INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION REFERENCES

THE RHYTHMIC BEATING of the heart is the result of action potentials initiated in the pacemaker of the heart, the sinoatrial (SA) node. Mathematical models of the electrical activity of the SA node of the rabbit (the species for which most data have been obtained) have been produced. The first models were produced by Yanagihara et al. (61) and Noble and Noble (49), and subsequent models were developed from the earlier models (13, 18, 59).

All the above models are of a typical SA node action potential. However, the SA node, functionally, anatomically, and electrophysiologically, is not homogeneous. In the rabbit the SA node measures ~8 mm × ~10 mm (5). In the vertical direction it is bounded by the superior and inferior venae cavae, and in the horizontal direction it is bounded by the crista terminalis (a thick bundle of atrial muscle) and the interatrial septum. The action potential is initiated in a small part of the SA node, the leading pacemaker site. Normally, the leading pacemaker site is approximately midway between the two venae cavae and 1-2 mm from the crista terminalis (3). This region is referred to as the center of the SA node. From the leading pacemaker site in the center, the action potential propagates to the periphery of the SA node and then onto the atrial muscle of the crista terminalis. Conduction toward the interatrial septum is blocked (3). The periphery of the SA node (the region of the SA node close to the crista terminalis) is referred to by some authors as perinode or transitional tissue. Although the principal function of the periphery of the SA node is to conduct the action potential from the leading pacemaker site in the center to the atrial muscle, the periphery does show pacemaker activity. In response to a variety of interventions, for example, autonomic nerve stimulation, the leading pacemaker site shifts from the center, and in many cases it shifts toward the periphery (53); the pacemaker activity of the periphery of the SA node is, therefore, important physiologically. Most work on regional differences in the SA node has been carried out on tissue around the leading pacemaker site midway between the venae cavae and has focused on peripheral-central differences (little is known about the tissue from the more superior and inferior regions and also toward the interatrial septum). There are important anatomic differences; for example, in the center the cells are smaller and have fewer and more poorly organized myofilaments than in the periphery (3). There are electrophysiological differences; these have been studied in the intact SA node or in small balls of tissue from different regions of the SA node. In the periphery the takeoff potential is more negative, the action potential upstroke velocity is higher, the action potential is shorter, the maximum diastolic potential (also resting potential in quiescent tissue) is more negative, and the intrinsic pacemaker activity is paradoxically faster than in the center (33). Ion channel block has different effects in the different regions: block of tetrodotoxin-sensitive Na⁺ current (i_Na), 4-aminopyridine (4-AP)-sensitive transient outward current (i_to), or hyperpolarization-activated current (i_f) has a greater effect in the periphery, whereas block of L-type Ca²⁺ current (i_Ca,L) or rapid delayed rectifying K⁺ current (i_K,r) has a greater effect in the center (6, 32, 33, 45). These differences in the response to ion channel block suggest regional differences in ionic currents. Single cells have not been isolated from different regions of the SA node to confirm this. However, we isolate single cells from the whole of the SA node and then distinguish between cells on the basis of cell capacitance (C_m), a measure of cell size, which is known to vary between the periphery and the center (see above) (27, 28, 36, 39). The action potential characteristics vary with C_m in a manner consistent with the regional differences (see above) (27). For example, in large cells with a high C_m (presumably from the periphery) the upstroke velocity is high, whereas in small cells with a low C_m (presumably from the center) the upstroke velocity is low (27). We have measured the density of some ionic currents; whereas the density of i_Ca,L is not significantly different in cells of different size, the densities of i_Na, i_to, i_K,r, i_K,s, and i_f are greater in larger cells (27, 28, 36, 39; i_K,r and i_K,s data from unpublished observations).

Models incorporating regional differences within the SA node have been developed (49, 60). However, the models were based on speculation because of the absence of data on regional differences in ionic currents. The aim of the present study was to develop, on the basis of the evidence reviewed above, biophysically detailed models of action potentials in the periphery and center of the rabbit SA node.

Glossary

4-AP	4-Aminopyridine
AM	Atrial muscle
APD	Action potential duration
CL	Spontaneous cycle length
Cm	Cell capacitance
Cma(x), Cms(x)	Capacitance of atrial muscle cell or SA node cell in one-dimensional model of intact SA node at distance x from center of SA node
dL, dT	Activation variables for iCa,L and iCa,T
dNaCa	Denominator constant for iNaCa
dV/dtmax	Maximum upstroke velocity of action potential
Da, Ds	Diffusion coefficient between atrial muscle cells or SA node cells in one-dimensional model of the intact SA node
EK,s	Reversal potential for iK,s
ENa, ECa, EK	Equilibrium potentials for Na+, Ca2+, and K+
ECa,L, ECa,T	Reversal potentials for iCa,L and iCa,T
F	Faraday's constant
FK,r	Fraction of activation of iK,r that occurs slowly
FNa	Fraction of inactivation of iNa that occurs slowly
fL, fT	Inactivation variables for iCa,L and iCa,T
gp, gc	Conductance of a current in peripheral or central SA node cell models
ga(x), gs(x)	Conductance of a current in atrial muscle cell or SA node cell in one-dimensional model of intact SA node at distance x from center of SA node
gNa	Conductance of iNa
gCa,L, gCa,T	Conductance of iCa,L and iCa,T
gto, gsus	Conductance of ito and isus
gK,r, gK,s	Conductance of iK,r and iK,s
gf,Na, gf,K	Conductance of Na+ and K+ components of if
gb,Na, gb,Ca, gb,K	Conductance of ib,Na, ib,Ca, and ib,K
h1, h2	Fast and slow inactivation variables for iNa
h	Net fractional availability of iNa
iNa	TTX-sensitive Na+ current
iCa,L, iCa,T	L- and T-type Ca2+ currents
ito, isus	Transient and sustained components of 4-AP-sensitive current
iK,r, iK,s	Rapid and slow delayed rectifying K+ currents
iK	Sum of iK,r and iK,s
if	Hyperpolarization-activated current
if,Na, if,K	Na+ and K+ components of if
ib,Na, ib,Ca, ib,K	Background Na+, Ca2+, and K+ currents
iNaCa	Na+/Ca2+ exchanger current
ip	Na+-K+ pump current
	Maximum ip
itot	Total ionic current in a cell
itota(x), itots(x)	Total ionic current in atrial muscle cell or SA node cell in one-dimensional model of intact SA node at distance x from center of SA node
ist	Sustained current
iK,ACh	ACh-activated K+ current
iK,ATP	ATP-sensitive K+ current
kNaCa	Scaling factor for iNaCa
Km,Na, Km,K	Dissociation constants for Na+ and K+ activation of ip
L	Length of string of SA node and atrial tissue in one-dimensional model of intact SA node
Ls	Length of string of SA node tissue in one-dimensional model of intact SA node
m	Activation variable for iNa
MDP	Maximum diastolic potential
n	Steady-state value of n
pa	General activation variable for iK,r
pa,f, pa,s	Fast and slow activation variables for iK,r
pi	Inactivation variable for iK,r
Q10	Fractional change in a variable with a 10°C increase in temperature
r	Activation variable for ito
R	Universal gas constant
q	Inactivation variable for ito
SA node, SAN	Sinoatrial node
t	Time
T	Absolute temperature
TOP	Takeoff potential
V	Membrane potential
Va, Vs	Membrane potential of atrial muscle cell or SA node cell in one-dimensional model of intact SA node
Va(x), Vs(x)	Membrane potential of atrial muscle cell or SA node cell in one-dimensional model of intact SA node at distance x from center of SA node
x	Distance from center of SA node in one-dimensional model of intact SA node
xs	Activation variable for iK,s
y	Activation variable of if
z	Valency of ion
[Na+]i, [Ca2+]i	Intracellular Na+, Ca2+, and K+
[K+]i	concentrations
[Na+]o, [Ca2+]o	Extracellular Na+, Ca2+, and K+
[K+]o	concentrations
n	Voltage-dependent opening rate constant of n
n	Voltage-dependent closing rate constant of n
NaCa	Position of Erying rate theory energy barrier controlling voltage dependence of iNaCa
n	Time constant of n
	Space constant

