Modelling the Effect of Gap Junctions on Tissue-Level Cardiac Electrophysiology

When modelling tissue-level cardiac electrophysiology, a continuum approximation to the discrete cell-level equations, known as the bidomain equations, is often used to maintain computational tractability. Analysing the derivation of the bidomain equations allows us to investigate how microstructure, in particular gap junctions that electrically connect cells, affect tissue-level conductivity properties. Using a one-dimensional cable model, we derive a modified form of the bidomain equations that take gap junctions into account, and compare results of simulations using both the discrete and continuum models, finding that the underlying conduction velocity of the action potential ceases to match up between models when gap junctions are introduced at physiologically realistic coupling levels. We show that this effect is magnified by: (i) modelling gap junctions with reduced conductivity; (ii) increasing the conductance of the fast sodium channel; and (iii) an increase in myocyte length. From this, we conclude that the conduction velocity arising from the bidomain equations may not be an accurate representation of the underlying discrete system. In particular, the bidomain equations are less likely to be valid when modelling certain diseased states whose symptoms include a reduction in gap junction coupling or an increase in myocyte length.     

Microdomain Effects on Transverse Cardiac Propagation

The effect of gap junctional coupling, sodium ion channel distribution, and extracellular conductivity on transverse conduction in cardiac tissue is explored using a microdomain model that incorporates aspects of the inhomogeneous cellular structure. The propagation velocities found in our model are compared to those in the classic bidomain model and indicate a strong ephaptic microdomain contribution to conduction depending on the parameter regime. We show that ephaptic effects can be quite significant in the junctional spaces between cells, and that the cell activation sequence is modified substantially by these effects. Further, we find that transverse propagation can be maintained by ephaptic effects, even in the absence of gap junctional coupling. The mechanism by which this occurs is found to be cablelike in that the junctional regions act like inverted cables. Our results provide insight into several recent experimental studies that indirectly indicate a mode of action potential propagation that does not rely exclusively on gap junctions.     

Microdomain Effects on Transverse Cardiac Propagation

The effect of gap junctional coupling, sodium ion channel distribution, and extracellular conductivity on transverse conduction in cardiac tissue is explored using a microdomain model that incorporates aspects of the inhomogeneous cellular structure. The propagation velocities found in our model are compared to those in the classic bidomain model and indicate a strong ephaptic microdomain contribution to conduction depending on the parameter regime. We show that ephaptic effects can be quite significant in the junctional spaces between cells, and that the cell activation sequence is modified substantially by these effects. Further, we find that transverse propagation can be maintained by ephaptic effects, even in the absence of gap junctional coupling. The mechanism by which this occurs is found to be cablelike in that the junctional regions act like inverted cables. Our results provide insight into several recent experimental studies that indirectly indicate a mode of action potential propagation that does not rely exclusively on gap junctions.     

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.     

A Computer Model of Engineered Cardiac Monolayers

Engineered monolayers created using microabrasion and micropatterning methods have provided a simplified in vitro system to study the effects of anisotropy and fiber direction on electrical propagation. Interpreting the behavior in these culture systems has often been performed using classical computer models with continuous properties. However, such models do not account for the effects of random cell shapes, cell orientations, and cleft spaces inherent in these monolayers on the resulting wavefront conduction. This work presents a novel methodology for modeling a monolayer of cardiac tissue in which the factors governing cell shape, cell-to-cell coupling, and degree of cleft space are not constant but rather are treated as spatially random with assigned distributions. This modeling approach makes it possible to simulate wavefront propagation in a manner analogous to performing experiments on engineered monolayer tissues. Simulated results are compared to previously published measured data from monolayers used to investigate the role of cellular architecture on conduction velocities and anisotropy ratios. We also present an estimate for obtaining the electrical properties from these networks and demonstrate how variations in the discrete cellular architecture affect the macroscopic conductivities. The simulations support the common assumption that under normal ranges of coupling strength, tissues with relatively uniform distributions of cell shapes and connectivity can be represented using continuous models with conductivities derived from random discrete cellular architecture using either global or local estimates. The results also reveal that in the presence of abrupt changes in cell orientation, local estimates of tissue properties predict smoother changes in conductivity that may not adequately predict the discrete nature of propagation at the transition sites.     

