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(Circulation Research. 1997;80:124-138.)
© 1997 American Heart Association, Inc.

Articles

Electrophysiologic Effects of Acute Myocardial Ischemia

A Mechanistic Investigation of Action Potential Conduction and Conduction Failure

Robin M. Shaw, Yoram Rudy

the Cardiac Bioelectricity Research and Training Center, Department of Biomedical Engineering, Case Western Reserve University, Cleveland, Ohio.

	Abstract

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A multicellular ventricular fiber model was used to determine mechanisms of slowed conduction and conduction failure during acute ischemia. We simulated the three major pathophysiological component conditions of acute ischemia: elevated [K⁺]_o, acidosis, and anoxia. Elevated [K⁺]_o was the major determinant of conduction, causing supernormal conduction, depressed conduction, and conduction block as [K⁺]_o was gradually increased from 4.5 to 14.4 mmol/L. Only elevated [K⁺]_o caused conduction failure when varied within the range reported for acute ischemia. Before block, depressed upstrokes consisted of two distinct components: the first to the fast Na⁺ current (I_Na) and the second to the L-type Ca²⁺ current (I_Ca(L)). Even in highly depressed conduction, excitability was maintained by I_Na, with conduction block occurring at 95% I_Na inactivation. However, because I_Ca(L) supported the later phase of the depressed upstroke, I_Ca(L) enhanced conduction and delayed block by increasing the electrotonic source current. At [K⁺]_o=18 mmol/L, slow action potentials generated by I_Ca(L) were obtained with 10% I_Ca(L) augmentation. However, in the presence of acidosis and anoxia, significantly larger (120%) I_Ca(L) augmentation was required. The depressant effect was due mostly to anoxic activation of outward ATP-sensitive K⁺ current, which counteracts inward I_Ca(L) and, by lowering the action potential amplitude, decreases the electrotonic current available to depolarize downstream cells. The simulations highlight the interactive nature of electrophysiological ischemic changes during propagation and demonstrate that both membrane changes and load factors (by downstream fiber) must be considered.

Key Words: myocardial ischemia • hyperkalemia • acidosis • anoxia • supernormal conduction • conduction failure

	Introduction

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Acute myocardial ischemia is implicated in many cases of fatal arrhythmias.^{1 2} The basis of ischemic arrhythmogenesis is alteration in the electrical properties of ventricular tissue, leading to changes in action potential conduction.^{3 4} Altered electrical properties are a result of the pathophysiological conditions of ischemia, which directly affect membrane ionic currents and intracellular and extracellular ionic concentrations.^{5 6} Therefore, there exist cause-and-effect relationships between ischemia modification of membrane currents and ionic concentrations and ischemia-related changes in action potential conduction. We investigated these cause-and-effect relationships to determine the ionic mechanisms of depressed conduction and development of conduction block during acute ischemia.

Our investigative tool is a theoretical multicellular fiber model that accounts for the major conditions of ischemia at the level of individual ionic currents and concentrations. The fiber is composed of LRd model cells^{7 8 9} interconnected by resistive pathways representing gap junctions. The major conditions of ischemia are elevated [K⁺]_o, acidosis, and anoxia. These conditions are applied homogeneously to the fiber in different combinations and with varying degrees of severity. Because the conditions are applied at the level of individual ionic processes, the relationship between ionic currents and action potential conduction can be readily studied.

We are interested in early (relative to onset of ischemia) depression of conduction velocity, , which relates mostly to changes in I_Na and in later, more severe depression of , which may still be I_Na dependent and yet may also involve I_Ca(L). Our investigation focuses on acute ischemia (first 10-15 minutes) before the occurrence of gap junction uncoupling and irreversible cell damage. Beginning within the first few minutes of perfusion block, in otherwise healthy tissue progressively decreases.¹⁰ Within 15 minutes of arrested perfusion, ventricular tissue becomes inexcitable, and conduction block ensues.¹¹ It is believed that early depression of is a result of [K⁺]_o-induced depolarization of V_rest, which decreases membrane excitability by lowering the availability of I_Na.¹² However, moderate [K⁺]_o elevation alone always causes increased (supernormal) ,^{13 14 15 16} whereas during ischemia supernormal conduction at the same [K⁺]_o is rarely reported.¹⁰ Therefore, other conditions of ischemia must act to slow conduction. Anoxia and acidosis are both reported to have depressant effects on ,^{17 18} but the ionic mechanisms of these effects are not well understood. By simulating the effects of ischemic conditions individually and in combination, we determine the ionic mechanism and relative contribution of each condition to conduction slowing.

During acute ischemia, velocity of highly depressed conduction just before conduction block slows to 10 to 20 cm/s. In this period, (dV_m/dt)_max is also highly depressed. It has been suggested that excitability and conduction near block are supported by a highly depressed I_Na,^{10 11 19} by I_Ca(L),²⁰ or by a combination of both.²¹ If conduction near block relies on depressed I_Na, then the mechanism of conduction is similar in principle to the mechanisms of normal conduction and of conduction with moderately depressed membrane. Instead, if conduction near conduction block relies on I_Ca(L), then an entirely different class of ionic channels is responsible for maintaining excitability and for supporting action potential propagation.²² I_Ca(L)-generated action potentials have been described in the literature as "slow action potentials" because of their slow upstroke [low (dV_m/dt)_max]^{23 24} Early support for I_Ca(L)-dependent conduction was expressed by Cranefield and colleagues,^{20 25} who observed slow conduction with elevated [K⁺]_o and catecholamines (I_Ca(L) agonists). Other investigators have provided evidence for the presence of both I_Na-dependent and I_Ca(L)-dependent conductions in ischemic tissue before conduction block.^{21 26} Biphasic action potential upstrokes have been reported,^{11 27} with presumed I_Na dependence of excitability and the early depolarization phase and with I_Ca(L) dependence of the secondary depolarization to peak potential.³ In the present study, we explore with high temporal resolution the respective roles of I_Na and I_Ca(L) in highly depressed conduction and in generation of the propagating action potentials. We determine whether slow I_Ca(L)-dependent conduction is possible in the acute ischemic range of elevated [K⁺]_o and explore the effects of acidosis and anoxia on this type of conduction.

	Materials and Methods

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The objective of the present study is to investigate changes in action potential propagation during acute myocardial ischemia. Propagation is simulated in a multicellular linear fiber into which cellular (membrane) ischemic changes are introduced at the level of ionic processes. Methods are summarized below.

Single Cell
To simulate the cellular electrical changes due to acute ischemia, the ionic and metabolic conditions of ischemia are introduced in the LRd model of a mammalian ventricular cell^{7 8} (Fig 1, top). In this model, the 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 [Ca²⁺]_i, are introduced. The model includes the recent development⁹ to account for the two components of the delayed rectifier K⁺ current, I_Kr and I_Ks. Detailed tables of equations governing the model are provided in References 7 through 9.

View larger version (62K): [in this window] [in a new window]   Figure 1. Schematic diagram of the updated LRd model and multicellular fiber model. Top, Dynamic LRd model. The time and voltage dependence of the major ionic channels are formulated with Hodgkin-Huxley–type formalism with additional [K+]o-dependent conductance of IK1 and IKr, [Ca2+]i-dependent conductance of IKs, and [Ca2+]i-dependent inactivation of ICa(L). Processes affected by the simulated ischemic conditions are identified by rectangular frames. ICa,b indicates Ca2+ background current; ICa(T), T-type Ca2+ current; INaCa, Na+-Ca2+ exchange current; Ip(Ca), Ca2+ pump in the sarcolemma; Ins(Ca), nonspecific Ca2+-activated current; IKp, plateau K+ current; INaK, Na+-K+ pump current; INa,b, Na+ background current; Irel, Ca2+ release from junctional sarcoplasmic reticulum (JSR); Itr, Ca2+ translocation from network sarcoplasmic reticulum (NSR) to JSR; Ileak, Ca2+ leakage from NSR to cytoplasm; and Iup, Ca2+ uptake from myoplasm to NSR. Bottom, Multicellular cardiac fiber composed of a series of LRd cells interconnected by resistive gap junctions. Fiber is 7 mm (70 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 concerning the LRd cell model, see References 7 through 9.

The ionic and metabolic conditions of acute ischemia that affect the cell electrophysiology were introduced as three separate components: (1) increase in extracellular K⁺, (2) intracellular and extracellular acidosis (decrease in pH), and (3) anoxia and metabolic blockade (decrease in [ATP]_i). [K⁺]_o is directly modified as a parameter of acute ischemia. Its values range from control of 4.5 to as much as 35 mmol/L, depending on the simulation. Most simulations are performed with [K⁺]_o15 mmol/L, the range within which conduction block occurs.

