J. Phyaiol (1978). 283, pp. 263-282 With 8 text-figure Printed in Great Britain

263

AN ANALYSIS OF THE CABLE PROPERTIES OF FROG VENTRICULAR MYOCARDIUM

BY R. A. CHAPMAN AND C. H. FRY* From the Department of Phy8iology, University of Leicemter, Leicemter, LEJ 7RH

(Received 23 November 1977) SUMMARY

1. The passive and active electrical parameters of frog ventricular myocardium have been measured. 2. The cytoplasmic resistivity has been determined by following changes in the resistance of a micro-electrode on penetration of a cell. 3. Unidimensional cable analysis using direct and alternating currents revealed the presence of a single time constant attributed to the surface membrane. 4. Longitudinal impedance measurements indicate that a second time constant is present in the intracellular pathway. 5. The results indicate that the resistance between cells is low so that action potentials can propagate from cell to cell by local circuits. 6. A three-dimensional cable analysis has also been carried out and compared to a simplified mathematical model which is presented in an Appendix and which closely approximates the experimental situation. INTRODUCTION

Mammalian ventricular trabeculae behave as linear cables when stimulated under certain conditions (e.g. Kamiyama & Matsuda, 1966; Sakamoto, 1969; Sakamoto, & Goto, 1970; Weidmann, 1970). Thus, despite their complex syncitial structure, an equivalent electrical network for these trabeculae can be derived by applying the equations of Hodgkin & Rushton (1946) to the measured results. When stimulating currents are applied by a point intracellular source, the tissue responds in a way more consistent with its syncitial structure (Tanaka & Sasaki, 1966). We have made measurements in frog ventricular tissue as it offers an interesting comparison due to its simpler structure. The frog ventricle has no T-system (Staley & Benson, 1968) offering an easier estimation of the surface cell membrane area. Furthermore, intercalated disks are not as prominent as in the mammalian heart, appearing as isolated regions (Marceau, 1904; Sj6strand, Andersson-Cedergren & Dewey, 1958). Barr, Dewey & Berger (1965), however, have demonstrated that electrical coupling between cells does occur. Preparations taken from frog heart have been extensively studied using the sucrose-gap voltage-clamp method but because the electrical properties have been * Present address and address for reprints: Cardiothoracic Institute, 2 Beaumont Street, London, WIN 2DX.

264 R. A. CHAPMAN AND C. H. FRY so poorly defined it is difficult to assess the quality of voltage-clamp in these preparations. Thus, in the light of the criticisms that have been made of this particular voltage-clamp method (Johnson & Lieberman, 1971) it seemed important to undertake a study of the electrical properties of frog cardiac tissue. The results that we have obtained are qualitatively similar to those observed in mammalian ventricular cells. METHODS

