An Official Journal of the American Heart Association BRIEF REVIEWS. Evidence from the Voltage Clamp and Extracellular K + - Selective Microelectrodes

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1 Circulation Research An Official Journal of the American Heart Association BRIEF REVIEWS JANUARY 1982 VOL. 50 NO. 1 K + Fluctuations in the Extracellular Spaces of Cardiac Muscle Evidence from the Voltage Clamp and Extracellular K + - Selective Microelectrodes IRA COHEN AND RICHARD KLINE UNDERSTANDING cellular cardiac electrophysiology is an important first step in developing the ability to predict and alter pathological conditions in the whole heart. Over the past 30 years, many cardiac electrophysiologists have attempted to analyze the membrane properties of cardiac muscle in a manner similar to the classic analysis of the squid axon (Hodgkin and Huxley, 1952). The upstroke (Weidmann, 1955; Dudel and Rudel, 1970; Colatsky and Tsien, 1979), the plateau (Beeler and Reuter, 1970; Noble and Tsien, 1969), and the diastolic depolarization (Noble and Tsien, 1968) have been interpreted by assuming a number of parallel ionic channels with time- and voltage-dependent conductances. Neurophysiologists have enjoyed significant experimental advantages over the cardiac electrophysiologist in pursuing the initial observations of Hodgkin and Huxley. First, they were able to control the ionic concentrations on either side of the membrane more effectively because of their ability to perfuse the squid axon internally (Baker et al., 1962). Second, selective blockers of the two timeand voltage-dependent channels (tetrodotoxin for the Na + channel and tetraethylammonium for the K + channel) were available. Third, the squid axon can be voltage clamped by inserting an axial wire From the Department of Physiology and Biophysics, HSC SUNY at Stony Brook, and Departments of Pharmacology and Medicine, Mount Sinai School of Medicine, CUNY, New York, New York. Dr. Cohen is affiliated with SUNY, and Dr. Kline with Mount Sinai. Ira Cohen is supported by HL 20558, an RCDA from the National Heart, Lung, and Blood Institute, and a grant from the American Heart Association in conjunction with its Dutchess County affiliate. Richard Kline is supported by Young Investigator Award HL from the National Heart, Lung, and Blood Institute and a grant from the American Heart Association. Address for reprints: Dr. Ira Cohen, Department of Physiology & Biophysics, Health Sciences Center, State University of New York at Stony Brook, New York down its center. This approach guarantees adequate longitudinal homogeneity of voltage during voltage clamp steps. Cardiac muscle could not be internally perfused; in fact, there are delays and difficulties in rapidly perfusing the extracellular spaces of many cardiac preparations (Lamb and McGuigan, 1965, 1968). Furthermore, the presence of small intercellular clefts increases the possibility that experimental manipulations designed to study the membrane conductances may alter the ionic concentrations in the narrow spaces facing much of the cardiac membrane surface (Baumgarten and Isenberg, 1977; McAllister and Noble, 1966; Cleeman and Morad, 1976; Cohen et al., 1976a; DiFrancesco et al, 1979). The extracellular ion concentrations are also likely to fluctuate during the normal physiological event, the action potential (Kline and Morad, 1976; Kronhaus et al., 1978). Further, these same restricted extracellular clefts slow the temporal resolution of the voltage clamp and reduce the spatial homogeneity of the membrane voltage due to the electrical series resistance they contribute (Johnson and Lieberman, 1970; Attwell and Cohen, 1977; Beeler and McGuigan, 1978; Schoenberg and Fozzard, 1979). Several additional features of cardiac muscle increase the possibility of ionic concentration fluctuations. First, the small cell size and high degree of infolding implies that a larger membrane surface faces narrow extracellular spaces. Second, the slow time course of cardiac membrane currents, and of the action potential itself, means that even small imbalances among the factors determining extracellular ion concentration can have significant effects because of the long time over which they are sustained. The purpose of this review is not so much to provide an exhaustive study of the evidence and

2 CIRCULATION RESEARCH VOL. 50, No. 1, JANUARY 1982 importance of extracellular K + fluctuations as to give the reader a conceptual framework for dealing with the substantial body of literature that does exist. In keeping with this aim, the review will be divided into two sections. In the first section we will discuss the relevance of K + fluctuations to the interpretation of voltage clamp studies. A simple model will be developed to help the reader interpret the experimental results that occur when K + depletion occurs during voltage clamp steps. This is followed by a short review of experimental results pertaining to K + accumulation or depletion with the voltage clamp. The second section of the review considers evidence for K + fluctuations in extracellular spaces from experiments with extracellular K + selective microelectrodes. In this section, as well, a model is included to help the reader integrate the experimental results. It is worth indicating at the outset that neither the voltage clamp nor ion-selective microelectrodes are ideal probes for changes in the extracellular K + concentration. The inadequacies of the voltage clamp method as applied to cardiac muscle (Attwell and Cohen, 1977; Beeler and McGuigan, 1978) often present uncertainty in the interpretation of an experimental result. The finite size of the extracellular K + electrode suggests that it is unlikely to be positioned without distortion in the narrowest of cardiac extracellular spaces where accumulation and depletion probably are most significant. Also, the absence of an intact vascular supply in any of the in vitro studies on cardiac muscle raises the interesting and important question of the significance of the observations presented below for normal physiological function (for some discussion of this point, see Boyett and Jewell, 1980). Nevertheless, there can be little doubt that, within these constraints, the results suggest a significant role for K + accumulation and depletion in narrow extracellular spaces in determining the outcome of many studies of cellular cardiac electrophysiology. Parti Introduction Traditionally, cardiac membrane currents studied with the voltage clamp have been divided into time-dependent and time-independent membrane current components. Many of the time-dependent currents have been described as Hodgkin-Huxley (HH) in nature (Noble and Tsien, 1968,1969; Beeler and Reuter, 1970; Weidmann, 1955; Dudel and Rudel, 1970; Colatsky and Tsien, 1979). Time-independent conductances are due to channels (or carriers) whose rate of ion transfer can depend on voltage but not on time. In cardiac membranes, there are thought to be a number of time-independent currents, including: (1) an inwardly rectifying K + current (McAllister and Noble, 1966); (2) an electrogenie Na + /K + exchange pump current (Gadsby and Cranefield, 1979; Eisner and Lederer, 1980); (3) a Ca 2+ -mediated K + current (Isenberg, 1975); (4) an outwardly rectifying K + current (McAllister and Noble, 1966; Vereecke et al., 1980); and (5) an inward background current (Peper and Trautwein, 1969; Bosteels and Carmeliet, 1972). The sum total of these currents rectify in the inward direction in the diastolic range of potentials, and in the outward direction positive to the plateau range of potentials. K + is thought to be a major current carrier in both regions of potential. Fluctuations in [K + ] in the extracellular spaces of cardiac muscle can create another type of timedependent membrane current. This current can flow through time-independent membrane channels or carrier molecules but, due to the changes in extracellular [K + ], the current flow becomes time dependent. Time-dependent changes in extracellular [K + ] can also affect the magnitude and time course of the time-dependent gated currents. Some possible effects of time dependent changes in [K + ] in a restricted extracellular space (RECS) on the time-independent and time-dependent, gated (HH) membrane currents are discussed below. A. The Concentration Trajectory In order to study the dependence of membrane current on RECS [K + ], it is necessary to know the time course of the [K + ] change. This time course is a complicated function of cleft-to-bulk diffusion of K +, ion pumping across the cell membrane, membrane K + conductances, and cleft geometry. Recent reviews modeling the cleft concentration change have appeared (Attwell et al., 1979a, 1979b). It is not our purpose to duplicate this analysis, but, rather, to use a simple example to illustrate the diversity of effects that RECS [K + ] changes can induce. For this limited purpose we will make two simplifying assumptions: (1) The restricted extracellular space is homogeneous, and the cardiac membrane is uniform everywhere (see Attwell et al., 1979). (2) The extracellular [K + ] changes in the RECS decays exponentially from 4 to 3 mm with TK = 1 sec during each hyperpolarizing voltage clamp pulse. Neither of these assumptions can be strictly valid (see Part II), but they do allow a simple look at what is certainly an extremely complicated picture. B. Linear K + Current Consider first the effects of the RECS on a membrane that contains only a linear K + current. The current voltage relationship for this membrane is calculated from Ohms law and is shown in Figure ID. IK = gk (V - V K ) where gk is the potassium conductance, V is the membrane voltage, and V K is the Nernst equilibrium potential for K + (also shown as the reversal potential for a K + -specific current).