Ischemic acidosis originates with intracellular proton accumulation. Extracellular acidosis follows by proton transfer across the sarcolemma. It is therefore physiologically necessary to apply both intracellular and extracellular acidic effects in a model of ischemia. Extracellular acidosis reduces availability (decreases maximum conductance) of I_Na,^{18 28} and intracellular acidosis reduces availability of I_Ca(L).^{29 30 31} In the model, the maximum conductances of I_Na and I_Ca(L) are varied over a wide range, depending on the severity of acidosis. At pH 6.5, I_Na and I_Ca(L) availability are both reduced 25%. This value is used to represent a case of "typical" acidosis. Additionally, extracellular acidosis causes a positive voltage shift of the I_Na kinetics and a decrease in [K⁺]_i that causes resting depolarization.^{32 33 34 35} Our condition of acidosis includes a positive 3.4-mV shift in I_Na kinetics and [K⁺]_i=125 mmol/L.

The direct electrophysiological effects of anoxia are modeled by introducing I_K(ATP) into the LRd model. Several formulations of I_K(ATP) have been developed.^{36 37 38 39 40} Our formulation of I_K(ATP), originally developed in Reference 38, is based on the following equation:

(E1)

where G_K(ATP) is channel conductance per cm² (39x10^-3 nS/cm²),⁴¹ n is the power of [K⁺]_o dependence (n=0.24, [K⁺]_o,normal=4.0 mmol/L),⁴² and [ATP]_i follows Hill-type formalism with k_½=114 µmol/L⁴¹ and H=2.⁴¹ I_K(ATP) is activated when [ATP]_i is reduced under ischemic conditions. Metabolic factors present during acute ischemia decrease I_K(ATP) sensitivity to [ATP]_i-based inactivation (increase in k_½). These factors include ADP (which was present in the preparation of Nichols et al⁴¹ ), intracellular acidosis⁴³ (double increase in k_½), and intracellular lactate⁴⁴ (triple increase in k_½). We modified k_½ to 250 µmol/L to account for these additional ischemic effects. Similar to variation of channel availability for the condition of acidosis, anoxia is studied over a wide range of [ATP]_i. When a "typical" effect of anoxia is represented, [ATP]_i=3 mmol/L is used. As "Results" will indicate, this is a conservative estimate of the effects of anoxia.

In addition to I_K(ATP), ATP dependence of the L-type Ca²⁺ channel has been introduced into the model. Irisawa and Kokubun⁴⁵ recorded an increase in I_Ca(L) when [ATP]_i was raised from 2.5 to 9 mmol/L, providing direct evidence of metabolic regulation of I_Ca(L).⁴⁶ Other groups have demonstrated ATP^{47 48 49} and ATP-related⁵⁰ regulation of I_Ca(L). The relationship between I_Ca(L) and [ATP]_i, like that of I_K(ATP), is sigmoidal and can be fit with Hill-type formalism as follows:

(E2)

where P_Ca(L),ATP is a fraction applied to total I_Ca(L). I_Ca(L) is otherwise computed as described before.^{7 8} Noma and Shibasaki⁴⁷ recorded the dependence of I_Ca(L) on [ATP]_i using guinea pig ventricular cells. We used a Hill-type fit (k_½=1.4 mmol/L and H=2.6) to the Noma and Shibasaki data for metabolic regulation of I_Ca(L). These parameters cause 12% I_Ca(L) reduction at [ATP]_i =3 mmol/L, which is very similar to the reduction of I_Ca(L) recorded by Ohya and Sperelakis,⁴⁸ who used vascular smooth muscle cells at similar [ATP]_i.

Multicellular Fiber
For studying propagation of the action potential, the theoretical fiber (Fig 1, bottom) used in the present study is composed of 70 serially arranged ventricular cells, each of LRd formulation. The axial current flow (second spatial derivative of voltage) is related to the temporal transmembrane current fluxes of the LRd model by the following differential equation^{51 52} :

(E3)

where I_j 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 µF/cm²), and R_i is the axial resistance per unit length (cm), which is composed of R_myo (200 cm) and R_g (3.0 cm²). The value of R_g=3.0 cm² is equivalent to gap junction conductance of 1.27 µS and represents a normal degree of cellular coupling. This value is maintained in the simulations, since during acute ischemia, gap junction uncoupling does not occur.^{53 54 55 56} The differential form of Equation 3 above is approximated by a finite difference scheme and solved by the Crank-Nicolson implicit method.⁵⁷ As discussed below, the solution converges for a spatial discretization of 100 µm (one cell length). Thus, the discretization element in the computations is x=100 µm and R_i=R_myo+R_g/x. R_o was neglected (the fiber is assumed to be in an extensive medium). No-flux (sealed ends) boundary conditions were used by setting dV_m/dx=0 at the beginning and end of the fiber. Stimulation and termination artifacts are restricted to within one space constant (10 cells) from each end. and all other parameters were taken from cell 20 to 50, which were completely free from these effects. Solutions for transmembrane currents were computed with the modified Euler method of Rush and Larsen.⁵⁸ A routine involving variable time stepping was implemented that tracked the propagating action potential and adjusted the computational time increment (t) according to the degree of membrane activity. Transmembrane currents in fiber regions that were about to experience an action potential upstroke or were within 20 ms of a previous upstroke were computed with t=2 µs. Transmembrane currents during the remainder of the action potential and during quiescent periods were computed with t=1 ms. Membrane voltage over the entire fiber was always computed with t=2 µs. Solutions computed with the variable t were within 1% of solutions computed with a constant t=2 µs.

For a continuous fiber (no gap junction discontinuities), x has to be 1/10 of the space constant, , in order for each patch to be equipotential and for the solutions to numerically converge.⁵⁹ In other words, x must be small enough so that variations in V_m across the patch can be neglected. For normal cardiac tissue, a typical is of the order of 1 mm (10 cells), and x=1 cell is an adequate discretization. The fact that contains several cell lengths reflects the tight coupling and low R_g under normal conditions. As R_i increases, decreases, because with increasing R_i, V_m varies faster with distance along the fiber and x must be made smaller to preserve the equipotential condition. However, for a discontinuous fiber the change in potential along the fiber occurs with increasing exclusivity across the gap junctions as R_g is raised. Thus, the anatomic discontinuities of a cardiac fiber result in smaller potential changes within a single cell, with most of the change occurring at gap junctions (Fig 14 of Reference 52 ). Therefore, the entire cell is expected to be close to equipotential, and x=cell length is expected to be a sufficient discretization for a wide range of R_g. To examine the range of R_g for which x=1 cell is an adequate discretization, we ran simulations for fibers with two discretization levels, 1 patch per cell (x=100 µm) and 21 patches per cell (x=4.76 µm). The simulations were conducted over a wide range of R_g from 0 to 50 cm² (note that R_g=3 cm² used in the simulations is included in this range). For each level of discretization, the action potential amplitude, , and APD at 90% repolarization were practically identical. Maximal variation of (dV_m/dt)_max, the most sensitive parameter, was only 2%. Therefore, spatial discretization of one cell length is adequate and justified for the simulations in this study.

	Results

Top Abstract Introduction Materials and Methods Results Discussion References

 
Influence of Acute Ischemia on
The three component conditions of acute ischemia (elevated [K⁺]_o, acidosis, and anoxia) are all expected to decrease . Questions remain about the relative influence of each condition on overall ischemic . In Fig 2, panel A (bold line) and panels B and C show computed for the condition of elevated [K⁺]_o, acidosis, and anoxia, respectively. The shaded boxes in each panel correspond to the ranges of each condition that are typically reported for acute ischemia. Elevated [K⁺]_o has the greatest influence on conduction. Beyond an initial increase in with slight increase in [K⁺]_o ("Supernormal Conduction," discussed below), rapidly decreases with further [K⁺]_o elevation. Conduction block occurred at [K⁺]_o>14.4 mmol/L, well within the ischemic range of [K⁺]_o elevation. In contrast, acidosis monotonically decreases (the implementation of acidosis involved a positive 3.4-mV shift of I_Na kinetics, [K⁺]_i=125 mmol/L, and reduced maximum conductances of I_Na and I_Ca(L) as shown on the abscissa of Fig 2B). At 50% reduction of I_Na and I_Ca(L) (a reduction that corresponds to low pH conditions, pH 6.0),³⁰ decreases 23% from 60 to 46.21 cm/s. Acidosis alone can cause conduction block (at 85% reduction of both I_Na and I_Ca(L)), but these reductions in I_Na and I_Ca(L) correspond to pH levels that are well below the level found with acute ischemia. The third ischemic condition, anoxia, causes reductions in [ATP]_i that open the I_K(ATP) channels. Within the ischemic range, reduced [ATP]_i alone does not contribute to slowing (Fig 2C). Reduction in [ATP]_i from 10 to 2 mmol/L reduces by only 2.5%. Further [ATP]_i reduction causes significant conduction slowing and block at [ATP]_i=0.4 mmol/L. These values of [ATP]_i, like the values of acidosis that cause conduction block, are beyond the range typically reported for acute myocardial ischemia.^{60 61}

View larger version (41K): [in this window] [in a new window]   Figure 2. Effect of ischemic conditions on conduction velocity, . The three conditions of ischemia were applied individually with increasing severity (elevated [K+]o, acidosis, and anoxia in the bold curves of panels A, B, and C, respectively). The shaded region in each panel corresponds to the range of condition values that occurs during acute ischemia. Note that only elevated [K+]o causes a biphasic change in and can cause conduction block when varied within the ischemic range. The other three lines in the same group of panel A show combined conditions of elevated [K+]o with anoxia ([ATP]i=3 mmol/L, short-dashed line), acidosis (25% reduction of INa and ICa(L), thin line), and all three ischemic conditions (dotted line). The separate, long-dashed line in panel A represents the combined conditions of elevated [K+]o with acidosis and extreme anoxia at [ATP]i=0.5 mmol/L. Stars indicate highest [K+]o for successful conduction under [K+]o elevation alone ([K+]o=14.4 mmol/L), for elevated [K+]o with acidosis and anoxia at [ATP]i=3 mmol/L ([K+]o=12.9 mmol/L), and for elevated [K+]o with acidosis and anoxia at [ATP]i=0.5 mmol/L ([K+]o=10.3 mmol/L). gNa and gCa(L) indicate maximum conductances of INa and ICa(L), respectively.