General. Ventricular trabeculae from the heart of the frog Rana pipiew, 1-3 mm long and 200-500 /Am diameter, were used for most experiments. Strips of ventricle, 5-10 mm long and 0-5-1 mm diameter were used for measurement of the resistance to the longitudinal flow of current and longitudinal impedance. Frogs obtained from the suppliers (Mogul-Ed, Oshkosh, Wisconsin) were kept at 4 0 until required. All preparations were bathed in a frog Ringer of the following composition: NaCl, 117 mM; KCl, 3 mm; NaHPO4, 0-8 mm; NaH2PO,, 0-2 mM; pH 7-3 (Chapman & Tunstall, 1971). CaCl2 was added from a stock solution to give a final Ca concentration of 0-1-1-0 mm. Experiments were performed over a temperature range of 20-25 'C, with less than 1 TC variation during any one experiment. Exponential plots of results were reduced to a straight line by a specially designed programme on a Wang 720 C programmable calculator that gave information about the slope, intercept with the ordinate and the correlation coefficient. Error function (erf) values were obtained from the Handbook of Mathematical Functions (Abramovitz & Stegun, 1968). Detailed information concerning the imaginary Bessel coefficient Ko was obtained from the U.S. National Bureau of Standards booklet (1952). Recordings. Differential intracellular recordings were made with conventional 3 m-KCl-filled micro-electrodes. A Burr-Brown FET-input instrumentation amplifier (3153/25) with input capacity neutralization, was used as a preamplifier. It had an input resistance of 1011 Q, a frequency response flat to 100 kHz and 3 db down at 150 kHz. A 20 MCI resistor was placed in parallel with the input to allow measurements of the micro-electrode resistance. Measurement of the cytoploamic residtivity. The resistance of a micro-electrode (REL) changes when it is placed in solutions of different resistivities (Schanne, 1966). REL was measured first with the micro-electrode tip in the bathing solution, then with the tip inside the cell and finally repeated with the tip in the bathing solution. If the two values of REL before penetration and after withdrawal were the same, it was considered that the micro-electrode had not become blocked or broken in the process of inserting it into the cell and that the results were therefore reliable. The micro-electrode was calibrated after these extracellular and intracellular measurements by recording REL in KCI solutions of different resistivities (other potassium salts yielded similar calibration curves). The input resistance, Rinp, is considered to be in series with RBL when the micro-electrode is intracellular. Rip was measured separately with two micro-electrodes, one for passing current, the other for measuring voltage. The two micro-electrode tips were separated by less than 5 ,sm, the smallest distance that could be measured. This value of R. was then subtracted from the intracellular value of REL and the cytoplasmic resistivity value was read off the calibration curve. Measurement of the resistance to the longitudinal flow of current. This was measured by the method of Weidmann (1970), where half of the preparation was inactivated with tetrodotoxin (TTX) (4 x 10-7 g/mn.). In separate experiments with ventricular strips this concentration of TTX was found to block propagated action potentials. Cable analysis. Measurement of the cable properties and the conduction parameters were performed in a dish with a single rubber partition similar to that described by other workers (Kamiyama & Matsuda, 1966; Sakamoto, 1969). The preparation was passed through a hole in the rubber partition slightly smaller than the diameter of the preparation. The rubber membrane thus separated the bulk bathing fluid into two chambers. Direct or alternating polarizing currents were passed between the two chambers through large Ag/AgCl electrodes with a 2-7 MQi series resistor to ensure constant-current conditions. No attenuation of the current passed by these electrodes was observed over the frequency range used.

GABLE PROPERTIES OF HEART

265

Longitudinal impedance measurement&. These were made in a three-compartment dish, with two rubber partitions enclosing a central chamber which was 5 mm wide. The preparation was pulled through the rubber partitions so that 0-5-1 mm protruded into each end chamber. The central chamber was filled with paraffin oil and the outer chambers were filled with Ringer solution. Pt black electrodes were placed in the outer chambers and the whole system was used as one arm of a Wien bridge (Wayne-Kerr Autobalance Universal Bridge, B642). Values of the sample capacitance, C, and conductance, G (= I/R-resistance) were measured over a frequency range 80 Hz -40 kHz. It was difficult to balance the bridge beyond these extremes. Drift of the bridge readings at any one frequency was serious enough to necessitate compensation. Readings were thus referred to a fixed frequency (1591-5 Hz, the internal source frequency of the bridge) after each frequency measurement. The source of such drift may have been heating of the preparation due to the flow of current. After completion of the measurements the preparation was cut from the central chamber leaving the holes occluded. The remaining capacitance was probably due to the dish and was important only at frequencies greater than 20 kHz. To calculate the impedance of the preparation the resistance and capacitance of the Pt black electrodes were subtracted from the values obtained with the preparation in plaoe at each frequency (see Schwan (1963) for details) by use of the eqns. (1) and (2).

Re -RP, = C.- 1/w2R.2Cp,

R = C

(1)

(2) where the subscript 8 refers to the bridge values and p to the electrode values. The unsubscripted values refer to the sample. w = 2ff times the measuring frequency. Finally the small capacitance of the chamber was subtracted from the sample capacity as they can be assumed to lie in parallel. The impedance of the preparation can be expressed in the form of eqn. (3) Z

=

(R+jX),

(3)

where X is the reactance and j the complex operator J -1. The results can be expressed in the form of an impedance locus diagram which gives a separate dispersion for each separate time constant in the system. These were calculated from the sample resistance and capacitance by the relations

RG

______

G2 + (wOC)2;

G- + (O(C)2

Intracellular application of current. Two independent micro-electrodes were used, one to inject direct current into a cell and the other to record the electrotonic potential at various interelectrode distances. The current electrode was filled with 2 M-K citrate and the amount of current passed was controlled by a guard resistor of 2 kMfl. The distance between the two micro-electrode tips was calculated from the depths of the micro-electrodes in the trabecula, their angle of penetration and the surface distance between the two micro-electrodes. The error in calculation of this distance was estimated at 5 pm. RESULTS

Determination of the cytoplasmic resi8tivity, Rc. The cytoplasmic resistivity, measured in twenty experiments by the single micro-electrode method, was calcalculated to be 282 + 19 Q cm (s.E. of mean) at 23 'C. Table 1 shows no significant difference between the results obtained with 1 M or 3 m-KCl-filled micro-electrodes

(P

>

0-1).