3 K + FLUCTUATIONS IN EXTRACELLULAR SPACES/Cohen and Kline A B -50 mv- -50 mv- -60mV -60mV -80 mv -80mV - loomv -loomv E 3 if 4 3 H C -50mV- -60mV 2-80mV -100mV TO V(mV) 4CM -1-1 FIGURE 1 The effects of a restricted extracellular space (RECS) on a linear current voltage relationship. The membrane K + conductance is assumed to be 0.1 msec. The Nernstpotential for K + is assumed to be 97.7 mv in 3 mm K +, and 90.1 mv in 4 mm K +. A: The currents in response to hyperpolarization for 6 seconds in the absence of a RECS for 4 mm bulk K +, KB- B: The currents in response to hyperpolarization for 6 seconds in the absence of a RECS in the presence of 3 mm KB- C: The resultant currents in response to hyperpolarization for 6 seconds in the presence of a RECS. D: The current voltage relationships for 4 and 3 mm extracellular potassium. The arrows indicate the movements on the current voltage relationships in the presence of a RECS. The instantaneous current is determined by the movement from 50 mv to the appropriate test potential ( 60, 80, or 100 mv) on the K B = 4 mm currentvoltage relationship. The time-dependent current is determined by the movement between the two current-voltage relationships at the test potential.

4 CIRCULATION RESEARCH VOL. 50, No. 1, JANUARY 1982 A B -50mV- -50 mv mv -60mV -80 mv -80mV -100 mv -loomv 8r < 2 < L c 50 mv- e mV -80mV -loomv T V(mV) -2 J FIGURE 2 TTie effects of a restricted extracellular space (RECS) on an inwardly rectifying current-voltage relationship that exhibits cross-over. A: The currents in response to hyperpolarization for 6 seconds in the absence of a RECS in 4 mm Kt,. B: The currents in response to hyperpolarization for 6 seconds in the absence of a RECS in 3 mm KB- C: The resultant currents in response to hyperpolarization for 6 seconds in the presence of a RECS. D: The current-voltage relationships in 4 and 3 mm KB- The arrows indicate the movement on the current-voltage relationships in the presence of a RECS. The instantaneous current is determined by the movement from 50 mv to the appropriate test potential ( 60, -80, or -100 mv) on the K B - 4 mm current-voltage relationship. The time-dependent current is determined by

