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(Circulation Research. 1997;81:727-741.)
© 1997 American Heart Association, Inc.

Articles

Ionic Mechanisms of Propagation in Cardiac Tissue

Roles of the Sodium and L-type Calcium Currents During Reduced Excitability and Decreased Gap Junction Coupling

Robin M. Shaw, , Yoram Rudy

From the Cardiac Bioelectricity Research and Training Center, Case Western Reserve University, Cleveland, Ohio.

Correspondence to Yoram Rudy, Director, Cardiac Bioelectricity Research and Training Center, 505 Wickenden Bldg, Case Western Reserve University, Cleveland, OH 44106-7207. E-mail yxr{at}po.cwru.edu

	Abstract

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Abstract In cardiac tissue, reduced membrane excitability and reduced gap junction coupling both slow conduction velocity of the action potential. However, the ionic mechanisms of slow conduction for the two conditions are very different. We explored, using a multicellular theoretical fiber, the ionic mechanisms and functional role of the fast sodium current, I_Na, and the L-type calcium current, I_Ca(L), during conduction slowing for the two fiber conditions. A safety factor for conduction (SF) was formulated and computed for each condition. Reduced excitability caused a lower SF as conduction velocity decreased. In contrast, reduced gap junction coupling caused a paradoxical increase in SF as conduction velocity decreased. The opposite effect of the two conditions on SF was reflected in the minimum attainable conduction velocity before failure: decreased excitability could reduce velocity to only one third of control (from 54 to 17 cm/s) before failure occurred, whereas decreased coupling could reduce velocity to as low as 0.26 cm/s before block. Under normal conditions and conditions of reduced excitability, I_Ca(L) had a minimal effect on SF and on conduction. However, I_Ca(L) played a major role in sustaining conduction when intercellular coupling was reduced. This phenomenon demonstrates that structural, nonmembrane factors can cause a switch of intrinsic membrane processes that support conduction. High intracellular calcium concentration, [Ca]_i, lowered propagation safety and caused earlier block when intercellular coupling was reduced. [Ca]_i affected conduction via calcium-dependent inactivation of I_Ca(L). The increase of safety factor during reduced coupling suggests a major involvement of uncoupling in stable slow conduction in infarcted myocardium, making microreentry possible. Reliance on I_Ca(L) for this type of conduction suggests I_Ca(L) as a possible target for antiarrhythmic drug therapy.

Key Words: cardiac excitation • gap junction • cardiac ion channels

	Introduction

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Although the time and voltage-dependent kinetics of the major ionic membrane currents of ventricular cells are well described, their functional role in propagation of the action potential is not fully understood. The major cardiac inward currents are the fast sodium current, I_Na, and the L-type calcium current, I_Ca(L). Typically, I_Na is responsible for excitability and conduction, and I_Ca(L) is responsible for maintaining the action potential plateau. We sought to elucidate the relative role of I_Na and I_Ca(L) in maintaining multicellular ventricular propagation under both normal physiological conditions and two pathophysiological conditions: reduced membrane excitability and reduced intercellular coupling.

Reduced membrane excitability, caused by reduced availability of I_Na, is present during conditions of acute myocardial ischemia,^{1 2 3} tachycardia,³ and treatment with class I antiarrhythmic agents.⁴ Reduced excitability lowers the safety factor for conduction (safety factor is a dimensionless parameter that indicates the margin of safety with which the action potential propagates relative to the minimum requirements for sustained conduction). However, the precise relationship between safety factor and excitability has not been studied: it is not known whether the safety factor drops monotonically with decrease in I_Na availability, nor has the steepness of this relationship been characterized. For instance, does a 50% reduction in excitability bring the membrane very close or only marginally close to conduction block? Safety factor considerations usually address I_Na only. At highly reduced membrane excitability, it is possible that I_Ca(L) contributes significantly to the safety factor. In this study we attempt to determine the contribution of I_Ca(L) to the safety factor and its role in conduction.

Similar to the condition of reduced membrane excitability, reduced intercellular coupling at gap junctions decreases velocity of the propagating action potential. However, the magnitude of velocity reduction and the effect on safety factor for conduction may be vastly different for the two conditions. Several theoretical^{5 6 7} and experimental^{8 9 10} studies have suggested that reduced coupling can cause major reductions in velocity before conduction block and that velocity in the context of reduced coupling can potentially be much slower than velocity in the context of reduced excitability. Despite slowed conduction, evidence tends to suggest that propagation across cells with reduced coupling occurs with a high safety factor.^{11 12} These reports of high propagation safety with low intercellular coupling are not universal, and there exists evidence to the contrary.⁹

Long intercellular propagation delays associated with highly reduced gap junction coupling also suggest a potential role for I_Ca(L) in supporting conduction.^{13 14} With long intercellular delays, it is feasible that the downstream neighboring cell remains unexcited and constitutes a current sink when the upstream depolarized cell (current source) is already in its plateau phase. During the plateau phase, I_Ca(L) is the inward source current of the upstream cell and could be important to sustain cell-to-cell conduction. The degree of coupling reduction necessary before I_Ca(L) becomes a major contributor to both conduction velocity and safety factor has not been determined. Also, the phenomenon of calcium-dependent ([Ca]_i) inactivation of I_Ca(L) suggests that there can be direct coupling between [Ca]_i and propagation safety. It was recently shown in neuronal preparations (rat ganglion cells) that calcium overload reduces safety factor for conduction.¹⁵

We characterized propagation in the setting of reduced membrane excitability and reduced gap junction coupling with simulations using detailed cellular ionic models and the appropriate intercellular coupling conditions of a multicellular fiber. The individual cells were represented by the dynamic Luo-Rudy (LRd) ventricular cell formulation,^{16 17 18} interconnected by resistive pathways representing gap junctions. In the context of this study, it should be emphasized that the LRd model incorporates important kinetic properties of I_Na and I_Ca(L). These include fast and slow inactivation processes of I_Na, an order-of-magnitude faster I_Ca(L) activation than in previous models (eg, Beeler and Reuter¹⁹ ), and calcium-dependent inactivation of I_Ca(L). Safety factor of conduction was formulated using a modified version of formulations previously introduced.^{9 12}

The results demonstrate that a decrease in membrane excitability (eg, due to ischemia, tachycardia, or treatment with antiarrhythmics) causes a monotonic decrease of the safety factor for conduction and a modest decrease of conduction velocity before the occurrence of block. In contrast, a decrease in intercellular coupling (eg, due to anisotropy of the myocardial structure, aging, elevated intracellular calcium, or infarction) can have a profound depressant effect on conduction velocity that is accompanied by a paradoxical augmentation of the safety factor for conduction. The interplay between membrane factors and gap junction factors has significant relevance to conduction under pathological conditions in which one pathophysiological change (reduced coupling) may compensate for another (reduced excitability) in terms of their opposite effects on propagation safety. Detailed studies of the ionic mechanisms of normal and pathological conduction identify the relative importance of I_Na and I_Ca(L) in supporting conduction under a given set of conditions. Such insight could be very helpful for informed treatment of cardiac arrhythmias.

	Materials and Methods

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
The LRd cell model and its use as the cellular element of a multicellular fiber have been described previously.^{16 18 20} The following is a brief description of the cell and fiber models and formulation of the safety factor of conduction (SF). In view of the strong emphasis in this study on the role of gap junctions in propagation and the detailed cable analysis required for SF computation, related methodology is presented in "Appendices 1 and 2." A description of simulation protocols is also included below.

Multicellular Fiber Model
For each cell in the fiber, the LRd model is used to compute ionic currents and concentration changes. In this model, the guinea pig–type ventricular action potential is numerically constructed on the basis of experimental data. Included in the model are the membrane ionic channel currents, represented mathematically by a Hodgkin-Huxley type formalism, as well as ionic pumps and exchangers. In addition, processes that regulate ionic concentration changes, especially dynamic changes of intracellular calcium concentration, are introduced. A diagram of the cell model is provided in Fig 1A, with the two major excitatory currents (I_Na and I_Ca(L)) highlighted. Detailed tables of equations governing the model are provided in References 16¹⁶ to 18.

