Modification of a cylindrical bidomain model for cardiac tissue.

Previous models based on a cylindrical bidomain assumed either that the ratio of intracellular and interstitial conductivities in the principal directions were the same or that there was no radial variation in potential (i.e., a planar front, delta Vm/delta rho = 0). This paper presents a formulation and the expressions for the intracellular, interstitial, extracellular, and transmembrane potentials arising from nonplanar propagation along a cylindrical bundle of cardiac tissue represented as a bidomain with arbitrary anisotropy. For unequal anisotropy, the transmembrane current depends not only on the local change of the transmembrane potential but also on the nature of the transmembrane potential throughout the volume.     

Current injection into a two-dimensional anisotropic bidomain.

A two-dimensional sheet of anisotropic cardiac tissue is represented with the bidomain model, and the finite element method is used to solve the bidomain equations. When the anisotropy ratios of the intracellular and extracellular spaces are not equal, the injection of current into the tissue induces a transmembrane potential that has a complicated spatial dependence, including adjacent regions of depolarized and hyperpolarized tissue. This behavior may have important implications for the electrical stimulation of cardiac tissue and for defibrillation.     

Glutamate inhibitors in the crayfish neuromuscular junction

1. The effects of chlorisondamine and TI-233 on the crayfish neuromuscular junction were investigated in order to compare the action of glutamate with that of the excitatory transmitter. 2. The glutamate-induced synaptic current was inhibited by both of these two drugs. Excitatory junctional potentials were significantly reduced by chlorisondamine, whereas they were increased by TI-233. 3. It is suggested that chlorisondamine and TI-233 are powerful non-competitive antagonists for glutamate. 4. A quantum analysis of extracellular EJPs demonstrated that chlorisondamine did not possess presynaptic action in the crayfish neuromuscular junction. Chlorisondamine shortened the decay phase of extracellular EJPs, and the decay was frequently fitted by a double exponential in relatively low concentrations. 5. Semilogarithmic plots of the decay phase of the glutamate current evoked by a short glutamate pulse were nearly linear, but they shifted from linearity to some extent in the presence of chlorisondamine, showing prolongation of the glutamate current tails. 6. When TI-233 was added to the bathing solution at a concentration of 0.1 mM, the quantum content of extracellular EJPs was increased by about two times, but the average unit size was not changed. 7. There was no change in the rise time and the decay phase of the glutamate potential in the presence of TI-233. 8. Pharmacological difference between glutamate responses and EJPs was revealed in the presence of chlorisondamine and TI-233. Unless this difference can be explicated with a reasonable explanation on the glutamate transmitter hypothesis, it is difficult to confirm that glutamic acid is an excitatory transmitter at the crayfish neuromuscular junction.     

High resolution magnetic images of planar wave fronts reveal bidomain properties of cardiac tissue

We magnetically imaged the magnetic action field and optically imaged the transmembrane potentials generated by planar wavefronts on the surface of the left ventricular wall of Langendorff-perfused isolated rabbit hearts. The magnetic action field images were used to produce a time series of two-dimensional action current maps. Overlaying epifluorescent images allowed us to identify a net current along the wavefront and perpendicular to gradients in the transmembrane potential. This is in contrast to a traditional uniform double-layer model where the net current flows along the gradient in the transmembrane potential. Our findings are supported by numerical simulations that treat cardiac tissue as a bidomain with unequal anisotropies in the intra- and extracellular spaces. Our measurements reveal the anisotropic bidomain nature of cardiac tissue during plane wave propagation. These bidomain effects play an important role in the generation of the whole-heart magnetocardiogram and cannot be ignored.     

Quantal potential fields around individual active zones of amphibian motor-nerve terminals

The release of a quantum from a nerve terminal is accompanied by the flow of extracellular current, which creates a field around the site of transmitter action. We provide a solution for the extent of this field for the case of a quantum released from a site on an amphibian motor-nerve terminal branch onto the receptor patch of a muscle fiber and compare this with measurements of the field using three extracellular electrodes. Numerical solution of the equations for the quantal potential field in cylindrical coordinates show that the density of the field at the peak of the quantal current gives rise to a peak extracellular potential, which declines approximately as the inverse of the distance from the source at distances greater than about 4 microm from the source along the length of the fiber. The peak extracellular potential declines to 20% of its initial value in a distance of about 6 microm, both along the length of the fiber and in the circumferential direction around the fiber. Simultaneous recordings of quantal potential fields, made with three electrodes placed in a line at right angles to an FM1-43 visualized branch, gave determinations of the field strengths in accord with the numerical solutions. In addition, the three electrodes were placed so as to straddle the visualized release sites of a branch. The positions of these sites were correctly predicted on the basis of the theory and independently ascertained by FM1-43 staining of the sites. It is concluded that quantal potential fields at the neuromuscular junction that can be measured with available recording techniques are restricted to regions within about 10 microm of the release site.     

