Simulation of propagation along an isolated skeletal muscle fiber in an isotropic volume conductor.

This paper describes a model of the frog skeletal muscle fiber that includes the effects of the transverse tubular system (T system) on propagation. Uniform propagation on an isolated fiber suspended in Ringer's solution or in air is simulated by placing the cylindrical fiber model in a concentric three-dimensional isotropic volume conductor. The current through the T system outlets at the sarcolemmal surface is comparable in magnitude to the sarcolemmal current density, but is of opposite polarity. When it is added to the sarcolemmal current, the resulting triphasic waveform has a 100% increase in the leading positive peak, a 50% reduction in the negative peak, and more than 60% reduction in the trailing positive peak. As a result the tubular output current causes a reduction in the conduction velocity, a decrease in the maximum rate of rise of the action potential. and an important modification of the extracellular potential. Compared to an isolated fiber in a large volume of Ringer's solution, uniform propagation within a 2-micron-thick volume conductor annulus is slowed down from 1.92 to 0.72 m/s, and the extracellular potential is increased from 1 to 108 mV peak to peak, in agreement with published experimental measurements.

Simulation of propagation in a bundle of skeletal muscle fibers: modulation effects of passive fibers.

Computer simulations are used to study passive fiber modulation of propagation in a tightly packed bundle of frog skeletal muscle fibers (uniform fiber radius of 50 microns). With T = 20 degrees C and a uniform nominal interstitial cleft width d = 0.35 microns, about 92% of the active fiber source current (Ima) enters the passive tissue as a radial load current (Iep) while the rest flows longitudinally in the cleft between the active and adjacent passive fibers. The conduction velocity of 1.32 m/s was about 30% lower than on an isolated fiber in a Ringer bath, in close agreement with experimental results. The peak-to-peak interstitial potential (phi epp) at the active fiber surface was 38 mV, compared to 1.3 mV for the isolated fiber. A uniform increase of d from 0.35 to 1.2 microns decreased phi epp from 38 to 25 mV, increased the velocity from 1.32 to 1.54 m/s, and decreased the maximum rate of rise of the action potential upstroke (Vmax) from 512 to 503 V/s. Increasing the phase angle of the passive fiber membrane impedence (Zm) increases the phase delay between lma and lep thereby increasing phi epp which in turn slows down propagation and increases V max.

Propagation on a central fiber surrounded by inactive fibers in a multifibered bundle model.

We studied uniform propagation on a central active fiber surrounded by inactive fibers in a multifibered bundle model lying in a large volume conductor. The behavior of a fully active bundle is considered in a companion paper. The bundle is formed by concentric layers of small cylindrical fibers (radius 5 microns), with a uniform minimum distance (d) between any two adjacent fibers, to yield a bundle radius of about 72 microns. Individual fibers are identical continuous cables of excitable membrane based on a modified Beeler-Reuter model. The intracellular volume fraction (fi) increases to a maximum of about 90% as d is reduced and remains unchanged for d < 0.01 micron. In the range of d < 0.01 micron, the central fiber is effectively shielded from external effects by the first concentric layer of inactive fibers, and a large capacitive load current flows across the surrounding inactive membranes. In addition, the fiber proximity produces a circumferentially nonuniform current density (proximity effect) that is equivalent to an increased average longitudinal interstitial resistance. The conduction velocity is reduced as d becomes smaller in the range of d < 0.1 micron, the interstitial potential becomes larger, and both the maximum rate of rise and time constant of the foot of the upstroke are increased. On the other hand, for d > 0.1 micron, there are negligible changes in the shape of the upstroke, and the behavior of the central fiber is close to that of a uniform cable in a restricted volume conductor. For d larger than about 1.2 microns, the active fiber environment is close to an unbounded isotropic volume conductor.

Interactions between adjacent fibers in a cardiac muscle bundle.