View larger version (59K): [in this window] [in a new window]   Figure 1. A, Dynamic Luo-Rudy ventricular cell model (LRd). The time and voltage dependence of the major ionic channels are formulated with Hodgkin-Huxley type formalism with additional [K]o-dependent conductance of IKl and IKr, [Ca]i-dependent conductance of IKs, and [Ca]i-dependent inactivation of ICa(L). JSR and NSR are junctional and network sarcoplasmic reticulum, respectively. The major excitatory currents, INa and ICa(L), are emphasized. INa, fast Na+ current; ICa(L), L-type Ca2+ current; IKl, inward rectifier K+ current; IKr, fast component of the delayed rectifier current; IKs, slow component of the delayed rectifier current. B, Multicellular cardiac fiber composed of a series of LRd cells interconnected by resistive gap junctions. Fiber is 7 to 10 mm (70 to 100 cells) long. Conduction is initiated by an external stimulus applied to the proximal end of the fiber (cell 1). Cellular parameters are studied for cells in the middle of the fiber and conduction velocity is computed between cells 20 and 50. For details of the LRd cell model see References 16 through 18.

The theoretical fiber (Fig 1B) is composed of serially arranged ventricular cells, each of LRd formulation. The temporal transmembrane current fluxes of the LRd model are related to spatial (axial) current flow by a finite difference approximation⁶ of the cable equation^{21 22 23}

(1)

where I_ion represents the individual membrane ionic current densities (µA/µF) of the LRd model, I_s is the stimulus current density (µA/µF), a is the radius of the fiber (11 µm), C_m is the membrane capacity (1 µA/µF), V_i^t is membrane potential at segment i and time t, x is the discretization element (µm), R_CG is the ratio between capacitive and geometrical areas (R_CG=2), and R_i is the axial resistance per unit length (cm). R_i is composed of myoplasmic resistance (R_myo=150 cm) and gap junction resistance (normal R_g=1.5 cm², which corresponds to a conductance of g_j=2.5 µs). Direct explicit solutions to Equation 1 would necessitate a very small time increment (t) for numeric convergence and could not guarantee stability. For faster computing with a larger t, the relation in Equation 1 can be expressed in implicit Crank Nicolson²⁴ form by replacing the second central difference with the average of the second central difference at time t and t+t.⁶ The Crank and Nicolson method is second-order accurate and is unconditionally stable. Its solution entails, at any given time step, simultaneous solution of membrane potential over the entire fiber once individual ionic currents are computed.²⁵ We always use the Crank and Nicolson implicit method to solve for the relation in Equation 1. Extracellular resistance, R_o, is neglected (the fiber is assumed to be in an extensive medium). No flux (sealed ends) boundary conditions are used by setting V_m/x=0 at the beginning and end of the fiber.

Spatial discretization and distribution of fiber resistivity require special consideration for a discontinuous fiber, as discussed in "Appendix 1." For computations in this report, x=100 µm (entire cell length) is used as the discretization element with R_i reflecting the lumped contribution of axial and gap junction resistance (R_i=R_myo+R_g/x). When intracellular detail is sought, a smaller x of 21 computation elements per cell (x=4.76 µm) is used with R_g concentrated at the edge elements of the cells. All simulations were tested for numeric convergence and accuracy.

Regardless of fiber discretization, the intracellular calcium transient [Ca]_i, intracellular calcium buffering, and sarcoplasmic reticulum (SR) calcium fluxes are always computed for the entire cell as one unit. When x is less than one cell length, sarcolemma calcium currents from each element are summed for net calcium entry into the cell. Unless otherwise specified, the middle element of a multi-element cell is always used for computation of upstroke velocity, dV_m/dt.

Safety Factor for Conduction
On the basis of the approach of others,^{9 12} we define the safety factor for conduction (SF) as the ratio of charge generated for the fiber by cell excitation to the minimal amount of charge required to cause the excitation. An SF >1 indicates that more charge was produced during cellular excitation than charge required to cause excitation. The fraction of SF >1 indicates the margin of safety. When SF falls <1, the charge requirements are not met and conduction fails. The equation used to compute SF is

(2)

where I_c is the capacitive current of the cell in question, I_out is the axial current between the particular cell and its downstream neighbor, I_in is the axial current that enters the particular cell from its upstream neighbor, and Q_m is the integral of transmembrane current over time. A, the integration limit for SF, is the period that the cell in question remains a sink, the period during which the membrane has consumed more charge than it has produced. The beneficial aspect of our approach is that SF, computed at a particular cell, takes into consideration charge lost and gained with respect to the entire fiber. This definition accounts for excitability changes due to charging time (ie, strength-duration) and for effects of changes in intercellular coupling. A detailed explanation of SF is provided in "Appendix 2."

Simulation Protocol
The theoretical fibers used in this study were between 70 and 100 cells (0.70 to 1.0 cm) in length. R_myo (myoplasmic resistivity) was 150 cm and g_j (gap junction conductance) varied between 0.005 and 2.5 µs (corresponding to resistance changes between R_g=760 and 1.5 cm², respectively). A variable time step was introduced that tracks the propagating action potential, using small time steps (t=0.005 ms) when membrane activity is high and large time steps (t=1.0 ms) when membrane activity is low. The variable time step algorithm sums the change in membrane voltage for nine cells in each direction from the cell under consideration. Each millisecond, the sum is evaluated for all cells of the fiber. If a cell's sum is greater than a preset threshold (20 mV) or the cell is within 15 ms of calcium release, then the cell's time step for the next millisecond is small. Otherwise the time step is large. The variable time step is used for membrane current computations only. Axial currents are always computed every 0.005 ms. The variable time step was always in effect, except where indicated.

The fiber was excited by applying a 0.5-ms superthreshold stimulus to cell 1 and, when multiple (n) elements per cell were used, to the middle element of cell 1. Stimulus strength (µA/µF) varied with n and g_j. For n=1 and g_j=2.5 µs, stimulus was -600 µA/µF. Results are independent of the stimulus used for distances greater than one space constant (about 9 cells) from the stimulus site because propagation depends on fiber axial current, not on the applied stimulus, for excitation. End effects were also restricted to within 9 cells from the fiber end.

	Results

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Propagation in a Multicellular Fiber
The theoretical fiber we used is composed of interconnected cell models rather than a membrane model with finite spatial dimensions. The difference is that individual cell models contain both membrane and intracellular processes. Excitation in the multicellular fiber by a propagating action potential results in both membrane and intracellular events (eg, the transmembrane action potential and the intracellular calcium transient). Fig 2A shows the profile of a computed action potential (AP) in space at time=t and time=t+3 ms. The distance covered in 3 ms is 1.6 mm (16 cells), which is observed in Fig 3 and can be computed from the product of the time period (3 ms) and conduction velocity (54 cm/s). At the head of the action potential there is a slow electrotonic depolarizing phase (AP "foot"). Slightly further upstream (to the left) cells are undergoing a sharp upstroke caused by the fast sodium current (I_Na) depolarizing local membrane. Further upstream (left) of the upstroke, late upstroke, and early plateau depolarization is occurring, which is due to the L-type calcium current (I_Ca(L)).

View larger version (23K): [in this window] [in a new window]   Figure 2. Action potential propagating down a multicellular fiber. A, Head of action potential traveling from left to right at time=t ms (dashed line) and 3 ms later (t+3, solid line). B, Corresponding spatial profile of the intracellular calcium transient at the two time instants. The calcium transient lags behind the action potential upstroke. The spatial spread of calcium is not as sharp as the action potential upstroke due to calmodulin and troponin buffering of free myoplasmic calcium.

View larger version (34K): [in this window] [in a new window]   Figure 3. Increase in intercellular conduction delay with decrease in gap junction coupling. Action potential upstrokes from the edge elements (see inset) of two adjoining cells for intercellular conductance of 2.5 µs (A) and intercellular conductance of 0.25 µs (B). Discretization of 21 patches per cell was used. For control coupling (A), intercellular conduction delay is approximately equal to intracellular (myoplasmic) conduction time. A 10-fold decrease in intercellular conductance (B) increases intercellular conduction time and decreases intracellular conduction time dramatically, resulting in gap junction dominance of overall conduction velocity.