The data for each isolated condition in Fig 2 (bold curve in each panel) demonstrate that [K⁺]_o elevation (Fig 2A) is the single largest cause of conduction slowing. We investigated the extent to which acidosis and anoxia in combination with [K⁺]_o cause additional conduction slowing. Three curves in Fig 2A show versus [K⁺]_o for the following combined conditions: anoxia ([ATP]_i=3 mmol/L) and elevated [K⁺]_o (short-dashed line), acidosis (25% reduction of I_Na and I_Ca(L)) and elevated [K⁺]_o (thin line), and all three ischemic conditions (dotted line). The biphasic change of versus [K⁺]_o is prominent in all four ([K⁺]_o alone and [K⁺]_o in combination) curves. Acidosis causes a general reduction in at all levels of [K⁺]_o, as evidenced from the separation between the lower two (with acidosis) and upper two (without acidosis) curves. Peak acidic of 61.7 cm/s (at [K⁺]_o=8 mmol/L) is only slightly higher than (60 cm/s) for control (nonacidic, [K⁺]_o=4.5 mmol/L) conditions. Therefore, compared with control conditions (ie, elevated [K⁺]_o alone), acidosis limits the supernormal phase of conduction. Propagation failed at [K⁺]_o=13.2 mmol/L under acidic conditions. Anoxia at [ATP]_i=3 mmol/L contributes minimally to conduction slowing over the entire range of [K⁺]_o elevations, except for highly elevated [K⁺]_o near conduction failure. At [K⁺]_o12 mmol/L for [K⁺]_o elevation alone and at [K⁺]_o11 mmol/L for [K⁺]_o elevation with acidosis, anoxia slows and causes earlier conduction failure.

To extend our consideration of anoxia, we also decreased [ATP]_i below that which is generally reported for myoplasmic concentrations during ischemia. The long-dashed line in Fig 2A contains changes in with elevated [K⁺]_o, acidosis, and anoxia at [ATP]_i=0.5 mmol/L. This extremely low level of [ATP]_i causes a relatively high availability of I_K(ATP) (20% availability at [ATP]_i=0.5 mmol/L versus 0.69% at [ATP]_i=3 mmol/L) and results in decreased at all levels of [K⁺]_o, with conduction block at [K⁺]_o>10.3 mmol/L.

Acidosis and anoxia (at [ATP]_i=3 mmol/L) affect different membrane currents, which produce the different effects on seen in Fig 2. The predominant effect of acidosis is to reduce I_Na conductance, thereby reducing membrane excitability. A fixed degree of acidosis (25% reduction of I_Na in Fig 2A) results in uniform slowing at all [K⁺]_o. The effect of acidosis adds to the effect of elevated [K⁺]_o, since both act to depress I_Na (elevated [K⁺]_o does so by depolarization-induced reduction of Na⁺ channel availability). In contrast, anoxia does not directly affect I_Na. It activates the outward K⁺ current (I_K(ATP)), which, at [ATP]_i=3 mmol/L, is small and increases linearly with membrane depolarization. For all but extreme values of elevated [K⁺]_o, I_Na is sufficiently available to overwhelm I_K(ATP). Therefore, subthreshold depolarization at most [K⁺]_o levels is unaffected by I_K(ATP) (at [ATP]_i=3 mmol/L). Only at highly elevated [K⁺]_o, when I_Na is small, can I_K(ATP) (at [ATP]_i=3 mmol/L) affect the depolarizing current to cause slow conduction and excitation failure. Thus, the effects of highly elevated [K⁺]_o on conduction are potentiated by anoxia, causing greater slowing and earlier block. As is explored later in this article, anoxia-induced block is a complicated multicellular event. At extremely low levels of [ATP]_i (ie, [ATP]i=0.5 mmol/L), I_K(ATP) is sufficiently activated to influence conduction at any [K⁺]_o. This degree of anoxia is extreme but is hypothesized to occur if [ATP]_i is compartmentalized between general myoplasmic and submembrane compartments.

Supernormal Conduction
The biphasic change of versus [K⁺]_o (Fig 2A) suggests a complicated relationship between local membrane excitability and propagation of excitation. An index of membrane excitability is (dV_m/dt)_max. Fig 3A contains computed (dV_m/dt)_max versus for two fibers, one subject to elevated [K⁺]_o (solid line) and the other subject to both elevated [K⁺]_o and acidic conditions (dotted line). (dV_m/dt)_max is shown for the middle cell of the 70-cell fiber, and is computed from the time of (dV_m/dt)_max between cells 20 and 50. Propagation was initiated by externally stimulating cell 1. Acidic conditions correspond to pH 6.5 (25% reduction of I_Na and I_Ca(L), a 3.4-mV shift in I_Na kinetics, and [K⁺]_i=125 mmol/L). Experimental recordings under elevated [K⁺]_o with and without acidosis, reported in a study by Kagiyama et al¹⁸ involving guinea pig papillary muscle, are provided for comparison in Fig 3B. It can be observed that as [K⁺]_o is raised from initially low values, (dV_m/dt)_max changes little (increases slightly then decreases) but monotonically increases to a maximum. Maximum occurs at [K⁺]_o=8.2 and 8.0 mmol/L for nonacidic and acidic fibers, respectively. The increase of at slightly elevated [K⁺]_o is known as supernormal conduction. At higher [K⁺]_o, reductions in (dV_m/dt)_max coincide with reductions in , until propagation block occurs. Propagation failed at [K⁺]_o>14.4 mmol/L for the nonacidic fiber and [K⁺]_o>13.1 mmol/L for the acidic fiber. Note in Fig 3 that the curve with acidic conditions is always contained within the control curve; ie, nonacidic (dV_m/dt)_max is greater than acidic (dV_m/dt)_max at large but is less than acidic (dV_m/dt)_max at reduced .

View larger version (18K): [in this window] [in a new window]   Figure 3. "Comma-shaped" relation between (dVm/dt)max and conduction velocity, , for nonacidic and acidic fibers. A, Theoretical results under nonacidic (solid line) and acidic (dotted line) conditions. Numbers next to symbols correspond to [K+]o (millimolar). peaked at [K+]o=8.2 mmol/L and [K+]o=8.0 mmol/L for nonacidic and acidic fibers, respectively. B, Data obtained under similar experimental conditions from guinea pig papillary muscle by Kagiyama et al.18 For both acidic (dotted line) and nonacidic (solid line) cases, peaked at [K+]o=9 mmol/L in the experiment.

The single-cell relationship between nonacidic (dV_m/dt)_max and acidic (dV_m/dt)_max is explored in Fig 4. Single-cell simulations were conducted here (Fig 4A) in order to investigate membrane effects, without the complicating effects of electrical loading by neighboring cells. The solid curve in Fig 4A corresponds to [K⁺]_o-induced (dV_m/dt)_max changes alone (control Na⁺ channels), and the dotted curve corresponds to [K⁺]_o-induced (dV_m/dt)_max changes in the presence of acidic Na⁺ channels (pH 6.5 with I_Na kinetics shifted by 3.4 mV) and acidic [K⁺]_i. Changes in V_rest on the abscissa for both nonacidic and acidic curves result from changes in [K⁺]_o. Stimulation strength for all levels of [K⁺]_o was adjusted to 10% above threshold and was always 0.5 ms in duration. Corresponding experimental data¹⁸ from guinea pig papillary muscle are provided for comparison in Fig 4B. It can be observed in Fig 4A that for the [K⁺]_o-induced changes alone, (dV_m/dt)_max plateaus initially and then monotonically decreases with further increase in [K⁺]_o and resting depolarization. The onset of (dV_m/dt)_max depression corresponds to resting Na⁺ channel inactivation. In the LRd model, Na⁺ channels in a resting membrane are 10% inactivated at V_m=-78.7 mV and 50% inactivated at V_m=-70.3 mV. As seen in Fig 4A, initially acidic (dV_m/dt)_max is depressed relative to the nonacidic case. However, crossover occurs at V_rest=-71.9 mV, and at more positive V_rest, the acidic upstroke is faster than the nonacidic upstroke. The initial upstroke depression of acidosis is due to reduced maximal Na⁺ channel conductance. However, acidosis shifts the Na⁺ channel inactivation process to more positive potentials (see "Materials and Methods"). Crossover occurs at a V_rest for which the depolarization-induced inactivation of nonacidic I_Na reduces the current more than the decreased maximum conductance of acidic I_Na. Because acidosis induces a positive shift in V_rest, the relationship between [K⁺]_o and V_rest changes under conditions of acidosis; thus, crossover of (dV_m/dt)_max at a given V_rest does not imply crossover at a given [K⁺]_o. In fact, if Fig 4 is redrawn as a function of [K⁺]_o, rather than V_rest, it would be observed that at any [K⁺]_o, nonacidic (dV_m/dt)_max is always greater than acidic (dV_m/dt)_max.

