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The Restitution Portrait. Introduction: Electrical restitution, relating action potential duration (APD) to diastolic interval (DI), was believed to determine the stability of heart rhythm. However, recent studies demonstrate that stability also depends on long-term APD changes caused by memory. This study presents a new method for investigation of rate- and memory-dependent aspects of restitution and for assessment of mapping models of APD. Methods and Results: Bullfrog ventricular myocardium was paced with a 'perturbed downsweep protocol.' Starting from a basic cycle length (BCL) of 1,000 ms, the tissue was paced until steady state was achieved, followed by single beats of longer and shorter cycle lengths. BCL was decreased by 50 to 100 ms and the process repeated. All APDs were plotted as a function of the preceding DI, which allowed simultaneous observation of dynamic, S1-S2, and two constant-BCL restitution curves in a 'restitution portrait.' Responses were classified as 1:1 (stimulus:response), transient 2:2, or persistent 2:2 (alternans) and were related to the slopes of the restitution curves. None of these slopes approached unity for the persistent 2:2 response, demonstrating that the traditional restitution condition does not predict alternans. The restitution portrait was used to evaluate three mapping models of APD. The models with no memory and with one-beat memory did not produce restitution portraits similar to the experimental one. A model with two-beat memory produced a qualitatively similar portrait. Conclusion: The restitution portrait allows a more comprehensive assessment of cardiac dynamics than methods used to date. Further study of models with memory may result in a clinical criterion for electrical instability. (J Cardiovasc Electrophysiol, Vol. 15, pp. 1-12, June 2004) [ABSTRACT FROM AUTHOR]

We investigate, both experimentally and theoretically, the period-doubling bifurcation to alternans in heart tissue. Previously, this phenomenon has been modeled with either smooth or border-collision dynamics. Using a modification of existing experimental techniques, we find a hybrid behavior: Very close to the bifurcation point, the dynamics is smooth, whereas further away it is border-collision-like. The essence of this behavior is captured by a model that exhibits what we call an unfolded border-collision bifurcation. This new model elucidates that, in an experiment, where only a limited number of data points can be measured, the smooth behavior of the bifurcation can easily be missed.

Spatial heterogeneity of repolarization can provide a substrate for reentry to occur in myocardium. This heterogeneity may result from spatial differences in action potential duration (APD) restitution. The restitution portrait (RP) measures many aspects of rate-dependent restitution: the dynamic restitution curve (RC), S1-S2 RC, and short-term memory response. We used the RP to characterize epicardial patterns of spatial heterogeneity of restitution that were repeatable across animals. New Zealand White rabbit ventricles were paced from the epicardial apex, midventricle, or base, and optical action potentials were recorded from the same three regions. A perturbed downsweep pacing protocol was applied that measured the RP over a range of cycle lengths from 1,000 to 140 ms. The time constant of short-term memory measured close to the stimulus was dependent on location. In the midventricle the mean time constant was 19.1 +/- 1.1 s, but it was 39% longer at the apex (P < 0.01) and 23% longer at the base (P = 0.03). The S1-S2 RC slope was dependent on pacing site (P = 0.015), with steeper slope when pacing from the apex than from the base. There were no significant repeatable spatial patterns in steady-state APD at all cycle lengths or in dynamic RC slope. These results indicate that transient patterns of epicardial heterogeneity of APD may occur after a change in pacing rate. Thus it may affect cardiac electrical stability at the onset of a tachycardia or during a series of ectopic beats. Differences in restitution with respect to pacing site suggest that vulnerability may be affected by the location of reentry or ectopic foci.

Many features of the sequence of action potentials produced by repeated stimulation of a patch of cardiac muscle can be modeled by a 1D mapping, but not the full behavior included in the restitution portrait. Specifically, recent experiments have found that (i) the dynamic and S1-S2 restitution curves are different (rate dependence) and (ii) the approach to steady state, which requires many action potentials (accommodation), occurs along a curve distinct from either restitution curve. Neither behavior can be produced by a 1D mapping. To address these shortcomings, ad hoc 2D mappings, where the second variable is a "memory" variable, have been proposed; these models exhibit qualitative features of the relevant behavior, but a quantitative fit is not possible. In this paper we introduce a new 2D mapping and determine a set of parameters for it that gives a quantitatively accurate description of the full restitution portrait measured from a bullfrog ventricle. The mapping can be derived as an asymptotic limit of an idealized ionic model in which a generalized concentration acts as a memory variable. This ionic basis clarifies how the present model differs from previous models. The ionic basis also provides the foundation for more extensive cardiac modeling: e.g., constructing a PDE model that may be used to study the effect of memory on propagation. The fitting procedure for the mapping is straightforward and can easily be applied to obtain a mathematical model for data from other experiments, including experiments on different species.