Spread of electrical excitation in Fig 2A initiates a chain of events that lead to calcium-induced calcium release (CICR) and a rise in intracellular calcium. Fig 2B shows the calcium transient in space for the same time instants as the AP profiles in Fig 2A. At each time instant (t and t+3) the AP upstroke (wave front) is about 25 cells ahead of the calcium transient. This spatial difference results from the period of calcium accumulation that triggers CICR¹⁶ and the dynamics of the SR calcium release process.

The coupling between membrane excitation and calcium release is bidirectional. Calcium accumulation from calcium entry through I_Ca(L) during the late upstroke initiates CICR. The resulting myoplasmic calcium transients decrease local I_Ca(L) via calcium-dependent inactivation. Under normal conditions, CICR is sufficiently delayed from the upstroke to not interfere with conduction. However, when conduction is slowed, CICR leading to calcium-dependent inactivation of I_Ca(L) may become an important factor.

Fig 3 shows action potential profiles in two neighboring cells under conditions of normal gap junction coupling (panel A) and a marginal (10-fold) decrease in coupling (panel B). For normal coupling (g_j=2.5 µs), the gap junction conductance between cells is equal to the myoplasmic conductance of the entire cell. The result is equal conduction time of 0.09 ms between cells as in crossing the cell length. On a macroscopic (many cells) scale, the result is an apparent uniform spread of excitation as seen in Fig 3A. With only 10-fold reduction in coupling (Fig 3B) a large (0.5 ms) conduction delay is introduced at each intercellular junction, while the entire cell depolarizes almost simultaneously. The almost simultaneous depolarization of the cell is due to increased confinement of depolarizing current to the cell when intercellular coupling is reduced, decreasing loading effects and current leakage out of the cell. Note (Fig 3B) that propagation under these conditions is nonuniform ("discontinuous"), with long delays at gap junctions. In fact, the macroscopic conduction velocity over many cells is determined by the gap junction delays rather than by the (negligible) time spent in traveling across individual cells. With even further reduction in coupling, the intercellular delay can be on the order of milliseconds, providing time for calcium released from the SR to contribute to I_Ca(L) inactivation, altering the balance of currents at the head of a traveling action potential. The issues of propagation down a fiber with reduced coupling and interaction between the calcium transient and I_Ca(L) are explored in later sections of this report. Principles observed in Figs 2 and 3 are that propagation is a multicellular phenomenon that involves interaction between membrane currents, gap junction properties, and intracellular ionic processes. The interactions between these multiple factors must be considered when propagation is analyzed and when mechanisms of normal and abnormal conduction are investigated.

Reduced Membrane Excitability Lowers Conduction Velocity and Propagation Safety, With Little Effect From I_Ca(L)
We investigated the effects of reduced membrane excitability (eg, due to acute ischemia or class I drugs) on conduction of the action potential impulse. The parameters used to characterize conduction with changes in excitability are conduction velocity (), safety factor for conduction (SF), maximum upstroke velocity ([dV_m/dt]_max), and peak sodium current (I_Na,max). These computed parameters are shown with different degrees of membrane excitability in Fig 4. Because excitability is determined by availability of fast sodium channels, membrane excitability is reduced by lowering ¯g_Na, the maximum conductance of I_Na. All four indices of conduction decrease monotonically with excitability reduction. This result is to be expected; lower sodium channel availability causes slower conduction of an action potential that has a slower upstroke and is less safe. Fig 4A shows that the minimum conduction velocity obtained before conduction failure is 17 cm/s, indicating that the minimum possible conduction velocity with reduction in excitability is only one third of control velocity (54 cm/s).

View larger version (25K): [in this window] [in a new window]   Figure 4. Conduction parameters for decreased membrane excitability. A, Conduction velocity (solid line) and safety factor of conduction (dashed line) versus % ¯gNa. B, Maximum upstroke velocity (solid line) and peak INa,max (dashed line) versus % ¯gNa. All four parameters decrease in a monotonic fashion with decreasing excitability. Conduction velocity decreases threefold to 17 cm/s before conduction fails. (dVm/dt)max and INa,max decrease in approximately linear fashion with % ¯gNa. INa,max is peak sodium current; (dVm/dt)max is maximum upstroke velocity of the action potential rising phase; ¯gNa is maximum conductance of INa and indicates membrane excitability. Horizontal bars indicate occurrence of conduction block.

Safety factor of conduction, also in Fig 4A, decreases slowly with excitability reduction, indicating a relative insensitivity to moderate changes in membrane excitability. It can be recalled from the discussion in "Methods" that SF is the ratio between charge gained by the fiber from cell excitation to charge required for cell depolarization. Our simulations reveal that reducing sodium channel availability only marginally reduces charge gained by the fiber from excitation (numerator of SF) because (1) capacitive charge gained (first numerator term) is proportional to peak membrane potential, which is not reduced significantly by reducing I_Na, and (2) subthreshold depolarization time is not greatly reduced, resulting in only small changes to the charge delivered downstream (second numerator term). Our simulations also indicate that charge required to depolarize the fiber (the denominator of SF) is not significantly increased. Therefore less than extreme reductions in I_Na availability do not constitute a great detriment to SF. At I_Na availability <11.25%, the cell membrane has difficulty reaching threshold, depolarizing charge requirements from upstream fiber start to increase dramatically (denominator), and SF drops precipitously toward 1. The rapid decrease of SF toward 1 in Fig 4C is indicative of the nonlinear "all or none" response of the cell membrane.

The data in Fig 4B suggest that I_Na dominates conduction for almost the entire range of membrane excitability. Maximum upstroke velocity, an indicator of the depolarizing current during the upstroke, follows I_Na,max at all levels of reduced excitability. However, as membrane excitability is reduced below 30% ¯g_Na, I_Na,max decreases faster than (dV_m/dt)_max, suggesting additional support from another inward current (I_Ca(L)). We also found that when I_Ca(L) is removed, conduction fails at slightly higher membrane excitability (¯g_Na=15% as compared with 11% when I_Ca(L) is present). Therefore we examined, in Fig 5, the role of I_Ca(L) in action potential conduction under conditions of depressed membrane excitability. In Fig 5A, SF is computed for a control fiber with all membrane currents intact (solid line) and for a fiber with I_Ca(L) removed (dashed line). For most values of membrane excitability, the presence of I_Ca(L) does not influence SF. At extreme reductions of excitability (<30% availability), I_Ca(L) augments SF slightly, resulting in conduction failure at the slightly lower value of % ¯g_Na.

View larger version (22K): [in this window] [in a new window]   Figure 5. Role of ICa(L) in conduction under reduced membrane excitability. A, Safety factor (SF) computed over a range of membrane excitability for fibers with (solid line) and without (dashed line) ICa(L). SF for the fiber without ICa(L) begins to diverge from SF with ICa(L) at 30% ¯gNa. B, High-resolution action potential upstrokes from the middle cell of the fiber at 20% ¯gNa with (solid line) and without (dashed line) ICa(L). Inset bar graph shows relative charge, Q (current integrated over time) generated by INa and ICa(L) in the period from the middle cell's (dVm/dt)max to (dVm/dt)max of the adjoining cell (marked by the thin vertical line at time=0.37 ms). This charge computation reflects ionic source charge during conduction. Even at severely reduced ¯gNa, charge contributed by INa is much larger than charge contributed by ICa(L). Horizontal bars in A indicate conduction failure (note that conduction cannot be sustained when SF<1; ie, when the minimal charge requirements are not met).

At extremely low membrane excitability, in the vicinity of ¯g_Na=20%, a slight positive influence of I_Ca(L) on SF is evident in Fig 5A. We investigated the mechanism by which I_Ca(L) increases the safety of highly depressed conduction. Fig 5B shows action potential upstrokes from the middle cell of the fiber computed with I_Ca(L) (solid line) and without I_Ca(L) (dashed line), both at 20% ¯g_Na. Time 0 is the time of (dV_m/dt)_max for the middle cell. The action potential upstrokes, as reflected in (dV_m/dt)_max, are slightly faster for the fiber computed with I_Ca(L) (46 versus 35 V/s). The difference in (dV_m/dt)_max is not due to different I_Na,max, which is actually higher for the fiber without I_Ca(L) (-48 versus -54 µA/µF, not shown), nor is the difference due to direct local action of I_Ca(L), which does not significantly activate at the negative voltage range (around -33 mV) at which (dV_m/dt)_max occurs. (dV_m/dt)_max is somewhat higher in the fiber with I_Ca(L) because I_Ca(L) increases slightly the action potential amplitude in the adjoining upstream cell (compare action potentials with and without I_Ca(L) in Fig 5). Higher action potential amplitude acts to increase the potential gradient and the electrotonic driving force between an excited cell and its downstream neighbor that is still undergoing depolarization. The result is a slight increase in axial current and somewhat greater axial charge delivery that contributes directly to the depolarization of the downstream cell.