A strand of cardiac muscle was modeled as a small bundle of individual fibers surrounded by a large volume conductor. The bundle is a uniform assembly of small identical cylindrical fibers, arranged as a series of concentric layers, and its behavior is examined in the presence (coupled bundle) or absence (uncoupled bundle) of transverse resistive coupling between adjacent fibers. Individual fibers are continuous cables of excitable membrane, with circumferential segmentation into 12 equal patches to make the membrane potential changes dependent upon the local interstitial potential. The minimum spacing (d) between adjacent fibers is used to modify the interstitial microstructural organization and the intracellular volume fraction (fi). When d is small enough (d < 0.01 micron), fi remains unchanged at its maximum of about 90%, the interstitial potential is large, the transverse interstitial resistance is high, and the proximity effect arising from the close juxtaposition of adjacent fibers is important. A surface fiber of the uncoupled bundle exhibits little sensitivity to changes in the interstitial microstructure, owing to the dominant influence of the external volume conductor, whereas the central fiber shows a large decrease in velocity, substantial waveshape modifications, and a large increase in interstitial potential as d is reduced. In the coupled bundle, all fibers adopt the same velocity during uniform propagation, owing to the strong transverse resistive coupling; when d is reduced in the range of d < 0.01 micron, the velocity and interstitial potential changes are less pronounced than in the uncoupled bundle. When d is large enough (d > 0.01 micron), the bundle behavior (coupled and uncoupled) approaches that obtained with a bidomain formulation.

Simulation of two-dimensional anisotropic cardiac reentry: effects of the wavelength on the reentry characteristics.

A two-dimensional sheet model was used to study the dynamics of reentry around a zone of functional block. The sheet is a set of parallel, continuous, and uniform cables, transversely interconnected by a brick-wall arrangement of fixed resistors. In accord with experimental observations on cardiac tissue, longitudinal propagation is continuous, whereas transverse propagation exhibits discontinuous features. The width and length of the sheet are 1.5 and 5 cm, respectively, and the anisotropy ratio is fixed at approximately 4:1. The membrane model is a modified Beeler-Reuter formulation incorporating faster sodium current dynamics. We fixed the basic wavelength and action potential duration of the propagating impulse by dividing the time constants of the secondary inward current by an integer K. Reentry was initiated by a standard cross-shock protocol, and the rotating activity appeared as curling patterns around the point of junction (the q-point) of the activation (A) and recovery (R) fronts. The curling R front always precedes the A front and is separated from it by the excitable gap. In addition, the R front is occasionally shifted abruptly through a merging with a slow-moving triggered secondary recovery front that is dissociated from the A front and q-point. Sustained irregular reentry associated with substantial excitable gap variations was simulated with short wavelengths (K = 8 and K = 4). Unsustained reentry was obtained with a longer wavelength (K = 2), leading to a breakup of the q-point locus and the triggering of new activation fronts.

The dynamics of sustained reentry in a ring model of cardiac tissue.

This paper describes the dynamics of circus movement around a fixed obstacle, using a one-dimensional continuous and uniform ring model of cardiac tissue to simulate sustained reentry. The membrane ionic current is simulated by a modified Beeler-Reuter formulation in which the kinetics of the fast sodium current were updated using more recent voltage-clamp data. Changes in the ring length are used to modify the dynamics of reentry. Reentry is stable if the ring length (X) exceeds a critical value (Xcrit) and complete block occurs if X is below a minimum (Xmin). Irregular sustained reentry is observed at intermediate ring lengths, as a narrow range of aperiodic reentry near Xcrit, and a larger range of quasi-periodic reentry at shorter ring lengths. The basic pattern of irregular reentry is an alternation between long and short cycle length, action potential duration (APD), diastolic interval (DIA), wavelength, and excitable gap. In aperiodic reentry cycle length variations are small, APD and DIA fluctuations are of medium amplitude, and conduction velocity over the whole pathway is essentially constant during successive turns. Much larger fluctuations in these various quantities occur during quasi-periodic reentry, and they increase in size as X approaches Xmin. The complexity of quasi-periodic reentry patterns is related to three factors: the slope of the APD versus DIA relation, which is greater than 1, the existence of a zone of slow conduction on the ring when the excitable gap becomes quite short, and the occurrence of triggered waves of secondary repolarization and excitability recovery. In the present model, quasi-periodic reentry with triggered secondary recovery covers most of the range of ring lengths, giving rise to sustained irregular reentry. There is very close agreement between our simulation results and experimental data obtained on rings of cardiac tissue.