View larger version (23K): [in this window] [in a new window]   Figure 4. (dVm/dt)max vs Vrest for nonacidic and acidic conditions. A, Single-cell (dV/dt)max as a function of Vrest. Changes in Vrest reflect changes in [K+]o. Solid curve corresponds to nonacidic membrane conditions, and dotted curve corresponds to acidic membrane conditions. Numbers near symbols indicate [K+]o. B, Similar experimental data from guinea pig papillary muscle during action potential propagation (Kagiyama et al18 ). Acidic conditions reflect respiratory acidosis, pH 6.5. Acidic (dVm/dt)max is initially smaller than nonacidic (dVm/dt)max, but crossover occurs at more positive Vrest (-72 mV in the theoretical curves [A] and -65 mV in the experimental curves [B]).

Relationships between (dV_m/dt)_max, , and V_rest in the multicellular fiber during propagation are shown in Fig 5. (dV_m/dt)_max (bold line) and (dotted line) are plotted as a function of V_rest (adjusted by altering [K⁺]_o), normalized to their control values at V_rest=-91.1 mV ([K⁺]_o=4.5 mmol/L). A slight increase of (dV_m/dt)_max is observed with initial depolarization of V_rest. It then decreases monotonically as was observed in Fig 4 for the single cell. also undergoes a biphasic increase then a decrease, but the initial increase is considerably steeper than that of (dV_m/dt)_max, and peak velocity occurs at a more depolarized membrane [peak values are at V_rest=-83.5 mV for (dV_m/dt)_max and V_rest=-76.0 mV for ].

View larger version (22K): [in this window] [in a new window]   Figure 5. Relationship between (dVm/dt)max and conduction velocity, , in the presence of [K+]o changes. Values for (dVm/dt)max (bold line), (dotted line), and 2 (dashed line) normalized to their respective values at Vrest=-91.1 mV ([K+]o=4.5 mmol/L). The product of (h·j)rest (where h and j are fast and slow inactivation parameters, respectively, of INa) and (thin line) approximates well (dVm/dt)max.

The initial relatively flat portion of the (dV_m/dt)_max curves in Figs 4 and 5 sheds light on the mechanism of supernormal conduction. (dV_m/dt)_max is only marginally affected by changes in V_rest as long as V_rest remains too negative to cause significant Na⁺ channel inactivation. In this range of V_m (V_rest<-81 mV for the nonacidic curve in Fig 4), V_rest depolarization brings the membrane closer to threshold without appreciable Na⁺ channel inactivation. The reduced difference between V_rest and V_thresh reduces t_thresh and increases the velocity of propagation, . Beyond this phase, (dV_m/dt)_max decreases, reflecting reduced membrane excitability due to Na⁺ channel inactivation. This, in turn, results in slowing of conduction.

A lesson from cable theory is that (dV_m/dt)_max should vary in proportion to ². The square of is shown in Fig 5 (dashed line), normalized to [K⁺]_o=4.5 mmol/L. Clearly, during the phase of supernormal conduction, (dV_m/dt)_max is not proportional to ². This suggests that either the proportionality relationship is not general and does not apply during the supernormal phase or that supernormal conduction is inconsistent with cable theory and cannot be described by this classic formalism. Because our solution is computed from the cable equations (and hence is consistent with cable theory), we attempted to rederive and generalize the relationship between (dV_m/dt)_max and to include the phenomenon of supernormal conduction. Changes in V_rest induce changes in that are inversely related to changes in t_thresh; peaks when t_thresh is at a minimum. We approximate this behavior by the following:

(E4)

Additionally, (dV_m/dt)_max is directly related to (h·j)_thresh (h is the fast inactivation gate and j is the slow inactivation gate of I_Na; see Reference 7). We assume that (to a good approximation) (h·j)_thresh is directly related to (h·j)_rest and inversely related to t_thresh (slower depolarization allows for greater inactivation). It follows that (dV_m/dt)_max varies directly with the product of (h·j)_rest and 1/t_thresh:

(E5)

Substituting Equation 4 into Equation 5 results in the following relationship between (dV_m/dt)_max and :

(E6)

The normalized plot of (h·j)_rest· versus V_rest is shown in Fig 5 (thin line) and clearly parallels the behavior of (dV_m/dt)_max. The correspondence between (h·j)_rest· and (dV_m/dt)_max holds for the acidic fiber as well (not shown). Note that the state of I_Na activation (activation gate, m) does not appear in the above relationships. This is because m does not vary appreciably over the ischemic range of V_rest.

Role of I_Ca(L) in Depressed Conduction
The data in Fig 5 demonstrate that I_Na parameters determine conduction even under depressed conditions near block. We are interested in the contribution of I_Ca(L) to the depressed upstroke and its role in determining propagation velocity during ischemia. Fig 6A shows computed versus V_rest for a fiber with an acidic Na⁺ channel, acidic [K⁺]_i, and four levels of I_Ca(L) conductance: 100%, 50%, 25%, and 0%. It can be observed that for most of the acute ischemic period, and propagation are independent of I_Ca(L). However, at highly depolarized V_rest immediately before conduction block, I_Ca(L) plays an increasingly important role (see inset). Propagation failed at [K⁺]_o=12.6, 12.8, 13.1, and 14.4 mmol/L for I_Ca(L) conductance at 0%, 25%, 50%, and 100% of maximum, respectively. At [K⁺]_o=12.5 mmol/L (V_rest=-61.1 mV), which was the highest [K⁺]_o for which propagation was successful for all levels of I_Ca(L) conductance, increased from 26 to 34 cm/s as I_Ca(L) was increased from 0% to 100%.

View larger version (20K): [in this window] [in a new window]   Figure 6. Effect of ICa(L) on depressed conduction. A, Conduction velocity, , for a range of Vrest induced by changes in [K+]o. Simulation is for acidic membrane conditions with four levels of ICa(L) conductance: 0%, 25%, 50%, and 100%. Note that ICa(L) contributes to only in highly depressed conduction, immediately before block. Inset is magnification of the highly depressed portion of the graph. B, Action potentials for the most depolarized Vrest (-61.1 mV) at which conduction was successful for all levels of ICa(L) conductance. Observe the influence of ICa(L) on the second half of the action potential upstroke and on amplitude of the action potential plateau.

The action potentials of the multicellular fiber corresponding to V_rest=-61.1 mV for the four levels of I_Ca(L) conductance are shown in Fig 6B. Time 0 corresponds to 2 ms before stimulation of the proximal end of the fiber (cell 1); action potentials are from the middle cell (cell 35). It can be observed, outside of the expected plateau changes, that the action potentials with reduced I_Ca(L) have slower upstroke phases. (dV_m/dt)_max is 17.4, 22.9, 26.1, and 30.6 V/s for the four levels of I_Ca(L) conductance (from 0% to 100%), which seems to suggest a direct role of I_Ca(L) in the highly depressed upstroke. However, (dV_m/dt)_max occurred at V_ms between -36.5 and -31.4 mV, which is below the range of appreciable activation of I_Ca(L)⁷ (I_Ca(L) is only 5% activated at V_m=-30 mV).

To resolve the above conflict regarding the role of I_Ca(L) during depressed conduction, in Fig 7 we plotted the upstrokes, the two major inward membrane currents (I_Na and I_Ca(L)), and I_axial for the action potentials in Fig 6B. Time 0 of Fig 7 is the time of (dV_m/dt)_max of cell 34, the cell immediately proximal to the cell investigated (cell 35). The arrows in Fig 7A indicate the occurrence of (dV_m/dt)_max for cell 35 of the 100% I_Ca(L) (filled arrow) and 0% I_Ca(L) (empty arrow) fibers. A comparison of panels A and C in Fig 7 reveals that the Ca²⁺ current activates appreciably well after (dV_m/dt)_max, too late to have a significant quantitative effect on the early upstroke. Note that the scale of I_Ca(L) (Fig 7C) is an order of magnitude smaller than that of I_Na (Fig 7B). It can be concluded that in terms of membrane currents, the depressed upstroke (dV_m/dt)_max is determined by I_Na (Fig 7B).

