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(Circulation Research. 1995;76:366-380.)
© 1995 American Heart Association, Inc.

Articles

The Stochastic Nature of Cardiac Propagation at a Microscopic Level

Electrical Description of Myocardial Architecture and Its Application to Conduction

Madison S. Spach, J. Francis Heidlage

From the Departments of Pediatrics and Cell Biology, Duke University Medical Center, Durham, NC.

Correspondence to Madison S. Spach, MD, Box 3475, Duke University Medical Center, Durham, NC 27710.

	Abstract

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

 
Abstract The object of this study is to present evidence that the myocardial architecture creates inhomogeneities of electrical load at the cellular level that cause cardiac propagation to be stochastic in nature; ie, the excitatory events during propagation are constantly changing and disorderly in the sense of varying intracellular events and delays between cells. At a macroscopic level, however, these stochastic events become averaged and appear consistent with a continuous medium. We examined this concept in a two-dimensional (2D) model of myocardial architecture by exploring whether experimentally observed _max variability reflected different patterns of intracellular excitation events and junctional delays. The patterns of _max variability at randomly chosen intracellular sites were similar experimentally and in the 2D model. The 2D cellular model produced marked variability in gap junction delays; however, on the average, different gap junctions were used for cell-to-cell charge flow during conduction in different directions. During longitudinal propagation (LP), the velocity increased from the proximal to the distal end of each myocyte, and _max was lowest proximally, increased to a maximum at the distal fourth of the cell, and decreased distally. Transverse propagation (TP) produced rapid intracellular conduction with variable intracellular excitation sequences. TP _max was greater than LP _max in most subcellular regions, but near the ends of some myocytes, a reversed "TP>LP _max" relation occurred. Total charge carried by the sodium current varied inversely with _max, demonstrating feedback effects of cellular loading on the subcellular sodium current and the kinetics of the sodium channels. The results suggest that the stochastic nature of normal propagation at a microscopic level provides a considerable protective effect against arrhythmias by reestablishing the general trend of wave-front movement after small variations in excitation events occur.

Key Words: stochastic propagation • myocardial architecture • discontinuous propagation • gap junction delays • intracellular conduction

	Introduction

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

 
The importance of a transiently blocked region in the genesis of a reentrant process is well accepted. The biophysical basis for such regions, however, has been limited almost entirely to mechanisms based on membrane ionic properties for a continuous medium, with the electrical effects of structural complexity being considered to be of minor importance.¹ Earlier, we demonstrated that block in a region could be "turned on" or "turned off" simply by changing the direction of an excitation wave front with respect to the axis of the fibers.^{2 3} Our interpretation of this observation (eg, Fig 5 of Reference 3³ ) was that there are recurrent microscopic discontinuities due to cellular interconnections that alter the apparent load a cell "sees" as the direction of propagation is changed, causing propagation to fail. These observations in 1981 led us to propose that at a microscopic level cardiac conduction is discontinuous in nature,² which provided a major departure from the long-held idea that cardiac muscle behaves as a continuous syncytium. Since then, computer models representing cells as one-dimensional (1D) uniform cylinders have demonstrated some of the effects of discreteness on propagation and have provided insight into the effects of discontinuities.^{4 5 6 7 8 9} We recently observed, however, that _max at a single site could be forced to have many values by causing a macroscopic planar wave front to propagate in multiple directions.¹⁰ This is an important adjunct to our earlier observation that a region of block was similarly sensitive to the direction of the excitation wave front.^{2 3}

View larger version (26K): [in this window] [in a new window]   Figure 5. Graphs showing the loading effects of cellular network on impulse transfer between two well-coupled myocytes. The two cells are shown in isolation (upper left) and within the cellular network (upper right); solid squares mark the membrane segments, one on each side of an interplicate junction, where transmembrane potential (Vm) and sodium current (INa) were recorded. The two myocytes were coupled by one plicate, two interplicate, and one combined plicate gap junction, which produced an effective gap junctional conductance value of 1.2 µS. Gap junctional delay (g.j. delay) was determined as the time difference in max of the two segments. A, A g.j. delay of 55 µs following a two times threshold stimulus to cell x with the cell pair in isolation. Stimulus symbol shows the shape and duration (0.5 ms) of the current pulse. B, A g.j. delay of 165 µs with the cell pair incorporated into the 2D cellular array. Macroscopic plane-wave transverse propagation (arrow) was initiated along a line located more than three resting space constants from the group of cells shown. C, Time course of INa in the isolated cell pair. D, Time course of INa during transverse propagation with the cell pair incorporated into cellular network. Cell x is cell 8 and cell y is cell a within the basic 33-cell unit shown in Fig 1C.

The purpose of the present study is to highlight the importance of irregularities in cell geometry and to introduce the concept that these irregularities, which are associated with irregularly distributed gap junctions,^{11 12 13} are responsible for load variations within individual cells as propagation direction is varied. To examine the implications of this concept in detail, it was necessary to determine whether conduction events within realistically shaped cells vary because of the different electrical load experienced by each segment of a cell. If so, intracellular variability of these events would represent a major departure from the general idea that spatially discrete cells are isopotential,¹⁴ which requires homogeneous intracellular excitation. To explore these ideas, we considered that it was important to first establish the features of normal myocardial architecture that should be added to available conduction models.^{4 5 6 7 8 9}

Details of the arrangement of cardiomyocytes, their irregular shapes, and the nonuniform distribution of their gap junctions have not entered into analyses of cardiac conduction up to this point. Without a realistic electrical model of cardiac architecture, there have been insurmountable experimental and theoretical problems in estimating the electrical load on a cell. The difficulty is that electrical load,¹⁰ like the effective coupling conductance between cells,¹⁵ is strongly dependent on the nonuniform topology of gap junctions. Thus, the only available approach we know of to study the effects of load variations in individual cells is to (1) experimentally analyze changes in _max as an index of load variations at single microelectrode impalement sites when propagation traverses each site from multiple directions¹⁰ and (2) develop a two-dimensional (2D) multicellular model that approximates the associated myocardial architecture.¹⁶

This combined approach would allow one initially to mimic experiments in a cellular model and to determine whether _max behavior in the model is similar to that of real tissue. However, variability in experimental results has a large number of potential sources, and using variability to justify the use of a particular model requires caution. Because many factors change _max, we had to make certain that the experimental changes we analyzed were caused by loading effects rather than by other factors. We believe that we were able to meet this criterion in a convincing manner for the following reasons: (1) The resting potential remained constant when _max changed at each impalement site in response to altering the direction of conduction. Thus, each observation site provided its own control for nonloading factors. (2) Differences in ionic currents, as well as technical differences, can cause _max variability at different sites. These causes were ruled out because some of the lowest _max values during longitudinal propagation (LP) occurred at the same site that had the highest _max values during transverse propagation (TP), and vice versa.¹⁰ (3) "Noise" in the _max values was small and stable in repeated recordings for a given propagation direction.¹⁰

In developing a 2D electrical model to approximate myocardial architecture, two considerations were important. First, because of the limitations of available recording techniques, detailed model predictions (activation times, _max) at multiple sites inside individual cells can only be validated experimentally by yet-to-be-developed recording methods. This limitation requires caution in interpreting any agreement between detailed model results and experimental results obtained with markedly less spatial resolution. Second, we chose to approximate the architecture of left ventricular epicardium (where the experiments were performed) because it has the greatest variety of cell shapes and sizes, along with the highest degree of anisotropic coupling of cells, that we have encountered.^{2 10 17} The distribution of cell shapes and the gap junctional arrangement chosen for one model, however, represent only one of an infinite number of possible arrangements. Thus, rather than focus on a specific structure, we examined the concept that the variable sizes and shapes of cardiomyocytes, along with the arrangement of their interconnections, have a major effect on microscopic conduction. For this analysis, we ignored effects at a slightly larger-sized scale, such as the arrangement of real myocytes into separate bundles at different depths in the ventricular wall. Our previous microscopic mapping had shown that large areas of the canine epicardium had no insulted boundaries,¹⁰ and histologically, collagenous structures did not divide the muscle mass into fascicles.¹⁷

The experiments extended our recent results in that there was a unique relation at each impalement site between LP _max and TP _max and with 180° shifts in conduction direction.¹⁰ We were unable to find a satisfactory explanation for the measured variability of _max based on available models.^{4 5 6 7 8 9} However, if conduction at a cellular level is recognized as a kind of stochastic process due to the effects of the complex myocardial architecture, the observed directional differences in _max at each site can be accounted for. We examined this concept in the 2D cellular model to explore whether the _max variability at random sites reflected variable patterns of excitation events within individual cells as well as in the delays across gap junctions for each direction of propagation.