5 K + FLUCTUATIONS IN EXTRACELLULAR SPACES/Cohen and Kline The current observed on voltage clamping from -50 to potentials of -60, -80, or -100 mv in the absence of a RECS is shown for 4 and 3 mm extracellular [K + ] in Figure 1, A and B. At each potential, there is an instantaneous inward jump in current. The magnitudes of the jumps in current (although not the absolute value of the currents) are equal at each test potential in the two different [K ]'s because the current voltage relationships are parallel. In the presence of a RECS, the concentration changes from 4 to 3 mm [K + ] during the hyperpolarizing voltage clamp pulses. Thus, Ohms law equation must be modified to take this into account. IK (t) = g K [V - V K (t)] The current is now time dependent because the driving force V K is changing during the voltage clamp pulses. The resultant membrane currents are shown in Figure 1C. The easiest approach to understanding these time-dependent changes in membrane current is to consider the trajectory of movement on the membrane current voltage relations. This approach to understanding accumulation and depletion currents was used effectively in an important analysis of K + fluctuations in the Purkinje fiber performed by Baumgarten and Isenberg (1977). Figure ID illustrates this trajectory of movement, first on one current voltage relationship giving the instantaneous current, and then from one relationship to another yielding the time-dependent current at a constant potential. Thus the change in RECS [K + ] will result in time-dependent currents from initially time-independent linear K + currents. These currents increase with time in an outward direction. In the next section, these membrane currents become more complicated when the current-voltage relationships are no longer linear and the currents rectify in an inward direction. C. Inward Rectifier with Cross-over Figure 2D shows hypothetical current-voltage relationships for time-dependent inwardly rectifying K + currents in 4 and 3 mm K +. Since the currentvoltage relationships are not parallel, extracellular [K + ] affects the membrane K + conductance in addition to the driving force on the K + currents. In this case, the [K + ] dependence of the membrane conductance results in a cross-over of the currentvoltage relationships in 3 and 4 mm [K + ]. At higher [K + ]'s, more outward current is required to depolarize the membrane positive to the cross-over potential, and more inward current is required to hyperpolarize the membrane to potentials negative to the cross-over potential (see figure legend 2 for applicable equations). Figure 2, A and B, shows the resultant membrane currents for voltage clamp steps from 50 to potentials of -60, -80, and -100 mv in 4 and 3 mm [K + ] in the absence of a RECS. There are two major differences between these results and those presented for the linear K + current of Figure 1. First, for progressively more negative test pulse potentials, the increment of inward current becomes proportionately larger. This is a consequence of the inward rectification in the current voltage relationships. Second, for the higher [K + ], 4 mm, the increment in membrane current for a given pulse potential is larger. This is a consequence of the [K + ] dependence of the membrane conductance. In the presence of a RECS, we again assume that the concentration changes from 4 to 3 mm during the voltage clamp pulses. The results for the inward rectifier with cross-over are shown in Figure 2C. There is an inward instantaneous jump at 80 and 100 mv, followed by a slow outward current movement with time. This is similar to the linear K + current at those potentials (see Fig. 1C). However, at 60 mv, the result is different. After the instantaneous inward jump, the current slowly becomes more inward with time. The origin of these currents is demonstrated by movements on and between the current-voltage relationships in Figure 2D. An inwardly rectifying K + current with crossover can reverse the direction of the time-dependent component of the current. A linear K + current cannot. This reversal of a depletion current was first recognized by Aimers (1972) for the T-tubular system in skeletal muscle. It can cause errors in the estimate of the driving force on and selectivity of a membrane current. It has been called a pseudoreversal potential. D. The Interpretation of a Hodgkin-Huxley Current in the Presence of K + Fluctuations Besides converting time-independent membrane currents into time-dependent currents, K + fluctuate movement between the two current-voltage relationships at the test potential. The equations used to generate the current voltage relationships in Figures 2D and 3D are given below: For negative slope: TK 2 = 2.8(exp[0.04(V - V K )J - l)/(exp[0.08(v k + 50)]) + exp([0.04[v- (V K + 50)]) (8 x w' v * nw ) For inward rectifier with cross-over: i Kl = IK.J2.8 + (0.2fV -(V K + 80)])/[1 - exp(-0.04fv- (V K + 80)])] where V is the membrane potential and V K is the potassium equilibrium potential, i^ is used as the ion transfer function in Part I, Section E, while J'AT, is used as the ion transfer function in Part I, Section D.

6 CIRCULATION RESEARCH VOL. 50, No. 1, JANUARY 1982 tions will also distort the time course and magnitude of conventional Hodgkin-Huxley (HH) currents. For an HH current, one must describe both the gating characteristic (the time course and voltage range over which the channel opens), and the voltage dependence of the conductance of the open channel (its rectification properties). The discussion that follows employs a formalism for the gated current appropriate for i K,, the Noble and Tsien (1968) description of the Purkinje fiber pacemaker current. The ionic nature of this current is presently in question, (see part I, F). However, the important points of the discussion do not rely on the ultimate outcome of future research on this subject. The aim is to convey a sense of the types of modifications that [K + ] fluctuations can impart to HH currents. The i K, formalism is convenient for this purpose; however, similar results would be obtained for i x (the plateau K + current) or i f (the new formalism for the Purkinje fiber pacemaker). Consider a gated K + current presumed to have all its channels open at 60 mv and all of them closed at -90 mv. The voltage clamp pulses are again from -50 to -60, -80, and -100 mv. The time constant of gating of this current is assumed to be 0.5 second at 60 mv, 2 second at 80 mv, and 0.5 second at 100 mv. This channel is an inward rectifier like those in Figure 2D, but also possesses negative slope (see Fig. 3D). This negative slope results in less K + current through the open channel at potentials positive to about 60 mv. This means that the channel conductance positive to 60 mv decreases at a more rapid rate than the driving force on K +, (V V k ), increases. The equations describing the ion transfer function for this channel are from Cohen et al. (1978) and are given in the legend to Figure 2. The currents in response to voltage clamp pulses from -50 to -60, -80, or -100 mv in the absence of a RECS are shown in Figure 3, A and B, for 4 and 3 mm [K + ]. Since the channels are assumed to be initially open at 50 mv, the voltage step to 60 mv results in no time-dependent current. However, since the ion transfer function shows negative slope (the instantaneous outward current declines at potentials positive to 60 mv), an instantaneous outward jump in membrane current is seen. For the pulses to 80 and 100 mv, the current decays exponentially from the initial value of the gating variable (assumed to be 1.0 at 50 mv) to its final value (1 at -60, 0.33 at -80 mv, and 0 at -100 mv). When a RECS exists, the ion transfer rate becomes time dependent. The resultant currents in response to clamp pulses to 60, 80, and 100 mv are shown in Figure 3C. The major differences created by the RECS are (1) at -60 mv, a timedependent current now exists; (2) at 80 mv, a significant deviation from exponentiality exists at early times with a slowing of the half time of decay (TI/O = 2 sec as compared to 1.39 sec in the absence of the RECS). The magnitude of the time-dependent current is also reduced; (3) at 100 mv, the time-dependent current again deviates from exponentiality. The half-time of decay is now faster than that in either solution (TI /2 = 0.25 second, compared with 0.35 second in either solution). The magnitude of the time dependent current is also increased. It is clear that the effects of K + depletion on an HH current carried by K + can be complicated. In a real membrane, both time-independent and timedependent K + currents exist. This is discussed in the next section. E. A Hodgkin-Huxley Channel in the Presence of a Time-Independent Inwardly Rectifying K + Channel The previous sections emphasized the difficulties in interpreting HH or time-independent inwardly rectifying K + currents when the [K + ] changes in a RECS. The currents are even more complicated when both are present. This is shown in Figure 4 for the same concentration trajectory. The clamp pulses are again from -50 mv to potentials of 60, -80, and -100 mv. Compared with the HH currents of Figure 3A and B, the time-dependent currents in Figure 4 show: time dependence at 60 mv; a biphasic current at 80 mv, first increasing outward, and then increasing inward; and a substantially larger current at -100 mv than that in Figure 3 (A or B). In the light of the substantial alterations in the voltage dependence, amplitude, and time course conferred by [K + ] fluctuations, it is apparent that no simple relationship exists between the measured time-dependent membrane current and the true Hodgkin-Huxley conductance. Evidence for K + Fluctuations from Voltage and Current Clamp Studies We mentioned earlier the inherent advantages experimeters on simpler preparations enjoyed over the cardiac electrophysiologist. However, even in the simpler excitable membrane preparations (squid axon, myelinated nerve, skeletal muscle), there have been reports of significant K + accumulation and depletion. Frankenhauser and Hodgkin (1956) found that K + accumulates beneath the surrounding Schwann cell sheath of the squid axon following repetitive activity. K + accumulation in the perinodal space at the node of Ranvier has also been reported (Dubois and Bergman, 1975). Skeletal muscle offers the extracellular space most analogous to cardiac muscle, with its long T-tubules offering a significant volume for restricted extracellular diffusion. The analysis by Aimers (1972) of K + fluctuation in the T-tubules serves as the model for most studies of similar depletion and accumulation processes in cardiac muscle. In multicellular preparations, some of the most

