Physiological Reviews

Basic Mechanisms of Cardiac Impulse Propagation and Associated Arrhythmias



Kléber, André G., and Yoram Rudy. Basic Mechanisms of Cardiac Impulse Propagation and Associated Arrhythmias. Physiol Rev 84: 431–488, 2004; 10.1152/physrev.00025.2003.—Propagation of excitation in the heart involves action potential (AP) generation by cardiac cells and its propagation in the multicellular tissue. AP conduction is the outcome of complex interactions between cellular electrical activity, electrical cell-to-cell communication, and the cardiac tissue structure. As shown in this review, strong interactions occur among these determinants of electrical impulse propagation. A special form of conduction that underlies many cardiac arrhythmias involves circulating excitation. In this situation, the curvature of the propagating excitation wavefront and the interaction of the wavefront with the repolarization tail of the preceding wave are additional important determinants of impulse propagation. This review attempts to synthesize results from computer simulations and experimental preparations to define mechanisms and biophysical principles that govern normal and abnormal conduction in the heart.


Irregularities of the heartbeat due to cardiac electric dysfunction are one of the most frequent causes of mortality and morbidity in the human population. Most of these cases of fatal arrhythmias occur in the setting of cardiac failure as an end-stage syndrome of a variety of cardiac diseases (231, 232).

The understanding of basic mechanisms of rhythm disturbances in the ventricles and atria has grown rapidly in the course of the last century. It became evident as early as 1912 that rapid repetitive excitation of cardiac tissue may arise from disturbances in impulse propagation termed circus movement reentry (222). In 1949, Coraboeuf and Weidmann (46, 47) recorded the first transmembrane action potential from cardiac tissue. Based on the principles of nerve excitation established by Hodgkin and Huxley (131), this discovery made it possible to relate important properties of membrane ion channels to the generation of the action potential. Later, evidence for electric coupling of cardiac cells through low-resistance gap junctions was established (356, 358, 359), a property that is crucial for the propagation of the cardiac impulse. The description of a Ca2+ inward current as a mediator of electromechanical coupling and the introduction of the patch-clamp method to characterize the properties of single membrane ion channels were additional key steps toward the understanding of the molecular mechanism of cardiac action potential generation and its propagation in the heart (233, 269, 270).

Over the past three decades, research employing reductionistic approaches to define the multitude of ion channels that contribute to transmembrane currents that generate the cardiac action potential has evolved rapidly. It became possible to define the relationship between gene expression and the structure and function of ion channels and gap junction proteins. Such approaches have made it possible to understand the effects of drug binding and mutations in the channel protein on the ionic current across the channel pore (213, 224).

The rapidly growing body of detailed scientific knowledge at the molecular and cellular levels has made it timely and important to integrate the isolated elements of knowledge into a system that resembles the electric behavior of cardiac tissue. The complex interactions among the many molecular events that determine the structural and functional cardiac phenotype exclude a simple and linear interpretation of phenomena observed at the tissue and whole heart levels in terms of molecular processes. It is a characteristic of complex and nonlinear systems that relatively small perturbations in elementary processes can have a major effect on the system behavior. This property makes it particularly important to integrate results obtained from reduced systems (e.g., single-channel or single-cell recordings) into models that take the integrated-system complexities into account. One approach for integrating multiple dynamic processes is the use of computer models, which are continuously refined through introduction of new experimental data. In the context of arrhythmia research, computer modeling has provided crucial insight into the contribution of cellular and molecular processes to cardiac electric function in health and disease.

This review focuses on the mechanisms of cardiac electric impulse propagation and arrhythmogenesis. It attempts to synthesize experimental observations and related computer simulations, in an effort to describe basic principles that underlie cardiac electric function.


Cardiac excitation involves generation of the action potential (AP) by individual cells and its conduction from cell-to-cell through intercellular gap junctions. The combination of a locally regenerative process (AP generation) with the transmission of this process through flow of electric charge is common to a broad class of reaction-diffusion processes (368). AP generation is accomplished through a complex interplay between nonlinear membrane ionic currents and the ionic milieu of the cell. A major thrust of basic research in cardiac electrophysiology in recent years has been the study of the structure, kinetic properties, regulation, and subtypes of the ionic channels and transporters that conduct these currents. The literature on this subject is very extensive, and the readers are referred to recent reviews of the topic (127, 227, 325, 326). The main subject of this article is the propagation of the AP in the multicellular cardiac tissue under physiological and pathophysiological conditions. However, because AP generation is a prerequisite for its propagation, we provide here a brief description of the processes that generate the AP and determine its properties.

For a single cardiac cell under space-clamp conditions, the following equation relates the transmembrane potential (Vm) to the total transmembrane ionic current (Iion) Math(1) where Cm is the membrane capacitance (1 μF/cm2) provided by the charge separation across the lipid bilayer (104, 130).

Equation 1 simply states that changes in Vm occur due to displacement of charge on the membrane capacitance by the movement of ions across the cell membrane. This movement occurs via voltage-gated ion channels, pumps, and exchangers, and Iion represents its sum total. Note that a negative Iion (inward flow of positive ions into the cell) produces a positive dVm/dt, which elevates (depolarizes) the membrane potential. A positive Iion indicates an outward flow of positive ions and acts to reduce (repolarize) the membrane potential by generating a negative dVm/dt. Importantly, in a single, space-clamped cell (357), generation of the AP results from the time-, voltage-, and concentration-dependent evolution of Iion, which represents the contribution of many ion-selective mechanisms for ion movement across the membrane. Figure 1 is a schematic diagram of a cardiac ventricular cell and its electrophysiological components. It also serves to illustrate a mathematical model of the mammalian cardiac ventricular cell [the Luo-Rudy (LRd) model] (88, 208, 209, 285, 305, 344, 345, 387) that computes the AP from membrane currents carried by ionic channels, pumps, and exchangers. This model accounts for dynamic changes in ionic concentrations during the AP (including Na+, K+, and Ca2+) and their effects on the membrane ionic currents and provides the basis for the quantitative description of AP generation given below.

fig. 1.

Schematic diagram of the dynamic Luo-Rudy (LRd) ventricular cell model. INa, fast sodium current; ICa(L), calcium current through L-type calcium channels; ICa(T), calcium current through T-type calcium channels; IKr, rapid delayed rectifier potassium current; IKs, slow delayed rectifier potassium current; Ito, transient outward current; IK1, inward rectifier potassium current; IK(ATP), ATP-sensitive potassium current; IKp, plateau potassium current; IK(Na), sodium-activated potassium current (activated under conditions of sodium overload); Ins(Ca), nonspecific calcium-activated current (activated under conditions of calcium overload); INa,b, sodium background current; ICa,b, calcium background current; INaK, sodium-potassium pump current; INaCa, sodium-calcium exchange current; IP(Ca), sarcolemnal calcium pump; Iup, calcium uptake from the myoplasm to the network sarcoplasmic reticulum (NSR); Irel, calcium release from junctional sarcoplasmic reticulum (JSR); Ileak, calcium leakage from NSR to the myoplasm; Itr, calcium translocation from NSR to JSR. Calmodulin and troponin are calcium buffers in the myoplasm. Calsequestrin is a calcium buffer in the JSR. The source code and further details of the LRd model can be found at under Research. [From Rudy (285).]

The AP, the corresponding intracellular calcium transient, and selected ionic currents that generate the AP and determine its morphology and duration are shown in Figure 2. Once activation threshold is reached, the fast inward sodium current (INa) depolarizes the membrane at a very fast rate (maximum dVm/dt = 393 V/s) and generates the fast AP upstroke. INa reaches a very large peak magnitude of –391 μA/μF in ∼1 ms and quickly inactivates. When the Vm upstroke reaches about –25 mV, the inward L-type calcium current [ICa(L)] activates and provides a depolarizing current that supports the AP plateau against the repolarizing action of the outward delayed potassium currents IKr (r = rapid) and IKs (s = slow). Note that ICa(L) exhibits a “spike and dome” morphology during the AP (71, 200). Its early peak of –4.92 μA/μF is reached in 2.74 ms and contributes very little to the rising phase of the ventricular AP, which is dominated by INa under normal conditions. It plays an important role in triggering Ca2+ release from the sarcoplasmic reticulum (SR) through the calcium-induced calcium release (CICR) mechanism (89) to generate the calcium transient and initiate contraction. The dome of ICa(L) maintains the plateau; it slowly declines as L-type calcium channels inactivate. The two repolarizing potassium currents, IKr and IKs, gradually increase during the plateau, shifting the balance of currents in the outward direction to repolarize the membrane towards its resting potential. The sodium/calcium exchanger (INaCa) is an electrogenic process with a 3 Na+:1 Ca2+ stochiometry (81, 267). Early during the AP it operates in its “reverse mode” to extrude Na+ from the cell, generating a small outward current (20, 31, 59, 162, 230). It then reverses direction and operates in its “direct mode” to extrude Ca2+, becoming a significant inward current that acts to slow repolarization during the late plateau and prolongs the AP duration (17, 71, 77, 80). Finally, there is a large increase (late peak) of the outward (inward rectifier) potassium current IK1, that dominates the late repolarization phase and the return of the membrane to its resting level. Note that the simulation of Figure 2 does not include the transiet outward current Ito. This current is not expressed in guinea pig ventricle or in endocardial cells of other species. In its presence, a “notch” is created in the AP following its peak upstroke, a process termed “phase 1 repolarization” (see Fig. 9 in Ref. 285).

fig. 2.

Major ionic currents during the action potential (AP). Shown are the AP (repeated at top of both columns for reference), the calcium transient (free calcium in the myoplasm during the AP; [Ca2+]i), and selected ionic currents that determine the AP morphology (current symbols are defined in Fig. 1). INa is also shown on an expanded time scale (inset). All quantities are simulated using the LRd model. The cell has reached steady-state during pacing at a constant cycle length (CL) of 1,000 ms. [From Rudy (285).]

According to Figure 2, it is clear that two ionic currents, INa and ICa(L), are the major contributors of depolarizing charge during the AP. The most important properties of these currents (represented in the LRd model and the simulations of Fig. 2) are 1) INa is characterized by fast activation and by fast and slow inactivation processes (18, 183, 186, 225, 293) and 2) ICa(L) is inactivated by both a fast Ca2+-dependent process and a slower voltage-dependent process (123, 159, 192, 307, 309, 385). The involvement of these currents in AP propagation is discussed in the following sections. It is important to realize that in many cardiac arrhythmias, such as in ventricular and atrial fibrillation, the head of a propagating wave interacts with the phase of repolarization of a preceding wave. Under such circumstances membrane ion channels affecting AP repolarization may become important determinants of impulse conduction, in addition to INa and ICa(L).

In addition to the LRd model, used in the simulations above, other theoretical models of cardiac cellular electric activity have been developed, including 1) ventricular myocytes (16, 236, 374), 2) atrial myocytes (55, 197, 238), 3) sinus node cells (366, 381), and 4) Purkinje cells (69, 218). In general, these models are formulated in the classical Hodgkin-Huxley scheme, which computes the whole cell AP from transmembrane currents generated by large ensembles of ion channels. Such models have proven to be very useful and continue to be so in many areas of cardiac electrophysiology, including simulations of AP generation by the single cell and its propagation in models of the multicellular tissue.

In recent years, a large body of knowledge has accumulated on the structure-function relationships of ion channels and their modification by genetic defects that are associated with cardiac arrhythmias (260, 261). Most of these data were obtained in expression systems (e.g., Xenopus oocyte), away from the physiological environment of cardiac cells where ion channels interact to generate the AP. Mathematical models can be used to integrate this information into the functioning cardiac cell to relate single-channel behavior to whole cell function. Importantly, such modeling approach can be used to link altered channel function due to mutation to the resulting cellular phenotype. This requires incorporation into the cell of Markov models that represent specific structural states of the channel (e.g., open, closed, inactivated) and their interdependencies (a major departure from the Hodgkin-Huxley scheme). A first example of this approach was a study of a sodium-channel mutation (ΔKPQ, a three-amino acid deletion) that affects the channel inactivation and is associated with a congenital form of the long-QT syndrome, LQT3 (40). The simulations showed that mutant channel reopenings from the inactivated state and channel bursting due to transient failure of inactivation generate a late sodium inward current during the AP plateau. This depolarizing current prolongs the AP duration, leading to the development of arrhythmogenic early afterdepolarizations (EADs) at slow pacing rates (consistent with the bradycardia-related arrhythmic episodes during sleep or relaxation in LQT3 patients, Ref. 299). Other examples include simulations of mutations in the HERG gene that encodes IKr, leading to the LQT2 type of the long-QT syndrome (41), and of a single sodium channel mutation (1795insD) that results in the seemingly paradoxical coexistence of two different phenotypes, the long-QT and the Brugada syndromes (42). So far, all examples involved integration of ion-channel function into the single, isolated cardiac cell. It is conceivable that in the near future, single-channel models will be utilized at a higher level of integration into the multicellular cardiac tissue. This is an exciting possibility that will help to relate propagation of the cardiac AP to the kinetic properties of single ion channels.


A. Continuous Propagation

The simplest model for AP propagation relates to the linear cellular structure. In this continuous chain of excitable elements, current flows from a depolarized cell to its less depolarized neighbors via intercellular resistive pathways known as gap junctions. This situation is different from the process of AP generation in an isolated, space-clamped cell (Eq. 1 and Fig. 2), where the ion current is used solely to change the charge on the membrane capacitance, Cm. The following equation, which is a general equation describing reaction-diffusion systems, relates the transmembrane current and the axial current that flows between cells in a linear cell chain Math(2) where a is the fiber radius and ri is its axial resistance per unit length. Note that Vm is a function of both time (t) and space (x), hence the use of the partial derivative symbol ∂ to indicate its first derivative in time (∂Vm/∂t) or its second derivative in space (∂2Vm/∂x2). Under space-clamp conditions Vm is constant in space, ∂2Vm/∂x2 = 0, and Equation 2 reduces to Equation 1 of the nonpropagated AP. The left side of Equation 2 is the total transmembrane current, consisting of the capacitive component Cm·∂Vm/∂t and the ionic component Iion. The right side computes the net gain or loss of axial current as it flows down the fiber. This equation, therefore, simply states the conservation principle that the net change in axial current must be accounted for by the current that crosses the membrane.

A comparison of Equations 1 and 2 shows that the proportionality between dVm/dt and Iion, which exists in the isolated single cell, is lost in the multicellular tissue. In the isolated cell the entire Iion is used to discharge the local membrane capacitance and therefore determines the rate of depolarization, dVm/dt. In this situation, dVm/dt can be taken as a measure for transmembrane current flow (357), and maximal flow of INa occurs at the time of the maximal rate of rise of the AP upstroke, dVm/dtmax. In contrast, in the multicellular model the charge generated by Iion during the depolarization of a given cell is divided between discharging the local membrane capacitance and depolarizing the membrane of downstream cells via the axial current. Also, the maximal inward flow of INa occurs later during the upstroke than dVm/dtmax. Therefore, the moment of occurrence of dVm/dtmax is not an accurate measure of local activation (96, 211, 321).

The resistance ri in Equation 2 corresponds to an average resistance of the intracellular space, which in reality is composed of cytoplasmic and gap junctional resistances (see below). This simplified model therefore considers cardiac tissue, similarly to nerve, as a medium with continuous diffusive properties, i.e., as an electric syncytium. An important theoretical property of an electric continuum is the fact that changes in ri or ionic depolarizing current flow have independent effects on propagation velocity, θ. Formally, θ is proportional to the square root of the maximal upstroke velocity, dVm/dtmax, and independently, inversely proportional to the square root of ri (132, 329, 350). Importantly, dVm/dtmax remains constant as ri is changed in a continuous medium.

The experimental application of a continuous model for cardiac propagation is limited to a structure where the multitude of individual cardiac cells can be lumped into a single “macroscopic model cell.” Therefore, it requires geometrically well-defined tissue (e.g., a cylindrical papillary muscle) as well as a regular network of electrically well-coupled cardiac cells devoid of major discontinuities. Such discontinuities can be introduced at the cellular scale by uncoupling (partial or complete) of gap junctions. On a macroscopic scale, they can be formed for instance by trabeculations or connective tissue layers. Representation of cardiac tissue by a continuous model has shown a close accordance between experimental results and theoretical expectations in the description of 1) macroscopic passive electric properties of cardiac muscle (169, 359), 2) the relationship between the change in AP upstroke velocity and macroscopic propagation (measured at a scale >1 mm) (33), and 3) the effect of moderate changes in cell-to-cell coupling (170) and of a change in the extracellular space resistance on conduction velocity (103).

B. Principles of Discontinuous Propagation

One obvious consequence of the specific cardiac structure is that propagating electric waves will interact with structural boundaries. Boundaries exist at the cellular level (cell membranes) as well as at the more macroscopic level (microvasculature, connective tissue barriers, trabeculation, Refs. 193a, 312). The basic biophysical principles that determine the interaction of propagating waves with such boundaries, independently of their scale, have been established many years ago. However, their crucial importance for explaining the mechanisms of arrhythmias has been recognized only relatively recently.

As a first, simple principle one may consider the situation where a wave propagates towards a boundary, as shown in Figure 3. As the front approaches the boundary, the axial current is reflected at the boundary. This situation is formulated as a “sealed end” boundary condition in the so-called cable theory (142). Collisions of APs with complete or partial boundaries not only increase the local velocity of conduction, but they also feed back on the mechanism of generation of the AP (321) and change the shape of the local extracellular electrogram (313). By reducing electrical load on cells proximal to the collision site, reflection of local axial current increases the rate of depolarization, dVm/dtmax, and concomitantly decreases maximal INa during the AP upstroke (321). This reduction is caused by fast depolarization to a high peak voltage, which decreases the electrochemical driving force and increases the inactivation rate of INa (352; Fig. 2F). In contrast to continuous conduction, there is an inverse relationship between activation of the depolarizing ion current and dVm/dtmax at a site of collision. The second principle of discontinuous conduction, as depicted in Figure 4, is related to the situation opposite to partial wave collision. There is dispersion of local current in the front of the propagating wave, because excitation of a small number of elements has to furnish current to excite a larger number of excitable elements downstream (current-to-load mismatch). This reduces the density of current per unit membrane area exciting the elements down-stream, and therefore locally slows the AP upstroke and reduces conduction velocity (96, 352). If the mismatch between the up- and downstream elements becomes too large, conduction is blocked (96, 352). There is a direct relationship between AP upstroke velocity (Fig. 4B) and INa conductance (Fig. 4C) at such sites, because Na+ channel inactivation occurs during the prolonged upstroke (96, 352). Despite the reduced conductance, peak INa just distal to the transition site is increased as a consequence of reduced Vm and the resulting increased electrochemical driving force (Fig. 4D), displaying an inverse relationship to the reduced local upstroke velocity (Fig. 4B). However, recent simulations demonstrated a decrease in local peak INa just distal to the site of increased electrical loading (caused by either an increased intercellular coupling or tissue expansion; Figs. 2C and 5C in Ref. 352). In these simulations (see also Fig. 30), a long conduction delay across the site of current-to-load mismatch resulted in a large degree of INa inactivation that was not overcompensated by an increase in electrochemical driving force. A third scenario, which is particularly relevant for understanding cardiac impulse propagation, is defined by the interaction of sites of partial collision with sites of current dispersion. Such interaction may occur at the cellular scale, due to the repetitive occurrence of gap junctions or at the more macroscopic scale, in branching or fibrotic tissue, and probably in the AV node. The principal role of repetitive discontinuities in conduction was described in 1982 by Joyner et al. (150, 153) and is illustrated in Figure 5. Figure 5 shows that in a structure with a constant lumped resistance per unit length (effective Ri), propagation velocity depends on the repartition into subelements of low (Rlow) and high (Rhigh) resistance values. At large values of Rhigh (i.e., a high degree of discontinuity), conduction is only maintained within a certain range, characterized by a match between the value of the low resistance elements, Rlow, the number N of Rlow elements, and the value of Rhigh which separates the clusters of low resistance elements. Propagation block occurs either at too large a number of elements N (i.e., if there is no or weak electrotonic interaction between the sites of high resistance) or if Rhigh is increased beyond a critical value that blocks axial current flow. Furthermore, the success of conduction depends on the excitability of the membrane. The match between the discontinuous resistive properties of an excitable network and the degree of excitability demonstrates a close interdependence between active electric properties (ion channels, transporters, and exchangers) and passive resistive properties (resistance of gap junctions, tissue structure) as explained in the subsequent sections. It also contradicts the assumption, often made intuitively, that the lower the degree of cell-to-cell coupling the higher the probability of occurrence of conduction block and arrhythmias. As discussed below, the match between the discontinuous resistive properties and the excitability of the tissue underlies the mechanism of very slow conduction in cardiac tissue.

fig. 3.