From eight determinations the input resistance, Rinp, obtained from the slope of the current-voltage relationship had a value of 320 + 120 kQ (s.E. of mean). The re8i8tance of the total intracellular pathway, Ri. One half of a ventricle strip

266 R. A. CHAPMAN AND C. H. FRY was perfused with TTX-containiing Ringer whilst the other half was perfused with normal Ringer. The two extracellular compartments were in communication at the centre of the strip. Fluid was drawn away here by a piece of cotton wool on the underside of the strip. The preparation was placed under paraffin oil so that the perfusing fluids formed a layer about 1t00 #om thick around the strip. Monophasic action potentials were recorded between an extracellular site at the inactivated end and an intracellular or extracellular site at the active end. The ratio of the two action potentials is equivalent to the ratio of the intracellular and extracellular resistances. From nine strips a ratio of 6-70 + 1-09 (s.E. of mean) was obtained. TABLE 1. Summarized results of the determination of the cytoplasmic resistivity obtained with micro-electrodes filled with 1 M or 3 M-KCl

Solution in micro-electrode 1 M-KCl 3 M-KCI All results

Temperature (OC) 24 22 23

Cytoplasmic resistivity (Q cm) 324 ± 30 (s.E. 7 expts.) 265 + 23 (s.E. 13 expts.) 282 + 19 (s.E. 20 expts.)

The specific resistance was obtained from the relative cross-sectional areas occupied by each resistive component. A figure of 25 % was used for the extracellular space of frog ventricular strips based on inulin space measurements by Niedergerke (1963) and an assumed density of 1-05. However, there was also a layer of Ringer of significant thickness surrounding the strip which would contribute to the total extracellular space. The thickness of this layer was measured under a binocular microscope and was of the order of 100 #m. The effective extracellular space is therefore about 44 % giving an intracellular to extracellular cross sectional ratio of 1-27: 1 (56 %:44 %). The specific resistivity of the fluid in the extracellular space was assumed to be that of Ringer and was measured as 69 fl at 25 'C.

Ri can be calculated from eqn. (6) (a.p.)l

rI

Ri.ao

(6) (a.p.)o ro Ro.aj' 0 where (a.p.)i are the intracellular and extracellular action potential heights respectively and a,,. are the cross-sectional areas of the spaces. Ri was calculated to have a value of 588 + 95 Q cm (S.E. of mean). Calculation of the intercellular junction resistance, Rd. If the total intracellular resistive pathway consists of two series components (the cytoplasm and intercellular junction resistances) then the two preceding measurements allow a calculation of the intercellular junction resistance to be made. Each individual cell is considered to be a cylinder 131 ,um long and 5,um diameter (von Skramlik, 1921) abutting at its ends to others of similar size and the whole end region is assumed to be the site of intercellular coupling. Subtracting the value of the cytoplasmic resistivity (282 Q cm) from the total intracellular resistance (588 K2 cm) yields a value of 4-0 Q cm2 for each intercellular coupling. Electrotonic potentials. Trabeculae were placed in a single partition dish and stimulated with hyperpolarizing current pulses via Ag/AgCl electrodes. Electrotonic potentials were recorded intracellularly at various distances from the partition. The amplitude of the steady-state electrotonic potential, on a logarithmic scale, was plotted against the distance from the partition (Fig. 1 A). The linear relationship

CABLE PROPERTIES OF HEART A

B

t (msec)

Pulse ht. (mV) 20 -5

4-

3

10 _

8

6

*

267

0

-

2

1

-

00

4

_

I I I I I I I I 100 200 300 400 500 600 100 200 300 400 500 600 Distance from stimulating partition (tim)

L

I

I

C X=027

X=055

X=080

X=1 10

X=1 60

5 mV[

20 msec

Fig. 1. Experiment for determination of the cable parameters A and Tm A is a semilogarithmic plot of the steady-state electrotonic potential, Vmax, against the distance from the partition. B shows the linear relationship between the time for the electrotonic potential to reach V,,,,/2 and the distance from the partition. C shows tracings of the electrotonic potentials used to construct the curves of A and B. The points are obtained from the one-dimensional cable equation

V.,t

=

V.0I e-I +

erf(X

-VT)]+ee1 -erf(

+VT)]}

where A and 7,, are expressed in terms of the dimensionless variables X (= x/A) and T (= t/Tr) - x is the distance from the stimulating partition. Experiment was performed at 25 0C in 1 mM-Ca2 .

