The “Mirror” Estimate: An Intuitive Predictor of Membrane Polarization during Extracellular Stimulation

Achieving controlled extracellular microstimulation of the central nervous system requires understanding the membrane response of a neuron to an applied electric field. The “activating function” has been proposed as an intuitive predictor of membrane polarization during stimulation, but subsequent literature raised several limitations of this estimate. In this study, we show that, depending on the space constant λ, the steady-state solution to the passive cable equation is theoretically well approximated by either the activating function when λ is small, or the “mirror” image of the extracellular potential when λ is large. Using simulations, we then explore the respective domain of both estimates as a function of λ, stimulus duration, fiber length, and electrode-fiber distance. For realistic λ (>50-100 μm), the mirror estimate is the best predictor for either long electrode-fiber distances or short distances (<20-30 μm) when stimulus durations exceed a few tens of microseconds. For intermediate distances, the mirror estimate is all the more valid that the stimulus duration is long and the fiber is short. We also illustrate that this estimate correctly predicts the steady-state membrane polarization of complex central nervous system arborizations. In conclusion, the mirror estimate can often be preferred to the activating function to intuitively predict membrane polarization during extracellular stimulation.     

Spatial distribution of cardiac transmembrane potentials around an extracellular electrode: dependence on fiber orientation.

Recent theoretical models of cardiac electrical stimulation or defibrillation predict a complex spatial pattern of transmembrane potential (Vm) around a stimulating electrode, resulting from the formation of virtual electrodes of reversed polarity. The pattern of membrane polarization has been attributed to the anisotropic structure of the tissue. To verify such model predictions experimentally, an optical technique using a fluorescent voltage-sensitive dye was used to map the spatial distribution of Vm around a 150-microns-radius extracellular unipolar electrode. An S1-S2 stimulation protocol was used, and vm was measured during an S2 pulse having an intensity equal to 10x the cathodal diastolic threshold of excitation. The recordings were obtained on the endocardial surface of bullfrog atrium in directions parallel and perpendicular to the cardiac fibers. In the longitudinal fiber direction, the membrane depolarized for cathodal pulses (and hyperpolarized for anodal pulses) but only in a region within 445 +/- 112 microns (and 616 +/- 78 microns for anodal pulses) from the center of the electrode (n = 9). Outside this region, vm reversed polarity and reached a local maximum at 922 +/- 136 microns (and 988 +/- 117 microns for anodal pulses) (n = 9). Beyond this point vm decayed to zero over a distance of 1.5-2 mm. In the transverse fiber direction, the membrane depolarized for cathodal pulses (and hyperpolarized for anodal pulses) at all distances from the electrode. The amplitude of the response decreased with distance from the electrode with an exponential decay constant of 343 +/- 110 microns for cathodal pulses and 253 +/- 91 microns for anodal pulses (n = 7). The results were qualitatively similar in both fiber directions when the atrium was bathed in a solution containing ionic channel blockers. A two-dimensional computer model was formulated for the case of highly anisotropic cardiac tissue and qualitatively accounts for nearly all the observed spatial and temporal behavior of vm in the two fiber directions. The relationships between vm and both the "activating function" and extracellular potential gradient are discussed.     

Analytical theory for extracellular electrical stimulation of nerve with focal electrodes. I. Passive unmyelinated axon.

The cable model of a passive, unmyelinated fiber in an applied extracellular field is derived. The solution is valid for an arbitrary, time-varying, applied field, which may be determined analytically or numerically. Simple analytical computations are presented. They explain a variety of known phenomena and predict some previously undescribed properties of extracellular electrical stimulation. The polarization of a fiber in an applied field behaves like the output of a spatial high-pass and temporal low-pass filter of the stimulus. High-frequency stimulation results in a more spatially restricted region of fiber excitation, effectively reducing current spread relative to that produced by low-frequency stimulation. Chronaxie measured extracellularly is a function of electrode position relative to the stimulated fiber, and its value may differ substantially from that obtained intracellularly. Frequency dependence of psychophysical threshold obtained by electrical stimulation of the macaque cochlea closely follows the frequency dependence of single-fiber passive response.     