Electroporation uses electric pulses to promote delivery of DNA and drugs into cells. This study presents a model of electroporation in a spherical cell exposed to an electric field. The model determines transmembrane potential, number of pores, and distribution of pore radii as functions of time and position on the cell surface. For a 1-ms, 40 kV/m pulse, electroporation consists of three stages: charging of the cell membrane (0-0.51 micros), creation of pores (0.51-1.43 micros), and evolution of pore radii (1.43 micros to 1 ms). This pulse creates approximately 341,000 pores, of which 97.8% are small ( approximately 1 nm radius) and 2.2% are large. The average radius of large pores is 22.8 +/- 18.7 nm, although some pores grow to 419 nm. The highest pore density occurs on the depolarized and hyperpolarized poles but the largest pores are on the border of the electroporated regions of the cell. Despite their much smaller number, large pores comprise 95.3% of the total pore area and contribute 66% to the increased cell conductance. For stronger pulses, pore area and cell conductance increase, but these increases are due to the creation of small pores; the number and size of large pores do not increase.

Novel methods need to be developed to detect electrical instability in children. The dynamical properties of action potential restitution play an important role in the development of instability leading to arrhythmias. A new method, the Restitution Portrait (RP), was developed to visualize and quantify these properties at the action potential level. Here, we apply the RP method using the activation-recovery interval (ARI) from the ECG to detect, in vitro, repolarization abnormalities in neonatal and preadolescent rabbit myocardium with drug-induced Long QT Syndrome (LQTS Type 1 or Type 2). The ECG was recorded during programmed endocardial pacing to record the RP. The ECG RP demonstrated significant changes in dynamical restitution components during drug-induced LQTS compared to baseline. This study shows that the ECG RP may be an important noninvasive diagnostic tool for detecting electrical instability in the young.

Electroporation, in which electric pulses create transient pores in the cell membrane, is an important technique for drug and DNA delivery. Electroporation kinetics is mathematically described by an advection-diffusion boundary value problem. This study uses singular perturbation to derive a reduced description of the pore creation transient in the form of a single integrodifferential equation for the transmembrane voltage Vt. The number of pores and the distribution of their radii are computed from Vt. The analysis contains two nonstandard features: the use of the voltage deviation to peel away the strong exponential dependence of pore creation upon the transmembrane potential, and the autonomous approximation of the pore evolution. Comparing the predictions of the reduced equation with the simulations of the original problem demonstrates that this analysis allows one to predict with good accuracy the number and distribution of pores as a function of the electric pulse strength.

Modern models of cardiac membranes are often highly complex and computationally expensive, particularly when used in long simulations of spatially extended models of cardiac tissue. Therefore, there is a need for simpler membrane models that preserve the features of the complex models deemed important. This communication describes an empirical procedure that was used to choose the parameters of the three-variable Fenton-Karma (FK3V) model to reproduce the restitution properties of the Courtemanche-Ramirez-Nattel model of atrial tissue. The resulting parameter values for the FK3V model and the sensitivity table for all its parameters are provided. Thus, this study gives insight into the behavior of the FK3V model and the effect of its parameters on restitution properties.

Restitution, the characteristic shortening of action potential duration (APD) with increased heart rate, has been studied extensively because of its purported link to the onset of fibrillation. Restitution is often represented in the form of mapping models where APD is a function of previous diastolic intervals (DIs) and/or APDs, A(n+1)=F(D(n),A(n),D(n-1),A(n-1),...), where A(n+1) is the APD following a DI given by D(n). The number of variables previous to D(n) determines the degree of memory in the mapping model. Recent experiments have shown that mapping models should contain at least three variables (D(n),A(n),D(n-1)) to reproduce a restitution portrait (RP) that is qualitatively similar to that seen experimentally, where the RP shows three different types of restitution curves (RCs) [dynamic, S1-S2, and constant-basic cycle length (BCL)] simultaneously. However, an interpretation of the different RCs has only been presented in detail for mapping models of one and two variables. Here we present an analysis of the different RCs in the RP for mapping models with an arbitrary amount of memory. We determine the number of variables necessary to represent the different RCs in the RP. We also present a graphical visualization of these RCs. Our analysis reveals that the dynamic and S1-S2 RCs reside on two-dimensional surfaces, and therefore provide limited information for mapping models with more than two variables. However, constant-BCL restitution is a feature of the RP that depends on higher dimensions and can possibly be used to determine a lower bound on the dimensionality of cardiac dynamics.