R. A. CHAPMAN AND C. H. FRY 268 indicates that one-dimensional cable analysis is applicable to this tissue in this arrangement. The space constant, A, obtained from these plots was 328+22 jzm (s.E. of mean, eleven trabeculae). The membrane time constant, Tm, can also be calculated from these electrotonic potentials because the half-maximum potential travels at a constant velocity of 2A/Tm (see Hodgkin & Rushton, 1946). Fig. 1 B shows a plot of the time taken to reach half-maximum amplitude against distance from the partition. The slope of the plot is Tm/2A. In seven trabeculae where A had been measured a value of 4-15 + 0 55 msec (s.E. of mean)-was obtained for Tm. A final test for the applicability of one-dimensional cable theory to this experimental situation is seen in Fig. 1 C where experimentally obtained electrotonic potentials are compared with the one-dimensional cable equation (Hodgkin & Rushton (1946), eqn. (4.1)). The continuous lines are traces of the recorded potentials. The derived values of A and Tm have been inserted into the cable equation and the points are the calculated voltages at various times. By a curve-fitting procedure applied to each electrotonic potential a value for Tm can be obtained if the distance from the partition as a fraction of A is known for each transient. From theoretical plots of the cable equation at different distances from the origin, the ratio of V/Vmax at time Tm can be calculated; from eighteen randomly selected electrotonic potentials a value of 3-7 + 2-0 msec (S.D. of an observation) was obtained for Tm. Seasonal varatiom in the space constant. A tendency of A to decrease with the period of time that the frogs had been stored at 4 0C in our laboratory was noted. This trend, equivalent to a decrease in A of 2-6 Cm/day, however, showed a low correlation coefficient of only 039.

The conduction parameters. The conduction velocity (0) measured in eleven trabeculae had a value of 13-0 + 1-2 cm/sec (s..E. of mean). The distance over which 0 was measured was never less than 500 /zm and was usually 1 mm or more. At smaller distances the variability increased probably reflecting the imperfect alignment of cells within the trabeculae along the longitudinal axis. The time constant from the foot of the action potential, Ta.p. was measured by plotting the rise of potential as its logarithm against time. From a total of sixty-two randomly selected action potentials from eleven trabeculae a value of 1-69 + 0'11 msec (s.E. of mean) was obtained for Ta.p.. Calculation of membrane resistance, Rm, and membrane capacitance, Cm. Estimates of the membrane parameters, Rm and Cm, can be calculated from the cable parameters A and Tm and the conduction parameters 0 and Ta.p.. Rm can be obtained from eqn. (7).

2R1A2

Rm

a

(7)

where a is the cell radius (2.5 1sm, Page & Niedergerke, 1972). Rm has a value of 5-06 kQ cm2, using the experimental values of R1 = 588 Q cm and A = 328 j#m. If the trabecula is treated as a one-dimensional cable, then Tm is given by eqn. (8)

Tm

=

CmRm.

For Tm = 3-7 msec then Cm = 0 73 #uF/cm2.

(8)

269 PROPERTIES ou s ll3J OF J' HEART JvX 26 Cm can also be calculated from eqn. (9) for the foot of the action potential (i.e. Cole & Curtis, 1938). Cm= 2R As (1 + T ) (9)

CABLE k~i./.ZX

in this case Cm is calculated to be 0.51 1tF/cm2. Pas8age of alternating current along the trabecula. The behaviour of the trabeculae to direct currents has shown that, under suitable methods of stimulation, a trabecula can be treated as a one-dimensional cable. This is shown in the ladder network of Fig. 2. Here Y1/dx is the membrane admittance per unit length which we have assumed to be a single time constant. Z2/dx is the intracellular impedance per unit length, which we have assumed to be purely resistive. The extracellular impedance has been assumed to be negligible as it would be for a trabecula immersed in a large volume of Ringer. Analysis of the responses of the trabeculae to sinusoidally varying current at different frequencies is another test of the applicability of one-dimensional cable equations to the tissue when current is passed by means of an isolating

Fig. 2. A one-dimensional cable represented by a ladder network of lumped impedances. Y1/dx represents the membrane admittance per unit length. Z2/dx represents the intracellular impedance per unit length. The extracellular impedance is assumed to be negligible. One purpose of the analysis is to determine whether Z2 has purely resistive or some reactive properties.