7 K + FLUCTUATIONS IN EXTRACELLULAR SPACES/Cohen and Kline - 60rm B 50mV 60mv ' -80mV -80mV -100mV mV 10 I 10 E < 2 M c D -50mV- 10 < 2 a. 2-80mV -100mV -100/ [K] B FIGURE 3 The effects of a restricted extracellular space (RECS) on a Hodgkin-Huxley (HH) current in the absence of other K + currents. A: The HH currents in response to hyperpolarization for 6 seconds in the absence of a RECS in 4 mm KB- Gating properties are given in the text. B: The currents in response to hyperpolarization for 6 seconds for an HH current in the absence of a RECS in 3 mm K B. C: The resultant currents in the presence of a RECS. D: The current-voltage relation of an inward rectifier with negative slope. See figure legend 2 for the equations generating this relation.

8 CIRCULATION RESEARCH VOL. 50, No. 1, JANUARY mv- 22 r 14 - e L -60mV -80mV -100mV FIGURE 4 The effects of a RECS on a HH current in the presence of an inwardly rectifying background current. The responses in Figures 2D and 3C are summed to give Figure 4. The assumption of a constant concentration trajectory is discussed in the text. impressive studies of [K + ] fluctuations have been performed on the central nervous system. Accumulations of up to 10 mm can be found during stimulation in cortex (Prince et al, 1973), hippocampus (Lewis and Schuette 1975), and spinal cord (Kriz et al., 1975). Larger accumulations of up to 70 mm have been found during epileptic seizures and spreading depression (Vyskocil et al., 1972; Prince et al., 1973.) It should come, then, as no surprise that similar evidence for K + fluctuations in narrow extracellular spaces exists also for cardiac muscle. A brief review of the available evidence from voltage clamp studies is provided below. Purkinje Fibers The first indications of contamination of the results of voltage or current clamp studies with K + accumulation or depletion were reported in McAllister and Noble reported that prolonged depolarizing current pulses in sheep Purkinje fibers activated a delayed K + conductance. However, contrary to expectation, on termination of the depolarizing current pulse, the potential did not repolarize rapidly to negative potential near V k but, instead, remained at a depolarized potential and repolarized only slowly over a period of many seconds. This result suggested substantial potassium accumulation in the extracellular spaces of the sheep Purkinje fiber during the long current clamp pulse, shifting the value of the potassium equilibrium potential to more positive values. Initial studies of the passive membrane properties of sheep Purkinje fibers by Fozzard (1966) showed that the capacitative decay following a voltage clamp step was fit by two exponentials. One of these exponentials represented capacitative charging through an extracellular series resistance. Its time constant was lengthened on reduction of the ionic strength of the bathing solution. This indicated the possible presence of narrow extracellular spaces. The anatomical studies of Hellam and Studt (1973) and Mobley and Page (1971) demonstrated that sheep Purkinje fibers possess very narrow intercellular clefts, frequently as small as angstroms in diameter. These clefts faced as much as 90% of the total membrane surface area but only comprised 0.2% of the fiber volume. Under such circumstances, small ion fluxes would be expected to have substantial effects on the ion gradients across the majority of the cell membranes (see Cohen et al., 1976b; Attwell et al., 1979). This expectation, based on anatomical studies, was followed by a variety of experimental measurements suggesting substantial K + accumulation and depletion in sheep Purkinje fibers. Cohen et al. (1976a) suggested that the reversal potential for the pacemaker current was negative to the value of VK+ based on the bulk [K + ]. They proposed a steady state depletion of K + in the clefts of the Purkinje fiber. The initial suggestion by Cohen et al. (1976) was followed by a detailed study of depletion currents in the presence and absence of external Na + (Baumgarten and Isenberg, 1977; Baumgarten et al., 1978). These studies suggested that the [K + ] changed continuously during application of a depolarizing or hyperpolarizing voltage clamp step, creating a current like that shown in Figure 4. This suggested that measurement of the gating and ion transfer properties of the pacemaker current would be badly contaminated by [K + ] fluctuations. Direct measurements of [K + ] fluctuations with K + -selective microelectrodes have now been made during voltage clamp pulses in the canine Purkinje fiber (Cohen et al., in press). The cleft [K + ] between plateau and diastolic potentials can change by nearly a factor of two. A Hodgkin- Huxley current has a time course at a given potential which must be independent of prehistory, and also independent of the electrochemical equilibrium of the ion carrying the current. Both these requirements have been violated in previously reported data. For example, Figure 4 of Noble and Tsien (1968) reports the effects of changing the bathing [K + ] on the reversal potential of the pacemaker current i K,. They pulsed regularly from -68 to -100 mv before, during, and after the solution change. A dramatic change in the time course of the tail current at 68 mv can be seen on

