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279:H397-H421, 2000.Am J Physiol Heart Circ PhysiolBoyettH. Zhang, A. V. Holden, I. Kodama, H. Honjo, M. Lei, T. Varghese and M. R.periphery and center of the rabbit sinoatrial nodeMathematical models of action potentials in the

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Mathematical models of action potentials in theperiphery and center of the rabbit sinoatrial node

H. ZHANG,1 A. V. HOLDEN,1 I. KODAMA,2 H. HONJO,2 M. LEI,1

T. VARGHESE,3 AND M. R. BOYETT11School of Biomedical Sciences, University of Leeds, Leeds LS2 9JT, United Kingdom;2Departments of Circulation and Humoral Regulation, Research Institute of Environmental Medicine,

Nagoya University, Nagoya 464-01, Japan; and 3Institute for Mathematics and Its Application,

University of Minnesota, Minneapolis, Minnesota 55455

Received 17 June 1999; accepted in final form 5 January 2000

Zhang, H., A. V. Holden, I. Kodama, H. Honjo, M. Lei,T. Varghese, and M. R. Boyett. Mathematical models ofaction potentials in the periphery and center of the rabbitsinoatrial node. Am J Physiol Heart Circ Physiol 279:H397H421, 2000.Mathematical models of the action po-tential in the periphery and center of the rabbit sinoatrial

(SA) node have been developed on the basis of publishedexperimental data. Simulated action potentials are consis-tent with those recorded experimentally: the model-gener-ated peripheral action potential has a more negative takeoffpotential, faster upstroke, more positive peak value, promi-nent phase 1 repolarization, greater amplitude, shorter du-ration, and more negative maximum diastolic potential thanthe model-generated central action potential. In addition, themodel peripheral cell shows faster pacemaking. The modelsbehave qualitatively the same as tissue from the peripheryand center of the SA node in response to block of tetrodotoxin-sensitive Na current, L- and T-type Ca2 currents, 4-ami-nopyridine-sensitive transient outward current, rapid andslow delayed rectifying K currents, and hyperpolarization-activated current. A one-dimensional model of a string of SA

node tissue, incorporating regional heterogeneity, coupled toa string of atrial tissue has been constructed to simulate thebehavior of the intact SA node. In the one-dimensionalmodel, the spontaneous action potential initiated in the cen-ter propagates to the periphery at 0.06 m/s and then intothe atrial muscle at 0.62 m/s.

heart; pacemaking; regional differences; computer modeling

THE RHYTHMIC BEATING of the heart is the result of actionpotentials initiated in the pacemaker of the heart, thesinoatrial (SA) node. Mathematical models of the elec-trical activity of the SA node of the rabbit (the species

for which most data have been obtained) have beenproduced. The first models were produced by Yanagi-hara et al. (61) and Noble and Noble (49), and subse-quent models were developed from the earlier models(13, 18, 59).

All the above models are of a typical SA node actionpotential. However, the SA node, functionally, anatom-ically, 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 horizontadirection it is bounded by the crista terminalis (a thickbundle of atrial muscle) and the interatrial septumThe action potential is initiated in a small part of theSA node, the leading pacemaker site. Normally, theleading pacemaker site is approximately midway between the two venae cavae and 12 mm from the cristaterminalis (3). This region is referred to as the center othe SA node. From the leading pacemaker site in thecenter, the action potential propagates to the peripheryof the SA node and then onto the atrial muscle of thecrista terminalis. Conduction toward the interatriaseptum is blocked (3). The periphery of the SA node(the region of the SA node close to the crista terminalisis referred to by some authors as perinode or transi-tional tissue. Although the principal function of theperiphery of the SA node is to conduct the action

potential from the leading pacemaker site in the centerto the atrial muscle, the periphery does show pacemaker activity. In response to a variety of interventions, for example, autonomic nerve stimulation, theleading pacemaker site shifts from the center, and inmany cases it shifts toward the periphery (53); thepacemaker activity of the periphery of the SA node istherefore, important physiologically. Most work on regional differences in the SA node has been carried outon tissue around the leading pacemaker site midwaybetween the venae cavae and has focused on peripheral-central differences (little is known about the tissue from the more superior and inferior regions andalso toward the interatrial septum). There are impor-

tant anatomic differences; for example, in the centerthe cells are smaller and have fewer and more poorlyorganized myofilaments than in the periphery (3)There are electrophysiological differences; these havebeen studied in the intact SA node or in small balls otissue from different regions of the SA node. In the

Address for reprint requests and other correspondence: M. R.Boyett, School of Biomedical Sciences, University of Leeds, LeedsLS2 9JT, UK (E-mail: [email protected]).

The costs of publication of this article were defrayed in part by thepayment of page charges. The article must therefore be herebymarked advertisement in accordance with 18 U.S.C. Section 1734solely to indicate this fact.

Am J Physiol Heart Circ Physio279: H397H421, 2000

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periphery the takeoff potential is more negative, theaction potential upstroke velocity is higher, the actionpotential is shorter, the maximum diastolic potential(also resting potential in quiescent tissue) is morenegative, and the intrinsic pacemaker activity is par-adoxically faster than in the center (33). Ion channelblock has different effects in the different regions:block of tetrodotoxin-sensitive Na current (iNa),

4-aminopyridine (4-AP)-sensitive transient outwardcurrent (ito), or hyperpolarization-activated current (if)has a greater effect in the periphery, whereas block ofL-type Ca2 current (iCa,L) or rapid delayed rectifyingK current (iK,r) has a greater effect in the center (6,32, 33, 45). These differences in the response to ionchannel block suggest regional differences in ionic cur-rents. Single cells have not been isolated from differentregions of the SA node to confirm this. However, weisolate single cells from the whole of the SA node andthen distinguish between cells on the basis of cellcapacitance (Cm), a measure of cell size, which isknown to vary between the periphery and the center(see above) (27, 28, 36, 39). The action potential char-acteristics vary with Cm in a manner consistent withthe regional differences (see above) (27). For example,in large cells with a high Cm (presumably from theperiphery) the upstroke velocity is high, whereas insmall cells with a low Cm (presumably from the center)the upstroke velocity is low (27). We have measuredthe density of some ionic currents; whereas the densityofiCa,L is not significantly different in cells of differentsize, the densities ofiNa, ito, iK,r, iK,s, and if are greaterin larger cells (27, 28, 36, 39; iK,r and iK,s data fromunpublished observations).

Models incorporating regional differences within theSA node have been developed (49, 60). However, the

models were based on speculation because of the ab-sence of data on regional differences in ionic currents.The aim of the present study was to develop, on thebasis of the evidence reviewed above, biophysicallydetailed models of action potentials in the peripheryand center of the rabbit SA node.

Glossary

4-AP 4-Aminopyridine AM Atrial muscle APD Action potential durationCL Spontaneous cycle lengthCm Cell capacitance

Cma

(x), Cms

(x) Capacitance of atrial muscle cellor SA node cell in one-dimen-sional model of intact SA nodeat distance x from center of SAnode

dL, dT Activation variables for iCa,L andiCa,T

dNaCa Denominator constant for iNaCadV/dtmax Maximum upstroke velocity of ac-

tion potentialDa, Ds Diffusion coefficient between atrial

muscle cells or SA node cells in

one-dimensional model of theintact SA node

EK,s Reversal potential for iK,sENa, ECa, EK Equilibrium potentials for Na

Ca2, and K

ECa,L, ECa,T Reversal potentials for iCa,L andiCa,T

F Faradays constant

FK,r Fraction of activation of iK,r thatoccurs slowly

FNa Fraction of inactivation ofiNa thatoccurs slowly

fL, fT Inactivation variables for iCa,Land iCa,T

gp, gc Conductance of a current in peripheral or central SA node celmodels

ga(x), gs(x) Conductance of a current in atriamuscle cell or SA node cell inone-dimensional model of intacSA node at distance x from cen

ter of SA nodegNa Conductance of iNagCa,L, gCa,T Conductance of iCa,L and iCa,Tgto, gsus Conductance of ito and isusgK,r, gK,s Conductance of iK,r and iK,sgf,Na, gf,K Conductance of Na

and K components of if

gb,Na, gb,Ca, gb,K Conductance of ib,Na, ib,Ca, andib,K

h1, h2 Fast and slow inactivation variables for iNa

h Net fractional availability of iNaiNa TTX-sensitive Na

currentiCa,L, iCa,T L- and T-type Ca

2 currents

ito, isus Transient and sustained components of 4-AP-sensitive current

iK,r, iK,s Rapid and slow delayed rectifyingK currents

iK Sum of iK,r and iK,sif Hyperpolarization-activated cur

rentif,Na, if,K Na

and K components of ifib,Na, ib,Ca, ib,K Background Na

, Ca2, and K

currentsiNaCa Na

/Ca2 exchanger currentip Na-K pump current

ip Maximum ip

itot Total ionic current in a cellitota (x), itot

s (x) Total ionic current in atrial muscle cell or SA node cell in onedimensional model of intact SAnode at distance x from centerof SA node

ist Sustained currentiK,ACh ACh-activated K

currentiK,ATP ATP-sensitive K

currentkNaCa Scaling factor for iNaCaKm,Na, Km,K Dissociation constants for Na

and K activation of ip

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L Length of string of SA node andatrial tissue in one-dimensionalmodel of intact SA node