Rapid pacing is an important tool for understanding cardiac arrhythmias. A recent experiment involving rapid pacing of sheep atria indicated that the initiation of atrial arrhythmias may be related to the 1:1/2:1 bistability. To elucidate the mechanism of this relation, this study applied the pacing protocol from the sheep study to an idealized model of the right atrium. The model included all major anatomical features, the sino-atrial node, and the regional differences in the action potential duration (APD). A pacing protocol was applied, in which the basic cycle length (BCL) was decreased in steps of 10 ms until the response switched to 2:1, then BCL was increased. The 1:1-to-2:1 transitions occurred at shorter BCLs than the 2:1-to-1:1 transitions yielding a global bistability window of 60ms. As in the sheep study, idiopathic waves were observed at BCLs within or near the bistability window. The model was used to quantify the types, prevalence, and persistence of idiopatic waves, study their initiation and termination, and relate them to the model components. The results demonstrate that idiopatic waveforms move with the shift of the bistability window and that they disappear when bistability is eliminated. Thus, this modeling study supports causal relationship between the 1:1/2:1 bistability and the initiation of arrhythmias.

Measuring cardiac action potentials at many sites within the ventricular wall is important for understanding cardiac arrhythmias; however, recording in the depth of the heart wall presents many difficulties. We have developed a multichannel optical mapping system for recording cardiac action potentials transmurally. Each channel uses a single small-diameter optical fiber to transmit and collect light from the cardiac tissue. Excitation light is supplied by low-power green lasers. Wavelength separation is performed with a dichroic mirror, and fluorescence is detected with a photodiode. We have recorded action potentials with an unfiltered signal-to-noise ratio (SNR) as high as 60:1 and a temporally filtered SNR as high as 200:1. The collection of fluorescence is optimized so that low excitation light intensity can be used, which increases the available recording time. Channels are modular and compact, and the system can be easily expanded to include additional channels, ratiometry or dual-dye mapping. In addition, the system is highly flexible and can be used for virtually any experiment from single cell recording to surface and transmural mapping of the whole heart.

Introduction: Electrical restitution, relating action potential duration (APD) to diastolic interval (DI), was believed to determine the stability of heart rhythm. However, recent studies demonstrate that stability also depends on long-term APD changes caused by memory. This study presents a new method for investigation of rate- and memory-dependent aspects of restitution and for assessment of mapping models of APD., Methods and Results: Bullfrog ventricular myocardium was paced with a "perturbed downsweep protocol." Starting from a basic cycle length (BCL) of 1,000 ms, the tissue was paced until steady state was achieved, followed by single beats of longer and shorter cycle lengths. BCL was decreased by 50 to 100 ms and the process repeated. All APDs were plotted as a function of the preceding DI, which allowed simultaneous observation of dynamic, S1-S2, and two constant-BCL restitution curves in a "restitution portrait." Responses were classified as 1:1 (stimulus:response), transient 2:2, or persistent 2:2 (alternans) and were related to the slopes of the restitution curves. None of these slopes approached unity for the persistent 2:2 response, demonstrating that the traditional restitution condition does not predict alternans. The restitution portrait was used to evaluate three mapping models of APD. The models with no memory and with one-beat memory did not produce restitution portraits similar to the experimental one. A model with two-beat memory produced a qualitatively similar portrait., Conclusion: The restitution portrait allows a more comprehensive assessment of cardiac dynamics than methods used to date. Further study of models with memory may result in a clinical criterion for electrical instability.