Effects of wavefront collision on the upstroke of the transmembrane action potential and the Na+ inward current. The values computed during uniform conduction (solid lines) are compared with the values computed at a collision site (dashed line). Top left: change of membrane potential (Vm) during action potential upstroke. Bottom left: maximal upstroke velocity of transmembrane action potential in V/s. Top right: Na+ inward current (INa). Bottom right: Na+ conductance (gNa). [From Spach and Kootsey (321).]

fig. 4.

Effect of wavefront dispersion on the upstroke of the transmembrane action potential and the Na+ inward current. A, inset: simulated 2-dimensional strand of excitable tissue emerging into a large area. Signals on AD are simulated from locations 1–11 shown on the inset during propagation from the small strand into the bulk area. Action potential upstrokes (A) and dVm/dt traces (B) show two components that are most prominent at the site of tissue expansion (signals 6). C: time course of Na+ conductance, gNa. D: time course of Na+ inward current, INa. Note that INa increases at the expansion site (site 6). [Modified from Fast and Kléber (96).]

fig. 30.

Safety factor and ionic mechanism of conduction in structurally inhomogeneous tissue. A–D: conduction along a fiber with inhomogeneous intercellular coupling. A: starting from the junction between cells 79 and 80, gap junction conductance (gj) is increased from 0.08 to 2.5 μS. B: action potentials (Vm); numbers indicate selected cells. C: safety factor (SF) along fiber (line graph); local charge contributions from INa (QNa) and ICa(L) (QCa) are shown in bar graph. D: peak values of INa (INa,max; solid line) and ICa(L) [ICa(L),max; dashed line] along fiber. E–H: propagation across an expansion site. E: fiber expansion (branching) is introduced at cell 80 and repeated twice with an expansion ratio (ER) of 2.3. F: action potentials; numbers indicate selected cells. G: line indicates SF along fiber; bars indicate QNa and QCa. H: INa,max (solid line) and ICa(L),max (dashed line) along fiber. [Modified from Wang and Rudy (352).]

fig. 5.

Principles of discontinuous propagation. Top: discontinuity is defined by a row of excitable elements (abscissa denotes element number) separated by resistors. N elements forming a group are interconnected by resistors of low value (Rlow). Each group of N elements is connected to the next group by a resistor of high value (Rhigh). The effective or overall longitudinal resistance (Ri) plotted in the bottom panel is equal to the average longitudinal resistance. Bottom: change of propagation velocity (θ) as a function of the Ri for 3 degrees of discontinuity. For simplicity, values are plotted in units of Ri/200. Case A: continuous case, θ2·α·1/Ri (see text and Ref. 329). Case B: moderate discontinuity, Rlow = 200 Ω/cm, Rhigh = 5,000 Ω/cm, the numbers on the trace denote the number of elements N in a group. Case C: marked discontinuity, Rlow = 200 Ω/cm, Rhigh = 10,000 Ω/cm, the numbers on the trace denote the number of elements N in a group. Note that case B behaves similarly to the fully continuous case A. In case C, the propagation velocity decreases with both a large and a small number of elements N and is optimal only in a region where there is a match between Rlow, Rhigh, and N. [Modified from Joyner (150).]

C. The Safety of Propagation

During AP propagation an excited cell serves as a source of electric charge for depolarizing neighboring unexcited cells towards their excitation threshold. The unexcited cells constitute an electric sink (load) for the excited cell. For propagation to succeed, the excited cell must provide sufficient charge to the unexcited cells to bring their membrane to excitation threshold. Once threshold is reached and AP generated, the load on the excited cell is removed, and the newly excited cell switches from being a sink to being a source for the downstream tissue, perpetuating the process of AP propagation. The safety factor for conduction (SF) is an important quantity that is related to the source-sink relationship and defines the success of AP propagation. Several approaches have been used to define this quantity (67, 194, 305). A formulation that has been recently introduced defines SF as the ratio of charge generated to charge consumed during the excitation cycle of a single cell in the tissue (305). It is computed using the following equation Math(3) Ic is the capacitive current of the cell in question and Iin and Iout are the axial currents in and out of the cell. The charge Q associated with each current is computed by the time integral of that current over an interval A during which the net membrane charge Qm is positive. Initially, Vm = Vrest and Qm = 0. As the cell begins to depolarize, it consumes charge (sink) and Qm increases to a positive peak value. It then decreases as the cell returns charge (source) to depolarize downstream cells. When Qm returns to zero, the cell has completed its sink-source cycle defined by the interval A|Qm >0. In Equation 3, the numerator is the sum of the charges that the cell generates for its own depolarization (Qc) and for the depolarization of downstream cells (Qout). The denominator (Qin) is the charge that the cell receives from the upstream tissue. SF >1 indicates that more charge is produced during cellular excitation than the charge required to cause the excitation. For linear systems with uniform cellular properties, SF >1 characterizes successful conduction. The factor of SF above 1 can therefore be taken to indicate the margin of safety of propagation (305). As evident from Equation 3, the SF is computed from the main parameters involved in propagation, namely, the integral of the currents involved in or produced by excitation. Although this approach is straightforward in mechanistic terms, the formalism given by Equation 3 does not compute the separate contribution of the different ion currents [INa and ICa(L)], the gap junctional resistance, or the cell dimensions to propagation safety. The contributions of these variables to SF have to be computed separately for each case as discussed in the following sections.

D. Two-Dimensional Propagation and Curvature

Thus far the discussion about the basic rules governing impulse propagation was based on parameters which determine linear propagation, such as propagation in one-dimensional tissue, e.g., a linear continuous cable or a linear cell chain, or similarly, propagation in two-dimensional tissue of a planar wavefront. Both deviation of waves at either functional (301, 337) or structural obstacles (97) cause the wavefront to turn and to assume a curved shape. This presence of curvature adds complexity to the discussion of cardiac impulse propagation and is especially relevant to the mechanism of arrhythmogenesis.

The velocity of the noncurved wavefront(θο) is, as mentioned in the above sections, determined by the passive and active properties of excitable tissue (97). If the excitation front is curving outward (convex), the conduction velocity is θ < θο. This is because the local excitatory current supplied by the cells in the front of a convex wave diverges into a larger membrane area downstream. Inversely, when the excitation front is curving inwards (concave), the excitatory current converges in front of the propagating wave producing a more rapid membrane depolarization. As a result, conduction velocity of a concave wavefront is greater than θο. In terms of biophysical mechanisms, these situations, which occur as changes in shape or geometry of excitation waves in two- or three-dimensional tissue, are equivalent to the principles of current-to-load mismatch and partial collisions governing discontinuous propagation in one-dimensional networks. Thus two-dimensional geometrical models of propagation share close similarities with the one-dimensional models described in the above section.

The degree of wavefront curvature (ρ) can be defined as the negative reciprocal of the local radius of curvature, r Math(4)

A quantitative expression for the dependence of conduction velocity on curvature in a continuous isotropic two-dimensional excitable medium can be derived analytically for small values of r. It was shown that for such conditions the velocity θ is given by the following equation (389) Math(5)

The coefficient D is determined by the passive properties of the medium, where D is equal to 1/CmSvRi where Cm is the specific membrane capacitance, Sv is the cell surface-to-volume ratio, and Ri is the intracellular resistivity.

Several important bioelectric events in the myocardium are related to wavefront curvature. A simple consequence of wavefront curvature is the dependence of propagation velocity in anisotropic myocardium on the mode of stimulation, as shown in Figure 6. In these experiments (171), stimulation of the myocardium by a linear array of electrodes was found to produce a higher propagation velocity in the direction of the main fiber axis (longitudinal velocity) than point stimulation producing elliptic spread. This difference can be explained by the dispersion of local excitatory current along the major axis of the ellipse where the wavefront is curved. In line with this phenomenon is the observation that the maximal upstroke velocity of the transmembrane AP is markedly lower at the tip of elliptic propagation spread. Since more depolarizing Na+ channels are activated at such sites (96), curved waves are also more sensitive than linear waves to drugs that block Na+ channels in the open channel state (139). It should be noted that most of the values for longitudinal conduction velocity have been obtained with point stimulation and are therefore probably underestimated.

fig. 6.

Effect of curvature on propagation. Left: stimulation of a perfused rabbit ventricular epicardial layer with a single electrode (point stimulation from black dot) produces a convex excitation front. Right: stimulation with a line of electrodes (line stimulation) produces an almost flat excitation front. Numbers correspond to activation times in milliseconds. Isochrone lines are shown at intervals of 3 ms. Average longitudinal velocity of curved wave is 13% slower than that of flat wave. [Modified from Knisley and Hill (171).]

The dependence of propagation velocity θ on the radius of curvature predicts that conduction is not sustained below a critically small radius, rc. This dependence is indeed observed in cardiac tissue and has important consequences for the understanding of impulse conduction in reentrant circuits and in tissue with a discontinuous structure. Spread of excitation from a point or focus in the tissue can only occur if the critical mass of simultaneously excited cells (pacemaker cells or tissue excited by point stimulation) forms a nucleus with a radius greater than or equal to rc. A similar requirement has long been recognized and formulated in the concept of “liminal length” for one-dimensional excitable strands (105). Accordingly, the critical amount of cells in two-dimensional tissue comprises a “liminal area.” In a two-dimensional computer model, the liminal area was calculated by Ramza et al. (266). They studied impulse initiation produced by a point current injection in a continuous, isotropic model described by the Beeler-Reuter ionic kinetics. The liminal area necessary to generate sufficient inward current during stimulation was determined as a function of the maximal sodium conductance (gNamax). At a level of excitability estimated to correspond to the adult ventricular myocardium, the radius of the liminal area, assumed to form a circle, was 200–250 μm. Experimentally, the extent of the liminal area was estimated from measurements of the stimulation threshold as a function of electrode size by Lindemans and co-workers (198, 199) and found to be 0.2 mm. It needs to be noted however that application of external current through a small stimulating electrode in anisotropic cardiac tissue produces a complex pattern of changes in transmembrane potential. This complexity, characterized by the presence of both depolarized and hyperpolarized areas (361, 365), is due to the bidomain nature of cardiac tissue, i.e., the presence of a restricted extracellular space whose resistance may be as large as the resistance of the intracellular space (169). The profile of membrane potential around a point stimulus pulse therefore assumes a so-called “dog-bone” shape (235, 364). Because this profile can be related to initiation of reentry, it will be discussed and illustrated (Fig. 36) in a subsequent section.

fig. 36.

Initiation of quadrifoil reentry by local stimulation. A: this panel depicts the potential distribution immediately after application of a stimulus pulse in the center of the field of vision. Light areas correspond to sites depolarized by the stimulus; darker areas correspond to sites hyperpolarized by the stimulus. The 2 dark lines mark the separation between hyper- and depolarized areas. The oblique line corresponds to the direction of the cardiac fibers in this epicardial preparation of a rabbit heart. Measurements were made from the fluorescence change of a voltage-sensitive dye. Note the correspondence between the fiber direction and the orientation of the hyper- and depolarized zones. B: isochronal map of excitation following stimulus (isochrone interval 3.8 ms, area 20 × 13.5 mm2). Excitation spreads along the fibers in the hyperpolarized zone, and a phase singularity is formed along the lines separating the zones of hyperpolarization and depolarization. Subsequently, the excitatory wavefronts enter the depolarized zones, and reentry occurs when the 4 circles close at the original stimulation site. [Modified from Lin et al. (196).]

Circulating excitation and reentry is associated with wavefronts interacting either with functional zones of block or structural obstacles. Both interactions involve dispersion of local current at convex wavefronts, an associated decrease in local conduction velocity, and a change in activation of local depolarizing current, as illustrated in Figure 4. Therefore, wavefront curvature is an important determinant of conduction slowing, conduction block, and reentrant arrhythmias, as discussed in later sections.


A. Macroscopic Anisotropic Propagation

It has long been known that the anisotropic architecture of most myocardial regions, consisting of elongated cells that are forming strands and layers of tissue, leads to a dependence of propagation velocity on the direction of impulse spread (273, 377). Experimentally determined transverse and longitudinal conduction velocities show a large variation of values with 1) specific differences among specific cardiac regions and 2) considerable variability within a given cardiac region, e.g., the atria or the ventricles. Numerous studies have been carried out to determine longitudinal and transverse conduction velocities in several specific regions of the heart (see Table 12–1 in Ref. 167). In the direction of the long cell axis, the highest velocity values (θL) are found in the specific ventricular conduction system (1.7–2.5 m/s, Refs. 57, 74, 283, 334), while the lowest values are measured in the ventricle (0.48–0.61 m/s, Refs. 28, 43, 155, 168, 273, 324, 336). The anisotropy ratio of propagation velocity (θLT) ranges from ∼10 in the crista terminals of the right atrium to ∼2.1 in the ventricles (167). Many of the values obtained experimentally for θL may have been slightly underestimated, because point stimulation produces elliptic spread with a convex longitudinal wavefront (see section about the effect of curvature) (171). In the following sections, the structural determinants of propagation velocity are discussed in detail.

B. The Structural Basis of Propagation at the Cellular Level

The anisotropic cellular structure of the myocardium is important for our understanding of both normal propagation and arrhythmogenesis. Structural anisotropy may relate to cell shape and to the cellular distribution pattern of proteins involved in impulse conduction such as gap junction connexins and membrane ion channels. Cardiomyocytes have an elongated shape in most regions of the heart, either with a “brick stone”-like (adult ventricular myocytes) or more fusiform (neonatal myocytes, sinoatrial node cells) cellular appearance. In the ventricular myocardium, comparison of individual cells among species and different regions of the heart shows a large variability in size with a relatively consistent length-to-width ratio (167).

The functional connections between cardiac cells, consisting of so-called gap junctions, vary in their molecular composition, degree of expression, and the distribution pattern, whereby each of these variations may contribute to the specific propagation properties of a given tissue in a given species. The reader is referred to several textbooks and review articles for a detailed appreciation of the biophysical and biological properties of gap junction proteins (see, e.g., Refs. 64, 288, 290, 367). Connexin 43 (Cx43) is the most abundant protein in the heart and in many other organs. Expression of Cx43 seems to be mostly restricted to the ventricle, atrium, and the specific ventricular conducting system (64, 120, 157, 239, 240, 340) while its presence is disputed in the sinoatrial node and in the atrioventricular node (11, 64, 239241, 333). Cx40 plays an important role in the atria, the atrioventricular node, and the specific ventricular conducting system (15, 39, 64, 120, 121, 289). Due to its large single-channel conductance, Cx40 is likely to contribute to a high propagation velocity in parts of the atria (crista terminalis) and the specific ventricular conducting system. While some studies showed expression of Cx45 in most myocytes (39), its role in impulse conduction in the ventricle is not fully clarified. A further, still not fully answered question relates to the functional consequences of colocalization of different connexins in gap junctions. Such colocalization may reflect heterotypic or heteromeric connexin formation with electric properties that are different from the properties of the corresponding homotypic or homomeric channels. While such formation has shown to produce a multitude of electric conductance states in vitro (34, 53, 126, 339, 342), their functional role in vivo still remains to be defined.

The pattern of gap junction location depends on the stage of heart development and can be remodeled by disease. Normal adult myocardial tissues show a preferential location of large gap junctions at the longitudinal cell ends and relatively small and less frequent junctions along the lateral borders (136, 289). Neonatal tissue in culture or in vivo (94, 120) and remodeled tissue in areas surrounding myocardial infarction (251, 256, 311) show a more regularly distributed and spaced localization of gap junctions of about equal size around the cell perimeter. Remodeling of gap junctions also occurs in early and later stages of ventricular hypertrophy and failure (252). In vitro, remodeling of gap junction by cAMP (61) and mechanical stretch (388) is followed by concomitant changes in conduction velocity. Connectivity, i.e., the average number of neighboring myocytes connected to an individual myocyte, has been taken as a parameter that reflects cell shape, gap junction distribution, and gap junction density. Connectivity seems relatively independent of cell size [almost identical values in dog (136) versus mouse ventricle (330)], while it varies largely between different regions of the heart and can change in disease. Thus connectivity amounts to 11 ± 3 cells in normal dog ventricle (289) versus 6.4 (289) in normal dog atrial crista terminalis and 6.5 (207) in surviving tissue within infarcted areas of dog ventricle, in accordance with the high degree of electric anisotropy in the latter tissues.

C. Cellular Parameters Affecting Normal Propagation

1. Role of cell size

The relationship between cell size (cell length at a constant cell radius), cell-to-cell coupling by gap junctions, and propagation was shown early by Joyner et al. (150). In this theoretical work on the discontinuous nature of conduction, the type (continuous vs. discontinuous) and the velocity of propagation depended not only on the magnitude of the intercellular resistance but also on the length of the low-resistance segments (representing cells) as outlined in detail in section iiiB. An appreciation of the respective contributions of cell size, cell shape, and the clustering of gap junctions to impulse propagation has been provided recently by Spach et al. (318). In this study, illustrated in Figure 7, the authors first simulated propagation in networks of 1) adult canine ventricular cells (column a) and 2) neonatal rat ventricular cells (column d), based on experimental data of cell shape, cell size, gap junction expression, gap junction distribution, and AP upstrokes. Subsequently, they created two virtual cellular networks, a first exhibiting the large cell size of the adult dog ventricle and the gap junction distribution of neonatal rat hearts (Fig. 7, column b) and a second showing the small cell size of the rat neonatal ventricular myocytes and the gap junction distribution pattern of the adult dog (Fig. 7, column c). The simulation allowed for separation of the effects of gap junctions distribution from the effect of cell size on propagation. As demonstrated in Figure 7A for transverse propagation, the distribution pattern of gap junctions had a relatively small effect on the average cell-to-cell delay of the electric impulse (as an indirect measure of conduction velocity), while cell size had a major effect. The important effect of cell size is also underlined by the experimental observation that average conduction velocity in neonatal isotropic cell cultures (being composed of nonaligned and nonelongated cells) is slower than longitudinal velocity in corresponding anisotropic cultures (composed of anisotropically aligned and elongated cells) (94). These studies underline the important role of cell size in determining the anisotropic conduction properties of myocardial tissue. It should be added that pathological changes in cell size (e.g., cell swelling) also involve changes in the interstitial (extracellular) volume that affect conduction velocity by modifying the interstitial resistance to current flow (103). The interstitial space is taken into account in bidomain models of cardiac tissue (129).

fig. 7.

Effect of cell size and distribution pattern of gap junctions on cell-to-cell propagation delay (A) and upstroke velocity of the action potential (B) during transverse propagation. Column a represents values simulated from a model of the normal adult dog heart cell with gap junctions located predominantly at the longitudinal ends. Column d represents values of the normal neonatal rat heart cell with uniformly spaced gap junctions around the cell perimeter. Column b corresponds to a virtual cell with the cell size of a dog myocyte and the gap junction pattern of a neonatal rat heart cell; accordingly, column c corresponds to a virtual cell with the cell size of a neonatal rat myocyte and the gap junction pattern of an adult dog myocyte. Note that cell size has a significantly larger effect than gap junction pattern on both parameters. [Modified from Spach et al. (318).]