	MODEL DEVELOPMENT

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION REFERENCES

Mathematical models of the action potential in peripheral and central cells of the rabbit SA node at 37°C were developed using experimental data from rabbit SA node preparations. New formulations for a number of ionic currents were developed on the basis of newly published data from rabbit SA node cells: i_Na, i_Ca,L, i_Ca,T, i_to, 4-AP-sensitive sustained outward current (i_sus), i_K,r, i_K,s, and i_f. Full details are given below. The models also include formulations for background currents (i_b,Na, i_b,Ca, and i_b,K), i_p, and i_NaCa; these formulations are similar to those in other models (13, 17, 25). The membrane potential is calculated using Eq. 2 (Table 1). The Glossary defines all abbreviations used. Formulations for ionic currents are shown in Tables 2-9. All parameter values are listed in Table 10. Differences in current densities between the peripheral and central SA node cell models are listed in Table 11. Initial values of variables used to run the models are listed in Table 12.

Model of i_Na

Activation and inactivation curves. Activation curves (corresponding to the steady-state value of m³) are shown in Fig. 1A. The filled squares show the activation curve based on data from Baruscotti et al. (2) from young rabbit SA node cells at room temperature, and the filled triangles represent data from Muramatsu et al. (43) from cultured rabbit SA node cells at 22-24°C (fits to the experimental data rather than the original data are shown). The solid line is the model-generated activation curve, which fits well with the data of Baruscotti et al. The dashed line is the activation curve from a model of a rabbit atrial cell (41). The general inactivation variable h is the weighted sum of h₁ and h₂ (Eq. 7, Table 2). F_Na is the fraction of inactivation that occurs slowly and is dependent on the membrane potential; h₁ and h₂ change with different time constants but have the same steady-state value. Inactivation curves (corresponding to the steady-state value of h) are also shown in Fig. 1A. The open squares show the inactivation curve based on data from Baruscotti et al. from young rabbit SA node cells, and the open triangles represent data from Muramatsu et al. from cultured rabbit SA node cells (fits to the experimental data rather than the original data are shown). There is a substantial difference between the data from the two groups. It is known that ion channels can change in culture, and this perhaps explains the difference. The model-generated inactivation curve (solid line) is closer to the data of Baruscotti et al. from young rabbit SA node cells. The dashed line is the inactivation curve from the model of a rabbit atrial cell (41).

View larger version (24K): [in this window] [in a new window]   Fig. 1.   TTX-sensitive Na+ current (iNa). A: activation (m3, filled symbols) and inactivation (h, open symbols) curves. B: time constant of activation (m). C: time constant of fast inactivation (h1). D: time constant of slow inactivation (h2). E: fraction of iNa inactivation that occurs slowly (FNa). F: simulated iNa during 10-ms voltage-clamp pulses to 55 to +40 mV (in 5-mV increments) from a holding potential of 60 mV (top) and current-voltage relationships for iNa (bottom). For the current-voltage relationships, iNa was measured as peak inward current. G: density of iNa (measured from the peak inward current during a pulse to 5 mV) plotted against cell capacitance (Cm). , Data from Honjo et al. (27) from rabbit sinoatrial (SA) node cells; dotted line, regression line; , values used in the peripheral (Cm = 65 pF) and central (Cm = 20 pF) SA node cell models.

Kinetics. Because of the rapid activation of i_Na, study of i_Na activation is difficult, and there are no data from rabbit SA node cells on the time constant of activation (_m). Data from the study of Brown et al. (8) on rat ventricular cells were used. Inasmuch as the experiments of Brown et al. were carried out at room temperature (22°C), a Q₁₀ of 1.7 (41) was used to correct the data for 37°C. The _m is plotted as a function of membrane potential in Fig. 1B, in which the circles show the temperature-corrected experimental data, and the solid line was generated by the model. In Fig. 1C, the time constant of fast inactivation (_h1) is plotted as a function of membrane potential. The formulation for _h1 was based on data from Muramatsu et al. (43) from cultured rabbit SA node cells (circles in Fig. 1C), Honjo et al. (27) from rabbit SA node cells (triangles), and Brown et al. from rat ventricular cells (squares). A Q₁₀ of 1.7 (41) was used to correct the experimental data (collected at 22°C) for 37°C (the temperature-corrected data are shown in Fig. 1C). In Fig. 1C, the solid line was generated by the model and the dashed line is from the model of a rabbit atrial cell (41). In Fig. 1D, the time constant of slow inactivation (_h2) is plotted as a function of membrane potential. The formulation for _h2 was based on data from Muramatsu et al. from cultured rabbit SA node cells (circles in Fig. 1D) and Brown et al. from rat ventricular cells (squares). A Q₁₀ of 1.7 (41) was used to correct the experimental data (collected at 22°C) for 37°C (the temperature-corrected data are shown in Fig. 1D). In Fig. 1D, the solid line was generated by the model and the dashed line is from the model of a rabbit atrial cell (41). In Fig. 1E, the fraction of slow inactivation (F_Na) is plotted as a function of membrane potential. The formulation for F_Na was based on data from Muramatsu et al. from cultured rabbit SA node cells. In Fig. 1E, the circles show experimental data, and the solid line was generated by the model. Over a wide range of membrane potentials, F_Na is ~10% of total inactivation of i_Na. Inasmuch as the density of i_Na is large in a peripheral SA node cell, ~10% of i_Na that inactivates slowly will contribute a substantial inward current during the early period of the action potential.