The quantitative role of I_Ca(L) in augmenting electrotonic driving force and downstream depolarization can be established by computing the charge (current integrated over time) generated by a cell membrane between its own (dV_m/dt)_max and (dV_m/dt)_max of its downstream neighbor (identified by a thin vertical line in the figure). This computation is a measure of charge generated by a source cell to depolarize its downstream neighbor. Note that I_Na and I_Ca(L) are integrated over a very short interval (to the thin vertical line) 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). During the short interval, I_Na is near its maximum value and I_Ca(L) is only beginning to activate. In the inset of Fig 5B, we show relative charge generated by I_Na and I_Ca(L) at 20% ¯g_Na. The computed charge ratio Q_Na:Q_Ca is 75:1. Thus it can be concluded that the predominant current responsible for maintaining conduction, even at 20% ¯g_Na, remains I_Na.

The results of this section demonstrate that despite a slight influence from I_Ca(L) in supporting axial charge delivery at extreme levels of membrane depression, in an otherwise well-coupled fiber excitability and conduction are determined by I_Na. Decreased excitability decreases all parameters of conduction, ultimately leading to conduction block. In the following section we evaluate the effect of decreased intercellular coupling while the membrane currents are not directly modified.

Reduced Intercellular Coupling Reduces Conduction Velocity and Has a Biphasic Effect on Propagation Safety, With Major Effect from I_Ca(L)
Propagation transverse to fiber orientation and propagation across areas of a healed infarct are examples of normal and pathophysiological propagation in which intercellular coupling is reduced. Fig 6 contains the parameters of propagation plotted against changes in intercellular coupling conductance, g_j. Conduction velocity decreases monotonically with reduction in intercellular coupling. This behavior is more dramatic but qualitatively similar to that of reductions in membrane excitability. However, as seen in both Fig 6A and 6B, safety factor for conduction and maximum upstroke velocity of the action potential do not decrease with decreasing levels of intercellular coupling. Both parameters display a biphasic behavior, increasing then decreasing, in response to reduction of g_j. Across the entire range of g_j, SF is greatest at g_j=0.023 µs and (dV_m/dt)_max is greatest at g_j=0.040 µs. Therefore, maximum values for both parameters occur when intercellular coupling is reduced by a factor of about 15 relative to its normal value of 2.5 µs.

View larger version (24K): [in this window] [in a new window]   Figure 6. Conduction parameters for decreased intercellular coupling. A, Conduction velocity (solid line) and safety factor for conduction (dashed line) versus gap junction conductance, gj. B, Maximum upstroke velocity (solid line) and INa,max (dashed line) versus gj. Although velocity decreases in a monotonic fashion with decreased coupling, the behavior of the safety factor and of (dVm/dt)max is biphasic (an increase followed by a decrease). Conduction velocity decreases 200-fold to 0.26 cm/s before block occurs. Safety factor and (dVm/dt)max increase initially, reflecting greater confinement of current for depolarizing local membrane (less downstream leak due to reduced gj). INa,max decreases slowly at first and then abruptly (its decrease is due to dynamic inactivation during slow depolarization to threshold). Near block, (dVm/dt)max and INa,max are identical, reflecting the almost isolated cell conditions of highly uncoupled tissue. Horizontal bars indicate conduction failure.

The slight monotonic decrease in I_Na,max with decrease in g_j, seen in Fig 6B, establishes that membrane currents are not the cause of the biphasic changes in SF and (dV_m/dt)_max. Instead, the changes of intercellular coupling affect source-load relationships in the fiber, which feed back on the action of membrane currents. Inward membrane current either acts to depolarize local membrane or generates axial current flow downstream to act on distant membrane. When coupling is reduced, less inward current is shunted downstream (less load), effectively increasing availability of inward current for local depolarization. As a result of increased availability of local depolarizing current, safety factor and (dV_m/dt)_max increase. The descent of safety factor and (dV_m/dt)_max at very low levels of coupling is due to reduced availability of I_Na source-current under these conditions. I_Na,max is small at very low levels of intercellular coupling due to a long subthreshold depolarization phase (slow charging process due to the small axial current when resistance between cells is large) that provides for dynamic inactivation of sodium channels before reaching their activation threshold. This results in reduced sodium channel availability at threshold and a small I_Na,max. Ultimately, for very low degrees of intercellular coupling, subthreshold dynamic inactivation of I_Na is not compensated for by conservation of current for local membrane due to the reduced cellular coupling. The result is decreased safety factor, decreased (dV_m/dt)_max, and with sufficiently reduced coupling, conduction block. The data in Fig 6B demonstrate that when coupling is extremely reduced, I_Na,max is almost the sole determinant of (dV_m/dt)_max (the two curves practically overlap at this range), as occurs in an isolated space-clamped cell (a situation that is approached by the highly uncoupled fiber).

The results of Fig 6 illustrate that altered intercellular coupling can cause major changes of the action of membrane currents even when the intrinsic properties (ie, density and gating) of these membrane currents remain unchanged. We introduced earlier the possibility that I_Ca(L) can affect conduction when coupling is reduced. This hypothesis is tested by computing the safety factor for conduction over a full range of intercellular coupling with and without the presence of I_Ca(L). Results are shown in Fig 7A that show that the two safety factor curves have the same biphasic shape. However, a striking difference between the curves is that conduction failed at g_j=0.0056 µs for the fiber with all ionic currents intact and at a much higher g_j=0.0197 µs for the fiber computed without I_Ca(L). The finding that the presence of I_Ca(L) can support conduction that otherwise fails over more than a threefold decrease in gap junction coupling strongly suggests that I_Ca(L) is a major determinant of propagation under conditions of highly reduced intercellular coupling and long propagation delays across gap junctions.

View larger version (25K): [in this window] [in a new window]   Figure 7. Role of ICa(L) in conduction when intercellular coupling is decreased. A, Safety factor (SF) computed over a range of intercellular coupling for fibers with (solid line) and without (dashed line) ICa(L). SF computed for the fiber without ICa(L) begins to diverge from SF with ICa(L) at a fivefold decrease in intercellular coupling (gj=0.5 µs). Conduction fails in the fiber without ICa(L) at three times the conductance for which block occurs when ICa(L) is present. B, High temporal resolution action potential upstrokes from the middle cell of the fiber at gj=0.020 µs with (solid line) and without (dashed line) ICa(L). Time 0 corresponds to time of (dVm/dt)max for each case. Inset bar graph shows relative charge, Q (current integrated over time), generated by INa and ICa(L) in the period from the middle cell's (dVm/dt)max to (dVm/dt)max of the adjoining cell (marked by thin vertical line at time=3.6 ms). The long intercellular delay extends into the plateau of the source-cell, during which ICa(L) is the major inward membrane current. At the reduced coupling of B, charge contributed by ICa(L) approaches that contributed by INa. At extreme levels of uncoupling, illustrated in C, charge contribution from ICa(L) exceeds that from INa by up to an order of magnitude.