2. Role of gap junctions

The role of gap junctional conductance in normal propagation has been investigated in linear cell chains and two-dimensional cellular networks, both theoretically and experimentally (93, 286, 305, 316). Figure 8 illustrates simulated propagation in a strand consisting of single cells. AP upstrokes are shown at the proximal and distal ends of an upstream cell (locations 1 and 2, respectively) and of its neighboring downstream cell (locations 3 and 4). In Figure 8A, gap junction coupling was normal (gj = 2.5 μS), resulting in a typical conduction velocity of 54 cm/s. In Figure 8B, coupling was moderately reduced (10-fold decrease of gj to 0.25 μS) without changing the intracellular myoplasmic resistivity (150 Ω/cm). For normal coupling, the gap junction conductance between cells was the same as the myoplasmic conductance of the entire cell. As a result, the time spent by the impulse in crossing the gap junction (∼0.1 ms) was the same as the conduction time across the cell length. Since the model cell is 100 μm long while a gap junction cleft is only ∼80 Å wide, this large difference in dimensions implies that propagation in a linear cell chain is discontinuous at the cellular level even in the state of normal cell-to-cell coupling. Very similar values were obtained in experiments using synthetic cell chains of neonatal rat ventricular cells. In these (smaller) cells the average cytoplasmic conduction time was 38 μs, compared with 80 μs across the gap junctions at the cell ends (93). The fact that cells are coupled in both the lateral and longitudinal directions diminishes the effect of the high resistance pathway represented by a single gap junction. Thus, at normal cell-to-cell coupling, it was shown that the junctional delay that was 50% of the total conduction time in linear cell chains (93) decreased to ∼20% in multicellular strands containing lateral junctions. This effect of lateral apposition of cells to render propagation significantly more homogeneous was explained by electrotonic current through the lateral gap junctions that acts to smooth inhomogeneities in the wavefront during longitudinal propagation (socalled lateral averaging”). Extrapolating these data to three dimensions and taking into account the three-dimensional connectivity of an average working myocyte suggests that three-dimensional propagation can be considered continuous under physiological conditions of normal gap junctional coupling.

fig. 8.

Effect of a reduction in gap junctional conductance on cell-to-cell propagation. AP upstrokes from the edge elements of neighboring cells are shown in A and B (see inset). A: gap junction conductance (gj) = 2.5 μS (normal coupling). B: gj = 0.25 μS (reduced coupling). For normal coupling (A), intercellular conduction delay at the gap junction (shaded) is approximately equal to intracellular (myoplasmic) conduction time. A 10-fold decrease in gap junction conductance (B) increases the intercellular delay and decreases intracellular conduction time dramatically, resulting in gap junction dominance of macroscopic conduction velocity. Bottom panel: a small segment of the theoretical model used in the simulations of A and B. The multicellular fiber is composed of 100-μm-long LRd cells, interconnected through gap junctions. [Modified from Shaw and Rudy (305).]

In most simulation studies relating the functional role of gap junctions to the process of electric impulse conduction, the gap junctional conductance is represented by the reciprocal value of a simple resistor. Based on double-voltage clamp experiments defining the kinetics and open states of single connexins, a dynamic model of gap junctional conductance was recently presented (346) and used to simulate propagation (128). It was shown that gap junctional conductance is not uniform in time, but increases moderately immediately after passage of the wavefront. It was suggested that such a mechanism, which directs the local circuit current involved in propagation, could play a role in partially uncoupled tissue, such as in myocardial ischemia or infarction.

3. Role of channel clustering

Similarly to the clustered localization of connexins in the membrane, ion channels participating in the process of impulse spread appear to be confined to specific areas in the cell membrane as well. Thus it has been reported that Na+ and K+ channels are concentrated close to gap junctions, (44, 216, 257, 275) while L-type Ca2+ channels colocalize with the invaginating t tubules.(112, 300). In the study by Spach et al. (318), clustering of Na+ channels close to gap junctions (212), simulated by selection of different values for maximal Na+ conductance at different cell locations, had no significant effect on propagation velocity. A recent study by Kucera et al. (179) that incorporated both the gap junctions and intercellular clefts showed that Na+ channel clustering at the junctions could facilitate conduction when gap junction coupling is greatly reduced (<10%).

D. Propagation and the Shape of the Cardiac Action Potential

Many studies related to the mechanism of propagation in two-dimensional networks addressed the question of the direction-dependent characteristics of impulse conduction and of related changes in the shape of the cardiac AP. The special importance of these direction-dependent mechanisms was underlined early on by the experimental demonstration of anisotropy-dependent initiation of reentry (70, 323). Although early studies were partially controversial (43, 323), it has now become clear that the shape of the transmembrane AP is dependent on the direction of impulse spread. This includes both the initial subthreshold phase, generated by the flow of electrotonic current into the local membrane capacitance (129, 317, 379), and the AP upstroke phase, determined by flow of electric charge through open Na+ and/or Ca2+ channels.

In cellular networks with relatively large cells, such as canine myocardium (average length, 122 μm; Ref. 207), the shape of the transmembrane AP is determined by the rules governing discontinuous conduction, as explained in section iiiB. A detailed simulation of two-dimensional anisotropic conduction is illustrated in Figure 9 (316). This theoretical model used different values for cell-to-cell resistances located at either cell ends or lateral cell borders. Each cell in the network was composed of up to 36 excitable elements. Moreover, the model mimicked the naturally occurring cell shapes and sizes of the canine ventricle (136). The effect of the discontinuities formed by the cell borders and the location-dependent presence of gap junctions produced distinct profiles of propagation velocity, INa, and of the steepness of the AP upstroke. It can be seen that conduction velocity is not homogeneous throughout the cell. Instead, there is a slight decrease of velocity where local current flowing through gap junctions is dispersed, i.e., beyond gap junctions; inversely, there is a local increase of conduction velocity as the impulse approaches the cell-to-cell connections, because of reflection of axial current by cell borders at these sites (“partial collision”). Similarly, the steepness of the transmembrane AP, reflected in the maximal upstroke velocity dVm/dtmax, is lowest at the dispersion sites beyond gap junctions and highest at partial sites of reflection of intracellular electrotonic current, i.e., before gap junctions (194, 316, 318). Similar spatial dependence relative to gap junction locations of conduction velocity and dVm/dtmax was simulated in a one-dimensional multicellular model, where each cell was discretized into 20 excitable elements (287). Since the spatial subcellular profiles of velocity and dVm/dtmax result from an interaction of the propagating wave with the discontinuous architecture of the cellular network, they depend in two- and three-dimensional tissue models on the direction of impulse spread.

fig. 9.

Subcellular heterogeneity of activation (A), dV/dtmax (B), and INa (C) in a network of simulated dog myocytes. Left graphs correspond to longitudinal propagation from left to right, and right graphs correspond to transverse propagation from top to bottom. Note the close direct correspondence between isochrone spacing and dV/dtmax and the inverse correspondence to INa during both transverse and longitudinal propagation. Immediately after passage of the wavefront through gap junctions, dV/dtmax and conduction velocity show low values and INa is high (sites of current dispersion) while the inverse situation is present before the passage of the waves through gap junctions (sites of partial collision). [Modified from Spach and Heidlage (316).]

The higher values measured for dVm/dtmax during propagation in the direction of the transverse cell axis is a consequence of the higher degree of discontinuity in this direction (150, 286, 318). Since the degree of discontinuity can be changed (see Fig. 5 and Ref. 150) by modifications in the low resistance segments (corresponding to the cell) and the high resistance segments (corresponding to the intercellular connections) both are predicted to affect not only conduction velocity but also dVm/dtmax of the AP upstroke. Figure 10 depicts dVm/dtmax in a one-dimensional chain of cells for a large range of intercellular coupling (305); dVm/dtmax for a continuous fiber is provided for comparison (in this continuous simulation gap junctions are not present and their conductance is contained in the lumped intracellular resistance of the continuous fiber). Consistent with the classical theory of AP propagation in continuous media (e.g., the squid axon, Ref. 132), dVm/dtmax for the continuous fiber is constant for all values of fiber conductance. Therefore, it solely reflects membrane properties (membrane excitability) in this case. In contrast, dVm/dtmax of the discontinuous multicellular chain shows strong dependence on the degree of intercellular coupling when gap junctions are present. In response to progressive decrease of gap junction coupling, dVm/dtmax displays a biphasic behavior, first increasing to a maximum value and then decreasing. The maximum value of dVm/dtmax occurs at a gap junction conductance of gj = 0.04 μS, when coupling is reduced by a factor of ∼60 from its normal value of 2.5 μS. The initial increasing phase reflects an increasing degree of discontinuity in the fiber with greater confinement of depolarizing charge to individual cells. The later decreasing phase, occurring at very high values of intercellular resistance, is due to effects of decreasing electrotonic current flow into the cells downstream and will be discussed in section vB.

fig. 10.

Effect of decreasing gap junctional conductance on the upstroke velocity (dVm/dt)max and peak INa. Maximum upstroke velocity (dVm/dt)max (solid line) and peak sodium current INa,max (short-dashed line) are shown as a function of intercellular coupling through gap junctions in a multicellular fiber. (dVm/dt)max for a continuous fiber (long-dashed line) is also shown for comparison (the continuous fiber does not contain discrete cells and gap junctions: “intercellular conductance” in this case is evenly spread along the fiber). INa,max decreases slowly at first and then abruptly towards conduction block (indicated by the black horizontal bar). The sharp decrease is due to dynamic inactivation of sodium channels during slow depolarization (small depolarizing current through low conductance gap junctions) to threshold. Discontinuous (dVm/dt)max increases initially due to greater local confinement of depolarizing current. It then decreases following the decrease of INa,max. Note that the continuous (dVm/dt)max remains constant, unaffected by changes in the fiber conductance. [Modified from Shaw and Rudy (305).]

In accordance with the theory of discontinuous conduction it has recently been shown that cell size affects the shape of the transmembrane AP and conduction velocity (318). Figure 7 illustrates the results of a theoretical study (columns a–d are described in detail in an earlier section) that assessed the effect of cell size and gap junction distribution pattern on cell-to-cell delays and dVm/dtmax. As shown in Figure 7, cell size was a major determinant of dVm/dtmax, while the gap junction distribution pattern had only a minor effect. Computation of the downstream load imposed on the excited cells (data not shown), which was taken as a measure of the degree of resistive discontinuity, showed that decreasing cell size also decreases the degree of discontinuity. These findings are likely to explain previous, seemingly controversial, experimental findings showing that dVm/dtmax is dependent on the direction of propagation relative to the axis of the cardiac strands in canine ventricle (large cells) (323), but independent of the direction of propagation in anisotropic rat ventricular cultures (small cells) (94).

E. Conduction and Cell-to-Cell Interaction Between Myocytes and Nonmyocytes

It is normally assumed that cell-to-cell coupling by gap junctions is confined to the myocyte compartment, while coupling is absent between myocytes and nonmyocyte cells, such as fibroblasts and vascular endothelial cells. Although it is undisputed that connective tissue layers act functionally as electric insulators, the above paradigm has recently been questioned in the context of electric phenomena observed after heart transplantation. Thus a number of clinical reports have indicated that unidirectional or bidirectional conduction may be established between recipient and donor hearts. Atrial conduction across the boundary between the recipient and donor hearts was observed 7 years after transplantation in 10% of the cases (9, 185, 193, 292).

Thus far, experimental proof of cell-to-cell coupling between myocytes and nonmyocytes has been obtained in vitro; direct proof in vivo is still lacking (93, 281, 282). Figure 11 depicts propagation of the electric impulse across a dense monolayer of neonatal rat myocytes. Within the field of vision two nonmyocyte cells are interspersed. Myocyte-to-nonmyocyte coupling is suggested by the fact that the local current circuits during propagation produce an electrotonic “action potential” in the nonmyocyte cell with a very slow upstroke. The fact that the inexcitable nonmyocyte cell only consumes local current but does not contribute to current flow produces a distinct local slowing of conduction. Until now, the types of gap junctions between myocytes and nonmyocytes have not been characterized in terms of biophysical properties or connexin composition. Their role in pathophysiological settings in vivo remains to be investigated.

fig. 11.

Effect of nonmyocyte cells on microscopic propagation in culture. A and B: isochronal maps of propagation (top to bottom in A; right to left in B), relative activation times are given in μs. The isochronal maps are superimposed on a drawing (exact reproduction of microscopic picture) that illustrates cell borders with the myocytes dark-shaded and nonmyocyte cells light-shaded. Note that nonmyocytes produce a local conduction delay, suggesting that they are electrically coupled to myocytes. C: action potentials (rising phase) from myocytes (top and bottom trace) and membrane potential from a nonmyocyte cell (middle trace). Note that the electrically coupled nonmyocyte cell shows a membrane potential of low upstroke velocity. D: map of maximal upstroke velocities dV/dtmax of all transmembrane action potentials showing a dip of low values at the location of the nonmyocyte cells. [Modified from Fast et al. (90).]

F. Determination of Local Activation From the Extracellular Electrogram

In many clinical and experimental settings, local activation times are determined from the shape of the local unipolar electrogram, or alternatively, from the bipolar electrogram, which corresponds to the voltage difference between two closely spaced unipolar leads. Classically, the moment of local activation is taken as the moment of the steepest negative deflection (“instrinsic deflection”) of the unipolar electrogram, which coincides with the steepest portion of the upstroke of the local AP (133). Although this relationship holds for tissue with continuous conduction, resistive discontinuities render the interpretation of the shape of the extracellular electrogram more complex (96, 211, 287). Figure 12 shows the simulated membrane potential and the corresponding extracellular potential (unipolar electrogram) at the surface a cell chain (287). For the case of normal cell-to-cell coupling, the membrane potential and the extracellular potential are smooth and do not reflect the underlying discrete structure of the multicellular fiber (Fig. 12A). With reduced cell-to-cell coupling (Fig. 12B), irregularities appear in both the membrane potential and the extracellular potential waveforms. The main negative deflection of the electrogram reflects activation of the local cell. In addition, the electrogram displays a positive hump in the early phase (Fig. 12, 1) and a negative notch in the terminal phase (Fig. 12, 2), which correspond to activation of the upstream and downstream neighboring cells, respectively. Thus, for normal cellular coupling, the extracellular electrogram is indistinguishable from that generated in a continuous syncytium. However, as cells become less coupled, the discontinuous nature of propagation and depolarization events of individual cells are revealed in the electrogram waveform. Under conditions of greatly reduced coupling, propagation delays at gap junctions are long. As a result, the depolarization events of neighboring cells are sufficiently separated in time to be reflected as distinct deflections in the extracellular potential waveform. The simulations of Figure 12 can be scaled to represent structural discontinuities on a more global level. Each excitable element can be thought of as representing a tightly coupled fiber bundle separated from neighboring bundles by connective tissue, except at scarce sites of coupling. This situation is characteristic of the inhomogeneous substrates associated with myocardial infarction and aging cardiac tissue, where increased separation between fiber bundles occurs due to ingrowth of connective tissue septa. Indeed, irregular “fractionated” electrograms, reflecting conduction delays between fiber bundles, are measured in these settings (70, 314, 315, 319).

fig. 12.

Relationship between the upstroke of the transmembrane action potential and the extracellular unipolar electrogram. Action potential (Vm) and corresponding extracellular potential (Φe) at the surface of the fiber are shown for different degrees of intercellular coupling. A: normal coupling. B: coupling is reduced by a factor of 50. B: 1 and 2 indicate irregularities in extracellular potential waveforms (see text). [Modified from Rudy and Quan (287).]


Slow conduction in the heart occurs as a normal physiological phenomenon across the atrioventricular node, where it serves to adjust the time sequence of atrial and ventricular contraction (217, 221), and in many pathophysiological settings associated with arrhythmogenesis. In the latter, the slowing of conduction is an important determinant of the size of reentrant circuits, since it scales the wavelength of excitation λ, which is given by the product Math(6) where θ is conduction velocity and tr equals the refractory period of the tissue. For example, a decrease of conduction velocity to 10 cm/s in the presence of a refractory period of 100 ms yields a wavelength of 1 cm, or if 1 cm corresponds to the circumference of the inner core of a circuit in reentrant excitation, to a circuit radius of 1.6 mm. Besides the scaling of reentrant circuits by conduction velocity, a further important issue relates to the change in the SF during the development of slow conduction. As will be shown, there is an essential difference in the alteration of SF between slow conduction due to a decrease in excitability and slow conduction due to cellular uncoupling or structural heterogeneity. This predicts an entirely different behavior of the stability of conduction patterns in the two cases.

A. Slow Conduction and Conduction Failure Due to Reduced Membrane Excitability

As stated earlier, AP propagation and conduction velocity in cardiac tissue are determined by source-sink relationships, which reflect the interplay between membrane factors (source) and tissue-structure factors (sink). In this section, we explore the properties and ionic mechanisms associated with conduction slowing and block caused by reduced membrane excitability.

The most important determinant of reduced excitability is the reduced availability of Na+ channels leading to a reduction of INa. This condition is present during acute ischemia (72, 168, 172, 341), tachycardia (341), certain electric remodeling processes (111, 262, 386), and treatment with class I antiarrhythmic agents (23, 33). It can also be a consequence of genetic mutations that result in loss of sodium channel function, as occurs in the Brugada syndrome (22, 380). Figure 13 (305) shows the simulated effects of progressive reduction of excitability (reduction in the membrane density of available sodium channels) on conduction velocity and the SF for conduction. As excitability decreases, less INa source current is generated, and both velocity and SF decrease monotonically. When SF falls below 1, conduction cannot be sustained anymore and failure occurs. The minimum possible conduction velocity prior to failure in this simulation is 17 cm/s, a value that is reduced by only a factor of 3 relative to control velocity (54 cm/s) at full membrane excitability. Thus reduced membrane excitability does not support very slow conduction; rather, it leads to a transition to abrupt conduction failure from conduction velocities that are relatively fast. This simulation is in close quantitative accordance with experimental studies. Thus conduction was found to be abruptly blocked by tetrodotoxin (TTX) and class I antiarrhythmic agents at velocities of ∼25 cm/s (33), by elevation of extracellular K+ at ∼10–35 cm/s (156, 168), and by acute ischemia at 20–30 cm/s (156, 168).

fig. 13.

Conduction during reduced membrane excitability. Conduction velocity (solid line) and safety factor (SF) of conduction (dashed line) versus sodium channel availability (%gNa). Both velocity and SF decrease monotonically as membrane excitability is reduced. Horizontal bars indicate conduction failure, which occurs when SF decreases below 1. The slowest conduction velocity attainable before failure is 17 cm/s. [From Shaw and Rudy (305).]

The abrupt onset of conduction failure can be explained by the SF curve of Figure 13. Initially, SF decreases very slowly and is relatively insensitive to moderate changes in membrane excitability (Na+ channel availability) because sufficient current is generated to reach excitation threshold. However, for extreme reduction in channel availability (INa availability <11%), the generated depolarizing charge is not sufficient to depolarize the membrane to excitation threshold, and SF drops precipitously toward 1. The rapid decrease of SF toward 1 (conduction block) beyond a critical value of reduced excitability reflects the nonlinear “all-or-none” behavior of the cell membrane and the threshold phenomenon that characterizes the excitation process.

