Simulation analysis of conduction of excitation in the atrioventricular node

J. theor. Biol. (1987) 126, 275-288

Simulation Analysis of Conduction of Excitation in the Atrioventricular Node SEIlCHI URUSHIBARA, MITSUO KAWATO,t KAZUO NAKAZAWA AND RYOJI SUZUKI

Department of Biophysical Engineering, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan (Received 21 April 1986, and in revised form 17 November 1986) The excitation conduction in the atrioventricular node was simulated based on the spatially discrete model of the heart proposed in an earlier paper (Kawato et al., 1986). We constructed a model system composed of the atrium, the atrioventricular node and the Purkinje fiber. Coupling coefficients between these tissues were quantitatively estimated from experimental data on size and membrane capacitance of the three kinds of cardiac cells. We found the following three important features in the simulated excitation conduction along the atrioventricular node. First, shape of action potential was found to be different at different locations of the atrioventricular node although the membrane properties were assumed uniform through the atrioventricular node. Our analysis suggests that the difference in the action potential waveforms observed by Paes de Carvalho & De Almedia (1960) can be ascribed to the electrical influences of the atrium and the His bundle on the atrioventricular node. Second, when the excitation wavefront invaded the atrioventricular node from the atrium, a step was observed in the depolarization phase of the action potential at the atrioventricular node neighboring with the atrium. Janse found a similar step in the real experiment (1969). It is revealed that this step is caused by termination of the junctional current which flows from the atrium to the atrioventricular node. Finally, we found that the conduction velocity measured near the boundary between the atrium and the atrioventricular node was lower than that in the middle part of the atrioventricular node, which is in accordance with the experimental observation by Scher et al. (1959).

1. Introduction The atrioventricular node (AV-node) which lies in between the atrium (A) and the Purkinje fiber (P)--ventricle (V) system, is the cardiac tissue which conducts the atrial excitation to the ventricle. It is known that a slow conduction of excitation in the AV-node facilitates the efficient pumping of the blood by creating a delay between the atrial and the ventricular contraction (Brockman, 1963). In an earlier paper (Kawato et al., 1986), the mechanisms of the slow conduction in the AV-node were investigated. One of them was the electrical influences of the Purkinje fiber on the AV-node excitation, since the Purkinje fiber lies adjacent to the AV-node. To investigate the conduction of excitation in the AV-node, therefore, the electrical interactions between the AV-node and neighboring tissues must be taken into account. The electrical influence of the Purkinje fiber was discussed in t To whom correspondence should be addressed. 275 0022-5193/87/110275 + 14 $03.00/0

© 1987 Academic Press Inc. (London) Ltd

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the preceding paper, but we did not consider those of the atrium. In this study, we constructed an AV model system connected with the atrium as well as with the Purkinje fiber. The following problems were investigated in this model. (i) Peas de Carvalho & De Almeida (1960) classified the AV-node into three regions referring to action potential patterns during the normal conduction and the retrograde conduction. The three regions were called the "AN (Atrio-Nodal)", "N (Nodal)", and "NH (Nodal-His)" regions, respectively. The AN region is located in the neighborhood of the atrium, the N region is in the center strip of the AV-node, and the NH region is in the neighborhood of the His bundle (HB). The differences in the action potential patterns among these three regions may be caused by different properties of an individual excitable membrane. Another possibility is that the difference of the action-potential patterns may be due to the electrical influences of the atrium and the HB even though the individual cell has a uniform property. In the present study, we examined the latter possibility by a simulation where membrane properties of the AV-node were assumed to be uniform. According to Janse (1969), the excitation which occurred in the atrium invaded the AV-node from the region termed crista terminalis (CT). He reported that in normal conduction, or in the case where current stimuli were applied to the CT, a step was observed in the depolarization phase of the action potential obtained from the AV-nodal cells on the border of the atrium. We examined whether or not this feature was also observed in our simulation. (ii) Scher et al. (1959) reported that the conduction velocity in the AV-node measured near the boundary between the atrium was lower than that in the middle of AV-node. We examined whether this phenomenon was reproduced in the simulation, and investigated its possible mechanisms. In the preceding paper, coupling coefficients between different kinds of tissues were not estimated quantitatively. In this study we estimated them from experimental data of dimensions and membrane capacitances of cardiac cells (see methods). Joyner et al. (1983) simulated the conduction of excitation in the model system which was composed of the Purkinje and the ventricular cells. However, they did not quantitatively estimate the strength of electrical couplings between cardiac cells. They regarded the cardiac tissues as a spatially continuous system, but we regarded them as a spatially discrete system (Kawato et al., 1986). We also investigated the conduction of excitation in the AV-node regarding the heart as a spatially discrete system. 2. Methods (A) S P A T I A L L Y D I S C R E T E M O D E L O F T H E H E A R T