The interplay between I_Na and I_Ca(L) during conduction with decreased intercellular coupling is further explored in Fig 7B, which contains an action potential upstroke at g_j=0.020 µs, computed with (solid line) and without (dashed line) I_Ca(L). Gap junction conductance of g_j=0.020 µs is used because it is slightly greater than the conductance at which I_Ca(L) becomes essential for conduction. As was explained in the context of reduced excitability, I_Ca(L) does not play a direct role in local excitation but, by maintaining a higher plateau, I_Ca(L) enhances the axial driving force and the resulting electrotonic source current. When decreased coupling causes long propagation delays, I_Ca(L), which is the major depolarizing current during late upstroke and early plateau, becomes extremely important. The role of I_Ca(L) in augmenting the driving force and the electrotonic current is confirmed by the postupstroke membrane potentials of Fig 7B. The action potential that included I_Ca(L) maintains an early plateau at significantly higher potential than the action potential without I_Ca(L). It is the higher plateau that increases the driving force, forcing more source current to the adjacent downstream cell. The bar graph inset in Fig 7B indicates the increasing role of I_Ca(L) in supporting conduction under conditions of decreased coupling. The bars, like those of Fig 5, show the relative charge contribution to sustain conduction from I_Na and from I_Ca(L) and are computed by integrating these currents over time, until the downstream neighboring cell is activated. Note that due to the long delay at the gap junction, the downstream neighbor is excited when the upstream source cell is well into its plateau (thin vertical line in Fig 7B). Therefore, I_Ca(L), which is active during the plateau, is integrated over a much longer time than in the case of depressed membrane (compare with Fig 5B), and I_Na, which is inactivated at the plateau, does not contribute additional charge beyond its early contribution during this interval. As a result, unlike the I_Na dominance conditions of reduced excitability (Fig 5), charge contributions from both currents are almost equal (Q_Na:Q_Ca(L)=1.47:1) for reduced coupling of g_j=0.020 µs. Therefore, I_Ca(L) under such conditions is almost as important as I_Na in sustaining conduction. With further reductions in coupling, charge contribution from I_Ca(L) needed to support conduction exceeds that from I_Na (ie, at g_j of 0.010, 0.006, and 0.0057 µs, Q_Na:Q_Ca(L) charge ratios are 0.81, 0.26, and 0.16, respectively; the case of g_j=0.006 µs is shown in Fig 7C). Therefore, in the situation of highly reduced coupling, I_Na is necessary to bring the membrane into the activation range of I_Ca(L), but the major source of depolarizing charge and the most significant current to sustain propagation is I_Ca(L).

Intracellular Calcium Can Influence Conduction
There are numerous pathological conditions during which ventricular cells experience an increase in intracellular calcium, a condition known as calcium overload. Because [Ca]_i has a direct effect on I_Ca(L) via calcium-dependent inactivation, it follows that changes in intracellular calcium may affect conduction. We explored the effect of elevated [Ca]_i on action potential propagation by comparing propagation in a fiber with control values of resting [Ca]_i=0.12 µM to a fiber with elevated resting [Ca]_i=0.25 µM. Considering that I_Ca(L) affects propagation when coupling is reduced, we computed the safety factor of each fiber over a full range of intercellular coupling with a protocol similar to that of Fig 7A. We found (not shown) that calcium overload begins to lower SF after a 100-fold reduction in g_j and causes conduction failure at slightly higher intercellular conductance (failure occurs at g_j=0.007 µs and g_j=0.0057 µs at elevated and normal [Ca]_i, respectively). Therefore, under extreme conditions, when dependence on I_Ca(L) to sustain conduction is a major factor, decreasing I_Ca(L) via elevated [Ca]_i will decrease propagation safety.

To confirm that [Ca]_i action on I_Ca(L) is the mechanism of decreased propagation safety under conditions of calcium loading, we examined, in Fig 8, the action potential upstroke and related calcium current that occur at low intercellular coupling (g_j=0.008 µs). Two fibers were used, one with normal [Ca]_i (solid line) and one with elevated [Ca]_i (dashed line). The action potential upstrokes are followed by a dip, which (as recognized from previous figures) is caused by the charging of the adjoining downstream cell. During the dip, I_Ca(L) reaches a maximum value and then decreases. The maximum values of I_Ca(L) are coincident with calcium release from the SR and the following decline is due mostly to calcium-dependent inactivation. I_Ca(L) decreases more for the cell with elevated [Ca]_i, reflecting greater calcium-dependent inactivation. Greater inactivation of I_Ca(L) causes (Fig 8A) faster action potential repolarization, decreased axial driving force, and longer charging period of the adjacent downstream cell. The inset of Fig 8B shows that the charge from I_Ca(L) needed to excite the adjoining cell is approximately the same for both fibers. Since I_Ca(L) is smaller in the overloaded fiber, longer intercellular charging time is needed, leading to decreased sodium channel availability (greater dynamic inactivation) and reduced SF.

View larger version (21K): [in this window] [in a new window]   Figure 8. Influence of calcium overload on ICa(L) at severely reduced intercellular coupling. A, Action potential upstrokes from the middle cell of the fiber computed for control resting [Ca]i (solid line) and calcium overload (dashed line). In both cases, intercellular coupling was reduced to gj=0.008 µs. Longer action potential foot and greater early-plateau repolarization are seen for the cell with elevated [Ca]i. B, L-type calcium current for the same two cases. There is significantly greater post-release inactivation of ICa(L) in the calcium-overloaded fiber. Similar total ICa(L) charge for the two cases (inset of B) indicates that reduced ICa(L) in the calcium-overloaded fiber is compensated for by a longer transjunctional charging time.

	Discussion

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
This study establishes two types of seemingly paradoxical behavior for propagation of excitation in multicellular cardiac fibers. The first is that as conduction velocity is slowed by decreased gap junction coupling, propagation safety is increased. Conduction in the milieu of decreased coupling (such as in a zone of infarction) is slow but safe. The second is that the functional role of excitation currents (ie, I_Na and I_Ca(L)) is determined more by passive structural factors external to the membrane than by intrinsic membrane factors. Decreased intercellular coupling, not decreased sodium channel availability (reduced membrane excitability), results in a major functional role for I_Ca(L) in propagation. The significance of these behaviors is discussed below.

Slow Conduction and Propagation Safety
Both reduction in membrane excitability and reduction in intercellular coupling reduce conduction velocity. However, the mechanisms by which each pathology causes conduction slowing are fundamentally different. Reduced excitability decreases availability of the depolarizing membrane current, I_Na, slowing membrane depolarization. Reduced coupling increases intercellular conduction time by limiting (reducing) electrotonic axial current flow. By also decreasing electrical load, decreased coupling confines current to the depolarizing cell, causing an enhancement of its depolarization.

Safety factor data for the two different pathologies are compiled with respect to conduction velocity in Fig 9. The data establish that during conduction slowing decreased coupling increases safety factor for conduction, whereas decreased excitability decreases safety factor. The opposite effects on safety factor are evident from the minimum conduction velocity obtained before block. Reduced excitability can reduce conduction velocity only threefold (to 17 cm/s) before block occurs, whereas uncoupling can reduce conduction velocity to an extremely low value of 0.26 cm/s before conduction fails.

View larger version (21K): [in this window] [in a new window]   Figure 9. Conduction velocity and safety factor (SF) with reduced excitability and reduced coupling. A, Summary of computed velocity for a fiber subjected to uniform reductions of excitability (dashed line) and a fiber subjected to uniform reductions of intercellular coupling (solid line). Minimum velocity attainable with reduced coupling is <1/200 of control, while minimum velocity with reduced excitability is only one third of control, before block occurs (indicated by asterisk). B, Computed safety factor for the two cases in A, reduced excitability (dashed line, empty circles) and reduced coupling (solid line, filled circles). Although both reduction in excitability and reduction in coupling reduce velocity, reduced excitability decreases SF, whereas decreased coupling increases SF. For this reason, reduced coupling can support much slower conduction velocity, as seen in A. Data are compiled from Figs 4 and 6. Conduction failure is indicated by asterisk.

There are differences in the literature as to whether conduction slowing with reduced coupling occurs with higher or lower safety. In a classic study, Spach et al⁸ found that cardiac conduction in the longitudinal (well-coupled) direction had higher conduction velocity but was less safe than conduction in the transverse (less-coupled) direction. Safety was evaluated by susceptibility to block with premature stimulation. Delgado et al⁹ observed the opposite, that conduction failed preferentially in the transverse direction. In their study, block was obtained by elevation of extracellular potassium concentration. A possible reason for the disparity between the two studies is the technique used to obtain block. Elevation of extracellular potassium concentration, unlike premature stimulation, alters the potassium reversal potential and depolarizes the resting membrane potential.^{26 27} As a result, not only do sodium channels inactivate, but distance between resting potential and threshold potential is reduced.^{27 28} In addition, conduction near block due to elevated extracellular potassium depends on the L-type calcium current to aid in excitation.²⁰ Therefore an evaluation of safety factor based on elevating [K]_o may include important contribution from I_Ca(L) and might differ from the safety factor evaluated by premature stimulation.