An important mechanistic question is whether the L-type calcium current contributes to propagation during slow conduction caused by INa suppression. [It will be shown in subsequent sections that ICa(L) is important in supporting slow conduction that involves long local conduction delays at uncoupled gap junctions or other structural discontinuities.] This issue is illustrated in Figure 14A (305), where SF is computed as a function of membrane excitability with (solid line) and without (dashed line) ICa(L) present in the LRd cell model. For most of the excitability range, ICa(L) does not influence the SF curve. At extremely suppressed INa (≤30% availability), ICa(L) augments SF slightly, resulting in a shift of conduction failure to a slightly more depressed membrane than occurs with INa alone. The contribution of ICa(L) to AP propagation under conditions of depressed excitability is further examined in Figure 14B. Shown is the Vm during the AP rising phase with (solid line) and without (dotted line) ICa(L), for severely depressed membrane (20% INa availability). The slightly higher amplitude of the AP when ICa(L) is present acts to increase the electrotonic driving force (voltage gradient) and, consequently, the depolarizing current to downstream tissue. The increased supply of depolarizing charge slightly augments the SF. Quantitatively, however, the effect is very small. The bar graph inset of Figure 14B shows the charge generated by INa and ICa(L) to depolarize the downstream tissue under these conditions of an extremely depressed membrane. Charge is computed by integrating current over time from dVm/dtmax of a source cell to dVm/dtmax of its downstream neighbor (identified by a thin vertical line in the Fig. 14B). This interval is very short because the downstream neighbor is activated early during the plateau of the upstream source cell (a long intercellular delay at the gap junction is not present since coupling is normal). During this short interval, INa is near its maximum and ICa(L) is only beginning to activate. As a result, the computed ratio of charge contribution QNa/QCa to propagation is 75:1, clearly indicating that INa predominantly maintains conduction even when its availability is greatly reduced under conditions of severely depressed membrane excitability. The contribution of ICa(L) to propagation can be increased in adult porcine heart by addition of epinephrine in the presence of decreased INa. Thus, if extracellular K+ concentration ([K+]o) is elevated to 20 mM and normal INa-dependent propagation is completely blocked, addition of epinephrine can produce slow, ICa(L)-dependent propagation with velocity of 10 cm/s in the longitudinal direction of the fiber axis (168), in close agreement with theoretical simulations (303). Similarly, spontaneous ICa(L)-dependent propagation can be demonstrated in neonatal rat myocyte strands if INa is inactivated in the presence of elevated [K+]o or by TTX (177, 278). The increase of ICa(L) in neonatal cultured myocytes occurs in the presence of epinephrine or serum in the culture medium (117, 206). ICa(L)-dependent propagation was elicited in an early study to model microreentry in canine Purkinje fibers (58).

fig. 14.

The role of the L-type Ca2+ current [ICa(L)] in propagation during reduced excitability. A: SF over a range of membrane excitability computed with (solid line) and without (dashed line) contribution from ICa(L). SF without ICa(L) begins to diverge from SF with ICa(L) below 30% sodium channel availability. Horizontal bars at SF = 1 indicate occurrence of conduction failure. B: AP upstrokes during conduction at greatly reduced membrane excitability (only 20% sodium channel availability), with (solid line) and without (dashed line) ICa(L). Inset: bar graph shows relative charge Q (current integrated over time) generated by INa (QNa) and by ICa(L) [QCa(L)] in the period from a given cell's excitation (0 ms) to excitation of its downstream adjoining cell (marked by thin vertical line at time = 0.37 ms). Clearly, even at this level of severe INa suppression, charge contribution by INa to support conduction far exceeds charge contribution from ICa(L), and conduction is predominantly maintained by INa. [From Shaw and Rudy (305).]

1. AP propagation in acute myocardial ischemia

Acute myocardial ischemia is a major cause of ventricular tachycardia and fibrillation. The pathophysiological conditions of ischemia affect membrane ionic currents and intracellular and extracellular ionic concentrations, leading to changes in membrane excitability and AP conduction (144, 148). The acute phase of ischemia (first 10–15 min) does not involve gap junction uncoupling and is due mainly to a change in membrane excitability (165, 170). Therefore, the changes in propagation and the resulting reentrant arrhythmias can be viewed as a typical example of an excitability-related mechanism. The major contributing factors during this phase are extracellular potassium accumulation, metabolic acidosis (decrease in intracellular and extracellular pH), and anoxia with a decrease in intracellular ATP levels ([ATP]i). These conditions are not uniform in the ischemic zone but undergo a change from the ischemic center to the border (52). Figure 15A shows the simulated relationship between membrane excitability (as reflected in dVm/dtmax) and conduction velocity during hyperkalemia alone (solid curve) and with added acidotic conditions (dashed curve) (303). Corresponding experimental results in a guinea pig papillary muscle preparation are provided in Figure 15B for comparison (156). As [K+]o is raised from an initial low value of 3 mM, dV/dtmax shows very little change, but velocity increases to a maximum. In this example, the maximum velocity occurs at [K+]o = 8.2 and 8.0 mM for hyperkalemia alone and with added acidosis, respectively. The phenomenon of increased conduction velocity at slightly elevated levels of [K+]o is due to a transient decrease in threshold requirements for INa activation and is an example of “supernormal conduction.” At this phase, elevation of [K+]o is associated with resting membrane depolarization in a range that remains too negative to cause significant Na+ channel inactivation, hence the almost constant dVm/dtmax. Further increases in [K+]o and the associated membrane depolarization lead to significant Na+ channel inactivation and reduced availability. The result is depressed membrane excitability [reduced (dVm/dt)max], reduced conduction velocity, and eventually the development of conduction block. For hyperkalemia alone, propagation fails at [K+]o >14.4 mM. With added acidotic effects which further suppress INa (384), failure occurs at [K+]o >13.1 mM. These values are well within the range of [K+]o elevation observed during acute ischemia (166, 168). The slowest velocity attainable just before block is ∼17 cm/s (hyperkalemia alone) and 27 cm/s (hyperkalemia and acidosis), values that are in close accordance with experimental results in the acutely ischemic porcine heart (168). Clearly, reduced membrane excitability during acute ischemia cannot support very slow conduction and favors the occurrence of conduction failure. The simulations also reveal that elevated [K+]o (rather than acidosis or anoxia) is the major determinant of conduction slowing during acute ischemia and that conduction depends mostly on INa even when [K+]o is greatly elevated and excitability is highly depressed (304). Contribution of ICa(L) to depressed conduction during hyperkalemia is minor. However, if ICa(L) is augmented by 10% during pure hyperkalemia or during complete ischemia (to simulate the effect of epinephrine), ICa(L)-supported propagation is successful even at [K+]o of 20 mM with a conduction velocity of 10 cm/s (303), in close agreement with experimental observations (168). A further typical feature of propagation across ischemic myocardium relates to the dependence of propagation on heart rate. If propagation at a given heart rate is blocked in the presence of elevated [K+]o due to INa inactivation, an abrupt decrease in stimulation rate restores propagation immediately, because of time-dependent recovery of INa from inactivation (113, 172). This illustrates that very small changes in the availability of Na+ channels will decide whether propagation occurs with substantial velocity (i.e., at ∼20 cm/s) or whether it is completely blocked. This rate-dependent rapid transition from block to propagation is responsible for the very dynamic changes in size and position of macro-reentrant circuits during acute ischemia (144, 146, 148).

fig. 15.

Effects of simulated ischemia on (dVm/dt)max and propagation velocity and “comma-shaped” relation between (dVm/dt)max and conduction velocity during hyperkalemia. Two cases are shown: hyperkalemia alone (nonacidotic) and hyperkalemia in the presence of acidosis (acidotic). A: theoretical results. Numbers next to solid circles correspond to [K+]o (in mM). Propagation fails at [K+]o = 14.4 mM in the nonacidic fiber and [K+]o = 13.1 mM in the acidic fiber. B: experimental data obtained under similar conditions from guinea pig papillary muscle (156). [From Shaw and Rudy (303).]

B. Slow Conduction Related to Reduced Cell-to-Cell Coupling

Modification of cell-to-cell coupling occurs in a number of physiological and pathophysiological conditions. Physiologically, cell-to-cell coupling may be reduced in the transverse direction to the main fiber axis relative to the longitudinal direction, in atrial and ventricular myocardium (136, 316, 318). Reduced intercellular coupling is also likely to contribute to slow impulse conduction in the atrioventricular node (219). In pathophysiological settings, rapid uncoupling occurs in acutely ischemic myocardium after the first 10–15 min of ischemia (170, 272), and cell-to-cell coupling is also modified in ventricular hypertrophy (255), ventricular failure (45, 220), and tissue surviving infarction (256). Mechanistically, modification of cell-to-cell coupling may occur as a consequence of acute changes in the average conductance of gap junctions secondary to ischemia, hypoxia, acidification, or increase in intracellular [Ca2+] (82, 102, 170, 228, 237, 272, 376), or it may be brought about by changes in expression or cellular distribution patterns of gap junctions (61, 388). In this section we describe the mechanistic principles and basic properties of conduction in a conducting medium characterized by reduced intercellular coupling.

Changes in conduction velocity and SF caused by reduction in cell-to-cell coupling in a simulated linear cell strand are shown in Figure 16 (305). Similarly to its behavior during reduced membrane excitability (see Fig. 13), conduction velocity decreases monotonically with reduction in intercellular coupling (286, 305). However, quantitatively, the slowing is more dramatic in the reduced coupling case. The changes in SF with uncoupling are opposite to the changes observed with a reduction in excitability. SF increases to a maximum as coupling is reduced and velocity is significantly slowed. SF is greatest at a level of gap junction coupling that is reduced by a factor of ∼100 (to 0.023 μS) relative to the normal value of 2.5 μS. At this level of coupling, conduction is very slow (∼1/15 of normal velocity) but, paradoxically, very robust, i.e., conduction safety is much higher than during normal conduction. Due to the high SF, extremely slow conduction velocities can be sustained in tissue with greatly reduced intercellular coupling. The minimum velocity before block is 0.26 cm/s, a 200-fold decrease from the normal value in well-coupled tissue. This simulated value is very close to values obtained experimentally in uncoupled synthetic rat myocyte strands (278). It should be mentioned that a very large reduction of intercellular coupling is required to cause major slowing of conduction velocity. A recent study demonstrated a very minor slowing of conduction with 43% reduction of Cx43 expression (331).

fig. 16.

Conduction during reduced intercellular gap junctional coupling. Conduction velocity (solid line) and safety factor (SF) for conduction (dashed line) versus gap junction conductance (gj). Velocity decreases monotonically as coupling is reduced. In contrast, SF first “paradoxically” increases to a maximum and then decreases sharply to the point of conduction block (horizontal bar at SF = 1). Conduction reaches a very slow velocity (0.26 cm/s) before block occurs. [From Shaw and Rudy (305).]

The increase of SF with reduced coupling is a manifestation of the discontinuous nature of conduction in this setting. As cells become less coupled, there is greater confinement of depolarizing current to the depolarizing cell with less electrotonic load and axial flow of charge to the downstream cells. As a result, individual cells depolarize with a high margin of safety, but conduction proceeds with long intercellular delays. Therefore, under conditions of reduced coupling, propagation is slow, robust, and discontinuous. In accordance with the mechanism of discontinuous conduction, illustrated by Figure 5, the decrease in gap junctional coupling is associated with 1) a change in the type of conduction from continuous to discontinuous, 2) a change in the shape of the AP rising phase, and 3) a change in the contribution of INa and ICa(L) to conduction.

The change in shape and the increase of the average upstroke velocity dVm/dtmax of the transmembrane AP rising phase during cell-to-cell uncoupling has been explained in detail in the paragraph describing discontinuous conduction. As was shown in Figure 10 for a linear simulated cell strand, the curve describing dVm/dtmax as a function of uncoupling is biphasic, first increasing to a maximum value and then decreasing. The maximum value of dVm/dtmax occurs at a gap junction conductance of 0.04 μS, when coupling is reduced by a factor of ∼60 from its normal value of 2.5 μS. INa,max does not follow a similar biphasic behavior and shows very little change (slight decline) as gj is reduced toward 0.04 μS, at which dVm/dtmax reaches its maximum value. This indicates that the biphasic change in dVm/dtmax is not caused by membrane currents; rather, it results from altered source-sink relationships in the fiber due to the changes in intercellular coupling. In other words, the deviation of dVm/dtmax from a constant (the continuous case) is indicative of the discontinuous nature of conduction. Inward transmembrane current acts to depolarize local membrane and to generate axial flow of current to the less depolarized down-stream tissue. When coupling is reduced (reduced electric load), less current is shunted downstream, effectively increasing the availability of charge for local depolarization. The result is a faster rate of depolarization, which is reflected in the increased dVm/dtmax. Beyond its maximum value, dVm/dtmax declines sharply with further reduction of coupling. It is clear from Figure 10 that the decline of dVm/dtmax accompanies a sharp decline of INa,max during this phase of extreme discontinuity. During this phase, the membrane charging process is very slow due to the small axial current when resistance between cells is large. The long subthreshold depolarization process provides time for sodium channel inactivation before they reach their activation threshold, resulting in reduced channel availability and a smaller INa. Eventually, channel inactivation and reduced INa is not compensated for by the conservation of charge for local depolarization due to the reduced coupling and propagation fails. In contrast to the simulations shown in Figure 10, experiments on the effect of cell-to-cell uncoupling in synthetic neonatal rat myocyte strands did not show an increase in dVm/dtmax with reduced level of coupling (278). This observation is in accordance with the finding that dVm/dtmax is also independent of the direction of anisotropic impulse spread in cultured networks of neonatal rat myocytes (94). The most likely explanation for this lack of effect is the smaller cell size of neonatal rat versus adult guinea pig myocytes (100-μm-long guinea pig myocytes are simulated in Fig. 10). The smaller cell size and the relatively short cytoplasmic pathway are predicted to render the function of such networks less discontinuous for a given state of cell-to-cell coupling, as illustrated in Figure 7 (318).

The change in activation properties with advanced cell-to-cell uncoupling, and consequently, the increasing discontinuity of conduction, can assume extreme forms as illustrated in Figure 17 taken from an experiment using arachidonic acid (10 mM) to uncouple a cultured synthetic strand of neonatal rat myocytes (278). During normal propagation (not shown) such strands are conducting at a normal velocity of 30–40 cm/s (93, 177, 278), and about equal time delays are measured along the cytoplasm and across the cell junctions (93). In an advanced stage of cell-to-cell uncoupling, clusters of APs are observed (Fig. 17B), which localize to one to three cells (Fig. 17C). Between these clusters, i.e., across the borders of individual cells, delays of up to several milliseconds are observed (note similarity with the theoretical stimulation of Fig. 8B). These experimental results, showing a kind of “saltatory conduction,” are in exact correspondence with theoretical simulations of the discontinuous behavior of uncoupled tissue (305) and explain the very slow propagation velocities (<1 cm/s) under conditions of highly reduced coupling.

fig. 17.

Action potential propagation during marked cell-to-cell uncoupling in a synthetic strand of neonatal rat myocytes. A: microphotograph of a segment of a synthetic neonatal rat myocyte strand. White circles denote position of each light-measuring diode. Strand width was 60 μm. B: action potentials, recorded by each diode in A, show clustering of upstrokes in groups. Within each group excitation is almost simultaneous; propagation delays of up to 5 ms occur between groups. C: superimposition of groups of almost simultaneous action potentials on cellular morphology. Note that 1–3 cells are excited almost simultaneously and that complete conduction block occurs in the cell marked by the rectangular hatching. Comparison of B and C shows that conduction delays are confined to cell borders. [Modified from Rohr et al. (278).]

The fact that the conduction delays between individual cells or group of cells markedly exceed the duration of the AP upstroke has important consequences regarding the role of the depolarizing currents, INa and ICa(L), in propagation of the AP (151, 181, 262, 305, 328). This is shown in Figure 18 (305), which illustrates the upstroke of a simulated AP in a cell (cell 1) during propagation in a linear cell strand. The time of depolarization of cell 1 is set to zero, and the delay between depolarization of cell 1 and the next cell downstream (cell 2) is indicated by a horizontal bar above the AP. In Figure 18B, the propagating AP upstroke and the early plateau is computed with (solid line) and without (dashed line) ICa(L). Intercellular coupling was decreased to 0.02 μS, i.e., to 100 times smaller than normal. This corresponds to a value that is slightly greater than the coupling at which conduction without ICa(L) fails. The identical upstrokes of the two APs indicate that ICa(L) does not play a direct role in local excitation. However, ICa(L) maintains the early plateau at significantly higher potentials. This, in turn, increases the voltage gradient and driving force to downstream tissue, forcing more source current to the adjacent depolarizing cell. In Figure 18B, inset, the bar graphs show the relative charge contribution from INa (QNa) and ICa(L) [QCa(L)] to sustain conduction. These charges are computed by integrating INa and ICa(L) over time from depolarization of cell 1 to depolarization of cell 2 (horizontal bar). At this level of reduced gap junction coupling, the downstream cell is activated when the upstream source cell is well into its plateau phase. During this phase, INa is already inactivated and ICa(L) is the major inward membrane current that generates a depolarizing charge. Consequently, charge contribution from ICa(L) becomes significant. In fact, the ratio QNa:QCa = 1.47:1 indicates an almost equal charge contribution from INa and ICa(L) and comparable importance of these currents in supporting conduction. A very similar role of ICa(L) was also described in impulse transfer between cell pairs (151, 181, 328). This finding is in sharp contrast to slow conduction during reduced membrane excitability (Fig. 14), where INa dominates conduction even when it is greatly suppressed. With further cell-to-cell uncoupling (Fig. 18C), ICa(L) becomes the major source of depolarizing charge (QNa:QCa = 0.26). Usually, the resting potential is considerably more negative than the ICa(L) activation threshold. Therefore, even when QCa > QNa and ICa(L) is the major contributor to depolarizing charge during conduction, INa is still needed to depolarize the membrane into the activation range of ICa(L).

fig. 18.

The role of the L-type Ca2+ current [ICa(L)] in propagation during reduced gap junctional conductance. A: a diagram showing direction of propagation. B: AP upstroke at greatly reduced gj = 0.020 μS with (solid line) and without (dashed line) ICa(L). The AP shown is for cell 1 (see diagram in A). Inset: bar graph shows relative charge contribution to conduction by INa (QNa) and ICa(L) [QCa(L)] in the period from excitation of cell 1 to excitation of cell 2 (3.6 ms, marked by the horizontal bar above the AP). The long intercellular delay extends into the plateau of the source cell (cell 1), during which ICa(L) is the major inward membrane current and source of depolarizing charge. Consequently, charge contribution from ICa(L) approaches the charge contribution from INa (bar graph). C: at extreme levels of uncoupling (gj = 0.006 μS), charge contribution from ICa(L) exceeds that from INa by up to an order of magnitude, and conduction depends strongly on ICa(L). D: SF over a range of gap junction coupling with (solid line) and without (dashed line) contribution from ICa(L). SF computed without ICa(L) begins to diverge from SF with ICa(L) at about fivefold decrease in coupling (gj = 0.5 μS). Conduction without ICa(L) fails at three times the conductance for which failure occurs when ICa(L) is present. [Modified from Shaw and Rudy (305).]

The changing role of ICa(L) in propagation with progressive cell-to-cell uncoupling is also reflected in the plot of Figure 18D that shows the SF in presence (solid line) and absence (dashed line) of ICa(L) over a full range of intercellular coupling. Both SF curves display a biphasic behavior (increase to a maximum followed by fast decrease toward conduction block). However, at reduced coupling the SF curves diverge and the SF without ICa(L) becomes significantly smaller. Thus, in the absence of ICa(L), conduction fails at a level of coupling that is three times higher than when ICa(L) is included. This finding strongly suggests that ICa(L) is a major contributor to conduction under conditions of greatly reduced intercellular coupling, when long propagation delays across gap junctions are present.