	Materials and Methods

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

 
The experimental techniques have been described in detail previously,^{2 10 18} and the construction of the 2D cellular model, along with validation of the model and its passive properties, is presented in a recent article.¹⁶ Where those methods are important to anisotropic propagation, they are included here.

Experimental Methods
Electrical Measurements
We studied in vitro preparations of the left ventricular epicardial surface of the hearts of 20 dogs weighing 11 to 24 kg. All experiments conformed to the guiding principles of the Declaration of Helsinki. Each dog was anesthetized with pentobarbital sodium (30 mg/kg IV). The hearts were rapidly excised, and a 3- to 5-mm-thick layer was removed and pinned to the floor of a 15-cm tissue bath maintained at 36°C to 37°C. The composition of the bathing solution was (mmol/L) NaCl 128, KCl 4.69, MgSO₄ 1.18, NaH₂PO₄ 0.41, NaHCO₃ 20.1, CaCl₂ 2.23, and dextrose 11.1. Four pairs of unipolar stimulus electrodes were positioned to produce macroscopic planar wave fronts during LP and TP. This arrangement provided a way to reverse the propagation direction 180° along either axis.¹⁰ The epicardial fiber orientation was confirmed by microdissection after each experiment.

Unipolar tungsten electrodes (50 µm diameter) were used to record extracellular waveforms (_e) at each intracellular impalement site. Impalements were achieved with glass microelectrodes that had tip impedances of 15 to 25 M. When the microelectrode tip had impaled a cell, _e and the intracellular action potential (_i) were recorded with a computer, which sampled each waveform at 62 500 Hz. To ensure against contamination by injury currents that might have occurred on removing the electrode, the impalement sites were alternated back and forth across the fibers.¹⁰

Continuous display of the action potentials confirmed that the resting potential did not change when the direction of conduction was altered. Thereby, each recording site provided its own control. Only action potentials with resting potentials between -80 and -83 mV were analyzed. The transmembrane potential (V_m) was obtained as the difference between _i and _e (V_m=_i-_e).² _max was analyzed for conduction along the longitudinal and transverse axes in 17 preparations and during 180° reversal of conduction along either axis in three preparations.

Duplicating the Geometry of Ventricular Myocytes for the 2D Cellular Model
The method of Jacobson¹⁹ was used to obtain disaggregated single myocytes from the anterior left ventricular subepicardium of three adult dogs. The cells were fixed in a solution of 1.5% glutaraldehyde in 0.08 mol/L phosphate buffer.²⁰ Small aliquots of cells were placed on a glass slide, and photographs at x840 were made with an inverted microscope by use of interference contrast.²¹ Thirty-three cells were chosen to fit the frequency distribution of the maximum lengths and maximum widths of 456 myocytes.¹⁰ Each of the 33 photomicrographs was digitized with a LaCie scanner and Macintosh computer. The CANVAS drafting program (Deneba Software) was used to trace the outline of each myocyte (Fig 1A). As illustrated for five cells of the model in Fig 1A, the irregularities produced by the intercalated disks were similar to those of isolated rat²² and canine¹⁶ ventricular myocytes.

View larger version (38K): [in this window] [in a new window]   Figure 1. Diagrams showing the formation of the 33-myocyte basic unit of the two-dimensional cellular model. A, Outlines of five myocytes and the manner in which they were fitted together to form a group of cells. The stippled areas next to the intercalated disks represent interplicate gap junctional membrane. The grids show the 10 µmx10 µm segments that represent the sarcolemmal membrane patches and the interior of each cell (depth, 11.3 µm). B, Arrangement of the three types of gap junctions (symbols) that electrically interconnect the five myocytes (a through e). C, Arrangement of 33 myocytes and the distribution of their intercellular connections (symbols). The 33-cell unit could be replicated longitudinally and vertically by fitting the ends and sides together to form cellular arrays of different sizes and shapes. The group of five myocytes (a through e) is highlighted within the 33-cell unit by marking each segment within these myocytes.

The Model
We modeled myocardial architecture as a complex pattern of connected cells. We chose an arrangement of gap junctions that was comparable to that observed experimentally, and we chose values of parameters that provided agreement with experimentally derived observations. Our goal was to explore issues of intercellular conduction with a model that provides data that are not accessible at this time with experiments. Our choice of model parameters is in no way presented as unique. Rather, our parameters permit us to demonstrate some of the possible interactions between cellular geometry, gap junctions, and propagation events. The major assumption was a strategic one—that the model provided a reasonable representation of cardiac architecture. All of the features of the model were constrained by available experimental information. However, there are no available data for the specific conductance value of each gap junction. Our constraint for the values of the gap junctions, therefore, was that the effective conductance between the cells of many isolated pairs from the model produced a mean value consistent with available experimental data.

Individual Myocytes
Within each cell, the sarcolemma and intracellular space were represented in a manner similar to the original isotropic 2D sheet model of Joyner et al.²³ The membrane was considered to be located on two surfaces separated by an intracellular space to provide membrane surface areas and cell volumes similar to those of real ventricular myocytes.^{24 25} Each cell was divided into segments that had a volume of 1130 µm³. The volume arises from two square membrane surfaces having dimensions of 10 µm² (x=10 µm), separated by a depth of 11.3 µm. The mean number of segments per cell was 35 (16 to 64 segments per cell). The total membrane area of each segment (including both surfaces) was assigned a value of 376 µm² to account for membrane added to a smooth-surfaced structure by the irregular surface of the sarcolemma of cardiac myocytes.²⁶ This area results in a membrane surface area–to–volume ratio of 0.33 µm^-1 for each cell; Page and McCallister²⁴ measured a ratio of 0.3 µm^-1 in rat ventricular myocytes.

The mean values of several parameters of the 33 individual cells of the model were as follows: cell membrane area was 13 183 µm² (Page and McCallister^{24 27} measured 14 759 µm² in rat left ventricular cells), cell volume was 39 654 µm³ (Bishop and Drummond²² measured 25 000 µm³ in rat ventricular myocytes), and input resistance of each cell in isolation (R_m, 10 to 20 k · cm²) was 75 to 151 M (Tseng et al²⁸ measured a mean value of 60 to 78 M in isolated canine ventricular myocytes).

Types of Gap Junctions
Fig 1A illustrates the formation of a small group of cells by fitting their borders together, and Fig 1B shows the manner in which the cells were electrically coupled. We adhered to the constraint that all of the gap junctional membrane was in the immediate region of the intercalated disks, as demonstrated for normal mammalian ventricular myocytes.^{29 30 31 32 33} This constraint resulted in approximating the gap junctional membrane by three types of junctions (symbols in Fig 1B): (1) Plicate junctions were in the plicate segment of the intercalated disk, which connects cells in an end-to-end manner with "fingerlike" processes of cell adhesion.³² (2) Interplicate junctions were in regions juxtaposed to the plicate segments of intercalated disks³² or at the lateral borders of the plicate segments.³⁰ (3) Combined plicate junctions were located at intercalated disks that occur as small steplike irregularities at selected points along the cell border. We use the term "combined plicate" junctions for these small but distinct disk areas. They were included to be complete in the representation. On the basis of measurements in 20 isolated ventricular myocytes, the small disks were 10% of the size of the "large" disks.