partition. For a single time constant model as described above the amplitude of the recorded potential, V, depends on the frequency, f, of the stimulating current, I, and the distance, x, from the point of stimulation according to eqn. (10) (cf. Tasaki & Hagiwara, 1957). V

T1

~A

An 2441l+ (27Tf7-m)2]' m/A=

(10) 10

where m ={1 + V/[I + (2lffTm)2]}

42

and IZI is the modulus of the impedance. Fig. 3 shows a plot of the intracellularly recorded peak-to-peak voltage versus frequency at two distances from the partition. The continuous lines were calculated from eqn. (10) using values of A, rm and RI previously determined and the plots were normalized around the experimental value of 1 Hz. The fit between eqn. (10) and the results suggests that a single time

R. A. CHAPMAN AND C. H. FRY 270 constant model is suitable to describe the properties of the trabecula under these conditions. Falk & Fatt (1964) have reported that the variation of IZI with frequency is a relatively insensitive test to distinguish between a one or a multi-time constant system. However, it is still useful to express the data in terms of a single time constant model which closely approximates to the present system. It will also be useful for techniques where such a model is often assumed, i.e. in the single and double sucrose-gap voltage-clamp arrangements. At frequencies where 27rf'rm > 1

-Jf~xirrim+1o212 lVo f .r1

+

(11)

so that a plot of log (V. Jf) vs. Jf should give a straight-line plot of slope -x . Jlrricm. Fig. 4 shows that a straight-line plot is obtained at frequencies where 27Tffrm > 1. The lines through the points were drawn by eye. With Ri 588 Q cm a value of 0 70 + 0*29 #zF/cm2 (S.E. of mean) is obtained for Cm. 40

-

_3*0

E

20

-

1.0

0

1.0

I

l

I

20

3-0

4.0

log freq. (Hz) Fig. 3. The frequency response of the inside-to-outside admittance measured at two different distances from the stimulating plane. * = 100l m, A = 260 jsm. The curves are plots of eqn. (10) using values of R, = 588 Q cm, A = 328 ,um and T,,, = 3-7 msec and normalized around the experimental value obtained at 1 Hz. Experiment performed at 20 0 in 1 mm-Ca2 .

Whenf = 0 eqn. (10) reduces to the familiar form for d.c. cable analysis. Thus, the quantity m is a measure of the shortening of the space constant with increasing frequency. The a.c. space constant, Aa c, is thus related to the d.c. space constant, Ad.c., according to eqn. (12). (12) Aa&c = Adc/M = in which the membrane resistance, rm, has been replaced by the membrane impedance,

(kzjm/ri)2

Z m.

Fig. 5 shows an experiment where Aa.c. was measured at different frequencies.

271 CAABLE PROPERTIES OF HEART The continuous line is a plot of 1r/m, calculated on the basis of a single time constant of value 3'7 msec and normalized around the experimental value of Aa.c. at 600 Hz. The longitudinal impedance. In this experimental arrangement current is constrained to flow along the intracellular pathway to a greater extent due to the large extra20r

CA c

Cu

Cu

-

.0

D

I-

101A

10)

a)

0 0

1 1 100

50 s,

freq. (Hz)

Fig. 4. A plot of eqn. ( 11) (see text for details) for calculation of the product Ri Cm, the same data and symbols are used as in Fig. 3. The straight lines are drawn by eye and only represent the points when 2nffm > 1. 500

400 E

: 300 c Cu C

0

uO'Cu

200

(I)

100

0 0

10

20

30

X freq. (Hz) Fig. 5. Decrease in A.C with increasing frequency of measurement. The line is a plot of 1/m with Tm = 3-7 msec and normalized around the value of A,,c. obtained at 600 Hz. Experiment performed at 20 0C in 1.0 mM-Ca2 .