In normal conduction, excitation invades the AV-node from the CT and passes it to the HB. Janse (1969) reported that many AV-cells seemed to exite the HB coorperatively. Hence, we represented the AV-node by a bundle composed of several one-dimensional fibers along which action potentials were conducted synchronously with each other from the CT to the HB. Each fiber was described by one-dimensional spatially discrete model proposed in the previous paper.

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277

Let V"(t) represent the membrane potential o f the nth cell, they then satisfy the following system of ordinary differential equations (Kawato et al., 1986) C,,, d V " / d t = l~"o~(V",Z")+d.(V~+~ + V " - ~ - 2 V " ) , dZ'/dt = F(V", Z").

(1)

A list of symbols used above is as follows.

v"(t) t

c,. Z ~ F d

the membrane potential of the nth cell at time t (mV) time (msec) specific membrane capacitance (t~F/cm 2) membrane ionic current of the nth cell as a function of V" and Z " ( ~ A / c m 2) gating variables of several ionic channels of the nth cell (no dimension) nonlinear vector function which determines the rate of change of Z n coupling coefficient between neighboring cells (mS/cm 2) (B) MODELS OF EXCITABLE MEMBRANE

The one-dimensional array of the A-cells is called as the A-line. Terms "AV-iine" and "P-line" are used in the same way. The present model is composed of the A-line, the AV-line and the P-line connected in this order, reflecting the real structure. Since the action potential of the HB was reported as being similar to that of the peripheral P-cell (HoiIman et al., 1959), we assumed that the membrane properties of the HB are the same as those of the P-cell and then, as the ionic current kinetics in eqn (1) within the P-line, we used the reconstruction model of action potential for the P-cell which was experimentally obtained by McAllister et al. (1975): MNT-model. For the AV-line, we used a reconstruction model of action potential derived experimentally for the AV-cell (Kokubun et al., 1982; Kokubun, 1983; K-model). Such a model for the A-cell, however, has not been obtained. Since the membrane properties o f the A-cell are qualitatively similar to those of the V-cell, we assumed that the reconstruction model of the V-cell (Beeler & Reuter, 1977: BR-model) can be used also for the A-cell as an approximation. (C) C O U P L I N G

COEFFICIENT

IN E A C H L I N E

The coupling coefficient d (mS/cm 2) in each line was determined from the apparent space constant AA (cm) and the membrane resistance R,, (lq cm 2) estimated from each reconstruction model, by using the following formula (Kawato et al., 1986).

{

1/d= Rm(cr-1)2/o "

(2)

ty = exp ( - I , . / A A ) .

(3)

Here, Ic represents a length o f a cell (cm). Parameter values (dimensions of the cells, AA, Rm) required in eqns (2) and (3) for the A, the AV, and the P cells are summarized in Table 1. The morphological data in this table are taken from Taniguchi et al. (1981) for the A and the AV-cells, and from Draper & Weidmann (1951) for

278

S. URUSHIBARA

ET

AL.