Virtual electrode effects in defibrillation

This modeling study demonstrates that a re-entrant activity in a sheet of myocardium can be extinguished by a defibrillation shock delivered via extracellular point-source electrodes which establish spatially non-uniform applied field. The tissue is represented as a homogeneous bidomain with unequal anisotropy ratios in the cardiac conductivities. Spiral wave re-entry is initiated in the bidomain sheet following an S1-S2 stimulation protocol. The results indicate that the point-source defibrillation shock establishes large-scale changes in transmembrane potential in the tissue (virtual electrodes) that are ‘superimposed’ over regions of various degrees of membrane refractoriness in the myocardium. The close proximity of large-scale shock-induced regions of alternating membrane polarity is central to the ability of the shock to terminate the spiral wave. The new wavefronts generated following anode/cathode break phenomena restrict the spiral wave and render the tissue too refractory to further maintain the re-entry. In contrast, shocks delivered via line electrodes establish, in close proximity to the electrode, changes in transmembrane potential that are of same-sign polarity. These shocks are incapable of terminating the re-entrant activation.     

Success and failure of biphasic shocks: results of bidomain simulations.

The mechanisms behind the superiority of optimal biphasic defibrillation shocks over monophasic are not fully understood. This simulation study examines how the shock polarity and second-phase magnitude of biphasic shocks influence the virtual electrode polarization (VEP) pattern, and thus the outcome of the shock in a bidomain model representation of ventricular myocardium. A single spiral wave is initiated in a two-dimensional sheet of myocardium that measures 2 x 2 cm(2). The model incorporates non-uniform fiber curvature, membrane kinetics suitable for high strength shocks, and electroporation. Line electrodes deliver a spatially uniform extracellular field. The shocks are biphasic, each phase lasting 10 ms. Two different polarities of biphasic shocks are examined as the first-phase configuration is held constant and the second-phase magnitude is varied between 1 and 10 V/cm. The results show that for each polarity, varying the second-phase magnitude reverses the VEP induced by the first phase in an asymmetric fashion. Further, the size of the post-shock excitable gap is dependent upon the second-phase magnitude and is a factor in determining the success or failure of the shock. The maximum size of a post-shock excitable gap that results in defibrillation success depends on the polarity of the shock, indicating that the refractoriness of the tissue surrounding the gap also contributes to the outcome of the shock.     

Mathematical modeling of the excitation process in myocardial tissue: influence of fiber rotation on wavefront propagation and potential field

In our macroscopic model the heart tissue is represented as a bidomain coupling the intra- and extracellular media. Owing to the fiber structure of the myocardium, these media are anisotropic, and their conductivity tensors have a principal axis parallel to the local fiber direction. A reaction-diffusion system is derived that governs the distribution and evolution of the extracellular and transmembrane potentials during the depolarization phase of the heart beat. To investigate frontlike solutions, the system is rescaled and transformed into a system dependent on a small parameter. Subsequently a perturbation analysis is carried out that yields zero- and first-order approximations called eikonal equations. The effects of the transmural fiber rotation on wavefront propagation and the corresponding potential field, elicited by point stimulations, are investigated by means of numerical simulations.     

A simulation study of the reaction of human heart to biphasic electrical shocks

Background

This article presents a study, which examines the effects of biphasic electrical shocks on human ventricular tissue. The effects of this type of shock are not yet fully understood. Animal experiments showed the superiority of biphasic shocks over monophasic ones in defibrillation. A mathematical computer simulation can increase the knowledge of human heart behavior.