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.     

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.     

An approximate solution to the periodic bidomain equations in one dimension

An approximate, computationally tractable solution is proposed for the potentials in the bidomain model with periodic intracellular junctions (the periodic bidomain model). This new approach is based on the one-dimensional rigorous spectral method described previously by Trayanova and Pilkington (IEEE Trans. Biomed. Eng., May 1993). The total solution to the one-dimensional periodic bidomain problem is decomposed in the spectral domain into solutions to (1) the single-fiber classical bidomain problem in which the intracellular conductivity value incorporates the average contribution from cytoplasm and junction and (2) the “junctional” potential problem due to the presence of junctions at discrete locations alone. Solving for the junctional term rigorously requires most of the numerical effort in the solution for the periodic bidomain potentials. Here the junctional potential is found approximately with little numerical effort. A comparison between the rigorous and the approximate solutions serves as a justification for the proposed approximate solution procedure. The procedure outlined in this paper is applicable to higher spatial dimensions where both tissue anisotropy and junctional inhomogeneities play a role in establishing the transmembrane potential distribution.     

Effect of nonuniform interstitial space properties on impulse propagation: a discrete multidomain model

This work presents a discrete multidomain model that describes ionic diffusion pathways between connected cells and within the interstitium. Unlike classical models of impulse propagation, the intracellular and extracellular spaces are represented as spatially distinct volumes with dynamic/static boundary conditions that electrically couple neighboring spaces. The model is used to investigate the impact of nonuniform geometrical and electrical properties of the interstitial space surrounding a fiber on conduction velocity and action potential waveshape. Comparison of the multidomain and bidomain models shows that although the conduction velocity is relatively insensitive to cases that confine 50% of the membrane surface by narrow extracellular depths (≥2 nm), the action potential morphology varies greatly around the fiber perimeter, resulting in changes in the magnitude of extracellular potential in the tight spaces. Results also show that when the conductivity of the tight spaces is sufficiently reduced, the membrane adjacent to the tight space is eliminated from participating in propagation, and the conduction velocity increases. Owing to its ability to describe the spatial discontinuity of cardiac microstructure, the discrete multidomain can be used to determine appropriate tissue properties for use in classical macroscopic models such as the bidomain during normal and pathophysiological conditions.     

A perturbation solution of the mechanical bidomain model

This research focuses on finding analytical solutions to the mechanical bidomain model for cardiac tissue. In particular, a perturbation expansion is used to analyze the equations, with the perturbation parameter being inversely proportional to the spring constant coupling the intracellular and extracellular spaces. The results indicate that the intracellular and extracellular pressures are not equal and that the two spaces can move relative to each other. This calculation is complicated enough to illustrate the implications of the mechanical bidomain model but is nevertheless simple enough to solve analytically. One application of the calculation is to the mechanical behavior of active cardiac tissue surrounding an ischemic region.     

An Analysis and Comparison of Convergence and Uniqueness of Time-Independent Bone Adaptation Models

A recently presented solution method for the bidomain model (Johnston et al. 2006), which involves the application of direct current for studying electrical potential in a slab of cardiac tissue, is extended here to allow the use of an applied alternating current. The advantage of using AC current, in a four-electrode method for determining cardiac conductivities, is that instead of using ‘close’ and ‘wide’ electrode spacings to make potential measurements, increasing the frequency of the AC current redirects a fraction of the current from the extracellular space into the intracellular space.

The model is based on the work of Le Guyader et al. (2001), but is able to include the effects of the fibre rotation between the epicardium and the endocardium on the potentials. Also, rather than using a full numerical technique, the solution method uses Fourier series and a simple one dimensional finite difference scheme, which has the advantage of allowing the potentials to be calculated only at points, such as the measuring electrodes, where they are required.