Excitability and repolarization in an ionic model of the cardiac cell membrane.

The main objective of this study was to investigate the possibility of expressing the activation and repolarization processes of a realistic ionic model of the myocyte membrane in terms of simplified dynamic equivalents. The modified Beeler-Reuter model (MBR) of the ventricular membrane was selected for this purpose because its action potential upstroke, plateau and selected for this purpose because its action potential upstroke, plateau and repolarization phase occur along sufficiently well separated timescales. The information on the MBR model dynamics was obtained by premature stimulation at various coupling intervals, under stable conditions of regular pacing at different cycle lengths. A general method was developed to study the threshold behavior of the system. As a first step, a pair of complementary threshold criteria was defined in terms of peak ionic current and time to repolarization in order to reliably distinguish between classes of sub-threshold and supra-threshold responses. Of the main conclusions is that the activation of the MBR model by short-duration stimuli (< 5 msec) can be accurately represented by a one-variable or a two-variable dynamic equivalent. In addition, because of the large surge of Na+ current at threshold, the recovery of excitability is essentially independent of the conditioning action potential waveform (no threshold-memory effect). Another major result pertains to the higher complexity of the repolarization process, stressing the critical role played by the activation dynamics of the secondary inward current. There is a substantial dependence of the action potential duration (APD) on the conditioning action potential waveform (APD-memory effect), and at least a three-variable model is necessary for a reasonable approximation.

Analysis of an iterative difference equation model of the cardiac cell membrane.

An iterative difference equation (DE) model based on a small set of functional characteristics derived from the modified Beeler-Reuter ionic model (MBR) of a space-clamped membrane of the ventricular myocyte was studied. These characteristics were expressed as functions of a common independent variable, the diastolic interval (DIA), to define threshold, Thr (DIA), latency, L (DIA), action potential duration, APD (DIA), and delayed excitability recovery, R(DIA), owing to a sub-threshold response. Memory effects, in the form of a dependence on the immediately proceeding action potential waveform, were not included in the current DE formulation. The entrainment bifurcation structure of the DE model was found to be very sensitive to the form of APD (DIA) but, with a suitable choice of parameter values, the model reproduced the successive stable rhythms of synchronization (1:1-->2:2-->2:1-->4:2) found in the MBR model with decreasing basic cycle lengths. The fine details of the transition between stable rhythms were seen to depend on the specific forms of APD (DIA) and R (DIA) chosen, and on the initial conditions at the onset of pacing. The minor deficiencies of the DE R (DIA) chosen, and on the initial conditions at the onset of pacing. The minor deficiencies of the DE model are attributable to the memory-free APD (DIA) used, and to uncertainties in the description of the MBR model behavior owing to the slow convergence toward a given stable rhythm. Overall, the iterative DE formulation based on the transient properties of an ionic model, as derived by premature stimulation, it is also adequate to describe the stable entrainment patterns of the ionic model.

Modeling a non-inactivating delayed rectifier cardiac current using voltage clamp data.