Methods

The research presented in this article was done with different models representing a three-dimensional wedge of ventricular myocardium. The electrophysiology was described with Priebe-Beuckelmann model. The realistic fiber twist, which is specific to human myocardium was included. Planar electrodes were placed at the ends of the longest side of the virtual cardiac wedge, in a bath medium. They were sources of electrical shocks, which varied in magnitude from 0.1 to 5 V. In a second arrangement ring electrodes were placed directly on myocardium for getting a better view on secondary electrical sources. The electrical reaction of the tissue was generated with a bidomain model.

Results

The reaction of the tissue to the electrical shock was specific to the initial imposed characteristics. Depolarization appeared in the first 5 ms in different locations. A further study of the cardiac tissue behavior revealed, which features influence the response of the considered muscle. It was shown that the time needed by the tissue to be totally depolarized is much shorter when a biphasic shock is applied. Each simulation ended only after complete repolarization was achieved. This created the possibility of gathering information from all states corresponding to one cycle of the cardiac rhythm.

Conclusions

The differences between the reaction of the homogeneous tissue and a tissue, which contains cleavage planes, reveals important aspects of superiority of biphasic pulses. ...     

Virtual electrodes in cardiac tissue: a common mechanism for anodal and cathodal stimulation.

Traditional cable analyses cannot explain complex patterns of excitation in cardiac tissue with unipolar, extracellular anodal, or cathodal stimuli. Epifluorescence imaging of the transmembrane potential during and after stimulation of both refractory and excitable tissue shows distinctive regions of simultaneous depolarization and hyperpolarization during stimulation that act as virtual cathodes and anodes. The results confirm bidomain model predictions that the onset (make) of a stimulus induces propagation from the virtual cathode, whereas stimulus termination (break) induces it from the virtual anode. In make stimulation, the virtual anode can delay activation of the underlying tissue, whereas in break stimulation this occurs under the virtual cathode. Thus make and break stimulations in cardiac tissue have a common mechanism that is the result of differences in the electrical anisotropy of the intracellular and extracellular spaces and provides clear proof of the validity of the bidomain model.     

Action potentials and variation potentials in sunflower: An analysis of their relationships and distinguishing characteristics

Sunflower plants ( Helianthus annuus L.) were given an electrical stimulus to the stem or a heat (flame)‐wound to a single leaf or a cotyledon. The resulting electrical activity was monitored with extracellular electrodes. An electrical stimulus applied to the stem frequently evoked an action potential (AP), but never a variation potential (VP). In contrast, a heat‐wound applied to a leaf virtually always elicited a VP, which was often accompanied by one or more superimposed spikes (putative APs). The kinetic parameters of the AP and the VP were investigated. The AP appears to propagate without decrement in velocity or magnitude, whereas the VP parameters decrease significantly with distance. The heat stimulus triggered rapid alterations in stem elongation/contraction, which preceded changes in electrical potential, indicating the transmission of a hydraulic signal. Light‐off and light‐on stimuli evoked negative‐ and positive‐going changes in extracellular electrical potential, respectively, corresponding to de‐ and hyper‐polarization of the plasma membrane. Membrane depolarization (extracellularly manifested as a VP) evoked by both the light‐off and heat‐wounding stimuli was able to trigger one or more APs. We interpret these results to suggest that APs are "genuine" electrical signals involving voltage‐gated ion channels or pumps, which can be evoked directly by electrical stimulation or indirectly by changes in membrane potential occurring during the VP or after the light‐off stimulus. In contrast, VPs appear to be a local (non‐transmissible) electrical consequence of the passage of a rapidly transmitted hydraulic signal in the xylem, presumably acting on mechanosensitive ion channels or pumps in adjacent living cells.     

