Limitations of approximate solutions for computing the extracellular potential of single fibers and bundle equivalents

Knösche T. Modelling the effect of ephaptic coupling on spike propagation in peripheral nerve fibres

Experimental and theoretical studies have shown that ephaptic coupling leads to the synchronisation and slowing down of spikes propagating along the axons within peripheral nerve bundles. However, the main focus thus far has been on a small number of identical axons, whereas realistic peripheral nerve bundles contain numerous axons with different diameters. Here, we present a computationally efficient spike propagation model, which captures the essential features of propagating spikes and their ephaptic interaction, and facilitates the theoretical investigation of spike volleys in large, heterogeneous fibre bundles. We first lay out the theoretical basis to describe how the spike in an active axon changes the membrane potential of a passive axon. These insights are then incorporated into the spike propagation model, which is calibrated with a biophysically realistic model based on Hodgkin-Huxley dynamics. The fully calibrated model is then applied to fibre bundles with a large number of axons and different types of axon diameter distributions. One key insight of this study is that the heterogeneity of the axonal diameters has a dispersive effect, and that a higher level of heterogeneity requires stronger ephaptic coupling to achieve full synchronisation between spikes.

Analytic modeling of conductively anisotropic neural tissue

The abdominal ganglion of the Aplysia californica is an established in vitro model for studying neuroelectric behavior in the presence of an applied electrical current and recently used in studies of magnetic resonance electrical impedance tomography (MREIT) which allows for quantitative visualization of spatially distributed current and magnetic flux densities. Understanding the impact the Aplysia geometry and anisotropic conductivity have on applied electromagnetic fields is central to intepreting and refining MREIT data and protocols, respectively. Here we present a simplified bidomain model of an in vitro experimental preparation of the Aplysia abdominal ganglion, describing the tissue as a radially anisotropic sphere with equal anisotropy ratios, i.e., where radial conductivities in both intra- and extra-cellular regions are ten times that of their polar and azimuthal conductivities. The fully three dimensional problem is validated through comparisons with limiting examples of 2D isotropic analyses. Results may be useful in validating finite element models of MREIT experiments and have broader relevance to analysis of MREIT data obtained from complex neural architecture in the human brain.

Analytic Modeling of Neural Tissue: I. A Spherical Bidomain

Presented here is a model of neural tissue in a conductive medium stimulated by externally injected currents. The tissue is described as a conductively isotropic bidomain, i.e. comprised of intra and extracellular regions that occupy the same space, as well as the membrane that divides them, and the injection currents are described as a pair of source and sink points. The problem is solved in three spatial dimensions and defined in spherical coordinates [Formula: see text]. The system of coupled partial differential equations is solved by recasting the problem to be in terms of the membrane and a monodomain, interpreted as a weighted average of the intra and extracellular domains. The membrane and monodomain are defined by the scalar Helmholtz and Laplace equations, respectively, which are both separable in spherical coordinates. Product solutions are thus assumed and given through certain transcendental functions. From these electrical potentials, analytic expressions for current density are derived and from those fields the magnetic flux density is calculated. Numerical examples are considered wherein the interstitial conductivity is varied, as well as the limiting case of the problem simplifying to two dimensions due to azimuthal independence. Finally, future modeling work is discussed.

Computational Modeling of Single Neuron Extracellular Electric Potentials and Network Local Field Potentials using LFPsim

Local Field Potentials (LFPs) are population signals generated by complex spatiotemporal interaction of current sources and dipoles. Mathematical computations of LFPs allow the study of circuit functions and dysfunctions via simulations. This paper introduces LFPsim, a NEURON-based tool for computing population LFP activity and single neuron extracellular potentials. LFPsim was developed to be used on existing cable compartmental neuron and network models. Point source, line source, and RC based filter approximations can be used to compute extracellular activity. As a demonstration of efficient implementation, we showcase LFPs from mathematical models of electrotonically compact cerebellum granule neurons and morphologically complex neurons of the neocortical column. LFPsim reproduced neocortical LFP at 8, 32, and 56 Hz via current injection, in vitro post-synaptic N_2a, N_2b waves and in vivo T-C waves in cerebellum granular layer. LFPsim also includes a simulation of multi-electrode array of LFPs in network populations to aid computational inference between biophysical activity in neural networks and corresponding multi-unit activity resulting in extracellular and evoked LFP signals.

Three dimensional transport lattice model for describing action potentials in axons stimulated by external electrodes

Conditions that stimulate action potentials in one or more nerves is of widespread interest. Axon and nerve models are usually based on two dimensional pre-specified lumped equivalents that assume where currents will flow. In contrast, here we illustrate creation of three dimensional (3D) system models with a transport lattice of interconnected local models for external and internal electrolyte and axon membrane. The transport lattice solves Laplace's equation in the extracellular medium and is coupled to the Hodgkin-Huxley model at local membrane sites. These space-filling models incorporate the geometric scale, which allows explicit representation of confined axons and external electrodes. The present results demonstrate feasibility of the basic approach. These models are spatially coarse and approximate, but can be straightforwardly improved. The transport lattice system models are modular and multiscale (spatial scales ranging from the membrane thickness of 5 nm to the axon segment length of 2 cm).