Ls Length of string of SA node tissuein one-dimensional model of in-tact SA node

m Activation variable for iNaMDP Maximum diastolic potential

n Steady-state value of npa General activation variable for

iK,rpa,f, pa,s Fast and slow activation vari-

ables for iK,rpi Inactivation variable for iK,rQ10 Fractional change in a variable

with a 10C increase in temper-ature

r Activation variable for itoR Universal gas constantq Inactivation variable for itoSA node, SAN Sinoatrial nodet TimeT Absolute temperatureTOP Takeoff potentialV Membrane potentialVa, Vs Membrane potential of atrial

muscle cell or SA node cell inone-dimensional model of intactSA node

Va(x), Vs(x) Membrane potential of atrialmuscle cell or SA node cell inone-dimensional model of intactSA node at distance x from cen-ter of SA node

x Distance from center of SA node

in one-dimensional model of in-tact SA nodexs Activation variable for iK,sy Activation variable of ifz Valency of ion[Na]i, [Ca

2]i, Intracellular Na, Ca2, and K

[K]i concentrations[Na]o, [Ca

2]o, Extracellular Na, Ca2, and K

[K]o concentrationsn Voltage-dependent opening rate

constant of nn Voltage-dependent closing rate

constant of nNaCa Position of Erying rate theory en-

ergy barrier controlling voltagedependence of iNaCan Time constant of n Space constant

MODEL DEVELOPMENT

Mathematical models of the action potential in peripheraland central cells of the rabbit SA node at 37C were devel-oped using experimental data from rabbit SA node prepara-tions. New formulations for a number of ionic currents weredeveloped on the basis of newly published data from rabbit

SA node cells: iNa, iCa,L, iCa,T, ito, 4-AP-sensitive sustainedoutward current (isus), iK,r, iK,s, and if. Full details are givenbelow. The models also include formulations for backgroundcurrents (ib,Na, ib,Ca, and ib,K), ip, and iNaCa; these formula

tions are similar to those in other models (13, 17, 25). Themembrane potential is calculated using Eq. 2 (Table 1). TheGlossary defines all abbreviations used. Formulations foionic currents are shown in Tables 29. All parameter valuesare listed in Table 10. Differences in current densities between the peripheral and central SA node cell models arelisted in Table 11. Initial values of variables used to run themodels are listed in Table 12.

Table 1. General equations

itot iNa iCa,L iCa,T ito isus iK,r iK,s if

ib,Na ib,Ca ib,K iNaCa ip (1

dV

dt

1

Cmitot (2

ENaRT

zFln

([Na]o

[Na]i

)(3

ECaRT

zFln ([Ca

2]o

[Ca2]i) (4

EKRT

zFln ([K

]o

[K]i) (5Glossary defines all abbreviations used in equations.

Table 2. TTX-sensitive Na current (iNa)

iNagNam3h[Na]o

F2

RT

e(VENa)F/RT 1

eVF/RT 1V (6

h (1 FNa)h1 FNah2 (7

FNa9.52 102e6.310

2 (V 34.4)

1 1.66e0.225(V63.7) 8.69 102 (8

m ( 11eV/5.46)1/3

(9

m0.6247 103

0.832e0.335(V 56.7) 0.627e0.082(V 65.01) 4 10 5 (10

dm

dt

mm

m(11

h11

1e(V 66.1)/6.4(12

h2 h1 (13

h13.717 106e0.2815(V 17.11)

1 3.732 103e0.3426(V 37.76) 5.977 104 (14

h23.186 108e0.6219(V 18.8)

1 7.189 105e0.6683(V 34.07) 3.556 103 (15

dh1

dt

h1 h1

h1(16

dh2

dt

h2 h2

h2(17

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Model of iNa

The iNa was thought to be absent in SA node cells, andmost previous models of the SA node action potential do notinclude iNa. However, recent experimental results show that

iNa is present and physiologically important (27, 33). Demiret al. (13) introduced iNa in their model of the rabbit SA nodeaction potential; the formulation for iNa was based on theexperimental data of Colatsky (11) from rabbit Purkinjefibers. However, on the basis of the formulation for iNa fromDemir et al., the time dependence ofiNa is different from thaseen experimentally in rabbit SA node cells (27). The classicformulation for the Na current assumes that the Na conductance is controlled by m3h, in which m is the activation

variable and h is the inactivation variable (26). Most previouformulations for cardiac iNa have used the same term (17)However, recent voltage-clamp experiments on rabbit SAnode cells (27, 43) and other cardiac cells have shown that thetime course of recovery from inactivation can be best fitted by

Table 5. 4-AP-sensitive currents (ito and isus)

ito gtoqr(V EK) (40

q1

1e(V 59.37)/13.1(41

q 10.1 103

65.17 103

0.57e0.08(V 49) 0.24 104e0.1(V 50.93) (42

dq

dt

q q

q(43

r1

1e(V 10.93)/19.7(44

r 2.98 103

15.59 103

1.037e0.09(V 30.61) 0.369e0.12(V 23.84)(45

dr

dt

r r

r(46

isus gsusr(V EK) (47

Table 6. Rapid delayed rectifying K current (iK,r)

pa (1 FK,r)pa,f FK,rpa,s (48

iK,r gK,rpapi(V EK) (49

pa,f1

1e(V 14.2)/10.6(50

pa,s pa,f (51

pa,f1

37.2e(V 9)/15.9 0.96e(V 9)/22.5(52

pa,s1

4.2e(V 9)/17 0.15e(V 9)/21.6(53

dpa,f

dt

pa,fpa,f

pa,f(54

dpa,s

dt

pa,spa,s

pa,s(55

pi1

1e(V 18.6)/10.1(56

pi 0.002 (57

dpi

dt

pipi

pi(58

Table 3. L-type Ca2 current (iCa,L)

iCa,LgCa,L[fLdL0.006

1e(V 14.1)/6](VECa,L) (18)

dL14.19(V 35)

e(V 35)/2.5 1

42.45V

e0.208V 1(19)

dL

5.71(V 5)

e0.4(V

5) 1

(20)

dL1

dL dL(21)

dL1

1e(V 23.1)/6(22)

ddL

dt

dL dL

dL(23)

fL3.12(V 28)

e(V 28)/4 1(24)

fL25

1e(V 28)/4(25)

fL1

fL fL(26)

fL1

1e(V 45)/5(27)

dfL

dt

fLfL

fL(28)

Table 4. T-type Ca2 currents (iCa,T)

iCa,T gCa,T dTfT(V ECa,T) (29)

dT 1,068e(V 26.3)/30 (30)

dT 1,068e(V 26.3)/30 (31)

dT1

dT dT(32)

dT1

1e(V 37)/6.8(33)

ddT

dt

dT dT

dT(34)

fT 15.3e(V 71.7)/83.3 (35)

fT 15e(V 71.7)/15.38 (36)

fT1

fT fT(37)

fT1

1e(V 71)/9(38)

dfT

dt

fTfT

fT(39)


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two exponentials; therefore, there are two components ofinactivation; hence, two inactivation variables are needed. Inthe present formulation, three variables are used to govern

the kinetics of iNa: m and h1 and h2, a fast and a slowinactivation variable. The equations for iNa are listed inTable 2.

Activation and inactivation curves. Activation curves (cor-responding to the steady-state value ofm3) are shown in Fig.1A. The filled squares show the activation curve based ondata from Baruscotti et al. (2) from young rabbit SA nodecells at room temperature, and the filled triangles representdata from Muramatsu et al. (43) from cultured rabbit SAnode cells at 2224C (fits to the experimental data ratherthan the original data are shown). The solid line is themodel-generated activation curve, which fits well with thedata of Baruscotti et al. The dashed line is the activationcurve from a model of a rabbit atrial cell (41). The generalinactivation variable h is the weighted sum ofh1 and h2 (Eq.

7, Table 2). FNa is the fraction of inactivation that occursslowly and is dependent on the membrane potential; h1 andh2 change with different time constants but have the samesteady-state value. Inactivation curves (corresponding to thesteady-state value of h) are also shown in Fig. 1A. The opensquares show the inactivation curve based on data fromBaruscotti et al. from young rabbit SA node cells, and theopen triangles represent data from Muramatsu et al. fromcultured 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 isknown that ion channels can change in culture, and thisperhaps explains the difference. The model-generated inactivation curve (solid line) is closer to the data of Baruscotti etal. from young rabbit SA node cells. The dashed line is theinactivation curve from the model of a rabbit atrial cell (41)

Kinetics. Because of the rapid activation ofiNa, study ofiNaactivation is difficult, and there are no data from rabbit SA

node cells on the time constant of activation (m). Data fromthe study of Brown et al. (8) on rat ventricular cells wereused. Inasmuch as the experiments of Brown et al. werecarried out at room temperature (22C), a Q10 of 1.7 (41) wasused to correct the data for 37C. The m is plotted as afunction of membrane potential in Fig. 1B, in which thecircles show the temperature-corrected experimental dataand the solid line was generated by the model. In Fig. 1 C, thetime constant of fast inactivation (h1) is plotted as a functionof membrane potential. The formulation for h1 was based ondata from Muramatsu et al. (43) from cultured rabbit SAnode cells (circles in Fig. 1C), Honjo et al. (27) from rabbit SAnode cells (triangles), and Brown et al. from rat ventricularcells (squares). A Q10 of 1.7 (41) was used to correct theexperimental data (collected at 22C) for 37C (the tempera

ture-corrected data are shown in Fig. 1C). In Fig. 1C, thesolid line was generated by the model and the dashed line isfrom the model of a rabbit atrial cell (41). In Fig. 1D, the timeconstant of slow inactivation (h2) is plotted as a function omembrane potential. The formulation for h2 was based ondata from Muramatsu et al. from cultured rabbit SA nodecells (circles in Fig. 1D) and Brown et al. from rat ventricularcells (squares). A Q10 of 1.7 (41) was used to correct theexperimental data (collected at 22C) for 37C (the temperature-corrected data are shown in Fig. 1D). In Fig. 1D, thesolid line was generated by the model and the dashed line isfrom the model of a rabbit atrial cell (41). In Fig. 1E, thefraction of slow inactivation (FNa) is plotted as a function omembrane potential. The formulation for FNa was based ondata from Muramatsu et al. from cultured rabbit SA node

cells. In Fig. 1E, the circles show experimental data, and thesolid line was generated by the model. Over a wide range ofmembrane potentials, FNa is 10% of total inactivation oiNa. Inasmuch as the density ofiNa is large in a peripheral SAnode cell, 10% ofiNa that inactivates slowly will contributea substantial inward current during the early period of theaction potential.