View larger version (20K): [in this window] [in a new window]   Figure 7. Origin of the action potential upstroke in highly depressed tissue. A, Expansion of action potential upstrokes from Fig 6B is shown. Time 0 corresponds to the time of (dVm/dt)max of cell 34, the cell immediately proximal to the cell reflected in the data (middle cell of fiber, cell 35). Solid arrow indicates time of (dVm/dt)max at cell 35 for 100% conductance of ICa(L). Open arrow indicates time of (dVm/dt)max at cell 35 for 0% ICa(L) conductance. B and C, INa (B) and ICa(L) (C) of cell 35 indicate that INa dominates the upstroke at the time of (dVm/dt)max for all four levels of ICa(L) conductance (0%, 25%, 50%, and 100%). Note the order of magnitude smaller scale of ICa(L) (C) compared with INa (panel B). D, Early net negative (into cell) axial current at cell 35 is attenuated for reduced ICa(L) conductance. This occurs well before activation of the local inward current, indicating an electrotonic influence. Iaxial is normalized to unit membrane capacitance so that direct comparison with membrane currents can be made. It is the electrotonic current supplied by an excited cell to its downstream neighbor during propagation and, by Ohm's law, is proportional to the potential gradient between these cells.

The differences in (dV_m/dt)_max for the different levels of I_Ca(L) conductance are caused by the effect that proximal (upstream) I_Ca(L) has on the local I_Na. Reduced I_Ca(L) reduces the peak V_m obtained during the second half of the upstroke (Fig 6B). This implies a reduced potential gradient in the direction of propagation, which, by Ohm's law, decreases the axial electrotonic current delivered to charge the membrane capacitance and excite the downstream cells. Note the significant difference of inward (negative) I_axial in Fig 7D for different I_Ca(L) conductances over the relative time window of -0.5 to 0.25 ms, the time of proximal excitation before local excitation and local (dV_m/dt)_max. A diminished I_axial causes a longer subthreshold depolarization phase (slower charging of membrane capacitance), which prolongs t_thresh, thereby reducing and (dV_m/dt)_max because of the increased dynamic inactivation of I_Na. These results suggest that in a highly depressed tissue, proximal I_Ca(L) influences the upstroke of adjoining depolarizing cells. I_Ca(L) augments a highly depressed Na⁺ upstroke indirectly, through its enhancing effect on the electrotonic source current, which depolarizes the membrane to threshold.

Transition to Slow (Ca²⁺-Dominated) Conduction
A result from Figs 2 through 6 is that [K⁺]_o elevation causes resting membrane depolarization, leading to decreased Na⁺ channel availability. [K⁺]_o elevation of >14.4 mmol/L (Fig 2) causes conduction block. At [K⁺]_o=14.4 mmol/L, the upstroke is still dependent on residual Na⁺ current. Unmodified Ca²⁺ current at [K⁺]_o>14.4 mmol/L is unable to sustain propagation. Enhanced Ca²⁺ current may, however, sustain slow propagation. We investigated the requirements for a transition from depressed Na⁺ upstrokes to I_Ca(L)-dominated upstrokes.

Fig 8 contains (top to bottom) computed V_m, its slope (dV_m/dt), and the inward currents I_Na and I_Ca(L) of four action potential upstrokes corresponding to conditions of [K⁺]_o=4.5, 10, 14, and 20 mmol/L (columns A through D, respectively). Data are shown for cell 35 of the 70-cell fiber. Ca²⁺ conductance was enhanced 100% at [K⁺]_o=20 mmol/L to greatly increase I_Ca(L) and facilitate propagation (Fig 8, column D). At [K⁺]_o=4.5 mmol/L and [K⁺]_o=10 mmol/L (Fig 8, columns A and B), I_Na clearly dominates the upstroke. I_Ca(L) contributes to membrane depolarization well after (dV_m/dt)_max has been reached and only after the majority of phase 0 depolarization has been completed. At [K⁺]_o=14 mmol/L, a condition close to conduction block, I_Na is highly depressed, and the upstroke becomes biphasic. Without I_Na, at [K⁺]_o=14 mmol/L the membrane would not depolarize sufficiently to activate I_Ca(L). However, although sufficient I_Na exists at [K⁺]_o=14 mmol/L; depolarization is slow enough to allow significant I_Ca(L) activation at V_m=-30 mV, relatively early during the upstroke. The two phases of depolarization, slightly apparent in V_m versus time (Fig 8, first graph in column C), are clearly seen in the plot of dV_m/dt (second graph). I_Na (third graph) is much slower to develop (the peak is wider) than that at [K⁺]_o=4.5 mmol/L and [K⁺]_o=10 mmol/L. Peak I_Na for [K⁺]_o=14 mmol/L is 5% of control value. Slowed depolarization allows for greater I_Ca(L) activation in the mid upstroke region, because its activation gate has a longer time to open. Note that peak I_Ca(L) is close in magnitude to peak I_Na under these conditions (Fig 8, bottom graph of column C). The two dV_m/dt peaks at [K⁺]_o=14 mmol/L coincide with peak I_Na and peak I_Ca(L), respectively. Dynamic load conditions prevent an exact correlation between dV_m/dt and local membrane current, which is possible in the single cell. At [K⁺]_o=20 mmol/L (Fig 8, column D), the transition to I_Ca(L) upstroke is complete. The pure I_Ca(L) upstroke is smooth, monophasic, and supported solely by I_Ca(L). Resting I_Na availability at this degree of membrane depolarization (V_rest=-51.3 mV) is zero.

View larger version (21K): [in this window] [in a new window]   Figure 8. Transition from INa-dominated to ICa(L)-dominated action potential upstrokes at increasing [K+]o. Vm, its time derivative (dVm/dt), and the excitatory currents INa and ICa(L) are shown (top to bottom) for upstrokes at [K+]o=4.5, 10, 14, and 20 mmol/L (A through D, respectively). ICa(L) was doubled for computations at [K+]o=20 mmol/L (D). INa dominates the upstrokes at [K+]o=4.5 and 10 mmol/L. A two-component upstroke is visible at [K+]o=14 mmol/L. At [K+]o=20 mmol/L, INa is fully inactivated, and ICa(L) is the only excitatory current.

It is clear from Fig 8, column D, that enhanced I_Ca(L) can support propagation. We are interested in the minimum degree of I_Ca(L) augmentation necessary to support conduction for both the fiber with elevated [K⁺]_o and the fiber with all three ischemic conditions. Therefore, we introduced a scaling factor, , to I_Ca(L). For every [K⁺]_o, we computed iteratively the lowest value of that was sufficient to maintain nondecremental conduction. Criterion for successful conduction was a <5% drop in (dV_m/dt)_max between cells 20 and 50. Results are shown in Fig 9. Two fibers were used: a control fiber (solid line) in which only [K⁺]_o was varied and an "ischemic" fiber (dotted line) with all three conditions—acidosis (pH 6.5), anoxia ([ATP]_i=3 mmol/L), and elevated [K⁺]_o. In the control fiber, the value was 1 at [K⁺]_o14.4 mmol/L. Therefore, as was shown in previous figures, propagation fails without augmented I_Ca(L) at this degree of [K⁺]_o elevation. It is surprising, however, that only a small degree of I_Ca(L) augmentation is required for the control fiber to sustain propagation beyond [K⁺]_o=14.4 mmol/L. For [K⁺]_o=18 mmol/L, for example, only 10% augmentation is required (=1.10). slowly increases with increasing [K⁺]_o, and the resulting membrane depolarization, reflecting the balance between resting I_Ca(L) inactivation and a closer proximity to I_Ca(L) activation threshold. Beyond a critical [K⁺]_o (30 mmol/L), the requirement to overcome I_Ca(L) inactivation dominates, and rises rapidly.

View larger version (20K): [in this window] [in a new window]   Figure 9. Degree of ICa(L) enhancement necessary to sustain conduction. is the minimum scaling factor to ICa(L) conductance required to sustain nondecremental conduction. Nondecremental conduction is defined as 95% conservation of (dVm/dt)max between cells 20 and 50. Control fiber corresponds to a fiber subjected to elevated [K+]o only. Ischemic fiber corresponds to fiber subjected to elevated [K+]o, acidosis, and anoxia. Shaded region highlights the zone of transition between INa-dominated upstrokes and ICa(L)-dominated upstrokes.