Electroporation, in which electric pulses create transient pores in the cell membrane, is becoming an important technique for gene therapy. To enable entry of supercoiled DNA into cells, the pores should have sufficiently large radii (>10 nm), remain open long enough for the DNA chain to enter the cell (milliseconds), and should not cause membrane rupture. This study presents a model that can predict such macropores. The distinctive features of this model are the coupling of individual pores through membrane tension and the electrical force on the pores, which is applicable to pores of any size. The model is used to explore the process of pore creation and evolution and to determine the number and size of pores as a function of the pulse magnitude and duration. Next, our electroporation model is combined with a heuristic model of DNA uptake and used to predict the dependence of DNA uptake on pulsing parameters. Finally, the model is used to examine the mechanism of a two-pulse protocol, which was proposed specifically for gene delivery. The comparison between experimental results and the model suggests that this model is well-suited for the investigation of electroporation-mediated DNA delivery.

We present an analytical solution for the electrical potential and field established by a six-needle array electroporation electrode, which is used in vivo for cancer treatment and DNA delivery. The analytical solution closely matches the numerical solution obtained with the finite element method: the mean error is less than 0.6%.

This study computes the contribution of the externally induced transmembrane potential to the energy of large, highly conductive pores. This work was undertaken because the pore energy formulas existing in the literature predict qualitatively different behavior of large pores: the original formula proposed by Abidor et al. in 1979 implies that the electrical force expanding the pore increases linearly with pore radius, while later extensions of this formula imply that this force decreases to zero for large pores. Starting from the Maxwell stress tensors, our study derives the formula for the mechanical work required to deform a dielectric body in an ionic solution with steady-state electric current. This formula is related to a boundary value problem (BVP) governing electric potentials and fields in a proximity of a pore. Computer simulations yield estimates of the electrical energy for pores of two different shapes: cylindrical and toroidal. In both cases, the energy increases linearly for pore radii above approximately 20 nm, implying that the electrical force expanding the pore asymptotes to a constant value for large pores. This result is different from either of the two energy formulas mentioned above. Our study traces the source of this disagreement to approximations made by previous studies, which are suitable only for small pores. Therefore, this study provides a better understanding of the energy of large pores, which is needed for designing pulsing protocols for DNA delivery.

We analyze a mathematical model of paced cardiac muscle consisting of a map relating the duration of an action potential to the preceding diastolic interval as well as the preceding action potential duration, thereby containing some degree of "memory." The model displays rate-dependent restitution so that the dynamic and S1-S2 restitution curves are different, a manifestation of memory in the model. We derive a criterion for the stability of the 1:1 response pattern displayed by this model. It is found that the stability criterion depends on the slope of both the dynamic and S1-S2 restitution curves, and that the pattern can be stable even when the individual slopes are greater or less than one. We discuss the relation between the stability criterion and the slope of the constant-BCL restitution curve. The criterion can also be used to determine the bifurcation from the 1:1 response pattern to alternans. We demonstrate that the criterion can be evaluated readily in experiments using a simple pacing protocol, thus establishing a method for determining whether actual myocardium is accurately described by such a mapping model. We illustrate our results by considering a specific map recently derived from a three-current membrane model and find that the stability of the 1:1 pattern is accurately described by our criterion. In addition, a numerical experiment is performed using the three-current model to illustrate the application of the pacing protocol and the evaluation of the criterion.

The Smoluchowski equation (SE), which describes the evolution of pores created by electric shocks, cannot be applied to modeling large and long-lived pores for two reasons: (1) it does not predict pores of radius above 20 nm without also predicting membrane rupture; (2) it does not predict postshock growth of pores. This study proposes a model in which pores are coupled by membrane tension, resulting in a nonlinear generalization of SE. The predictions of the model are explored using examples of homogeneous (all pore radii r are equal) and heterogeneous (0

Polarization of individual cells ("sawtooth") has been proposed as a mechanism for field stimulation and defibrillation. To date, the modeling work has concentrated on the myocardium with periodic spatial structure; this paper investigates potentials arising in cardiac fibers with random spatial structure. Ten different random fibers consisting of cells with varying length (l(c) = 100 +/- 50 microm), diameter (d(c) = 20 +/- 10 microm), thickness of extracellular space (t(e) = 1.18 +/- 0.59 microm), and junctional resistance (R(j) = 2 +/- 1 M(omega)) are studied. Simulations demonstrate that randomizing spatial structure introduces to the field-induced potential (V(m)) a randomly varying baseline, which arises due to polarization of groups of cells. This polarization appears primarily in response to randomizing t(e); R(j), l(c), and d(c) have less influence. The maximum V(m) increases from 3.5 mV in a periodic fiber to 20.5+/-4.7 mV in random fibers (1 V/cm field applied for 5 ms). Field stimulation threshold E(th) decreases from 6.9 to 1.59 +/- 0.43 V/cm, which is within the range of experimental measurements. Thresholds for normal and reversed field polarities are statistically equivalent: 1.59 +/- 0.43 versus 1.44 +/- 0.41 V/cm (p value = 0.453). Thus, V(m) arising due to random structure of the myocardium may play an important role in field stimulation and defibrillation.