TABLE 1

Dimensions, apparent space constants, membrane resistances, coupling coefficients and coupling resistances within the A-line, the A V-line and the P-line. For the value of Rg of the P-line, see section 2(E)

A AV P

Dimension (i.tm) length x width x height

AA (p.m)

100x 12x 12 100x 10x 10 100x 30 (diameter)

610 450 1960

Rm

l l c m 2)

d (mS/cm 2) Rg (xl061-l)

6-0x 103 1-48x 104 4.46x 104

6.2 1-4 8.6

3.36 17.9 0-51

the P-cell. Values of AA were taken from the experimental data of Bonke (1973) for the A-line, De Mello (1977) for the AV-line, and Fozzard (1966) for the P-line. Table 1 also shows the estimated coupling coefficient d, and the values of the resistance of a gap junction Rg (Kf~) which were determined from d and the dimensions of the cells by the following equation (Kawato et al. 1986)

d = 1/(RgS~) = 1/(2gg(w+h)lc).

(4)

Here, Sc denotes the excitable membrane area of a cell (cm2), and w and h represent width and height of a cell (cm). The values o f Rg will be used in section 2(E). The estimated value d - - 8 . 6 for the P-line was not large enough to reproduce the experimentally obtained conduction velocity of the peripheral P-fiber (Kawato et al., 1986). However, the simulated conduction velocity (0.7 m/see) was similar to that of the HB (1.1 m/sec; Lazzara et al., 1975). (D) NUMBER OF CELLS IN EACH LINE The length of the path along which the excitation wave-front crossed the AV-node was estimated 2 to 3 mm (Paes de Carvalho & De Almeida, 1960; De Flice & Challice, 1969). Therefore, if the cell length o f the AV-cell is assumed 100 i~m, then the AV-line is composed of 20 to 30 AV-cells. In this study, the AV-line was composed o f 25 AV-cells. The number of cells of the A-line or the P-line was chosen so that the electric current threshold for excitation of the A-line or the P-line was saturated, which changes as a function of the number of cells. In the previous study (Kawato et al., 1986) we found that the AV-line, which can excite the A-line composed of 40 cells, can also excite the A-line composed o f any number of cells, using a simulation experiments in which the number of A-cells was systematically changed and the electric current threshold for excitation was measured. Hence we set the number o f cells in the A-line to be 40 in order to save computer resources. Here we repeated the same procedure to determine the sufficient number of cells in the P-line so that its electrical effect is realistically taken into account. We found that the same 40 is sufficient for the P-line. We can expect that the model AV-line connected to 40 A and P-cells simulates well the excitation conduction in the AV-node which is connected with much larger number of the two kinds of cells.

SIMULATION (E) COUPL1NG

OF ATRIOVENTRICULAR

COEFFICIENTS

BETWEEN

279

CONDUCTION

DIFFERENT

KINDS

OF TISSUES

In this subsection, we will determine the coupling coefficients between different kinds of lines from the specific membrane capacitance, the excitable membrane area, the area of junctional m e m b r a n e of each kind of cell and the coupling resistance between different kinds of cells. In order to determine the coupling coefficients between different tissues, we need to make an assumption about how different cell types make a boundary. Fig. l(a) illustrates our assumptions about structures o f boundaries a m o n g three different tissues. We assumed the so called direct transition; that is, there is no cell of intermediate size between different tissues and different cells are apposed directly with each other at the boundary. Actually, an electron microscopic study showed the direct transitions (Shimada, 1984). Figure l(b) shows an equivalent circuit to the "direct transition" model at the boundary of the tissue-1 c o m p o s e d o f cells -1 and the tissue-2 composed of cells-2. $1, S2, A~, A2 and Rg denote the areas o f excitable membrane, the areas of junctional m e m b r a n e of cell-1 (0)

.EAv (b) ½

AI

tissuel

cells

tissue 2

FIG. 1. (a) A morphological model about the transitions from the atrium to the AV-node and from the AV-node to the His bundle. (b) The equivalent circuit to the model shown in (a).