Conductance Value of Each Gap Junction
We modeled each gap junction as a pure ohmic resistance. We know of no experimental data that allow one to assign a specific conductance value (g_j) for each gap junction within the region of contact between the borders of two cells. The only experimental data available for this purpose are effective g_j [g_j(eff)] values, which represent the collective conductance of all open connexons between two cells of an isolated pair. Weingart³⁴ measured a mean g_j(eff) of 0.58 µS in rat ventricular myocyte pairs, and Kieval et al³⁵ found a mean g_j(eff) of 1.24 µS in "normal" rabbit ventricular myocyte pairs. Based on these experimental data, our aim was to (1) make the multicellular model manageable by arranging three types of gap junctions in a pattern consistent with the morphological constraints and (2) assign a g_j value to each type of gap junction so that the net effect of all gap junctions would produce a mean g_j(eff) value between 0.58 and 1.24 µS for the cells of isolated pairs. We proceeded step by step until we arrived at the following g_j value for each type of gap junction (Fig 1B): plicate junction, 0.5 µS; interplicate junction, 0.33 µS; and combined plicate junction, 0.062 µS.

Two features of the model should be emphasized as to the lack of uniqueness in making such an electrical representation of myocardial architecture: (1) The distribution of cell shapes and the types of gap junctions of the model represent only one of an infinite number of possible arrangements. (2) There is no unique arrangement of g_j values that produces a desired g_j(eff) between two cells of an isolated pair—any assignment of g_j values to multiple gap junctions is an approximation at best. As pointed out by Kameyama³⁶ in his original measurement of g_j(eff) values in isolated cell pairs, it is "almost impossible to estimate the accurate contact area by electron microscopy combined with the rj [where rj is 1/g_j(eff)] measurements."

In view of the lack of uniqueness of a multicellular electrical model of myocardial architecture, we considered the following to be important features that provided a reasonable and realistic approximation of normal electrical coupling between myocytes in ventricular muscle: (1) The individual g_j values of the three types of gap junctions produced a mean g_j(eff) of 0.77 µS between the cells of isolated cell pairs of the 2D model (n=50 isolated pairs), which is within the range of experimentally measured mean g_j(eff) values in isolated cell pairs.^{34 35} (2) The g_j values and the arrangement of the gap junctions provided 70% of the gap junctional conductance in the interplicate areas and 30% in the plicate areas. These values are consistent with the distribution of gap junctional membrane in canine ventricular myocytes.³³

A comparison of experimental and cellular model estimates of mean g_j(eff) in isolated cell pairs might not be valid if the types of cellular apposition (end to end versus side to side) were different in the experimental cell pairs and in isolated cell pairs of the model. In his experiments, Kameyama³⁶ measured the length of cell contact by light microscopy in isolated cell pairs and plotted these values against the g_j(eff) values. He found no correlation between the two and concluded that there was "little or no quantitative difference" in the g_j(eff) values of end-to-end versus side-to-side types of apposition.

We mimicked the experiment of Kameyama³⁶ by using our 2D model to isolate 82 cell pairs. To identify the type of apposition between the cells of each pair, we used the criteria of Luke and Saffitz³⁷ for the percentage of lateral border overlap; ie, >25%, side to side (n=46); <25%, end to end (n=36). Like Kameyama, we found no correlation between the type of apposition between cells and g_j(eff) in the isolated pairs. Therefore, the model results are consistent with Kameyama's conclusion that g_j(eff) of an isolated cell pair is not related to the length of contact between the borders of cells as measured by light microscopy.³⁶

Coupling Single Cells Together to Form 2D Multicellular Arrays
To form a basic unit of the multicellular model, the 33 cells were fitted together as their irregular shapes and variation in size allowed (Fig 1C). Plicate junctions connected the cells longitudinally, and interplicate and combined plicate junctions connected the cells laterally (Fig 1B and 1C). This arrangement resulted in each cell being connected to an average of 6 other cells, which compares with an average of 9.1 cells, to which individual ventricular myocytes were connected in the study of Hoyt et al.³² Based on electron micrographs of Sommer and Johnson,³⁸ the distance between the borders of two cells was considered to be so small (0.15 µm) that it was ignored. The cells at the upper and lower borders of the 33-cell unit and those at the ends of the unit were modified slightly by adding or removing a segment (10 µmx10 µm). This adjustment provided a way to join together multiple 33-cell units to form large cellular arrays of different sizes and shapes.

The Calculations
The propagation of excitation through the multicellular array was studied by modeling membrane depolarization and early repolarization, which made it unnecessary to model the plateau and repolarization phases of the action potential. Thereby, computer time was greatly reduced. The numerical analysis techniques have been widely used.^{4 5 6 7 8 9} It is their application to the specific 2D cellular model that is new. The method used was an extension of the Crank-Nicolson approach³⁹ as first applied to a 2D sheet of cardiac muscle by Joyner et al.²³

The Hodgkin-Huxley model⁴⁰ with Ebihara-Johnson kinetics⁴¹ was used to approximate the fast sodium current (I_Na) of the sarcolemmal membrane by the following equation:

(1)

where _Na is the maximal sodium conductance, m and h are gating parameters, and V_Na is the sodium equilibrium potential. We used a value of 28 mS/cm² for _Na and a value of 33.45 mV for V_Na. We approximated a repolarization, or leakage, current (I_R) by the following equation:

(2)

where _R is the repolarization conductance (0.05 mS/cm²), V_m is the transmembrane potential, and V_R is the equilibrium potential of the repolarization current. V_R was set to the value of the resting potential (-80 mV).

Representing the 2D Geometry of Single Myocytes
The 10 µmx10 µm segments (Fig 1A) within the cells were laid out in a rectangular array of rows and columns. Each segment could be connected to as many as four adjacent segments, and the values for the resistive connections in the four directions were specified separately for each segment. The cells were spatially discretized in two dimensions (x=y=10 µm); eg, each membrane segment corresponded directly to one of the grid squares shown in Fig 1A. The equivalent electrical circuit (Fig 2) within the borders of each cell was that of a 2D sheet with a membrane capacitance of 1.0 µF/cm². The surfaces of the cells were assumed to be exposed to a large volume conductor that had negligible resistance compared with that of intracellular (cytoplasmic) space, which had a resistivity of 250 · cm. The segments within a cell were interconnected with low resistances similar to the axial resistivity for a 2D isotropic continuous sheet as performed by Joyner et al.²³ In Fig 1A, the thin grid lines inside the cell boundaries correspond to the locations of the low-resistance connections [r_i(x) and r_i(y)] in the circuit diagram of Fig 2. To simulate cellular boundaries, adjacent segments were isolated from one another by specifying no cytoplasmic interconnections in the appropriate directions, except where connections were placed to represent the three types of gap junctions (Fig 2).

View larger version (37K): [in this window] [in a new window]   Figure 2. Diagram showing the electrical equivalent circuit of three interconnected myocytes of the two-dimensional cellular model. Each membrane patch corresponded to a 10 µmx10 µm intracellular segment in Fig 1A. In each patch, during depolarization there was an outward capacitive current (Ic) in parallel with an inward ionic current (Iion), which was approximated by the kinetics of the fast sodium current by using the Ebihara-Johnson model.41 The unconnected terminals of each membrane patch represent the outside surfaces of the patches, which were shunted to ground. Intracellularly, the patches were interconnected with low resistances [ri(x) and ri(y)] to produce an isotropic sheet. The boundaries between cells (diagonal lines) were represented by the absence of cytoplasmic interconnections in the appropriate direction. Electrical coupling between myocytes was achieved through three resistors that represented three types of gap junctions, each of which had a constant conductance: plicate (0.5 µS), interplicate (0.33 µS), and combined plicate (0.062 µS).

The net membrane currents were calculated as the sum of the capacitive and ionic channel currents for each segment. We used Gauss-Seidel iteration (Strang⁴² ) with 0.1 µV for the convergence criterion, which is small compared with values given in the literature, eg, the 5-µV value used by Roth⁴³ in his bidomain model. However, we found that with the 2D cellular model, convergence values as large as 1 µV produced artifacts in the computed waveforms; eg, using 1 µV instead of 0.1 µV for the convergence criterion changed some of the values of _max by >0.5%.