The effect of simplifying assumptions in the bidomain model of cardiac tissue: application to ST segment shifts during partial ischaemia

In this study various electrical conductivity approximations used in bidomain models of cardiac tissue are considered. Comparisons are based on epicardial surface potential distributions arising from regions of subendocardial ischaemia situated within the cardiac tissue. Approximations studied are a single conductivity bidomain model, an isotropic bidomain model and equal and reciprocal anisotropy ratios both with and without fibre rotation. It is demonstrated both analytically and numerically that the approximations involving a single conductivity bidomain, an isotropic bidomain or equal anisotropy ratios (ignoring fibre rotation) results in identical epicardial potential distributions for all degrees of subendocardial ischaemia. This result is contrary to experimental observations. It is further shown that by assuming reciprocal anisotropy ratios, epicardial potential distributions vary with the degree of subendocardial ischaemia. However, it is concluded that unequal anisotropy ratios must be used to obtain the true character of experimental observations.

Finite element modeling of electromagnetic signal propagation in a phantom arm

Improving the control of artificial arms remains a considerable challenge. It may be possible to graft remaining peripheral nerves in an amputated limb to spare muscles in or near the residual limb and use these nerve-muscle grafts as additional myoelectric control signals. This would allow simultaneous control of multiple, degrees of freedom (DOF) and could greatly improve the control of artificial limbs. For this technique to be successful, the electromyography (EMG) signals from the nerve-muscle grafts would need to be independent of each other with minimal crosstalk. To study EMG signal propagation and quantify crosstalk, finite element (FE) models were developed in a phantom-arm model. The models were validated with experimental data collected by applying sinusoidal excitations to a phantom-arm model and recording the surface electric potential distribution. There was a very high correlation (r > 0.99) between the FEM data and the experimental data, with the error in signal magnitude generally less than 5%. Simulations were then performed using muscle dielectric properties with static, complex, and full electromagnetic solvers. The results indicate that significant displacement currents can develop (> 50% of total current) and that the fall-off of surface signal power varies with how the signal source is modeled. Index Terms-Control, electromyography (EMG), finite element (FE), modeling, prosthesis.

The importance of anisotropy in modeling ST segment shift in subendocardial ischaemia

In this paper, a simple mathematical model of a slab of cardiac tissue is presented in an attempt to better understand the relationship between subendocardial ischaemia and the resulting epicardial potential distributions. The cardiac tissue is represented by the bidomain model where tissue anisotropy and fiber rotation have been incorporated with a view to predicting the epicardial surface potential distribution. The source of electric potential in this steady-state problem is the difference between plateau potentials in normal and ischaemic tissue, where it is assumed that ischaemic tissue has a lower plateau potential. Simulations with tissue anisotropy and no fiber rotation are also considered. Simulations are performed for various thicknesses of the transition region between normal and ischaemic tissue and for various sizes of the ischaemic region. The simulated epicardial potential distributions, based on an anisotropic model of the cardiac tissue, show that there are large potential gradients above the border of the ischaemic region and that there are dips in the potential distribution above the region of ischaemia. It could be concluded from the simulations that it would be possible to predict the region of subendocardial ischaemia from the epicardial potential distribution, a conclusion contrary to observed experimental data. Possible reasons for this discrepancy are discussed. In the interests of mathematical simplicity, isotropic models of the cardiac tissue are also considered, but results from these simulations predict epicardial potential distributions vastly different from experimental observations. A major conclusion from this work is that tissue anisotropy and fiber rotation must be included to obtain meaningful and realistic epicardial potential distributions.

Modeling of surface myoelectric signals--Part I: Model implementation

The relationships between the parameters of active motor units (MU's) and the features of surface electromyography (EMG) signals have been investigated using a mathematical model that represents the surface EMG as a summation of contributions from the single muscle fibers. Each MU has parallel fibers uniformly scattered within a cylindrical volume of specified radius embedded in an anisotropic medium. Two action potentials, each modeled as a current tripole, are generated at the neuromuscular junction, propagate in opposite directions and extinguish at the fiber-tendon endings. The neuromuscular junctions and fiber-tendon endings are uniformly scattered within regions of specified width. Muscle fiber conduction velocity and average fiber length to the right and left of the center of the innervation zone are also specified. The signal produced by MU's with different geometries and conduction velocities are superimposed. Monopolar, single differential and double differential signals are computed from electrodes placed in equally spaced locations on the surface of the muscle and are displayed as functions of any of the model's parameters. Spectral and amplitude variables and conduction velocity are estimated from the surface signals and displayed as functions of any of the model's parameters. The influence of fiber-end effects, electrode misalignment, tissue anisotropy, MU's location and geometry are discussed. Part II of this paper will focus on the simulation and interpretation of experimental signals.