Simulated current. Figure 1F shows simulated iNa fromthe peripheral SA node cell model during depolarizing voltage-clamp pulses as well as the current-voltage relationshipofiNa from the model (solid line and filled squares). The opencircles show the experimental data of Honjo et al. (27) from arabbit SA node cell with a Cm of 54.5 pF.

Table 7. Slow delayed rectifying K current (iK,s)

iK,s gK,sxs2(V EK,s) (59)

EK,sRT

Fln ([K

]o 0.12[Na]o

[K]i 0.12[Na]i) (60)

xs14

1e(V 40)/9(61)

xs eV/45 (62)

xsxs

xs xs(63)

xs1

xs xs(64)

dxs

dt

xsxs

xs(65)

Table 8. Hyperpolarization-activated current (if)

if if,Na if,K (66)

if,Na gf,Nay(V ENa) (67)

if,K gf,Ky(V EK) (68)

y e(V 78.91)/26.62 (69)

y e(V 75.13)/21.25 (70)

yy

y y(71)

y1

y y(72)

dy

dt

yy

y(73)

Table 9. Background, pump, and exchanger currents

ib,Na gb,Na(V ENa) (74

ib,K gb,K(V EK) (75

ib,Ca gb,Ca(V ECa) (76

iNaCa kNaCa[Na]i

3[Ca2]oe0.03743VNaCa

[Na]o3[Ca2]ie

0.0374V(NaCa 1)

1 dNaCa([Ca2]i[Na

]o3 [Ca2]o[Na

]i3)

(77

ip ip ( [Na]i

Km,Na [Na]i)

3

( [K]o

Km,K [K]o)

21.6

1.5e(V 60)/40(78


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Models of iCa,L and iCa,T

The equations for iCa,L are listed in Table 3. Activation andinactivation curves [corresponding to the steady-state valuesof the activation variable (dL) and the inactivation variable(fL)] are shown in Fig. 2A. The squares and circles show datafrom Hagiwara et al. (22) and Fermini and Nathan (19),respectively, from rabbit SA node cells at 3637C, and thesolid lines were generated by the model. For the time con-stants of activation and inactivation, we followed Demir et al.(13) and used the data of Nilius (46) from guinea pig SA nodecells after temperature correction (from 25 to 37C with a Q10of 2.3). Figure 2C shows simulated iCa,L from the peripheral

SA node cell model during depolarizing voltage-clamp pulsesFigure 2D shows current-voltage relationships for iCa,L (circles); the open circles show data from Hagiwara et al. from arabbit SA node cell, and the filled circles (and solid line) showdata from the peripheral SA node cell model.

The equations for iCa,T are listed in Table 4. Activation and

inactivation curves [corresponding to the steady-state valuesof the activation variable (dT) and the inactivation variable(fT)] are shown in Fig. 2B. The squares, circles, and trianglesshow data from Hagiwara et al. (22), Fermini and Nathan(19), and Lei et al. (38), respectively, from rabbit SA nodecells at 37C. In the model the activation and inactivationcurves (solid lines) were computed using equations (Eqs. 33and 38, Table 4) formulated by Lei (35) and Lei et al. (38based on their experimental data from rabbit SA node cells at

Table 10. Parameter values

Peripheral SA Node Cell Model Central SA Node Cell Model

Ratio Absolute value Normalized value Absolute value Normalized value

Cm 65 pF 20 pFdNaCa 0.0001 0.0001ECa,L 46.4 mV 46.4 mVECa,T 45 mV 45 mV

gNa 1.2 10

6

S 1.85 10

8

S/pF 0 S 0 S/pF gCa,L 6.59 102 S 1.0 103 S/pF 0.58 102 S 2.90 104 S/pF 3.45

gCa,T 1.39 102 S 2.14 104 S/pF 0.43 102 S 2.14 104 S/pF 1

gto 36.49 103 S 5.6 104 S/pF 4.91 103 S 2.5 104 S/pF 2.33

gsus 1.14 102 S 1.8 104 S/pF 6.65 105 S 3.3 106 S/pF 54.55

gK,r 1.60 102 S 2.46 104 S/pF 7.97 104 S 3.99 105 S/pF 6.17

gK,s 1.04 102 S 1.6 104 S/pF 5.18 104 S 2.59 105 S/pF 6.17

gf,Na 0.69 102 S 1.05 104 S/pF 0.0548 102 S 0.27 104 S/pF 3.93

gf,K 0.69 102 S 1.05 104 S/pF 0.0548 102 S 0.27 104 S/pF 3.93

gb,Na 1.89 104 S 2.9 106 S/pF 5.8 105 S 2.91 106 S/pF 1

gb,Ca 4.3 105 S 6.61 107 S/pF 1.32 105 S 6.62 107 S/pF 1

gb,K 8.19 105 S 1.3 106 S/pF 2.52 105 S 1.3 106 S/pF 1

ip 0.16 nA 2.46 103 nA 4.78 102 nA 2.46 103 nA 1kNaCa 0.88 10

5 nA 1.36 107 nA 0.27 105 nA 1.36 107 nA 1[Na]o 140 mM 140 mM[Na]i 8 mM 8 mM[Ca2]o 2 mM 2 mM

[Ca2]i 0.0001 mM 0.0001 mM[K]o 5.4 mM 5.4 mM[K]i 140 mM 140 mMKm,K 0.621 0.621Km,Na 5.64 5.64NaCa 0.5 0.5

*Ratio of normalized conductance or current in peripheral to central SA node cell model.

Table 11. Current densities in theSA node cell models

Current Density, pA/pF

Ratio*Potential,

mVPeriphery Center

iNa 78 0 10

iCa,L 11.15 3.23 3.45 0iCa,T 5.60 5.60 1 10ito 36.0 15.45 2.33 50isus 29.45 0.54 54.55 50iK,r 5.97 0.97 6.17 10iK,s 4.08 0.66 6.17 40if 13.2 3.36 3.97 110ib,Na 0.43 0.43 1 100ib,K 0.002 0.002 1 100ib,Ca 0.13 0.13 1 100iNaCa 0.013 0.013 1 100ip 0.098 0.098 1 100

*Ratio of current densities in peripheral to central SA node cellmodel.

Table 12. Initial values

Peripheral SA Node Cell Model Central SA Node Cell Mode

V, mV 64.35 51.44

m 0.124h1 0.595h2 5.250 10

2

dL 8.450 102 5.912 102

fL 0.987 0.825dT 1.725 10

2 0.106fT 0.436 0.119y 5.280 102 3.775 102

r 1.970 102 3.924 102

q 0.663 0.358n 7.670 102 5.700 102

pa,f 0.400 0.470pa,s 0.327 0.637pi 0.991 0.965


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37C. In the model we used equations (Eqs. 3032 and 3537,Table 4) formulated by Hagiwara et al. (22) for the timeconstants of activation and inactivation (dT and fT) inrabbit SA node cells. Figure 2C shows simulated iCa,T duringdepolarizing voltage-clamp pulses. Figure 2D shows current-

voltage relationships for iCa,T (squares); the open squaresshow data from Hagiwara et al. from a rabbit SA node cell,and the filled squares (and solid line) show data from theperipheral SA node cell model.

Model of 4-AP-Sensitive Current

Previous models of the SA node action potential did notincorporate ito. However, ito is now known to be present in therabbit SA node and to play an important role (6, 28, 39). Theito is known to be blocked by 4-AP. In rabbit SA node cells,4-AP blocks a transient outward current as well as a sus-tained outward current. It is unclear whether the transientand sustained components represent two phases of one cur-rent or two separate currents (28). We chose to treat the twocomponents as separate mathematical entities: ito and isus.Honjo et al. (28) found no difference in the activation curvesfor ito and isus, and therefore in the model we used the sameactivation variable (r) for ito and isus. Of course, the inacti-

vation variable (q) only governs ito. The equations for ito andisus are listed in Table 5.

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, filledsquares, and filled diamonds show data from Honjo et al. (28from rabbit SA node cells with capacitances of 63.4, 34.5, and20.3 pF at 25C; the activation curves are for the sum of itand isus. The filled hexagons show data from Lei et al. (39from rabbit SA node cells at 35C; the activation curve is fo

ito only. The filled circles show data from Giles and vanGinneken (21) from rabbit crista terminalis cells at 20.5Cbecause of the method used, the activation curve is for thesum of ito and isus (if the latter was present). The solid lineshows the activation curve generated by the model. Inactivation curves for ito only (corresponding to the steady-state

value ofq) are also shown in Fig. 3A. Inactivation curves areshown from Honjo et al. (28) from rabbit SA node cells (opentriangles, open squares, open diamonds: data from cells withcapacitances of 63.4, 47.1, and 23.6 pF, respectively), Lei etal. from rabbit SA node cells (open hexagons), Giles and vanGinneken from rabbit crista terminalis cells (open circles)and the present model (solid line). The model-generated

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 slowinactivation (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 of60 mV (top) and current-voltagerelationships for iNa (bottom). For the current-voltage relationships, iNa was measured as peak inward current. G:

density ofiNa (measured from the peak inward current during a pulse to 5 mV) plotted against cell capacitance(Cm). E, Data from Honjo et al. (27) from rabbit sinoatrial (SA) node cells; dotted line, regression line; s, values usedin the peripheral (Cm 65 pF) and central (Cm 20 pF) SA node cell models.