272 R. A. CHAPMAN AND C. H. FRY cellular resistance of the oil gap. The passage of current over the surface membrane near the rubber membranes gives one time constant to the system which shows as a discrete dispersion on a - X vs. R plot. The purpose of the following experiments is to see if such a model is adequate or if other reactive components are present. The longitudinal impedance, Z, has been shown by Cole & Baker (1941) to be proportional to the square root of the membrane impedance, Zm, as in eqn. (13) Zm (= rm+jxm) OC Z2 (= R2-X2+j2RX). (13) Fig. 6 shows a Z and Z2 plot of a single time constant system. For Tm = 3 7 msec the characteristic frequency, f*, has a value of 43 Hz as in eqn. (14) wO*r = 2irf*r = 1. (14) 43 Hz

0 50

> 025

0 0

025

0.50 R

0.75

1 0

0*4

02

0

0

025

050 075 1.0 R Fig. 6. Impedance-locus diagrams of a single time constant system. Part A is for a simple CR circuit. This is equivalent to Z2 in eqn. (13). Part B is for a cable and is equivalent to Z in eqn. (13). The example has a time constant of 3-7 msec (characteristic frequency 43 Hz) similar to that of the surface membrane.

Fig. 7 A shows a typical longitudinal impedance complex-plane plot for a ventricular strip measured from 80 Hz to 40 kHz. Over such a frequency range one major dispersion is evident, although at lower frequencies a further dispersion may be present. It is unlikely that the major dispersion is due to the surface membrane as the characteristic frequency is too large. The low frequency dispersion is more likely to be due to the surface membrane. If this is so the effect of the surface membrane at high measuring frequencies will be negligible. The high frequency

-CABLE PROPERTIES OF HEART 273 locus can thus be calculated from data obtained at these high frequencies by comparison with theoretical longitudinal impedance loci. The phase angle, 0, can be obtained from the high frequency limb as it approaches the R axis at an angle of 0/2'. The result of removing the low frequency dispersion is shown in Fig. 7B. In order to obtain information about the elements contributing to such a locus, an impedance locus was constructed from the above data, according to eqn. (13). Fig. 7 C shows the result of such a conversion; the two limbs of the locus approach the R axis at an angle of '0. The capacitance, C, of the system contributing to such a dispersion is given by eqn. (15) )*T A

A

=

RE*C = 1, 1K

5000 -0 _.__

(15) .*400

_*__

200

_._

10 K 0

33,

L.96--

58 60

70

80

I

82

R (()x 103)