This paper describes a new parameter estimation method applicable to experimental voltage-clamp records. The method is based on the Hodgkin-Huxley (HH) representation of a generic non-inactivating delayed rectifier current (IK) which can be assimilated to the delayed rectifier potassium current of cardiac cells. The model involves a single gating variable of activation (chi) of degree (lambda chi). Its parameters include the voltage-dependent steady-state characteristic (chi infinity), time constant tau chi, the degree lambda chi as a positive integer, and the maximal conductance gK. The method is based on linear optimization. It implements a series of least-squares minimization steps to calculate a first estimate of each model parameter, followed by global minimization to obtain final estimates. The required data, in the form of ionic current responses, correspond to standard voltage-clamp protocols. The effects of noise are minimized by avoiding the use of the time derivative of IK in the calculations. Simulated voltage-clamp data using either a HH model or a five-state Markov chain (MC) model served two purposes: (i) to test the performance of the HH parameter estimation method, and (ii) to study the suitability of the HH model to reproduce data generated by models other than HH. A nominal MC model was obtained by fitting its current responses to those of the HH model. Rate constants of the nominal MC model were then modified and voltage-clamp current responses were generated. Excellent results were obtained with HH and nominal MC data. Data sets generated by a 20% change in the rate constants of the nominal MC model showed that the closed-state rate constants have only a limited influence on the HH parameter estimates, whereas changes in the closed-to-open rate constants produce substantial effects. Nevertheless, a given MC data set can be fitted quite closely by a HH model. In the light of these simulation results it is indicated that an hybrid HH-MC representation of IK data would be more flexible than a straight HH model by removing some of the constraints between the rate constants, and less cumbersome than a straight MC model by substantially reducing the number of parameters to be estimated.

A model study of extracellular stimulation of cardiac cells.

Point source extracellular stimulation of a myocyte model was used to study the efficacy of excitation of cardiac cells, taking into account the shape of the pulse stimulus and its time of application in the cardiac cycle. The myocyte was modeled as a small cylinder of membrane (10 microns in diameter and 100 microns in length) capped at both ends and placed in an unbounded volume conductor. A Beeler-Reuter model modified for the Na+ dynamics served to simulate the membrane ionic current. The stimulus source was located on the cylinder axis, close to the myocyte (50 microns) in order to generate a nonlinear extracellular field (phi e). The low membrane impedance associated with the high frequency component of the make and break of the rectangular current pulse leads to a current flow across the membrane and an abrupt change in intracellular potential (phi i). Because the intracellular space is very small, phi i is nearly uniform over the length of the myocyte and the membrane potential (V = phi i-phi e) is governed by the applied field phi e. There is then a longitudinal gradient of membrane polarization which is the inverse of the gradient of extracellular potential. With an anodal (positive) pulse, for instance, the proximal portion of the myocyte is hyperpolarized and the distal portion is depolarized. Based on this principle and considering the voltage-dependent activation/inactivation dynamics of the membrane, it is shown that a cathodal (negative) pulse is the most efficacious stimulus at diastolic potentials, an anodal current is preferable during the plateau phase of the action potential, and a biphasic pulse is optimal during the relative refractory phase. Thus a biphasic pulse would constitute the best choice for maximum efficacy at all phases of the action potential.

Estimation of the steady-state characteristics of the Hodgkin-Huxley model from voltage-clamp data.

In a companion paper in this issue we show that all the parameters and functions of the Hodgkin-Huxley (HH) model can be calculated in a unique and optimal manner from voltage-clamp peak current data when the steady-state activation (x infinite) and inactivation (z infinite) characteristics are known. Assuming that x infinite and z infinite can be adequately expressed by a Boltzmann equation with two parameters, the present paper describes an optimization procedure to estimate these parameters from peak current data without any constraint on the time constants of activation and inactivation. The required voltage-clamp data are the peak ionic current value (Ip) and its time of occurrence (tp), as provided by two complementary voltage-clamp protocols involving, in each case, a single fixed value of clamp potential. The performance of the procedure was very good with simulated medium- or high-resolution data as it was then possible to determine with confidence the degrees of the gating variables. The performance was also very good with low-resolution data, provided that the degrees of the gating variables were chosen correctly. Good results were also obtained in the presence of Gaussian noise. On the other hand, estimates of x infinite and z infinite based on normalization of peak current measurements always give uncertain results that are likely to be incorrect in a number of circumstances. It is concluded that the HH model can be a useful tool for the interpretation of voltage-clamp peak current data when a reasonable database is available.

On the interpretation of voltage-clamp data using the Hodgkin-Huxley model.