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inactivation curve is close to the data of Lei et al. from rabbitSA node cells at 35C.

Kinetics. The time constant of activation ofito and isus (r)is shown in Fig. 3B. The circles show data from Giles and vanGinneken (21) from rabbit crista terminalis cells. Inasmuchas the experimental data were collected at 24C, a Q10 of 2.18

(41) was used to correct the data for 37C (the temperature-corrected data are shown in Fig. 3B). The solid line wasgenerated by the model. Figure 3C shows the time constantof inactivation ofito (q). The squares and triangles show datafrom Honjo et al. (28) from rabbit SA node cells for the fastand slow components of inactivation [data collected at 25Ccorrected for 37C with a Q10 of 2.18 (41)]. The circles showdata from Giles and van Ginneken from rabbit crista termi-nalis cells [data collected at 20.5C corrected for 37C witha Q10 of 2.18 (41)]. The solid line was generated by the model.

Simulated current. Figure 3D shows simulated 4-AP-sen-sitive current (ito isus) from the peripheral SA node cellmodel during depolarizing voltage-clamp pulses. The cur-

rents are similar to 4-AP-sensitive currents in rabbit SA nodecells (28, 39). Current-voltage relationships for ito are shownin Fig. 3E. The open circles show data from Lei et al. (39from rabbit SA node cells, and the filled circles (and solidline) show data from the peripheral SA node cell model.

Model of iK,r

Recent experiments have shown that iK in rabbit SA nodecells (29, 37, 51) can be separated into two kinetically different components, iK,r and iK,s. A formulation for the delayedrectifying K current with two components, iK,r and iK,s, wasconstructed; the equations are listed in Tables 6 and 7Equations for iK,r take the general form suggested by Shibasaki (56) with an activation variable (pa) and inactivation

variable (pi).Activation and inactivation curves. Activation and deacti

vation ofiK,r in rabbit SA node cells have double-exponentiatime courses (37, 51). To model this, we have used two

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 sym-bols) curves for iCa,T; error bars, SE. C: simulatediCa,L (300-ms voltage-clamp pulses to 30 to 40mV in 10-mV increments from a holding potential of40 mV) and iCa,T (300-ms voltage-clamp pulses to70 to 10 mV in 10-mV increments from a holding

potential of 80 mV). D: current-voltage relation-ships for iCa,L (circles) and iCa,T (squares); iCa,L andiCa,T were measured as peak inward current. E:density of iCa,L (measured from the peak inwardcurrent during a pulse to 0 mV) plotted against Cm.E, Data from Honjo et al. (27) from rabbit SA nodecells; s, values used in the peripheral (Cm 65 pF)and central (Cm 20 pF) SA node cell models.


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activation variables: a fast activation variable (pa,f) and aslow activation variable (pa,s). The general activation vari-able (pa) is the weighted sum of the fast and slow activation

variables (Eq. 48, Table 6). In rabbit SA node cells, experi-mental data have shown no distinct dependence of the frac-tion of inactivation that occurs slowly (FK,r) on membranepotential, and the ratio of the slow to the fast component ofactivation ofiK,r is 2:3 (37). In the model, FK,r is assumed to

be constant with a value of 0.4. Figure 4A shows activationcurves corresponding to the steady-state value of pa (weassume that pa,f and pa,s are the same and equal to pa). Thetriangles represent data from Lei and Brown (37) from rabbitSA node cells at 37C (fit to the experimental data ratherthan the original data is shown). The solid line was generatedby the present model and is close to the data of Lei andBrown. In a model of a guinea pig ventricular cell, Noble et el.(50) assumed that the steady-state values ofpa,f and pa,s aredifferent, and in the right-hand part of Fig. 4A the longdashed and short dashed lines show the dependence of pa,fand pa,s, respectively, from the model of Noble et al. onmembrane potential.

Inactivation variable. Figure 4A shows inactivation curves(corresponding to the steady-state value of pi). In the modethe inactivation curve (solid line) was computed using anequation (Eq. 56, Table 6) formulated by Ito and Ono (29, 51based on their data (circles) from rabbit SA node cells a33C. The dashed line is from the model of a guinea pig

ventricular cell (50). There appears to be a nearly 18-mVshift in the inactivation curve in rabbit SA node cells com

pared 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 modethe time constants (solid lines) were computed using equations (Eqs. 52 and 53, Table 6) formulated by Ono and Ito (51based on their experimental data (circles) for rabbit SA nodecells. The dashed lines in Fig. 4, B and C, show the timeconstants used for the guinea pig ventricular cell model oNoble et al. (50). It appears that pa,f and pa,s are larger inguinea pig ventricular cells than in rabbit SA node cells. Noexperimental data are available on the relationship betweenthe time constant of inactivation (pi) and potential in the

Fig. 3. Transient and sustained components o4-aminopyridine (4-AP)-sensitive currents (ito andisus). A: activation (r, filled symbols) and inactivation (q

, open symbols) curves. B: time constant o

activation (r). C: time constant of inactivation (q)D: simulated 4-AP-sensitive current during 200-m voltage-clamp pulses to 70 to 60 mV (in 10-mVincrements) from a holding potential of 80 mV(top, voltage-clamp protocol; bottom, current). Ecurrent-voltage relationship for ito (measured apeak outward current at the start of the pulse andthe current at the end of the pulse; currents havebeen normalized to the maximum current at 60mV); error bars, SE. F and G: densities of ito (F

measured as the difference between the peak 4-APsensitive outward current during a 200-ms pulse to50 mV from a holding potential of 80 mV and thecurrent at the end of the pulse) and isus (G, measured as the 4-AP-sensitive current at the end of a200-ms pulse to 50 mV from a holding potential o80 mV) plotted against Cm. E, Data from Lei et al(39) from rabbit SA node cells; dotted lines, regression lines; s, values used in the peripheral (Cm 65pF) and central (Cm 20 pF) SA node cell models.


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rabbit SA node. Here we assume that pi is independent ofthe membrane potential and has a fixed value of 0.002 s, assuggested by Ono and Ito.

The iK,r has been consistently observed to reverse directionat the K equilibrium potential (EK) (51); therefore, only K

is assumed to carry iK,r, and the reversal potential is equalto EK.

Simulated current. Figure 4F shows simulated iK,r fromthe peripheral SA node cell model during depolarizing volt-age-clamp pulses. Figure 4Hshows the current-voltage rela-tionship of iK,r (iK,r measured at the end of the pulse). Theopen circles show data from Ito and Ono (29) from rabbit SAnode cells, and the filled circles (and solid line) show datafrom the peripheral SA node cell model. The current-voltagerelationship from the model is roughly similar to the exper-imental data.

Model of iK,s

The slow sigmoidal activation of iK,s is modeled by

squaring a gating variable (xs). Figure 4D shows the acti vation curve (corresponding to the steady-state value oxs

2). The circles show data from Lei and Brown (37) fromrabbit SA node cells at 37C, and the solid line is theactivation curve generated by the model. There are limitedexperimental data available for the time constant of theactivation (xs) of iK,s in rabbit SA node cells [triangle inFig. 4E (37)]. Instead, we used equations (Eqs. 61, 62, and64, Table 7) formulated by Heath and Terrar (24) based ontheir data (circles) from guinea pig ventricular cells a3536C, as shown in Fig. 4E; in Fig. 4E the solid line wasgenerated using the model. The iK,s has been observed toreverse at voltages positive to EK, which suggests that the

Fig. 4. Rapid and slow delayed rectifying currents (iK,r and iK,s). A: activation (pa, filled symbols) andinactivation (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 constantof activation ofiK,s. Fand G: simulated iK,r (F) and iK,s (G) during 1-s pulses applied to potentials between 50and 40 mV (in 10-mV increments) from a holding potential of 60 mV. Hand I: current-voltage relationshipsfor iK,r (H) and iK,s (I). Currents were measured at the end of the voltage-clamp pulses. J and K: densities ofiK,r (J, measured as the peak 3 M E-4031-sensitive tail current after a 1-s pulse to 10 mV from a holdingpotential of50 mV) and iK,s (K, measured as the peak 3 M E-4031-insensitive tail current after a 1-s pulseto 40 mV from a holding potential of 50 mV) plotted against Cm. E, Data from M. Lei (unpublishedobservations) from rabbit SA node cells; dotted lines, regression lines; s, values used in the peripheral (Cm 65 pF) and central (Cm 20 pF) SA node cell models.


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iK,s channel is permeable to an ion in addition to K (23).

We assumed that iK,s is permeable to a small extent toNa. The reversal potential of iK,s is shown in Table 7.Figure 4G shows simulated iK,s from the peripheral SAnode cell model during depolarizing voltage-clamp pulses.Figure 4Ishows a current-voltage relationship for iK,s fromthe peripheral SA node cell model.