Electrical interactions via the extracellular potential near cell bodies

Ephaptic interactions between a neuron and axons or dendrites passing by its cell body can be, in principle, more significant than ephaptic interactions among axons in a fiber tract. Extracellular action potentials outside axons are small in amplitude and spatially spread out, while they are larger in amplitude and much more spatially confined near cell bodies. We estimated the extracellular potentials associated with an action potential in a cortical pyramidal cell using standard one-dimensional cable theory and volume conductor theory. Their spatial and temporal pattern reveal much about the location and timing of currents in the cell, especially in combination with a known morphology, and simple experiments could resolve questions about spike initiation. From the extracellular potential we compute the ephaptically induced polarization in a nearby passive cable. The magnitude of this induced voltage can be several mV, does not spread electrotonically, and depends only weakly on the passive properties of the cable. We discuss their possible functional relevance.

Effects of bath resistance on action potentials in the squid giant axon: myocardial implications

This study presents a simplified version of the quasi-one-dimensional theory (Wu, J., E. A. Johnson, and J. M. Kootsey. 1996. A quasi-one-dimensional theory for anisotropic propagation of excitation in cardiac muscle. Biophys. J. 71:2427-2439) with two components of the extracellular current, along and perpendicular to the axis, and a simulation and its experimental confirmation for the giant axon of the squid. By extending the one-dimensional core conductor cable equations, this theory predicts, as confirmed by the experiment, that the shapes of the intracellular and the extracellular action potentials are related to the resistance of the bath. Such a result was previously only expected by the field theories. The correlation between the shapes of the intracellular and the extracellular potentials of the giant axon of the squid resembles that observed during the anisotropic propagation of excitation in cardiac muscle. Therefore, this study not only develops a quasi-one-dimensional theory for a squid axon, but also provides one possible factor contributing to the anisotropic propagation of action potentials in cardiac muscle.

The extracellular potential of a myelinated nerve fiber in an unbounded medium and in nerve cuff models

A model is presented for the calculation of single myelinated fiber action potentials in an unbounded homogeneous medium and in nerve cuff electrodes. The model consists of a fiber model, used to calculate the action currents at the nodes of Ranvier, and a cylindrically symmetrical volume conductor model in which the fiber's nodes are represented as point current sources. The extracellular action potentials were shown to remain unchanged if the fiber diameter and the volume conductor geometry are scaled by the same factor (principle of corresponding states), both in an unbounded homogeneous medium and in an inhomogeneous volume conductor. The influence of several cuff electrode parameters, among others, cuff length and cuff diameter, were studied, and the results were compared, where possible, with theoretical and experimental results as reported in the literature.

Computer simulation of epicardial potentials using a heart-torso model with realistic geometry

Previous cardiac simulation studies have focused on simulating the activation isochrones and subsequently the body surface potentials. Epicardial potentials, which are important for clinical application as well as for electrocardiographic inverse problems studies, however, have usually been neglected. This paper describes a procedure of simulating epicardial potentials using a microcomputer-based heart-torso model with realistic geometry. Our heart model developed earlier is composed of approximately 65,000 cell units which are arranged in a cubic close-packed structure. An action potential waveform with variable in duration is assigned to each unit. The heart model, together with the epicardial surface model constructed recently, are mounted in an inhomogeneous human torso model. Electric dipoles, which are proportional to the spatial gradient of the action potential, are generated in all the cell units. These dipoles give rise to a potential distribution on the epicardial surface, which is calculated by means of the boundary element method. The simulated epicardial potential maps during a normal heart beat and in a preexcited beat to mimic Wolff-Parkinson-White (WPW) syndrome are in close agreement with those reported in the literature.

Action potentials of curved nerves in finite limbs

Previous simulations of volume-conducted nerve-fiber action-potentials have modeled the limb as semi-infinite or circularly cylindrical, and the fibers as straight lines parallel to the limb surface. The geometry of actual nerves and limbs, however, can be considerably more complicated. This paper presents a general method for computing the potentials of fibers with arbitrary paths in arbitrary finite limbs. It involves computing the propagating point-source response (PPSR), which is the potential arising from a single point source (dipole or tripole) travelling along the fiber. The PPSR can be applied to fibers of different conduction velocities by simple dilation or compression. The method is illustrated for oblique and spiralling nerve fibers. Potentials from oblique fibers are shown to be different for orthodromic and antidromic propagation. Such results show that the straight-line models are not always adequate for nerves with anatomical amounts of curvature.