On the electrotonic spread in cardiac muscle of the mouse

As an appropriate model which can simulate the cardiac working muscle with respect to the passive electrical spread, a lattice whose sides have linear cable properties is presented, and the passive potential spread on the model is mathematically analyzed in the fiber direction. Distribution of electrotonic potential in the fiber direction was measured with a pair of intracellular microelectrodes in the cardiac muscle fiber of mouse. By employing “pencil type” microelectrodes potential distribution in the transverse direction within a fiber was also measured. This transverse effect was differentiated from the longitudinal potential distribution. A tonically applied potential at any point of a cell interior spreads continuously in a manner described by a Bessel function. Using appropriate electrical and morphological parameters the experimental results proved to fit the curve obtained from numerical calculation on the model. The apparent length constant obtained for smaller distances (less than 100 μ) from the current source was 70 μ, and it increases as the distance becomes larger. At a point inside the fiber the resistance to the extracellular fluid ranged from 200 to 600 KΩ. The influence of coupling resistance between current and recording electrodes on the measurement of electrotonic potential was examined for small interelectrode distance.     

How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle

The bidomain model, which describes the behavior of many electrically active tissues, is equivalent to a multi-dimensional cable model and can be represented by a network of resistors and capacitors. For a two-dimensional sheet of tissue, the intracellular and extracellular conductivity tensors can be visualized as two ellipses. For any pair of conductivity tensors, a coordinate transformation can be found that reduces the extracellular ellipse to a circle and aligns the intracellular ellipse with the coordinate axes. The eccentricity of the intracellular ellipse in this new coordinate system is an important parameter. It can have two special values: zero (in which case the tissue has equal anisotropy ratios) or one (in which case the tissue is comprised of one-dimensional fibers coupled through the two-dimensional extracellular space). Thus the bidomain model provides a unifying framework within which the electrical behavior of a wide variety of nerve and muscle tissues can be studied.When the anisotropy ratios in the intracellular and extracellular domains are not equal, stimulation with an anode always causes depolarization of some region of tissue. An analogous effect occurs in models that describe one-dimensional fibers, in which an activating function determines the site of stimulation. Experiments indicate that cardiac muscle does not have equal anisotropy ratios. Therefore, models developed to describe stimulation of axons may also help in understanding stimulation of two- or three-dimensional cardiac tissue, and may explain the concept of anodal stimulation of cardiac tissue through a virtual cathode.     

Roles of electric field and fiber structure in cardiac electric stimulation.

This study investigated roles of the variation of extracellular voltage gradient (VG) over space and cardiac fibers in production of transmembrane voltage changes (DeltaV(m)) during shocks. Eleven isolated rabbit hearts were arterially perfused with solution containing V(m)-sensitive fluorescent dye (di-4-ANEPPS). The epicardium received shocks from symmetrical or asymmetrical electrodes to produce nominally uniform or nonuniform VGs. Extracellular electric field and DeltaV(m) produced by shocks in the absolute refractory period were measured with electrodes and a laser scanner and were simulated with a bidomain computer model that incorporated the anterior left ventricular epicardial fiber field. Measurements and simulations showed that fibers distorted extracellular voltages and influenced the DeltaV(m). For both uniform and nonuniform shocks, DeltaV(m) depended primarily on second spatial derivatives of extracellular voltages, whereas the VGs played a smaller role. Thus, 1) fiber structure influences the extracellular electric field and the distribution of DeltaV(m); 2) the DeltaV(m) depend on second spatial derivatives of extracellular voltage.     

Signal transduction of a tissue interaction during embryonic heart development.