The calculations were performed in 2D networks that contained 700 to 2455 myocytes (24 500 to 85 000 segments, respectively). The shape of each array was arranged to extend 8 (resting space constants) in the direction of plane-wave propagation and 1.5 along an axis perpendicular to the direction of propagation. The longitudinal space constant (_L) was 1.3 mm (in agreement with experimental data^{44 45} ), and the transverse space constant (_T) was 0.4 mm (no experimental data available for _T¹⁶ ). These dimensions of the 2D arrays were used because they prevented end effects⁴⁶ at the boundaries of the 2D model from influencing the computed results. Macroscopic plane-wave LP was initiated near the right or left border of the model by an intracellular current stimulus two times threshold along a line perpendicular to the long axis of the cells. To produce macroscopic planar wave fronts during TP, excitation was initiated near the top or bottom of the model by a current stimulus two times threshold along a line parallel to the longitudinal axis of the cells. To ensure that the stimulus current did not influence the results, the area of observation was located more than three resting space constants from the stimulus line. Increasing the stimulus to four times threshold did not change the results.

Data Output and Analysis
We placed 300 to 600 "observation points" at various segments located at the center of the 2D cellular array. The values of each variable were initially computed at 1-µs intervals in all segments. Printed output for each observation site consisted of the value of _max, the time at which _max occurred (activation time), and the areas of the sodium conductance and the I_Na curves.

To determine the excitation sequence within each cell, the time of _max (±1 µs) for each segment was formatted with each value printed at the location of its corresponding segment on an outline of each cell. Intracellular isochrones were drawn by hand, with linear interpolation used where necessary to produce equal time intervals between the isochrones. The time required to excite the sarcolemmal membrane throughout each myocyte was determined as the difference between the earliest and latest time of _max within a given cell. Gap junctional delays were determined as the time difference in _max at segments on each side of a junction. Student's t test with paired observations was used to evaluate whether the _max means were different during LP and TP and whether the junctional delay and peak value of the transjunctional voltage for each type of gap junction were different during LP and TP. Values of P<.01 were considered significant. ANOVA was used when comparisons were made of the time delays and peak voltages across the three types of gap junctions. We used univariate regression to determine the relation between _max and the total I_Na and the area of the sodium conductance curve.

	Results

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

 
Experimental Nonuniformity of _max and Predictions of 2D Cellular Model
_max During LP and TP
Fig 3A shows histograms of representative experimental _max values obtained at 20 impalement sites in a typical ventricular epicardial preparation. Paired observations showed that TP _max was significantly greater than LP _max (P<.001), as found previously.^{2 10 18} However, there was considerable variation in the values of _max from site to site during both LP and TP, and the variation was greater during TP than LP. The histograms (Fig 3A) demonstrated overlap of some of the TP _max values with some of the LP _max values. A histogram of the paired TP-LP differences in _max at each site showed that most of the TP-LP _max values were positive, as expected (Fig 3A, TP-LP). However, at some sites TP _max was lower than LP _max, as indicated by a few negative values.

View larger version (38K): [in this window] [in a new window]   Figure 3. Representative experimental and two-dimensional (2D) cellular model max results at different sites during longitudinal propagation (LP) and transverse propagation (TP). A, The histograms illustrate the max values (dV/dtmax in volts per second) at 20 different impalement sites in one ventricular epicardial preparation. Macroscopic plane-wave conduction was produced in one direction along the longitudinal axis and in one direction along the transverse axis of the fibers (arrows). The takeoff potentials were similar (-80 to -83 mV) at all impalement sites. The experimental mean±SD max values were 155±18 V/s for TP and 121±13 V/s for LP; LP max varied from 93 to 139 V/s, and TP max varied from 110 to 181 V/s. In the histogram at the bottom (TP-LP), LP max was subtracted from TP max at each impalement site to give the paired differences. B, The histograms illustrate the max values at 20 randomly chosen segments during macroscopic plane-wave LP (left to right) and TP (top to bottom) in the 2D cellular model. The drawing at the top shows the locations of the 20 segments (solid squares) within a group of 12 cells. The cellular model mean max values were 181±13 V/s for TP and 162±10 V/s for LP. The (TP-LP) histogram at the bottom represents the difference in the paired TP max and LP max values at each of the 20 segments.

We next tested whether the experimentally observed variations of _max were consistent with results derived from the cellular model. Although each microelectrode impalement site was in a different cell in the experiments, the location of the microelectrode tip within each cell was unknown. To reproduce this feature of the experiment in the 2D cellular model, we mimicked the movement of the microelectrode by marking 20 randomly chosen points within a rectangle. The rectangle then was superimposed on an outline of a group of 12 cells, and the segments underlying the randomly chosen 20 sites were identified (Fig 3B, top). These 20 sites in the cellular model produced TP _max values that were significantly greater than LP _max values (P<.001). Histograms of each _max group (Fig 3B) were qualitatively similar to the those of the experimental _max values: (1) There was considerable variation of _max from segment to segment during LP and TP. (2) The variation in TP _max was greater than that of LP _max. (3) Some of the TP _max values overlapped the LP _max values. (4) TP _max was lower than LP _max in a few segments, as indicated by the few negative values in the histogram of the paired TP-LP differences in Fig 3B.

Reversal of Conduction Along the Longitudinal and Transverse Axes of the Fibers
When the direction of conduction was reversed 180° along the same axis of the ventricular epicardial fibers, _max changed appreciably at most impalement sites (n=30). Representative changes are shown in Fig 4A. The magnitude of the change in _max was not significantly different for reversing the direction of conduction during LP versus TP (P>.05). The mean change in _max for 180° reversals along both axes was 18±11 V/s. However, _max changed only slightly (<3 V/s) at two sites with reversal of conduction along the longitudinal axis and at three sites with reversal along the transverse axis of the fibers (Fig 4A, LP 6 and TP 3). An increase or decrease in _max with reversals was site specific; ie, the same 180° reversals produced increases in _max at some sites and decreases in _max at other sites.

View larger version (29K): [in this window] [in a new window]   Figure 4. Plots comparing experimental (A) and two-dimensional (2D) cellular model (B) changes in max produced by reversing the direction of longitudinal and transverse propagation (LP and TP, respectively). A, Representative experimental max values (dV/dtmax in volts per second) are shown for six microelectrode impalement sites during reversal of the direction of LP (top) and for six different impalement sites during reversal of the direction of TP (bottom). The relative locations of the impalement sites are indicated by the diamond-shaped squares on each drawing representing the preparation. B, Representative changes in max at six segments of the 2D cellular model during reversal of LP (top) and at six different segments during reversal of TP (bottom). In the five myocytes above each panel, solid squares mark the segments at which the max values were obtained. The arrows and corresponding symbols denote each direction of conduction for the experimental (A) and 2D cellular model (B) max values.

To determine whether similar changes in _max occurred with reversing the direction of LP and TP in the cellular model, we recorded _max values at 30 segments chosen randomly as before. Representative results are shown in Fig 4B for six LP sites and six different TP sites. When the direction of conduction was reversed during LP and TP, _max changed in a way that was qualitatively similar to that observed experimentally: (1) _max increased or decreased appreciably at most segments when the direction of conduction was reversed along either axis of the cells. (2) Increases or decreases in _max occurred independent of a specific direction of conduction along a given axis. (3) The magnitude of the change in _max was not significantly different for reversing the direction of LP versus TP (P>.05). (4) The mean difference in _max for opposite directions of LP and TP was 12±8 V/s. (5) Small changes in _max (<3 V/s) occurred in a few segments (Fig 4B, LP 5 and TP 3).

Conduction Events at a Microscopic Level in the 2D Cellular Model
We considered the foregoing patterns of _max variability at randomly selected sites to be similar in the 2D cellular model and in the experimental preparations. Therefore, we proceeded to use the 2D cellular model to explore whether the variation in _max reflected different patterns of excitation events within individual cells and different patterns in the delays across gap junctions.