It is worthwhile to emphasize the magnitude of the safety factor changes caused by reduced excitability versus reduced coupling. Reduced excitability causes a monotonic reduction in SF, lowering it by 60% (from 1.6 to 1), before causing conduction failure. In contrast, reduced coupling increases SF by 200% before causing a drop in SF and block. This is evidence that, in terms of safety factor for conduction, membrane excitability properties are less influential than passive fiber properties. A slight decrease in coupling could compensate for a major decrease in membrane excitability. For example, the ischemic combination of acidosis (pH=6.5) and hyperkalemia ([K]_o=10 mmol/L) decreases resting I_Na availability by 60%,²⁰ lowering SF from 1.6 to 1.4. Merely decreasing gap junction coupling by a factor of three will compensate for such ischemic decrease in safety. Uncoupling is typically considered to be an attempt of the myocardium to isolate damaged cells.²⁹ However, because reduced coupling improves the safety factor for conduction, a certain degree of uncoupling may be an important tissue-level response to ensure conduction despite depressed membrane excitability.

Roles of I_Na and I_Ca(L) in Propagation
Under all conditions other than reduced intercellular coupling, I_Na is the major ionic current that determines excitability. This includes slow conduction due to depressed membrane. When intercellular coupling is even marginally reduced, we found that conduction relies on I_Ca(L) as well as I_Na. Conduction block occurred, with all membrane currents intact, at gap junction conductance g_j=0.0057 µs. When I_Ca(L) availability was reduced to zero, conduction block occurred at g_j=0.0197 µs, three times the intercellular conductance that caused block in the presence of I_Ca(L). The important role of I_Ca(L) under conditions of decreased coupling is also evidenced by the total charge generated to excite an adjoining cell. At control conditions (g_j=2.5 µs) the charge ratio Q_Na:Q_Ca is 166:1, indicating I_Na dominance. At g_j=0.020 µs, the ratio decreases to 1.47:1, reflecting equal roles of I_Na and I_Ca(L) in conduction. The charge ratio decreases further with further decreases in coupling, and at g_j=0.0057 µs, Q_Na:Q_Ca is 0.16:1 indicating I_Ca(L) dominance and an order of magnitude greater charge contribution by I_Ca(L) than by I_Na.

The effect of I_Ca(L) at low intercellular coupling has been studied in isolated cell pairs (between two real cells, a real cell and a model cell, and two model cells) by Joyner and colleagues.^{13 30} They determined that if the leader cell is unable to excite a follower cell due to limited intercellular conductance or size related source-load mismatch, I_Ca(L) enhancement in the leader cell could successfully restore excitation in the follower cell. The converse was also true, I_Ca(L) inhibition in the leader cell could block otherwise successful follower cell excitation. For all conditions, I_Ca(L) was effective when intercellular delay was on the order of 5 ms (equivalent to conduction velocity of about 2 cm/s).

In the multicellular fiber, we found that I_Ca(L) influenced safety factor for conduction when g_j was as high as 0.5 µs (one fifth of control), which causes an intercellular conduction delay of only 0.3 ms. As g_j decreases and the conduction delay increases, the downstream cell depolarizes when the upstream source cell is "deeper" into the plateau phase of its action potential. At this phase, I_Ca(L) is the depolarizing current and the major contributor of depolarizing charge to the downstream cell (note that charge involves integrating the current over time). Therefore, with longer intercellular conduction delays, more of the depolarizing charge delivered to the downstream cell is generated by I_Ca(L). Two- and three-dimensional structural anisotropy as well as nonmyocyte inhomogeneities (eg, connective tissue septae) may cause local conduction delays even when global conduction velocity is not drastically reduced. For instance Spach and Heidlage,^{31 32} using a detailed two-dimensional cellular model with realistic cell geometry and gap junction distribution, observed microscopic transcellular conduction delays on the order of 1 ms. Rohr and colleagues^{14 33 34} found, with cultured rat heart cells patterned into a narrow strand opening into a large rectangular area (large load), that I_Ca(L) block caused conduction failure when activation delay at this opening was in the millisecond range. Conversely, when block of conduction from the small strand to the large area preexisted, I_Ca(L) enhancement with Bay K 8644 reestablished successful conduction.^{33 34} Fast and colleagues^{35 36 37} have observed in cultured rat cell monolayers that nonuniformities on different-size scales (eg, uneven gap junction distribution, connective tissue sheets, nonexcitable vascular cells) may represent predilective sites for conduction block.³⁵ On the basis of these studies and our results, I_Ca(L) may then be a major component of microscopic propagation and of the safety factor for conduction at these sites even when overall macroscopic conduction velocity is not significantly reduced. I_Ca(L) is likely to play an important role in any situation when propagation is associated with long local conduction delays (eg, tissue expansion at Purkinje muscle junctions, increase in load on a wave front while circulating around a pivot point during reentry).

There is strong experimental evidence that the electrophysiological characteristics of conduction within a healed infarct zone are the result of reduced intercellular coupling. Fractionated electrograms, observed in infarcted regions,^{10 38} are associated with separation and altered orientation of myocardial fibers.^{10 39} In these cases slowed activation and very slow conduction are likely a result of decreased intercellular coupling.³⁸ On the basis of our simulations (Fig 9), the very slow velocities (<1 cm/s) that are measured in healed infarcts and that make reentry in a small region ("microreentry") possible must involve long local conduction delays and cannot be supported by depressed excitability (which cannot support velocity slower than 17 cm/s). Slow and discontinuous conduction within surviving muscle of epicardial infarcts has been identified as a mechanism of reentrant arrhythmias.^{40 41 42 43} The newly recognized importance of I_Ca(L) in highly discontinuous propagation that involves long conduction delays identifies the L-type calcium channel as a possible target for antiarrhythmic therapy under such conditions.

With our discussion focusing on the role of I_Ca(L), we hasten to emphasize that I_Na, even with reduced availability, is always required for conduction. I_Ca(L) activates between V_m=-40 and -30 mV.¹⁶ I_Na is still needed to bring the membrane potential to within the range of I_Ca(L) activation. [K]_o elevation, and consequent depolarization of resting membrane potential, is the only scenario we know in which I_Ca(L) may be the sole excitatory current.²⁰

Intracellular Calcium Transients and Propagation
Through the calcium-induced calcium release process, excitation-contraction coupling involves coupling between sarcolemmal calcium flux via I_Ca(L) and intracellular calcium transients. That is, intracellular calcium responds to sarcolemmal currents. In the reverse direction, I_Ca(L) and other currents (eg, I_NaCa, I_Ks) are modulated by [Ca]_i. The simulations of this study indicate a possible functional role for I_Ca(L) modulation by [Ca]_i in propagation of the cardiac action potential.

The L-type calcium current can be inactivated by free intracellular calcium ions.⁴⁴ Sipido et al⁴⁵ have shown in guinea pig cells that I_Ca(L) inactivation is proportional to [Ca]_i. Grantham and Cannell⁴⁶ found in guinea pig cells (using the I_Ca(L) formulation from the LRd model) that I_Ca(L) behavior during an action potential cannot be reconstructed correctly without calcium-dependent inactivation. Lüscher et al⁴⁷ showed that replacement of calcium ions by strontium ions (also a divalent positive charge carrier) prevented action potential conduction failure in dorsal root ganglion cells, suggesting that [Ca]_i-dependent inactivation of I_Ca(L) can affect conduction (strontium causes minimal inactivation of I_Ca(L)). More recently, the same team¹⁵ found that flash photolytic liberation of a calcium buffer, during trains of action potentials, which partly failed to invade the cell body, immediately lowered intracellular [Ca]_i and restored safe action potential propagation. Also, the velocity of the propagated action potential was reduced when intracellular calcium was increased by flash photolysis.¹⁵

Our results demonstrate that elevated intracellular calcium can both decrease conduction velocity and lower the safety factor for conduction during highly discontinuous ventricular propagation, which depends on I_Ca(L). Doubling the free intracellular calcium concentration (to 0.25 µmol/L) resulted in conduction failure at g_j0.007 µs, 23% higher than the conductance at which it otherwise occurred. Although, for the uniform fiber, reduction of gap junction coupling by two orders of magnitude is needed before the effects of [Ca]_i on propagation are apparent, the principle of [Ca]_i modulated conduction may have much greater significance in conduction across tissue inhomogeneities where large source-load mismatches exist and cause very long conduction delays.³⁴ By studying the direct response of I_Ca(L) to changes in [Ca]_i, we established that the ionic mechanism of [Ca]_i modulated propagation is, as suggested by Lüscher et al,^{15 47} calcium-dependent inactivation of I_Ca(L).