The ischemic fiber requires greater I_Ca(L) enhancement. At [K⁺]_o=12.9 mmol/L, =1 (Fig 9). At [K⁺]_o=18 mmol/L, increases to 2.20 (120% increase), significantly larger than the 10% increase for the control fiber with elevated [K⁺]_o alone. Some of the augmentation required for the ischemic fiber is due to acidic reduction of I_Ca(L) maximum conductance and the somewhat decreased I_Ca(L) channel availability caused by lowered [K⁺]_i (which depolarizes the membrane slightly). However the more significant factor that impairs conduction is the presence of I_K(ATP) in the ischemic fiber. For instance, 25% a priori I_Ca(L) reduction due to acidosis and anoxia (in the absence of I_K(ATP)) requires 33% enhancement to return I_Ca(L) to control nonacidic levels. That the ischemic fiber with [K⁺]_o=18 mmol/L required 120% augmentation of I_Ca(L) to sustain conduction means that most ischemic depression is due to anoxia and I_K(ATP), not acidosis.

It is interesting to note that during all except highly depressed Na⁺ upstrokes, I_K(ATP) at [ATP]_i=3 mmol/L does not affect conduction (Fig 2A, anoxia). Yet under conditions of highly depressed I_Na and for all I_Ca(L)-dominated upstrokes, activation of the I_K(ATP) outward current at 3 mmol/L [ATP]_i is sufficient to block conduction. The influence of anoxia-activated I_K(ATP) on slow I_Ca(L)-controlled conduction is examined in Fig 10. Fig 10A contains computed V_ms for cells 5, 10, 15, and 20 of two fibers, both responding to a stimulus at cell 1. The control fiber (solid lines) was computed under conditions of [K⁺]_o=18 mmol/L, with I_Ca(L) augmented by 10%. The anoxic fiber (dashed lines) was also computed with [K⁺]_o=18 mmol/L and 10% I_Ca(L) enhancement, but with the additional contribution of I_K(ATP) at [ATP]_i=3 mmol/L. As expected, minimal I_Ca(L) augmentation is sufficient to sustain nondecremental conduction in the control fiber. Action potentials in the first 10 cells of the control fiber are influenced by the stimulus. By cell 15, action potentials in the fiber settle to their stimulus-independent form and continue to propagate in a stable mode. In contrast, propagation in the anoxic fiber is decremental, and by cell 20, it is clear that the anoxic fiber cannot support conduction.

View larger version (27K): [in this window] [in a new window]   Figure 10. Influence of IK(ATP) on ischemic conduction. Computed action potential upstrokes (A) for cells 5, 10, 15, and 20 of a fiber with [K+]o=18 mmol/L and 10% ICa(L) enhancement (control, solid lines) and a similar fiber with the added condition of [ATP]i=3 mmol/L (anoxia, dashed lines). Iion for the same cell is shown for the control and anoxic fibers in panels B and D, respectively. For the control fiber, cells 5 and 10 reflect the stimulus influence that was applied at cell 1. Cell 15 of the control fiber (B) reflects the steady state, stable, biphasic (outward then inward) shape of Iion typical of nondecremental conduction. For the ischemic fiber, by cell 15 Iion never becomes inward, a clear sign of conduction failure.

I_K(ATP) is an outward current that decreases the action potential amplitude and causes early repolarization, decreasing APD. Decreased amplitude and decreased APD are clearly evident in the anoxic action potentials of Fig 10A. However, at [ATP]_i=3 mmol/L, the direct effect of I_K(ATP) on the local depolarizing membrane is not sufficient to cause the observed differences between the control and anoxic action potentials. For instance, in cell 5 of the anoxic fiber (chosen because it is away from the stimulus but still produces an action potential), I_K(ATP) never exceeds 1.5 µA/µF, which is an order of magnitude smaller than peak I_Ca(L). Therefore, quantitatively, outward I_K(ATP) does not have significant local effect compared with inward I_Ca(L). Similarly, in an isolated cell, I_K(ATP) at [ATP]_i=3 mmol/L causes only marginal APD shortening, yet there is an 80% difference in APDs between cell 5 of the control fiber and cell 5 of the anoxic fiber. That I_K(ATP) causes a dramatic change in action potentials between the two fibers suggests that the influence of I_K(ATP) is much greater than its local effect.

We learned from the previous section that I_Ca(L) affects propagation by influencing the electrotonic source current. Similarly, I_K(ATP), by decreasing the action potential amplitude and, consequently, the potential gradient in the direction of propagation, can limit the electrotonic current flow to downstream cells. The influence of I_K(ATP) on downstream cells is best understood by comparing the total transmembrane ionic current of corresponding cells of the control and anoxic fibers. Fig 10 contains I_ion for cells 5, 10, and 15 of the control fiber (panel B) and for cells 5, 10, 15, and 20 of the anoxic fiber (panel C). I_ion of cell 5 of the control fiber has a brief outward (positive) phase followed by a large slow inward phase. The outward phase is due to outward K⁺ current (mostly I_K1), which counteracts depolarization due to large axial current from upstream cells and from the stimulus applied to cell 1. As I_axial-induced depolarization at cell 5 brings the membrane to the range of I_Ca(L) activation, I_ion changes polarity and becomes inward, reflecting dominance of I_ion by inward I_Ca(L). Thus, the membrane switches from sink (consuming current delivered electronically) to source (generating its own I_ion). The initial outward phase of I_ion is prolonged in cells 10 and 15 of the control fiber because of the loss of the stimulus current. By cell 15 of the control fiber, I_ion reaches a stable shape, indicating nondecremental steady state propagation.

In contrast to the control fiber, I_ion for the anoxic fiber never achieves steady state. At cell 5, I_ion for the anoxic fiber (Fig 10C) is very similar to I_ion for the control fiber (Fig 10B). This is because I_Ca(L) is sufficiently activated by the strong stimulus and overwhelms I_K(ATP). However, I_K(ATP) reduces the action potential amplitude at cell 5 (Fig 10A). The result is less electrotonic source current available to depolarize adjacent downstream cells. At cell 10 of the anoxic fiber, an inward I_Ca(L) response still develops but with much a diminished magnitude. Note also that the initial upstroke phase of cell 10, when I_ion is outward, is prolonged compared with the corresponding cell of the control fiber, reflecting slower depolarization due to diminished electrotonic depolarizing current. Consequently, cell 10 is depolarized even less than cell 5, further reducing electrotonic current for downstream depolarization. By cell 15, electrotonic current formed upstream is insufficient to depolarize the membrane sufficiently to activate I_Ca(L), and the membrane never switches from sink to source. As a result, propagation is clearly decremental, leading ultimately to propagation failure as confirmed by the marginal depolarization seen in cell 20. Thus, I_K(ATP), like reduced I_Ca(L), exerts its influence by decreasing the electrotonic source current available for depolarizing downstream cells.

	Discussion

Top Abstract Introduction Materials and Methods Results Discussion References

 
The present study explores the mechanisms of conduction changes caused by the three major components of acute ischemia: elevated [K⁺]_o, acidosis, and anoxia. These ischemic conditions were applied at the level of ionic currents, thereby providing insights into mechanisms at the level of ionic processes. We found that [K⁺]_o is the single greatest determinant of propagation during acute ischemia. Changes in [K⁺]_o cause large variations in , and elevated [K⁺]_o alone can cause conduction block when it is varied within the ischemic range. In contrast, neither acidosis nor anoxia in the range of ischemic values can cause failure of conduction in the absence of elevated [K⁺]_o. In Fig 11, we summarize our findings regarding and ionic currents responsible for conduction over a range of V_rest. The range of V_rest reflects membrane depolarization due to elevated [K⁺]_o during acute ischemia. Under conditions of [K⁺]_o elevation alone, slight depolarization of V_rest causes supernormal conduction. At further V_rest depolarization, conduction is depressed and, ultimately, blocked. During both supernormal and depressed conduction, the upstroke is dominated by I_Na. Conduction block occurs when I_Na is almost fully inactivated, although I_Ca(L) can delay the onset of block by increasing the electrotonic source current. At conditions near conduction block, biphasic upstrokes occur (the shaded region in Fig 11); the first phase is due to peak I_Na, and the second is due to I_Ca(L) (see Fig 8). For the ischemic fiber (acidosis and anoxia included), slow action potentials due to I_Ca(L) alone are possible only with major I_Ca(L) enhancement. Acidosis causes [K⁺]_o-independent depression of I_Na, which can eliminate the supernormal conduction phase of hyperkalemia. Acidosis-induced reductions in I_Na and I_Ca(L) and anoxia-induced increases in outward currents (I_K(ATP)) result in conduction block at less depolarized V_rest (lower [K⁺]_o; compare shaded regions in top and bottom diagrams of Fig 11). Below is a detailed discussion of these observations.

View larger version (21K): [in this window] [in a new window]   Figure 11. Summary of ionic current dependence and type of conduction at different levels of Vrest. Characteristics of propagating action potential under elevated [K+]o only (top) and combined ischemic conditions (bottom) over a range of [K+]o-induced changes in Vrest are shown. Shaded zone corresponds to region in which biphasic upstrokes are apparent. The bold line in each shaded zone indicates the least depolarized Vrest at which conduction will fail without ICa(L) enhancement.