B

K ~~~~~1

B 5000,

400

~~~~~~~200

10K 0

3066~

60 R

C

8000

-

70 (jx 1 03) 0

0 K
or a potential measured on the £ axis will be Z2/o times greater than a potential measured at an equidistant point perpendicular to this axis. Thus, some of the observed scatter seen in the results at any one interelectrode distance will be due to this anisotropy. DISCUSSION

Membrane parameters. The membrane capacity has been estimated between 0 50 and 0*70 isF/cm2 and is similar to but slightly lower than estimates from experiments with mammalian ventricular preparations (e.g. Sakamoto, 1969; Sakamoto & Goto, 1970; Weidmann, 1970). However, these authors did not take into account infolding of the membrane due to the T-tubular system, absent in the frog myocardium, which would lead to an over-estimation of Cm. Page & Niedergerke (1972) have described invaginations on the surface of frog ventricular cells which will also increase the effective surface area in this tissue. Thus, as emphasized by Weidmann (1970), reliable measurements of membrane parameters cannot be made until the topology of the surface membrane is properly understood. The same problems apply to the estimation of the specific membrane resistance, Rm, but the value of 5-06 kQcm2 is again similar to those found in mammalian preparations. The membrane resistance in the hearts of summer frogs may be larger because Brown, Noble & Noble (1976) have shown that the space constant in summer frogs is twice that of winter frogs. The intracellular pathway. The intracellular pathway has been divided into two series components for the purpose of this analysis, the intracellular contents and the junctions between neighbouring cells. The cytoplasmic resistivity has been measured by following changes in the resistance of a micro-electrode, REL, when it was transferred from outside to inside a cell. It is likely that the observed changes in REL were due mainly to the surrounding fluid entering the electrode tip so that changes in REL reflected differences in the resistivity of the surrounding fluid. The value of the measured input resistance, Rinpq is likely to be an underestimation of the true value because of some possible separation of the current and voltage micro-electrode tips; we could not estimate distances less than 5 /m. It is therefore important to assess the contribution of R1np on the change of REL on penetration of a cell. The value of 320 k Q for Rinp was generally less than 5 % of the observed changes in REL on penetration. This value was similar to 200-600 kfQ quoted by Tanaka & Sasaki (1966) using double-barrelled micro-electrodes where there was no electrode separation. With a model similar to theirs we estimate that our value of Rinp should not exceed 1 MQ. This would reduce the cytoplasmic resistivity by about 15 %. The fact that there was no significant difference between the cytoplasmic resistivity as determined with 1 M or 3 M KCl-filled micro-electrodes suggests that any effect of RIp was small. This is so because the absolute value of REL and the change of REL on penetration of a cell are much larger with 1 M-KCl-filled microelectrodes. A further source of error arises from the possibility that the micro-electrode

277 CABLE PROPERTIES OF HEART becomes blocked or blunted on penetration. This possibility was minimized by comparing the values of REL before and after penetration of the cell while in the extracellular phase. Similarity of the two values would indicate that damage to the micro-electrode was minimal. The value of 282 Q cm for R, at 23 0C may be compared with a value of 230 Q cm calculated from data given by Weidmann (1970) for mammalian ventricle at 37 0C. If a temperature coefficient of 1-37 per 10 'C change in temperature is assumed for Re (Hodgkin & Nakajima, 1972, for skeletal muscle) then at 37 'C a value of 179 Q cm is obtained for frog ventricle. Schanne, Thomas & Ceretti (1966) obtained a value of 120 Q cm for rat atrial cells with a technique similar to that used in this work. Our results suggest that electrical current between contiguous cells flows via low resistance electrical couplings. The space constant is long (328 jtm) compared with the length of a single cell (131 gim). Electrotonic interaction was observed at distances of nearly 300 /sm even with a point intracellular source of current, in which the voltage distribution would decline more quickly than with a planar current source. Such a distance would include at least two intracellular junctions. The value of 328 ,tm for the space constant, A, is comparable with values from mammalian myocardium (e.g. Sakamoto & Goto, 1970; Weidmann, 1970; Bonke, 1973) especially as A is proportional to the square root of the fibre radius for onedimensional spread. The frogs used were winter adapted and the value is similar to that obtained by Brown et al. (1976) in frog atrial preparations. The value of 4 0 2 cm2 for the specific intercellular junctional resistance is based on a much simplified model of the junctions. It is close to the value derived from a similar model for mammalian cardiac muscle (Weidmann, 1966). These results indicate that adjoining cells in the heart are electrically coupled and that the action potential is propagated by local circuits. Brink & Barr (1977) have obtained a comparable value for the specific resistance of the septum in the median giant axon of the earthworm. Theoretical studies by Heppner & Plonsey (1970) have also shown that for two cells separated by a gap of 80 A (similar to the gap width of toad desmosomes, see Nayler & Merrillees, 1964) electrical transmission can occur if the two opposing membranes have a resistance of less than 4 Q cm2. A true value for the specific resistance could be calculated if the ratio of the surface area of the coupling sites to the total surface membrane area was known, but no such studies have yet been made. The model used for the above calculation means that the junctional sites occupy nearly 1 % of the total surface area. Any decrease in such a percentage would decrease the specific resistance. The high frequency dispersion observed in the longitudinal impedance measurements resembles those observed in other multicellular preparations, i.e. Purkinje fibres (Freygang & Trautwein, 1970), moth myocardium (Stibitz & McCann, 1974) and smooth muscle (Ohba, Sakamoto, Tokuno & Tomita, 1976). The capacity contributing to this dispersion is unlikely to be due to a significant portion of the surface membrane capacity lying in series with a large resistance. This situation is found in skeletal muscle (Falk & Fatt, 1964) and Purkinje fibres (Fozzard, 1966), where square pulse analysis gave a spuriously large value for Cm because this additional surface membrane was not taken into account. In frog myocardium, the