General Effects of Cellular Network
Gap junctional delays and depolarization events influenced by cellular load. To establish whether differences in electrical load alter junctional delays between normally coupled cells, we first created a minimal load by isolating a pair of myocytes with a g_j(eff) value of 1.2 µS (Fig 5, top). When one of the cells was excited with a threshold stimulus of 0.5 ms, there was prolonged latency of V_m at -44 to -45 mV before simultaneous activation of both cells occurred (not shown). Throughout the myocytes, _max varied between 268 and 275 V/s, the high values reflecting a minimal electrical load. The absence of a junctional delay was similar to the experimental result of Weingart and Maurer,⁴⁷ who used a threshold stimulus. When the stimulus was increased to two times threshold, however, a junctional delay of 55 µs occurred (Fig 5A), and _max varied between 289 and 321 V/s.

When the same cell pair was incorporated into the 2D cellular network, cell x was connected to seven cells and cell y was connected to six cells. During TP, _max in both cells decreased to values between 165 and 179 V/s, and the junctional delay increased to 165 µs (Fig 5B). Despite the significant intercellular conduction delay, the action potential upstrokes maintained a smooth contour (Fig 5B), as occurs experimentally during TP.² The effects of loading on I_Na and the kinetics of the sodium channels were demonstrated by the following changes in the segments on each side of the gap junction (Fig 5, top): (1) In the isolated cell pair, total I_Na averaged 46 µcoulombs (µC)/cm² (Fig 5C), and in the cellular network, total I_Na increased to 105 µC/cm² (Fig 5D). (2) In the isolated cell pair total sodium conductance averaged 2.4 mS · ms/cm², and it increased to 3.05 mS · ms/cm² (27% increase) in the cellular network (not shown).

Effects of myocardial architecture on the spatial distribution of depolarization. To explore directional differences in the microscopic distribution of depolarization, we plotted V_m at 5-µs intervals along one row of segments during LP (Fig 6, top) and along one column of segments during TP (Fig 6, bottom). Depolarization extended approximately one resting space constant during both LP and TP (_L=1.3 mm, _T=0.4 mm¹⁶ ). The macroscopic conduction velocities produced a TP-to-LP velocity ratio of 0.31, with a TP velocity of 0.15 m/s and an LP velocity of 0.48 m/s, values that agree with experimental data in ventricular muscle.^{2 10 48}

View larger version (21K): [in this window] [in a new window]   Figure 6. Spatial distribution of depolarization within two-dimensional cellular network during longitudinal and transverse propagation (LP and TP, respectively). Top, The spatial depolarization curve during LP (arrows) is a plot of the transmembrane potential (Vm) values at each consecutive segment along the horizontal solid line overlying the cellular network. The values of Vm were obtained at the same instant of time. Bottom, The spatial depolarization curve during TP (arrows) is a similar plot of the Vm values at each consecutive segment along the vertical solid line overlying the network of cells. In each spatial depolarization curve, the bold solid portion represents the change of Vm within cells, and the dashed lines represent the step changes in Vm at the gap junctions and cell borders. In the TP spatial depolarization curve, the superimposed solid line (Exp.) is the best fit of a single exponential to the Vm values between -20 and -80 mV (r=.99).

The LP spatial pattern of depolarization approximated a smooth curve with large changes of V_m inside each cell and small V_m discontinuities at the connections between cells (Fig 6, LP). The LP pattern was similar to the spatial potential wave front demonstrated by Rudy and Quan⁵ in a 1D cable with intercalated disks at 100-µm intervals. However, during TP the pattern was just the opposite—large discontinuities of V_m occurred between cells, and V_m showed little change across the interior of each cell (Fig 6, TP). The overall effect of the irregularities of V_m as a function of distance was well described as a single exponential process (r=.99) over many cells in the foot of the spatial action potential, as shown for TP in Fig 6. Thus, at the macroscopic level, the discrete changes in V_m become averaged and appear consistent with the continuous (exponential) approximation of the passive spread of currents in cardiac bundles.^{44 48}

What Is the Nature of Excitation Spread Through the Cellular Network?
The marked directional differences in the spatial depolarization patterns of V_m in Fig 6 suggest that there should be accompanying directional differences in the temporal patterns of activation spread. Consequently, we examined LP and TP by recording the times of _max (±1 µs) in each of the segments comprising 16 myocytes located at the center of an array of 700 cells. The sensitivity of excitation spread to the boundaries of the individual myocytes was best revealed by perspective plots, which provided a view of the multidimensional spatial distribution of the activation times (Fig 7). To simplify the presentation, representative results are shown for the five myocytes highlighted within the 33-cell unit in Fig 1B and shown at the top of Fig 7.

View larger version (34K): [in this window] [in a new window]   Figure 7. Perspective plots of activation spread through a network of five myocytes within the two-dimensional cellular model. During longitudinal and transverse propagation (LP and TP, respectively), the time of max was determined at each of the segments (eight rows of longitudinal marks) within the five myocytes, which formed a group 80 µm wide and 300 µm long as shown at the top. A, Longitudinal propagation in a left-to-right direction. B, Transverse propagation in a bottom-to-top direction. In each perspective plot, the activation (Act.) times along each of the eight rows of segments are plotted as function of distance along the longitudinal axis of the cells. The hatched steps denote delays across plicate gap junctions. In both panels, the letters a through e are included to identify each myocyte with the activation times along the corresponding rows.

Longitudinal propagation. Step increases of activation time (discontinuities) occurred in the region of the end-to-end connections between cells (Fig 7A). However, the major increases in activation time occurred along the sarcolemmal membrane within each myocyte. The overall process produced a predominantly smooth pattern of excitation spread, which is consistent with the results of Fast and Kléber⁹ for longitudinal conduction in cultured strands of neonatal cells.

A major feature of LP was that the locations of the propagation discontinuities along the longitudinal axis corresponded to the irregular distribution of the plicate junctions. These irregular longitudinal local delays at the end-to-end connections of myocytes produced asynchrony of excitation in different portions of myocytes located side by side. Thus, superimposed on the overall smooth process of LP, the nonuniformly distributed longitudinal and lateral discontinuities of activation spread reflected the irregular shapes of the cells (Fig 7A).

Transverse propagation. As shown in Fig 7B, there were large lateral "jumps" in activation time between cells, while within each myocyte there was almost simultaneous activation of the sarcolemma. Also, along the longitudinal axis of the cellular network there were a few prominent step increases in activation time in the region of the plicate gap junctions (Fig 7B, steps connecting cells c and e). The lateral jumps in activation time coincided with the lateral borders of the underlying cells, and the quite variable longitudinal discontinuities of activation time corresponded to the irregular distribution of the plicate junctions. A few sites displayed prominent longitudinal discontinuities of activation time that were due to the asynchrony of activation of two irregularly shaped cells connected end to end by plicate gap junctions. This asynchrony occurred because the lateral border of the earliest activated cell extended further in the direction of the approaching wave front than did the lateral border of the adjoining cell, which was activated later (Fig 7B, cells c and e). However, only small differences in activation time occurred across the end-to-end connections of most cells, eg, the small activation time discontinuity between cells a and d. Therefore, the pattern of transverse excitation spread was quite different from that of LP. TP occurred as large jumps in activation time between the lateral borders of juxtaposed cells, and within individual myocytes there was almost simultaneous activation of the entire sarcolemmal membrane.

A general conclusion of the results shown in Fig 7 is that during LP and TP, plane waves do not occur at a microscopic level because of the disruption of the excitation wave by the irregularly located cell boundaries and the associated irregularly distributed gap junctions.

Impulse Transfer Across Gap Junctions
The foregoing results generate two related questions: (1) What are the conduction delays across each type of gap junction during LP and TP? (2) What are the associated maximum voltage differences across each type of gap junction? To answer these questions, we first determined the time of impulse transfer across each gap junction in the original group of 16 cells located at the center of the model. Next, we obtained the peak voltage difference across each gap junction (peak V_j) in the five cells of Fig 7. V_j was calculated every 20 µs by subtracting the V_m waveforms at segments on each side of a junction, and the largest absolute value was saved as peak V_j.