Simulated current. Figure 1F shows simulated i_Na from the peripheral SA node cell model during depolarizing voltage-clamp pulses as well as the current-voltage relationship of i_Na from the model (solid line and filled squares). The open circles show the experimental data of Honjo et al. (27) from a rabbit SA node cell with a C_m of 54.5 pF.

Models of i_Ca,L and i_Ca,T

View larger version (18K): [in this window] [in a new window]   Fig. 2.   L- and T-type Ca2+ currents (iCa,L and iCa,T). A: activation (dL, filled symbols) and inactivation (fL, open symbols) curves for iCa,L. B: activation (dT, filled symbols) and inactivation (fT, open symbols) curves for iCa,T; error bars, SE. C: simulated iCa,L (300-ms voltage-clamp pulses to 30 to +40 mV in 10-mV increments from a holding potential of 40 mV) and iCa,T (300-ms voltage-clamp pulses to 70 to +10 mV in 10-mV increments from a holding potential of 80 mV). D: current-voltage relationships for iCa,L (circles) and iCa,T (squares); iCa,L and iCa,T were measured as peak inward current. E: density of iCa,L (measured from the peak inward current during a pulse to 0 mV) plotted against Cm. , Data from Honjo et al. (27) from rabbit SA node cells; , values used in the peripheral (Cm = 65 pF) and central (Cm = 20 pF) SA node cell models.

The equations for i_Ca,T are listed in Table 4. Activation and inactivation curves [corresponding to the steady-state values of the activation variable (d_T) and the inactivation variable (f_T)] are shown in Fig. 2B. The squares, circles, and triangles show data from Hagiwara et al. (22), Fermini and Nathan (19), and Lei et al. (38), respectively, from rabbit SA node cells at ~37°C. In the model the activation and inactivation curves (solid lines) were computed using equations (Eqs. 33 and 38, Table 4) formulated by Lei (35) and Lei et al. (38) based on their experimental data from rabbit SA node cells at 37°C. In the model we used equations (Eqs. 30-32 and 35-37, Table 4) formulated by Hagiwara et al. (22) for the time constants of activation and inactivation (_dT and _fT) in rabbit SA node cells. Figure 2C shows simulated i_Ca,T during depolarizing voltage-clamp pulses. Figure 2D shows current-voltage relationships for i_Ca,T (squares); the open squares show data from Hagiwara et al. from a rabbit SA node cell, and the filled squares (and solid line) show data from the peripheral SA node cell model.

Model of 4-AP-Sensitive Current

Activation and inactivation curves. Activation curves (corresponding to the steady-state value of the activation variable, r) are shown in Fig. 3A. The filled triangles, filled squares, and filled diamonds show data from Honjo et al. (28) from rabbit SA node cells with capacitances of 63.4, 34.5, and 20.3 pF at 25°C; the activation curves are for the sum of i_to and i_sus. The filled hexagons show data from Lei et al. (39) from rabbit SA node cells at 35°C; the activation curve is for i_to only. The filled circles show data from Giles and van Ginneken (21) from rabbit crista terminalis cells at ~20.5°C; because of the method used, the activation curve is for the sum of i_to and i_sus (if the latter was present). The solid line shows the activation curve generated by the model. Inactivation curves for i_to only (corresponding to the steady-state value of q) are also shown in Fig. 3A. Inactivation curves are shown from Honjo et al. (28) from rabbit SA node cells (open triangles, open squares, open diamonds: data from cells with capacitances of 63.4, 47.1, and 23.6 pF, respectively), Lei et al. from rabbit SA node cells (open hexagons), Giles and van Ginneken from rabbit crista terminalis cells (open circles), and the present model (solid line). The model-generated inactivation curve is close to the data of Lei et al. from rabbit SA node cells at 35°C.

View larger version (22K): [in this window] [in a new window]   Fig. 3.   Transient and sustained components of 4-aminopyridine (4-AP)-sensitive currents (ito and isus). A: activation (r, filled symbols) and inactivation (q, open symbols) curves. B: time constant of activation (r). C: time constant of inactivation (q). D: simulated 4-AP-sensitive current during 200-ms voltage-clamp pulses to 70 to +60 mV (in 10-mV increments) from a holding potential of 80 mV (top, voltage-clamp protocol; bottom, current). E: current-voltage relationship for ito (measured as peak outward current at the start of the pulse and the current at the end of the pulse; currents have been normalized to the maximum current at +60 mV); error bars, SE. F and G: densities of ito (F, measured as the difference between the peak 4-AP-sensitive outward current during a 200-ms pulse to +50 mV from a holding potential of 80 mV and the current at the end of the pulse) and isus (G, measured as the 4-AP-sensitive current at the end of a 200-ms pulse to +50 mV from a holding potential of 80 mV) plotted against Cm. , Data from Lei et al. (39) from rabbit SA node cells; dotted lines, regression lines; , values used in the peripheral (Cm = 65 pF) and central (Cm = 20 pF) SA node cell models.

Kinetics. The time constant of activation of i_to and i_sus (_r) is shown in Fig. 3B. The circles show data from Giles and van Ginneken (21) from rabbit crista terminalis cells. Inasmuch as the experimental data were collected at 24°C, a Q₁₀ of 2.18 (41) was used to correct the data for 37°C (the temperature-corrected data are shown in Fig. 3B). The solid line was generated by the model. Figure 3C shows the time constant of inactivation of i_to (_q). The squares and triangles show data from Honjo et al. (28) from rabbit SA node cells for the fast and slow components of inactivation [data collected at 25°C corrected for 37°C with a Q₁₀ of 2.18 (41)]. The circles show data from Giles and van Ginneken from rabbit crista terminalis cells [data collected at ~20.5°C corrected for 37°C with a Q₁₀ of 2.18 (41)]. The solid line was generated by the model.