R. A. CHAPMAN AND C. H. FRY 278 value of 0-7 ,tF/cm2 for Cm, obtained by square pulse analysis, is much closer to the measured value for mammalian heart when the total surface membrane area has been accounted for (Mobley & Page, 1972). Another possibility favoured by some authors is to place this longitudinal capacitance at the junctions between cells (see Ohba et al. 1976). If such a situation is assumed to exist in frog myocardium, the data of Table 2 yield values of 0 93 Q cm2 and 370 ,tF/cm2 for the junctional resistance and capacitance. This resistance is lower than the value obtained from the difference between the total longitudinal resistance and the cytoplasmic resistance. The supposed capacity is very large and to be explained would require an electrolytic capacitor, perhaps formed by a space where ions can accumulate or be depleted. Fatt (1964) has proposed a similar large capacitance at the mouths of the T-tubular system in skeletal muscle. In two experiments changes in the osmolarity of the bathing solution did not affect the longitudinal capacity (C. H. Fry, unpublished), a result that would not be expected if the junctional spaces are accessible to the bulk extracellular solution. Other possible explanations for this large longitudinal capacitance exist. It is possible that the area over which capacitative coupling between cells occurs may be larger than the area over which resistive coupling occurs. It is also possible that the connective tissue surrounding the cells may have capacitative properties (but see Freygang & Trautwein, 1970). The time constant of the junctional complex is about 10 % of the surface membrane time constant, a larger proportion than observed in the other tissues (see Ohba et al. 1976). However, the d.c. and a.c. unidimensional cable analyses revealed only one time constant in the inside-to-outside admittance and it can be concluded that such analyses are not sensitive enough to observe a second time constant of this size. A similar a.c. cable analysis applied to skeletal muscle by Tasaki & Hagiwara (1957) also revealed only a single time constant, although a second component was later identified by FaLk & Fatt (1964). Consequences for voltage-clamp studies. Several recent studies both theoretical and experimental have questioned the quality of the voltage control achieved during a voltage clamp applied by means of a sucrose-gap to multicellular preparations like frog heart muscle (Johnson & Lieberman, 1971; Ramon, Anderson, Joyner & Moore, 1975; Poindessault, Duval & LUoty, 1976; Tarr & Trank, 1976). The reports indicate that a large extracellular resistance in series with the cell membrane is responsible for the loss of voltage control during the passage of inward current. The relative smallness of the space constant found in our experiments as well as its steep dependence on the frequency of the applied current indicate that spatial control during voltage-clamp may also be difficult to achieve for all but the delayed outward currents. We would like to thank the M.R.C. for financial support and Dr Paul Fatt for much helpful

discussion. APPENDIX

Derivation of the equation describing three-dimensional current flow from a point source of current in a syncitial system. There are two extremes for a syncitial system. Either the side connexions are so far apart that only flow along the fibre axis is

279 CABLE PROPERTIES OF HEART important (see the solution by Eisenberg & Johnson, 1970, for this case) or they are so close together that there is an equal probability that current will flow in any direction. The latter case will produce a series of spherical or ellipsoidal isopotentials radiating from the source of current (i.e. the whole system is treated as a continuum). It has been assumed that the intercalated disks form the region of electrical connexion between cells and as they occur widely over the cell surface the latter model may be more realistic. Consider, in the first instance, a sphere of purely resistive conducting fluid of conductivity C- (i.e. 1/ri -the total intracellular resistance) which is homogenous and isotropic throughout. With no current loss from the system V2V = 0. However, we must consider the situation where there is current loss from the system to a zero potential infinite in extent. This loss is proportional to the voltage and area. Assuming the isopotential has an infinitesimally small thickness, then V.i = -ikV, (A 1) i = oE, (A 2) E = grad V = dv/dp (VV), (A 3) where V is the divergence (div) of the current vector, i flowing through a medium of conductivity, a-, and k is a measure of the current loss. Thus,

divgrad V = V.VV = V2V (A4) so that Laplace's equation becomes V2V = (k/o) V = l' V. (A 5) If we make the further assumption that V is only a function of p (the distance from the point of stimulation), so that there is no angular dependence, then Laplace's equation in spherical co-ordinates reduces to d

(2d)

=

k1V

(A 6)

or

d2V 2 dV

dP dpl-+ p T--k'V

=

0.

With the boundary condition that as p -%- o then V 0 the voltage at any point p is given by , V (P) (A 7) Up where VR is the potential at some small distance R, for practical purposes the outer wall of the micro-electrode (note that when there is no current loss, i.e. V' = 0, then the solution reduces to V (p) = VRR/p, obtained by solution V2V = 0). A similar but more exact solution has been published by Jack, Noble & Tsien (1975), with a representation for the constant i', which produces an identical voltage distribution. It has been demonstrated that conduction in one axis- in cardiac muscle may be different from the other two axes (Sano, Takayama & Shimamoto, 1959; Clerc,

R. A. CHAPMAN AND C. H. FRY 1976). Thus the isopotentials, may be pictured as a series of prolate spheroids. This anisotropy is conveniently described by the conductivity tensor, eqn. (A 8) £ 0 0 C. ° U/cm (A 8) Tij =

280

which will modify eqn. (A 7) according to the axis of measurement, so that in Cartesian co-ordinates d2V d2V d2V +=k V. (A 9) dy2

Ex

If the following transformations are made x

,

Zr

=

XX, x, y

y

d2V+ d2V

d2V

z-z

x

then eqn. (A 9) becomes kV

(A 10)

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