Ischemic Versus [K⁺]_o-Induced Conduction Changes
Supernormal conduction in hyperkalemic solutions is well established. However, although increased during early acute ischemia has been observed,^{10 62 63} the phenomenon is not universally accepted. Transient increases in excitability, a response related to slight V_rest depolarization, is more commonly reported during the early stages of ischemia. For instance, Elharrar et al⁶⁴ found a decrease in excitation threshold in the ischemic zone during the first 3 minutes following coronary occlusion. The increase in excitability was attributed to increased [K⁺]_o.⁶⁴ Similarly, Coronel et al⁶⁵ found a transient decrease in diastolic stimulation threshold within the first 4 minutes of coronary occlusion at an average [K⁺]_o of 6 mmol/L.

It is likely that [K⁺]_o-induced reduction in excitation threshold, under full ischemic conditions, does not always result in supernormal conduction. If so, other factors in addition to hyperkalemia must be present during ischemia to reduce from its supernormal values that are due to hyperkalemia alone. It has been suggested as a possibility that acidosis^{66 67} and other metabolic factors⁵³ cause reductions in intercellular coupling during ischemia, which would lead to reduction in . Yet it has been established that intercellular coupling remains stable during the reversible stage of acute ischemia and that sharp decreases in coupling are coincident with ischemic contracture, the secondary rise of [K⁺]_o, and irreversible cellular damage.^{53 54 55 56} Therefore, membrane factors (not gap junction factors) must play a role in reducing velocity during the acute ischemic stage. Veenstra et al⁶⁸ found that hyperkalemia alone ([K⁺]_o=8 mmol/L) increased ventricular , acidosis (pH 6.8) alone reduced , and anoxia (95% N₂/5% CO₂) alone had no effect on conduction. When acidosis and hyperkalemia were combined, remained unchanged at slow pacing and decreased at a fast pacing rate.⁶⁸

As discussed in "Materials and Methods," acidosis reduces Na⁺ and Ca²⁺ channel conductance and introduces a depolarizing shift in Na⁺ channel membrane kinetics. Acidosis also induces a slight depolarization of V_rest (via altered [K⁺]_i), which increases the probability of Na⁺ channel inactivation. In our simulations of isolated hyperkalemia, conduction increased from 60 cm/s at [K⁺]_o=4.5 mmol/L to a peak of 68 cm/s at [K⁺]_o=8.2 mmol/L (Fig 2A). At the same [K⁺]_o but under additional conditions of acidosis, reached only 61 cm/s (Fig 2A). Therefore, our results suggest that acidic effects on the Na⁺ channel alone can counterbalance [K⁺]_o-induced supernormal conduction.

Role of I_Ca(L) in I_Na-Supported Conduction
It is not known with certainty whether depressed ischemic upstrokes in ventricular myocardium are supported by a depressed I_Na or by I_Ca(L). The experimental data tend to support Na⁺-dominated depressed upstrokes and I_Na failure as the mechanism of conduction block.^{10 19 69} The results reported in the present study strengthen the conclusion that I_Na is responsible for maintaining propagation in ischemic tissue, with a caveat. The general expression (dV_m/dt)_max(h·j)_rest· (see Equation 6) makes direct use of I_Na inactivation parameters to describe upstroke and conduction velocities. When the fiber was subjected to conditions of elevated [K⁺]_o, acidosis, and anoxia, conduction block always occurred within the vicinity of complete I_Na inactivation (the lowest I_Na availability that still supported conduction was 5%, obtained with [K⁺]_o elevation to 14.4 mmol/L). Ventricular membranes reach V_thresh and generate an action potential well before significant activation of I_Ca(L),⁷ and contribution to (dV_m/dt)_max from I_Na is more than an order of magnitude greater than the contribution from I_Ca(L) (Fig 7).

The caveat is that I_Ca(L) can assist the Na⁺-supported propagation through its effect on the electrotonic source current. In highly depressed conduction, I_Ca(L) increases action potential amplitude and duration, which by Ohm's law increases electrotonic current flow and accelerates excitation of adjoining unexcited tissue. Through this electrotonic mechanism, maintenance of I_Ca(L) availability in regions of highly depressed conduction results in faster and delayed onset of conduction block (Fig 6A). These findings are consistent with the observations that in acutely ischemic regions of canine hearts, catecholamine release (catecholamines are I_Ca(L) agonists) from stimulation of the stellate ganglia has caused increased .⁷⁰ In the later stages of ischemia, catecholamines may mediate recovery of V_rest and return of electrical excitability.^{17 71} It is possible that during early ischemia moderate catecholamine release aids conduction by enhancing I_Ca(L), which, in turn, improves conduction of the traveling wave front through its electrotonic effect (Fig 6).

In conclusion, our results suggest that I_Ca(L) does not contribute directly (as an excitatory current) to the early upstroke of even the most depressed propagating action potentials during acute ischemia. However, I_Ca(L) facilitates conduction by supporting the plateau of the action potential, thereby generating a potential gradient in the direction of propagation. The potential gradient, through Ohm's law, generates an I_axial that helps to depolarize the downstream cells and to sustain conduction.

I_Ca(L)-Supported Conduction
Although normal (not enhanced) I_Ca(L) could not support conduction when I_Na was completely depressed, only marginal I_Ca(L) enhancement was necessary for slow I_Ca(L)-supported conduction in the presence of hyperkalemia alone. For example, at [K⁺]_o=18 mmol/L, only 10% I_Ca(L) enhancement was needed for nondecremental conduction (Fig 9). The action potential upstroke under these conditions is smooth and monophasic, supported solely by I_Ca(L). The experimental precedent for slow I_Ca(L)-supported conduction is classic. Cranefield²⁰ has loosely translated Trautwein and Schmidt's⁷² observation on the effect of epinephrine (an I_Ca(L) agonist) to initiate slow action potential conduction as follows: "Every cardiac electrophysiologist knows that adding a drop of epinephrine to the tissue bath may bring an apparently dead fiber back to life." It should be noted that 10% augmentation of I_Ca(L) could fall within the physiological range of "no enhancement." In fact, I_Ca(L)-supported propagation has been observed experimentally in purely hyperkalemic preparations (guinea pig) in the absence of catecholamines. Slow I_Ca(L)-supported conduction in cardiac tissue under complete ischemic conditions is, however, more difficult to obtain. Additional ischemic conditions of acidosis (25% reduction of I_Ca(L)) and anoxic activation of I_K(ATP) ([ATP]_i=3 mmol/L) at [K⁺]_o=18 mmol/L required a 120% augmentation of I_Ca(L) for Ca²⁺-dependent conduction to be sustained (Fig 9). As shown in Fig 10, marginal outward current contribution by I_K(ATP) is sufficient to cause conduction block. The spatial heterogeneity of acutely ischemic tissue with multiple inexcitable regions that constitute current sinks puts a great demand on the excitatory current. Therefore, it is unlikely that substantial Ca²⁺-dominated slow propagation occurs during acute ischemia without major catecholamine involvement.

Effect of I_K(ATP) Over a Range of [ATP]_is
The role of I_K(ATP) in ischemic electrophysiology remains the subject of considerable speculation. Major activation of the current requires [ATP]_i that is two orders of magnitude less than normal or ischemic myoplasmic concentrations.⁷³ However, it has been shown that fractional (0.6%) I_K(ATP) activation by [ATP]_i in the millimolar range can influence cellular electrophysiology by shortening APD,^{38 40 41 60} reflecting the high membrane density of I_K(ATP) channels (similar to that of I_Na). O'Rourke et al⁷⁴ have also shown that metabolic oscillations can cause oscillations in membrane current via regulation of I_K(ATP). Our results indicate that I_K(ATP), when responding to a typical myoplasmic [ATP]_i that occurs during acute ischemia (ie, 3 mmol/L), can facilitate the occurrence of conduction block (ie, shift its occurrence to slightly lower [K⁺]_o) during I_Na-supported excitation. Also, successful I_Ca(L)-supported conduction, in the presence of ischemic I_K(ATP), requires major enhancement of I_Ca(L).