An analysis of the relationships between subthreshold electrical properties and excitability in skeletal muscle

Skeletal muscle activation requires action potential (AP) initiation followed by its sarcolemmal propagation and tubular excitation to trigger Ca(2+) release and contraction. Recent studies demonstrate that ion channels underlying the resting membrane conductance (G(M)) of fast-twitch mammalian muscle fibers are highly regulated during muscle activity. Thus, onset of activity reduces G(M), whereas prolonged activity can markedly elevate G(M). Although these observations implicate G(M) regulation in control of muscle excitability, classical theoretical studies in un-myelinated axons predict little influence of G(M) on membrane excitability. However, surface membrane morphologies differ markedly between un-myelinated axons and muscle fibers, predominantly because of the tubular (t)-system of muscle fibers. This study develops a linear circuit model of mammalian muscle fiber and uses this to assess the role of subthreshold electrical properties, including G(M) changes during muscle activity, for AP initiation, AP propagation, and t-system excitation. Experimental observations of frequency-dependent length constant and membrane-phase properties in fast-twitch rat fibers could only be replicated by models that included t-system luminal resistances. Having quantified these resistances, the resulting models showed enhanced conduction velocity of passive current flow also implicating elevated AP propagation velocity. Furthermore, the resistances filter passive currents such that higher frequency current components would determine sarcolemma AP conduction velocity, whereas lower frequency components excite t-system APs. Because G(M) modulation affects only the low-frequency membrane impedance, the G(M) changes in active muscle would predominantly affect neuromuscular transmission and low-frequency t-system excitation while exerting little influence on the high-frequency process of sarcolemmal AP propagation. This physiological role of G(M) regulation was increased by high Cl(-) permeability, as in muscle endplate regions, and by increased extracellular [K(+)], as observed in working muscle. Thus, reduced G(M) at the onset of exercise would enhance t-system excitation and neuromuscular transmission, whereas elevated G(M) after sustained activity would inhibit these processes and thereby accentuate muscle fatigue.     

Experimental study of the conducted action potential in cardiac Purkinje strands

Conduction velocity is a complex physiological process that integrates the active and passive properties of the excitable cell. The relations between these properties in determining the conduction velocity are not intuitively obvious, and models have been used frequently to illustrate important relationships. To study the relationships of important parameters and to evaluate commonly used models, we changed conduction velocity experimentally in sheep cardiac Purkinje strands by reducing extracellular Na systematically. Cable analyses were also performed to obtain passive membrane and cable properties. Resting membrane resistance and capacitance did not change, nor did core resistance. Active properties measured in addition to conduction velocity included maximal upstroke velocity, action potential height, time constant of the foot, peak inward current, and upstroke power. With reduction in extracellular Na, all of these parameters of the action potential changed nonlinearly and not in direct proportion to the change in conduction velocity. The only simple relation found was a linear relationship between maximal upstroke velocity and peak inward current, normalized by the capacity of the foot. Models based on the cable equation and the wave equation offer a basis for quantitative analysis of conduction, and these data can be used to test the models.     

The velocity of conduction in nerve fiber and its electric characteristics

The theory developed in this paper shows that the propagation of spike potential along a nerve fiber and the conduction of an electric wave along an inert inorganic conductor follow a common quantitative relationship. This result gives further support to the belief that propagation of excitation is an electrical process. The basic idea of the theory is derived from the consideration that velocity has, by its mathematical definition, a local meaning; conduction in a nerve is completely determined by the local characteristics of the latter, as well as those of the wave. The final formula derived does not make use of any other field of science beyond the fundamental principles of electricity. It gives the conduction velocity in terms of the electric characteristics of the fiber and of the duration of the spike potential. The formula is in agreement with the known dependence of the conduction velocity on various parameters characterizing the axon. The computed velocity agrees with the measured ones on the squid giant axon, crab nerve axon, frog muscle fiber and Nitella cell. The membrane inductance appears as a velocity controling agent which prevents also a possible distortion of the spike potential during conduction. The structural meaning of the electric characteristics of the axon membrane is discussed from the viewpoint of the diffusion theory. A formula for the velocity of spread of the electrotonus is also derived.     

Membrane Characteristics of the Canine Papillary Muscle Fiber

Passive and active responses to intracellular and extracellular stimulation were studied in the canine papillary muscle. The electrotonic potential produced by extracellular polarization with the partition chamber method fitted the time course and the spatial decay expected from the cable theory (the time constant, 3.3 msec; the space constant, 1.2 mm). Contrariwise, spatial decay of the electrotonic potentials produced by intracellular polarization was very short and did not fit the decay curve expected for a simple cable, although only a small difference of time course in the electrotonic potentials produced by intracellular and extracellular polarizations was observed. A similar time course might result from the fact that when current flow results from intracellular polarization, the input resistance is less dependent on the membrane resistance. The foot of the propagated action potential rose exponentially with a time constant of 1.1 msec and a conduction velocity of 0.68 m/sec. The membrane capacity was calculated from the time constant of the foot potential and the conduction velocity to be 0.76 µF/cm². The responses of the papillary muscle membrane to intracellular stimulation differed from those to extracellular stimulation applied with the partition method in the following ways: higher threshold potential, shorter latency for the active response, linearity of the current-voltage relationship, and no reduction in the membrane resistance at the crest of the action potential during current flow.     