9 K + FLUCTUATIONS IN EXTRACELLULAR SPACES/Cohen and Kline changing from 2.7 to 4 mm [K + ] o. The half time of decay of the tail current decreases by more than a factor of two at the higher potassium concentration. A dependence of the tail current on the prehistory of the test pulse potential is observed in Figure 1 of Cohen et al. (1976b). The holding potential was -78 mv and the preparation was both depolarized to -74 mv and hyperpolarized to -82, -84, and -86 mv. The half time of decay of the current recorded at the holding potential (which should be independent of test pulse potential) decreased by more than a factor of two for the most hyperpolarized test pulse as compared to that for the most depolarized test pulse. Possibly the most significant report involving the importance of accumulation and depletion of K + in sheep Purkinje fiber function concerns the ionic nature of the pacemaker current (Difrancesco, 1980; Difrancesco and Noble, 1980). These investigators suggested that the current reversal near the potassium equilibrium potential is not due to a K + -specific current, as postulated by Noble and Tsien (1968), but rather to the superimposition of a depletion current (of type 2C) upon a Na + /K + current activated by hyperpolarization. Evidence in favor of the hypothesis comes from application of Ba 2+ which removes background K + permeability, K + depletion currents, and also the current reversal near VK. Some difficulties with the new hypothesis have been reported (see Cohen and Falk, 1980; Cohen et al., in press). Nevertheless, if correct, K + fluctuations play an important role in Purkinje fiber automaticity. Atrial Muscle Many of the voltage clamp studies on atrial muscle concentrated on analyzing the plateau potassium currents (i x ) and their role, along with that of the slow inward current, in generating pacemaker activity and controlling the action potential duration. The analysis of the i x current system has involved the separation of a number of current components, all activating over the same voltage range. The analysis has usually been performed by the method of exponential splitting (Noble, 1976). The assumptions involved in such a procedure are the exponentiality of the early time course of each of the component currents, and their complete independence from each other. The difficulty is that, in many cases, three exponentials are needed to describe the decaying tail current accurately. Noble (1976) has presented convincing evidence that the slowest of these three components is due to the decay of K + accumulated during the test pulse in the extracellular spaces of the frog atrial trabeculae. Maughan (1973) varied the length and amplitude of the depolarizing test pulse, showing that some of the faster components of the tail currents may also be due to the decay of K + accumulation. Although this accumulation can be reduced by choosing trabeculae of narrow diameter, it cannot be eliminated for significantly positive or prolonged depolarizing pulses. In fact, it is thought to be responsible for a pseudo-reversal similar to that illustrated in Figure 3C (Noble, 1976). Since this accumulation of potassium will affect the time course and magnitude of the I x currents carried by K +, the efficacy of the exponential splitting procedure has been questioned (Attwell et al., 1979). Ventricular Muscle McGuigan (1974) presented the first detailed analysis of the difficulties that accumulation could create in the interpretation of the current-voltage relationships of sheep and calf ventricular muscle, although the existence of depletion on hyperpolarizing voltage clamp pulses was suspected earlier (Maughan et al., 1973). McGuigan found that much, but not all, of his experimental data could be explained if he assumed appreciable accumulation of extracellular K + near much of the myocardial membrane on depolarizing the membrane potential. The amount of K + accumulation increased with the duration of the depolarizing voltage clamp pulse. He supported this idea by comparing instantaneous current-voltage relations induced by experimental changes in the bulk [K + ] with the current-voltage relations induced by prolonged depolarizations. These current-voltage relations were not identical, but substantial similarities were observed. Convincing evidence for accumulation of K + in ventricular muscle was reported by McDonald and Trautwein (1978) for cat ventricular trabeculae. They were investigating a plateau conductance with significant K + selectivity. They found that the prolonged depolarizations necessary to investigate the channel kinetics produced changes in the reversal potential for the current. They suggested that the change in reversal potential was due to the extracellular accumulation of K + during the depolarizing voltage clamp pulses. This accumulation precluded detailed analysis of the properties of this K + conductance. More direct measurements of accumulation and depletion of K + during voltage clamp protocols in frog ventricular strips were made by Cleeman and Morad (1976, 1979a, 1979b). Using the single sucrose gap technique and an extracellular K + selective microelectrode, they showed convincing changes in the extracellular K + concentrations, both as a function of time during voltage clamp pulses to a given potential, and as a function of the potential (very little accumulation occurred in the negative slope region of the membrane current-voltage relationship where the K + efflux would be expected to be small). Conclusions Membrane currents recorded by the voltage clamp technique in atrial, ventricular, and Purkinje preparations have been interpreted in terms of the Hodgkin-Huxley formalism. Implicit in the appli-

10 FLUCTUATIONS IN EXTRACELLULAR SPACES/Cohen and Kline 11 0 mv 80 1 Mi n FIGURE 5 K + electrode trace (V K : double barrel electrode) during rapid stimulation shows summation of the accumulations during single action potentials generating slow accumulation of K* in the extracellular space. The lower trace shows depolarization of the resting potential during the same time period. The accumulation subsides and the membrane repolarizes when stimulation stops. The bracket indicating a 2 trim [K*] change represents an 8.5-mV response of the K* electrode. Elevation of I mm corresponds to the first 4.6 mv of this change. Thus the fk*] scale is not completely linear within the bracket. magnitude of K + accumulation at a constant stimulation rate (Kline and Morad, 1978; Kunze, 1977). C. Active Transport Exists to Restore Potassium to the Intracellular Compartment Even if a significant net K + efflux occurs during trains of stimulation, a steady state of elevated K () cannot exist in the absence of intracellular K + gradients. Such a steady state would imply a continuous net loss of K + from the intracellular compartment to the extracellular space and thence to the bathing solution. This ultimately must result in elimination of the transmembrane K + gradient. Therefore, an active transport process restoring K + to the intracellular compartment must exist. Kunze (1977) did in fact find that during prolonged stimulation, elevated K o eventually declined to pre-stimulus levels. Following such rapid stimulation, Kunze (1977) found periods of extracellular K + depletion (K,, undershoot). These depletions were larger for longer trains of stimuli at higher rates, both of which might be expected to increase Na +, (Cohen and Fozzard, 1979). Since the K,, levels were below pre-stimulus values during the K,, undershoot, the undershoot could not be due to stimulation of the external pump site by K,,, but must be due to persistent Na +, elevation. To test for the role of active processes in this K,, undershoot, Kunze (1977) performed a number of experimental interventions whose aim was to alter the activity of the active transport process. These interventions included exposure to deoxygenated Tyrode's solution, application of blocking doses of the cardiac glycoside ouabain, substitution of Li* for Na +, and removal of perfusate K +. All these maneuvers resulted in an elimination or significant reduction in the magnitude of the K o undershoot, and also prolonged the period of time during which stimulation maintained elevated levels of K o. It is useful to think of the point at which K, has subsided to control levels during rapid stimulation as a steady state. At this point, there is no net K + efflux from the extracellular space to the bathing solution, since K,, is not elevated. This also implies a balancing of active transport and passive membrane efflux of K +, even though both rates are likely to be different from their initial values (the efflux should be elevated due to the increased time spent at the plateau, whereas the pump rate is probably elevated due to Na + loading). On reduction of the stimulation rate, the K + efflux should be markedly reduced, while the Na + /K + pump rate returns only slowly to resting levels as Na + i is reduced. This necessitates a period of K o undershoot. In summary, there exists substantial evidence for each of the preconditions for measuring single and cumulative K,, fluctuations in the extracellular spaces of cardiac muscle. At this point, it is worth considering whether these necessary conditions are sufficient to produce the observed experimental results. To achieve this aim, we introduce a model of the frog ventricular trabecula. We will show, from known magnitudes of K + efflux and active pumping rate and a detailed consideration of the anatomy of the preparation, that it is possible to reproduce many of the experimental results commented on above. D. A Computer Model of K«Fluctuations in Frog Ventricular Trabeculae The Histology of Frog Ventricular Trabeculae and the Computer Model Correlates Page and Niedegerke (1972) provided a detailed histological description of the frog ventricular trabeculae. Briefly, a strip is organized into fiber bundles or trabeculae with mean cross-sectional areas of about 3000 junr (~60 /im diameter). Individual fibers are 3-5 jum in diameter, and are closely packed within the bundle. The average separation between fibers, the interfiber space (IFS), is about 150 angstroms. Each bundle is surrounded by a sheath that is several microns thick. Underneath the sheath is the subendothelial space (SES). This varies between 1 and 5 /xm in thickness. The extracellular space outside the trabeculae, labeled the extra-trabecular space (ETS), is contiguous with the bulk solution. The fraction of the total strip volume represented by each of these compartments is the following: interfiber space (IFS)-1%, subendothelial space (SES)-10%, extra-trabecular space (ETS)-15%, and intracellular space (ICS)-74%. The model includes two extracellular compartments (SES and ETS) obtained from the histological description and also divides the preparation into radial shells. Accumulation occurs in the IFS and SES spaces during single beats or short voltage clamp steps. During trains of activity, the K + would be expected to move continually from the SES