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and cell-2 and the coupling resistance between them, respectively. Let A~ > A2, that is cell-1 is thicker than cell-2, then electric current flow from cell-1 to cell-2 is ( v, - v 2 ) / G

(5)

where, VI and V2 are the membrane potentials of cell-1 and cell-2, respectively. The electric current per unit area of the excitable membrane of cell-2 provided by cell-1 through the gap junction is

( V, - V2)/(GS~).

(6)

Therefore, the coupling coefficient from line-1 to line-2 (d12) equals

d,2 = 1/(R~$2).

(7)

On the other hand, A1/A2 cells-2 attach to the single cell-l, on an average, as shown in Fig. l(b). So, the total electric current flow with which the cells-2 provide cell-1 is

( A J A2)( V2- Vi)/ gg.

(8)

Hence, the coupling coefficient from line-2 to line-1 (d21) is

d2~ = ( A , / A2)/ ( RgS~).

(9)

Since values of Rg in the regions of transition have not been measured experimentally, we assumed an intermediate value between those of the two neighboring tissues. That is, we chose 9-0 x 106f~ at the boundary between the atrium and the AV-node and 4-0 x 10 6 ~ between the AV-node and the HB from Rg values described in Table I (see discussion). Here, we explain in detail how Rg in the P-line was determined as shown in Table 1. The electrical influences of the AV-node on the P-cell depend on the value of membrane capacitance per unit area of the P-cell because the effect of the intercellular current depends on it. In the P-cells, Fozzard (1966) found that the capacity component charged at the early phase of the depolarization was 2.4 I~F/cm 2, and the residual capacity component connected to the extracellular space in series with a resister was charged very slowly. Because the early depolarization phase of the action potential is most important in the study o f excitation conduction, we neglected the slow capacitance component. There are still two possibilities about the value of the real specific membrane capacitance. One is that the specific membrane capacitance of the P-cell is 2.4 p . F / c m 2 and the excitable membrane area is equal to the surface area calculated regarding the P-cell as a simple cylinder (cylindrical surface area). The other is that the real specific membrane capacitance is 1.0 ixF/cm 2 and the area of excitable membrane is 2.4 times as large as the cylindrical surface area. The differential equations which describe the membrane potential are exactly the same in both cases. However, for calculation of Rg for the P cells by using eqn (4), we need to specify the real surface area o f excitable membrane. Because the membrane capacitances per unit area of other excitable membranes are usually close to 1.0 ~ F / c m 2, we assumed the latter case. Therefore, the excitable membrane area (So) used for the calculation of Rg in section 2(c) was

SIMULATION OF ATRIOVENTRICULAR C O N D U C T I O N

281

2-4 times as large as the cylindrical surface area. Incidentally, it was reported that the surface area of membrane of the P-cell was 11.5 times as large as the cylindrical surface area (Mobley & Page, 1972). Figure 2 summarizes the parameter values determined from the above procedures. The numerals with rightward arrows represent the rightward coupling coefficients at transition boundaries. Those in three boxes are the coupling coefficients within each line. Those with leftward arrows are the leftward coupling coefficients at transition boundaries. The numbers of cells contained in each line are shown lower most. Repetitive electric constant current stimuli with duration of 10 ms were applied at 400 ms intervals at the left end in the case o f normal direction of excitation conduction of the A-AV-P system and at the right end for retrograde conduction. To minimize the artifact, stimulus intensities were chosen 30 ixA/cm 2 for the normal conduction and 20 i~A/cm 2 for the retrograde conduction, which were slightly above the thresholds for successful propagation of excitation. N u m e r i c a l methods were described in the preceding p a p e r (Kawato et al., 1986).

3. Results

(A) THE THREE REGIONS OF THE AV-NODE The five phases of a typical cardiac action potential are schematically illustrated in Fig. 3. In the AN region, transition from the resting (diastolic) period (phase 4) to the onset of the depolarization (phase 0) in the action potential was adrupt in normal conduction but it was smooth in retrograde conduction (Paes de Carvalho & De Almeida, 1960). The smooth transition from the phase 4 to the phase 0 is referred to as "smooth onset" in this paper. Correspondingly, the rapid transition from the phase 4 to the phase 0 is referred to as "abrupt onset" hereafter. In the N region, the smooth onset was observed both in the normal and in the retrograde conduction. In the N H region, the smooth onset was not observed in the normal conduction but was not observed in the retrograde conduction. This is the opposite situation to the AN-region.

