INTRODUCTION

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THE PROCESS OF CARDIAC EXCITATION involves generation of the action potential (AP) by excitatory membrane processes and its propagation in the architecturally complex structure of cardiac tissue. Success or failure of AP conduction and its velocity are determined by the interplay between membrane processes and structural properties of the tissue. Although structural inhomogeneities are present in normal myocardium (e.g., Purkinje-muscle junctions, branching of fibers, connective tissue septa, and blood vessels), they are greatly enhanced by aging and pathology (4, 12, 24, 35, 37). For example, the substrate associated with myocardial infarction contains islands of surviving myocardium interconnected by narrow strands and regions with reduced intercellular coupling through gap junctions. In addition to structural inhomogeneities, membrane heterogeneities can also be present, including nonuniformities of ion-channel densities caused by remodeling (1, 6, 10, 25, 38, 40) and regional changes of excitability caused by nonuniform spread of ischemic conditions [most importantly, hyperkalemia (17)]. In the diseased heart, membrane heterogeneities and structural inhomogeneities usually coexist and interact, affecting propagation in a very complicated manner.

Understanding the complex phenomena that occur during propagation of the AP in the inhomogeneous myocardium requires an understanding of the principles and mechanisms that govern such propagation at the cellular and ion-channel levels. Recent experimental studies of impulse transmission in cell-pair preparations (14, 19) and in patterned cell cultures (18, 28) have provided important insights into such processes. Theoretical simulations of propagation in models of inhomogeneous cardiac fibers (7, 8, 13, 15, 16, 31) have demonstrated the asymmetry associated with inhomogeneities of structure and of membrane properties as well as their possible role in the development of unidirectional block. In this study, we used a detailed model of the ventricular myocyte (22, 38, 41) in a multicellular fiber to determine the ionic mechanism of propagation through regions of structural inhomogeneities (nonuniform intercellular coupling through gap junctions and tissue expansion) and through regions of altered membrane properties associated with ischemia (reduced sodium channel availability and hyperkalemia). The use of a detailed myocyte model that represents correctly the kinetics of membrane ionic currents on the basis of recent single-cell and single-channel data is essential for a mechanistic study of the ionic basis of these phenomena. Importantly, correct kinetics of the sodium current (I_Na; fast activation, fast and slow inactivation) and the L-type calcium current [I_Ca(L); activation that is an order of magnitude faster than that in models used in earlier studies (2, 21), voltage- and calcium-dependent inactivation that requires computation of the dynamic calcium transient] must be used because these currents are responsible for AP propagation under the conditions investigated here. In this study, we focused on the following issues: 1) a quantitative evaluation of conduction safety (the safety factor, SF) and its dependence on location relative to the regions of inhomogeneous properties; 2) the location-dependent contributions of I_Na and I_Ca(L) to SF and propagation; 3) the location-dependent kinetics of I_Na and I_Ca(L) during conduction; and 4) the location-dependent effects of modulation of I_Na and I_Ca(L) by blocking agents, calcium overload, and hyperkalemia. This study also investigated the interplay between various inhomogeneities (e.g., tissue expansion and reduced gap junction coupling) and its integrated effect on conduction. The study was previously reported in abstract form (39).

	METHODS

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Multicellular fiber model. The theoretical fiber used in this study (Fig. 1A) consists of 160 cells, each 100 µm in length, connected through gap junctions as previously described (29, 30, 32, 34). Each cell in the fiber is represented by the Luo-Rudy dynamic (LRd) model of a mammalian ventricular myocyte (22, 33, 38, 41) (Fig. 1B). In this model, the AP is numerically constructed from ionic processes formulated on the basis of experimental data obtained mostly from the guinea pig. The model also accounts for processes that regulate dynamic ionic concentration changes. These include concentrations of sodium, potassium, and calcium ions. I_Na is characterized by fast activation and by fast and slow processes of inactivation. I_Ca(L) is inactivated by both voltage- and calcium-dependent processes.