This paper describes a method to extract membrane model parameters from experimental voltage-clamp records. The underlying theory is based on two premises: (1) the membrane dynamics can be described by a Hodgkin-Huxley (HH) model, and (2) the most reliable data provided by voltage clamp experiments are peak current (Ip) measurements. First, the steady-state characteristics of activation (x infinity) and inactivation (z infinity) must be estimated, and it is shown that Ip data provided by standard voltage-clamp stimulation protocols are sufficient for this purpose for the case of well-separated activation (tau x) and inactivation (tau z) time constants, tau x << tau z. Next, we propose a test (R test) to establish the suitability of the HH model to represent the data. When the HH model is applicable (successful R test), the procedure yields the degree of the gating variables, a range of maximum membrane conductance (g) values, and a tau x/tau z ratio that relates x infinity and z infinity to the Ip data. When additional information is available, such as the time of occurrence of Ip or an estimate of tau z from the late portion of the ionic current response, one can narrow down the value of g and estimate all the HH parameters and functions. Otherwise, when the R test is not successful, one can conclude that x infinity and z infinity have been incorrectly estimated because tau x and tau z are not sufficiently separated or that the HH model is not applicable to the data.

A model study of electric field interactions between cardiac myocytes.

The transmission of excitation via electric field coupling was studied in a model comprising two myocytes abutted end-to-end and placed in an unbounded volume conductor. Each myocyte was modeled as a small cylinder of membrane (10 microns in diameter and 100 microns in length) capped at both ends. A Beeler-Reuter model modified for the Na+ current dynamics served to simulate the membrane ionic current. There was no resistive coupling between the myocytes and the intercellular junction consisted of closely apposed pre- and post-junctional membranes, separated by a uniform cleft distance. The membrane current crossing the prejunctional membrane during the action potential upstroke tends to flow out of the cleft, but it is partly prevented from doing so by the shunt resistance constituted by the cleft volume conductor. The prejunctional upstroke gives rise to a pulse of positive potential within the cleft which induces a small capacitive current across the post-junctional membrane to yield a small positive change in the intracellular potential in the post-junctional cell. The net result is an hyperpolarization of the post-junctional cleft membrane and a slight depolarization of the rest of the cell membrane since the extracellular potential outside of the cell is zero. The magnitude of this depolarization is quite small for a flat junctional membrane and it can be increased by membrane folding and interdigitation, so as to increase the junctional membrane area by a factor of 10 or more. Even then the post-junctional depolarization does not reach threshold when the extracellular potential around the post-junctional cell is effectively zero. Threshold depolarization occurs in the presence of a large decrease of post-junctional load, by increasing the junctional membrane capacitance and/or decreasing the volume of the post-junctional cell. Assuming that the normal resistive coupling between two cardiac myocytes is 1-4 M omega, our model study indicates that electric field coupling would then be about two orders of magnitude smaller. However, substantial enhancement of the efficacy of electric field transmission was observed in the case of cells with substantial junctional membrane folding.

Modeling the dynamic features of the electrogenic Na,K pump of cardiac cells.

The purpose of this paper is to examine the dynamic features of the electrogenic Na,K pump of cardiac cells, based on a comparative analysis of a mechanistic model and an ad hoc mathematical description of the Na,K pump. Both representations are incorporated into a modified version of the Beeler-Reuter model for the ventricular membrane, and the resulting action potential models are studied under conditions of repetitive stimulation at steady rates between 0 and 3 Hz. The two Na,K pump representations have nearly identical steady-state characteristics of sensitivity to internal Na+ concentration, external K+ concentration, and membrane potential. Rapid voltage-dependent transient pump currents are present in the mechanistic model, while they are absent in the ad hoc mathematical description we used. The stimulation results show that a sizable peak of pump current caused by the action potential upstroke in the mechanistic model affects phase 1 repolarization, and that this effect is relatively independent of the stimulation rate. The pump current generated by our ad hoc mathematical description is constant during the action potential and does not affect directly the repolarization time course. While the two Na,K pump models show similar pumping efficiency at low stimulation rates, the mechanistic pump is more efficient at high rates of activity. In essence, the distinctive features of the mechanistic model are due to an energy barrier expressing the voltage dependence of the translocation step of the mechanism, and to the redistribution of the intermediates of the biochemical reactions during activity. In comparison, the ad hoc mathematical description exhibits a fixed dependence of the pump current on voltage and ionic concentrations.