The membrane switch from INa supported conduction to mostly ICa(L) supported conduction when intercellular coupling is greatly reduced is an excellent example of the intimate interaction between the passive network properties (the multicellular network) and excitatory membrane currents during conduction. It shows that altered network properties (in this case reduced gap junction coupling) can cause major changes of the action of membrane currents even when the intrinsic properties (i.e., density and gating) of these currents remain unchanged. The modulation of the ionic currents that generate conduction by the passive network in which conduction occurs constitutes a feedback mechanism from the sink (load) to the source of excitation.

The simulations in Figure 18 were conducted using the LRd model of the guinea pig ventricular myocyte. The transient current Ito is not expressed in guinea pig ventricle and was not considered in these simulations. Experiments in coupled cell pairs from species that express Ito [rabbit atrium (353) or rabbit ventricle (137)] demonstrated facilitation of intercellular AP transmission across a large intercellular resistance when Ito was reduced by 4-aminopyridine block or fast pacing. Being an early plateau repolarizing current, Ito counteracts the depolarizing effect of ICa(L) and acts to inhibit conduction. Its suppression shifts the balance of currents in the depolarizing direction, thereby facilitating conduction in the presence of long local delays.

C. Slow Conduction Related to Tissue Structure

The discussion in the previous sections revealed a clear difference between the effects of reductions in excitability and in cell-to-cell coupling on SF and propagation velocity. Especially it revealed that very slow conduction (velocity <1 cm/s) can only occur at extreme levels of uncoupling. The observations that conduction in the center of the atrioventricular node (24, 25, 221, 244, 338) and in infarct scars (65, 66) is extremely slow have suggested that there might be a third mechanism of slow conduction related to specific tissue structures. In studies on the electrophysiology of the atrioventricular node it was observed early on that the node has a complex anatomy with so-called “dead end pathways” emerging from the center of the node. These pathways do not participate directly in atrioventricular or ventriculoatrial impulse transmission and are activated later than the central portion of the node. One type of dead end pathway consists of atrial overlay fibers terminating in the base of the septal leaflet of the tricuspid valve; another dead end pathway branches off the central node and extends posteriorly along the tricuspid orifice (8, 145, 265, 338, 378). In the setting of myocardial infarction, very long transition times (suggesting very slow conduction) were observed in the presence of APs with normal upstrokes. It was postulated that the long effective conduction times were due to zig-zagging of conduction in infarct scars that contain complex branching structures of viable myocardium (66).

An experimental and theoretical model to investigate the role of branching tissue structures in slow conduction was recently developed by Kucera and co-workers (177, 180). This structure is depicted in Figure 19A, the corresponding conduction profile is shown in Figure 19B. The principle of slow conduction is due, in analogy to the situation of cell-to-cell uncoupling, to a high degree of discontinuity. In contrast to an uncoupled cell strand (where the high resistance junctions alternate with the low cytoplasmic resistance of the cells), the high degree of discontinuity is achieved by large tissue segments (consisting of a segment with side branches) alternating with small tissue segments having a small tissue mass (connecting segments without branches). In correspondence with the rules of discontinuous conduction, two interacting processes determine the conduction properties of this structure. As conduction moves from the interconnecting segment to the branching segment, a small mass of cells has to excite the large mass in the branches. As a consequence, local current is dispersed; the side branches only activate after a delay and the macroscopic conduction velocity in the main strand decreases (so-called “pull effect”). Once the large mass in the side branches is excited, it can deliver excitatory current to the next interconnecting segment to drive propagation forward. This latter process (so-called “push effect”) is similar to partial collision and serves to maintain a high margin of safety. With an appropriate length of the side branches, ICa(L)-dependent conduction (INa completely inactivated) can be sustained at a minimum velocity as low as 1 cm/s. Rendering only the side branches inexcitable by local application of TTX produces conduction block in the entire structure and demonstrates the role of the side branches in conduction safety. Theoretical computation of the SF has confirmed the crucial role of the branching structures in maintaining the very slow conduction (180). The phenomenological observations made in the atrioventricular node and in infarct scars together with the experimental studies in cell cultures and the theoretical simulations underline the role of interactions between structure and ionic currents during AP propagation. Therefore, the classical concept stating that slow conduction in the atrioventricular node is solely due to flow of “slow” Ca2+ inward current requires reexamination. In cases where conduction delays above a critical “threshold” are imposed by a discontinuous structure, the flow of INa, as determined by the kinetic properties of the Na+ channels, is too rapid to provide depolarizing charge for a sufficiently long duration, and flow of the slower ICa(L) is also necessary for sustaining propagation. Thus there is a mutual interdependence between the passive tissue structure and the role and efficacy of ionic membrane currents.

fig. 19.

Very slow conduction due to a discontinuous tissue structure. A: structure created in synthetic strand of neonatal rat myocytes showing a main horizontal strand from which vertical side branches emerge (the ends of these branches are not shown). B: conduction profile along the main strand (oblique profile, solid squares) and in the two side branches (horizontal profiles, solid circles) shown in A. The oblique profile shows a staircase-like pattern with a delay corresponding to the segment approaching the branch point and relatively fast conduction at and immediately beyond each branching point. The side branches draw depolarizing charge from the main strand (pull effect) and are activated almost simultaneously to form a large source that facilitates excitation of the next linear segment (push effect). C: conduction velocity as a function of branch length at elevated extracellular [K+] [ICa(L)-dependent conduction]. At branch length = 0, a situation corresponding to a linear strand (without branches), the minimum sustained propagation velocity is between 15 and 20 cm/s; it decreases with increasing branch length to <1 cm/s. [Modified from Kucera et al. (177).]


A. General Principles

In the previous sections, the discussion of slow conduction and conduction failure in cardiac tissue was focused on phenomena associated with uniform change of an electric parameter, either a decrease in excitability or in cell-to-cell coupling. However, in most instances conduction slowing or block occurs at particular locations within the tissue while more robust conduction is maintained at other sites. This heterogeneity of impulse conduction is a key phenomenon for the initiation of circulating excitation and, if additional conditions are fulfilled, the occurrence of reentrant arrhythmias. In essence, local source-sink relationships determine the formation of conduction heterogeneities (including unidirectional block) and provide conditions for the development of slow conduction, unidirectional block, and reentry.

B. Asymmetry of Membrane Excitability: the Vulnerable Window for Unidirectional Block

1. Interaction of electric excitation with a preceding excitation wave in an electrically homogeneous medium

It is important to realize that unidirectional block can occur in tissue with homogeneous and uniform electric properties, because of the functional electric asymmetry of the cardiac AP. The effects of this asymmetry can be enhanced by intrinsic dispersion of repolarization due to inhomogeneities of the tissue electric properties (147). If local cardiac excitation interacts with the repolarization phase (tail) of a preceding wave, a unidirectional block (UB) can develop (264, 302, 327). This interaction may arise as a result of 1) application of an external stimulus, 2) spontaneous initiation of a premature beat, or 3) coexistance of interacting wavefronts, e.g., during multiple wavelet reentry. The critical or vulnerable window within which UB occurs can be characterized as a time interval or window (TW), or alternatively, it can be represented as a distance in the space domain (SW) or as a range of membrane potentials in the voltage domain (VW). Outside this window it is impossible to induce UB (or, equivalently, unidirectional conduction). The vulnerable window during the relative refractory period of a propagating AP is depicted in Figure 20 (284). When a premature stimulus is applied outside this window, the induced AP either propagates or blocks in both directions; specifically, a stimulus that is applied too early fails to induce a propagating AP in either direction, while a late stimulus (to the right of the window) results in bidirectional conduction. For a stimulus within the window, the membrane generates a critical sodium current, giving rise to an AP that propagates incrementally in the retrograde direction (to the right in Figure 20) but blocks in the antegrade direction (left in Fig. 20) following a short distance of decremental conduction in this direction (264). The asymmetry in conduction occurs because in the retrograde direction the tissue is progressively more recovered as the distance from the window increases in this direction, while in the antegrade direction the tissue is progressively less excitable as the distance from the window increases. Figure 21 (302) provides examples of premature stimulation of a linear cell strand (see Fig. 8) and the APs elicited by responses to differently timed stimuli. Bidirectional block (Fig. 21A) or bidirectional conduction (Fig. 21C) is shown for stimuli occurring outside the window. When the stimulus falls within the window (Fig. 21B), UB develops and conduction is successful only in the retrograde direction. Figure 22 (264) shows the response to premature stimulation in the vulnerable window in greater detail and demonstrates its graded nature. In the antegrade direction, Vm (Fig. 22A) increases with distance from the window, as do dVm/dtmax and the velocity of conduction (data not shown). In contrast, these parameters decrease in the antegrade direction until propagation fails (see Vm in Fig. 22B). The incremental AP response in the retrograde direction reflects the underlying progressive recovery from inactivation of sodium channels, gNa (Fig. 22C). Thus gNa (Na+ channel conductance) recovers slowly from curve 1 (0.78 mS/cm2) to curve 2 (2.4 mS/cm2) in the retrograde direction. In the antegrade direction, gNa decreases sharply from curve 3 (0.62 mS/cm2) to curve 4 (0.18 mS/cm2), reflecting an abrupt decrease of sodium channel availability (Fig. 22D). This asymmetry of channel availability and membrane excitability results in the development of UB. The cause and effect relationship between UB and reentry implies that the size of the vulnerable window provides an index of the vulnerability to the development of reentrant arrhythmias. The window in the time domain TW is a convenient working definition of such vulnerability. For a large TW, the time interval during which unidirectional block can be induced is long. Therefore, the probability that a premature stimulus (e.g., during clinical electrophysiological evaluation in the catheterization laboratory, or a naturally occurring extrasystolic stimulus in the diseased heart) will fall inside the window and induce reentry is high. In contrast, very precise timing of a premature stimulus is required to induce reentry in a small window, and the probability of such an event is low. In normal tissue, TW is very small (<1 ms), and inducibility of unidirectional block and reentry is negligible (264, 302).

fig. 20.

Schematic representation of the vulnerable window during the refractory period of a propagating action potential. TW, SW, and VW represent the vulnerable window in the time domain, space domain, and voltage domain, respectively. [From Rudy (284).]

fig. 21.

Propagation or block of a wavefront in the repolarizing tail of a preceding wave. A: bidirectional block occurs when the premature stimulus is applied 198 ms after initiation of the conditioning action potential and results from an inability of the premature stimulus to excite the fiber in either direction. B: unidirectional block at 199 ms results from successful retrograde, but not antegrade, excitation. C: bidirectional conduction at 200 ms, when both retrograde and antegrade fiber segments are excited. The bottom trace in each panel is the membrane potential immediately prior to the premature stimulus. [From Shaw and Rudy (302).]

fig. 22.

Properties of propagation induced by a premature stimulus in the vulnerable window. Antegrade decremental propagation (right) and retrograde incremental propagation (left) is induced. Vm, membrane potential; gNa, sodium conductance which determines membrane excitability. The parameters are displayed at increasing distances (curves 1–5) from the site of premature stimulation in both directions. Note different gNa scales in left and right panels. [Modified from Quan and Rudy (264).]

The width of the vulnerable window may be affected by changes in several electrophysiological properties, such as the availability of Na+ channels for depolarization, cell-to-cell coupling (264, 302), and repolarizing K+ currents (327). For normal intercellular coupling of 2.5 μS, TW is only ∼0.4 ms; with a uniform decrease of intercellular coupling to 0.09 or 0.025 μS, TW increases to 3.7 and 30 ms, respectively.

Figures 20, 21, 22 show that electric asymmetry in time and space can exist in completely homogeneous cardiac tissue as a result of the passage of a (primary) impulse. Thus this asymmetry depends on the history of tissue excitation; it does not require heterogeneity of tissue properties and is therefore termed “functional heterogeneity.” However, any intrinsic asymmetry in electric properties of the tissue (e.g., in ion channel expression, excitability or refractory periods, gap junction coupling) may act to widen the vulnerable window and increase the probability of unidirectional block (264). Several mechanisms may underlie physiological electric heterogeneity in normal cardiac tissue and its accentuation in pathophysiological settings.

2. Local heterogeneity of excitability

Occurrence of asymmetrical UB in a region of depressed and heterogeneous excitability was postulated as early as 1895 by Engelmann in skeletal muscle (86). Subsequently, a variety of techniques were applied to produce local depression of excitability and conduction block [hyperkalemia (57, 297), focal cooling (73, 355), crushing, and application of depolarizing current (73, 297, 355)]. The importance of an asymmetry in excitability for the generation of UB is illustrated in Figure 23. The asymmetry in these experiments, produced by local cooling and crushing, led to an abrupt rise in the threshold for excitation in one direction and to a more gradual rise in the other. In between these zones, an inexcitable gap was located. An impulse propagating towards the gap in the retrograde direction, facing an abrupt transition in excitability, elicited an AP on the other side of the gap through electrotonic transmission, if the gap was sufficiently short. In contrast, the impulse was blocked at the same gap during antegrade conduction, because the gradual decrease of excitability reduced the AP amplitude and the driving force for electrotonic transmission. Thus conduction failed when the wavefront encountered the least depressed site first and was successful in the direction where it encountered the most depressed site first. Interestingly, Engelmann (87) postulated as early as 1896 that impulses are conducted more easily from a rapidly conducting tissue to a slowly conducting tissue than in the opposite direction.

fig. 23.

Asymmetric depression of excitability producing unidirectional block. Top panel: a gradient of increasing injury is produced by a crushing probe. The degree of shading on the Purkinje fiber indicates the increasing degree of injury. Bottom panel: during antegrade conduction, the line C-X-B with the arrow from left to right illustrates the change in amplitude of the antegrade wavefront. X represents the point of transition between partially excitable cells and inexcitable cells; the zone from X to Y is inexcitable (shaded box on the baseline). Antegrade conduction along C towards X is decremental, and the electrotonic transmission of the impulse across the X-Y region (segment B) fails to excite the site Y, where transition from the inexcitable, injured to the normal zone is abrupt. During retrograde conduction, the electrotonic transmission of the impulse from Y to X (segment A) is sufficient to excite site X; beyond site X conduction is incremental in segment C. This explains successful retrograde and failing antegrade conduction. [Modified from Waxman et al. (355).]

Acute ischemia represents a pathophysiological setting where local heterogeneity in excitability, leading to a widening of the vulnerable window, is a key factor for the generation of unidirectional block and arrhythmias. As discussed in the section on conduction slowing, the prolonged recovery from inexcitability associated with elevation of extracellular [K+], hypoxia, and acidification decreases the availability of Na+ channels for repetitive rapid excitation (113). Consequently, both the absolute and relative refractory periods become markedly prolonged; they exceed the duration of the previous AP and lead to a marked widening of the vulnerable window (144, 148). Importantly, regional ischemia is a setting where extracellular accumulation of [K+] occurs with a gradient from the center to the border of the ischemic zone, resulting in marked local inhomogeneities (4851). The fact that a local gradient in [K+]o and consequently in resting membrane potential are present produces a marked inhomogeneity in the local recovery of the Na+ channels from inactivation, which, as shown in Figure 22, determines the vulnerable window (148, 173).

3. Local heterogeneity of refractoriness

Local dispersion of refractory periods is a normal feature of ventricular myocardium. In the normal dog ventricle and at physiological heart rates, the differences between refractory periods amount to ∼40 ms (143, 147). Many experimental interventions have been used to increase the instrinsic heterogeneity of repolarization, and subsequently, to elicit arrhythmias by premature stimulation: lowering local temperature, sympathetic stimulation, ischemia, application of chloroform, cardiac steroids, or high doses of quinidine (124, 184, 349). During the past two decades research about the mechanisms of heterogeneity of repolarization has rapidly evolved, and several causes have been delineated: 1) intrinsic heterogeneity in cell types, that express a different spectrum of repolarizing K+ channels, late Na+ channels, and Na+/Ca2+ exchange current in the ventricles (10); 2) specific inherited mutations of Na+ and K+ channels leading to prolongation of the QT interval and an increased incidence of sudden death (the long QT syndrome) (260, 261); 3) drug-induced, acquired long QT syndrome owing to interference of drugs with IKr (160, 224, 274, 382, 383); and 4) electrical remodeling with increased heterogeneity of repolarization in the setting of ventricular hypertrophy and failure (21, 154, 259) and in the setting of myocardial infarction (258). The common denominator of these changes resides in the local widening of the vulnerability window and an increased probability for the generation of unidirectional block and reentry, as described above. Most of the heterogeneities in repolarization have been defined in the time domain; investigations about the spatial extension of these gradients are relatively scarce but may be crucial for the initiation of reentry (4).

C. Conduction Heterogeneities and Unidirectional Block Due to Discontinuities in Tissue Structure

In most regions of the heart, conduction does not occur as a continuous process; rather, propagating electric waves interact with tissue structures at all levels, in the complex and trabeculated structures of the atria (320, 322), the atrioventricular junction (8, 145, 338), the network of the specific ventricular conducting system (243, 313), and the ventricles (193a). In midmural layers of the ventricles, structural discontinuities exist in the form of thin tissue planes or sheets interconnected by small trabecula (193a). In addition to the structural discontinuities in the normal heart, propagation is expected to interact with connective tissue septa characteristic of aging, infarcted, hypertrophic, and failing myocardium. The role of tissue discontinuities in conduction and formation of conduction block has been recognized and studied in seminal work many years ago (149, 243, 320, 322). The increasing resolution of electric mapping techniques and the possibility to elucidate biophysical mechanisms by combining theoretical simulations with experimental work have allowed a more precise analysis of the role of tissue structure in arrhythmogenesis over the past 10 years.

The three most frequently studied elements of tissue geometry are 1) a simple linear connective tissue structure with a sharp end, which constitutes a pivot point for a turning wavefront (70, 294, 296, 375), 2) a small bridge (isthmus or “gate”) within a connective tissue structure connecting two large regions of myocardial tissue (35, 36), and 3) an abruptly changing tissue geometry (95, 96, 280, 281, 352). While each of these structures may play a distinct role in the generation of conduction block and arrhythmias, similar biophysical laws describe the interaction of propagation waves with such structures, as described for discontinuous conduction in a previous section of this article. Since the discontinuities brought about by cellular elements like gap junctions follow the same laws as well, one can conclude that the differences between propagation phenomena at the cellular versus the tissue level is only a matter of scale. A further consequence of this is that the principal role of complex tissue structures in propagation can be simulated with relatively simple models and that scales can be interchanged, as will be discussed below.

A frequent form of tissue heterogeneity is due to lateral separation of cells caused by interspersed connective tissue. Because the connective tissue is aligned with the cell strands, the wavefront collides or interacts with such resistive obstacles only during transverse propagation, while longitudinal propagation is relatively continuous (97). Figure 24 illustrates propagation of a wave around the end of a resistive obstacle, a so-called pivoting point. In Figure 24A, consecutive isochrone lines are depicted before and during the curving of the wave around the obstacle; Figure 24B shows the extensions of the corresponding electric phases (resting phase, refractory phase, and phase of full excitation) at the instant defined by the marked (asterisk) isochrone. Propagation around the pivot point produces a radius of curvature that cannot assume an infinitely small value but is limited by the requirements of critical curvature, defined by Equations 4 and 5. As a consequence, a tip of a spiral wave is formed that is detached from the anatomical structure and follows a trajectory determined by the critical curvature radius rc. At the spiral tip three phases of excitation fuse into a single point termed “phase singularity” or “wave break.” These phases are as follows: 1) the nonexcited zone between the spiral tip and the pivot point; 2) the fully excited state, which forms the head of the wavefront and the phase of depolarization (“wave head”); and 3) the phase of repolarization, which forms the tail of the wave (“wave back” or “wave tail”). Phase singularities play a crucial role in the understanding of spiral wave initiation and dynamics and will be discussed more extensively in the following sections. The phase singularity is a general phenomenon of spiral waves in excitable media (118, 188, 189, 201, 229, 306, 372). In anisotropic tissue containing obstacles that are aligned with the fibers, the local slowing of conduction at pivoting points has two mechanisms: it is due 1) to the fact that the direction of impulse spread during the turn is oriented perpendicular to the fiber direction and 2) to curvature with concomitant dispersion of local current at the wavefront (97). To separate the influence of fiber direction from the role of curvature at the pivoting point, artificial obstacles (injury by a laser beam) were produced in a perpendicular direction to fiber orientation. In this situation, fiber direction per se produced a low velocity along the obstacle and a higher velocity transverse to the obstacle. Nevertheless, excitation curving around the pivoting point was still slower than excitation along the obstacle, demonstrating that curvature is the major determinant of conduction velocity at such points (116). The slowing of propagation at pivoting points is an important factor determining the behavior of circulating excitation and reentry and may help to explain the effect of antiarrhythmic drugs on arrhythmias that involve structural obstacles (60, 139).

fig. 24.