View larger version (58K): [in this window] [in a new window]   Fig. 1.   Multicellular inhomogeneous cardiac fiber. A: fiber is composed of 160 serially arranged ventricular cells represented by dynamic Luo-Rudy (LRd) model. Cells are interconnected by resistive gap junctions. Stimulus is applied to cell 0 or cell 159. Axial current into a cell (Iin) and out of a cell (Iout) are depicted in 1 cell. Structural inhomogeneities are introduced starting from cell 80 and include increased gap junction conductance (represented symbolically by thicker lines) or expansion and branching of fiber (represented symbolically by doubled cells). Membrane inhomogeneities are introduced in middle section of fiber (see text). B: LRd ventricular cell model. Fast sodium current (INa) and L-type calcium current [ICa(L)] are major excitatory currents that play a role in conduction and are emphasized. For definitions of all other currents and parameters and for details of LRd model, see Refs. 22, 33, 38, and 41.

SF for conduction. SF is defined as the ratio of charge generated to charge consumed during the excitation cycle of a cell in the fiber. SF > 1 indicates that more charge was produced during cellular excitation than the charge required to cause the excitation. The fraction of SF that is >1 indicates the margin of safety. When SF falls below 1, the cell contributes less charge to the fiber than it receives. In a homogeneous fiber, SF is constant along the fiber (except for cells with stimulus or end effects), and SF < 1 results in conduction failure. A detailed discussion of SF in the context of a homogeneous fiber can be found in Ref. 34.

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	RESULTS

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Inhomogeneity of intercellular coupling. Spatial nonuniformity in the degree of intercellular coupling through gap junctions is an important structural property of cardiac tissue. In the following simulations (Fig. 2), a fiber containing 160 cells (cell 0 to cell 159) is used. Inhomogeneity of intercellular coupling is introduced by increasing the gap junction conductance, g_j, from 0.08 to 2.5 µS, starting at the junction between cells 79 and 80. The geometric dimensions remain uniform (radius = 11 µm) throughout the entire fiber. In Fig. 2, A-C, left, the fiber is stimulated at cell 0 and propagation is from the low- to the high-conductance fiber (arrow). Conduction velocity along the poorly coupled fiber is 10 cm/s; it accelerates to 55 cm/s in the well-coupled fiber. There is a long conduction delay of ~12 ms at the transition between cells 78 and 79 (Fig. 2A) because cell 79 receives small current from cell 78 and loses large current to cell 80. This delay is much longer than the intercellular delays in the uniform segments of the fiber (1 ms between poorly coupled cells, 0.18 ms between well-coupled cells). Cells just beyond the transition display a high foot potential of long duration. The poorly coupled fiber (Fig. 2B) has a higher SF value (2.73) than the well-coupled fiber (1.60) because of a reduced load and reduced loss of charge from a depolarizing cell to its neighbors (34). SF decreases sharply as the AP approaches the transition region; it reaches a minimum value of 0.98 at cell 79 (note that this value is <1). At neighboring cells, SF is 2.56 (cell 78) and 1.20 (cell 80). Thus the transition site is the "Achilles' heel" of propagation. The data in Fig. 2B indicate the charge contribution from I_Na (Q_Na) and I_Ca(L) (Q_Ca) to support conduction. Along the homogeneous segments of the fiber, I_Na plays the major role in sustaining propagation (Q_Na:Q_Ca = 10 in the poorly coupled fiber; Q_Na:Q_Ca = 105 in the well-coupled fiber). Note that the relative contribution of I_Ca(L) is greater in the poorly coupled fiber, in agreement with previous observations of Shaw and Rudy (34). For cells near the transition region, the charge contribution from I_Ca(L) exceeds that from I_Na (Q_Na:Q_Ca = 0.35 at cell 78) as depicted in Fig. 2B. The large Q_Ca results from the long conduction delay across the transition zone. Throughout this delay, cells proximal to the transition supply depolarizing charge to cells distal to the transition. During most of the delay, the proximal cells are in their plateau phase, when I_Ca(L) is the source of depolarizing charge (because of its fast inactivation, the charge contribution from I_Na is negligible beyond 1 ms). Figure 2C shows peak values of I_Na and I_Ca(L) along the fiber. I_Na is much smaller just beyond the transition as a result of inactivation during the prolonged, elevated foot potentials associated with the transmission delay across the inhomogeneity (Fig. 2A). Sodium channel availability is given by the product h · j of the two inactivation gates (h = fast, j = slow) of I_Na. In the homogeneous segments, away from the transition zone, h · j is 0.97 (poorly coupled segment) or 0.99 (well-coupled segment); it is reduced to 0.6 in the zone of transition at cell 79. Peak I_Ca(L) is sharply increased (from 16.5 to 30.5 µA/µF) in cells just proximal to the site of transition. This increase reflects an increase in driving force on I_Ca(L) caused by reduced plateau potentials in these cells (e.g., cell 78 in Fig. 2A). The lower plateau is caused by the large load on these cells during the slow depolarization of cells distal to the transition site. Because of the long transmission delay, distal cells still depolarize to threshold and consume charge from the proximal fiber when proximal cells are deep into the plateau phase of their AP, "pulling down" the plateau potential.