Model of if

The if is a mixed current and is carried by Na and K. In

our formulation, if has two components, if,K and if,Na. Theequations are listed in Table 8. Figure 5A shows activationcurves (corresponding to the steady-state value of y). In Fig.5A, the triangles and squares show data from Liu et al. (42)from rabbit SA node cells at 3536C, the circles show datafrom van Ginneken and Giles (57) from rabbit SA node cellsat 3033C, and the solid line was generated by the model.Figure 5B shows the time constant of activation of if (y). InFig. 5B, the triangles and circles show data from Liu et al.and van Ginneken and Giles, respectively, from rabbit SAnode cells, and the solid line was generated by the model.Figure 5C shows simulated if from the peripheral SA nodecell model during hyperpolarizing voltage-clamp pulses. Fig-ure 5D shows current-voltage relationships for i

f

. The opentriangles show data from Honjo et al. from a rabbit SA nodecell at 36C with a Cm of 44.7 pF, and the open circles showdata from Honjo et al. from a smaller rabbit SA node cell at

36C with a Cm of 20.2 pF. The filled circles (and solid lineshow data from the peripheral SA node cell model (Cm 65pF), and the filled squares (and solid line) show data from thecentral SA node cell model (Cm 20 pF). The threshold ofiin the model is 40 mV, which is in the range of that seenexperimentally in rabbit SA node cells (1416, 57) but morepositive than that seen by Honjo et al.

Regional Differences in Ionic Current Densities

The variation in electrical activity from the periphery tothe center of the SA node (as described in the introductioncould be the result of a gradual decrease in the electrotonicinfluence of the surrounding atrial muscle (with more hyperpolarized diastolic potentials) from the periphery to the center. However, experiments using ligated or dissected smalpieces of tissue from different regions of the rabbit SA node(in which the large mass of surrounding atrial muscle iremoved) also show marked differences in the electrophysiological characteristics of the tissue from the periphery andcenter of the SA node (33). Immunocytochemical studies haveshown that atrial cells are intermingled with SA node cells inthe periphery of the SA node in several species (52). It hasbeen conjectured (58) that the electrophysiological propertiesof pacemaker cells in the SA node are roughly uniformthroughout the SA node, and the apparent regional differences in electrical activity in the intact SA node are the resultof a progressive decrease in the percentage of intermingling

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 holdingpotential of40 mV (top, voltage-clamp protocol; bottom, if). D: current-voltage relationships for if; ifwas measuredas the increase in inward current from the beginning to the end of the pulses. E: density of if (measured duringpulse to 110 mV) plotted against Cm. E, data from Honjo et al. (27) from rabbit SA node cells; dotted lines,regression lines; s, values used in the peripheral (Cm 65 pF) and central (Cm 20 pF) SA node cell models.


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atrial cells toward the center, giving rise to a progressivedecrease in their hyperpolarizing influence from the periph-ery toward the center. However, we have little evidence ofsuch intermingling in the rabbit SA node (12). Furthermore,in single SA node cells from the rabbit, we record a range ofelectrical activities similar to that observed in the differentregions of the intact SA node (27). Therefore, we believe thatthe properties of rabbit SA node cells are not uniform, assuggested by others (58). The size of cells varies from the

periphery to the center of the rabbit SA node: cells are largein the periphery and small in the center (in the center, thecells are 8 m diameter and 2530 m long) (3). We haveshown that the electrical activity of rabbit SA node cells iscorrelated with cell size (as measured by Cm), and, as ex-pected, large cells have properties characteristic of the pe-riphery of the SA node and small cells have properties char-acteristic of the center (see the introduction). The densities ofa number of ionic currents are dependent on cell size (see theintroduction and below). In construction of the models, weassumed that a peripheral cell is large in size, with a Cm of 65pF, and a central cell is small in size, with a Cm of 20 pF.These values ofCm represent the approximate maximum andminimum values measured experimentally for rabbit SAnode cells (27, 28, 36, 39).

Ionic current densities for the peripheral and central SAnode cell models are listed in Table 11. For some but not allcurrents, the relationship between current density and Cmhas been determined for rabbit SA node cells, and the finalpanels of Figs. 15 show the densities of various currentsplotted against Cm. The open circles are experimental data(27, 28, 39; iK,r and iK,s data from unpublished observations),and the filled squares are the chosen values for the periph-eral and central SA node cell models. The experimentallymeasured densities ofiNa, ito, isus, iK,r, iK,s, and if are signif-icantly correlated with Cm and are larger in cells with higherCm (Figs. 1 and 35) (27, 28, 36, 39; iK,r and iK,s data fromunpublished observations). Values chosen for the densities ofthese currents for the peripheral and central SA node cellmodels are within the experimental range and are greater in

the peripheral SA node cell model. Although the density ofiCa,L measured experimentally is not significantly correlatedwith Cm (27), it was necessary to increase the density ofiCa,Lin the peripheral SA node cell model compared with that inthe central SA node cell model (Fig. 2E). Despite this, thechosen values for the density of iCa,L in the peripheral andcentral SA node cell models are reasonably consistent withthe experimental values (Fig. 2E).

During the development of the models, there were noexperimental data available on the relationship between thedensities ofiK,r and iK,s and Cm for rabbit SA node cells. Somedata were available: Lei and Brown (37) had reported thatthe ratio of the amplitude ofiK,r to iK,s tails after a 1-s pulseto 40 mV from a holding potential of40 mV was 1:0.30.4,and, therefore, in the peripheral and central SA node cell

models, the ratio was set at 1:0.3. Although there were nodirect measurements of the density of iK,r in rabbit SA nodecells with different values of Cm, various indirect lines ofevidence suggested that the density of iK,r is greater inperipheral than in central SA node cells. First, althoughcomplete block of iK,r by 1 M E-4031 abolishes pacemakeractivity in tissue from the periphery and center of the rabbitSA node, partial block of iK,r by 0.1 M E-4031 abolishespacemaker activity in central but not peripheral tissue (32).One explanation for this different response to partial block ofiK,r is that the density ofiK,r is greater in the periphery of theSA node (32). Second, an immunocytochemical study exam-ined the distribution of ERG (channel protein responsible for

iK,r) in the ferret SA node (7). If the anatomy of the SA nodein the ferret is similar to that of the rabbit, the data from thestudy support the possibility that the density ofiK,r is greaterin the periphery of the SA node: the labeling of ERG was littlein the intercaval region (where the center of the SA node islocated in the rabbit at least) and substantial close to thecrista terminalis (where the periphery of the SA node ilocated in the rabbit at least). Finally, in the models operipheral and central SA node action potentials, it wanecessary to increase the density of iK,r in the peripheral celmodel compared with that in the central cell model to obtainsome of the characteristic differences between the two regions (see Fig. 10). After the development of the modelsexperimental measurements of the densities of iK,r and iK,became available (Fig. 4, J and K, open circles) from ourlaboratory in rabbit SA node cells (unpublished observations). Although the densities ofiK,r and iK,s in the peripheraand central SA node models were set before the experimentadata became available, they are within the experimentarange obtained from rabbit SA node cells.

In the case of currents for which there are no data concerning current density and Cm, the current density wasassumed to be the same in peripheral and central SA node

cells.

Intracellular Ionic Concentrations

Previous SA node models have included computation oconcentrations of intracellular Na and Ca2 (13, 18, 49, 59)In the present models, all intracellular ion concentrations areassumed to be constant (see Table 10 for values). There are

various reasons for this: 1) little is known about intracellularNa in the SA node, and there is only one report aboutintracellular Ca2 in the rabbit SA node (40); 2) nothing isknown about differences in intracellular Na and Ca2 handling between the periphery and center of the SA node; 3intracellular Na changes only slowly over several minutes

and over a few beats intracellular Na

is approximatelyconstant; 4) buffering intracellular Ca2 with 1,2-bis(2aminophenoxy)ethane-N,N,N,N-tetraacetic acid-AM or reducing the amplitude of the intracellular Ca2 transient withryanodine does not abolish spontaneous activity in rabbit SAnode cells (40), although it can produce a decrease in the rateof spontaneous action potentials (40); and 5) much of our dataconcerning the dependence of electrical activity and the density of ionic currents on Cm was obtained from rabbit SA nodecells in which intracellular Ca2 was buffered with EGTA(27, 28). Because of points 1 and 2, inclusion of intracellularNa and Ca2 handling in the models would be based onspeculation to a large degree, and, because of points 3 5inclusion of intracellular Na and Ca2 handling in themodels is not essential for reasonably accurate calculation oelectrical activity (over a few beats at least). Inclusion ointracellular Ca2 handling may be important if inwardiNaCa triggered by the intracellular Ca

2 transient plays asignificant role in electrical activity: the slowing of spontaneous activity by 1,2-bis(2-aminophenoxy)ethane-N,N,N,Ntetraacetic acid-AM or ryanodine could indicate such a rolebut even in these cases the slowing may be the result ofactions on other currents (40); the role of inward iNaCa triggered by the intracellular Ca2 transient in the SA node istherefore, unclear. Despite these arguments that the inclusion of intracellular Na and Ca2 handling is not crucialfuture models will have to address this shortcoming.


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Comparison of the Models of Peripheral and Central SANode Action Potentials With Action Potentials MeasuredExperimentally From the Rabbit SA Node

After construction of the models as described above, theconfiguration of the simulated action potentials was com-pared with the configuration of action potentials measuredexperimentally from rabbit SA node cells (with different

values ofCm) at 35C (27) and from small balls of tissue from

the periphery and center of the rabbit SA node at 32C (33).The experimental data from rabbit SA node cells with vari-ous values ofCm at 35C (27) are similar to the experimentaldata from small balls of tissue from the periphery and centerof the rabbit SA node at 32C (33), although the actionpotential is shorter and the rate of spontaneous activity isfaster in the single cells. We assume that these differencesare the result of the higher temperature at which the exper-iments on single cells were conducted; for this reason, whenmodel and experimental data were compared, more impor-tance was placed on the data from single cells. In addition tothis, the response of the simulated peripheral and centralaction potentials to block of various currents was calculatedand compared with the experimentally measured response ofperipheral and central action potentials to various blockers.