Our results (Fig 2A) also indicate that extreme reduction of [ATP]_i can cause a much higher level of I_K(ATP) availability (ie, 20% channel availability at [ATP]_i=0.5 mmol/L). Under such extreme conditions, I_K(ATP) can depress conduction at any [K⁺]_o and cause conduction block at a relatively low [K⁺]_o. Direct support for extreme reduction of [ATP]_i during ischemia is lacking. However, several theories have been proposed that support a greater role for I_K(ATP) than indicated by the myoplasmic ATP concentrations. One possibility is that ATP is compartmentalized into bulk myoplasmic and submembrane compartments, with the submembrane concentration being significantly lower than the bulk concentration during ischemia and other forms of metabolic deprivation.^{75 76} Recently, the compartmentalization hypothesis was questioned by Priebe et al,⁷⁷ who provided evidence that submembrane ATP concentrations are not significantly different from general myoplasmic concentrations and that highly localized ATP deprivation may occur in relation to Na⁺-K⁺ pump sites. Other metabolic substrates such ADP, H⁺, and lactate, which are present during ischemia, also decrease I_K(ATP) sensitivity to ATP,^{37 44 73 78} causing greater activation at higher [ATP]_i. In our studies, we increase k_1/2 from 114 to 250 µmol/L to account for the effect of the additional substrates. We also used an ATP concentration that is on the low end ([ATP]_i=3 mmol/L) of values reported to occur during acute ischemia.⁶⁰ The simulation of extremely low [ATP]_i in Fig 2A constitutes a predictive study that suggests major effects of I_K(ATP) on conduction under conditions of severe metabolic deprivation. Further experimental studies are needed to evaluate this possibility and to examine local intracellular ATP changes in the context of ischemia and other metabolic insults.

It should also be emphasized that the depressant effect of anoxia on excitability and velocity is fundamentally different from that of elevated [K⁺]_o and acidosis. Acidosis directly reduces Na⁺ channel availability. Elevated [K⁺]_o, through its effect on V_rest, also causes reduction of Na⁺ channel availability. In contrast, anoxia in cardiac tissue has no direct effect on I_Na. By activating I_K(ATP) (a repolarizing current that increases with membrane depolarization), anoxia delays attainment of threshold, which decreases (dV_m/dt)_max (by dynamic Na⁺ channel inactivation) and increases the likelihood of block.⁷⁹ Also, in multicellular preparations, anoxic I_K(ATP) acts to decrease electrotonic source current flow to adjoining cells by decreasing action potential amplitude, which also delays attainment of threshold (Fig 10). When I_Na is already reduced by effects such as elevated [K⁺]_o, anoxia can slow conduction further and cause conduction block under conditions when propagation is otherwise maintained.

Supernormal Conduction and Cable Theory
Two distinct phases characterize the relationship between (dV_m/dt)_max and over the [K⁺]_o-induced range of V_rest during acute ischemia (Figs 3 and 5). A phase of supernormal conduction, during which rises above its normal value while (dV_m/dt)_max varies only slightly, is followed by a phase during which and (dV_m/dt)_max descend in concert. If conduction along the fiber maintains a constant velocity (ie, conduction is nondecremental), then cable theory predicts the following relationship between temporal changes of V_m and :

(E7)

where C_m is membrane capacity. Equation 7 makes no assumption about the nature of the ionic current (I_j); however, it also precludes a general explicit relationship between and (dV_m/dt)_max. Polynomial and exponential models of the action potential upstroke^{80 81 82} have been used with the cable equations to determine a direct proportionality between (dV_m/dt)_max and ². This velocity-square relationship is a staple of classic cable theory. However, it is clear from Fig 5 that during supernormal conduction, (dV_m/dt)_max is not proportional to ². Therefore, we explored the possibility that an alternative relationship between (dV_m/dt)_max and exists during elevated [K⁺]_o-induced supernormal conduction.

A major advantage of the upstroke formulation in terms of ionic-based models over simplified exponential and polynomial models is the ability of the ionic models to adapt the membrane excitation threshold and current-voltage profiles to changes in V_rest. Fig 5 and the accompanying derivation suggest that (dV_m/dt)_max varies as the product of and (h·j)_rest, ie, (dV_m/dt)_max(h·j)_rest·. This relationship is general and holds for all values of V_rest, including the phase of supernormal conduction. At this phase, is determined by the difference between V_rest and V_thresh, since (h·j)_rest (Na⁺ channel availability) does not vary with V_rest. Since the membrane depolarizes with increasing [K⁺]_o, V_rest moves toward V_thresh, t_thresh is reduced, and increases. Note, however, that if depolarization starts from a higher V_rest, (h·j)_thresh is less inactivated because of a shorter t_thresh. This explains the slight initial elevation of (dV_m/dt)_max during the supernormal phase (Fig 3A). For more positive V_rest, beyond the value for which attains its maximum, there is appreciable inactivation of I_Na, and the dependence of on V_rest is determined by (h·j)_rest. At this range, is proportional to (h·j)_rest, and the general relationship reduces to the well-known form: (dV_m/dt)_max². Buchanan et al¹⁴ found in guinea pig papillary muscle that, except for [K⁺]_o-induced supernormal conduction, the proportionality between (dV_m/dt)_max and ² holds for a full spectrum of altered Na⁺ channel conductance (ie, by tetrodotoxin, class I antiarrhythmic agents, and increased stimulation frequency). If (h·j)_rest could be estimated in the guinea pig preparations of Buchanan et al,¹⁴ it would be interesting to determine whether their measured data fit the more general relationship during supernormal conduction.

In summary, the phenomenon of supernormal conduction has puzzled investigators over many years on two accounts: (1) the physiological implication and mechanism and (2) the theoretical framework that describes this phenomenon (whether it falls within the framework of classic cable theory).^{6 14} In the present study, we tried to provide mechanistic insight into this phenomenon and to investigate its theoretical framework. We also attempted to relate the theoretical framework to the physiological mechanism. This effort resulted in two very important conclusions: (1) the mechanistic basis for supernormal conduction is, as suggested by experimental studies,⁸³ an interplay between (h·j)_rest and distance from V_rest to V_thresh and (2) the supernormal phenomenon falls within the framework of classic cable theory. By quantitatively introducing (h·j)_rest, it was possible to include supernormal conduction and its mechanism in the classic theory of propagation.

Relationship to Ischemic Arrhythmogenesis
Ventricular arrhythmias occur in two distinct phases during acute ischemia⁸⁴ : phase 1a arrhythmias occur between 2 and 10 minutes from the onset of ischemia, and phase 1b arrhythmias occur between 12 and 30 minutes from the onset of ischemia. The pathophysiological conditions of elevated [K⁺]_o, acidosis, and anoxia are present during phase 1a arrhythmogenesis. Mapping experiments have revealed that 1a arrhythmias are due to reentry,^{3 84 85} whose initiation requires slowed conduction and the presence of unidirectional block.^{3 86 87} As shown in the present simulations, ischemic conduction slowing is caused by acidic and hyperkalemic reductions in membrane excitability (Figs 3 and 5). These reductions, as suggested by the rapidity of electrical change, are likely to be heterogeneous within the affected tissue. Heterogeneity of membrane excitability, caused directly by acidosis and hyperkalemia and indirectly by anoxic shortening of APD, increases the vulnerable window for unidirectional block and reentry.^{88 89} Therefore, conditions for induction of reentry (ie, slowed conduction and unidirectional block) develop as a result of acute ischemic changes in membrane properties, and ischemic tissue can provide the substrate for the development of reentrant arrhythmias.

The ischemic conditions used in the present study were applied homogeneously over the entire fiber. Recent data suggest the importance of inhomogeneities in the initiation of ischemic arrhythmias.⁹⁰ The baseline homogenous studies contained here set the stage for a detailed study of the role of superimposed spatial inhomogeneities in ischemic arrhythmogenesis. These should include inhomogeneous distribution of extracellular K^{+91 92 93 94 95 96} and inhomogeneous increase of extracellular (interstitial) resistance induced by ischemia.^{96 97}

	Selected Abbreviations and Acronyms

 

= scaling factor for ICa(L) = space constant = conduction velocity (dVm/dt)max = maximum upstroke velocity of the action potential (h·j)rest = Na+ channel availability at rest potential (h·j)thresh = Na+ channel availability at threshold APD = action potential duration Iaxial = intracellular axial current ICa(L) = L-type Ca2+ current Iion = total transmembrane ionic current IK(ATP) = ATP-sensitive K+ current IK1 = inward rectifier K+ current IKr = fast component of the delayed rectifier K+ current IKs = slow component of the delayed rectifier K+ current INa = fast Na+ current k½ = half maximum activation LRd model = dynamic Luo-Rudy ventricular cell model Rg = gap junction resistance Ri = axial resistance Rmyo = myoplasmic resistance Ro = extracellular resistance tthresh = depolarization time to threshold Vm = membrane potential Vrest = membrane potential at rest Vthresh = threshold potential

	Acknowledgments

 
This study was supported by National Institutes of Health grants HL-49054 and HL-33343 (National Heart, Lung, and Blood Institute). We thank Xiaoqin Zou and Jinglin Zeng for help in developing the multicellular fiber model and for many helpful discussions and suggestions. We also wish to thank Robert Harvey and Matthew Levy for valuable discussions.

	Footnotes

 
Reprint requests to Prof Yoram Rudy, Department of Biomedical Engineering, Case Western Reserve University, Cleveland, OH 44106-7207.

Previously published in abstract form for the 69th Scientific Sessions of the American Heart Association, New Orleans, La, November 10-13, 1996 (Circulation. 1996;94[suppl I]:I-306).

Received May 9, 1996; accepted October 8, 1996.

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