View larger version (39K): [in this window] [in a new window]   Fig. 2.   Conduction along a fiber with inhomogeneous intercellular coupling. Top: starting from junction between cells 79 and 80, gap-junction conductance (gj) is increased from 0.08 to 2.5 µS. Left: stimulus is applied to cell 0. Right: stimulus is applied to cell 159 and direction of propagation is reversed. A and D: action potential (Vm); numbers indicate selected cells. B and E: safety factor (SF) along fiber; local charge contributions from INa (QNa) and ICa(L) (QCa) are shown. C and F: peak values of INa (INa max; solid line) and ICa(L) [ICa(L) max; dashed line] along fiber.

View larger version (36K): [in this window] [in a new window]   Fig. 3.   Unidirectional block caused by inhomogeneous coupling. Top: in this simulation, gj in poorly coupled segment is further decreased to 0.07 µS. Other conditions are same as in Fig. 2. A and C: action potential; numbers indicate selected cells. B and D: SF along fiber.

View larger version (35K): [in this window] [in a new window]   Fig. 4.   Unidirectional block caused by reduced INa or ICa(L). Top: fiber of Fig. 2 is used in these simulations. Stimulus is applied at cell 0. Left: INa is reduced to 40% (60% block). Right: ICa(L) is reduced to 70% (30% block). A and C: action potential; numbers indicate selected cells. B and D: SF along fiber.

Tissue expansion. Another important structural property of cardiac tissue is geometric nonuniformities that involve local expansion and branching of fibers. In the simulation shown in Fig. 5, expansion through branching is introduced starting from cell 80 and repeated twice with ER = 2.3. Gap junction conductance is homogeneous throughout at g_j = 0.5 µS. The fiber is stimulated from cell 0, and propagation is into the branching fibers. Qualitatively, the behavior is very similar to that caused by increased gap junction coupling in Fig. 2 (both expansion and increased coupling present an increased electrical load to the propagating wave front). SF decreases sharply as the AP approaches the transition region; it reaches a minimum value of 0.73 (<1) at cell 80. In this region, I_Ca(L) is a very important depolarizing current and provides more depolarizing charge than I_Na (Q_Na:Q_Ca = 0.85 for the cell just before the expansion site, Fig. 5B). As in the case of increased coupling, there is a long delay across the transition region, a reduction of I_Na, and an augmentation of I_Ca(L) [peak I_Ca(L) increases from 11 to 34 µA/µF in cells proximal to the site of transition, Fig. 5C]. For this tissue structure, either 15% I_Ca(L) block or 25% I_Na block causes conduction failure into the branching fibers. In contrast, when propagation is in the reverse direction (stimulus is applied to cell 159), SF is high everywhere and propagation is supported with a high margin of safety by I_Na with only negligible contribution from I_Ca(L).