If there were differences, we adjusted the densities of appro-priate currents until the simulated and experimental resultswere more similar. Although the current densities in themodels were adjusted in this manner, the final current den-sities had to be consistent with current densities measuredexperimentally (the scope for change was, therefore, limited).The models finally developed do not produce action potentialsthat are exact matches of any one experimental recording;instead, the action potentials have characteristics within therange observed experimentally and behaviors to ion channelblock within the range observed experimentally.

One-Dimensional Model of the SA Node andSurrounding Atrial Muscle

On the basis of the models developed for peripheral and

central SA node cells, a one-dimensional, partial differentialequation, multicellular model for the SA node and atriumwas developed. The equations for the one-dimensional modelare listed in Table 13. In the model the multicellular SA nodeand atrium are modeled as a string of tissue with a length (L)of 12.6 mm; of this the string of SA node tissue has a length(Ls) of 3 mm [similar to the distance from the center of the SAnode to the atrial muscle in the rabbit (3)], and the string ofatrial tissue has a length of 9.6 mm. It could be argued thata two-dimensional model of the SA node and atrium as usedby others would be more appropriate. However, the one-dimensional model is computationally more efficient, and,furthermore, in a radially symmetrical, two-dimensionalmodel of a circular SA node within a concentric atrium, thebehavior of a string of tissue extending from the center of the

SA node to atrial muscle will be no different from the one-dimensional model described here, because there will be nonet current flow laterally (i.e., at a tangent to the propagat-ing wavefront of the action potential) in the one- or thetwo-dimensional model. Within the string of SA node tissue,we assume that Cm changes from 20 pF (Cm in the central SAnode cell model) to 65 pF (Cm in the peripheral SA node cellmodel) exponentially ( Eq. 80, Table 13) and ionic currentconductances are functions of Cm (Eq. 81, Table 13). In themodel, single atrial cells are represented by the Earm-Hilge-mann-Noble equations (48). Electrotonic interactions withinthe tissue are modeled by the diffusive interactions of mem-brane potentials (Eqs. 82 and 83, Table 13). We used nonflux

boundary conditions for both ends of the model (Eqs. 79 and84, Table 13). Ds and Da scale the conduction velocity of theaction potential in the SA node and the atrial muscle. Theconduction velocity for near-planar waves is 0.0010.1 m/in the SA node and 0.30.8 m/s in the atrium (20). We set

Ds at 0.6 cm2 /s, which gives a conduction velocity of 0.06m/s in the SA node, and Da at 1.25 cm2 /s, which gives aconduction velocity of 0.62 m/s in the atrial muscle. Couplingat the junction of the SA node and atrial muscle is by thediffusion coefficient (Ds).

Computational Methods

The models were coded in C and Fortran languages. The

programs were run on Indigo 2, Silicon Graphics machineswith an IRIS 6.0 operating system. A fourth-order RungeKutta-Merson numerical integration method (55) was used tosolve the ordinary differential equations. The time step was0.1 ms, which gives a stable solution of the equations andmaintains the accuracy of the computation of membranecurrent and potential. One-dimensional partial differentiaequations were solved by an explicit Euler method with athree-node approximation of the Laplacian operator (55)with a time step of 0.1 ms and a space step of 0.1 mm for SAnode tissue and 0.32 mm for atrial muscle. The chosen timeand space steps are sufficiently small for a stable and accurate solution.

RESULTS

Peripheral and Central SA Node Action Potentials

Figure 6 shows the action potentials generated usingthe models of a peripheral cell (Cm 65 pF) and acentral cell (Cm 20 pF) at 37C from the SA node othe rabbit at fast (Fig. 6A) and slow (Fig. 6B) timebases. For comparison, Fig. 6 also shows action poten-tials recorded experimentally from rabbit SA nodepreparations: Figure 6C shows action potentials at afast time base recorded from small balls of tissue fromthe periphery and center of the SA node at 32C, andFig. 6D shows action potentials at a slow time base

Table 13. One-dimensional modelof the intact SA node

For x 03 mm

dVs

dn

x 0

0 (79

Cms (x) 20

1.07(x 0.1)

Ls

[1 0.7745e(x 2.05)/(0.295)

]

(65 20) (80

gs(x)[65Cm

s (x)]gc (Cms 20)gp

65 20(81

dVs(x)

dt

1

Cms (x) [itots (x)Ds

d2Vs(x)

dx2 ] (82For x 312.6 mm

dVa(x)

dt

1

Cma (x) [itota (x)Da

d2Va(x)

dx2 ] (83dVa

dn

xL

0 (84


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recorded from single cells with Cm of 57.5 and 22.0 pFat 35C. The simulated action potentials are similar tothose recorded experimentally. The action potential ofthe peripheral model has a more negative takeoff po-tential, a more rapid upstroke, a more positive peakvalue, a greater amplitude, a shorter duration, and a

more negative maximum diastolic potential than theaction potential of the central model. Furthermore, thespontaneous activity of the peripheral model is fasterthan that of the central model. All these are character-istic differences seen experimentally between smallballs of tissue from the periphery and center of therabbit SA node (31, 33) or large and small rabbit SAnode cells (27) (Fig. 6, C and D). The action potentialfrom the peripheral model has an early rapid phase ofrepolarization (phase 1) after the action potential up-stroke. Such an early rapid phase of repolarizationafter the action potential upstroke can be observed

frequently in the periphery of the intact SA node (butnot in the center) and in small balls of tissue from theperiphery (but not from the center) (6, 34, 45) (Fig. 6C)

Figure 6, EH, compares the characteristics of thesimulated action potentials with the average characteristics of action potentials recorded experimentally

from rabbit SA node cells by Honjo et al. (27) at 35CThe open circles in Fig. 6, EH, show experimentameasurements of the takeoff potential, maximum upstroke velocity, maximum diastolic potential, and cyclelength (time between successive spontaneous actionpotentials) of rabbit SA node cells plotted against CmIn all cases, there are significant correlations of the variables with Cm (27). In Fig. 6, E H, the filledsquares show corresponding values from the peripheral and central models. In all cases, the model valuesare close to those recorded experimentally; the changeswith Cm are also comparable to those seen experimen

Fig. 6. Simulated peripheral and central SA node ac-tion potentials. A: simulated action potentials at a fasttime base. B: simulated action potentials at a slow timebase. C: action potentials recorded from small balls ofperipheral and central rabbit SA node tissue ( balls Band D) at a fast time base (unpublished observations).D: action potentials recorded from rabbit SA node cellswith Cm of 57.5 and 22.0 pF at a slow time base (un-published observations). E H, takeoff potential (TOP;

E), maximum upstroke velocity (dV/dtmax,F), maximumdiastolic potential (MDP, G) and cycle length (CL, H)plotted against Cm. E, Data from Honjo et al. (27) fromrabbit SA node cells; dotted lines, regression lines; s,values computed from the peripheral (Cm 65 pF) andcentral (Cm 20 pF) SA node cell models.


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tally. There is also a significant correlation betweenaction potential amplitude and Cm in rabbit SA nodecells [action potential amplitude is greater in largercells (27)], and once again corresponding values fromthe peripheral and central models are comparable (notshown). Although Honjo et al. observed no significantcorrelation between action potential duration and Cmin rabbit SA node cells, there is a difference in action

potential duration between the peripheral and centralmodels (Fig. 6A). However, the action potential dura-tions in the peripheral and central models are stillcomparable to the data from rabbit SA node cells (notshown), and, furthermore, in small balls of tissue fromthe periphery and center of the rabbit SA node, asignificant regional difference in action potential dura-tion is observed (Fig. 6C) (5) similar to that betweenthe peripheral and central models (Fig. 6A).

The takeoff potential, maximum upstroke velocity,peak value of the action potential, action potentialamplitude, maximum diastolic potential, and cyclelength in the peripheral and central models are com-parable to experimental data obtained from small balls

of tissue from the periphery and center of the rabbit SAnode (33). However, in the case of the data from thesmall balls of tissue (at 32C), the action potentialduration and cycle length are longer than in the pe-ripheral and central models (37C) and also longerthan in single cells (obtained at 35C) (27). This may bethe result of the lower temperature of 32C used in thework on the small balls of tissue (33).

Figure 7 shows changes in various ionic currents

(iNa, iCa,L, ito, iK,r, iCa,T, iK,s, and if) during the actionpotentials generated by the peripheral and centramodels. Because of the regional differences in ionicurrent densities and Cm described above, the amplitudes of the ionic currents are different in the peripheral and central models. The changes in iCa,L, iK,r, andif 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 nodecells as measured with the action potential-clamp technique (62).

Effect of Block of Ionic Currents

In the absence of voltage-clamp data from single cellisolated from the different regions of the SA nodevoltage-clamp data from single SA node cells of different size (as measured by Cm) have been used as described above. However, this relies on the assumptionthat Cm is a good indicator of the origin of the cell in theintact SA node. A second source of data concerning

regional differences within the SA node has been smalballs of tissue from the different regions. In this casethe origin of the tissue is not in question. Although it isnot possible to obtain voltage-clamp data from smalballs of tissue from different regions, information abouregional differences in ionic currents can be inferred byinvestigating the effects of ion channel blockers. In thissection, the effects of block of various ionic currents inthe models (37C) are compared with the effects ob

Fig. 7. Simulated peripheral (ACand central (DF) SA node action potentials (A and D) and underlying ionicurrents (B and E: iNa, iCa,L, ito, andiK,r; C and F: iCa,T, iK,s, and if).