Conclusions
The study emphasizes that reduced membrane excitability (eg, during acute ischemia) cannot support very slow conduction and is mostly associated with conduction failure. In contrast, slow conduction due to gap junction decoupling is safe, and very slow velocities can be supported by such structural changes (eg, due to infarction).

Our results suggest that the passive myocardial structure has a major effect on the role of membrane currents in propagation of the cardiac action potential. We find it striking that a nonmembrane change (ie, decreased intercellular coupling) can cause the membrane to switch to a different process (calcium current instead of sodium current) as the major mechanism for supporting conduction. A control fiber relies on I_Na to sustain conduction and, except for extreme circumstances, a fiber with depressed membrane excitability still relies on I_Na for action potential conduction. However, reduced gap junction coupling (a nonmembrane structural change) changes the dynamics of membrane ionic currents during the action potential and switches the membrane to an I_Ca(L)-dependent mode of operation. This finding illustrates the strong bidirectional interaction between the membrane (source) and passive myocardial structure (sink) during action potential propagation.

In addition to its direct feedback influence on membrane dynamics, reduced gap junction coupling modulates the effects of interaction between intracellular ionic processes (eg, the calcium transient) and membrane ionic currents (eg, I_Ca(L)). In this study, [Ca]_i-dependent inactivation of I_Ca(L) was shown to influence propagation only when long conduction delays at gap junctions were involved (the condition for I_Ca(L) involvement in sustaining conduction). This example emphasizes even more the highly interactive nature of action potential propagation in cardiac tissue.

The fact that I_Ca(L) plays an important role in conduction only where propagation is very slow and highly discontinuous is encouraging from the perspective of antiarrhythmic drug therapy. Such conduction requires a pathological substrate (eg, infarcted region), implying that conduction depends on I_Ca(L) only where the pathology exists. In healthy myocardium with normal intercellular coupling, I_Ca(L) is not needed to maintain propagation of excitation. With appropriate levels of universally applied I_Ca(L) block, conduction will be modified in the vulnerable infarcted region only, without affecting the normal myocardium, thereby ensuring specific drug action. In the same context of antiarrhythmic drug therapy, the study demonstrates that the intervention should consider the underlying pathophysiological state of the myocardium. For example, slow conduction during most phases of acute ischemia involves only I_Na, whereas slow conduction in a healed infarct involves long local delays and therefore I_Ca(L). Consequently, the targets for intervention (drug treatment, genetic modification) might be very different in treating propagation-related arrhythmias (eg, reentry) in the context of different pathophysiological conditions.

View larger version (18K): [in this window] [in a new window]   Figure 10. Relationship of fiber currents used to compute safety factor of propagation. A, Coupled cardiac cell with equivalent current network. Im indicates total transmembrane current; Iin, axial current into the cell; Iout, axial current out of the cell. Im is equivalent to the sum of ionic, Iion, and capacitive, Ic, currents. Note that Im is outward (positive) during electrotonic charging from upstream fiber but becomes inward (negative) when Iion is activated. B, Behavior of transmembrane charge, Qm, (dotted line, equal to integral of Im over time) during the action potential upstroke (Vm, solid line). C, Response of alternate formulations of safety factor (SF) to changes in membrane excitability. Safety factor computed with the definitions of Leon and Roberge12 (SFLeon, solid line) and Delgado et al9 (SFDel, dotted line) for the middle cell of a 70 cell fiber with normal intercellular coupling and varying sodium channel availability (excitability). Note that both definitions suggest that safety factor increases with decreasing membrane excitability and that propagation failure occurs for both definitions with SF>1 (in fact, for a correct formulation of SF, conduction should fail when SF falls below 1, in the shaded area).

View this table: [in this window] [in a new window]   Table 1. Action Potential Characteristics for Different Fiber Discretizations and Different Degrees of Intercellular Coupling

	Acknowledgments

 
This work was supported by the National Institutes of Health grants HL-49054 and HL-33343 (National Heart, Lung, and Blood Institute).

	Footnotes

 
Previously published in abstract form (Biophys J. 1997;72:A110).

	Appendix 1

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Spatial computational discretization and distribution of fiber resistivity require two separate considerations, one of numeric accuracy (convergence) and one of anatomic correctness. For a continuous fiber, numeric accuracy requires that the spatial discretization element (x) be no longer than one tenth of the fiber space constant so that the entire element is essentially equipotential.⁷ However, a ventricular fiber is discontinuous, consisting of intracellular myoplasm of low resistivity separated by intercellular gap junctions of (comparatively) high resistivity. Under normal coupling conditions, resistance over the entire 100 µm of myoplasm is equal to the resistance across the gap junction,^{48 49} which occupies no more than 80 Å. To select a x that is both numerically accurate and reflects the discontinuous structure of a cardiac fiber, we computed key physiological parameters for different values of gap junction resistance (R_g) at different levels of spatial discretization (x). Results are shown in the Table, which is a compilation of peak membrane potential (V_m,peak), maximum upstroke velocity ([dV_m/dt]_max), action potential duration at 90% repolarization ([APD]₉₀), and conduction velocity (), all computed for an action potential traveling down a 7-mm fiber. Simulations were repeated for three types of fiber discretization: (1) a discontinuous fiber of 21 computational elements per cell length (x=4.76 µm) with R_g confined to the edge elements of the cell (representing the local gap junction resistance); this high level of discretization ensures convergence and provides a gold standard for propagation in a discontinuous fiber; (2) a continuous fiber also discretized to 21 elements/cell length (x=4.76 µm) but with R_g evenly distributed across the entire cell; this provides an accurate model of propagation in a continuous fiber; and (3) a continuous fiber discretized to computational elements equal to the entire cell length (x=100 µm) with R_myo and R_g lumped together at each discretization point. All simulations were performed with implicit Crank Nicolson²⁴ solution of the cable equation. When the fiber is discontinuous, the partial differential equations are modified for the prejunctional and postjunctional nodes to match boundary conditions and account for R_g. The Crank Nicolson solution and discretization of a discontinuous fiber are detailed clearly in Reference 6. The computed parameter values are shown together with percent error relative to the truly discontinuous, highly discretized (21 elements/cell) fiber. Cellular data were computed at the middle element of cell 35 in the discontinuous fiber and at the same location in the continuous fibers. Conduction velocity was computed from the time difference between the occurrence of maximum upstroke velocity in cells 20 and 50.

The Table reveals that for the highly discretized discontinuous fiber, (dV_m/dt)_max (the most sensitive parameter of computational accuracy) increases with increasing intercellular resistance (R_g), whereas for the highly discretized continuous fiber, (dV_m/dt)_max is constant. Conduction velocity for both fibers decreases with increasing R_g. These results are expected [in the continuous fiber, eg, an axon, (dV_m/dt)_max is determined only by the membrane excitability; in the discontinuous fiber, cellular current confinement due to gap junctions acts to modify (dV_m/dt)_max⁵ ] and establish the necessity of using a discontinuous fiber model for simulating propagation in the multicellular myocardial tissue. A very important observation, based on the data in the Table, is that the fiber that is formulated as continuous but with a large x=100 µm (equal to the entire cell length of the discontinuous fiber) approximates very closely the accurate discontinuous fiber with 21 elements per cell (x=4.76 µm). The greatest error of computed parameters is for (dV_m/dt)_max at R_g=1.5 cm², which differs by <1% (actual error is 0.61%) from the accurate value obtained with the highly discretized discontinuous fiber.