The Table shows that each type of gap junction had a different mean junctional delay during LP versus TP. Also, there was considerable variation in the values for each type of gap junction, especially during TP. During LP the mean junctional delay decreased from plicate to interplicate and combined plicate gap junctions (P<.001). During TP, the opposite occurred; the mean junctional delay increased in going from plicate to interplicate and combined plicate junctions (P<.001). The Table also shows that the mean peak transjunctional voltage was also different for each type of gap junction during LP and TP, with a considerable range of peak V_j values for each type of junction. During LP the mean peak V_j value decreased in going from plicate to interplicate and combined plicate gap junctions (P<.001). Contrariwise, during TP the mean peak V_j value increased in going from plicate to interplicate and combined plicate junctions (P<.001). The significance of the directional differences in driving force is that despite the considerable variations in peak V_j for each type of junction, on the average, different gap junctions are being used for cell-to-cell charge flow during propagation in different directions.

View this table: [in this window] [in a new window]   Table 1. Junctional Time Delay and Peak Voltage Across Gap Junctions During Longitudinal and Transverse Plane-Wave Conduction

Excitation Spread Within Individual Myocytes
Longitudinal propagation. During LP, the mean time to excite all of the sarcolemmal membranes within each myocyte was 226±78 µs (range, 68 to 348 µs, n=11). Representative intracellular excitation sequences during LP are presented in Fig 8A, which shows isochrones within each of the five myocytes previously analyzed as a network. Except for slight bending at intercalated disks near the ends of the myocytes, the isochrones maintained a vertical orientation throughout each cell. However, within each myocyte the isochrones shifted farther apart as excitation moved from the area where the action potential entered the cell to the area where it exited the cell. Consequently, the major intracellular feature of LP was that conduction was slower in the proximal part and faster in the distal part of each myocyte. These subcellular events produced an alternating sequence of slower and faster conduction along the path of longitudinal conduction throughout the cellular network.

View larger version (59K): [in this window] [in a new window]   Figure 8. Intracellular excitation sequences during "four-way" conduction along the longitudinal and transverse axes of the myocytes. A, Isochrones within each of the five myocytes (a through e) during propagation in both directions (arrows) along the longitudinal axis of the cells. The isochrones are separated by 4 µs. B, Intracellular isochrones during propagation in both directions (arrows) along the transverse axis of the cells. The isochrones are separated by 3 µs. Cells a through e are the same five myocytes of Fig 7.

On viewing left-to-right conduction at the top of Fig 8A, the question arises as to whether the regions of slower conduction were related to the reduced cross-sectional areas within some of the irregularly shaped myocytes. For example, slower conduction occurred in the smaller proximal regions of cells a and c, whereas faster conduction occurred in the larger distal regions of these cells. Reversing the direction of longitudinal conduction shows that the subcellular differences in the speed of conduction were not caused by variations in cross-sectional area within the myocytes. With right-to-left conduction, the pattern of isochrone spacing remained similar, but the polarity was reversed within each myocyte (Fig 8A, ). Thus, despite subcellular variations in cross-sectional area, the proximal part of each myocyte with respect to the direction of LP remained the region of slowest conduction, and the distal part of each myocyte remained the region of fastest conduction.

Transverse propagation. The major subcellular feature of transverse conduction was the rapidity with which excitation spread throughout each myocyte. During TP, the mean intracellular conduction time was 21±10 µs (range, 8 to 39 µs) for the same cells in which the mean intracellular conduction time during LP was 226 µs (P<.001). Another major difference was that during TP the pattern of intracellular excitation spread was different in each myocyte (Fig 8B). Within the same myocyte, the isochrones were oriented in different directions, and the pattern within each myocyte changed drastically when the direction of conduction was reversed along the transverse axis of the cells (Fig 8B). Collisions occurred in a few cells. For example, when the direction of TP was from top to bottom, collisions (asterisks) occurred inside cells a and c (Fig 8B, ). However, when the direction of TP was reversed (Fig 8B, ), a collision occurred only in cell b. We did not find a myocyte that demonstrated an intracellular collision during both directions of TP.

_max Variations Within Individual Myocytes
Fig 9 shows representative results of the distribution of _max within three myocytes for four directions of conduction. Along the short transverse axis of each cell there was little change in _max within each subcellular area. Therefore, in Fig 9 each intracellular distribution of _max is shown along a line representing consecutive segments between the ends of each myocyte. As can be seen, there was one general pattern of varying _max values within all myocytes during conduction in every direction. A distinct maximum occurred near the center of each myocyte, and there were distinct minima near the ends of each myocyte.

View larger version (40K): [in this window] [in a new window]   Figure 9. Distribution of max within three myocytes during longitudinal and transverse propagation (LP and TP, respectively). In each graph, the values of max are plotted as a function of distance (in micrometers) along a line of consecutive segments extending between the ends of each myocyte. The outline of each myocyte with the accompanying intracellular excitation sequence for each direction of TP (isochrones separated by 3 µs) is presented above each graph. A similar cellular outline with the accompanying intracellular excitation sequence for each direction of LP (isochrones separated by 4 µs) is presented below each graph. A, Top-to-bottom TP and left-to-right LP. B, Bottom-to-top TP and right-to-left LP. Myocytes a, c, and e are those shown in previous figures.

Longitudinal propagation. The intracellular location of the _max maximum and the relative values of the two _max minima were systematically different for each direction of LP. During LP in the left-to-right direction (Fig 9A), _max was lowest in the proximal (left) part of each myocyte, where intracellular conduction was slowest. _max increased to its maximal value between the middle and distal fourth of each cell. In the distal (right) part of each myocyte, _max decreased, although conduction was fastest in this region. During LP, the _max minima at the distal ends of the myocytes had higher values than did the minima at the proximal ends (P<.001). The fluctuating values of _max within each myocyte resulted in an alternating sequence of lower and higher _max values along the longitudinal axis of the network of cells.

In comparing the model results to the experimental _max observations, it is reasonable to assume that in the experiments the tip of the microelectrode varied randomly in its intracellular location relative to the ends of each impaled myocyte. According to the model results, at different subcellular locations within different myocytes, the values of _max would be different because of the fluctuations of _max within the individual cells. Thereby, the cellular model results were consistent with the experimental variety of _max values observed at different impalement sites during LP (Fig 3A).

When the direction of LP was reversed (Fig 9B), the same subcellular pattern of _max occurred in each cell but with reversed polarity. The reversal of polarity changed the intracellular locations of the _max maximum and the two relative minima, which altered _max at almost every segment within each myocyte. Consequently, the systematic changes in _max of the individual segments behaved in the same manner as the experimental changes in _max (Fig 4A). However, when the directionally different LP curves of each myocyte were superimposed (not shown), there was little difference in the _max values at a few segments within each myocyte where the LP _max curves for each direction crossed each other (eg, cells a, c, and e of Fig 9). Thus, the different _max intracellular patterns produced little change in _max at a few sites, consistent with the experimental observation in Fig 4A (top).

Transverse propagation. In contrast to LP, during TP there was considerable cell-to-cell variation in the _max maximum and the two minima (Fig 9). The mean _max value within almost every myocyte was greater during TP than during LP. However, the TP _max minima near the ends of each myocyte were often lower than the LP _max maximum located near the center of each cell. Consequently, there was considerable overlap of TP and LP _max values when comparing values from different subcellular areas of the same or different myocytes. The different TP _max values within each myocyte (Fig 9) thereby were consistent with the experimental results of Fig 3A in the following ways: (1) TP _max varied more than LP _max at multiple impalement sites. (2) There was overlap of some of the TP _max values with the LP values.

When the paired values of _max were compared at each segment, however, TP _max was greater than LP _max throughout most cells. This relation within each subcellular region is consistent with the experimentally paired results of Fig 3A (TP-LP), which show a predominant TP>LP relation. Near the ends of a few myocytes, however, there was reversal of the usual TP-LP _max relation (eg, myocyte e in Fig 9A and myocyte c in Fig 9B). These areas near the ends of a few myocytes may provide a subcellular basis for the experimental result that at a few impalement sites, TP _max was less than LP _max (Fig 3A, TP-LP).

