates. These coordinates are computed by using the diffusion method implemented in Comsol by solving Laplace’s equation, −grad(U) = 0, where the gradient of the solution U forms the first basis vector of the coordinate system. Thereby the anisotropy follows the shape of the white matter fiber bundles. The bipolar electrode model and the stimulation parameters are based on the in vivo model in Badstubner, 2013 . Results In this model, an anisotropic conductivity following curvilinear coordinates is implemented in the corticofugal pathway (CP). In Fig. 1 the periprosthetic potential either for the fully isotropic or partially anisotropic model is compared. Here, the relative difference of the potential Φ is the difference between a baseline model, with a homogeneous conductivity in all structures, and both heterogeneous models. The relative difference varies up to 15% and 60% in the isotropic and anisotropic case, respectively. Conclusion Our results suggest that neglecting an anatomically realistic anisotropy would lead to incorrect assessments of the neuronal activation.

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.

Effects of electrode displacement in high-definition transcranial direct current stimulation: A computational study.

In order to understand better the ways in which cortical excitability is linked to target brain areas, this study describes the effects of focalized high-definition transcranial direct current stimulation (HD-tDCS), and investigates the way in which these effects persisted after the stimulus electrodes were displaced from the target area. We constructed a 3D volume conduction model of an anatomically realistic head that is ideal for HD-tDCS, as well as compartmental models of layer 3 and layer 5 pyramidal neurons. Using extracellular approaches, we observed stimulus-induced electric fields and simulated neuronal responses by combining stimulus-induced potential fields with pyramidal neuronal models coupled with the head model. We found that the stimulus-induced electric fields were focused on the hand-knob when the electrodes were placed directly above the target region; further, the neuronal responses varied, such that the upper parts of the dendrites were hyperpolarized, while the soma and axons were depolarized. The magnitude of the electric fields, as well as the maximum polarizations at each compartment decreased according to the displacement of the electrodes from the target area. Considerable excitability at the target area within the range of 5 mm displacement between electrodes and the target area was shown by means of stimulus-induced electric fields and membrane polarization. In conclusion, using detailed computational approaches, we discovered the ways in which excitability in the target area persisted even with increased distance from the active electrode.