The new alternating current model, which includes intracellular capacitance, is used with a particular four-electrode configuration, to show that the potential measured is affected by changes in fibre rotation. This is significant because it indicates that it is necessary to include fibre rotation in models, which are to be used in conjunction with measuring arrays that are more complex than those involving simply surface probes or a single vertical probe.     

A solution method for the determination of cardiac potential distributions with an alternating current source

A recently presented solution method for the bidomain model (Johnston et al. 2006), which involves the application of direct current for studying electrical potential in a slab of cardiac tissue, is extended here to allow the use of an applied alternating current. The advantage of using AC current, in a four-electrode method for determining cardiac conductivities, is that instead of using 'close' and 'wide' electrode spacings to make potential measurements, increasing the frequency of the AC current redirects a fraction of the current from the extracellular space into the intracellular space. The model is based on the work of Le Guyader et al. (2001), but is able to include the effects of the fibre rotation between the epicardium and the endocardium on the potentials. Also, rather than using a full numerical technique, the solution method uses Fourier series and a simple one dimensional finite difference scheme, which has the advantage of allowing the potentials to be calculated only at points, such as the measuring electrodes, where they are required. The new alternating current model, which includes intracellular capacitance, is used with a particular four-electrode configuration, to show that the potential measured is affected by changes in fibre rotation. This is significant because it indicates that it is necessary to include fibre rotation in models, which are to be used in conjunction with measuring arrays that are more complex than those involving simply surface probes or a single vertical probe.     

Quantal current fields around individual boutons in sympathetic ganglia.

The release of a quantum of neurotransmitter from an active zone of a bouton is accompanied by the flow of extracellular current that creates a potential field about the site of transmitter action beneath the bouton. It is shown theoretically that the density of the field at the peak of the quantal current gives rise to an extracellular potential that declines to values of less than 5 microV at 1.3 microm distance in the circumferential direction around the neuron and equally rapidly in the radial direction away from the neuron. A loose-patch electrode placed over a bouton distorts the quantal field about the bouton and calculations show that under current-clamp conditions, potentials of over 40 microV can be recorded with an electrode of tip diameter 2 microm, provided the separation between the tip and the neuron's surface is about 0.1 microm. Quantal release recorded from visualized boutons on rat monopolar pelvic ganglion cells with loose-patch electrodes is in agreement with the properties of the quantal potential field given in the theoretical analysis.     

An Extended Bidomain Framework Incorporating Multiple Cell Types

The muscular layers within the walls of the gastrointestinal tract contain two distinct cell types, the interstitial cells of Cajal and smooth muscle cells, which together produce rhythmic depolarizations known as slow waves. The bidomain model of tissue-level electrical activity consists of single intracellular and extracellular domains separated by an intervening membrane at all points in space and is therefore unable to adequately describe the presence of two distinct cell types in its conventional form. Here, an extension to the bidomain framework is presented whereby multiple interconnected cell types can be incorporated. Although the derivation is focused on the interactions of the interstitial cells of Cajal and smooth muscle cells, the conceptual framework can be more generally applied. Simulations demonstrating the feasibility of the proposed model are also presented.     

Identification of transmitter release sites in the motor nerve terminal

A model was produced of generation of postsynaptic current following release of a quantum of neurotransmitter from the nerve ending, whereby the law of current density attenuation is defined as j=I/r^b (A), where I is current density at the generation site and j stands at distance r from that site. Coefficient b was shown experimentally to be close to 1 using extracellular techniques of signal recording. Assuming that sites of signal generation and transmitter release are spatially identical, a new technique for determining the coordinates of the transmitter release site in the motor nerve terminal is suggested. This consists of measuring uniquantal signal amplitude by means of three extracellular microelectrodes spaced 5–10 µm apart. We were able to establish, by producing "spatial pictures" of transmitter release based on analysis of several hundred signals in the frog cutaneous pectoris muscle, that release sites are arranged in groups running diagonally to the nerve ending. These groups are thought to reflect transmitter release in active zones of the nerve ending. Advantages, disadvantages, and inaccuracies of the method are identified.S. V. Kurashov Medical Institute, Ministry of Public Health of the RSFSR, Moscow. V. I. Ulyanov-Lenin University, Kazan'. Translated from Neirofiziologiya, Vol. 22, No. 3, pp. 309–318, May–June, 1990.     

Electric and magnetic fields from two-dimensional anisotropic bisyncytia.