Propagation of a wave around a pivot point. A: isochronal map of activation spread with an interval of 5 ms. Horizontal black line is an anatomical obstacle. Note that 1) the isochrones assume a spiral shape, and 2) the tip of the isochrones follows a path that is detached from the pivot point, because propagation fails beyond a critical value of curvature. B: snapshot of activation at the moment marked by the asterisk in A. The black color shows the excited area defined by the activation of INa. The gray color shows the area in the refractory state as defined by inactivation of INa. Point P marks the phase singularity or wave break (for explanation, see text). The dashed line t shows the trajectory of the wave tip with the radius rc. The simulation was made in a model using Luo-Rudy ionic kinetics. The maximal sodium conductance was reduced to 6.6 mS/cm2 to better visualize the inner zone of functional conduction block. [Modified from Fast and Kléber (97).]

Interaction with obstacles produces characteristic changes in the shape of the transmembrane AP and the extracellular electrograms. This is shown at the microscopic level from experiments carried out in anisotropic cell cultures (Fig. 25) (90). During longitudinal impulse spread, interaction of the electric wave with the extracellular cleft is minor and the upstrokes of the APs are monotonic. However, transverse spread is associated with collision of the wavefront at the obstacle and curving of the wavefront around the pivoting point of the obstacle. Close to the pivoting point a crowding of the isochrones indicates a marked slowing of conduction and a discontinuous spread of excitation. The discontinuities are visible form the multiple “humps” of the AP upstroke.

fig. 25.

Heterogeneity of propagation produced by anisotropic microscopic barriers. A: phase-contrast image of a cell culture (neonatal rat myocytes) with the overlaid diode array (30 μm per diode). Action potential upstrokes are measured at each diode location. The numbers 1–10 on the diode array correspond to the locations of the signals shown in D and E. In B and C, the location of these signals is indicated by the gray vertical shading. The 2 clefts in the central area (outlined in white in A and marked by striations in B and C) form a narrow isthmus of 40 μm. Activation maps of longitudinal and transverse conduction are shown in B and C, respectively. Numbers denote separation of isochrones by 100 μs. Selected recordings of action potential upstrokes during longitudinal and transverse conduction are shown in D and E, respectively. Discontinuities in the action potential upstrokes and slowing of conduction occur at the expansion site during transverse propagation (C and E) while longitudinal propagation (B and D) is continuous. [Modified from Fast et al. (90).]

Spach and co-workers (319, 322, 323) have shown that the presence of dense, repetitive discontinuities in canine and human atrial trabecula determines the direction-dependent behavior of conduction. Thus premature impulses generated in atrial trabecula with anisotropic discontinuous structure were blocked earlier in the longitudinal than in the transverse direction (319, 322, 323), while no differences were observed in anisotropic tissue devoid of structural discontinuities (Fig. 26). The higher “safety” of transverse versus longitudinal conduction was used to postulate a new mechanism of “anisotropic reentry” (323). This interpretation is in line with SF considerations made for discontinuous conduction at the cellular level and in branching tissue, as explained in previous sections. Therefore, it represents an example of the “scale independence” of the rules governing discontinuous conduction (a further example will be discussed below). The fact that other studies failed to demonstrate a higher safety of conduction transverse to fiber direction (67, 68, 168, 294) may be tentatively explained by the critical amount of fibrosis necessary to produce the phenomena observed in the canine and human atrial trabecula.

fig. 26.

Direction-dependent block of discontinuous anisotropic conduction in human and canine atrial bundles. Measurements were obtained in a uniform anisotropic pectinate muscle of a 12-year-old child (A), in a nonuniform anisotropic pectinate muscle of a 62-year-old man (B), and a nonuniform anisotropic muscle (crista terminalis) of an adult dog (C). As shown in the inset, two electrode pairs were placed in longitudinal (LP) and transverse (TP) directions on the muscle bundles, and conduction times (ms/mm interelectrode distance) were obtained from analyzing the unipolar extracellular electrograms. Solid circles in the graphs represent longitudinal propagation, and open circles represent transverse propagation. Each preparation was stimulated at a basic rate, and premature action potentials were introduced at decreasing intervals, A1-A2. In the uniform anisotropic bundle (A), conduction times became longer with the shortening of the A1-A2 intervals, and block occurred in both directions at the same interval. In the nonuniform cases (B and C), block occurred first in the longitudinal direction at a premature interval of 325 (B) and 310 ms (C). [Modified from Spach and Josephson (319).]

Other studies of discontinuous propagation and conduction block were conducted in models containing the aforementioned isthmus or “gate,” or a geometrical expansion of the tissue (35, 95). At an isthmus, the propagating wave has to cross a narrow excitable path, and the wavefront immediately beyond the isthmus becomes curved. A decrease in the width of the isthmus leads to a decrease in the radius of the curved wavefront beyond the isthmus and eventually to conduction block at the critical curvature (35). This demonstrates that the principle of critical curvature can also be applied to conduction block caused by structural discontinuities and that no basic mechanistic differences exist between functional (due to transient refractoriness) and structural (due to tissue geometry) conduction block when curvature of wavefront determines its behavior. Due to its critical nature, the success of propagation at an isthmus is sensitive to small changes in depolarizing INa. Thus an increase in the rate of stimulation causes block at an isthmus width that still conducts the impulse at a lower rate.

The abrupt tissue expansion, shown from a simulation study in Figure 27 has been used in several studies as a prototype of a tissue discontinuity, and the results of experimental studies are in close accordance with computer simulations. The first and obvious property of such an asymmetric tissue structure is the formation of unidirectional conduction block at a critical strand width (96, 280). Due to the current-to-load mismatch, the wavefront assumes a curved shape and propagation blocks immediately beyond the geometrical expansion at a critical strand width, while conduction is maintained in the opposite direction (unidirectional block).

fig. 27.

Effect of changes in sodium channel conductance (gNamax) on discontinuous conduction. Discontinuous conduction is simulated across an abrupt tissue expansion, from a narrow strand of width h into a large bulk area (inset). The critical width of h, hc, at which conduction block occurs, is plotted as a function of gNamax of the excitable elements in the narrow strand. [Modified from Fast and Kléber (95).]

Because geometrical expansions can be fabricated in vitro in synthetic tissue cultures, they have been used to study the behavior of discontinuous conduction extensively. Especially, the dependence of unidirectional block on 1) the geometry of the transition (95, 277, 352), 2) cell-to-cell coupling at the transition (95, 277, 352), and 3) the flow of INa and ICa(L) at the transition (96, 280, 352) was analyzed by theoretical simulations and experiments using high-resolution optical mapping of transmembrane potential. Similarly to the situation at the isthmus or gate (35), the critical strand width, at which the propagation into the expansion fails, is sensitive to changes in INa (95). If this dependence is extrapolated to all types of discontinuities, it predicts that unidirectional conduction block due to the interaction of a propagating wave with tissue structure is sensitive to all interventions that modify INa, such as heart rate per se and drugs that inhibit INa. Furthermore, the degree of discontinuity also determines the ionic current involved in propagation at this site, in analogy with the state of advanced cell-to-cell uncoupling illustrated in Figure 18. This is shown in Figure 28, which demonstrates that nifedipine, a blocker of ICa(L), can produce unidirectional block at a site of a tissue expansion. Also, unidirectional block can be reversed to bidirectional conduction if ICa(L) is increased by BAY K 8644 (276). From the perspective of antiarrhythmic therapy, the fact that ICa(L) plays an important role in conduction in heterogeneous or poorly coupled substrate but not in the homogeneous and well-coupled myocardium might be of relevance. It suggests that drugs that target ICa(L) can modify conduction in a structurally complex pathological substrate (e.g., healed infarct), where propagation-related arrhythmias such as reentry are likely to arise, without affecting the normal myocardium. For example, slow conduction during most phases of acute ischemia involves only INa (303), while in a healed infarct it involves long local delays and therefore ICa(L) (352). The involvement of ICa(L) in anisotropic reentry was recently investigated in the experimental model of the peri-infarction zone in vivo, and shown to be more complex than suggested by the theoretical and experimental in vitro studies discussed above. Thus application of BAY Y 5959 to enhance ICa(L) produced conduction block in the center of the remodeled peri-infarction zone, while it enhanced conduction only at more peripheral sites (37). This finding was taken to indicate that the effect of BAY Y 5959 to increase intracellular Ca2+ and to uncouple the cells in the center of the peri-infarction zone dominated over the effect to enhance ICa(L).

fig. 28.

Role of L-type Ca2+ current in discontinuous conduction, represented by an abrupt tissue expansion. Experiments were carried out in patterned cell cultures of neonatal rat myocytes. Top panels: antegrade conduction from narrow strand across abrupt expansion into large bulk area. Bottom panels: retrograde conduction from large bulk area into narrow strand. A: sketch of cell culture. Region of local perfusion of narrow strand is symbolized by area containing horizontally aligned pattern. Dots represent location of measuring diodes. B: optically measured action potential upstrokes from the bulk area (thick lines) and the strand (thin lines). For control, a delay of ∼2 ms occurs during antegrade conduction because of current-to-load mismatch at the expansion; the delay is absent during retrograde conduction. C: local superfusion with nifedipine [ICa(L)blocker] produces antegrade conduction block at the expansion, while retrograde conduction remains unaffected. This demonstrates that presence of ICa(L) is necessary for successful conduction across the expansion site. [Modified from Rohr and Kucera (276).]

In addition to ionic current flow [INa vs. ICa(L)], conduction across an abrupt tissue expansion depends on the state of cell-to-cell coupling, as shown in both theoretical and experimental work (95, 277, 352). Interestingly, partial cellular uncoupling favors conduction across a geometrical expansion and can revert unidirectional conduction block to bidirectional conduction. As illustrated in Figure 29, the increase of cell-to-cell resistance in a simulated geometrical expansion has two effects, depending on whether the resistance is increased along the axis of propagation (x-resistance) or transverse to the axis of propagation (y-resistance) (95). The increase of the x-resistance produces, as expected, a general slowing of propagation. The increase of the y-resistance decreases curvature at the transition site, determining success or failure of conduction. This effect is consistent with the notion that reduced cell-to-cell coupling can facilitate the driving of a large mass of tissue by a small mass of pacemaking cells (e.g., the sinus node) (152). It indicates that propagation across tissue with changing architecture requires an optimal match between the macroscopic structure and cell-to-cell coupling.

fig. 29.

Role of cell-to-cell coupling in discontinuous conduction. Discontinuous conduction is simulated across an abrupt tissue expansion, from a narrow strand of width h into a large bulk area (inset). A: the critical width of h, hc, at which conduction block occurs is plotted as a function of R, the specific resistance of the simulated gap junctions between the excitable elements of both the narrow strand and the bulk region. B: the critical width hc at which conduction block occurs is plotted as a function of Ry, the specific gap junctional resistance between the excitable elements of the bulk in the y direction, and of Rx, the specific resistance of the simulated gap junctions in the x direction. If the resistance is changed globally (A), the decrease of hc as a function of increasing R shows that cell-to-cell uncoupling can protect propagation from being blocked at the expansion. B demonstrates that the effect shown in A is due to the increase of the coupling resistance in the y direction (decrease of local current dispersion and current-to-load mismatch). [Modified from Fast and Kléber (95).]

D. One-Dimensional Versus Two-Dimensional Simulations of Discontinuous Propagation: Consequences of Scale Independence

In many instances in the previous sections, the scale independence was discussed as a prominent feature of the rules governing cardiac impulse propagation. For instance, involvement of INa and ICa(L) was discussed to play a role in both discontinuities caused by cell-to-cell uncoupling in linear strands and by changes in macroscopic tissue geometry. This raises the question about the applicability of relatively simple computer models of one-dimensional cell strands to more complex propagation phenomena occurring in two and three dimensions. Such a simplification might be particularly important in situations where the simulation of the electric behavior of a single cellular element requires sophisticated and laborious simulations of, for instance, the molecular behavior of single ion channels. Comparison of the one-dimensional simulation of propagation in a fiber containing a resistive discontinuity (abrupt change in cell-to-cell coupling at a given site along the fiber) with simulation of a two-dimensional phenomenon of propagation into a geometrical discontinuity (fiber branching) is shown in Figure 30 (352). The left panels of Figure 30, A–D, show a simulated cell strand containing 160 cells (cell 0 to cell 159). Inhomogeneity of coupling is introduced by increasing gap junction conductance from 0.08 to 2.5 μS starting at the junction between cells 79 and 80. The fiber is stimulated at cell 0, and propagation is from left to right, i.e., from the less coupled to the well-coupled tissue. The AP encounters a long conduction delay (∼12 ms) between cells 78 and 79, as it approaches the heterogeneous transition site (Fig. 30B). The delay is long because cell 79 receives small current from cell 78 through the poor coupling between these cells while, at the same time, it loses large current to cell 80 and the downstream well-coupled portion of the fiber which constitutes a large electric load due to its large conductance. The transition site constitutes a location of source-sink mismatch, which is reflected in the SF index. As the AP approaches the transition region, SF (line graph in Fig. 30C) decreases sharply from a value of 2.73 (its value in the poorly coupled fiber) and reaches a minimum value of 0.98 at cell 79. It recovers to a value of 1.20 in cell 80 and slowly increases to 1.60 away from the transition zone. The bar graphs (Fig. 30C) provide a quantitative measure of the charge contribution from INa and ICa(L) to support conduction. In the homogeneous segments of the fiber, away from the transition zone, INa plays the dominant role in sustaining conduction. In cells near the transition region, ICa(L) becomes an important participant in conduction; in fact, its charge contribution exceeds that of INa. The large contribution from ICa(L) is a consequence of the long conduction delay across the transition region. During most of the delay ICa(L) is the sole source of depolarizing charge from proximal to distal cells (because of its fast inactivation, charge contribution from INa is negligible beyond 1 ms). Figure 30D shows peak values of INa and ICa(L) along the fiber. Just distal to the transition region INa is greatly reduced (has a smaller negative magnitude) as a result of inactivation during the prolonged foot at depolarized potentials associated with the transmission delay. In contrast, there is a striking augmentation of peak ICa(L) in cells proximal to the transition zone. Due to the large load on these cells from slowly depolarizing cells distal to the transition site, their plateau potentials are reduced (“pulled down” by the load, see cell 78 in Fig. 30A). The reduced potential increases the driving force on ICa(L) and results in a larger current. Interestingly, ICa(L) is augmented exactly where it is needed as a source for sustaining conduction (where QCaQNa). This phenomenon can be viewed as a feedback, compensatory response of the tissue. The two-dimensional geometric nonuniformity due to local branching of fibers is shown in the right panels of Figure 30, E–H. Comparison with the model in the left panels of Figure 30 shows that the two models are qualitatively equivalent, since both constitute an increase of electric loading on the propagating AP. Indeed, SF decreases sharply as the AP approaches the region of expansion (Fig. 30G). In this region, ICa(L) is a major contributor of depolarizing current and provides more depolarizing charge than INa to support conduction (QNa:QCa = 0.85). As in the case of increased coupling, there is a long delay across the transition zone (Fig. 30F), a reduction of INa, and an augmentation of ICa(L) (Fig. 30H). When the direction of propagation is reversed (not shown), SF is high across the transition region in both simulations, since the convergence of fibers or the decrease in cellular coupling constitutes a reduction of electric load. Conduction in this direction is robust and supported by INa with only negligible contribution from ICa(L) in both models.


At the completion of normal cardiac excitation, the electric wave becomes extinct and the subsequent excitation cycles originate from a pacemaker impulse. The fact that the physiological excitation waves vanish spontaneously is due to the long duration of refractoriness in cardiac tissue compared with the duration of the excitation period. In pathological settings, excitation waves can be blocked in circumscribed areas, rotate around these zones, and reenter the site of original excitation in repetitive cycles. This mechanism, termed reentry or circus movement, is known to be responsible for a number of clinically important tachyarrhythmias. The first observation of this phenomenon was made already at the beginning of the last century. Ever since, cardiac scientists have tried to unravel the extremely complex mechanisms of cardiac reentrant arrhythmias and to associate these arrhythmias with the structurally and functionally altered substrate in cardiac disease. The initially simple concept of reentry has undergone many revisions, and complexity has been added. Especially, recent experimental findings and computer simulations have allowed relating the biophysical properties of reentrant circuits to the molecular changes associated with disease. Traditionally, reentry has been divided into two types: “anatomical” reentry if there was a distinct relationship of the reentry pathway to the underlying tissue structure, or “functional” if reentrant circuits occurred at random locations without a clearly defined anatomical circuit. While this distinction has a historical background and is useful for didactic purposes, both the anatomical and functional forms may coexist in a given pathological setting and share many common basic biophysical mechanisms. Therefore, circus movement and reentry will be discussed from a historical perspective in the first part of this section, and the mechanisms of initiation, maintenance, and extinction of reentry will be discussed from a unified perspective for all forms in the second part.

A. From Anatomic to Functional Reentry

A first demonstration that circus movement can occur in excitable tissue was provided by Mayer in experiments on the jellyfish Sychomedusa cassiopea (214, 215). He cut subumbrella tissue into rings and stimulated these at one (arbitrary) point. Premature stimuli delivered during an appropriate period induced unidirectional block with a single wave of contraction propagating in only one direction. The wave circled around the central hole of the ring, returned to and reexcited the site of initiation. In such a way, wave rotation, as observed by eye from local contractions, could continue for many days at a constant rate. The location of wave initiation did not play any role in the maintenance of circulating excitation. These experiments inspired Mines (222, 223) to investigate circus movement in cardiac muscle. Mines reproduced Mayer's observations using ringlike preparations of atrial and ventricular tissue from various animals and suggested that circus movement would be responsible for cardiac tachyarrhythmias in humans. He predicted reentrant excitation to occur in hearts with accessory atrioventricular connections. This prediction was confirmed many years later in patients with Wolff-Parkinson-White (WPW) syndrome (75, 135).

