Phase-Plane analysis of cardiac action potentials

J. ELECfROCARDIOLOGY, 1 (1), 31-42, 1968

Phase-Plane Analysis of Cardiac Action Potentials" BY NICK SPERELAKIS AND H. KEM.'ETH SHUMAKERt

SUMMARY

INTRODUCTION

Phase-plane recordings of action potentials were made from frog and cat ventricular muscle and cultured chick heart cells. In the phase plane, membrane potential (V) is plotted against its first time derivative (V); exponential functions can be identified as linear regions and their rate constants and equilibrium potentials measured. Two linear regions (a and b) occurred before the rising phase of the action potential reached the inflection point, and they intersected at about the expected threshold potential. A third linear region (c) occurred after the inflection point, and extrapolated to a potential near +10 mv rather than to the sodium equilibrium potential. The mean values for the maximum rate of rise of the action potentials (+ Vmax) were 90 and 130 v/ sec in frog and cat ventricular muscle (2SoC), respectively, and 10 v/sec in cultured heart cells (35°C). Linear correlations were found, within limits, between the magnitude of the action potential and + Vmax as the resting potential was varied by polarizing current and by K+ depolarization. An analysis for net ionic current derived from cable theory gives either inward current before threshold or large outward current after threshold, depending on whether region b or a, respectively, is taken as the initial passive region. However, since in cultured pacemaker cells, region a corresponded to part of the pacemaker potential, and since pacemaker cells fired action potentials without electrotonic spread ofcurrent from adjacent cells, region b must represent an active process beyond threshold.

Phase-plane methods have been used in two ways for the analysis of action potentials. In one use, a multi-dimensional mathematical system is reduced to a two-dimensional plane, as for example, reduction of the V, 111, II, and n parameters of the Hodgkin-Huxley equations to two variables s,6,7. In a second use, action potentials are displayed by plotting membrane potential. V, against its first time derivative, V, thus giving several advantages. First, in the phase plane, exponential processes appear as straight lines whose slope is equal to the rate constant of the process, and the point where dV/ dt ~ 0 gives the equilibrium potential", For example, the phase-plane trajectory during the rising phase of the skeletal muscle action potential was linear extending nearly to the inflection point, and threshold was assumed to occur at the point of departure from linearity; a second quasilinear segment occurring after the inflection point extrapolated to about the sodium equilibrium potentiaJl2-1s. Second, the compressed regions of the voltage-time recording are expanded maximally in the phase plane, giving a more sensitive indication of changes in the shape of the action potential. For example, alterations in the phase plane of dystrophic skeletal muscle were found which were not detectable in the conventional voltage-time plots 3 •4• Third, the phaseplane technique is also useful for the determination of membrane ionic current 12,13 because of the inherent difficulties of applying voltage-clamp techniques to cardiac muscle. Therefore, the phaseplane method was used to determine the ionic currents and the threshold potential. To aid in the analysis, pacemaker cultured heart cells were examined because the cell fires without propagation from adjacent cells and the threshold potential can be easily identified 1o,1l,2o.

* From the Department of Physiology, Western Reserve University School of Medicine, Cleveland, Ohio. Received Nov. 8, 1967. Supported by a grant from the U. S. Public Health Service (HE-HISS). This work was done by Dr. Sperelakis during his tenure of an Established Investigatorshipof the American Heart Association. His present address is: Department of Physiology, University of Virginia, Charlottesville, Va. A preliminaryaccount of some of this work has appeared». Requests for reprints should be addressed to Dr. Sperelakis. t Predoctoral Fellow of the U. S. Public Health Service(5-FI-GM-24, 513-02).

METHODS The ventricles from frog and cat hearts were isolated and opened so that microelectrode penetration could be made on the inner surface. The composition in 111M of the Tyrode solution used

32

SPEREAlKIS AND SHUMAKER

to bathe the cat ventricles was: 137 NaCl, 2.7 KCl, 1.8 CaCI2, 11.9 NaHCOs, and 0.42 NaH 2P04 ; the composition of the solution used to bathe the frog ventricles was: 100 NaCl, 2.0 KCl, 1.3 CaCh, 0.73 MgCh, 8.7 NaHCO s, and 0.31 NaH 2PO J • In some cases, the experiment was begun in K+-free Tyrode solution and then the external K+ concentration was increased by the addition of small volumes of 0.375 M KCI. The preparation of cultured heart cells and the techniques for intracellular recording have been described'"-". The ventricles from 8 day chick embryonic hearts were trypsin-dispersed and the cells cultured 3 to 21 days. The cultured cells generally consisted of a loose monolayer network of flattened, ribbon-like cells attached to the bottom of the culture dish. In all experiments, the cultures were maintained at 35°C. The capillary microelectrodes (flint glass) were filled with 3 M KCl, and had tip diameters of about 0.5 JJ. and resistances of 20 to 50 MQ. The reference electrode was an agar-Tyrode salt bridge immersed in the bath. Each electrode was connected to a calomel half-cell. Potentials were recorded with a capacity-neutralized, electrometerinput d-e preamplifier. The microelectrode was mounted in a bridge circuit (Fig. 1) which, when balanced prior to penetration, allowed passage of current while simultaneously recording membrane potentials", Constant-current pulses were applied across the bridge (points B and D) with a squarewave stimulator and stimulus isolation unit (SIU). Another stimulator was used to stimulate the muscles through external electrodes. The recorded potentials were filtered, buffered,