Reversing the direction of TP produced a wide variety of changes in the _max maximum and the two minima within each myocyte (Fig 9). The only constant relation we found was that the value of the _max maximum varied in relation to the manner by which excitation was initiated via different inputs in each myocyte. The multiple inputs were nonuniformly distributed along the myocytes because of the irregular topology of the gap junctions associated with the arrangement of cells of multiple shapes and sizes (Fig 8B). Consequently, when the direction of TP was reversed, the arrangement of the input gap junctions changed markedly for each myocyte. The _max maximum within each myocyte had its greatest value when intracellular excitation was initiated almost simultaneously at two widely separated input areas (Fig 9A). When the direction of TP was reversed, the value of the _max maximum decreased in these myocytes in association with intracellular excitation being initiated predominantly at one input area (Fig 9B).

With opposite directions of TP, however, there was no consistent relation between the input or output gap junction areas and the values of the two _max minima near the ends of each myocyte. In some cells, _max was lowest at the input area and higher at the output area (Fig 9B, cell a). In numerous myocytes, however, _max at the output areas was lower than _max at the input areas of the same myocyte (Fig 9A, cell a; Fig 9B, cell e). These different responses within different myocytes produced a wide variety of changes in the values of the two _max minima near the ends of each myocyte when the direction of TP was reversed. Thus, the TP directional differences in the values of the _max maximum and the two minima within each myocyte were similar to the experimental result that reversing the direction of TP altered the value of _max at almost every impalement site (Fig 4A). However, when the directionally different TP curves of each myocyte were superimposed (not shown), there was little difference in the _max values at a few segments within some myocytes (eg, cells a and e in Fig 9). These minimal differences occurred primarily toward the ends of these myocytes, where the TP _max curves for each direction approximated or crossed one another. Thus, the different _max intracellular patterns produced little change in _max at a few sites in some cells, consistent with the experimental observation in Fig 4A (bottom).

	Discussion

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

 
The results presented here add up to a novel concept of cardiac propagation—at a microscopic (cellular) level, cardiac propagation is stochastic in nature. Instead of being orderly, stable, and uniform as it appears at the larger macroscopic (tissue) size, propagation at a microscopic level is seething with change and disorder in the sense of continuously varying excitatory events and delays between cells. A unique feature of the stochastic nature of cardiac propagation is that electrical boundaries produced by myocardial architecture create inhomogeneities of electrical load that affect conduction inside individual cells and influence conduction delays across gap junctions. This process produces discontinuous propagation as a primary reflection of the nonuniformities of electrical load because of the irregular arrangement of the cellular borders and the associated nonuniform distribution of their interconnections.

At this point, it is not a question of "either/or" with regard to whether discontinuous versus continuous propagation occurs in cardiac muscle. Rather, the results show that discontinuous propagation produces excitatory events that are stochastic in nature at a microscopic size scale, and at a macroscopic level, these stochastic events become averaged and appear consistent with a continuous medium, as has been depicted experimentally.^{48 49 50} Our view implies an important synthesis for the future—establishing a new relation between discontinuous and continuous propagation, which should provide the missing link between ion channel activity and conduction events that lead to normal heart beats or reentrant arrhythmias. Such a relation is a reflection of the central limit theorem,⁵¹ which provides a path from discontinuous events at a microscopic level to smoothed (averaged) events at a macroscopic level.

A major implication of the stochastic nature of microscopic conduction is that small input changes may produce large changes in the events of propagation. For example, simply changing the direction of conduction produces considerable change in the excitatory events of the action potential and in the gap junctional delays, and the excitatory events feed back on one another from cell to cell. Consequently, we suggest that the most fundamental consequence of the stochastic nature of normal propagation is that it provides a major protective effect against arrhythmias by reestablishing the general trend of wave-front movement after small variations in excitation events occur. When there is a decrease in diversity at a very small size scale, such as occurs when there are regularly repeating relatively isolated groups of cells, larger fluctuations of load than occur normally can develop and be distributed over more cells. The myocardial architecture may therefore fail to reestablish a smoothed wave front and become proarrhythmic. Relatively independent groups of cells are known to be produced by the loss of side-to-side connections,⁵² and in such bundles unidirectional block and anisotropic reentry can occur in the absence of repolarization inhomogeneities.¹ Such anisotropic phenomena do not occur in bundles with intact side-to-side connections between all of the cells.⁵² Consequently, to develop a complete picture of the conduction mechanisms of reentrant arrhythmias, it will be necessary to learn more about how the excitatory currents are affected by normal and abnormal cellular loading with feedback effects on the kinetics of the ionic channels.

Normal Feedback Effects of Cellular Loading on the Excitatory I_Na
To make certain that the intracellular load variations that alter _max also affect the kinetics of I_Na inside individual cells, we analyzed the pattern of total I_Na generated by each segment within 16 cells of the 2D cellular model. The pattern of I_Na was the same as that of _max, with a reciprocal relation between the two (LP, r=.78 and P<.001; TP, r=.87 and P<.001; n=564 segments). The typical intracellular relation between excitation spread, _max, and I_Na is illustrated for a single myocyte in Figs 10 and 11 for LP and TP, respectively. During LP, I_Na was greatest in the proximal part of the myocyte and less in the distal part, with the least I_Na in a region located between the middle and distal fourth of the cell. During TP, I_Na was greatest near the ends of the myocyte and lowest in the central area of the cell (Fig 11C). The subcellular variations in I_Na were also linked to variations in the kinetics of the sodium channels; the total sodium conductance was proportional to total I_Na in each segment (r=.99, P<.001, n=564). These considerations may be important in studies of variations of the density of sodium channels in Purkinje cells as demonstrated by Makielski et al⁵³ and in rat papillary muscle by Antoni et al.⁵⁴

View larger version (0K): [in this window] [in a new window]   Figure 10. Intracellular two-dimensional spatial relations of excitation spread, max, and sodium current (INa) within a single myocyte during longitudinal propagation. The relations are shown for myocyte c shown in previous figures. A, Map of the intracellular excitation sequence. Act. indicates activation. B, Map of the distribution of dV/dtmax. C, Map of the total charge (in nanocoulombs per square centimeter) carried by INa. The isovalue map of the distribution of INa (area INa) was constructed from the values of the total depolarizing INa of each segment (nC/cm2) within the myocyte. The opposite patterns of panels B and C are due to the reversed relation between max and INa in each membrane segment.

View larger version (0K): [in this window] [in a new window]   Figure 11. Intracellular two-dimensional spatial relations between excitation spread, max, and the sodium current (INa) within a single myocyte during transverse propagation. The relations are shown for the same myocyte shown in Fig 10. A, Map of the intracellular excitation sequence. Act. indicates activation. B, Map of the distribution of dV/dtmax. C, Map of the total charge (in nanocoulombs per square centimeter) carried by INa.

A question arises as to whether the results would be qualitatively different if we had used an ionic membrane current model other than the Ebihara-Johnson⁴¹ representation of the sodium channel kinetics. The qualitative answer to this question is no, since the primary effect of the nonuniform discontinuities of resistance is to produce variable loading effects on the sarcolemma with consequent variations in I_Na from site to site. To confirm this answer, we replaced the Ebihara-Johnson ionic model with the more comprehensive Luo-Rudy model⁵⁵ of the membrane depolarization and repolarization currents and repeated the simulations. Although the _max values were greater with the Luo-Rudy model, both ionic models produced the same relative differences of _max in each segment of 16 cells for LP and TP (r=.99, P<.001, n=564 paired segments). As with the Ebihara-Johnson ionic model, the Luo-Rudy model produced a reciprocal relation between _max and total I_Na (LP, r=.85 and P<.001; TP, r=.90 and P<.001), and the subcellular patterns were the same as in Figs 10 and 11. Therefore, we concluded that for the "normal" I_Na conditions approximated in the present study, the results were qualitatively the same for these two widely used models of the sodium channel kinetics.^{41 55}

Limitations of the 2D Cellular Model
Available models of cardiac conduction have provided important insights thus far.^{4 5 6 7 8 9} However, we know of no other conduction model results that are consistent with the experimentally demonstrated variations of _max during "four-way" conduction in anisotropic cardiac muscle. In the most advanced available 2D model of Leon and Roberge,⁸ the resistive equivalents of myocardial architecture are represented in an averaged manner by continuous 1D parallel cables connected side to side by regularly spaced resistors of the same value. That model produces values of _max at all sites that are considerably greater during TP than LP. The lack of overlapping values of _max contrasts with the experimental results during LP and TP (Fig 3). Also, the Leon-Roberge model produces practically no change in _max (<3.7 V/s) at different sites during longitudinal conduction along the continuous cables, nor does _max change at a single site when the direction of LP is reversed. Therefore, we conclude that incorporating the details of the arrangement of cardiomyocytes, their irregular shapes, and the associated irregular topology of the gap junctions is important in the analysis of conduction at a microscopic level.