An analytical study of the physiology and pathology of the propagation of cardiac action potentials

Many cardiac diseases are caused by the abnormal propagation of electrical waves. Previous experimental and modelling work is reviewed, then a detailed study of the mathematics of cardiac propagation is presented. Pathologies are examined in the context of the models by varying parameters in the models to mimic different pathological states. Ionic models of cells are simplified to form analytically tractable models of the propagation of electrical cardiac waves. The roles that sodium channel activation and inactivation play in determining the conduction velocity are studied in detail, and the roles of resting potential currents in conduction block are calculated. The effect of curvature on the conduction velocity is examined, and the conditions in which curvature leads to conduction block and fibrillation are discussed. Hyperkalaemia (important during ischaemia) is modelled, and the model correctly describes the bi-phasic relation between propagation velocity and extracellular potassium.     

Influence of dynamic gap junction resistance on impulse propagation in ventricular myocardium: a computer simulation study

The gap junction connecting cardiac myocytes is voltage and time dependent. This simulation study investigated the effects of dynamic gap junctions on both the shape and conduction velocity of a propagating action potential. The dynamic gap junction model is based on that described by Vogel and Weingart (J. Physiol. (Lond.). 1998, 510:177-189) for the voltage- and time-dependent conductance changes measured in cell pairs. The model assumes that the conductive gap junction channels have four conformational states. The gap junction model was used to couple 300 cells in a linear strand with membrane dynamics of the cells defined by the Luo-Rudy I model. The results show that, when the cells are tightly coupled (6700 channels), little change occurs in the gap junction resistance during propagation. Thus, for tight coupling, there are negligible differences in the waveshape and propagation velocity when comparing the dynamic and static gap junction representations. For poor coupling (85 channels), the gap junction resistance increases 33 MOmega during propagation. This transient change in resistance resulted in increased transjunctional conduction delays, changes in action potential upstroke, and block of conduction at a lower junction resting resistance relative to a static gap junction model. The results suggest that the dynamics of the gap junction enhance cellular decoupling as a possible protective mechanism of isolating injured cells from their neighbors.     

Electrophysiological interaction through the interstitial space between adjacent unmyelinated parallel fibers.

The influence of interstitial or extracellular potentials on propagation usually has been ignored, often through assuming these potentials to be insignificantly different from zero, presumably because both measurements and calculations become much more complex when interstitial interactions are included. This study arose primarily from an interest in cardiac muscle, where it has been well established that substantial interstitial potentials occur in tightly packed structures, e.g., tens of millivolts within the ventricular wall. We analyzed the electrophysiological interaction between two adjacent unmyelinated fibers within a restricted extracellular space. Numerical evaluations made use of two linked core-conductor models and Hodgkin-Huxley membrane properties. Changes in transmembrane potentials induced in the second fiber ranged from nonexistent with large intervening volumes to large enough to initiate excitation when fibers were coupled by interstitial currents through a small interstitial space. With equal interstitial and intracellular longitudinal conductivities and close coupling, the interaction was large enough (induced Vm approximately 20 mV peak-to-peak) that action potentials from one fiber initiated excitation in the other, for the 40-microns radius evaluated. With close coupling but no change in structure, propagation velocity in the first fiber varied from 1.66 mm/ms (when both fibers were simultaneously stimulated) to 2.84 mm/ms (when the second fiber remained passive). Although normal propagation through interstitial interaction is unlikely, the magnitudes of the electrotonic interactions were large and may have a substantial modulating effect on function.     

A discrete formalism for the computation of extracellular potentials

To facilitate the computation of field potentials generated by a population of neurons a discrete formalism of the physical laws governing propagation of current in a conductive medium (volume conductor theory) is proposed. The formalism is used in combination with the compartmental model of Rall, which models the membrane activity of a neuron taking into account the electrical as well as the geometrical properties of a neuronal membrane. The direct objective is to use computer programs based on this combination in order to simulate field potentials caused by synchronous excitation of the granule cells of the fascia dentata of the hippocampus stimulated by means of the afferent fibers (perforant path) from the entorhinal cortex. To demonstrate the validity of the formalism the extracellular field of an action potential propagating in an axon has been modelled. The extracellular action potential shows a two or threephasic character which is dependent on the direction of propagation of the membrane activity.     

A Note on the Local Current Associated with the Rising Phase of a Propagating Impulse in Nonmyelinated Nerve Fibers

To extend our recent paper dealing with the cable properties and the conduction velocity of nonmyelinated nerve fibers (Bull. Math. Biol. 64, 1069; 2002), the behavior of the local current associated with the rising phase of a propagating action potential is discussed. It is shown that the process of charging the membrane capacity by means of the local current plays a crucial role in determining the velocity of nerve conduction. The symmetry of the local current with respect to the boundary between the resting and active regions of the nerve fiber is emphasized. It is noted that there are several simple quantitative rules governing the intensities of the capacitive, resistive and total membrane currents observed during the rising phase of an action potential.     