and differentiated. The two operational amplifiers of a Tektronix Type 0 plug-in unit were used for buffering and differentiation. The input capacitor, Cin, and feedback resistor, Rj, of the differentiator were adjusted to give the appropriate gain according to the equation: V o = -RICin' dVin/dt, where V: and Vi« are the output and input voltages, respectively, and RIC,'n is the gain. Since the input impedance of an operational amplifier is just that of the leading element because of the virtual ground at the grid, the differentiator has an input impedance of 1/2 7f' fe,'n, wherefis the signal frequency. Therefore, at high frequencies the differentiator would cause loading of the preamplifier and distortion of the action potential if a buffer were not used; the buffer amplifier (unity gain) supplied a constant input impedance of I MQ to prevent such loading. Since differentiation intensifies high-frequency noise, the signal was filtered before differentiation in some experiments. Optimal filtering was obtained by adjusting a capacitor substitution box until there was maximum filtering, but without significant attenuation of the differentiated signal. Photographs containing much noise (e.g., see Figs. 3, 4, 6, 8) are from experiments in which no filtering was done. However, even when filtering was done, the calculated 3 db point (0.707V) is greater than 850 cps, equivalent to a rise time of less than 0.3 rnsec, which is much less than the rise time of the cardiac and skeletal muscle action potentials. A sinewave input signal of 100 rnv peak to peak and at frequencies of 100 and 1000 cps was used to calibrate the system!', The potential, V, taken from the buffer output, OIFFERE\TI:'·:~

BUFfER

",

I.n

ocor-oio p'

SI.u.

~

~

CJoiA"'''E:LB

VERT A

liERT B

. HC"'Z A

HO",Z.

tNT. Tlr.!EI

BASE

OUAL. TIME BASE OSCILLOSCOPE

L---==:"':'=:"':::':~'::';":'''::'''':'''---'

Fig. 1. Schematic of bridge circuit and apparatus for phase-plane recording of cardiac action potentials. See details in text.

33

CARDIAC PHASE PLANE

was applied to the vertical deflecting plate of both beams of a Tektronix 565 dual time-base oscilloscope (Fig. 1). The horizontal drive for channel A was the internal time-base generator of channel A; that for channel B was produced by the first derivative of the membrane potential (V) obtained from the differentia tor. Thus, channel A displayed the variation of V as a time function (voltage-time plot), and channel B displayed Vas a function of V (phase-plane plot). In experiments with frog sartorius muscles, because of the brevity of the action potential, it was desirable to have the phase-plane beam on dim while the muscle was inactive to prevent overexposure of the camera film. A positive voltage pulse (of variable duration), supplied by Tektronix (160 series) pulse and waveform generators, was synchronized with the action potential (by adjusting a variable delay) and applied to the grid of the cathode-ray tube to brighten the phaseplane trajectory during the action potential. THEORY I. Meaning of linear regions oj the phase plane. Linear regions of the phase-plane plot (V plotted against dV/dt) correspond to exponential regions of the voltage-time plot because for an exponential process:

(1) (2)

V -

v.

=

Al'

dV dt

where k is the rate constant, V; is the equilibrium potential, and A is a constant. Equation 2 states that dV/dt is a linear function of V with a slope of k and that the intercept on the voltage axis gives V o - Thus, a linear region of the phase plane indicates an exponential process with a slope equal to the rate constant, and the point where dV/dt = 0 gives the equilibrium potential-"; dV/ dt equals zero at r = 0 for a positive exponential process (+k), and at I = co for a negative exponential process (-k) . II. Ionic current analysis from the phase plane. An equation can be derived from the HodgkinHuxley cable equation? which allows determination of ionic current (Ii) from the phase-plane plot l 2 :

(4)

d 2V 2R,Cm e'1. d V dt'!. = --a-- ~/t

+ 2R

iCm

e'!. .ls. Cm

a

(4a) (5)