During early cardiac development, progenitors of the valves and septa of the heart are formed by an epithelial-mesenchymal cell transformation of endothelial cells of the atrioventricular (AV) canal. We have previously shown that this event is due to an interaction between the endothelium and products of the myocardium found within the extracellular matrix. The present study examines signal transduction mechanisms governing this differentiation of AV canal endothelium. Activators of protein kinase C (PKC), phorbol myristate acetate (PMA) and mezerein, both produced an incomplete phenotypic transformation of endothelial cells in an in vitro bioassay for transformation. On the other hand, inhibitors of PKC (H-7 and staurosporine) and tyrosine kinase (genistein) blocked cellular transformation in response to the native myocardium or a myocardially-conditioned medium. Intracellular free calcium concentration ([Ca2+]i) was measured in single endothelial cells by microscopic digital analysis of fura 2 fluorescence. Addition of a myocardial conditioned medium containing the transforming stimulus produced a specific increase in [Ca2+]i in "competent" AV canal, but not ventricular, endothelial cells. Epithelial-mesenchymal cell transformation was inhibited by pertussis toxin but not cholera toxin. These data lead to the hypothesis that signal transduction of this tissue interaction is mediated by a G protein and one or more kinase activities. In response to receptor activation, competent AV canal endothelial cells demonstrate an increase in [Ca2+]i. Together, the data provide direct evidence for a regional and temporal regulation of signal transduction processes which mediate a specific extracellular matrix-mediated tissue interaction in the embryo.     

Effects of elevated extracellular potassium on the stimulation mechanism of diastolic cardiac tissue

During cardiac disturbances such as ischemia and hyperkalemia, the extracellular potassium ion concentration is elevated. This in turn changes the resting transmembrane potential and affects the excitability of cardiac tissue. To test the hypothesis that extracellular potassium elevation also alters the stimulation mechanism, we used optical fluorescence imaging to examine the mechanism of diastolic anodal unipolar stimulation of cardiac tissue under 4 mM (normal) and 8 mM (elevated) extracellular potassium. We present several visualization methods that are useful for distinguishing between anodal-make and anodal-break excitation. In the 4-mM situation, stimulation occurred by the make, or stimulus-onset, mechanism that involved propagation out of the virtual cathodes. For 8-mM extracellular potassium, the break or stimulus termination mechanism occurred with propagation out of the virtual anode. We conclude that elevated potassium, as might occur in myocardial ischemia, alters not only stimulation threshold but also the excitation mechanism for anodal stimulation.     

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.     

The topology of defibrillation

We describe how Art Winfree's ideas about phase singularities can be used to understand the response of cardiac tissue with a random preexisting pattern of reentrant waves (fibrillation) to a large brief current stimulus. This discussion is organized around spatial dimension, beginning with a discussion of reentry on a periodic ring, followed by reentry in a two-dimensional planar domain (spiral waves), and ending with consideration of three-dimensional reentrant patterns (scroll waves). In all cases, we show how reentrant activity is changed by the application of a shock, describing conditions under which defibrillation is successful or not. Using topological arguments we draw the general conclusion that with a generic placement of stimulating electrodes, large-scale virtual electrodes do not give an adequate explanation for the mechanism of defibrillation.     

Potential and current distributions in a cylindrical bundle of cardiac tissue.

The intracellular and interstitial potentials associated with each cell or fiber in multicellular preparations carrying a uniformly propagating wave are important for characterizing the electrophysiological behavior of the preparation and in particular, for evaluating the source contributed by each fiber. The aforementioned potentials depend on a number of factors including the conductivities characterizing the intracellular, interstitial, and extracellular domains, the thickness of the tissue, and the distance (depth) of the field point from the surface of the tissue. A model study is presented describing the extracellular and interstitial potential distribution and current flow in a cylindrical bundle of cardiac muscle arising from a planar wavefront. For simplicity, the bundle is considered as a bidomain. Using typical values of conductivity, the results show that the intracellular and interstitial potential of fibers near the center of a very large bundle (greater than 10 mm) may be approximated by the potentials of a single fiber surrounded by a limited extracellular space (a fiber in oil), hence justifying a core-conductor model. For smaller bundles, the peak interstitial potential is less than that predicted by the core-conductor model but still large enough to affect the overall source strength. The magnitude of the source strength is greatest for fibers lying near the center of the bundle and diminishes sharply for fibers within 50 microns of the surface.     