In this initial 2D electrical description of myocardial architecture, we have attempted to include the maximum plausible degree of cell-to-cell coupling of irregularly shaped cardiomyocytes of variable sizes. The approximation to real structure has been based on duplication of the irregular shapes of isolated ventricular myocytes and on the approximate distribution of gap junctions as described by Hoyt et al.³² As much as possible, we have adhered to the constraint that each parameter is based on available experimental data, thereby reducing the assumptions to a minimum. However, a complete model of cardiac conduction will require representation of the effects of restricted extracellular space, as done in "bidomain" models⁵⁶ of conduction at the larger macroscopic size scale. In available bidomain models of cardiac muscle, however, intracellular connectivity is represented as the averaged effect of many cells, which produces a single value of _max at all sites along any single axis.⁵⁶ Consequently, a minimally complete model of anisotropic conduction will require a combination of the bidomain averaged representation of restricted extracellular space plus the irregular arrangement of cardiomyocytes with their irregular topology of the gap junctions. A final obvious additional component in a more complete three-dimensional model is the incorporation of multiple layers of cellular arrays.

The present study has focused on the effects of myocardial architecture in the presence of normal intercellular coupling and normal kinetics of I_Na. We considered it necessary to develop a 2D cellular model of "normal" myocardium as a basis for future exploration of the arrhythmogenic effects of myocardial architecture.¹ An important feature of the results is that when the stochastic microscopic events are averaged, they predict events with values similar to those measured experimentally at the larger macroscopic level, eg, smoothly shaped isochrones,¹⁰ exponential rise of the spatial foot of the action potential,⁴⁹ and a 3.1 LP-to-TP velocity ratio.^{10 48} The implication of these macroscopic predictions is not related to any unique parameter of the 2D cellular model. Rather, the results suggest that 2D cellular models that use different features of cellular geometry and connectivity may produce different macroscopic results. Thereby, electrical descriptions of myocardial architecture may provide a way to develop expressions^{57 58 59 60} to characterize different types of myocardial architecture as to their antiarrhythmic and proarrhythmic effects.

View larger version (25K): [in this window] [in a new window]   Figure 12. Effects of nonuniform arrangement of gap junctions in a one-dimensional (1-D) vs two-dimensional (2-D) model. In the 1-D model (left column), the cross-sectional area is the same as a row of segments in the 2-D cellular model. The membrane ionic kinetics41 are the same in both models. The junctional conductance and the distances between successive junctions in the 1-D model match those of the plicate junctions in the 2-D cellular model. The max values and activation (Act.) times are graphed for consecutive 10-µm segments in the 1-D model and for consecutive segments along the dashed longitudinal line through four cells at the center of the 2-D cellular model. The 1-D model consisted of 100 cells, and the 2-D array contained 700 cells. Within the curves of each graph, the dashed lines represent the gap junctions encountered during left-to-right conduction.

	Acknowledgments

 
This study was supported by US Public Health Service grant HL-50537 and the North Carolina Supercomputing Center. We wish to express our appreciation to Dr C. Frank Starmer for his many helpful discussions.

	Appendix 1

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

 
It is not clear how realistic it is to generalize 1D results to a 2D model with irregularly shaped cells and a nonuniform arrangement of cell-to-cell connections. Propagation in a 1D model generalizes only to plane-wave excitation in a 2D model, which can be approximated at a macroscopic level in multidimensional ventricular muscle.¹⁰ However, plane waves do not occur at a microscopic level, because the excitation wave is broken up by the irregular topology of the cell borders and nonuniform arrangement of the connections between cardiomyocytes (Fig 7). We found that some of the observations from a 1D model can be carried over to a 2D model. However, delays at the end-to-end connections, nonuniformities of intracellular conduction velocity, and _max variations in the 1D and 2D models were qualitatively different. Because these differences likely are important in the arrhythmogenic effects of nonuniform electrical loading,^{1 2 3} we present a brief comparison of the results of a 1D model and the 2D cellular model with irregularly spaced plicate gap junctions.

In a previous 1D model with junctions 100 µm apart, Rudy and Quan⁶¹ computed the times of normalized _max and peak I_Na and demonstrated that junctional delays were reflected in the computed unipolar extracellular waveforms. To probe the role of irregular cell topology, we used a similar 1D model but adjusted the membrane properties and cross-sectional area to be identical to one row of segments of the 2D cellular model. Also, the distances between junctions in the 1D model varied between 80 and 160 µm to match the nonuniform arrangement of the plicate gap junctions (g_j=0.5 µS) along one row of the 2D cellular model.

In both the 1D and 2D model, _max varied within cells, with the highest _max located near each output junction (Fig 12, panels A and B). However, the influence of cell length on _max differed in the two models. In the 1D model, the highest _max value within a cell increased as the cell length increased (r=.91, P<.01). The lowest _max value, which occurred at the input junction, was inversely related to the length of the cell (r=.90, P<.01). In the 2D model, there was no relation between cell length and the highest or lowest _max values. An additional 2D effect was to reduce the magnitude of change of _max that occurred in the 1D cells.

Increases in activation time in the 1D model occurred primarily at the junctions, with little time required for intracellular conduction (Fig 12C). Minimal to no variations in the conduction velocity occurred within the 1D cells, as shown by the slopes of the intracellular activation time curves. The delay across each input gap junction was related to the length of its cell (r=.90, P<.001); ie, the delay was greatest at the input junction of the 160-µm cell and smallest at the input junction of the 80-µm cell. There was no relation between the delay at each output junction and the length of its cell. The average junctional delay was 3.7 times the average intracellular conduction time (194 and 52 µs, respectively, in Fig 12C).

Conduction was different in the 2D cellular model (Fig 12D). The major increases in activation time occurred intracellularly, and the gap junctional delays were shorter. There was no relation between cell length and delays at the input or output junctions; eg, the smallest input junctional delay occurred with the greatest distance between junctions (160 µm). The average junctional delay was 0.65 that of the average intracellular conduction time (91 and 139 µs, respectively, in Fig 12D). The slopes of the activation time curves varied within each 2D cell, indicating that conduction was slower near the input and faster near the output gap junctions.

In summary, the predictable relation between "cell length" and input junctional delays and _max behavior in the 1D model disappeared in a network of irregularly shaped 2D cells. The effects of nonuniformly distributed junctions in the 2D model had an angular dependence on conduction, varying between the transverse and longitudinal axes. This angular dependence was not evident in extrapolations from the 1D model. Additionally, some features of _max behavior during TP in the 2D model did not occur in the 1D model. During TP in the 2D model, the maximal _max value usually was located at the center of each cell (Fig 9), and _max was higher at the input junctions than at the output junctions of some cells (Fig 9A, cell a).

It is interesting to note that there were some important extensions of the 1D results to a 2D medium that are valid. However, when the 2D medium is composed of irregularly shaped cells, the extrapolation of some results becomes problematic. Here, we have shown that it is not realistic to generalize at a microscopic level the results of a 1D model (_max, activation sequences) to a 2D model in the presence of irregularly shaped cells.

Received May 9, 1994; accepted December 1, 1994.

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