(Sa) (6) (7)

m=

d[dV/df] dV Ii] -+C;

dV [dV m-=k dt p dt

-!.! C m

= dV

dt

IJ

[.!.!.!. kp

where e is the conduction velocity, (l is the fiber radius, R, is the myoplasmic resistivity, C m is the membrane capacitance, and III is the slope of the phase plane at any point. Since ionic current must be zero (I, = 0) during the initial passive part of the phase plane trajectory before threshold (although dV/dl is finite), and assuming Cm to remain constant, equation 7 reduces to (mp/k p - 1) = 0, where m p is the initial slope of the phase plane. Therefore, since k p = l1l p, k p is the slope of the initial passive .segment of the phase plane. III. Meaning ojarea ofthe phase plane. Since the dimensions of the phase-plane plot are volts (V) and volts/sec (VI I), and since charge, Q = V· C; = I·t, the following is t~ue:

(8)

Area/cycleJ= V'

~ sec

= .Q. .

c,

~ t

V I watts = C = m

c;-

Therefore, the area of the phase plane in V2/sec multiplied by the membrane specific capacitance (farads/ems) yields the energy expenditure per each action potential cycle (power) per unit area of membrane. RESULTS I. Phase-plane measurements. Frog and cal cardiac muscle. A typical cardiac action potential of frog ventricle recorded simultaneously in voltage-time plots (V, I) and in phaseplane plots (V, V) is shown in Figure 2. The rising phase of the cardiac action potential usually pro-

34

SPERELAKIS AND SHUMAKER

duced a phase-plane trajectory with three linear or quasilinear regions: two regions (a and b) preceeding the inflection point and a third region (c) following the inflection point. At the inflection point, the first derivative was maximum (+ Vrnax ) . Extrapolation of region c to the voltage axis gave an intercept near zero potential, a value consider-

.,.

...

EIIOa

.

.,

ably lower than the expected sodium equilibrium potential, E~a. The intersection of region a and region b gave a value very near the expected threshold potential (Vr). The rate constants of the linear segments are given in Table I. The mean values of+Vnux were 90 v/sec for frog ventricle and 130 v/sec for cat papillary muscle at 25°C (Table I). Action potentials and phase-plane records from impalements in two frog ventricles are shown in Figure 3 (A-B, and C-D). The rising phase of the action potentials are shown at fast sweep speeds in Band D for comparison with + Vmax • The small deflection of the

•

Vm

'm y )

-20 -

-4 0-

..

-

-I' - ' 00

E.

j

>

'0

::0

I 33

V (vo l "

., I

.. ~

60

u

-1mv

/ I OC )

Fig. 2. Simultaneous recording of a frog ventricular action potential in voltage-time and phase-plane coordinates. Assumed values for the K+ equilibrium potential (EK) and the sodium equilibrum potential (Ex.) are indicated. The maximum rate of rise (iT max), and three linear regionsof the phase plane (a, b, and c) during the rising phase of the action potential are labelled. Region a and region b intersect at approximately the expected threshold potential (VT), and region c extrapolates to considerably below Ex•.

Fig. 3. Typical action potentials and phase-plane records from two penetrations in frog ventricle (A-B and C-D). In Band D the rising phase of the action potentials are shown at fast sweepspeeds. In A and C, the gain of the phase planeis lO-fold larger to display the phase-plane contour during repolarization.

TABLE 1 Summary of phase plane measurements, and comparison of the measured values of k« and k o with the calculated value of k p Cardiac Muscle

+ iT max (v/sec) - iTmax

(v/sec)

k. (sec I) kb (seer") k c (sec-I) R, (fl-cm) em (IlI/cm') 8 (cm/sec) a (Il)

Calc. k p (sec'<)

Frog (25°C)

Cat (25°C)

Cultured Chick (WC)

Frog Skeletal Muscle (15°C)

90 (50-140) 1.5 (I.l-2. I) 750 (250-1250) 2000 (1000-2800) -1700

130 (60-160) 1.5 (0.9-2.2) 650 (25G-1250) 1500 (1000-2100) -1400

10 (5-80) 0.5 (0.3-1.5) 100 (50-400) 250 (l5G-1000) -300

350 (90-430) 80 (3G-IIO)

200 10 10 5 800

200 10 15 8 1125

200 10 5 5 200

4700 (3100-5100) -2600 250 6 70 40 3,675

35

CARDIAC PHASE PLANE

phase-plane during repolarization could be seen only at higher gains; therefore, in A and C the gain was IO-fold larger than that in Band D. Repolarization of the action potential is shown at slow speed in A and C for comparison with the maximum rate of repolarization (- Vm • x ) . - Vm • x was approximately 1.5 vfsec in both frog and cat ventricle (fable 1). The repolarization rate remained relatively constant during experimental procedures producing large changes in the depolarization rate. Cultured Heart cells. To aid the analysis for membrane ionic currents, the phase-plane analysis was applied to cultured ch ick heart cells because the threshold potential is more readily established in pacemaker cells 10 •20 • A valid current analysis requires that ionic current be zero until nearly threshold (local excitatory response) when the flow -of inward ionic current should begin. Typical records from the cultured cells are shown in Figure 4 Ae-C; C depicts the action potential shown in B at fast sweep speed. The phase-plane trajectories were remarkably congruous during a series of action potentials; this was true for slowrising (D) or fast-rising (E) action potentials from pacemaker and driven cells, respectively. Occasionally, the phase plane spontaneously changed without a detectable shift in resting potential or spike magnitude (F). In G, a +2.7 na depolarizing current pulse was applied to initiate firing in a quiescent. cell (dormant pacemaker); the initial action potential generally had a larger Vrnax than the succeeding impulses in the train. There