An orthotropic viscoelastic model for the passive myocardium: continuum basis and numerical treatment

This study deals with the viscoelastic constitutive modeling and the respective computational analysis of the human passive myocardium. We start by recapitulating the locally orthotropic inner structure of the human myocardial tissue and model the mechanical response through invariants and structure tensors associated with three orthonormal basis vectors. In accordance with recent experimental findings the ventricular myocardial tissue is assumed to be incompressible, thick-walled, orthotropic and viscoelastic. In particular, one spring element coupled with Maxwell elements in parallel endows the model with viscoelastic features such that four dashpots describe the viscous response due to matrix, fiber, sheet and fiber-sheet fragments. In order to alleviate the numerical obstacles, the strictly incompressible model is altered by decomposing the free-energy function into volumetric-isochoric elastic and isochoric-viscoelastic parts along with the multiplicative split of the deformation gradient which enables the three-field mixed finite element method. The crucial aspect of the viscoelastic formulation is linked to the rate equations of the viscous overstresses resulting from a 3-D analogy of a generalized 1-D Maxwell model. We provide algorithmic updates for second Piola–Kirchhoff stress and elasticity tensors. In the sequel, we address some numerical aspects of the constitutive model by applying it to elastic, cyclic and relaxation test data obtained from biaxial extension and triaxial shear tests whereby we assess the fitting capacity of the model. With the tissue parameters identified, we conduct (elastic and viscoelastic) finite element simulations for an ellipsoidal geometry retrieved from a human specimen.     

Virtual electrode polarization in the far field: implications for external defibrillation

We recently suggested that failure of implantable defibrillation therapy may be explained by the virtual electrode-induced phase singularity mechanism. The goal of this study was to identify possible mechanisms of vulnerability and defibrillation by externally applied shocks in vitro. We used bidomain simulations of realistic rabbit heart fibrous geometry to predict the passive polarization throughout the heart induced by external shocks. We also used optical mapping to assess anterior epicardium electrical activity during shocks in Langendorff-perfused rabbit hearts (n = 7). Monophasic shocks of either polarity (10-260 V, 8 ms, 150 microF) were applied during the T wave from a pair of mesh electrodes. Postshock epicardial virtual electrode polarization was observed after all 162 applied shocks, with positive polarization facing the cathode and negative polarization facing the anode, as predicted by the bidomain simulations. During arrhythmogenesis, a new wave front was induced at the boundary between the two regions near the apex but not at the base. It spread across the negatively polarized area toward the base of the heart and reentered on the other side while simultaneously spreading into the depth of the wall. Thus a scroll wave with a ribbon-shaped filament was formed during external shock-induced arrhythmia. Fluorescent imaging and passive bidomain simulations demonstrated that virtual electrode polarization-induced scroll waves underlie mechanisms of shock-induced vulnerability and failure of external defibrillation.     

A multiaxial constitutive law for mammalian left ventricular myocardium in steady-state barium contracture or tetanus.

The constitutive law of the material comprising any structure is essential for mechanical analysis since this law enables calculation of the stresses from the deformations and vice versa. To date, there is no constitutive law for actively contracting myocardial tissue. Using 2,3-butanedione monoxime to protect the myocardium from mechanical trauma, we subjected thin midwall slices of rabbit myocardium to multiaxial stretching first in the passive state and then during steady-state barium contracture or during tetani in ryanodine-loaded tissue. Assuming transverse isotropy in both the passive and active conditions, we used our previously described methods (Humphrey et al., 1990a) to obtain both passive and active constitutive laws. The major results of this study are: (1) This is the first multiaxial constitutive law for actively contracting mammalian myocardium. (2) The functional forms of the constitutive law for barium contracture and ryanodine-induced tetani are the same but differ from those in the passive state. Hence, one cannot simply substitute differing values for the coefficients of the passive law to describe the active tissue properties. (3) There are significant stresses developed in the cross-fiber direction (more than 40 percent of those in the fiber direction) that cannot be attributed to either deformation effects or nonparallel muscle fibers. These results provide the foundation for future mechanical analyses of the heart.