View larger version (23K): [in this window] [in a new window]   Fig. 5.   Propagation across an expansion site. Top: fiber expansion (branching) is introduced at cell 80 and repeated twice with an expansion ratio (ER) of 2.3. A: action potential; numbers indicate selected cells. B: line indicates SF along fiber; bars indicate QNa and QCa. C: INa max (solid line) and ICa(L) max (dashed line) along fiber.

View larger version (19K): [in this window] [in a new window]   Fig. 6.   Reduced coupling compensates for tissue expansion. Top: expansion is such that propagation would have failed at transition region if intercellular coupling were gj = 0.5 µS along entire fiber (ER = 3.1). In this simulation, intercellular coupling beyond expansion site is decreased from 0.5 to 0.03 µS. Reduced load caused by reduced gj compensates for increased load caused by expansion, and propagation is successful despite a long delay across transition region (A) and a local reduction of SF (B). A: action potential; numbers indicate selected cells. B: SF along fiber.

View larger version (11K): [in this window] [in a new window]   Fig. 7.   Maximum expansion ratio (ERmax) increases with reduced coupling. ERmax for which propagation is successful is shown as a function of decreasing gj. Expansion is continuous ("fanning out") with same ER at all locations.

Inhomogeneities of membrane excitability. As described in the introduction, nonuniformities of membrane properties are a common occurrence in cardiac tissue. In the following simulations, the effects of regional reduction of excitability (Fig. 8) or regional elevation of extracellular potassium (Fig. 9) on SF and conduction are shown. The inhomogeneities are introduced in a central compartment of a three-compartment fiber. Figure 8 shows SF for propagation in a fiber in which excitability of the central compartment (cells 40-100) is suppressed by making sodium channels unavailable for excitation (g_Na = 0). Propagation is studied at two levels of intercellular coupling, normal (g_j = 2.5 µS) and reduced (g_j = 0.05 µS). For both degrees of coupling, SF decreases in the central (depressed) compartment but stays above 1, and propagation across this compartment is successful (Fig. 8A). Note that SF for the poorly coupled case is higher than for the well-coupled case; in the depressed region the SF values are 1.4 and 1.1, respectively. The conduction velocity in the central compartment is ~10 cm/s (well-coupled case) and 1.8 cm/s (poorly coupled case). In the absence of I_Na, excitation and propagation in the depressed region are solely supported by I_Ca(L). The APs in this region are characterized by a slow rising phase generated by I_Ca(L) (compare with APs outside this region, Fig. 8A). The dependence on I_Ca(L) suggests that a reduction of this current should have a major effect on SF and conduction through the central compartment. Figure 8B shows that 30% block of I_Ca(L) is sufficient to reduce SF below 1 and to cause conduction failure in this region.

View larger version (23K): [in this window] [in a new window]   Fig. 8.   SF along a fiber with a central region of suppressed excitability. Central compartment is from cell 40 to cell 100. Excitability is suppressed by reducing sodium channel availability to zero (gNa = 0). A: in presence of normal ICa(L) (100% gCa). B: same as A, except that ICa(L) is reduced by 30% [70% gCa(L)]. SF is shown for normal (thin line) and reduced (thick line) gj. Action potentials from each region are shown in A.

View larger version (22K): [in this window] [in a new window]   Fig. 9.   SF along a fiber with a central region of elevated extracellular potassium concentration ([K+]o). Central compartment is from cell 50 to cell 100. A: normal [K+]o = 4.5 mM. Moderate elevation to [K+]o = 8.5 mM causes increase of SF for both values of gap junction coupling (gj = 0.05 µS, thick line; gj = 2.5 µS, thin line). B: further increase of [K+]o to 13.5 mM causes reduction of SF in hyperkalemic region, where conduction is mostly supported by ICa(L). C: at [K+]o = 14 mM, SF drops below 1 and propagation fails.