Propagation of activation in cardiac muscle.

The hypothesis of local circuit current flow underlying propagation of activation in cardiac muscle has been extensively documented by one-dimensional and two-dimensional simulation studies. The assumptions of spatially uniform membrane capacitance and membrane ionic properties yield simulation results that are in good agreement with experimental observations in healthy cardiac muscle, thereby indicating that differences in propagation velocity and action potential upstroke between longitudinal and transverse directions can be explained solely on the basis of anisotropic intercellular coupling. Two-dimensional model studies of anisotropic propagation have also stressed the more efficient charging of the membrane capacitance and higher safety factor of propagation in the transverse direction. These conditions favor the occurrence of longitudinal unidirectional block and the initiation of reentry via transverse propagation. The authors simulated rotating waves initiated by properly phased transverse and longitudinal plane waves in a two-dimensional sheet model. Sustained propagation requires a minimum anisotropy ratio, corresponding to a velocity ratio of about 4:1. It was found, for uniform anisotropy, that the central focus wandered slightly. A higher anisotropy ratio favors a more stable rotating pattern and a more restricted movement of the central focus.

Structural complexity effects on transverse propagation in a two-dimensional model of myocardium.

A thin sheet of cardiac tissue was modeled as a set of resistively coupled excitable cables with membrane dynamics described by the modified Beeler Reuter model. Transverse connections have a resistance Rn and are regularly distributed with a spacing delta on any given cable, to provide alternating input and output junctions. Flat wave longitudinal propagation corresponds to propagation along a single continuous cable since all units of the network are functionally isolated due to the absence of transverse current flow. Events on a given cable during flat transverse propagation include electrotonic spread of potential from input to output junctions, action potential initiation at input junctions, and collision at output junctions. The propagating two-dimensional transverse wavefront is an undulating transmembrane potential surface with highs at the input junctions and lows at the output junctions. The action potential upstroke is also modulated in a periodic manner with minimum and maximum Vmax at the input and output junctions respectively. Thus, the network is capable of a diversity of dynamic behavior spatially distributed in relation to the specific pattern of transverse connections chosen. Overall, the behavior of the network model is in good agreement with available structural and electrophysiological data on myocardium. In addition, this network topology allows to handle more easily parameters governing propagation and to avoid very large matrices which are costly in computational effort and overall computer time.

Directional characteristics of action potential propagation in cardiac muscle. A model study.

Propagation of an elliptic excitation wave front was studied in a two-dimensional model of a thin sheet of cardiac muscle. The sheet model of 2.5 x 10 mm consisted of a set of 100 parallel cables coupled through a regular array of identical transverse resistors. The membrane dynamics was represented by a modified Beeler-Reuter model. We defined the charging factor (CF) to represent by a single number the proportion of input current used to charge the membrane locally below threshold and showed that CF is inversely correlated with the time constant of the foot of the action potential (tau foot) during propagation on a cable. A safety factor of propagation (SF) was also defined for the upstroke of the action potential, with SF directly correlated with the maximum rate of depolarization (Vmax) and, for cablelike propagation, with propagation velocity. Propagation along the principal longitudinal axis of the elliptic wave front is cablelike but, in comparison with a flat wave front, transverse current flow provides a drag effect that somewhat reduces the propagation velocity, Vmax, SF, and CF. With a longitudinal-to-transverse velocity ratio of 3:1 or more, the wave front propagating along the principal transverse axis is essentially flat and is characterized by multiple collisions between successive pairs of input junctions on a given cable; Vmax, SF, and CF are larger than for longitudinal propagation, but CF is no longer correlated with tau foot. There are transient increases in propagation velocity and Vmax with distance from the stimulation site along both principal axes until stablized values are achieved, and a similar transient decrease in tau foot. Away from the principal axes, the action potential characteristics change progressively along the elliptic wave front.(ABSTRACT TRUNCATED AT 250 WORDS)

A model study of stability and oscillations in the myocardial cell membrane.