The goal of this study was to collect a comprehensive set of data that related lethal effects of electric fields to the duration of the pulse. Electric pulses of different strengths and durations were applied to a suspension of HEp-2 cells (epidermoid carcinoma of the human larynx) using a six-needle electrode array connected through an autoswitcher to a square wave generator. Pulse durations varied from 50 micros to 16 ms and the ranges of electric field were adjusted for each duration to capture cell viabilities between 0% and 100%. After pulsation, cells were incubated for 44 h at 37 degrees C, and their viability was measured spectrophotometrically using an XTT assay. For each pulse duration (d), viability data were used to determine the electric field that killed half of the cells (E50). When plotted on logarithmic axes, E50 vs. d was a straight line, leading to a hyperbolic relationship: E50=const/d. This relationship suggests that the total charge delivered by the pulse is the decisive factor in killing HEp-2 cells.

Realistic simulations of the effects of strong shocks on cardiac muscle require solving the bidomain model, a continuum representation of cardiac tissue by a system of two reaction-diffusion equations. For two- and three-dimensional problems, the computations tend to take a prohibitively long time. This study develops a computationally efficient and accurate approximation of the bidomain model: a "reduced bidomain" model. The approximation is based on the fact that during a strong shock, the extracellular field in the muscle changes only slightly and, therefore, can be approximated by an activating function, following the concept introduced by Rattay (Rattay, F. Analysis of models for external stimulation of axons. IEEE Trans. Biomed. Eng. 33:974-977, 1986). The activating function used here is time-dependent and is computed using an iterative algorithm. The results show that in two spatial dimensions, the "reduced bidomain" model, as implemented in this study, cuts the computational cost by two orders of magnitude while preserving most properties of the "full bidomain" model. It faithfully represents the spatial pattern and the temporal development of the muscle polarization. Consequently, relative errors in the "defibrillation" threshold, the strength of the weakest shock that terminates all electrical activity within 100 ms, are below 10%.

Introduction: Studies of the electrical dynamics of cardiac tissue are important for understanding the mechanisms of arrhythmias. This study uses high-frequency pacing to investigate the dynamics of sheep atria., Methods and Results: A 504-electrode mapping plaque was affixed to the right atrium in six sheep. Cathodal pacing stimuli were delivered to the center of the plaque. Pacing period (Tp) was decreased from 275 +/- 25 msec to 75 +/- 25 msec and then increased to 230 +/- 70 msec in steps of either 5 or 10 msec. In all 21 trials in six sheep, the atrium responded 1:1 at longer Tps and 2:1 at shorter Tps. As Tp was decreased, the response switched to 2:1 at a particular Tp. Conversely, as Tp was increased, the response switched back to 1:1 at a particular Tp. Over 21 trials, the 1:1-to-2:1 and 2:1-to-1:1 transitions occurred at 119.5 +/- 18.8 msec and 130.0 +/- 19.1 msec, respectively. This hysteretic behavior yielded bistability windows, 10.5 +/- 7.2 msec wide, wherein 1:1 and 2:1 responses existed at the same Tp. In 15 trials and in all animals, idiopathic wavefronts emanating from outside the mapped region passed through the mapped region. In 13 of those trials, the idiopathic wavefronts occurred at Tps within the bistability window or within 35 msec of its upper or lower limit., Conclusion: Bistability windows and idiopathic wavefronts were observed and found to be correlated with each other, suggesting a connection between bistability and arrhythmogenesis.