Simulated current. Figure 3D shows simulated 4-AP-sensitive current (i_to + i_sus) from the peripheral SA node cell model during depolarizing voltage-clamp pulses. The currents are similar to 4-AP-sensitive currents in rabbit SA node cells (28, 39). Current-voltage relationships for i_to are shown in Fig. 3E. The open circles show data from Lei et al. (39) from rabbit SA node cells, and the filled circles (and solid line) show data from the peripheral SA node cell model.

Model of i_K,r

Activation and inactivation curves. Activation and deactivation of i_K,r in rabbit SA node cells have double-exponential time courses (37, 51). To model this, we have used two activation variables: a fast activation variable (p_a,f) and a slow activation variable (p_a,s). The general activation variable (p_a) is the weighted sum of the fast and slow activation variables (Eq. 48, Table 6). In rabbit SA node cells, experimental data have shown no distinct dependence of the fraction of inactivation that occurs slowly (F_K,r) on membrane potential, and the ratio of the slow to the fast component of activation of i_K,r is 2:3 (37). In the model, F_K,r is assumed to be constant with a value of 0.4. Figure 4A shows activation curves corresponding to the steady-state value of p_a (we assume that p_a,f and p_a,s are the same and equal to p_a). The triangles represent data from Lei and Brown (37) from rabbit SA node cells at 37°C (fit to the experimental data rather than the original data is shown). The solid line was generated by the present model and is close to the data of Lei and Brown. In a model of a guinea pig ventricular cell, Noble et el. (50) assumed that the steady-state values of p_a,f and p_a,s are different, and in the right-hand part of Fig. 4A the long dashed and short dashed lines show the dependence of p_a,f and p_a,s, respectively, from the model of Noble et al. on membrane potential.

View larger version (26K): [in this window] [in a new window]   Fig. 4.   Rapid and slow delayed rectifying currents (iK,r and iK,s). A: activation (pa, filled symbols) and inactivation (pi, open symbols) curves for iK,r; error bars, SE. B: fast time constant of activation of iK,r (pa,f). C: slow time constant of activation of iK,r (pa,s); error bars, SE. D: activation curve for iK,s. E: time constant of activation of iK,s. F and G: simulated iK,r (F) and iK,s (G) during 1-s pulses applied to potentials between 50 and +40 mV (in 10-mV increments) from a holding potential of 60 mV. H and I: current-voltage relationships for iK,r (H) and iK,s (I). Currents were measured at the end of the voltage-clamp pulses. J and K: densities of iK,r (J, measured as the peak 3 µM E-4031-sensitive tail current after a 1-s pulse to 10 mV from a holding potential of 50 mV) and iK,s (K, measured as the peak 3 µM E-4031-insensitive tail current after a 1-s pulse to +40 mV from a holding potential of 50 mV) plotted against Cm. , Data from M. Lei (unpublished observations) from rabbit SA node cells; dotted lines, regression lines; , values used in the peripheral (Cm = 65 pF) and central (Cm = 20 pF) SA node cell models.

Inactivation variable. Figure 4A shows inactivation curves (corresponding to the steady-state value of p_i). In the model the inactivation curve (solid line) was computed using an equation (Eq. 56, Table 6) formulated by Ito and Ono (29, 51) based on their data (circles) from rabbit SA node cells at 33°C. The dashed line is from the model of a guinea pig ventricular cell (50). There appears to be a nearly 18-mV shift in the inactivation curve in rabbit SA node cells compared with that in guinea pig ventricular cells.

Kinetics. The time constants of the fast and slow activation variables (_pa,f and _pa,s) are bell-shaped functions of membrane potentials, as shown in Fig. 4, B and C. In the model the time constants (solid lines) were computed using equations (Eqs. 52 and 53, Table 6) formulated by Ono and Ito (51) based on their experimental data (circles) for rabbit SA node cells. The dashed lines in Fig. 4, B and C, show the time constants used for the guinea pig ventricular cell model of Noble et al. (50). It appears that _pa,f and _pa,s are larger in guinea pig ventricular cells than in rabbit SA node cells. No experimental data are available on the relationship between the time constant of inactivation (_pi) and potential in the rabbit SA node. Here we assume that _pi is independent of the membrane potential and has a fixed value of 0.002 s, as suggested by Ono and Ito.

Simulated current. Figure 4F shows simulated i_K,r from the peripheral SA node cell model during depolarizing voltage-clamp pulses. Figure 4H shows the current-voltage relationship of i_K,r (i_K,r measured at the end of the pulse). The open circles show data from Ito and Ono (29) from rabbit SA node cells, and the filled circles (and solid line) show data from the peripheral SA node cell model. The current-voltage relationship from the model is roughly similar to the experimental data.

Model of i_K,s

Model of i_f

View larger version (19K): [in this window] [in a new window]   Fig. 5.   Hyperpolarization-activated current (if). A: activation curves. B: time constant of activation (y). C: simulated if during 300-ms pulses to potentials from 50 to 120 mV (in 10-mV increments) from a holding potential of 40 mV (top, voltage-clamp protocol; bottom, if). D: current-voltage relationships for if; if was measured as the increase in inward current from the beginning to the end of the pulses. E: density of if (measured during pulse to 110 mV) plotted against Cm. , data from Honjo et al. (27) from rabbit SA node cells; dotted lines, regression lines; , values used in the peripheral (Cm = 65 pF) and central (Cm = 20 pF) SA node cell models.

Regional Differences in Ionic Current Densities

Ionic current densities for the peripheral and central SA node cell models are listed in Table 11. For some but not all currents, the relationship between current density and C_m has been determined for rabbit SA node cells, and the final panels of Figs. 1-5 show the densities of various currents plotted against C_m. The open circles are experimental data (27, 28, 39; i_K,r and i_K,s data from unpublished observations), and the filled squares are the chosen values for the peripheral and central SA node cell models. The experimentally measured densities of i_Na, i_to, i_sus, i_K,r, i_K,s, and i_f are significantly correlated with C_m and are larger in cells with higher C_m (Figs. 1 and 3-5) (27, 28, 36, 39; i_K,r and i_K,s data from unpublished observations). Values chosen for the densities of these currents for the peripheral and central SA node cell models are within the experimental range and are greater in the peripheral SA node cell model. Although the density of i_Ca,L measured experimentally is not significantly correlated with C_m (27), it was necessary to increase the density of i_Ca,L in the peripheral SA node cell model compared with that in the central SA node cell model (Fig. 2E). Despite this, the chosen values for the density of i_Ca,L in the peripheral and central SA node cell models are reasonably consistent with the experimental values (Fig. 2E).