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served experimentally in small balls of tissue from therabbit SA node (32C).

Effect of block of iNa. Figure 8, A and B, shows theeffect of block of iNa on the action potential of theperipheral and central models. Figure 8, C and D,shows the effect of block ofiNa by 20 M TTX on actionpotentials recorded from peripheral and central balls ofrabbit SA node tissue (33). Qualitatively, the models

behave in the same way as the tissue. In both cases(simulation and experiment), blocking iNa had no effecton the central action potential. In the simulation, atleast, this is because there is no iNa in the centralmodel. In contrast, blocking iNa had various effects onthe peripheral action potential: 1) the takeoff potentialwas shifted to a more positive value; 2) the maximumupstroke velocity was reduced (in the example of ex-perimental data the maximum upstroke velocity wasreduced from 100 to 5 V/s, and in the simulation it wasreduced from 60 to 8 V/s; after block of iNa, in simula-

tion and experiment, the upstroke velocity in the pe-riphery was approximately the same as that in thecenter); and 3) as a result of the change in the takeofpotential, spontaneous activity was slowed. In themodel at least, the effects of block of iNa can be explained by the important role of iNa in the upstroke othe peripheral action potential. The effect of block ofiNa on spontaneous activity was less in the periphera

model than that seen experimentally in peripheratissue (Fig. 8); this difference may be the result of thefaster spontaneous activity of the peripheral mode(which, in turn, may be the result of the model beingdeveloped for 37C and the experimental data beingobtained at 32C). In the peripheral model, block of iNalso reduced the peak value of the action potential, theaction potential duration, and the maximum diastolicpotential (Fig. 8A); these changes are also seen experimentally in small balls of tissue from the periphery ofthe rabbit SA node (Fig. 8C) (33). In conclusion, in

Fig. 8. Effect of block of iNa on peripheral (A and C)and central (B and D) SA node action potentials. Aand B: data from peripheral and central SA node cellmodels. C and D: data from Kodama et al. (33) fromsmall balls of peripheral and central rabbit SA nodetissue (balls A and D). Top panels: membrane poten-tial; bottom panels: rate of change of membranepotential (dV/dt). Data are shown under control con-ditions and after block of iNa (TTX). In experiments,20 M TTX was used to block iNa.


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simulation and experiment, pacemaking in the periph-ery, unlike that in the center, is sensitive to block ofiNa.

Effect of block of iCa,L. Figure 9, A and B, shows theeffect of the block of iCa,L on the action potential of theperipheral and central models. Figure 9, C and D,shows the effect of block of iCa,L by 2 M nifedipine onaction potentials recorded from peripheral and central

balls of rabbit SA node tissue (33). Qualitatively, themodels behave in the same way as the tissue. In bothcases (simulation and experiment), block of iCa,L abol-ished the action potential in the center of the SA node:the membrane potential settled at 45 mV in theexperiment (Fig. 9D) and at 42 mV in the simulation(Fig. 9B). In the periphery, block of iCa,L had differenteffects on electrical activity. In both cases (simulationand experiment), block of iCa,L 1) shortened the actionpotential, 2) increased the pacemaking rate [presum-

ably as a consequence of the shortening of the actionpotential; in the simulation, block ofiCa,L caused a 22%increase in the pacemaking rate; in experiments, 2 Mnifedipine caused a 21 1% (SE) increase in thepacemaking rate in ball A (33)], 3) decreased the maximum upstroke velocity, and 4) decreased the peakvalue of the action potential. In the simulation, block ofiCa,L caused a decrease in the maximum upstroke ve

locity from 60 to 40 V/s and a decrease in the peak othe action potential from 27 to 8 mV. In the experiment shown in Fig. 9C, on application of 2 M nifedipine, the maximum upstroke velocity was decreasedfrom 82 to 75 V/s and the peak of the action potentiawas decreased from 22 to 6 mV. In conclusion, insimulation and experiment, only pacemaking in thecenter is abolished on block of iCa,L.

Effect of block of iCa,T. Hagiwara et al. (22) reportedthat block of iCa,T by 40 M Ni

2 produced a 515%

Fig. 9. Effect of block ofiCa,L on peripheral (A and Cand central (B and D) SA node action potentials. A andB: data from peripheral and central SA node cell models. C and D: data from Kodama et al. (33) from smalballs of peripheral and central rabbit SA node tissue(balls A and D). Top panels: membrane potential; bottom panels: rate of change of membrane potential. Dataare shown under control conditions and after block oiCa,L (nifedipine). In experiments, 2 M nifedipine waused to block iCa,L.


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increase in the cycle length in rabbit SA node cells, andLei et al. (38) reported a similar effect in rabbit SAnode cells. No information is available from single cellsor small balls of tissue on whether the effect of block ofiCa,T differs in peripheral and central SA node tissue.Because there is also no information available onwhether the density of iCa,T differs in peripheral andcentral SA node cells, the density of iCa,T is assumed to

be the same in the two cell types (see MODEL DEVELOP-MENT). In the peripheral and central models, block ofiCa,T caused a small increase in cycle length (of 4 and19%, respectively) similar to that reported experimen-tally.

Effect of block of 4-AP-sensitive current. Figure 10,AC, shows the effect of block of 4-AP-sensitive cur-rent (ito and isus) on peripheral and central actionpotentials. 4-AP has also been shown to block iK,rpartially (J. Hancox, unpublished observations). Inthe simulation of the effect of 4-AP, iK,r was blockedby 10%. Figure 10, A and B, shows the effect of blockof 4-AP-sensitive current on the action potential ofthe peripheral and central models. Figure 10C shows

the effect of 5 mM 4-AP on action potentials recordedfrom peripheral and central balls of rabbit SA nodetissue (6). Qualitatively, the models behave in thesame way as the tissue. In both cases (simulationand experiment), block of 4-AP-sensitive currentcaused a prolongation of the action potential in theperiphery 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). Insimulations, block of 4-AP-sensitive current caused a50% increase in action potential duration in theperipheral model and a 21% increase in the centramodel. Block of 4-AP-sensitive current also causedan increase in the peak value of the action potentialan increase in cycle length in the periphery, and adecrease in cycle length in the center. In experiments, 4-AP caused a 28 3% increase in cyclelength in peripheral tissue and a 5 2% decrease incycle length in central tissue (Ref. 6; data for balls Aand D are quoted). In simulations, 4-AP caused a24% increase in cycle length in the peripheral mode

Fig. 10. Effect of block of 4-AP-sensitive current andiK,r on peripheral and central SA node action poten-tials. A and B: effect of block of 4-AP sensitivecurrent on the peripheral (A) and central (B) SAnode cell models. C: effect of 5 mM 4-AP on smallballs of peripheral (top) and central (bottom) rabbitSA 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 theperipheral (D) and central (E) SA node cell models.F: effect of 0.1 M E-4031 [a low dose of E-4031expected to block 50% ofiK,r (32)] on small balls ofperipheral (top) and central (bottom) rabbit SA nodetissue [balls B and F; from Kodama et al. (32)]. Dataare shown under control conditions and after blockof 4-AP-sensitive current (4-AP) or iK,r (E-4031).


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and a 3% decrease in cycle length in the centralmodel. As discussed above, in experiments on therabbit SA node, in some cases, after the upstroke ofthe action potential in the periphery there can be aninitial rapid phase of repolarization (i.e., phase 1)(however, not in the experimental recording shownin Fig. 10C), and in such cases this was abolished by4-AP (6). In the peripheral model, the initial rapid

phase of repolarization was also abolished by block of4-AP-sensitive current (Fig. 10A). In summary, sim-ulations and experiments show that 4-AP-sensitivecurrent plays a major role in action potential repo-larization, and its role varies regionally.

In simulations in which ito and isus were blocked only(i.e., iK,r was not blocked by 10%), the changes weresimilar: in the peripheral model, the action potentialduration was increased by 49% (rather than by 50%),the maximum diastolic potential was changed by 1mV (rather than by 0.5 mV), and the cycle length wasincreased by 26% (rather than by 24%); in the centralmodel, the action potential duration was increased by19% (rather than by 21%), the maximum diastolic

potential was changed by 0.9 mV (rather than by5.7 mV), and the cycle length was decreased by 1.5%(rather than by 3%).

Effect of block of iK,r . Figure 10, D and E, shows theeffect of the block of 50 and 100% iK,r (simulating theeffects of low and high doses of E-4031) on the actionpotential of the peripheral and central models. Figure10F shows the effect of partial block (perhaps 50%)(32) of iK,r by 0.1 M E-4031 on action potentials

recorded from peripheral and central balls of rabbit SAnode tissue (33). Qualitatively, the models behave inthe same way as the tissue. In both cases [simulation(Fig. 10, D and E) and experiment (not shown; see Ref32)], complete block of iK,r (in experiments, by 1 ME-4031) caused cessation of spontaneous activity in theperiphery and center. After complete block of iK,r, themembrane 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. 32data for ball D), 33 mV in the peripheral model, and30 mV in the central model. This shows that in theperiphery and center, iK,r is important for pacemakingThe iK,r is responsible for generating the maximumdiastolic potential, and thus when iK,r is blocked, themembrane during diastole is depolarized and spontaneous activity ceases.