Mines defined one of the basic requirements for initiation of reentrant excitation to be the establishment of unidirectional conduction block, a phenomenon discussed in the previous sections. The basic properties of anatomical reentry according to Mines (222) are illustrated in Figure 31. In this type of reentry, the excitation wave propagates around a relatively large inexcitable obstacle so that the revolution time exceeds the absolute and the relative refractory periods. Mines (222) realized that the initiation and maintenance of reentry was dependent both on conduction velocity θ and refractory period tr. Thus, as long as the extension of the refractory zone behind the excitation wave, the so-called wavelength of excitation λ Math(6) was smaller than the entire length l of the anatomically defined reentrant pathway, a zone of excitable tissue, the so-called excitable gap, existed between the tail of the preceding wave and the head of the following wave. In essence, circus movements containing an excitable gap are stable with respect to their frequency of rotation and can persist at a constant rate for hours. In the case where λ is longer than l, the excitation wave becomes extinct when it encounters not-yet-recovered inexcitable tissue. A special case is present in the intermediate situation, when the head of the following wavefront meets the partially refractory tail of the preceding wavefront, λ ∼ l. This situation is characterized by unstable rotation periods and complex dynamics of the rotating AP, as discussed in the subsequent sections. The hypothesis stating that reentry can occur without involvement of an anatomical obstacle was first proposed by Garrey in 1924 (110). Fifty years later, with the development of the multielectrode mapping technique, the experimental proof that such forms of reentry can occur was provided by Allessie and co-workers (35, 310) in isolated atrial tissue of the rabbit. The observation that late premature stimuli induced continuous elliptic spread of the electric impulse was taken to indicate that the atrial tissue contained no major morphological obstacle that would force the propagation into predetermined pathways. However, application of a critically timed premature stimulus induced a rapid tachycardia caused by macroscopic reentry (4).

fig. 31.

Model of fixed anatomical reentry presented by Mines in 1913 (222). Excited, refractory tissue is marked in black; relatively refractory tissue is marked by dots; white ring segments are fully excitable. A: unidirectional block produces excitation that propagates only clockwise. Because the wavelength of excitation is longer than the length of the ring, the wavefront collides with its own refractory tail and becomes extinct. B: the wavelength is shorter and the tissue is excitable at the point of wave origin when the original wave has traveled along a full circle; this enables reentry to occur.

To explain the properties of a single functional reentrant circuit Allessie et al. (5) formulated the “leading circle” concept (Fig. 32). It was postulated that during wave rotation in a tissue without inexcitable obstacles, the wavefront impinges on its refractory tail and travels through partially refractory tissue. The interaction between the wavefront and the refractory tail determines the properties of functional reentry. The “leading circle” was defined as “the smallest possible pathway in which the impulse can continue to circulate,” and “in which the stimulating efficacy of the wavefront is just enough to excite the tissue ahead which is still in its relative refractory phase” (5). Because the wavefront propagates through partially refractory tissue, the conduction velocity is reduced. The velocity value and the length of the circuit depend on the excitability of the partially refractory tissue and on the stimulating efficacy of the wavefront, which is determined by the amplitude and the upstroke velocity of the AP and by the passive electric properties of the tissue (e.g., gap junctional conductance). The partially refractory tissue determines the revolution time period, and a fully excitable gap does not exist in the leading circle. After the pioneering experiments of Allessie and co-workers in atrial muscle, functional reentrant circuits with activation patterns of varying complexity were observed in both atrial and ventricular muscle. It was found that circulating waves often appear in pairs, whereby two wavefronts rotate in opposite directions. This type of activation pattern was called “figure-8” or “double-loop” reentry (84, 85, 119). Because the functional reentrant pathways are not tied to anatomical structures, they can change their location and size. The activation can become even more complex when several rotating wavefronts are present in cardiac muscle (so called “random” reentry, Ref. 134). Thus coexisting reentrant circuits were observed during stable atrial fibrillation (6) and during ventricular fibrillation in ischemic hearts (144, 146).

fig. 32.

Functional reentry. Top right: isochronal activation map during the tachycardia (interval between isochrones = 10 ms). Bottom right: schematic representation of the leading circle that is the source of the electrotonic potentials in the core where functional conduction block prevails. Left: transmembrane potentials recorded during the tachycardia from 7 sites. The locations of these sites are indicated in the isochronal map. Note that sites 3 and 4, located in the Z-shaped core, receive electrotonic input twice during each cycle, when the wavefront passes at electrodes 2 and 5 that are located in the leading circle. [Modified from Allessie et al. (5).]

The leading circle concept was based on properties of impulse propagation in a one-dimensional tissue that forms a closed pathway (e.g., a ring). The concept was a major breakthrough in the understanding of the mechanisms of reentrant excitation. However, it became evident that these considerations alone do not fully describe wave rotation in two- and three-dimensional cardiac tissue. As discussed in previous sections, propagation of two- and three-dimensional waves also depends on wavefront curvature, a property that is not present in one-dimensional preparations. Because the maximal velocity of a convex, rotating wavefront can never exceed the velocity of a flat front and the period of rotation remains constant in a stable rotating wave, the velocity has to decrease from the periphery (where the highest value corresponds to linear velocity) to the center of a rotating wave. As a consequence, any freely rotating wave in an excitation-diffusion system has to assume a spiral shape. The spiral wave concept was used in the generic theory of excitable media to describe rotating waves in a variety of systems of biological, chemical, and physical origin. One of the most extensively studied examples is the Belousov-Zhabotinsky (BZ) reaction (229). In this chemical reaction, malonic acid is reversibly oxidized by bromate in the presence of ferroin. In this process, ferroin changes its color from red to blue and then back to red, which allows visual observation of the reaction. Figure 33 demonstrates a rotating wave in a thin two-dimensional layer of the BZ reaction. In the center of the rotating wave (core), the tip of the wave moves along a complex trajectory and radiates waves into the surrounding medium. The term rotor initially described the rotating source, and the “spiral wave” defined the shape of the wave emerging from the rotating source (373). In many publications this difference has been blurred, and terms used in the literature include “rotors,”“vortices,” and “reverberators.” In addition to the BZ reaction (369), spiral waves were also observed in other excitable media (see also Ref. 248) including neural tissue [depression waves in the retina (118) and cerebral cortex (306)], intracellular calcium signaling systems [Xenopus laevis oocytes (188), cardiac myocytes (201), and amoebae colonies (332)]. In the heart, spiral waves have been implicated in the generation of cardiac arrhythmias for a long time (13, 122, 301, 369). Both two-dimensional spiral waves and three-dimensional spiral waves, the socalled “scroll waves,” have been implicated in the mechanisms of reentry in atrial and ventricular tachycardia and fibrillation (247, 249, 371).

fig. 33.

Belousov-Zhabotinsky reaction. Shown is the spiral wave in the chemical Belousov-Zhabotinsky reaction. [Modified from Müller et al. (229).]

B. Initiation of Reentry

The original observation of Mayer (214) that appropriately timed local stimulation of excitable rings cut from subumbrella tissue produced unidirectional block implicated that such tissue has asymmetric instrinsic electric properties. The causes for unidirectional block in cardiac tissue have been discussed in detail in previous sections. Importantly, they involve not only anatomical asymmetry of the underlying tissue structure, but also functional asymmetry of excitability that may occur if a propagating excitation wave or a premature stimulus encounters the tail of a preceding wavefront (see Figs. 20, 21, 22).

In anatomical reentry, the criteria for initiation are simple. An impulse propagating unidirectionally will follow the fixed anatomical path and reenter, if the path length exceeds the length of the excitation wave. Thus the definition of the two criteria for initiation of anatomically fixed reentry are straightforward: 1) formation of unidirectional block (see previous sections) and 2) λ < l. The situation is more complex in those forms of reentry, where the circus movement is only partially determined by anatomical obstacles or where no anatomical obstacles are present at all. In the classical experiment of Allessie (4), a rapid tachycardia was induced in rabbit atrial tissue by application of a critically timed premature stimulus. Figure 34 (4) shows an activation map during regular pacing (basic beat, A), during the premature beat that initiated the tachycardia (B), and during the first cycle of tachycardia (C). Figure 34D shows a map illustrating the distribution of refractory periods measured during regular basic rhythm and showing local gradients of tr <30 ms. Normal excitation in Figure 34A propagated in all directions from the central stimulating electrode, indicating gross electric homogeneity under these basic conditions. The premature wave propagated into the areas with shorter refractory periods and was blocked in the direction where refractory periods were longer. The line of conduction block extended across the center of the preparation along a distance of ∼5 mm. Excitation propagated in two directions around the line of block, and the two wavefronts merged behind the line of block. At this time, the tissue proximal to the site of block had recovered from the premature excitation and could be reexcited by the merged wavefront. The original area of conduction block broke up into two new areas, and two wavefronts propagated around them in opposite directions: one clockwise and the other counterclockwise. Subsequently, one wave became extinct at the border of the preparation leaving a single reentrant circuit.

fig. 34.

Initiation of functional reentry by premature stimulation. A: isochronal activation map of basic beat (pacing interval 500 ms) initiated by central stimulation (dot) in isolated rabbit atrial muscle. Activation times (ms) are given relative to the stimulus onset. B: map of premature beat (coupling interval 56 ms). Bars indicate conduction block. C: first cycle of the reentrant tachycardia. D: refractory periods measured during basic rhythm (in ms). [Modified from Allessie et al. (4).]

1. Initiation of reentry through interaction of wavefronts with anatomical or functional obstacles

While experiments of the type shown in Figure 34 illustrate that reentry can occur without an obvious anatomically fixed reentrant circuit, they do not exclude the possibility that local heterogeneities of cellular electric properties and tissue architecture contribute to the formation of unidirectional conduction block. It is important to note that local refractory periods in the multicellular tissue also depend on the local state of cell-to-cell coupling and the microarchitecture of the tissue (144). Formation of spiral waves at a sharp obstacle was described in a computer model (97, 250) and in the experiments on the chemical BZ reaction (1). An important role of tissue microarchitecture in the initiation of reentry can be derived from Figure 24 (97). The minimal radius rc (or maximal curvature) that sustains propagation and determines the trajectory of the phase singularity in this situation depends on the degree of excitability and cell-to-cell coupling (389). In case of reduced excitability or decreased cell-to-cell coupling, rc can become so large that the returning wavefront encounters excitable tissue and detaches from the pivoting point. Formally, this criterion is met when the pivoting trajectory becomes longer than the wavelength of excitation (rp × 2π > λ). Experimentally, this was shown on the epicardial surface of sheep ventricular muscle, where narrow obstacles were created with sharp cuts (36). Reduction of tissue excitability caused by application of TTX or by stimulation at high rates, caused detachment of wavefronts from the sharp obstacles and initiation of spiral waves, a phenomenon called “vortex shedding.” These experimental observations suggest a role of potential importance for vortex shedding in cardiac arrhythmias.

In analogy to the initiation of spiral waves at the end of an anatomical obstacle, one can imagine replacement of this area of permanent block by a segment of transiently refractory tissue in an electric medium with homogeneous electric properties. This concept was already formulated in 1948 by Selfridge (301) in his study on atrial flutter and fibrillation and later in computer simulations (337) of reentry. A refractory segment of the tissue can form a temporary obstacle (see also Fig. 22) and force a propagating wave to deviate from its course and circumvent the inexcitable segment. Once this segment recovers from inexcitability, the wave can penetrate this region and establish a reentrant circuit rotating around a central zone of functional conduction block. The concept of transient refractoriness as a mechanism for initiating reentry can be extended to the interaction of a prematurely elicited wave with the wave of basic excitation, since refractoriness is an inherent property of tissue located immediately behind a propagating wavefront. It is predicted that this interaction will produce reentry in a homogeneous electric medium if the timing of interaction between the two waves is appropriate, analogous to the initiation of unidirectional block in the vulnerable window shown in Figures 21 and 22. This is illustrated in Figure 35, where reentry is produced in a simulated rectangular sheet containing 650 electrically coupled excitable elements (337). Figure 35 shows a premature impulse that is blocked through interaction with the tail of the preceding excitation wave and produces reentry by propagating around the refractory zone. Similarly to Figure 33, the isochrones assume a spiral shape. In summary, initiation of reentry has been studied in both anatomically determined reentrant circuits and in electrically homogeneous tissue where reentry is completely functional. In both cases, the trajectory of a point on the reentrant wavefront must exceed the spatial extent of refractoriness along this trajectory.

fig. 35.

Initiation of spiral wave reentry: isochronal maps. Top left: linear propagation from top to bottom of a wavefront after simultaneous stimulation of sites A–E (line stimulation). Top right: after the initiation of a wavefront from top to bottom (see A), a second stimulus is applied simultaneously at sites F–H. This wave is blocked in the antegrade direction (below electrodes F–H) by the tail of the preceding wavefront but travels retrogradely and sets up a reentrant circuit. Activation during the subsequent reentry cycles is depicted in the bottom left (2nd cycle) and bottom right panels (3rd cycle). [Modified from van Capelle and Durrer (337).]

2. Initiation of reentry by electric field shocks

Interactions between extracellularly applied electric fields and cardiac tissue have been studied extensively to understand the mechanism of ventricular and atrial defibrillation. The basic mechanism underlying the induction of reentry by electric field shocks relates to the specific passive properties of cardiac muscle: 1) its anisotropic geometry, 2) its bidomain nature, and 3) discontinuities in tissue architecture (connective tissue septa, trabeculation). In a ventricular trabeculum, the current flowing during propagation or during application of an extracellular stimulus is almost equally distributed between the extra- and intracellular spaces due to the high electric resistance of the cardiac intramural extracellular compartment (169, 359). Similarly, an extracellular field applied to a linear cardiac bidomain structure produces voltage gradients of similar magnitudes in the intra- and extracellular spaces (103, 169). The ratio of the extra- to intracellular space resistances depends on the anisotropic structure of the myocardium (43). Therefore, the voltage gradients that develop in the extracellular space during field stimulation depend on the direction of the field relative to the longitudinal axis of the cardiac cells. An important consequence of the anisotropic structure of the myocardium is that current flow generated by point or field stimulation produces complex voltage gradients that depend on fiber direction, on electrotonic interactions between regions with different fiber angles, and on the waveform of the applied stimulus. For point stimulation, the requirements of the “liminal length” of an excitatory source (105) determine the locations around an electrode where self-sustained propagation is initiated. The flow of stimulating current produces regions of membrane polarization termed “virtual electrodes”; a depolarized region represents a “virtual cathode” and a hyperpolarized region represents a “virtual anode” (364). In general, virtual electrodes that extend around a point of central stimulation assume a so-called “dog bone” shape, whereby a hyperpolarized zone is created in the region along the main axis of the cardiac cells and depolarization is observed along the transverse fiber axis in the case of cathodal break stimulation (i.e., the voltage change at the end of a cathodal pulse) (234, 361, 363365).

Several forms of reentry may be initiated by interaction of electric field shocks with the passive electric tissue properties. First, so-called secondary sources, i.e., local changes in membrane potential, may be created at sites of tissue curvature (335), trabeculations (114, 115), or local resistive obstacles (100). At such sites new propagating wavefronts can be created (100, 115), which may interact with existing wavefronts to produce reentry. Second, reentry can be initiated through interaction of an electric field with propagating wavefronts in cardiac tissue without major discontinuities, as first shown by Frazier et al. (109). In their experiments, an electric field was created perpendicularly to a propagating wavefront. The interaction produced different responses depending on the site of the interaction with the repolarizing wave. At a “critical point” a phase singularity was created and a spiral wave started to rotate around a zone of block. The “critical point” concept was recently refined by high-resolution optical mapping of transmembrane potential (79). Thus wave breaks or phase singularities were shown at sites of interaction between regions rendered refractory by the shocks (prolongation of the AP by a depolarizing shock effect) and sites where shocks produced new excitation waves (reexcitation by hyperpolarizing shock effect).

An interesting formation of phase singularities leading to quatrefoil reentry was observed in cases where the stimulating electrode producing basic propagating excitation (S1 stimulus) was identical to the electrode delivering a premature (S2) shock and attributed to the dog bone shape of the virtual electrode, as illustrated in Figure 36 (196). In Figure 36, application of S2 to an electrode placed on the surface of a rabbit heart produced a dog bone-shaped virtual electrode, characterized by hyperpolarization along the long fiber axis during the shock pulse and depolarization transverse to the fiber axis. While hyperpolarization during a critical repolarization phase increased excitability, the depolarizing effect along the transverse fiber axis lengthened the AP and produced prolonged refractoriness. Therefore, the voltage change at the end of S2 (break pulse) produced excitation and propagation in the longitudinal direction of the fibers, which turned around to excite recovered tissue in the transverse direction, forming four reentry loops (Fig. 36). In this case, phase singularities formed where the S2 waves encountered tissue that recovered from inexcitability along the lines of separation between the depolarized and hyperpolarized regions.

C. Excitable Gaps in Reentrant Circuits

The basic properties of reentry along a fixed anatomical pathway are illustrated in Figure 31. As aforementioned, a gap of excitable tissue exists between the head of the reentrant wavefront and the tail of the preceding wavefront, if λ < l. A fully excitable gap can be defined as the circuit segment in which the tail of the preceding wave does not affect the head and the velocity of the following wave (absence of head-tail interaction). A partially excitable gap is defined as the zone where the rotating wave can be captured by local stimulation in the presence of head-tail interaction. While the excitable gap denotes a length of a segment within the reentrant circuit, the fully or partially excitable period denotes the time during which a circuit's segment is fully or partially excitable, respectively. The relationship between the excitable gap and the excitable period can be complex if the velocity of propagation changes within the reentry circuit (26, 116, 248, 253). The existence and the extent of an excitable gap in a reentrant circuit is important for several reasons: 1) it enables modulation of the frequency of a reentrant tachycardia by a locally applied stimulus or by field stimulation, 2) it can determine the effects of drugs on the reentrant circuit, and 3) it may be exploited to terminate a tachycardia. This is illustrated schematically in Figure 37, showing that a premature impulse enters the reentrant circuit at the end of the relative refractory period and propagates in both antegrade (orthodromic) and retrograde (antedromic) directions (B). In the retrograde direction, the premature wave collides with the circulating wave, and both waves annihilate. In the antegrade direction, the premature wave continues to propagate. As a result, the arrhythmia is reset, i.e., propagation continues to circulate with the former frequency but with a phase shift caused by the penetrating impulse. The time from the resetting stimulus to the next excitation of the stimulus site by the new antegrade wave is termed return cycle. When (Fig. 37C) the premature impulse enters the reentrant circuit early enough in the relative refractory period, it fails to propagate in the antegrade direction because it encounters absolutely refractory tissue. In the retrograde direction, it meets increasingly recovered tissue and is able to propagate until it meets the circulating wave and terminates the arrhythmia. When the heart is paced at a regular rate that is faster than the rate of the wave rotation, the situation depicted in Figure 37B may be perpetuated: every paced impulse blocks the circulating wavefront but also enters the antegrade pathway and maintains the circulating wave. Therefore, the arrhythmia follows the pacing rate, but on stopping the pacing, the original rhythm resumes. This phenomenon is called “transient entrainment” (12, 26, 108, 210, 347).

fig. 37.

Scheme of stimulation of an excitable gap. A: fixed anatomical reentry with the wavefront head (arrow) propagating clockwise. The relative refractory period is hatched, and the white circuit segment illustrates the fully excitable gap. B: a stimulus applied during full excitability of the stimulation site propagates antidromically (A) until it collides with the tachycardia wavefront, as well as orthodromically (O). The orthodromic excitation can change the tachycardia return cycle depending on the interaction with the tail of the preceding front and/or the conduction properties of the tissue (for details see text). C: the stimulus is applied during the partial refractory period of the preceding wave. It is blocked in both directions and terminates the tachycardia.

1. Relation between excitable gaps and structure

The existence of an excitable gap is undisputed in fixed anatomical reentry and has been demonstrated in clinical settings (VT in postmyocardial infarction, Ref. 7) and in experimental atrial flutter (29, 106) or ventricular tachycardia in the presence of a fixed anatomical obstacle (19, 27, 32). Many discussions focused on the question of an excitable gap in spiral wave reentry in continuous excitable media (373) and in mixed forms of reentry. The discussions about this topic are complicated by the fact that many reentrant arrhythmias, which occur in tissue with a seemingly continuous structure (regularly spaced and shaped isochrone patterns during normal rhythm), may actually have a morphological microscopic component (intrinsic tissue architecture, discontinuous gap junction expression, Refs. 26, 116, 248, 253, 254, 256, 314, 319) that affects the tissue electric properties (62, 248). The term functional reentry as it is used in the literature therefore refers to the absence of macroscopic obstacles during normal propagation but does not directly compare to theoretical simulations that use a continuous electric syncytium to model cardiac tissue.