A

AV

2"8 =

7-9 ~.,

3-4 40

P

6"3 25

40

FIG. 2. The parameters used in the simulation. The numerals represent, from top, the rightward coupling coefficientsin the transitional regions, the coupling coefficientsin the lines, the leftward coupling coefficients in the transitional regions and the numbers of cells of the lines, respectively.

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S. URUSHIBARA E T A L .

1

4 0 ~ ~ I~ Apd I

D

FIG. 3. The five phases of typical action potential: phase 0: the rapid depolarization phase; phase 1: the early repolarization phase; phase 2: the plateau phase; phase 3: the rapid repolarization phase; phase 4: the diastolic phase.

We simulated the normal and the retrograde excitation conduction in our model system. Figure 4(a) shows three action potential waveforms recorded from an A-cell located at the left end of the model, from an AV-cell located at the center of the AV-line and from a P-cell located at the fight end of the model, in the normal conduction. Figure 4(b) shows action potentials recorded from the same cells in the retrograde conduction. It must be noted that the smooth onset was not observed in the action potentials of the P or the A-cell both in the normal and the retrograde conduction. But for the AV-cell, the smooth onset was observed both in the normal and in the retrograde conduction. Figure 5(a), (b) and (c) show action potentials of the three AV-cells which were located at the left end of the AV-line (neighboring with the A-line), at the center of the AV-line and the right end of the AV-line (neighboring with the P-line), respectively. The left three action potentials of Fig. 5 were obtained in the normal conduction and the fight three action potentials in the retrograde conduction. In (a) the abrupt onset was observed in the normal conduction and the smooth onset in the retrograde conduction. On the contrary, in (c) the smooth onset was observed in the normal conduction and the abrupt onset in the retrograde conduction. In (b) the smooth onset was observed both in the normal and the retrograde conduction as already shown in Fig. 4. Therefore, the characteristics of the action potentials recorded from the three model AV-cells--(a), (b) and (c)--are in good agreement with the experimental data obtained from the AN, N and NH regions of the AV node (Paes de Carvalho & De Almeida, 1960). Summarizing the simulated results shown in Fig. 5, we can state that the abrupt onset within the AV-line is observed only from the cells near the border with the A-line or the P-line and only when the excitation wavefront invaded the AV-line through that border. This is well accounted for by properties of inherent shapes of action potentials of the three cardiac cells and electrical couplings between them as follows. Let us consider the normal conduction case first. When the excitation

SIMULATION OF ATRIOVENTRICULAR CONDUCTION

(a)

283

105

50 5~

o

>-50

;;

400 Time (msec)

0

80(

t

S"m.-"t 1 I A (b)

E

50

•

AV

P

105 53 1

0

g g 2-50 400 Time (msec)

P

AV

800

A

FIG. 4. The simulated action potentials of the A-cell, the AV-cell, and the P-cell in the normal conduction (a) and in the retrograde conduction (b). The three cells locate at the left end o f the A-line (with the numeral 1), the center of the AV-line (53) and the right end of the P-line (105) in the A - A V - P system.

wavefront propagated within the A-line, the action potentials of A-cells exhibited abrupt onsets as shown in Fig. 4(a) because of their inherent membrane dynamics. Because the junctional current flowing between the neighboring cells has the effect of making action potential waveforms of the two cells similar, the waveform of the AV-cell at the boundary with the A-line resembles that of the A-cell. Therefore, the waveform of the AV-cell at the boundary is very similar to that of the A-cell and exhibits the abrupt onset (Fig. 5(a), normal). The same discussion applies to the case of the retrograde conduction if we replace the A-line by the P-line (Fig. 5(c), retrograde). Because the middle part of the AV-line is relatively isolated from electrical effects of the A-line or the P-line, action potentials recorded there retain intrinsic patterns of the isolated AV-cell which have the marked smooth onset (Fig. 5(b)). At the boundary where the excitation wavefront leaves the AV-line, the AV-cell must provide the following cells of deep resting potential (large diastolic potential) with a large amount of electric current to excite them. Consequently, the rate of