Partial block ofiK,r has different effects on the actionpotential in the periphery and center. In simulation(iK,r blocked by 50%) and experiment (iK,r partiallyblocked by 0.1 M E-4031), partial block of iK,r abolished the action potential in the center but not in the

periphery (Fig. 10, DF). In both cases (simulation andexperiment), in the periphery, partial block of iK,r 1increased action potential duration, 2) decreased maximum diastolic potential, and 3) increased cycle lengthThis shows that the periphery of the SA node is moreresistant to block ofiK,r than the center. Kodama et al(32) proposed that this is the result of a greater densityof iK,r in peripheral cells than in central cells. In themodels at least, the greater resistance of the periphera

Fig. 11. Effect of block of if on peripheral (A and Band central (C and D) SA node action potentials. Aand C: data from peripheral and central SA node celmodels. B and D: data from Nikmaram et al. (45from small balls of peripheral and central rabbit SAnode tissue (balls A and D). Data are shown undecontrol conditions and after block of if (Cs

). In

experiments, 2 mM Cs

was used to block if.


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model to partial block of iK,r is indeed the result of agreater density of iK,r in the peripheral model: in theperipheral model, if the density of iK,r was reduced tothat in the central model, 50% block of iK,r abolishedspontaneous activity (not shown).

Effect of block of iK,s. Complete block ofiK,s had littleeffect on the pacemaker activity of the peripheral andcentral models: there was a change in cycle length of

0.3 and 1% in the peripheral and central models. M. Leiand P. Kohl (unpublished observations) observed thatblock ofiK,s by 50 M 293B had no significant effect onthe electrical activity of rabbit SA node cells: cyclelength was 256 24 and 270 26 (SE) ms (n 5)under control conditions and in the presence of 293B,respectively.

Effect of block of if. Figure 11, A and C, shows theeffect of block ofifon the action potential of the periph-eral and central models. Figure 11, B and D, shows theeffect of block of if by 2 mM Cs

on action potentialsrecorded from peripheral and central balls of rabbit SAnode tissue (45). Qualitatively, the models behave inthe same way as the tissue. In both cases (simulation

and experiment), block of if slowed spontaneous activ-ity, and the slowing was greater in the periphery. Inexperiments, block of if caused a 24 2% increase incycle length in peripheral tissue and a 7 2% increasein central tissue (Ref. 44, data for balls A and D),whereas in simulations, block of if caused a 34% in-crease in the cycle length in the peripheral model andan 8% increase in the central model. In the simula-tions, the greater effect of block of if on the peripheralmodel can be explained by the greater density of if inthe peripheral model: in the peripheral model, if thedensity of if was reduced to that in the central model,block of if caused only a 14% increase in the cycle

length. Resting potentials. In simulation and experiment,blocking iNa and iCa,L (by 3 M TTX and 6 M nifedi-pine, respectively, in experiments) or blocking iK,r (by 1M E-4031 in experiments) abolished spontaneous ac-tivity in the periphery and center (32). After block ofspontaneous activity, SA node cells settle at their rest-ing potential. Figure 12 shows the maximum diastolicpotential as well as the resting potentials in the pe-ripheral and central models (Fig. 12A) and peripheraland central balls of rabbit SA node tissue (Fig. 12B)(32); the two sets of data are comparable. In simulationand experiment, whereas there was a substantial dif-ference between the periphery and center in the max-

imum diastolic potential, there was a smaller differ-ence between the two in the resting potential whenspontaneous activity was terminated by block of iNaand iCa,L and little difference between the two in theresting potential when spontaneous activity was ter-minated by block of iK,r. It was previously suggestedthat this showed that the difference in maximum dia-stolic potential between the periphery and center wasthe result of a difference in the density of iK,r betweenthe two (32). Consistent with this, in the models, if thedensity ofiK,r in the peripheral model was set to be thesame as that in the central model, the difference in the

maximum diastolic potential was abolished (noshown; the maximum diastolic potential was 42 mV)

One-Dimensional Model of the Intact SA Node

Many of the properties of the intact SA node, forexample, conduction, conduction disturbances, suppression of the SA node by the surrounding atriamuscle, and pacemaker shift, are the result of themulticellular nature of the SA node and the electrotonic interaction between the SA node and the atria

muscle surrounding the SA node. The intact SA nodeand the surrounding atrial muscle represent a complexstructure, and an anatomically accurate model of theintact SA node is outside the scope of this studyHowever, the behavior of the intact SA node tissue canbe qualitatively simulated by a simple model that ignores the complex structure of the SA node and idealizes the tissue as a one-dimensional string of tissueextending from the center of the SA node to the periph-ery and then onto atrial muscle (Fig. 13A). Within thestring of SA node cells, Cm is assumed to change from20 to 65 pF in an exponential fashion, and densities of

Fig. 12. Resting potentials in the periphery and center of the SAnode. A: data from peripheral (solid bars) and central (hatched barsSA node cell models. B: data from Kodama et al. (32) from small ballof peripheral (solid bars; balls A and B) and central (hatched barsballs DF) rabbit SA node tissue. Values are means SE. MDPmaximum diastolic potential; RP(TTX/Nif), resting potential afterblock 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 M TTX, 6 Mnifedipine (Nif), and 1 M E-4031, respectively.


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ionic currents are functions of Cm. Electrotonic cou-pling between cells is determined by diffusion coeffi-cients (Ds and Da). The values of the diffusion coeffi-cients were chosen to obtain appropriate conduction velocities within the SA node and atrial muscle (seeMODEL DEVELOPMENT). Ds will also influence the effectivespace constant () of the string of SA node tissue, and,

therefore, was measured to determine whether the value of in the model is similar to that measuredexperimentally. A 2-ms current pulse was injected intoan SA node cell 1 mm from the junction of the SA nodewith the atrial muscle. The current was injected duringdiastole to depolarize the cell at the point of injectionby 17.5 mV. A similar technique has been used exper-imentally (4). The amplitude of the depolarization de-clined away from the point of injection in an exponen-tial manner with a of 380 m; this compares with of310520 m measured experimentally in the rabbit SAnode by Bleeker et al. (4).

Figure 13B shows action potentials computed usingthe one-dimensional model of the intact SA node. Action potentials from various points along the string otissue are shown. The junction of the SA node andatrium is at 3 mm. Figure 13B shows that spontaneousaction potentials were first initiated in the center of theSA node (highlighted by the arrow) and then propa-gated to the periphery of the SA node and onto the

atrial muscle. This is similar to the activation sequenceseen experimentally (3). The cycle length in the one-dimensional model of the intact SA node is 350 ms; thisis greater than the cycle length in the peripheral (160ms) or central (330 ms) model. However, the cyclelength in the one-dimensional model of the intact SAnode is comparable to that observed experimentally inthe intact SA node of the rabbit at 37C: 348 50 ms(30). Furthermore, at 33C the cycle length in theintact SA node of the rabbit (586 ms) is greater thanthe cycle length in isolated peripheral (298 8 ms) orcentral (331 13 ms) tissue (Ref. 31, data for balls Aand D). The SA node conduction time (time for theaction potential to conduct out of the SA node) is 45 mssimilar to that seen experimentally (1). Figure 13Cshows superimposed action potentials at a fast timebase from the one-dimensional model: from the centerof the SA node and the atrial muscle. Figure 13, D Fshows various action potential parameters (activationtime, i.e., the time taken for the action potential topropagate from the leading pacemaker site in the center, maximum upstroke velocity, peak value of theaction potential, and maximum diastolic potentialplotted against the distance from the center of the SAnode. The model data are comparable to experimentadata (31). In summary, the one-dimensional model othe intact SA node shows a range of behaviors compa

rable to that of the intact SA node of the rabbit.

DISCUSSION

From the modeling, it is concluded that in the periphery compared with the center of the SA node 1) thetakeoff potential is more negative and the maximumupstroke velocity is higher as a result of a higherdensity of iNa, 2) the action potential can have a notchas a result of a higher density of ito, 3) the actionpotential is short as a result of higher densities of iK,and 4-AP-sensitive current, 4) the maximum diastolicpotential is more negative principally as a result of ahigher density ofiK,r, and 5) the spontaneous activity is

faster as a result of higher densities ofiNa and ifas welas the shorter action potential.After the pioneering work of Noble (47), much effor

has been devoted to the development of mathematicamodels of cardiac cells based on voltage-clamp dataWith the development of computing power and resources, these models, together with information aboutthe structure of cardiac tissue, the electrical couplingbetween cells, cell orientation, and the heterogeneity inthe electrophysiological properties of cardiac cells, arebeing used to develop biophysically accurate and detailed mathematical models of cardiac tissues (54); the

Fig. 13. One-dimensional model of the SA node. A: schematic dia-gram of part of the one-dimensional model. SAN, SA node; AM, atrialmuscle; Ds and Da, diffusion coefficients. B: action potentials re-corded at regular intervals along the one-dimensional model fromthe center of the SA node (at 0 mm) to the junction of the SA nodewith 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 theaction potential (the leading pacemaker site). C: action potentialsrecorded from a central SA node cell (1) and an atrial cell (2) in theone-dimensional model. D F: activation time (D), maximum up-stroke velocity (dV/dtmax, E), and peak value of the action potentialand maximum diastolic potential (F) plotted against the distancefrom the center of the SA node.


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aim is to develop a model of the whole heart. It is hopedthat the models developed in the present study ofaction potentials in the periphery and center of the SAnode can be used to develop a two- or three-dimen-sional model of the intact SA node, which then can beused in the development of a whole heart model.

Comparison With Previous ModelsThe models developed have a structure similar to

previous SA node models (13, 49, 59). However, be-cause of recent experimental findings, we have b

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