As a step towards an improved understanding of cardiac arrhythmias caused by abnormal automaticity, we perform a stability analysis of a Hodgkin-Huxley model of the myocardial cell membrane (modified Beeler-Reuter, MBR). The bifurcation structure of the model is obtained as a function of three parameters: the intensity of an applied constant current; the potassium equilibrium potential representing the accumulation of K+ ions in the external medium; and the maximum conductance of the slow inward current mimicking the local application of catecholamines on the membrane. For a range of parameter values, the model exhibits either stable automaticity or bistability between two quiescent states or between a quiescent state and an oscillatory state. These transformations of the bifurcation structure are shown to depend on the interrelationship between three elements: the activation of the slow inward current, the region of high slope conductance of the time-independent potassium current functions, and the slow variables controlling the activation of the potassium current and the inactivation of the slow inward current. Reduced two- and three-dimensional models are shown to reproduce the main stability properties of the full MBR model and to facilitate the understanding of its dynamic behavior. The onset of instability and the oscillatory features of the MBR model are in good agreement with relevant experimental results, and possible sources of disagreement on certain points are discussed.

Estimation of fractional changes in peak gNa, -gNa, ENa, and h infinity (V) of cardiac cells from Vmax of the propagating action potential.

Fractional changes in the peak sodium conductances of the cardiac cell membrane during the action potential are often estimated from fractional changes in Vmax. The present model study shows, in reasonable accord with experimental evidence, that this approach is valid for propagating action potentials provided that the membrane capacitance does not change and that the nonsodium current is small at the time of Vmax. When the maximum conductance of the sodium channel (gNa) and the sodium equilibrium potential (ENa) are varied independently of one another, fractional changes in either of them can be predicted from fractional changes in Vmax if a reasonable estimate of the initial value of ENa is available. Manipulations which modify the resting membrane potential without changing gNa allow to calculate fractional changes in the steady-state Na+ inactivation [h infinity (V)] when ENa is known. Simulation runs were carried out for a continuous cable and a discontinuous cable with either a low (1 omega.cm2) or a high (10 omega.cm2) junctional resistance. The predictions of the model are valid in the discontinuous cable provided that the recording point remains strictly the same throughout the series of measurements. Because the high-resistance discontinuous cable provides conditions which reduce further the nonsodium current at the time of Vmax, the accuracy of the predictions are better in this case. It is concluded that properly designed experimental approaches based on Vmax measurements can yield important information on manipulations affecting gNa, ENa, and h infinity (V) during propagation, and that a better accuracy is possible in cardiac muscle when measurements are made during transverse propagation.

Simulation and experimental studies of the factors influencing the frequency spectrum of cardiac extracellular waveforms.

Spectral analysis of electrocardiographic signals has been proposed as a tool to detect features reflecting cardiac diseases, such as ventricular hypertrophy, myocardial infarction, and a predisposition to sustained ventricular tachycardia. The lack of a theoretical basis to address this question prompted the authors to undertake a simulation study using a bidomain volume conductor model of a strip of cardiac tissue, combined with Fourier analysis, and electrograms recorded from an isolated right atrial canine preparation. In the crista terminalis, the bandwidth of the normal electrogram was 840 +/- 200 Hz (mean +/- SD) during longitudinal propagation and 660 +/- 370 Hz during transverse propagation. During premature stimulation, signal bandwidth and propagation velocity increase with the coupling interval. In the model, a linear combination of Vmax and propagation velocity values allows simulation of the various features of premature excitation. Vmax is the major determinant of the high-frequency content of the signal. An important decrease in the high-frequency content of electrograms occurs when the recording electrode is moved away from the preparation or the simulation model; at distances larger than 1-5 mm, the bandwidth levels off to a value of 50-120 Hz. Partial blockade of axial current flow in the direction of propagation due to microscopic discontinuities and variable activation delays at these discontinuities may be the cause of fragmented activity in necrotic myocardium, which is associated with a reduced bandwidth. Thus, short- and long-term effects of ischemia followed by infarction, such as decreased propagation velocity, decreased action potential upstroke, and fragmentation, tend to decrease the electrocardiographic bandwidth.