Cardiac tissue can be considered macroscopically as a bidomain, anisotropic conductor in which simple depolarization wavefronts produce complex current distributions. Since such distributions may be difficult to measure using electrical techniques, we have developed a mathematical model to determine the feasibility of magnetic localization of these currents. By applying the finite element method to an idealized two-dimensional bisyncytium with anisotropic conductivities, we have calculated the intracellular and extracellular potentials, the current distributions, and the magnetic fields for a circular depolarization wavefront. The calculated magnetic field 1 mm from the tissue is well within the sensitivity of a SQUID magnetometer. Our results show that complex bisyncytial current patterns can be studied magnetically, and these studies should provide valuable insight regarding the electrical anisotropy of cardiac tissue.     

A sensitivity study of conductivity values in the passive bidomain equation

There is a complex interplay between the four conductivity values used in the bidomain equation and the resulting electric potential distribution in cardiac tissue arising from subendocardial ischaemia. Based on the three commonly used experimentally derived conductivity data sets, a non-dimensional formulation of the passive bidomain equation is derived, which gives rise naturally to several dimensionless conductivity ratios. The data sets are then used to define a parameter space of these ratios, which is studied by considering the correlation coefficients between different epicardial potential distributions.From this study, it is shown that the ratio of the intracellular longitudinal conductivity to the intracellular transverse conductivity is the key parameter in explaining the differences between the epicardial potential distributions observed with these three data sets.     

Probabilistic secretion of quanta and the synaptosecretosome hypothesis: evoked release at active zones of varicosities, boutons, and endplates.

A quantum of transmitter may be released upon the arrival of a nerve impulse if the influx of calcium ions through a nearby voltage-dependent calcium channel is sufficient to activate the vesicle-associated calcium sensor protein that triggers exocytosis. A synaptic vesicle, together with its calcium sensor protein, is often found complexed with the calcium channel in active zones to form what will be called a "synaptosecretosome." In the present work, a stochastic analysis is given of the conditions under which a quantum is released from the synaptosecretosome by a nerve impulse. The theoretical treatment considers the rise of calcium at the synaptosecretosome after the stochastic opening of a calcium channel at some time during the impulse, followed by the stochastic binding of calcium to the vesicle-associated protein and the probability of this leading to exocytosis. This allows determination of the probabilities that an impulse will release 0, 1, 2,... quanta from an active zone, whether this is in a varicosity, a bouton, or a motor endplate. A number of experimental observations of the release of transmitter at the active zones of sympathetic varicosities and boutons as well as somatic motor endplates are described by this analysis. These include the likelihood of the secretion of only one quantum at an active zone of endplates and of more than one quantum at an active zone of a sympathetic varicosity. The fourth-power relationship between the probability of transmitter release at the active zones of sympathetic varicosities and motor endplates and the external calcium concentration is also explained by this approach. So, too, is the fact that the time course of the increased rate of quantal secretion from a somatic active zone after an impulse is invariant with changes in the amount of calcium that enters through its calcium channel, whether due to changes consequent on the actions of autoreceptor agents such as adenosine or to facilitation. The increased probability of quantal release that occurs during F1 facilitation at the active zones of motor endplates and sympathetic boutons is predicted by the residual binding of calcium to a high-affinity site on the vesicle-associated protein. The concept of the stochastic operation of a synaptosecretosome can accommodate most phenomena involving the release of transmitter quanta at these synapses.     

The mechanism of β-bungarotoxin action. I. Modification of transmitter release at the neuromuscular junction

The protein, β-bungarotoxin, a presynaptic neurotoxin isolated from the venom of the snake Bungarus multicinctus, is known to inhibit mitochondrial function. Within 30 min after adding the toxin to a rat diaphragmphrenic nerve preparation, the quantal content increased tenfold and the frequency of miniature endplate potentials increased fourfold. No increase in miniature endplate potential frequency was seen in the absence of extracellular calcium. Since mitochondria may be involved in regulating intracellular calcium levels, the rate at which the transmitter release is turned off was studied by measuring delayed release in the presence and absence of toxin. Delayed release is elevated about eightfold by the toxin. If delayed release is due to residual calcium, as has been hypothesized, these data may be explained if the toxin does not alter the amount of calcium which enters the terminal, but rather the rate at which that calcium is removed. Alternatively, a calcium-dependent modification of the release process itself might be produced. The eventual reduction in transmitter output did not appear to result from depletion of the terminal of releaseable packets of transmitter, but does require extracellular calcium.