+

+ v.n.x

was a large range (5-80 v/scc) in the of cultured cells, even at a constant temperature of 35°C. The mean value for ms x was 10 v/sec, and that for - Vmax was O.S v/ sec. There was a slight deflection of the phase plane during the pacemaker potential, and a sharp deflection occurred when the membrane potential reached threshold. The region of the phase plane produced by the pacemaker potential corresponds to region a of Figure 2, whereas the linear region of the phase plane occurring after threshold corresponds to region b. Thus, region b of the phase plane must represent an active process and should not be used for determination of k p which, because of its derivation, must come from a passive region. The intersection of regions a and b should yield the equilibrium potential for the active process, i.e., the threshold potential VT • Skeietal muscle. The phase-plane trajectories of skeletal muscle (frog sartorius) action potentials were examined for comparison with those from cardiac muscle. In most of the observations, region a was either absent or obscured by region b (Fig . 5). However, in several cases (e.g., see Fig. 10), there was a definite region a present; it is possible that this region represented an end-plate potential although most impalements Wereat considerable distance from the nerve endings. In skeletal muscle kept at ISoC to reduce the force of contraction, the mean ms :< was 3S0 v/sec (range of 90-430) and - Vma:< ranged between 30

+V

+V

.0

Fig. 4. Phase-plane records and action potentials from cultured chick heart cells (3G-35°C).

Fig. 5. Recordings from a single impalement in frog sartorius muscle at 15cC showing effects of electrotonic polarization. The voltage and time calibrations apply to all photos. The V calibration in A (60 v/scc) applies to A and B and that in C (150 v/sec) applies to C-E. A: Control without current. B: a +7.3 na depolarizing current pulsewas applied. C: Control without current, lower gain. D-E: Hyperpolarizing current pulses of -12.3 (D) and - 21.8 na (E) were applied.

36

SPERELAKIS AND SHUMAKER

A

c

B

1 60

mv

E

D

F

1 60

mv

Fig. 6. Recordings taken from a single impalement in a frog ventricle showing effects of electrotonic polarization. The calibrations apply to all photos. In each photograph, a control recording with no current pulse applied is superimposed with a recording taken during application of electrotonic current (bridge off balance). A-C: With progressive increase in depolarizing current of +4.5 (A), +6.3 (B), and +8.4 na (C), there was a concomitant decrease in Vmax' D-F: With progressive increase in hyperpolarizing current of -8.4 (D), -12.3 (E), and -16.5 na (F), there was a concomitant increase in Vmn.

and 110 v/sec. At temperatures ranging between 20 and 25°C, Jenerick!' found mean values for frog sartorius fibers in phosphate buffer of 495 v/sec for Vm " ", 135 v/sec for - Vm m and 8700 sec- 1 for k p (k b) .

+

n. Variations ill resting potential. Electrotonic current polarization. The effects of electrotonic current on action potentials from frog sartorious muscle (Fig. 5) were similar to those obtained with cardiac muscle. The control without current is shown in A; Vmax was about 150 v/sec. In B, a depolarizing current pulse of 7.3 na caused a significant decrease in action

+

+

+

potential magnitude and in both Vma :< and - Vm ,:< ; Vmax diminished to about 90 v/sec. In C, a control phase plane without polarizing current is shown at reduced gain. Hyperpolarizing current pulses of -12.3 na (D) and - 21.8 na (E) produced large increases in Vma :", - Vmax , and action potential magnitude. These increases with hyperpolarization were greater than those obtained in cardiac muscle. Electrotonic current pulses were also used to produce changes in resting membrane potential of frog and cat ventricular muscle. Figure 6 shows a typical experiment of this type in cat ventricle in which the control and experimental phase planes

+

+

TABLE 2 Effect of electrotonic polarization on + Vmax in frog and cat ventricular muscle.

Polarization Depolarizing current (0.9-
Species

Number of observations categorized by change in + V max* Decreased Unchanged Increased

Total No. of Observations

Frog

25 (66%)

IO (26%)

3 (8%)

38

Cat

28 (80%)

2 (6%)

5 (14%)

35

Frog

28 (24%)

42 (36%)

46 (40%)

116

Cat

28 (29%)

22 (23%)

46 (48%)

96

* The numbers in parentheses indicate the percentage of the total

number of observations.