	DISCUSSION

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The important role of tissue architecture in determining the excitatory behavior of excitable tissues has been recognized and investigated early on in the context of neural excitation (see e.g., Ref. 11). As mentioned in the introduction, there is increasing appreciation of the importance of architectural heterogeneities and inhomogeneous membrane properties in determining conduction in the heart and, in particular, in arrhythmias that are associated with electrophysiological remodeling of cardiac tissue. The complexity of the myocardium has dictated a reductionist approach to the study of principles and cellular mechanisms that govern AP propagation in cardiac tissue. Important insights have been obtained recently from elegant experimental studies of AP propagation in one-dimensional tissue cultures (18, 28) and of AP transmission in cell pairs (14, 19). In this study, we used a one-dimensional theoretical model to provide quantitative characterization of AP conduction in cardiac tissue that contains structural and membrane inhomogeneities and to investigate its underlying mechanism at the level of membrane ion channels. The use of a one-dimensional model allowed us to 1) conduct the simulations at the subcellular scale in a reasonable computing time, 2) define and compute a location-dependent quantitative measure of conduction in terms of the SF, and 3) use a physiologically based detailed model of the myocyte that represents all important membrane ionic currents and dynamic changes of intracellular ionic concentrations. This level of detail is necessary for a meaningful study of the ionic mechanisms of conduction, quantitative computation of the contribution from different ion channels [e.g., I_Na, I_Ca(L)] in various locations along the inhomogeneous tissue, and location-dependent characterization of the kinetic behavior of these channels along the tissue. The principles and cellular-scale mechanisms that are determined in this model apply to cellular phenomena that govern conduction in the two- and three-dimensional myocardium at the microscopic scale. Of course, more global phenomena (e.g., wave-front curvature) require higher-dimensional models (3, 7, 8, 20). However, even such phenomena involve cellular-scale processes and can be better understood on the basis of insights and principles obtained from lower-dimensional studies (see below). Higher-dimensional models are still constrained by the computational magnitude of the problem and are forced to sacrifice spatial resolution (e.g., assume a syncytial tissue that does not resolve actual cells and gap junctions), to use tissue of small dimensions in which boundary effects influence the behavior, and to use simplified cellular models such as FitzHugh-Nagumo (9, 23), Beeler-Reuter (2), or Luo-Rudy phase 1 (21) that greatly limit the ability to study underlying ionic mechanisms [the role of I_Ca(L) in conduction could not have been studied with any of these models]. The work presented here could facilitate the development of detailed higher-dimensional models by guiding the formulation of quantitative measures such as SF (defining and computing SF in 2 and 3 dimensions is not a trivial extension of the 1-dimensional case) and by providing a sound basis for simplifications such as the use of less detailed myocyte models in higher-dimensional tissue simulations.

In a previous study (34), we characterized the ionic mechanism of conduction in a uniform fiber in which SF is constant and independent of position and must be 1 for conduction to succeed. The present study extends the previous work to include inhomogeneities in structure and in membrane properties as they exist in myocardial tissue. With the inclusion of these properties, SF becomes location dependent. In fact, propagation can succeed even if SF < 1 locally across a small region of structural change, with the AP "jumping" across this region to resume safe conduction (SF > 1) in the distal segment of the fiber.

This paper also provides a direct and quantitative evaluation of the relative importance of I_Na and I_Ca(L) in supporting AP propagation across a structural heterogeneity. In the uniform segments of the fiber, away from the transition zone, propagation is supported by I_Na with negligible contribution from I_Ca(L). I_Na is characterized by fast activation and fast inactivation; it generates a large depolarizing current over a short period of time (~1 ms). In the presence of long delays across structural heterogeneities, however, propagation relies on the slower I_Ca(L) for a sustained source of depolarizing charge (the delays are an order of magnitude longer than the time constant of I_Na inactivation). I_Na is still required, but only to depolarize the membrane to the threshold of I_Ca(L) activation. Interestingly, just proximal to the transition region where I_Ca(L) is needed as a source, it is augmented. This behavior is consistent with cell-pair experiments (19) in which the leader cell generated a greater calcium current than the follower cell during AP clamp protocol. This phenomenon can be viewed as a feedback, compensatory response of the fiber and serves as an example of the interplay between the "passive" structural components and "active" membrane components of cardiac tissue. In this case, the structural inhomogeneity dictates reliance of conduction on I_Ca(L) and the membrane responds by augmenting this current. The augmentation results from an increased driving force secondary to reduced plateau potential of the heavily loaded cells proximal to the transition zone [we verified in our simulations that kinetic changes of the activation/inactivation gates of I_Ca(L) do not contribute to this augmentation]. The reliance of inhomogeneous propagation on I_Ca(L) suggests that modulation of this current can substantially alter this type of conduction. It has been shown experimentally that I_Ca(L) suppression by nifedipine (14, 26) leads to conduction block under such circumstances. Alternatively, I_Ca(L) enhancement by BAY K 8644 or isoproterenol (14, 26) facilitates conduction across structural inhomogeneities. It should be added that modulation of I_Ca(L) can also occur indirectly in the physiological system of the cell. For example, calcium overload could reduce I_Ca(L) through calcium-dependent inactivation and block conduction, or a large transient outward current (I_to) could repolarize the plateau potential to augment I_Ca(L) by increasing its driving force.