Effect of diameter on membrane capacity and conductance of sheep cardiac Purkinje fibers

Membrane electrical properties were measured in sheep cardiac Purkinje fibers, having diameters ranging from 50 to 300 mum. Both membrane capacitance and conductance per unit area of apparent fiber surface varied fourfold over this range. Membrane time constant, and capacitance per unit apparent surface area calculated from the foot of the action potential were independent of fiber diameter, having average values of 18.8 +/- 0.7 ms, and 3.4 +/- 0.25 muF/cm2, respectively (mean +/- SEM). The conduction velocity and time constant of the foot of the action potential also appeared independent of diameter, having values of 3.0 +/- 0.1 m/s and 0.10 +/- 0.007 ms. These findings are consistent with earlier suggestions that in addition to membrane on the surface of the fiber, there exists a large fraction of membrane in continuity with the extracellular space but not directly on the surface of the fiber. Combining the electrical and morphological information, it was possible to predict a passive length constant for the internal membranes of about 100 mum and a time constant for chaning these membranes in a passive 100-mum fiber of 1.7 ms.     

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.     

A reliability analysis of cardiac repolarization time markers

Only a limited number of studies have addressed the reliability of extracellular markers of cardiac repolarization time, such as the classical marker RT_eg defined as the time of maximum upslope of the electrogram T wave. This work presents an extensive three-dimensional simulation study of cardiac repolarization time, extending the previous one-dimensional simulation study of a myocardial strand by Steinhaus [B.M. Steinhaus, Estimating cardiac transmembrane activation and recovery times from unipolar and bipolar extracellular electrograms: a simulation study, Circ. Res. 64 (3) (1989) 449]. The simulations are based on the bidomain - Luo-Rudy phase I system with rotational fiber anisotropy and homogeneous or heterogeneous transmural intrinsic membrane properties. The classical extracellular marker RT_eg is compared with the gold standard of fastest repolarization time RT_tap, defined as the time of minimum derivative during the downstroke of the transmembrane action potential (TAP). Additionally, a new extracellular marker RT90_eg is compared with the gold standard of late repolarization time RT90_tap, defined as the time when the TAP reaches 90% of its resting value. The results show a good global match between the extracellular and transmembrane repolarization markers, with small relative mean discrepancy (?1.6%) and high correlation coefficients (?0.92), ensuring a reasonably good global match between the associated repolarization sequences. However, large local discrepancies of the extracellular versus transmembrane markers may ensue in regions where the curvature of the repolarization front changes abruptly (e.g. near front collisions) or is negligible (e.g. where repolarization proceeds almost uniformly across fiber). As a consequence, the spatial distribution of activation-recovery intervals (ARI) may provide an inaccurate estimate of (and weakly correlated with) the spatial distribution of action potential durations (APD).     

A model of slow conduction in bullfrog atrial trabeculae.

The success or failure of the propagation of electrical activity in cardiac tissue is dependent on both cellular membrane characteristics and intercellular coupling properties. This paper considers a linear arrangement of individual bullfrog atrial cells that are resistively coupled end to end to form a cylindrical strand. The strand, in turn, is encased by an endothelial sheath that provides a restricted extracellular space and an ion diffusion barrier to the outer bathing medium. This encased strand serves as an idealized model of an atrial trabeculum. Excitable membrane characteristics of the atrial cell are specified in terms of a Hodgkin-Huxley type of model that is quantitatively based on single-microelectrode voltage clamp data from bullfrog atrial myocytes. This membrane model can simulate the behavior of normal cells as well as of ischemic cells that exhibit depressed electrophysiological behavior (e.g., decreased resting potential, upstroke velocity, peak height, and action potential duration). Depressed activity can be easily simulated with variation of a single model parameter, the gain of the Na+/K+ pump current (INaK). Intercellular coupling properties are specified in terms of a lumped resistive T-type network between adjacent cells. The atrial strand model provides a means for studying the theoretical aspects of slow conduction in a "hybrid" strand that consists of a central region of cells having abnormal membrane or coupling properties, flanked on either side by normal atrial cells. Both uniform and discontinuous conduction are simulated by means of appropriate changes in the coupling resistance between cells. In addition, by varying either the degree of depressed electrical activity or the intercalated disc resistance in the central zone of the strand, slow conduction or complete conduction block in that region is demonstrated. Since the cellular model used in this study is based on experimental data and closely mimics both the atrial action potential and the underlying membrane currents, it has the potential to (1) accurately represent the current and voltage wave-forms occurring in the region of intercalated discs and (2) provide detailed information regarding the mechanisms in intercellular current spread in the region of slow conduction.     

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

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