37

CARDIAC PHASE PLANE EFFECT OF ELECTROTONIC CURRENT ON

Vma • • Z~

% CHANGE

IN

Vmo•

."

.., .~

~

.~

~

.~

-~

~

-~

-8

.~

-4

.:

0

.Z

,4

.6

"''''''1 Fig. 7. ElTect of electrotonic current on the maximum rate of rise (1'max) of action potentialsfrom frog ventricle.The percent decreasein Vmax is plotted downward and the increase,upward. The current was varied from a hyperpolarizing current of - 25 na to a depolarizing current of +6 na. The data were averaged from all frog experimentsand the number of observations averagedfor each point isindicated on the graph. Depolarization usually caused a decrease in l'max, whereas hyperpolarization produced more variable changes. POLARIZING CURREN1IXIO·'

are present in each photograph. Depolarizing current decreased Vm a:", as shown in Ar-C for +4.5, +6.3 lind +8.4 na, respectively. Hyperpolarizing current generally increased + VIIl!lX' as shown in D-F for -8.4, -12.3, and -16.5 na, respectively; however, this effect was not as consistent or as prominent as that produced by depolarization. In the frog and cat experiments combined, + .Vm • x increased during hyperpolarizing pulses in 43 % of the observations, lind decreased during depolarizing pulses in 73 %. These data are summarized in Table 2. Vmax proIn Figure 7, the percent change in duced by electrotonic current polarization in frog ventricle has been averaged for each current step between - 25 na hyperpolarizing current and t- 6.3 na depolarizing current; the encircled numbers represent the number of observations averaged. Fewer observations were made at the larger current steps, and therefore they should be weighted less heavily. It can be seen that small depolarizing currents produced a decrease in Vmax. In contrast, hyperpolarization produced more erratic results and required larger currents for an Vm..x with electrotonic effect. The decrease in depolarization and the increase often observed with hyperpolarization may be due to a change in the availability of sodium carriers or channels as a function of the shift in membrane potential", Potassium depolarization. To complement the

+

+

+

+

Fig. 8. ElTect of potassium depolarization in three experiments on frog ventricular muscle (A-C, D-F, G-I). The time calibration in A, D, and G apply to A-C, D-F, and G-I, respectively. The voltagecalibrations apply to all photos. The 1T calibration was changed as depolarization progressed and is given in each photograph. As [K+]o was progressively increased in each experiment,the celldepolarized,action potential duration decreased, and + l'max decreased drastically. current-polarization experiments the external K+ concentration, [K+]o, was elevated to produce depolarization. The results of three typical K+·depolarization experiments on frog ventricular muscle are shown in Figure 8 (A--c, D-F, G-I). In all cases there was a large decrease in VIIl!lX as the membrane depolarized. For example, Vm• x decreased from 66 v/sec in A ([K+]oof 2 11IM) to 1.4 v/sec in C ([K+]o of 30 11IM); note the change in gain in each photo. Similar decreases occurred in the experiments illustrated in D-F and G-I. Elevated [K+]o produced the same effects as electrotonic depolarization (decrease in t- Vm • x , overshoot, and spike magnitude) but, in addition, caused a decrease in action potential duration. Both elevated [K+]o and depolarizing current appeared to produce the decrease in Vm • x by a change in membrane potential. Figure 9 (solid lines) shows the correlation between + Vm3x and the action potential magnitude in 3 experiments; the statistical data from these experiments are given in Table 3, where r is the correlation coefficient and SIl is the standard error of estimate. Note the steeper slope for the cat (higher Vm 3X for a given magnitude of action potential). The intercept on the abscissa predicts loss of excitability at an action potential magnitude of approximately 50 III\'; cessation of firing occurred generally at about 50 to 60 III~" In some cases, injury of the impaled cell by the microelectrode may result in

+

+

+

+

38

SPERELAKIS AND SHUMAKI:R

TABLE 3

Statistical correlation of action potential magnitude with during K+ depolarization.

+ fT

+V

m ax

Experiment

Frog 1

Frog 2

Cat

N

21 60-110 5-85

83 45-120 5-65

135 4G-1l5 5-135

0.78

0.71 19 v/sec

0.97

Magnitude (Range) (mv) + V max (Range) (v/sec) Duration (Range) (msec) r Sv

17

v/sec

9 v/sec

max

and with duration

Duration Frog 2 Frog 3

19 51-121

33 51-121

70--580 0.85 61 msec

50--510 0.84 71 msec

fROG

140

VlNlRlCLE 120

CULTURED

fROG

CHICK HEART

SKELETAL

600

.'

100

'00

iz 0

!;i

80

400 ~

at eo

;:: 300 ~

~

..

0

-'

0

~ Z

00

200

~

.. u

20

100

\.

....

/'\. :

S-f'Olfl

t",-"",.

\'. ..