Spatial- and frequency-domain ring source models for the single-muscle fibre action potential

In the paper, single-fibre models for the extracellular action potential are developed that will allow the potential to be evaluated at an arbitrary field point in the extracellular space. Fourier-domain models are restricted in that they evaluate potentials at equidistant points along a line parallel to the fibre axis. Consequently, they cannot easily evaluate the potential at the boundary nodes of a boundary-element electrode model. The Fourier-domain models employ axial-symmetric ring source models, and thereby provide higher accuracy than the line source model, where the source is lumped into a line source at the centre of the fibre. In the paper, new spatial models are developed based on elliptic integrals. These models employ axial-symmetric ring source models, and therefore are more accurate than the line source model. In the analysis, dual transform pairs are identified. Numerical examples including anisotropy show that the spatial models require extreme care in the integration procedure owing to the singularity in the weighting functions. With adequate sampling, the spatial models can evaluate extracellular potentials with high accuracy.

Computer simulations of three-dimensional propagation in ventricular myocardium. Effects of intramural fiber rotation and inhomogeneous conductivity on epicardial activation

Three-dimensional membrane-based simulations of action potential propagation in ventricular myocardium were performed. Specifically, the effects of the intramural rotation of the fiber axes and inhomogeneous conductivity on the timing and pattern of epicardial activation were examined. Models were built, with approximately 400,000 microscopic elements arranged in rectangular parallelepipeds in each model. Simulations used the nonlinear Ebihara and Johnson membrane equations for the fast sodium current. Constructed models had histological features of ventricular myocardium. All models were anisotropic. In a subset of the models, an abrupt intramural rotation of the fiber axes was included. This feature was also combined with randomly distributed inhomogeneous conductivity and regions of high transverse resistance to represent nonuniform anisotropy in a further subset of the models. Epicardial stimuli were applied for each simulation. Three-dimensional activation patterns and epicardial isochron maps were constructed from the simulations. We noted that the rotation of fiber axes accelerated epicardial activation distant from the stimulus site. The inhomogeneous conductivity caused regional acceleration and deceleration of activation spread. We also noted features of epicardial activation that resulted from the fiber rotation, and the inhomogeneous conductivity corresponded to that observed in maps from experimental animals.

Examination of the choice of models for computing the extracellular potential of a single fibre in a restricted volume conductor

The paper compares the rigorous and the conventional approximate line source solution of Laplace's equation used to evaluate the potential of a single cylindrical fibre. Particular attention is given to the solutions for a radially restricted circular cylindrical volume conductor. The effect of the extent of the volume conductor b on the difference between the potentials evaluated according to the different models is examined. For values of b larger than 10 times the fibre radius, the relative difference is less than 1 per cent and the values of b around 2 times the fibre radii, the error reaches as much as 17 per cent.

Simulation of propagation along a cylindrical bundle of cardiac tissue--II: Results of simulation

Previous evaluations of the cylindrical bidomain model of a bundle of cardiac tissue, have been obtained by using an analytic function for the transmembrane potential and assuming the activating wavefront through the bundle cross section is planar. In this paper, nonlinear membrane kinetics are introduced into the bidomain membrane and equal anisotropy ratios are assumed, permitting the transmembrane potential to be computed and its behavior examined at different depths in the bundle and for different values of conductivity and bundle diameters. In contrast with single fiber models, the bundle model reveals that the shape of the action potential is influenced by tissue resistivities. In addition, the steady-state activation wavefront through the cross-section perpendicular to the long axis of the bundle is not planar and propagates with a velocity that lies between that of a single fiber in an unbounded volume and a single fiber in a restricted extracellular space. In general, the bundle model is shown to be significantly better than the classical single fiber model in describing the behavior of real cardiac tissue.

Effect of intracellular anisotropy on electrical source determination in a muscle fibre

Expressions are available for describing, quantitatively, the source associated with an action potential propagating along an excitable fibre. For a nerve fibre one such expression defines an equivalent volume dipole density function tau(x) = -delta/delta x (sigma i phi i (x) - sigma e phi e (x)) ax (where x is the axial co-ordinate, i is the intracellular and e the extracellular region, sigma i and sigma e are isotropic conductivities, phi the potential at the membrane, while axial symmetry is assumed), and this source fills the intracellular region. This source, as distinct from transmembrane current formulations, lies in a uniform, isotropic, extracellular, medium. Consequently, for a fibre bundle a simple superposition of sources, all lying in a uniform, isotropic, extracellular space, can be accomplished. However, for muscle fibres the presence of non-conducting myofibrils causes the intracellular space to be anisotropic. The paper describes the modification in the aforementioned expressions for the case of longitudinal and transverse propagation and extrapolation to an arbitrary angle of propagation. The resultant source continues to be expressed relative to a uniform, isotropic, extracellular medium.