284

S. U R U S H I B A R A E T A L .

Retroqrode

Normol (o)

0

0

-50

-50 0

I

200

400

0

2OO

0

2OO

400

F

200

400

400

(b)

~50

ol/'X,

0 -50 I

0

20O

400

(c)

50 I-

0

200

400

0

Time (msec)

FIG. 5. The simulated action potentials at the left end (a), at the center (b) and the right end (c) of the AV-line. The three waveforms on the left side were obtained during the normal conduction, and those on the right side during the retrograde conduction.

rise of depolarizing potential at the boundary was smaller than the middle part, and the clear smooth onset was observed (Fig. 5(c), normal and 5(a), retrograde).

(B) T W O PHASES O F D E P O L A R I Z A T I O N OF" AV-CELL N E I G H B O R I N G WITH T H E A - L I N E

It has been widely observed that the depolarization of the AV-nodal cells on the border of the atrium consists of two phases, the first rapidly rising phase and the subsequent slowly rising phase. The two phases might be caused by sodium ionic current and calcium ionic current respectively. We examined another possibility, that is, the two rising phases are caused by electrical interactions. Figure 6(a) shows the action potential waveform obtained from the AV-cell neighboring with the A-line in the normal conduction, which is the same figure as Fig. 5(a) left. Figure 6(b) plots the junctional current which flows into the AV-cell from the A-line, and

285

SIMULATION OF ATRIOVENTRICULAR CONDUCTION (a)

(b) R"

E

g so

5O

~2

) o

'E ®25

g

"6

~ -50

g -'I I I I I I I I I I I I I I I 400 800

g

0

(c)

&-

o

400

800

(d) ~--,

I ."

I-

-s~1~---IOFI k 0

I I I I I I

~

I

!i.. rNo 1 I I I I

II II

400

I I I I

800 Time

°p,, g -5

I'

.~

:i~,,,

°-10

J m I

0

I m i

I i'~m

40O

I I m i

I J

800

(msec)

FIG. 6. (a) The simulated action-potential waveform of the AV-cell at the left end of the AV-line neighboring with the A-line. (b) The junctional current which flows from the A-line to the AV-cetl. (c) and (d) Five ionic currents of the AV-celh sodium current (iN~), delayed outward current carried by potasium ions (iK), time independent current (it), slow inward current carded by both sodium and calcium ions (/5) and hyperpolarization activated current showing multi-ionic property (ih).

Fig. 6(c) and (d) show 5 ionic currents of the AV-cell. The phase of the depolarization can be divided into the two phases intercepted by a step, which is in accordance with the real data. This step in the depolarization was not observed in the cells of the N or NH region (Fig. 5). As clearly seen in Fig. 6(b), (c) and (d), the rapidly rising phase of depolarization was caused by the junctional current (b) and the inward sodium current (iNa in (c)) while the slowly rising phase was caused by the slow inward current (is in (d)). Since the maximum amplitude of the junctional current was about 6-fold larger than that of the sodium current, the rapidly rising phase was caused mainly by the junctional current. The step in the depolarization phase coincided with the termination of the junctional current (compare (a) and (b)). Figure 7 shows the waveforms of cells No. 40, 41, . . . , 46, from left respectively, in the normal conduction. Cell No. 40 belongs to the A-line and cells No. 41, 42, . . . , 46 to the AV-line. Compare the waveform of cell No. 40 with that of cell No. 41. During the phase 0 of action potential of A-cell No. 40, it provided the neighboring AV-cell No. 41 with a large amount of electric current via the gap junction and the membrane potential of the ceil; No. 41 also depolarized quickly. Once the action potential of A-cell No. 40 came into phase 1 after its peak, AV-cell No. 41 kept depolarizing only slowly by its own ionic current (mainly by slow inward current) because the intercellular current flow was small or even negative (compare Figs. 6(b) and 7). Consequently, the step occurs when the membrane potential of the A-cell which is neighboring the AV-line reaches its maximum between the phase 0 and the phase 1. It is concluded that one of possible mechanisms which cause the step is the sudden termination of the junctional current which is

286

E T AL.