ACTION POTENTIAL MAGNITUDE (m,)

Fig. 9. Correlations of Vmax (solid lines)and action potential duration (broken lines) with action potential magnitude during K+ depolarization. The regions of the regression lines which did not fall in the observed range are dotted. The correlation coefficient (r) are given for each line. Each frog ventricle was recycled three times; i.e., the [K+]o was varied up and down three times. The statistical data for these experiments are listed in Table 2. slight underestimation of the action potential magnitude. These data show that action potential magnitude, or more correctly resting potential, has a major influence on li'max. Less Nar carriers or channels may become available as the membrane is depolarized", Figure 9 (broken lines) shows the correlation of action potential duration with magnitude during depolarization by high [K+]o. The correlation curves in these experiments indicate that, at the value of action potential magnitude where excitability is lost (50-60 IIIV), the duration should be between 50 and 100 msec. The last action potential elicited before loss of excitability usually fell in this range. The duration of the action potential always de-

+

I

/ I

Fig. 10. Phase-planeanalysisfor ionic current. The currents obtained from phase-plane plots of action potentials from frog ventricle (25°C), cultured chick heart (35°C) and frog skeletalmuscle(15°C) are shown. Solid lines indicate the current produced when k p is determined from region a; dotted lines when k p is determined from region b. creased with K+ depolarization, whereas the duration was either unaffected or, in the case of cultured cells'", actually increased by electrotonic depolarization. This indicates that the action potential duration is not a function of membrane potential and spike magnitude per se, but more likely is influenced by s«. That is, with K+ depolarization, gK is increased, whereas with electrotonic depolarization, s« is either unaffected or decreased (anomalous rectification)",

III. Current analysis. The result of applying the ionic current equation derived from cable theory (equation 7) to the

CARDIAC PHASE PLANE

rismg phase of the cardiac action potential is shown in Figure 10. The capacitance values assumed were 10 Ilf/cm2 in frog and cat ventricle and cultured chick heart cells, and 6 Ilf/cm2 in skeletal muscle. In the current analysis of skeletal muscle, the phase plane shown was chosen because it had a slow region a for comparison with cardiac action potentials. If the initial slope (region a) is selected for k p (as the derivation of the equation requires), the ionic currents produced are shown by the solid lines. In all cases, a positive (outward) current occurs after threshold. This outward current was relatively large compared to the peak inward current: a) for frog ventricular fibers the peak outward current was +0.50 ma] cm2 at a membrane potential of -47 mv compared to a peak inward current of -1.25 mafcm2 at - 20mv, b) for cultured chick heart cells, the peak outward current was +2.5 ma/cm2 at - 35 mv compared to an inward peak of -1.30 mafcm2 at +5 mv, and c) for frog skeletal muscle fibers, the outward peak was +0.88 ma/cm 2 at -43 mv compared to an inward peak of -1.5 mafcm' at +5 mv. If the slope of region b is selected for k p , the ionic currents produced are indicated by the dotted lines. In this case, a small negative (inward) current occurs before threshold in all three types of cells: a) for frog ventricle, an inward current peak of -0.033 ma/cm2 at -70 mv compared to 11 second larger inward peak of -0.76 ma/cm2 at -20 mv, b) for cultured chick heart cells, an inward peak of -0.0065 mafcni: at -47 mv compared to a second larger inward peak of -0.083 ma/cm2 at +5mv, and c) for frog skeletal muscle, an inward peak of -0.03 ma/em' at - 60 mv compared to a second inward peak of -0.8ma/cm2 at 5 mv. The peak inward current before threshold is about 5 % of that after threshold. The two currents (solid and broken lines) intersect at the inflection point of the action potential where the slope of the phase plane (m) is equal to zero; the value of current at the inflection point is equal to - C m Vmax (equation 7). Although the peak inward current occurs at about the same value of membrane potential in both current analyses, both analyses seem to be incorrect because neither current, i.e., an outward ionic current after threshold or inward ionic current before threshold, should occur.

+

DISCUSSION The findings in skeletal muscle3,4,12-1S indicate that the phase-plane trajectory departs from the