During the development of the models, there were no experimental data available on the relationship between the densities of i_K,r and i_K,s and C_m for rabbit SA node cells. Some data were available: Lei and Brown (37) had reported that the ratio of the amplitude of i_K,r to i_K,s tails after a 1-s pulse to +40 mV from a holding potential of 40 mV was 1:0.3-0.4, and, therefore, in the peripheral and central SA node cell models, the ratio was set at 1:0.3. Although there were no direct measurements of the density of i_K,r in rabbit SA node cells with different values of C_m, various indirect lines of evidence suggested that the density of i_K,r is greater in peripheral than in central SA node cells. First, although complete block of i_K,r by 1 µM E-4031 abolishes pacemaker activity in tissue from the periphery and center of the rabbit SA node, partial block of i_K,r by 0.1 µM E-4031 abolishes pacemaker activity in central but not peripheral tissue (32). One explanation for this different response to partial block of i_K,r is that the density of i_K,r is greater in the periphery of the SA node (32). Second, an immunocytochemical study examined the distribution of ERG (channel protein responsible for i_K,r) in the ferret SA node (7). If the anatomy of the SA node in the ferret is similar to that of the rabbit, the data from the study support the possibility that the density of i_K,r is greater in the periphery of the SA node: the labeling of ERG was little in the intercaval region (where the center of the SA node is located in the rabbit at least) and substantial close to the crista terminalis (where the periphery of the SA node is located in the rabbit at least). Finally, in the models of peripheral and central SA node action potentials, it was necessary to increase the density of i_K,r in the peripheral cell model compared with that in the central cell model to obtain some of the characteristic differences between the two regions (see Fig. 10). After the development of the models, experimental measurements of the densities of i_K,r and i_K,s became available (Fig. 4, J and K, open circles) from our laboratory in rabbit SA node cells (unpublished observations). Although the densities of i_K,r and i_K,s in the peripheral and central SA node models were set before the experimental data became available, they are within the experimental range obtained from rabbit SA node cells.

In the case of currents for which there are no data concerning current density and C_m, the current density was assumed to be the same in peripheral and central SA node cells.

Intracellular Ionic Concentrations

Comparison of the Models of Peripheral and Central SA Node Action Potentials With Action Potentials Measured Experimentally From the Rabbit SA Node

One-Dimensional Model of the SA Node and Surrounding Atrial Muscle

Computational Methods

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION REFERENCES

Peripheral and Central SA Node Action Potentials

View larger version (25K): [in this window] [in a new window]   Fig. 6.   Simulated peripheral and central SA node action potentials. A: simulated action potentials at a fast time base. B: simulated action potentials at a slow time base. C: action potentials recorded from small balls of peripheral and central rabbit SA node tissue (balls B and D) at a fast time base (unpublished observations). D: action potentials recorded from rabbit SA node cells with Cm of 57.5 and 22.0 pF at a slow time base (unpublished observations). E-H, takeoff potential (TOP; E), maximum upstroke velocity (dV/dtmax, F), maximum diastolic potential (MDP, G) and cycle length (CL, H) plotted against Cm. , Data from Honjo et al. (27) from rabbit SA node cells; dotted lines, regression lines; , values computed from the peripheral (Cm = 65 pF) and central (Cm = 20 pF) SA node cell models.

Figure 6, E-H, compares the characteristics of the simulated action potentials with the average characteristics of action potentials recorded experimentally from rabbit SA node cells by Honjo et al. (27) at 35°C. The open circles in Fig. 6, E-H, show experimental measurements of the takeoff potential, maximum upstroke velocity, maximum diastolic potential, and cycle length (time between successive spontaneous action potentials) of rabbit SA node cells plotted against C_m. In all cases, there are significant correlations of the variables with C_m (27). In Fig. 6, E-H, the filled squares show corresponding values from the peripheral and central models. In all cases, the model values are close to those recorded experimentally; the changes with C_m are also comparable to those seen experimentally. There is also a significant correlation between action potential amplitude and C_m in rabbit SA node cells [action potential amplitude is greater in larger cells (27)], and once again corresponding values from the peripheral and central models are comparable (not shown). Although Honjo et al. observed no significant correlation between action potential duration and C_m in rabbit SA node cells, there is a difference in action potential duration between the peripheral and central models (Fig. 6A). However, the action potential durations in the peripheral and central models are still comparable to the data from rabbit SA node cells (not shown), and, furthermore, in small balls of tissue from the periphery and center of the rabbit SA node, a significant regional difference in action potential duration is observed (Fig. 6C) (5) similar to that between the peripheral and central models (Fig. 6A).

The takeoff potential, maximum upstroke velocity, peak value of the action potential, action potential amplitude, maximum diastolic potential, and cycle length in the peripheral and central models are comparable to experimental data obtained from small balls of tissue from the periphery and center of the rabbit SA node (33). However, in the case of the data from the small balls of tissue (at 32°C), the action potential duration and cycle length are longer than in the peripheral and central models (37°C) and also longer than in single cells (obtained at 35°C) (27). This may be the result of the lower temperature of 32°C used in the work on the small balls of tissue (33).

Figure 7 shows changes in various ionic currents (i_Na, i_Ca,L, i_to, i_K,r, i_Ca,T, i_K,s, and i_f) during the action potentials generated by the peripheral and central models. Because of the regional differences in ionic current densities and C_m described above, the amplitudes of the ionic currents are different in the peripheral and central models. The changes in i_Ca,L, i_K,r, and i_f during the action potential are comparable in amplitude and time course to the changes in the three currents during the action potential in rabbit SA node cells as measured with the action potential-clamp technique (62).

View larger version (21K): [in this window] [in a new window]   Fig. 7.   Simulated peripheral (A-C) and central (D-F) SA node action potentials (A and D) and underlying ionic currents (B and E: iNa, iCa,L, ito, and iK,r; C and F: iCa,T, iK,s, and if).