11 12 CIRCULATION RESEARCH VOL. 50, No. 1, JANUARY 1982 through the sheath surrounding the trabeculae and eventually cause accumulation in the ETS. The time constant of diffusional equilibration for K + across the sheaths was estimated from histology to be about 0.22 to 2.9 seconds (Page and Niedegerke, 1972). The time course of decay of the ETS accumulation is determined by the size of the preparation, and has a time constant on the order of tens of seconds to minutes. Diffusion in the ETS space would dominate the K,, record during trains of activity or long voltage clamps (Kline and Morad, 1976; Cleeman and Morad, 1976). A schematic of the model is shown in Figure 6 (see Kline, 1975, for more details). A description of the individual compartments in the model is given in the figure legend. The Model Function In order to understand the functioning of the model, the actual physiological events that underly the accumulation will be described and related to the arithmetic operations of the model. At the start of each beat, the membrane depolarizes. This increase in driving force results in a continuous net efflux of K + through the cell membrane. The K + efflux causes accumulation in the interfiber spaces (IFS). As the gradient develops in the clefts of the fiber bundle (IFS), K + starts to diffuse into the SES space with a flux proportional to the steepness of the gradient, and to the cross-sectional area presented by the ends of clefts adjoining the SES. The gradient in the cleft spaces of the fiber bundle is dependent on, among other things, the K + flux per unit area of membrane, the total membrane area of Cylindrical Preparation Trabecula Sheath I.C.S. all the cells divided by the IFS volume, the square of the tortuosity of the clefts, and the square of the diameter of the fiber bundle, and is inversely dependent on the diffusion coefficient of K + in the extracellular fluid. To the extent that the pump rate responds to increases in K,,, K* uptake increases as K,, increases. At some point, flux from the IFS into the SES equals the total membrane efflux (the passive K + currents minus the active uptake), and a stable radial distribution of K,, is established within the bundle. This distribution is altered by time-dependent K + currents or by changes in SES K,, levels that alter the IFS gradient and induce new relative differences in the effects of K, on pump rate and passive currents. The millimolar increases in K,, levels detectable in the SES with K + ISE's are a manifestation of the gradients existing in the clefts (IFS). The simulation starts with the measured changes in the SES because fluxes from the intracellular space to the IFS are not directly measurable and the IFS to SES fluxes are the experimentally obtained variable. As K o increases in the SES space, it diffuses through the sheath to the larger surrounding ETS space. (In the model, SES diffuses only into the ETS in its radial shell). The time constant of the sheath, along with the time dependence of the net K + currents and pump fluxes, determines the shape of the beat-to-beat accumulations. The time course of the single action potential decay of accumulation is dependent on the decay of SES K,, elevation into the ETS space as the concentration gradient in the cleft collapses due to a rapid decline in K + efflux on repolarization. Wedge FIGURE 6 The schematic for the computerized diffusion model shows a crosssection of a cylindrical preparation. Assuming axial and angular homogeneity, a wedge-shaped section is examined and divided into four compartments which transect four equal radial segments. These compartments represent extra-trabecular space (ETS) and are shown in the "Model" (lower right). One trabecula from the cylindrical preparation is artistically represented (lower left). The small curved arrow represents diffusion across the sheath and is also shown in the comparable location in the model as the small curved arrow between SES and ETS. The small straight arrows between (ICS) (intracellular space) and SES represent the outward transmembrane K* movement. The long arrow through the 4 ETS compartments and the compartment "B" (bath) represents the radially outward flow of K" through the spaces outside the trabeculae to the bathing solution.