Electroporation is becoming an increasingly important tool for introducing biologically active compounds into living cells, yet the effectiveness of this technique can be low, particularly in vivo. One way to improve the success rate is to optimize the shock protocols, but experimental studies are costly, time consuming, and yield only an indirect measurement of pore creation. Alternatively, this study models electroporation in two geometries, a space-clamped membrane and a single cell, and investigates the effects of pulse duration, frequency, shape, and strength. The creation of pores is described by a first order differential equation derived from the Smoluchowski equation. Both the membrane and the cell are exposed to monophasic and biphasic shocks of varying duration (membrane, 10 micros-100 s; cell, 0.1 micros-200 ms) and to trains of monophasic and biphasic pulses of varying frequency (membrane, 50 Hz-4 kHz; cell, 200 kHz-6 MHz). The effectiveness of each shock is measured by the fractional pore area (FPA). The results indicate that FPA is sensitive to shock duration only in a very narrow range (membrane, approximately 1 ms; cell, approximately 0.25 micros). In contrast, FPA is sensitive to shock strength and frequency of the pulse train, increasing linearly with shock strength and decreasing slowly with frequency. In all cases, monophasic shocks were at least as effective as biphasic shocks, implying that varying the strength and frequency of a monophasic pulse train is the most effective way to control the creation of pores.

This study expands a previously developed model of a single cell electroporated by an external electric field by explicitly accounting for the ionic composition of the electroporation current. The previous model with non-specific electroporation current predicts that both the transmembrane potential V(m) and the pore density N are symmetric about the equator, with the same values at either end of the cell. The new, ion-specific case predicts that V(m) is symmetric and almost identical to the profile from the non-specific case, but N has a profound asymmetry with the pore density at the hyperpolarized end of the cell twice the value at the depolarized end. These modeling results agree with the experimentally observed preferential uptake of marker molecules at the hyperpolarized end of the cell as reported in the literature. This study also investigates the changes in intracellular ionic concentrations induced around an electroporated single cell. For all ion species, the concentrations near the membrane vary significantly, which may explain the electrical disturbances observed experimentally after large electric shocks are delivered to excitable cells and tissues.

This study develops a model for a single cell electroporated by an external electric field and uses it to investigate the effects of shock strength and rest potential on the transmembrane potential V(m) and pore density N around the cell. As compared to the induced potential predicted by resistive-capacitive theory, the model of electroporation predicts a smaller magnitude of V(m) throughout the cell. Both V(m) and N are symmetric about the equator with the same value at both poles of the cell. Larger shocks do not increase the maximum magnitude of V(m) because more pores form to shunt the excess stimulus current across the membrane. In addition, the value of the rest potential does not affect V(m) around the cell because the electroporation current is several orders of magnitude larger than the ionic current that supports the rest potential. Once the field is removed, the shock-induced V(m) discharges within 2 micros, but the pores persist in the membrane for several seconds. Complete resealing to preshock conditions requires approximately 20 s. These results agree qualitatively and quantitatively with the experimental data reported by Kinosita and coworkers for unfertilized sea urchin eggs exposed to large electric fields.

Introduction: Defibrillation shocks, when delivered through internal electrodes, establish transmembrane potentials (Vm) large enough to electroporate the membrane of cardiac cells. The effects of such shocks on the transmembrane potential distribution are investigated in a two-dimensional rectangular sheet of cardiac muscle modeled as a bidomain with unequal anisotropy ratios., Methods and Results: The membrane is represented by a capacitance Cm, a leakage conductance g(l) and a variable electroporation conductance G, whose rate of growth depends exponentially on the square of Vm. The stimulating current Io, 0.05-20 A/m, is delivered through a pair of electrodes placed 2 cm apart for stimulation along fibers and 1 cm apart for stimulation across fibers. Computer simulations reveal three categories of response to Io: (1) Weak Io, below 0.2 A/m, cause essentially no electroporation, and Vm increases proportionally to Io. (2) Strong Io, between 0.2 and 2.5 A/m, electroporate tissue under the physical electrode. Vm is no longer proportional to Io; in the electroporated region, the growth of Vm is halted and in the region of reversed polarity (virtual electrode), the growth of Vm is accelerated. (3) Very strong Io, above 2.5 A/m, electroporate tissue under the physical and the virtual electrodes. The growth of Vm in all electroporated regions is halted, and a further increase of Io increases both the extent of the electroporated regions and the electroporation conductance G., Conclusion: These results indicate that electroporation of the cardiac membrane plays an important role in the distribution of Vm induced by defibrillation strength shocks.