Effect of Block of Ionic Currents

Effect of block of i_Na. Figure 8, A and B, shows the effect of block of i_Na on the action potential of the peripheral and central models. Figure 8, C and D, shows the effect of block of i_Na by 20 µM TTX on action potentials recorded from peripheral and central balls of rabbit SA node tissue (33). Qualitatively, the models behave in the same way as the tissue. In both cases (simulation and experiment), blocking i_Na had no effect on the central action potential. In the simulation, at least, this is because there is no i_Na in the central model. In contrast, blocking i_Na had various effects on the peripheral action potential: 1) the takeoff potential was shifted to a more positive value; 2) the maximum upstroke velocity was reduced (in the example of experimental data the maximum upstroke velocity was reduced from 100 to 5 V/s, and in the simulation it was reduced from 60 to 8 V/s; after block of i_Na, in simulation and experiment, the upstroke velocity in the periphery was approximately the same as that in the center); and 3) as a result of the change in the takeoff potential, spontaneous activity was slowed. In the model at least, the effects of block of i_Na can be explained by the important role of i_Na in the upstroke of the peripheral action potential. The effect of block of i_Na on spontaneous activity was less in the peripheral model than that seen experimentally in peripheral tissue (Fig. 8); this difference may be the result of the faster spontaneous activity of the peripheral model (which, in turn, may be the result of the model being developed for 37°C and the experimental data being obtained at 32°C). In the peripheral model, block of i_Na also reduced the peak value of the action potential, the action potential duration, and the maximum diastolic potential (Fig. 8A); these changes are also seen experimentally in small balls of tissue from the periphery of the rabbit SA node (Fig. 8C) (33). In conclusion, in simulation and experiment, pacemaking in the periphery, unlike that in the center, is sensitive to block of i_Na.

View larger version (24K): [in this window] [in a new window]   Fig. 8.   Effect of block of iNa on peripheral (A and C) and central (B and D) SA node action potentials. A and B: data from peripheral and central SA node cell models. C and D: data from Kodama et al. (33) from small balls of peripheral and central rabbit SA node tissue (balls A and D). Top panels: membrane potential; bottom panels: rate of change of membrane potential (dV/dt). Data are shown under control conditions and after block of iNa (TTX). In experiments, 20 µM TTX was used to block iNa.

Effect of block of i_Ca,L. Figure 9, A and B, shows the effect of the block of i_Ca,L on the action potential of the peripheral and central models. Figure 9, C and D, shows the effect of block of i_Ca,L by 2 µM nifedipine on action potentials recorded from peripheral and central balls of rabbit SA node tissue (33). Qualitatively, the models behave in the same way as the tissue. In both cases (simulation and experiment), block of i_Ca,L abolished the action potential in the center of the SA node: the membrane potential settled at 45 mV in the experiment (Fig. 9D) and at 42 mV in the simulation (Fig. 9B). In the periphery, block of i_Ca,L had different effects on electrical activity. In both cases (simulation and experiment), block of i_Ca,L 1) shortened the action potential, 2) increased the pacemaking rate [presumably as a consequence of the shortening of the action potential; in the simulation, block of i_Ca,L caused a 22% increase in the pacemaking rate; in experiments, 2 µM nifedipine caused a 21 ± 1% (SE) increase in the pacemaking rate in ball A (33)], 3) decreased the maximum upstroke velocity, and 4) decreased the peak value of the action potential. In the simulation, block of i_Ca,L caused a decrease in the maximum upstroke velocity from 60 to 40 V/s and a decrease in the peak of the action potential from +27 to +8 mV. In the experiment shown in Fig. 9C, on application of 2 µM nifedipine, the maximum upstroke velocity was decreased from 82 to 75 V/s and the peak of the action potential was decreased from +22 to +6 mV. In conclusion, in simulation and experiment, only pacemaking in the center is abolished on block of i_Ca,L.

View larger version (26K): [in this window] [in a new window]   Fig. 9.   Effect of block of iCa,L on peripheral (A and C) and central (B and D) SA node action potentials. A and B: data from peripheral and central SA node cell models. C and D: data from Kodama et al. (33) from small balls of peripheral and central rabbit SA node tissue (balls A and D). Top panels: membrane potential; bottom panels: rate of change of membrane potential. Data are shown under control conditions and after block of iCa,L (nifedipine). In experiments, 2 µM nifedipine was used to block iCa,L.

Effect of block of i_Ca,T. Hagiwara et al. (22) reported that block of i_Ca,T by 40 µM Ni²⁺ produced a 5-15% increase in the cycle length in rabbit SA node cells, and Lei et al. (38) reported a similar effect in rabbit SA node cells. No information is available from single cells or small balls of tissue on whether the effect of block of i_Ca,T differs in peripheral and central SA node tissue. Because there is also no information available on whether the density of i_Ca,T differs in peripheral and central SA node cells, the density of i_Ca,T is assumed to be the same in the two cell types (see MODEL DEVELOPMENT). In the peripheral and central models, block of i_Ca,T caused a small increase in cycle length (of 4 and 19%, respectively) similar to that reported experimentally.

Effect of block of 4-AP-sensitive current. Figure 10, A-C, shows the effect of block of 4-AP-sensitive current (i_to and i_sus) on peripheral and central action potentials. 4-AP has also been shown to block i_K,r partially (J. Hancox, unpublished observations). In the simulation of the effect of 4-AP, i_K,r was blocked by 10%. Figure 10, A and B, shows the effect of block of 4-AP-sensitive current on the action potential of the peripheral and central models. Figure 10C shows the effect of 5 mM 4-AP on action potentials recorded from peripheral and central balls of rabbit SA node tissue (6). Qualitatively, the models behave in the same way as the tissue. In both cases (simulation and experiment), block of 4-AP-sensitive current caused a prolongation of the action potential in the periphery and center. In experiments, 4-AP increased action potential duration by 66 ± 4% in peripheral tissue and by 25 ± 5% in central tissue (Ref. 6; data from balls A and E are quoted). In simulations, block of 4-AP-sensitive current caused a 50% increase in action potential duration in the peripheral model and a 21% increase in the central model. Block of 4-AP-sensitive current also caused an increase in the peak value of the action potential, an increase in cycle length in the periphery, and a decrease in cycle length in the center. In experiments, 4-AP caused a 28 ± 3% increase in cycle length in peripheral tissue and a 5 ± 2% decrease in cycle length in central tissue (Ref. 6; data for balls A and D are quoted). In simulations, 4-AP caused a 24% increase in cycle length in the peripheral model and a 3% decrease in cycle length in the central model. As discussed above, in experiments on the rabbit SA node, in some cases, after the upstroke of the action potential in the periphery there can be an initial rapid phase of repolarization (i.e., phase 1) (however, not in the experimental recording shown in Fig. 10C), and in such cases this was abolished by 4-AP (6). In the peripheral model, the initial rapid phase of repolarization was also abolished by block of 4-AP-sensitive current (Fig. 10A). In summary, simulations and experiments show that 4-AP-sensitive current plays a major role in action potential repolarization, and its role varies regionally.