12 10 CIRCULATION RESEARCH VOL. 50, No. 1, JANUARY 1982 cation of these equations is the independence of the time course of current flow from the history of potential or current flow. However, recorded membrane currents do demonstrate dependence on history (possibly due to extracellular [K + ] fluctuations). Because of this dependence, the analysis has since centered on means of analyzing the membrane currents in the presence of measurable amounts of K + accumulation and depletion. This analysis has often sought to present a self-consistent picture. To avoid this problem, voltage clamp studies will have to be performed on preparations of simpler geometry in which the contamination caused by accumulation and depletion processes is small. Potassium ion fluctuations do occur in many cardiac preparations and may well be important in normal and pathologic electrophysiology. Therefore, another aim of future experimenters should be to examine the role played by this K + accumulation and depletion in such important processes as automaticity and the control of the action potential duration (see next section). Part II Introduction In Part I we discussed voltage clamp evidence for cleft K + accumulation and showed the effects such accumulation would have on voltage clamp currents generated by gated and ungated membrane conductances in the presence of a RECS. We assumed, for simplicity, that the extracellular [K + ], K o, fell during diastole with a single exponential time course. In the sections that follow, we will examine some of the features that determine the actual time course of K o (no longer assumed spacially uniform) during and after the action potential, and also review results from studies in which K o fluctuations have been directly measured with ion-selective electrodes (ISE). In order for K o fluctuations to both occur and sum in the extracellular spaces of cardiac muscle during trains of action potentials, three conditions are necessary. First, the K + efflux during the action potential must be greater than that at the resting potential. Second, complete decay of the K o fluctuation to resting levels cannot occur before the end of diastole. Third, for a steady state to be reached, there must be an active transport process pumping the K + that has left, back into the cells. Diffusion can reduce the level of accumulation to bath levels of [K + ], KB, but it cannot restore the lost K + to the intracellular space. (However, during the voltage clamp protocols, the current electrode can act as a source of K + for the intracellular compartment.) Experimental evidence supporting the existence of these preconditions is discussed below. A. Net K + Efflux Occurs during the Action Potential In the first section of this review we discussed membrane current-voltage relationships that displayed inward rectification, and even one that possessed negative slope. If the negative slope is significant, it would be possible for a K + current to be reduced below resting levels during the plateau, even though a larger driving force for K + efflux exists. A decrease in membrane conductance at plateau potentials has been reported by Weidmann (1951) and more recently by Haas and Kern (1966) and Vereecke et al. (1980). It is, therefore, not surprising that initially there was some question about the time during the cardiac cycle that net efflux of K + occurred or even, indeed, if it occurred (Lamb and McGuigan, 1968; Langer and Brady, 1966). Experimental evidence now exists, suggesting net efflux of K + during both the plateau (phase 2), and rapid repolarization (phase 3) of the action potential. This evidence is available in several cardiac tissue types: in rabbit SA node (Kronhaus et al., 1979); in myocardium [frog ventricle (Kline and Morad, 1976) and rabbit atrium (Kunze, 1977)]; and in large canine Purkinje fibers (Kline et al., 1980). All the above studies were performed using K + ISE's in the extracellular space. These electrodes can respond to changes in K + activity with time constants as fast as 5 msec (Neher and Lux, 1973). B. The Net K + Efflux Can Sum in the Extracellular Space with Repetitive Stimulation The existence of net potassium efflux during the action potential does not necessarily imply that K o will increase with rapid stimulation. During the diastolic interval, diffusion and active transport processes could reduce K o to initial levels. However, if K o does not completely decay during diastole, residual elevation of K o will result in progressive summation. It has been known for some time that ventricular muscle depolarizes on rapid beating (Niedegerke and Orkand, 1966; Reiter and Stickel, 1968). Both of these groups suggested that this depolarization could be due to accumulation of K +. This sustained accumulation with activity was verified for frog ventricle (Kline and Morad, 1976), guinea pig ventricle (Kline, Cohen and Barton, unpublished observations), canine ventricle (Kupersmith, Dickerman, and Kline, unpublished observations), and rabbit atrium (Kunze, 1977). These changes in K o have been correlated with rate-induced depolarization (see Fig. 5). The magnitude of the accumulation of K + is a function of several factors. First, increases in stimulation rate, which increase the fraction of time spent at depolarized potentials, and decrease the fraction of time during diastole increase the amount of accumulation for a given period of stimulation. Second, factors that increase the accumulation per beat (for example, prolonging the action potential by decreasing the temperature) or decreasing the rate at which Ko decays (by blocking active transport or prolonging the diffusion distance by using thicker strips of muscle) all tend to increase the

13 K + FLUCTUATIONS IN EXTRACELLULAR SPACES/Cohen and Kline 13 For rapid beating, progressive summation of ETS accumulations occurs with a time course equal to the slow radial diffusion time for the entire preparation. K + moves from one ETS space to another between the radial shells. During prolonged undershoots following beating, the gradient analysis above is reversed. Transmembrane K + movement shows net influx which reverses the IFS gradient, eventually depleting SES and then ETS spaces. All K + diffusion occurs due to gradients in K + activity in the extracellular fluid and is proportional to the magnitude of the gradient. Total fluxes are also dependent on the cross-sectional area available to diffusion, (in the model, this basic law of diffusion is represented in all diffusion between compartments). For increased tortuosity and larger radii, diffusion takes longer and is larger [varying with the radius squared (Kline and Morad, 1978)]. With the stimulation of uptake processes, which are dependent on levels of K o (more pumping for larger K o ), the effect is to speed the apparent diffusion equilibration times and reduce the total accumulation at equilibrium (Kline and Morad, 1978). All these features of K o fluctuations are included within the model and correspond to observations made with K + ISE's in physiological preparations (see, for example, sample record, Fig. 5). The Model Output A pure diffusion model can reproduce the following features: (1) larger accumulations are seen deeper in the preparation but accumulations at all depths have approximately equal time constants (see panel A of Fig. 7); (2) larger and slower accumulations are seen for larger preparations (compare panels A and B of Fig. 7); (3) the envelope of the ETS and SES accumulations are nearly equal for a typical sheath time constant (see Fig. 7, panel A); (4) accumulation in slowly beating fibers during single action potentials are about equal at any radial depth and are comparable to measured values (see Fig. 8); and, accumulations due to beat to beat fluctuations sum during rapid beating (SES traces of Fig. 7). In addition to reproducing these qualitative effects, the model is quantitatively consistent with the histological findings and estimates of currents and pump fluxes obtained from other methodologies (Keenan and Niedergerke, 1967; Kline, 1975). FIGURE 7 Output of computerized diffusion model is shown as mm K,, vs. time. K,, values at the start of each simulation are set at 3 mm for all compartments. Vertical calibration bracket represents a 6mM increase in K o. Upper traces (panel A) show the concentration changes in the SES compartments for the four radial shells during a period of rapid beating (30 beats/min; action potential duration set at 800 msec). Compartments 1 through 4 correspond to the smallest to the largest accumulations. Fluctuations in the trace show beat to beat changes in K,,. Lower traces (panel A) show the K,, changes in the four ETS compartments during the same period of simulated beating. Again, smaller accumulations represent K,, changes in the more superficial shells. (When the equations are solved exactly, there are some deviations from single exponential behavior with a radially independent time constant. Howver, these deviations probably are not experimentally detectable.) Panels B and C show only the SES fluctuations in K,, for the four shells. In B, strip diameter is reduced from 750 fim (value set in panel A) to 550 ixm. In panel C, the panel A output is now altered by the addition of pump uptake fluxes. For simplicity, the Na + /K + pump was activated by assuming an elevation in [Na*7; at the start of the simulation. This elevation was sufficient to generate a K + inward flux of 1.8 pmol/sec per cm 2 at 3 mmk o, the initial [Na*]i was then not allowed to vary during the course of simulation. The pump rate was assumed to depend on K,, linearly. In reality, [Na*j, rises slowly during stimulation and then decays following stimulation. Models of this type should provide similar results, and can be easily tested once appropriate data becomes available. The trabecula sheath time constant for alt panels is set at 3 seconds. SES changes are appropriate for representing steady net K + passive currents (during the entire plateau) of 1 \iamp/cm 2 membrane area.