Allessie et al. (3) showed an excitable period of ∼30 ms during atrial tachycardia caused by functional reentry. Application of a premature impulse during the first half of this period resulted in termination of the tachycardia, while stimulation during the second half produced resetting (3). The latter observation, indicating that this type of reentry involved a critical “head-tail” interaction in the inner (shortest) pathway of the rotating source (leading circle reentry, Ref. 5), suggested the presence of a partial excitable gap. The presence of an excitable gap in functional reentry was also suggested from theoretical considerations by Winfree (370) and demonstrated in the simulations of Pertsov et al. (248) in rotors occurring in two-dimensional continuous networks, although it was not specified whether fully or partially excitable gaps were present. Excitable gaps were not only shown in single rotors in theoretical and experimental models, but also experimentally and clinically during ventricular and atrial fibrillation (2, 76, 161, 163, 245). In the fibrillating pig heart, rapid pacing showed local capture of ventricular myocardium and local phase locking of ventricular complexes with burst stimuli (161).

Many studies were carried out to specify the nature and the importance of excitable gaps during reentry in the anisotropic myocardium (7, 116, 125, 248, 253, 295, 360). It should be pointed out that there are no major differences in the principles that govern the behavior of rotors and spiral waves in isotropic and anisotropic media as long as the media are continuous, i.e., devoid of discontinuities in passive or active electric properties (167, 248, 373). In theoretical simulations using continuous media, the transition from isotropy to anisotropy is achieved by the introduction of a direction-dependent scaling factor that “stretches” the dimensions of the circuit along the fiber axis relative to its transverse axis. As a consequence, the propagation velocity, the shape of the turning waves, and the extensions of excitable gaps become direction dependent, while the periods of wave rotation and the excitable periods remain unchanged. In addition, the APs in continuous anisotropic media show regular upstrokes that do not reflect discontinuous electric properties (absence of multiphasic upstrokes) (248). The situation is different if propagation is anisotropic and discontinuous due to longitudinally aligned structural obstacles that typically occur in aged myocardium and in hypertrophy (90, 96, 314). The contribution of these structural components to the properties of reentry might vary among experimental models and may explain the different responses of reentrant circuits to premature stimulation, as discussed below.

2. Entrainment: resetting of reentry by external stimuli

Entrainment of reentry by external stimuli was originally defined in the clinical setting as “an increase in the rate of a tachycardia to a faster pacing rate, with resumption of the instrinsic rate of the tachycardia upon either abrupt cessation of pacing or slowing of pacing beyond the instrinsic rate of the tachycardia” (348) and taken to indicate an underlying reentrant mechanism. Clinically, the electrograms resulting from fusion of intrinsic tachycardia waves and stimulated waves were used to define the forms of entrainment (348). Experimentally, it was shown that different types of reentry responded in various ways to premature stimulation. In fixed anatomical reentry, resetting is the rule, and the return cycles may show prolongation or be of normal duration, depending on the prematurity of the resetting impulse and the presence of a fully or only partially excitable gap. Similar patterns of return cycles, suggesting the presence of a fully or partially excitable gap, were observed in anisotropic reentry in a postmyocardial infarction model, where a major structural component is present. In this type of anisotropy, variations in conduction velocity and in the width of the excitable gap were nonuniform with marked conduction slowing and shortening of excitable periods at the pivot points due to curvature (97, 116). In anisotropic reentry with a relatively minor structural component, the tachycardia pathway could be invaded by premature impulses, but the reduced excitability in the partially excitable gap caused marked conduction slowing and prevented resetting (295).

3. Effects of drugs on reentry circles

In a simple concept, the effects of drugs on reentry can be understood by their respective action on the front of a circulating wave (head) and/or on the repolarizing tail. In functional reentry, defined by a “leading circle,” it was observed early on that shortening of the refractory period by a muscarinic agonist decreased the tachycardia cycle length (5). Because the dimensions of the line of functional block and the period of revolution are a consequence of head-tail interaction in such circuits, shortening the refractory period and λ resulted in a decrease of cycle length. Consistent observations were made in experimental acute atrial fibrillation (AF), where lengthening the wavelength of excitation terminated the tachycardia (164, 268, 351, 354). In contrast, shortening the refractory period by muscarinic agonists had no effect on atrial tachycardia in fixed anatomical reentry, while inhibition of INa by TTX with consequent slowing of conduction increased the cycle length. Generally, this concept suggests that reentry with a partially excitable gap and mainly functional components will react to drugs that affect either repolarization or depolarization of the reentrant waves, while fixed anatomical reentry with a large excitable gap will react to drugs that affect depolarization and decrease conduction velocity (preferentially at pivot points).

Recent observations in experimental chronic AF, a type of tachycardia with a substantial anatomical component, have shown that this concept requires refinement (360). Figure 38 (360) illustrates the effect of a so-called “type 1C” drug (with an inhibitory effect on INa and IKr, see Fig. 1) on the electric parameters characterizing AF. Application of this drug lengthened the cycle length of the tachycardia, due to a slowing in average conduction velocity, and consequently widened the excitable gap and lengthened the excitable period. Importantly, no effect on the excitation wavelength was observed. Also, it was shown that conduction block and conversion of atrial flutter in dogs by procainamide occurred preferentially at sites of slow conduction (298). Wijffels et al. (360) have proposed a variety of antiarrhythmic mechanisms that could be associated with the conduction slowing and the widening of the excitable gap. Among these explanations, the existence of curved wavefronts around pivot points may play a crucial role. Thus drugs that block INa cause preferential conduction slowing at pivot points (242). Since curved wavefronts represent sites where dispersion of local depolarizing currents takes place, the SF for propagation at such sites is reduced, and smaller levels of INa block are sufficient to interrupt conduction at these locations.

fig. 38.

Effect of a type 1C drug on persistent atrial fibrillation. The main effects of the drug are to prolong the excitable period and the spatial excitable gap. This is due to a major slowing of conduction velocity (CVAF, probably preferentially at pivot points), while the prolongation of the refractory period (RPAF) is of minor importance. The atrial fibrillation cycle length (AFCL) is lengthened until conversion to sinus rhythm occurs. [Modified from Wijffels et al. (360).]

D. Head-Tail Interaction and Instability of Reentrant Circuits

1. Instability of rotation in fixed anatomical reentry

The shape of the anatomical obstacle determines the path of a reentrant wave in fixed anatomical reentry. Therefore, instability of anatomical reentry is confined to variations of the rotating period and wavelength of excitation. This instability is characterized by the wavefront invading the repolarizing phase of the preceding wave resulting in oscillations of the rotating period. A first type of instability arises from the dependence of conduction velocity on the excitability of the membrane during the relative refractory period of the preceding wave. If a wavefront invades the partially refractory tail of the preceding wave, acting to shorten the tachycardia period, AP amplitude and wavefront strength decrease. As a result, the wavefront retreats from the refractory tail and the tachycardia period increases again. In turn, the retraction will cause the wave to propagate in more recovered tissue and accelerate. Thus alternating deceleration and acceleration of the wave is a first mechanism explaining oscillations of the tachycardia cycle length. Oscillations in cycle length based on dynamic changes in velocity were observed in an experimental model of an anatomically fixed reentrant circuit (308). Small initial perturbations in the cycle length initiated either instability with an amplification of the rotation-period oscillations and termination, or damped oscillations. The instability was critically affected by the dependence of conduction time in a segment of the circuit on the excitation interval. When this dependence was steep (negative slope <–1) instability and eventually block occurred, while damping was observed at less negative slopes.

In analogy to alternation of conduction velocity, the rotation period can oscillate with an alternation in AP duration and refractory period. A change in AP duration in a reentry circle produces a change in the diastolic interval, and vice versa, since the cycle length equals the sum of the AP duration and the diastolic interval. This mutual interdependence presents an important source of instability of reentrant circuits. In simple models of rotating excitation, the type of oscillations was dependent on the slope of the restitution curve of AP duration. Damped oscillations were observed with a slope <1, while instability was produced with a slope >1 (343). In real tissue, oscillations of the rotation period are more complex and may be aperiodic (141, 343). Frame et al. (107) showed that oscillations of cycle length, diastolic interval, AP duration, and conduction velocity were often irregular. Similar complex oscillations were reproduced in a computer model of reentry in a fixed anatomical pathway with homogeneous electric properties (138). In the heart, these oscillatory patterns are influenced by heterogeneities in cellular properties and tissue structure, as discussed below.

2. Instability of rotation in functional reentry

In functional reentry, head-tail interaction is not only a determinant of the rotation period but also of rotor stability. This has been shown in theoretical simulations using the so-called “cellular automata model” and the FitzHugh-Nagumo algorithm to compute the behavior of an AP. In both types of simulations, the degree of interaction determined the movement of the spiral wave core, indicating that the findings are model independent (1, 175, 176). In continuous electric media, the movement of the rotor tip was predicted from the relationship between the radius of curvature and the wavelength of excitation, as illustrated on Figure 39. If the trajectory of the wave tip (given by 2π rc, see Fig. 24) was longer than λ, then the head-to-tail interaction was minor, and the inner core of the spiral wave, characterized by the phase singularity, rotated around a circular core of nonexcited tissue (1, 175, 176). If the length of the rotor trajectory was similar to the wavelength of excitation (2π rp ∼ λ) instability of the movement of the rotor resulted in a meandering trajectory of the phase singularity, and a stable “Z-shaped” line of block resulted if λ >> 2π rp. In normal myocardium, at a normal state of excitability λ >> 2π rp, and a Z-shaped line of block has been found indeed in many experimental studies (83, 109, 295, 310).

fig. 39.

Instability of the spiral wave core. A–D show the trajectory of the core of a spiral wave as a function of increasing wavelength λ (relative numbers are given below the trajectories). Note 2 stability states: a stable rotation around an inexcitable core for a very short λ, and a movement along a Z-shaped line of block for a long λ. At intermediate stages the trajectory is unstable. [Modified from Fast et al. (92).]

3. Wave splitting

Splitting or break-up of a mother wave into two or more daughter waves is an important instability of freely rotating waves in excitable media. In the heart, wave splitting has been implicated as a crucial process in the mechanism of multiple wave reentry (226) and is considered important for the explanation of the mechanism of ventricular fibrillation. With the use of the Beeler-Reuter model (16), it was found that a single spiral wave could spontaneously break up soon after initiation and generate multiple excitation wavelets (56, 195). This effect was shown to be dependent on INa (78) and ISi (corresponding to the slow inward calcium current) (54). Spontaneous break-up into multiple waves was also reported for models of cardiac excitation such as the LRd model (38, 263) and for generic models of excitable media (140, 158, 246). In several studies, the restitution of AP duration (APD) was shown to be a key determinant of wave break up, and an increase in the slope of the APD restitution curve (APD vs. the previous diastolic interval) was considered to promote break-up (101, 158, 263). It has been suggested that spontaneous wave break-up due to steep APD restitution is responsible for the transition from ventricular tachycardia to ventricular fibrillation. Thus drugs that reduce the slope of the APD restitution curve should also prevent the induction of ventricular fibrillation or convert fibrillation into a periodic rhythm (271). This has been extended to postulate that a slope of the restitution curve >1 is a global indicator of risk for arrhythmogenesis (174). While APD restitution is certainly a key factor in determining stability of reentry, other parameters also affect the breakup of reentrant waves. Restitution of conduction velocity (CV) is another fundamental property of cardiac tissue. The CV restitution curve relates local wavefront velocity to the preceding diastolic interval (DI) at the same location. In analogy to APD restitution, CV restitution also affects dynamically the degree of head-tail interaction in a spiral wave and therefore influences its stability and possibility of break-up (101).

E. Effect of Heterogeneity in Active and Passive Electric Properties on Reentrant Circuits

It should be emphasized that APD and CV restitution curves are good predictors of spiral wave stability in homogeneous and electrically continuous models of cardiac tissue. In reality, cardiac tissue is not a homogeneous electric syncytium; it contains many structural discontinuities and heterogeneities of both cellular (membrane) properties and structural components (e.g., gap junction distribution). As described in previous sections, such elements (which become more important in substrate remodeled by pathology) are major determinants of conduction safety, CV, formation of conduction block, and spatial dispersion of repolarization. Therefore, in the heart, stability of spiral waves and other forms of reentry depend on these multiple factors and their complex interactions. Such factors, to the extent that they have been studied, are discussed in this section.

1. Drift of spiral waves

Drift of spiral waves has been shown to result from both a gradient of excitability and a gradient of repolarization. In computer simulations of a linear gradient of excitability in a continuous electric medium, spiral waves drifted consistently in the direction of the region exhibiting lower excitability and velocity. In these simulations the clockwise and counterclockwise rotors drifted at different angles relative to the velocity gradient vector (248). Drift of spiral waves was also observed in excitable tissue that exhibited a gradient of repolarization (91, 92, 99). Figure 40 is taken from an experiment (98) where the lower part of the preparation was superfused with quinidine to prolong repolarization. A spiral wave, elicited in the center of the preparation, failed to penetrate deeply into the zone of prolonged refractoriness and moved along the line defining the steep gradient of repolarization (between quinidine-perfused and normally perfused tissue), until it became extinct at the border of the preparation. A consequence of the rotor movement was that the period of excitation of the tissue was shorter in front than behind the rotating excitation wave, a phenomenon known as the Doppler effect in physical wave theory. As shown in Figure 40D, the Doppler effect produced a 50% difference in excitation period (120 vs. 180 ms) in the preparation. Doppler effects were confirmed in experiments assessing spiral wave drift due to gradients of repolarization (62, 248) and also described theoretically and experimentally in spiral wave drift due to a gradient of excitability (248).

fig. 40.

Drift of a spiral wave and the Doppler effect. A and B: isochronal activation maps showing initiation (A) and the first rotation cycle (B) of a spiral wave in an isolated epicardial preparation. An abrupt gradient in refractory period was created by separate superfusion of the upper half of the preparation with quinidine-containing solutions. Dashed line shows the border of inhomogeneity with longer refractoriness in the upper part. The asterisk shows the location of the stimulating electrode. C: trajectory of the spiral wave tip after initiation (S1) and during 3 subsequent cycles of spiral wave rotation (V1-V3). Arrow indicates direction of drift. D: excitation intervals measured along the gradient in refractoriness during cycle V2. Excitation intervals in front of the spiral wave are significantly shorter than intervals behind the spiral wave (Doppler effect). [From Fast and Pertsov (99).]

2. Anchoring of spiral waves

The finding of drift of spiral waves towards a region of decreased excitability prompted a computer simulation where 4 elements of an array of 96 × 96 elements were rendered inexcitable. As illustrated in Figure 41, this changed the behavior of the spiral wave from a nonstationary regime to a stationary regime (248). This phenomenon, called “anchoring” of spiral waves, has been observed experimentally in several studies (62, 63, 248). While anchoring was generally attributed to electric heterogeneity, it was not possible based on the experimental observation to decide whether this heterogeneity was due to a gradient in excitability or a local resistive obstacle. Importantly, however, the simulation study of Pertsov et al. (248) has shown that the size of the anchoring element can be very small relative to the extension of the wavefront. This may explain why many forms of reentry that are classified as functional, such as reentry in superfused rabbit atria, in epicardial layers of the rabbit ventricle and in the epicardial border zone of myocardial infarction, often exhibit stable or partially stable circuits. Both structural inhomogeneities that are inherent to normal myocardium or to remodeled myocardium in disease are likely to cause anchoring, rendering spiral waves stationary.

fig. 41.

Anchoring of a spiral wave at a site of inexcitability. Top: snapshot of a simulated spiral wave, illustrating the position of the wave tip. Small quadrangle corresponds to 4 elements rendered inexcitable (in a grid of 96 × 96 elements). Numbers denote number of cycles after initiation of the simulation; the locations of the numbers mark the positions of the wave tip in subsequent rotations. Note that the core of the wave is approaching the quadrangle until it becomes anchored during cycle 12. Bottom: visualization of anchoring by a so-called “frame-stack” plot. The entire plane of rotation is projected to a horizontal line, and the projection is plotted on the y-axis as a function of time from top to bottom. This allows for visualization of the spiral wave core (bifurcation points of white lines) that moves slowly from right to left during the first 12 cycles until its position remains unchanged (anchoring) during the subsequent cycles, in correspondence with the top panel. [From Pertsov et al. (248).]

3. Heterogeneities in ion current flow, ion accumulation, and channel expression

It is well known that ion channels are expressed with varying densities in different regions of the heart (14, 30, 202205). Recent theoretical simulations and experiments have shown that such gradients can affect spiral waves and wave splitting in ventricular fibrillation. Because rotors turn around a nonexcited core of varying shape (see Fig. 39), ion channels that determine membrane potential in the nonexcited core, especially IK1 (see Fig. 1) that is largely responsible for the stability of resting membrane potential, can exert a repolarizing effect on the tissue in the vicinity of a phase singularity. As a consequence, tissue with a high density of IK1 can produce rotors with a shorter period than tissue with a relatively lower density. This mechanism, which was demonstrated in computer simulations, has been taken to explain the observation that a single stable rotor in the epicardial zone of the left ventricle (high IK1 density) acted as a source for fibrillatory conduction and fibrillation in the remainder of the ventricles (291). Furthermore, the stability of reentrant circuits may also be influenced by intra- and extracellular ion accumulation that evolves in time as a consequence of the rapid and repetitive excitation of the tissue. In normal myocardium, rapid stimulation is associated with an increase in intracellular Na+ (88, 190, 191) and with accumulation of extracellular K+ (182). Similar accumulation of intracellular Na+ or extracellular K+ during a tachycardia is predicted to affect the AP upstroke, CV, and repolarization. As shown by Hund et al. (138) in a theoretical model of fixed anatomical reentry, such changes of ion concentrations can affect the stability of the rotation period. In these simulations, the effect of an increase in intracellular Na+ concentration during evolution of a tachycardia depended on the degree of head-tail interaction. For weak head-tail interaction, intracellular Na+ accumulation dampened oscillations and stabilized the dynamics of the reentrant AP. In contrast, for strong head-tail interaction, it augmented oscillations in the tachycardia cycle length and reduced stability. Increase of extracellular K+ concentration also increased oscillations and destabilized reentry, as a consequence of a prolonged tail of refractoriness caused by delayed recovery from inactivation of INa (113, 304). This finding suggests that extracellular K+ accumulation during ischemia may have a destabilizing effect on reentry and may contribute to its break-up into multiple reentrant circuits and disintegration into a fibrillatory state.

In summary, the interaction between the head of a wavefront and the tail of the preceding wavefront is an important determinant of the rotation period of a reentrant AP and of the stability of reentry and spiral wave excitation. In principle, instability of rotors can result solely as a consequence of this interaction in an electrically continuous and fully homogeneous medium. However, heterogeneities, which are inherent to cardiac tissue (ion channel expression and function, cell-to-cell connectivity, tissue structure) and are typically accentuated by remodeling in disease, are also primary determinants of reentrant waves and their dynamic properties with important consequences to cardiac arrhythmias.


We thank our many colleagues, friends, and students who contributed immeasurably to the insights that we tried to communicate in this review; they have added the joyful element of friendship to our work. We honor Professors Silvio Weidmann and Robert Plonsey who introduced us to the fascinating field of cardiac bioelectricity.

Preparation of this review was made possible by support from the Swiss National Science Foundation, the Swiss University Conference, grants from the Swiss Heart Foundation (to A. G. Kléber), and National Heart, Lung, and Blood Institute Grants R37-HL-33343 and RO1-HL-49054 (to Y. Rudy).


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