The reason that x=100 µm (the entire cell length) is an adequate discretization is different for simulating discontinuous propagation at different values of gap junction coupling resistance. When R_g is very small (R_g0), the fiber is effectively continuous with large axial conductance (large space constant). Under such conditions, x=100 µm is approximately one tenth of the space constant and serves as an adequate discretization that insures convergence and accuracy. When R_g is relatively high (ie, R_g 5 cm²), the gap junctions are the primary sites of potential changes and the entire cell becomes effectively isopotential. Under these conditions, x=100 µm (the cell) is accurate because the requirement that the discretization element be isopotential is fulfilled. The greatest error for the x=100 µm discretization is expected in the middle range of R_g, when R_g is large enough to reduce the space constant () so that x>/10, yet it is not so high that the cell becomes isopotential. The greatest error within this region (0.61% at R_g=1.5 cm²) is small enough for accurate computation. For this reason we use x=100 µm in our simulations (it translates into tremendous savings in computing time, which is 5% of that for the highly discretized fiber with 21 elements per cell). When knowledge of variation of dV_m/dt and other parameters within the cell is required, x is reduced to subdivide the cell with desired resolution. Otherwise x=100 µm is an adequate discretization for the multicellular discontinuous cardiac fiber. All our simulation protocols were checked for accuracy and convergence with 21 elements per cell but in general we use x=100 µm, with R_i reflecting the lumped contribution of axial and gap junction resistance (R_i=R_myo+R_g/x).

	Appendix 2

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Our definition of safety factor is based on the net membrane charge (Q_m) that is generated over the course of an action potential for any cell in the fiber. Q_m quantifies a balance between charge produced and charge consumed by the cell membrane. When Q_m is positive, the cell membrane has consumed more charge than it has produced, constituting a charge sink for the fiber. When Q_m is negative, the membrane has functioned as a net charge source for the fiber.

Over the course of an entire action potential, a cell membrane alternates between being a sink and a source. The currents involved and the changes in Q_m are illustrated in Fig 10. As a propagating action potential approaches a cell (from the left in Fig 10A), a positive voltage gradient is established between the cell and its upstream neighbor. The voltage gradient causes axial current flow, I_in, into the cell. A smaller gradient is formed between the cell and its downstream neighbor, causing axial current, I_out, to leave the cell. Since the system is conservative, the difference between I_in and I_out must cross the membrane as the transmembrane current, I_m, which charges the membrane capacitance to cause depolarization. The charge, Q_m, consumed by the membrane is given by the time integral of I_m. As the membrane continues to depolarize, contribution to I_m from axial current decreases. This decrease is due to increased voltage gradient between the cell and its downstream neighbor and decreased voltage gradient between the cell and its upstream neighbor, acting to increase I_out and decrease I_in and, consequently, to decrease I_m=I_in-I_out. At the same time, inward membrane ionic currents (ie, I_ion, mainly from the fast sodium current) activate and become the major source of depolarizing current. As a result, total transmembrane current, I_m, becomes negative causing Q_m (the integral of I_m) to decrease. In Fig 10B, this is indicated by a decreasing Q_m beyond the midpoint of the upstroke. The second half of the upstroke can be viewed as the period during which the membrane replaces the charge it initially consumed to reach its activation threshold. When Q_m returns to zero, which occurs near peak membrane potential (Fig 10B), the membrane has restored the charge it consumed and represents neither a net sink nor a net source to the fiber.

The return of Q_m to zero is an appropriate time to analyze the margin of safety because it marks the instant that membrane excitation is complete and the membrane is no longer in debt. The total charge the cell consumed from the fiber is the time integral of I_in, or Q_in. The total charge that the cell contributed to the fiber is the time integral of I_out, or Q_out. A third important charge is the capacitive (potential) energy stored in the membrane as a result of excitation, energy that will be contributed to the fiber as the membrane repolarizes back to its rest potential. The capacitive charge of the membrane, Q_c=C_m · (V_m-V_rest), is equal to the time integral of the membrane capacitive current, I_c. Therefore, safety factor, which is a ratio of total charge produced (Q_out+Q_c) to charge consumed (Q_in), is formulated as

(3)

(evaluated from the start of membrane depolarization to the point when Q_m returns to zero, ie, over the domain Q_m>0).

In formulating the safety factor of propagation, we borrowed heavily from two definitions that exist in the literature.^{9 12} Delgado et al⁹ defined safety factor as the total axial charge provided to a cell divided by the axial charge necessary to sustain propagation. Their safety factor is therefore based on the margin of extra charge provided to a cell, after a threshold amount has been provided:

(4)

where I_in is the axial current into the cell, as defined earlier. SF_Del is a quantitative formulation of the intuitive understanding of safety factor (the fraction of charge delivered to the cell beyond the membrane threshold is the propagation safety). However, there are two difficulties when computing safety factor with this definition. The first difficulty is with the integration limit of the denominator. Inward charge is computed from the beginning of axial current flow to "threshold." Threshold is determined by severing the fiber at the junction between the cell in question and its upstream neighbor at different time points until the earliest time is found at which the severed fiber can sustain conduction. A difficulty with this concept of "threshold" is that in a multicellular fiber the cell never functions independently. Propagation after severing the fiber and forming a sealed end will have different velocity and conduction parameters than propagation in the intact fiber.

A second more significant difficulty with SF_Del is that it does not account for axial charge that leaves the cell to downstream fiber (I_out). By not accounting for I_out, SF_Del implicitly assumes that all axial current that enters the cell remains in the cell and charges the local membrane. This assumption is not valid under most conditions. If, for instance, membrane excitability is reduced, the slope of the action potential upstroke will be slower below and above threshold. Greater charging time will cause both numerator and denominator of SF_Del to increase, the numerator more so because it integrates I_in over a longer time period. The result will be a nonphysiological increase in SF_Del with decreased membrane excitability. This can be seen in Fig 10C (dotted line). Increased safety factor with decreased excitability is indicative that the propagation safety is not computed correctly.

The other formulation of safety factor for multicellular cardiac propagation was developed by Leon and Roberge.¹² These authors took a local, cellular approach to safety factor that consisted of the ratio of charge generated by inward I_ion (Q_ion) to total membrane charge generated by inward I_m (Q_m). The mathematical formulation of the Leon and Roberge safety factor is:

(5)

The middle and right hand sides of SF_Leon indicate the equivalence between total transmembrane current (I_m) and the sum of ionic and capacitive currents (I_ion+I_c). Therefore, SF_Leon is based on the efficient use of local ionic current (I_ion) for local depolarization (I_c). Safety factor for a segment of membrane will increase, due to a smaller denominator, when a large portion of its ionic charge is used for membrane depolarization. For an isolated cell, load effects are zero, (I_ion=-I_c), and SF_Leon is infinite.

A difficulty with SF_Leon is the integration limit of the denominator. I_m is integrated only when it is negative, or inward. I_m becomes inward after excitation threshold is reached, when I_ion is greater than the requirements of depolarization. By computing I_m only above threshold, SF_Leon does not account for the charging requirements of the membrane before reaching threshold. Like SF_Del, this limitation can cause error in the computed safety factor. For instance, when membrane excitability is reduced, I_ion as a source of fiber charge is reduced. When I_ion is reduced, the integral of I_ion+I_c (the denominator of SF_Leon) is reduced by a greater fraction than the integral of I_ion alone (the numerator of SF_Leon) because, in general, the same absolute reduction (reduced Q_ion) of a smaller value (Q_m=Q_ion+Q_c; Q_ion and Q_c are of opposite sign) causes a greater fractional reduction of this value. As a result, SF_Leon will increase when membrane excitability is reduced. This behavior can be observed for an actual simulation (Fig 10C, solid line), where, similar to SF_Del, SF_Leon increases with decreasing membrane excitability.

In summary, SF_Del does not account for downstream loading effects while SF_Leon does not account for charge supplied by upstream fiber. The definition of SF developed here takes these factors into account. The response of our formulated SF to changes in excitability is shown in Fig 4 and discussed in the corresponding text. That SF decreases with reduction in membrane excitability and drops to unity just before propagation fails support the validity of its formulation. More detailed discussion of SF behavior during changes in membrane excitability and gap junction coupling are presented in the "Results" and "Discussion" sections of this report.

Received February 13, 1997; accepted August 21, 1997.

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