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Department of Physiology, University of Bern, Switzerland; and the Cardiac Bioelectricity Research and Training Center, Departments of Biomedical Engineering, Physiology and Biophysics, and Medicine, Case Western Reserve University, Cleveland, Ohio
ABSTRACT I. INTRODUCTION II. ACTION POTENTIAL GENERATION BY THE SINGLE CARDIAC CELL III. FUNDAMENTAL BIOPHYSICAL PRINCIPLES OF PROPAGATION A. Continuous Propagation B. Principles of Discontinuous Propagation C. The Safety of Propagation D. Two-Dimensional Propagation and Curvature IV. ACTION POTENTIAL PROPAGATION IN CARDIAC CELLULAR NETWORKS: RELATIONSHIP BETWEEN STRUCTURE AND FUNCTION A. Macroscopic Anisotropic Propagation B. The Structural Basis of Propagation at the Cellular Level C. Cellular Parameters Affecting Normal Propagation 1. Role of cell size 2. Role of gap junctions 3. Role of channel clustering D. Propagation and the Shape of the Cardiac Action Potential E. Conduction and Cell-to-Cell Interaction Between Myocytes and Nonmyocytes F. Determination of Local Activation From the Extracellular Electrogram V. MECHANISMS OF SLOW CONDUCTION A. Slow Conduction and Conduction Failure Due to Reduced Membrane Excitability 1. AP propagation in acute myocardial ischemia B. Slow Conduction Related to Reduced Cell-to-Cell Coupling C. Slow Conduction Related to Tissue Structure VI. MECHANISMS OF UNIDIRECTIONAL CONDUCTION BLOCK A. General Principles 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 2. Local heterogeneity of excitability 3. Local heterogeneity of refractoriness C. Conduction Heterogeneities and Unidirectional Block Due to Discontinuities in Tissue Structure D. One-Dimensional Versus Two-Dimensional Simulations of Discontinuous Propagation: Consequences of Scale Independence VII. BASIC PRINCIPLES OF CIRCULATING EXCITATION AND REENTRY A. From Anatomic to Functional Reentry B. Initiation of Reentry 1. Initiation of reentry through interaction of wavefronts with anatomical or functional obstacles 2. Initiation of reentry by electric field shocks C. Excitable Gaps in Reentrant Circuits 1. Relation between excitable gaps and structure 2. Entrainment: resetting of reentry by external stimuli 3. Effects of drugs on reentry circles D. Head-Tail Interaction and Instability of Reentrant Circuits 1. Instability of rotation in fixed anatomical reentry 2. Instability of rotation in functional reentry 3. Wave splitting E. Effect of Heterogeneity in Active and Passive Electric Properties on Reentrant Circuits 1. Drift of spiral waves 2. Anchoring of spiral waves 3. Heterogeneities in ion current flow, ion accumulation, and channel expression
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ABSTRACT |
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I. INTRODUCTION |
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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.
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II. ACTION POTENTIAL GENERATION BY THE SINGLE CARDIAC CELL |
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For a single cardiac cell under space-clamp conditions, the following equation relates the transmembrane potential (Vm) to the total transmembrane ionic current (Iion)
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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.
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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).
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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 (
KPQ, 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.
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III. FUNDAMENTAL BIOPHYSICAL PRINCIPLES OF PROPAGATION |
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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
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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, 
. Formally, 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.
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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
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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(o) 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 < o. 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 o. 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 (
) can be defined as the negative reciprocal of the local radius of curvature, r
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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 is given by the following equation (389)
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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.
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The dependence of propagation velocity 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.
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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.
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IV. ACTION POTENTIAL PROPAGATION IN CARDIAC CELLULAR NETWORKS: RELATIONSHIP BETWEEN STRUCTURE AND FUNCTION |
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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 (L) 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 (L/T) 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 L 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, 239–241, 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
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).
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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 
/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.
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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.
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.
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·1/Ri (see text and Ref. 