39

resting value with an initial passive linear region which extends to a threshold potential just preceding the inflection point of the action potential. In most of the present experiments, the action potentials from skeletal muscle also had only a single exponential process before the inflection point. However, if there were really two exponential processes present, but with similar rate constants (k", = k b), they would not necessarily be resolved. The Hodgkin-Huxley equations theoretically predict a passive exponential foot of the action potential", and this has been observed experimentallyH.22.2s. The idea that threshold occurs near the inflection point was first advanced by Cole and Curtiss, They presumed, as a result of their a.c, impedance studies on squid axon, that the foot of the monophasic action potential up to nearly the point of inflection represents a purely passive discharge into the region from the preceding active region. Schmitt" also assumed threshold to be at the inflection point and deduced the value of membrane capacitance necessary for passive capacitive current to account for all the membrane current up to the inflection point. This hypothesis that threshold potential is near the inflection point requires a rather large depolarization of approximately 50 mv to reach threshold. However, Cole, et aU found that a depolarization of only 5-10 mv was needed for nerve excitation in digital computer computations of the Hodgkin-Huxley equations. Jenerick'! also found that the degree of depolarization at the inflection point in skeletal muscle was larger than the threshold potential at the point of stimulation. In cardiac muscle, there were generally two linear regions on the phase plane prior to the inflection point: passive region a (membrane resistance constant), and region b perhaps active (Na" conductance increasing as a function of membrane potential). Determination of the equilibrium potentials is complicated when several exponential processes start at different times. For a positive exponential process (+k), the equilibrium potential occurs at t = 0 for the process; hence the equilibrium potential for the passive region a is the resting potential. In determining the equilibrium potential for region b, however, the point where t = 0 for that process can only be approximated since the second exponential process does not begin at the same time that passive depolarization begins. Therefore, the equilibrium potential for region b (the threshold potential if region b

40

SPERELAKIS AND SHUMAKER

is active) could be the value obtained by extrapolation to the voltage axis, or it could be the potential at the intersection of linear segments a and b. In either case, this interpretation that region b is active gave a threshold potential for cardiac muscle which seems to be more reasonable than if threshold were near the inflection point. Furthermore, in true pacemaker cells which fire action potentials without spread of current from adjacent cells, threshold potential is well established as being near the point where the pacemaker potential (slow diastolic depolarization) gives rise to the rapid upstroke of the action potentiapo.20. Therefore, in such pacemaker cells,region b ofthe phase plane must have been active, and threshold could not be near the inflection point, but rather is near the intersection of regions a and b. Van der Kloot and Dane" also found that the initial exponential foot of the action potential for frog ventricular muscle (mean rate constant of 344 sec.-I) terminated after a relatively small depolarization of only about 13 mv and at a potentiallevel close to the expected threshold potential. The observation that the quasilinear region c following the inflection point does not extrapolate to the sodium equilibrium potential in phase-plane plots from frog cardiac muscle could be explained if other ions beside Na" carried current near the peak of the action potential. This interpretation is consistent with recent findings in frog ventricular muscle that magnitude of the action potential overshoot was markedly less sensitive to lowering of [Nav], than predicted from the Nernst relationship and was sensitive to [Ca++]ol8 and that tetrodotoxin, which blocks Na" activation in nerve, suppressed the rate of rise of the action potential without affecting the overshoots. In cat ventricular muscle and cultured chick heart cells, region c extrapolated more closely to the expected EXa. In skeletal muscle, regionc does extrapolate to near the expected Elia13, which indicates that Na" carries most of the inward current during the entire rising phase of the action potential. The measured phase-plane slopes for frog and cat ventricular muscle ranged from 250 to 1250 seC l for region a and from 1000 to 2800 sec:" for region b. Calculated values for k p , assuming R, = 200 n·cm, em = 10 Jlf/cm 2, radius equal to 5 X 10-4 em (frog) and 8 X 10-4 em (cat), and e = 10 cm/sec (frog) and 15 cm/sec (cat), were 800 sec l in frog ventricle and 1125 sec- l in cat ventricle (Table 1). In the cultured cells, the slope

of region b ranged from 150 to 1000 seer"; in pacemaker cells, region a had a slope approximately lOa-fold lower than that of region b, whereas in driven cells, region a had a comparatively large slope of about 100 sec-l. The calculated value for k p , assuming R, = 200 Q-Cln, C m = 10 Jlf/cm 2, radius = 5 X 10- 4 CIIl, and e = 5 em/sec, was 200 sec-l. It is obvious from Table 1 that comparisons of the measured values for k« and k b with the calculated values for k p can not be used to distinguish whether or not region a or b is passive. The rate constant measured for region a in frog ventricular muscle is somewhat larger than that of 344 sec- l (range of 222 to 625) reported by Van der Kloot and Dane", Current analysis using equation 7 requires that cable theory be applicable to the tissue and that the value for k; be measured from a passive region of the phase plane where ionic current is zero (Ii = 0), which most likely is the initial region a. The results of such a current analysis are shown in Fig. 10. When the value of k p is obtained from region a, a large outward current appears during region b (after threshold). This does not agree with present concepts of ionic theory which require that, after attaining threshold potential, there be an inward current during region b. When k p is obtained from region b, region a yields a small inward ionic current; the presence of such inward current before threshold also does not agree with ionic theory. Jerierick'! and Evans & Schottelius3 •4 found no correspondence to region a in their phase-plane records from skeletal muscle, and therefore they used the slope corresponding to b to determine the value of k p ; in their analyses, no ionic current occurred until the phase-plane trajectory departed from linearity just prior to the inflection point (assumed to be VT ) . The experiments with cultured pacemaker cells show that region b is an active region, and therefore should not be used to determine k p • Since pacemaker cells are not activated by propagation 20, k p has no meaning. Furthermore, conduction velocity within the confines of one cardiac cell, pacemaker or nonpacemaker, must be very fast and the whole surface of the membrane fires nearly simultaneously, because of the evidence that there is little or no fall-off of electrotonic potential within one cell but there is a sharp falloff into contiguous cellsl6. Therefore, passive region a of the phase-plane could be related to a junctional transmission process operating between