Certain results of this study can be further generalized to apply at different scales of architecture such as coupling between multicellular fiber bundles or between multicellular regions of surviving tissue in an infarct. Because the effects of tissue heterogeneities are highly asymmetrical (decreased SF in the direction of increased electrical load and increased SF in the opposite direction), they generate conditions that favor the development of unidirectional block and reentry. Moreover, geometric nonuniformities are not necessarily associated with anatomic inhomogeneities. For example, a rotating wave front in a reentry pathway assumes a large curvature around pivot points and, at these locations, serves as a source to a large mass of tissue ("fanning-out" effect). Similarly, at the tip region of a spiral wave, where curvature is large, a small wave front (source) supplies current to a large sink, a situation that is very similar to our simulations of tissue expansion. The simulations suggest that in such regions (and only there) I_Ca(L) plays an important role in excitation and conduction. Importantly, in the heart, different geometric nonuniformities usually coexist. For example, a reentrant wave front that turns around its pivot point from the longitudinal tissue direction (along fibers) into the transverse direction experiences both expansion and a reduced level of coupling because of anisotropy (36). Figure 6 shows that an increase in SF caused by reduced coupling in a region of expansion can compensate for a reduction in SF caused by the expansion and can restore successful conduction. This result suggests that greater curvature can be supported by a wave front when it rotates in a direction of reduced coupling such as into the transverse direction. Figure 7 shows that within a broad range of gap junction conductances, uniformly reduced intercellular coupling can facilitate AP propagation in a continuously expanding tissue [a similar behavior was observed experimentally (27)]. It is possible that such a phenomenon can act to increase SF in regions of high curvature, stabilizing conduction and facilitating sustained reentry and spiral wave activity in smaller regions of the myocardium than otherwise possible. Because reduced coupling is a property of electrophysiologically remodeled substrates (e.g., a healed infarct), this process might contribute to arrhythmogenicity in this setting. Of course, these predictions should be examined experimentally and in higher-dimensional models of cardiac tissue.