S. URUSHIBARA

50

0- 1 0 mV

--o

o >

- 5 5 mV -50

J 25

0

1 50 Time (msec)

FIG. 7. The early phases of the action potentials of cells No. 40 to 46. Cell No. 40 belongs to the A-line and cells No. 41 to 46 belong to the AV-line, The upper broken line represents the membranepotential level o f - 1 0 mV and the lower the level of-35 mV. p r o v i d e d b y the atrium, w h e n the action p o t e n t i a l of the atrial cells starts to decrease a n d come into the early r e p o l a r i z a t i o n phase. (C) COb,~r)UCTION VELOCITY OF ACTION POTENTIALS We e x a m i n e d the c o n d u c t i o n velocities o f the excitation w a v e f r o n t in various parts of the model system. Figure 8(a) shows the time w h e n the excitation reached each cell after we started a p p l y i n g the electrical- c u r r e n t stimuli at the left e n d o f the A - A V - P system. T h e c o n d u c t i o n time ( o r d i n a t e ) was defined as when the m e m b r a n e p o t e n t i a l o f a certain cell r e a c h e d - 1 0 mV d u r i n g the d e p o l a r i z a t i o n phase. This definition of the c o n d u c t i o n time was the same as Irisawa et al. (1971). Figure 8(b) is the same as Fig. 8(a) except that the p r o p a g a t i o n time was defined 120

120

(o)

E

E

g

g g

~ 4o

(b)

e I1

Q.

J I

I

40

80 Cell No.

120

f I

I

40

80

120

Cell No,

FiG. 8. (a) The time when excitation reaches each cell as a function of the cell number. The origin of time was defined when current stimulus was applied at the left end of the A-AV-P system, and the propagation time was defined when the membrane potential reached -10 mV. (b) The same plot as (a) except that the propagation time was defined when the membrane potential of each cell reached -35 inV.

SIMULATION

OF ATRIOVENTRICULAR

CONDUCTION

287

as the time when the membrane potential of each cell reached -35 mV. Since the cells of No. 41 to 65 belong to the AV-line, the propagation time along the AV-line was 61 ms in both definitions (see Figs. 8(a) and (b)). This value agreed well with the experimental result (60 ms) reported by Hoffman et al. (1959). One of the characteristics of the curve shown in Fig. 8(a) was its upwards convexity at the left part of the AV-line neighboring with the A-line. Correspondingly, the conduction velocity measured near the boundary between the A-line and the AV-line (3.9 cm/sec) was lower than that in the middle part of the AV-line (5 cm/sec). This simulated results correspond well to experimental observations by Scher et al. (1959). Incidentally the curve was steeper and the conduction velocity was lower at the right part of the AV-line neighboring with the P-line than in the middle part. This is because the AV-cells in this region must provide the P-line with a large amount of electric current flow to excite it. This effect was described in the previous paper. On the other hand in Fig. 8(b) there was no upward convexity near the border between the A and AV-lines. But its incline was steeper in the right part of the AV-line than that in the middle part, as in Fig. 8(a). If action potential does not change its waveform as it propagates the AV-line, the curve in Fig. 8 must be linear as in the A-line or in the P-line away from the AV-line. In Fig. 7, the upper broken line represents the level of - 1 0 mV (reference of Fig. 8(a)) and the lower one the level of -35 mV (reference of Fig. 8(b)). As seen in this figure, the waveforms of action potentials recorded from the AV-cells were quite different. The time difference between when the potential of cells No. 41 and No. 42 reached - 1 0 mV was clearly longer than between cells No. 45 and No. 46. This time delay became shorter rightward along the AV-line. The upward convexity in Fig. 8(a) reflects this result. On the other hand, the time delays with respect to the level of -35 mV are not so different for different pairs of neighboring cells, which is in accordance with the results shown in Fig. 8(b). Generally, the conduction velocities can have different values depending on the level of membrane potentials to which we refer for the conduction time when the waveforms change along the path of excitation conduction. It might be worthwhile to emphasize that the slow conduction at the left end of the AV-line observed in Fig. 8(a) can be ascribed neither to high coupling resistance, small inward current, nor electrical load and so on. The apparent slow conduction was caused by electrical interactions between the different membrane dynamics of the A and the AV-cells via the gap junctions. 4. Discussion In this simulation, the electrical resistance of the gap junction between the A and the AV-cell was chosen 9.0 x 106 f~ and that between the AV and the P-cell 4.0 x 106 f~ rather ad hoc (but see section 2(E) and Table 1). However, we obtained essentially the same results as described here for other choices of coupling resistances as far as both the normal and the retrogrades conduction occurred successfully. So our results do not critically depend on specific values of the resistance of the gap junction at the boundaries of different tissues.