CARDIAC PHASE PLANE

contiguous cells; the fact that region a is exponential need only mean that the foot of the junctional potential is exponential. Therefore, unrestricted cable theory may not apply to cardiac cells and could account for failure of the current analysis. REFERENCES 1. Cole, K. S., Antosiewicz, H. A., and Rabinowitz, P.: Automatic computation of nerve excitation. J. Soc. Indust. Appl, Math. 3: 153-172, 1955. 2. Cole, K. S., and Curtis, H. J.: Electric impedance of the squid giant axon during activity. J. Gen. Physio!. 22: 649-670,1939. 3. Evans, T. C., Jr., and Schottelius, B. C.: Phase portraits of dystrophic and nondystrophic mouse muscle-fiber action potentials. Am. J. Physio!. 208: 724-731, 1965. 4. Evans, T. C., Jr., and Schottelius, B. C.: Phase portraits of normal mouse muscle-fiber action potentials in high calcium. Am. J. Physiol, 208: 732-736, 1965. 5. Fitzhugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1: 445-466, 1961. 6. Fitzhugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Diophys. 17: 257, 1955. 7. Fitzhugh, R.: Threshold and plateaus in the Hodgkin-Huxley nerve equations. J. Gen. Physiol, 43: 867-896, 1960. 8. Hagiwara, S., and Nakajima, S.: Tetrodotoxin and manganese ion: effects on action potential of the frog heart. Science 149: 1254-1255, 1965. . 9. Hodgkin, A. L., and Huxley, A. E: A qu~ntlta­ tive description of membrane current and Its application to conduction and excitation in nerve. J. Physio!. 117: 500-544,1952. 10. Hoffman, B. E, and Cranefield, P. E: Electrophysiology ofthe Heart. New York, McGraw-Hill, 1960. II. Hoshiko, T., and Sperelakis, N.: Prepotentials and unidirectional propagation in myocardium. Am. J. Physiol, 201: 873-880, 1961. 12. Jenerick, H.: An analysis of the striated muscle fiber action current. Biophys. J. 4: 77-91. 1964.

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13. Jenerick, H.: Ionic currents in membrane of active muscle fiber. Nature 191: 1074-1076, 1961. 14. Jenerick, H.: Phase plane trajectories of the muscle spike potentia!. Biophys. J. 3: 363-377, 1963. 15. Jenerick, H.: The effects of calcium on several electrical properties of muscle membrane. In Proceedings of the First National Biophysics Congress, New Haven, Yale University Press, 1957, pp. 377-389. 16. Lehmkuhl, D., and Sperelakis, N.: Electrotonic spread of current in cultured chick heart cells. J. Cell. Compo Physiol, 66: 119-133, 1965. 17. Lehmkuhl, D., and Sperelakis, N.: Transmembrane potentials of trypsin-dispersed chick heart cells cultured in vitro. Am. J. Physiol, 205: 12131220,1963. 18. Orkand, R. K., and Niedergerke, R.: Heart action potential: dependence on external calcium and sodium ions. Science 146: 1176-1177, 1964. 19. Schmitt, O. H.: Dynamic negative admittance components in statically stable membranes. In Electrochemistry in Biology and Medicine (T. Shedlovsky, editor), New York, John Wiley and Sons, Inc., 1955, p, 91. 20. Sperelakis, N., and Lehmkuhl, D.: Effect of current on transmembrane potentials in cultured chick heart cells. J. Gen. Physio!. 47: 895-927, 1964. 21. Sperelakis, N.: Electrophysiology of cultured chick heart cells, in Electrophysiology ami Ultrastructure of the heart, edited by T. Sano, V. Mizuhira, and K. Matsuda.. Bunkodo Co., Tokyo, 1967. 22. Tasaki, 1., and Hagiwara, S.: Capacity of muscle membrane. Am. J. Physiol, 188: 423-429, 1957. 23. Taylor, R. E.: Cable theory. In Physical Tecllll!ques ill Biological Research (W. L. Nastuk, editor), New York, Academic Press, 1963, pp, 219-262. 24. Weidmann, S.: The effects of the cardiac membrane potential on the rapid availability of the sodium carrying system. J. Physio!. 127: 213-224, 1955. 25. Van der Kloot W. G., and B. Dane.: Conduction of the action potential in the frog ventricle. Science, 146: 74-75, 1964.