View larger version (28K): [in this window] [in a new window]   Fig. 10.   Effect of block of 4-AP-sensitive current and iK,r on peripheral and central SA node action potentials. A and B: effect of block of 4-AP sensitive current on the peripheral (A) and central (B) SA node cell models. C: effect of 5 mM 4-AP on small balls of peripheral (top) and central (bottom) rabbit SA node tissue [balls A and D; from Boyett et al. (6)]. D and E: effect of block of iK,r by 50 and 100% (equivalent to low and high doses of E-4031) on the peripheral (D) and central (E) SA node cell models. F: effect of 0.1 µM E-4031 [a low dose of E-4031 expected to block ~50% of iK,r (32)] on small balls of peripheral (top) and central (bottom) rabbit SA node tissue [balls B and F; from Kodama et al. (32)]. Data are shown under control conditions and after block of 4-AP-sensitive current (4-AP) or iK,r (E-4031).

Effect of block of i_K,r. Figure 10, D and E, shows the effect of the block of 50 and 100% i_K,r (simulating the effects of low and high doses of E-4031) on the action potential of the peripheral and central models. Figure 10F shows the effect of partial block (perhaps ~50%) (32) of i_K,r by 0.1 µM E-4031 on action potentials recorded from peripheral and central balls of rabbit SA node tissue (33). Qualitatively, the models behave in the same way as the tissue. In both cases [simulation (Fig. 10, D and E) and experiment (not shown; see Ref. 32)], complete block of i_K,r (in experiments, by 1 µM E-4031) caused cessation of spontaneous activity in the periphery and center. After complete block of i_K,r, the membrane potential settled at 35 ± 2 mV in rabbit peripheral SA node tissue (Ref. 32, data for ball A), 32 ± 2 mV in rabbit central SA node tissue (Ref. 32, data for ball D), 33 mV in the peripheral model, and 30 mV in the central model. This shows that in the periphery and center, i_K,r is important for pacemaking. The i_K,r is responsible for generating the maximum diastolic potential, and thus when i_K,r is blocked, the membrane during diastole is depolarized and spontaneous activity ceases.

Effect of block of i_K,s. Complete block of i_K,s had little effect on the pacemaker activity of the peripheral and central models: there was a change in cycle length of 0.3 and 1% in the peripheral and central models. M. Lei and P. Kohl (unpublished observations) observed that block of i_K,s by 50 µM 293B had no significant effect on the electrical activity of rabbit SA node cells: cycle length was 256 ± 24 and 270 ± 26 (SE) ms (n = 5) under control conditions and in the presence of 293B, respectively.

Effect of block of i_f. Figure 11, A and C, shows the effect of block of i_f on the action potential of the peripheral and central models. Figure 11, B and D, shows the effect of block of i_f by 2 mM Cs⁺ on action potentials recorded from peripheral and central balls of rabbit SA node tissue (45). Qualitatively, the models behave in the same way as the tissue. In both cases (simulation and experiment), block of i_f slowed spontaneous activity, and the slowing was greater in the periphery. In experiments, block of i_f caused a 24 ± 2% increase in cycle length in peripheral tissue and a 7 ± 2% increase in central tissue (Ref. 44, data for balls A and D), whereas in simulations, block of i_f caused a 34% increase in the cycle length in the peripheral model and an 8% increase in the central model. In the simulations, the greater effect of block of i_f on the peripheral model can be explained by the greater density of i_f in the peripheral model: in the peripheral model, if the density of i_f was reduced to that in the central model, block of i_f caused only a 14% increase in the cycle length.

View larger version (27K): [in this window] [in a new window]   Fig. 11.   Effect of block of if on peripheral (A and B) and central (C and D) SA node action potentials. A and C: data from peripheral and central SA node cell models. B and D: data from Nikmaram et al. (45) from small balls of peripheral and central rabbit SA node tissue (balls A and D). Data are shown under control conditions and after block of if (Cs+). In experiments, 2 mM Cs+ was used to block if.

Resting potentials. In simulation and experiment, blocking i_Na and i_Ca,L (by 3 µM TTX and 6 µM nifedipine, respectively, in experiments) or blocking i_K,r (by 1 µM E-4031 in experiments) abolished spontaneous activity in the periphery and center (32). After block of spontaneous activity, SA node cells settle at their resting potential. Figure 12 shows the maximum diastolic potential as well as the resting potentials in the peripheral and central models (Fig. 12A) and peripheral and central balls of rabbit SA node tissue (Fig. 12B) (32); the two sets of data are comparable. In simulation and experiment, whereas there was a substantial difference between the periphery and center in the maximum diastolic potential, there was a smaller difference between the two in the resting potential when spontaneous activity was terminated by block of i_Na and i_Ca,L and little difference between the two in the resting potential when spontaneous activity was terminated by block of i_K,r. It was previously suggested that this showed that the difference in maximum diastolic potential between the periphery and center was the result of a difference in the density of i_K,r between the two (32). Consistent with this, in the models, if the density of i_K,r in the peripheral model was set to be the same as that in the central model, the difference in the maximum diastolic potential was abolished (not shown; the maximum diastolic potential was 42 mV).

View larger version (16K): [in this window] [in a new window]   Fig. 12.   Resting potentials in the periphery and center of the SA node. A: data from peripheral (solid bars) and central (hatched bars) SA node cell models. B: data from Kodama et al. (32) from small balls of peripheral (solid bars; balls A and B) and central (hatched bars; balls D-F) rabbit SA node tissue. Values are means ± SE. MDP, maximum diastolic potential; RP(TTX/Nif), resting potential after block of spontaneous activity by block of iNa and iCa,L; RP(E-4031), resting potential after block of spontaneous activity by block of iK,r. In experiments, iNa, iCa,L, and iK,r were blocked by 3 µM TTX, 6 µM nifedipine (Nif), and 1 µM E-4031, respectively.