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There was a discrepancy about the action potential duration (Apd of Fig. 3) of the AV-cell between the simulation (240 msec) and the experimental results by Janse (140 msec). The following two factors might explain this discrepancy. (1) In the real heart, A p d of the ventricle is longer than that o f the atrium. Since we used the reconstruction model of the ventricle for the atrium, Apd o f the AV-cell neighboring with the A-cell was prolonged. (2) The period of stimulus in the experiment o f Janse was 250 msec and was shorter than that used in the simulation (400 msec). It is known that if current stimuli were applied to the A-cell more frequently, its Apd became shorter (Hoffman & Suckling, 1953). If Apd of the A-cells are shorter, then Apd of the AV-cells become also shorter because of the electrical interactions. Consequently, the discrepancy of Apd between the experiment and the simulation might be explained, in part, by the different stimulus frequency. From the results described in the preceding section, it can be concluded that even if the m e m b r a n e properties of the atrioventricular node were uniform, the action potential waveforms of the AV-celis at different locations are different because of the electrical interactions with the A-line and the P-line. Further, the simulated action-potential shapes and the propagation speeds were in good accordance with experimental data even in various minute points. This, along with the following morphological studies, strongly supports the above theoretical viewpoint. De Flice & Challice (1969) could not correlate the physiological subdivisions (AN, N, N H regions) with the morphological characteristics of the AV-cells. We thank Prof. H. Irisawa for reading the manuscript. We also express our gratitude to Mr A. Yamanaka for discussing this work on several occasions. REFERENCES BEELER, G. W. & REUTER, H. (1977). Z Physiol. 268, 177. BONKE, F. I. M. (1973). Pfliigers Arch. 339, 1. BROCKMAN,S. K. (1963). Am. Z Physiol. 204(4), 597. DE FLICE, L. J. & CHALLICE, C. E. (1969). Circ. Res. 24, 457. DE MELLO, W. C. (1977). t~iigers Arch. 371, 135. DRAPER, M. H. ~¢. WEIDMANN, S. (1951). J. Physiol. 115, 74. FOZZARD, H. A. (1966). J. Physiol. 182, 255. HOFFMAN, B. F., PALS DE CARVALHO,A., DE MELLO,W. C. & CRANEFIELD,P. F. (1959). Circ. Res. 7,11. HOFFMAN, B. F. & SUCKLING, E. E. (1953). Am. J. Physiol. 173, 312. IRISAWA, H., CALDWELL,W. M. & WILSON,M. F. (1971). Jpn. J. Physiol. 21, 15. JANSE, M. J. (1969). Orc. Res. 25, 439. JOYNER, R. W., PICONE,J., VEENSTRA,R. & RAWLING,D. (1983). Circ. Res. 53, 526.

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