14 14 CIRCULATION RESEARCH VOL. 50, No. 1, JANUARY mm 1 mm 0 mv ±80 l mm 1 SEC FIGURE 8 The left panel shows the computer output of K o fluctuations for a single beat, using the same modeling parameters as in panel A of Figure 7. Top traces are SES output and bottom traces are ETS output. Only compartments in shells 1 and 4 are shown, since deviations at different radial depths are small and occur late in diastole for the parameters chosen. (Higher K o occurs in shell 4, the deeper shell in both cases.) The amount of ETS accumulation and the decay time of the falling phase of the SES K o are dependent on the sheath time constant, as well as the ratio of the SES and ETS volumes. Physiological recordings of K o fluctuations during three single beats are shown in the right panel. The top traces show action potentials for intracellular microelectrode recording measured simultaneously with double-barrel K + electrode recordings in frog ventricle (bottom trace). Pump flux* can be added to the model by considering two different sources of pump activation, Na + i and K o. If the pump is activated by Ko alone, it is possible to reproduce both a decrease in total accumulation and an increase in the rate of equilibration of both accumulation and decay of K +. However, one cannot reproduce the K o undershoot. If the model also includes a pump parameter for Na + i which "loads" during beating and decays following beating, one can reproduce the time course and magnitude of accumulation and also generate the undershoot (see panel C, Fig. 7). E. Physiological Effects of Ko Fluctuations It is well known that levels of bulk [K + ] markedly affect the resting potential, action potential duration, conduction velocity, and automaticity of cardiac muscle. We have shown in detail how changes in K o can occur during action potentials and affect membrane currents recorded during voltage clamp steps. We have also reviewed some of the experimental evidence from voltage clamp and ISE studies suggesting the presence of extracellular K + accumulation in in vitro cardiac muscle. In this section, we will discuss several instances in which K o fluctuations appear experimentally linked to resultant effects on membrane properties. It has been suggested by Weidmann (1956) and Carmeliet (1955) that accumulation occurring during the plateau was instrumental in bringing about Pump flux is modeled between the IFS and SES, not between the more physiological intraeellulaj space and IFS, because this is the experimentally obtained variable. repolarization of the cardiac action potential. By one such scheme (Carmeliet, 1955), residual K o could shorten the following action potential when it was activated earlier in diastole, for example, during an abrupt change in beating rate or an extrasystole. The higher levels of K o at the start of the second action potential could result in more outward current earlier in the plateau. Alternans in K o levels during such trials have now been measured with extracellular K + ISE's and the findings are consistent with these suggestions (Kline and Morad, 1978; Cleeman and Morad, 1979; Kline and Kupersmith, 1980). Stimulation of the vagus has dramatic effects on automaticity. In addition, net K + efflux should accompany the increase in potassium conductance induced by acetylcholine in the sinoatrial node (Hutter and Trautwein, 1955). Recent studies have demonstrated a post-vagal accumulation of K + in the rabbit sinoatrial node (Kronhaus et al., 1978). The decay of the accumulation is correlated in time to the decline of a phase of secondary slowing of the nodal rate (Spear et al., 1979). K + fluxes also have an additional role. The pumping of K + back into cells is associated with membrane currents. Vassalle (1970) predicted that, after a period of prolonged rapid stimulation, electrogenie currents associated with active transport of K + could suppress automaticity. There now exists substantial evidence for an electrogenic Na + /K + pumping mechanism in atrium (Glitch et al., 1978), ventricle (Daut and Rudel, 1980), and Purkinje fibers (Gadsby and Cranefield, 1979; Eisner and Lederer, 1980). The pumping mechanism, when

15 K + FLUCTUATIONS IN EXTRACELLULAR SPACES/Cohen and Kline 15 stimulated by Na + i; can deplete extracellular K + (Kunze, 1977). Kline et al. (1980) have now correlated the period of depletion with slowing deactivating membrane currents. General Conclusions K + fluctuates in the extracellular spaces of cardiac muscle during both voltage clamp pulses and normal beating. As shown in the first part of the review, these K + fluctuations will affect the time dependence of membrane currents. This distortion can be so severe that the measured time dependence may bear little resemblance to the gating kinetics. K + ISE's provide a direct measure of K + accumulation in larger extracellular spaces (see Part II). They have demonstrated significant increases in K o under a variety of experimental conditions in cardiac muscle in vitro. The future aims seem clear. For voltage clamp studies, it is necessary to seek preparations with little ion accumulation or depletion so that true membrane properties can be analyzed. Furthermore, experiments with K + ISE's can characterize the properties of K o fluctuations in the larger extracellular spaces under both physiological and pathological conditions. If we achieve these aims, we will be one step closer to predicting the effects of changes in K o on the basis of a knowledge of uncontaminated membrane properties. Acknowledgments We would like to thank Robin Falk and Dr. William Van der Kloot for constructive criticism of the manuscript. References Aimers W (1972) The decline of potassium permeability during extreme hyperpolarization in frog skeletal muscle. J Physiol (Lond) 225: Attwell D, Cohen I (1977) The voltage clamp of multicellular preparations. 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