One-Dimensional Model of the Intact SA Node

View larger version (44K): [in this window] [in a new window]   Fig. 13.   One-dimensional model of the SA node. A: schematic diagram of part of the one-dimensional model. SAN, SA node; AM, atrial muscle; Ds and Da, diffusion coefficients. B: action potentials recorded at regular intervals along the one-dimensional model from the center of the SA node (at 0 mm) to the junction of the SA node with the atrial muscle (3 mm) and into the atrial muscle (up to 12.6 mm from the center of the SA node). Arrow, point of initiation of the action potential (the leading pacemaker site). C: action potentials recorded from a central SA node cell (1) and an atrial cell (2) in the one-dimensional model. D-F: activation time (D), maximum upstroke velocity (dV/dtmax, E), and peak value of the action potential and maximum diastolic potential (F) plotted against the distance from the center of the SA node.

Figure 13B shows action potentials computed using the one-dimensional model of the intact SA node. Action potentials from various points along the string of tissue are shown. The junction of the SA node and atrium is at 3 mm. Figure 13B shows that spontaneous action potentials were first initiated in the center of the SA node (highlighted by the arrow) and then propagated to the periphery of the SA node and onto the atrial muscle. This is similar to the activation sequence seen experimentally (3). The cycle length in the one-dimensional model of the intact SA node is 350 ms; this is greater than the cycle length in the peripheral (160 ms) or central (330 ms) model. However, the cycle length in the one-dimensional model of the intact SA node is comparable to that observed experimentally in the intact SA node of the rabbit at 37°C: 348 ± 50 ms (30). Furthermore, at 33°C the cycle length in the intact SA node of the rabbit (586 ms) is greater than the cycle length in isolated peripheral (298 ± 8 ms) or central (331 ± 13 ms) tissue (Ref. 31, data for balls A and D). The SA node conduction time (time for the action potential to conduct out of the SA node) is 45 ms, similar to that seen experimentally (1). Figure 13C shows superimposed action potentials at a fast time base from the one-dimensional model: from the center of the SA node and the atrial muscle. Figure 13, D-F, shows various action potential parameters (activation time, i.e., the time taken for the action potential to propagate from the leading pacemaker site in the center, maximum upstroke velocity, peak value of the action potential, and maximum diastolic potential) plotted against the distance from the center of the SA node. The model data are comparable to experimental data (31). In summary, the one-dimensional model of the intact SA node shows a range of behaviors comparable to that of the intact SA node of the rabbit.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION REFERENCES

From the modeling, it is concluded that in the periphery compared with the center of the SA node 1) the takeoff potential is more negative and the maximum upstroke velocity is higher as a result of a higher density of i_Na, 2) the action potential can have a notch as a result of a higher density of i_to, 3) the action potential is short as a result of higher densities of i_K,r and 4-AP-sensitive current, 4) the maximum diastolic potential is more negative principally as a result of a higher density of i_K,r, and 5) the spontaneous activity is faster as a result of higher densities of i_Na and i_f as well as the shorter action potential.

After the pioneering work of Noble (47), much effort has been devoted to the development of mathematical models of cardiac cells based on voltage-clamp data. With the development of computing power and resources, these models, together with information about the structure of cardiac tissue, the electrical coupling between cells, cell orientation, and the heterogeneity in the electrophysiological properties of cardiac cells, are being used to develop biophysically accurate and detailed mathematical models of cardiac tissues (54); the aim is to develop a model of the whole heart. It is hoped that the models developed in the present study of action potentials in the periphery and center of the SA node can be used to develop a two- or three-dimensional model of the intact SA node, which then can be used in the development of a whole heart model.

Comparison With Previous Models

Achievements of the New Models

The fact that the densities of i_Na, 4-AP-sensitive current (i_to and i_sus), i_K,r, i_K,s, i_f, and possibly i_Ca,L (see MODEL DEVELOPMENT) decrease with a reduction of C_m and, therefore, possibly from the periphery to the center raises the possibility that the density of all currents decreases from the periphery to the center in the same way. Beyond the center (toward the interatrial septum) is a region of conduction block (see the introduction). Block of conduction is known to be the result of the loss of excitability rather than poor coupling between cells (4). It is possible that the loss of excitability is the result of a further decrease in the density of currents beyond the center (67).

Limitations of the Models

Model of the Intact SA Node

The intrinsic pacemaker activity of peripheral tissue from the rabbit SA node is faster than that of central tissue (33). Similarly, the pacemaker activity of large rabbit SA node cells is faster than that of small cells (27). However, in the intact SA node, the spontaneous action potential is first initiated in the center of the SA node (31). This is the result of the electrotonic suppression of the periphery of the SA node by the surrounding more hyperpolarized atrial muscle (the center of the SA node, being further from the atrial muscle, is less suppressed). The principal evidence for this is that if the atrial muscle is cut away from the rabbit SA node, the leading pacemaker site shifts to the periphery and the cycle length decreases on average from 350 to 290 ms (30). In the simulations, comparable behavior is observed. The spontaneous activity of the peripheral SA node cell model is faster than that of the central cell SA node model (Fig. 6), but in the one-dimensional model of the intact SA node it is the center that is the leading pacemaker site (Fig. 13). As in experiments, in the one-dimensional model, if the string of SA node tissue is uncoupled from the atrial tissue, the leading pacemaker site shifts to the peripheral cells and the cycle length decreases from 350 to 160 ms (not shown). Similar behavior has also been seen in earlier models of the intact SA node (60, 66).

The one-dimensional model of the rabbit SA node is already proving to be a useful tool: in preliminary studies we have shown pacemaker shift in the model, obtained data that suggest that i_Na in the periphery of the SA node helps the SA node drive the atrial muscle surrounding it, obtained data on the basis of which we have put forward a hypothesis to explain the deterioration of the SA node with age, and tested a hypothesis to explain the region of conduction block (63-65, 67). Although the one-dimensional model is a computationally efficient method of exploring the behavior of the intact SA node, it is only an approximation of the intact SA node, and in the future a more realistic model will have to be constructed, for example, to explore the asymmetric conduction of action potentials from the leading pacemaker site within the SA node.