Reduced membrane excitability reflects reduced availability of I_Na channels for fast membrane depolarization. As shown in Fig. 8A, propagation is possible through a region in which I_Na is made completely unavailable (100% I_Na block) and is accomplished by I_Ca(L)-generated APs at a velocity of 10 cm/s. The intervention of directly blocking I_Na does not alter the membrane rest potential and has no significant effect on I_Ca(L) during the AP (as verified by the simulations). Consequently, I_Ca(L) is fully available for membrane depolarization in the absence of I_Na. In contrast, I_Na reduction in hyperkalemia (Fig. 9) is accompanied by membrane depolarization, which acts to reduce the driving force on I_Ca(L) during the early phase of AP depolarization. Consequently, I_Ca(L) is reduced and conduction fails when [K⁺]_o exceeds 13.6 mM despite a finite (albeit small) residual I_Na availability at this concentration. If we augment I_Ca(L) by a factor of 2.1 [in ischemic myocardium, I_Ca(L) is enhanced by catecholamine release], I_Ca(L)-supported propagation is successful even at a [K⁺]_o of 36 mM with a conduction velocity of 14 cm/s. The possibility of propagation under complete I_Na block (with 22 µM TTX) has been demonstrated recently in experiments conducted in linear strands of cultured ventricular myocytes (28). The minimum conduction velocity measured in these experiments was 10.0 ± 2.0 cm/s, in excellent agreement with the simulated 10 cm/s velocity (Fig. 8A) under similar conditions. In the same experimental preparation, a velocity of 14.9 ± 3.4 cm/s was measured at an increased [K⁺]_o of 30 mM, in good agreement with our simulated velocity of 14 cm/s at a [K⁺]_o of 36 mM. The need to augment I_Ca(L) in the simulation to obtain successful conduction at such a high [K⁺]_o suggests that I_Ca(L) density is greater in the cultured neonatal rat cells (used in the experiments) than in the LRd cell model under control conditions (the LRd cell model is based on guinea pig data). In a guinea pig papillary muscle preparation (17), a minimum velocity of 10 cm/s was recorded at a [K⁺]_o of 17 mM. It should be mentioned that in a previous simulation study in a homogeneous fiber (34), conduction failed in the absence of I_Na (100% I_Na block), and transition to propagation of I_Ca(L)-supported action potentials was not achieved. This behavior seems contradictory to the results of Fig. 8A, in which I_Ca(L)-supported AP propagates through a region of complete I_Na block. In the absence of I_Na, the threshold for excitation is determined by the threshold for I_Ca(L) activation that occurs at a much higher membrane potential. Therefore, more charge is required to depolarize the membrane to threshold. In the homogeneous fiber (34) a current stimulus of 600 µA/µF over a 0.5-ms duration supplied sufficient charge to depolarize the membrane to the threshold of I_Na activation but not to that of I_Ca(L) activation. In the inhomogeneous fiber (Fig. 8A), the same stimulus excites the proximal segment of the fiber, where I_Na is available. Excitation of this segment, in turn, provides depolarizing current to the depressed segment over a much longer duration than that of the external stimulus, thereby generating sufficient charge to depolarize its membrane to the threshold of I_Ca(L) activation. We feel that this scenario better represents the physiological situation in cardiac tissue, where excitation is an intrinsic process and APs provide the excitatory stimuli.

The preceding discussion makes the point that a depressed membrane cannot support very slow conduction in cardiac tissue. Even when transition to I_Ca(L)-supported conduction occurs, the velocity cannot decrease below 10 cm/s before SF drops below 1 and conduction fails. These results support a similar conclusion of our previous study in a homogeneous fiber (34). In contrast, SF increases as velocity decreases because of reduced intercellular coupling at gap junctions (34) and reaches a maximum value of ~3 when gap junction coupling is reduced 100-fold. As a consequence, reduced coupling can support extremely slow conduction with <1 cm/s velocities. The minimum simulated velocity under such conditions is 0.26 cm/s [compare with 0.25 cm/s measured experimentally in the linear strand when gap junctions were uncoupled by palmitoleic acid (28)]. It follows that reduced coupling must play an important role in situations in which very slow conduction is measured [e.g., in infarcted myocardium (5)] and is a necessary condition for the development of sustained reentry loops in a small volume of cardiac tissue ("microreentry"). In the heart, depressed membrane and reduced intercellular coupling can coexist. The simulations of Figs. 6 and 7 demonstrate the possibility that one type of structural inhomogeneity (reduced coupling) can compensate for the reduction in SF caused by another type of structural inhomogeneity (expansion) and can restore successful conduction. We investigated the possibility of a similar phenomenon in the presence of reduced membrane excitability. We found (Figs. 8A and 9B) that reduced coupling (reduced load) augments SF that is reduced because of a depressed membrane (reduced source). This phenomenon could preserve conduction during acute ischemia and could be either antiarrhythmic, by preventing block, or proarrhythmic, by supporting very slow conduction that facilitates reentry. However, at high levels of membrane depression (Figs. 8B and 9C), the augmentation is small and insufficient to delay the onset of conduction failure.