Arrhythmia as a Result of Poor Intercellular Coupling in the Sinus

17 Pages·2001·0.52 MB·English

J. theor. Biol. (2001) 211, 201}217
doi:10.1006/jtbi.2001.2339, available online at http://www.idealibrary.com on
Arrhythmia as a Result of Poor Intercellular Coupling in the Sinus Node:
A Simulation Study
P OG *-, B W ? G O *
ER STBORN JOG RN OHLFART AND UNNAR HLED N
*Departments of Mathematical Physics, ‚und ;niversity, S-22100 ‚und, Sweden and
?Clinical Physiology, ‚und ;niversity, S-221 00 ‚und, Sweden
(Received on 12 June 2000; Accepted in revised form on 24 April 2001)
Thee!ects of reducedintercellularcouplingin the sinus nodewere investigatedby meansof
simulations.Coupling was reduced both uniformly, and by introducing localized interaction
blocks. In either case, model sinus node element networks typically splitted into frequency
domains. These were de"ned as groups of neighbour elements which all attained the same
mean "ring frequency.In systems, simulatingthe vicinityof an impulse outlet to the atrium,
thesinusnodeelementsoftensplittedintotwodomains,oneslowly"ringjustinsidetheoutlet,
and one normally "ring large domain in the sinus node interior. This two-domain situation
was analysed using a two-element system. Wenckebach conduction and advanced (m:1) exit
blocks were seen, together with more odd block patterns and slow chaotic rhythms. The
two-domain situation appeared also when two discrete outlets were considered. The slow
domainsaroundeachoutletsynchronizedviatheatrium.However,ifthereweresomedegree
ofexitblockthroughoneoftheoutletsonly,brady-tachylikerhythmscouldbesimulateddue
toare-entrantcircuitincludingbothsinusnodeandatrialtissue.Inconclusion,poorcoupling
in the sinus node seems to be su$cient to produce most arrhythmias in the sick sinus
syndrome
(2001AcademicPress
1. Introduction et al., 1992). In this syndrome, sinus rhythm is
mostoftenslowerthannormal,buttheremaybe
In this study, we investigated which clinically
paroxysmalattacksoftachycardiaoratrial"bril-
signi"cant cardiac arrhythmias can appear be-
lation. The latter situation is called the brady-
cause of poor intercellular coupling in the sinus
tachysyndrome.Theslowrhythmsmaybeeither
node (SN). We aimed to obtain a qualitative
periodic or aperiodic. By a periodic rhythm we
picture, not to give parameter values at which
mean a sequence of "ring intervals that "nally
di!erent phenomena appear.
repeats itself. The periodic rhythms are called
Patientssu!eringfromarrhythmiaoriginating
sino-atrial (SA) exit blocks. These include, for
in the SN area are said to have the &&sick sinus
example,Wenckebachconductionand advanced
syndrome’’(Rubensteinet al.,1972; Ferrer,1973;
(m:1) block. Slow aperiodic rhythms are called
Kaplan et al., 1973; Jordan et al., 1978; Delon
sinus arrest rhythms.
There is convincing evidence that the cells in
the SN entrain their frequencies because of the
-Authortowhomcorrespondenceshouldbeaddressed.
E-mail:[email protected] electric coupling which is mediated by gap
0022}5193/01/150201#17$35.00/0 ( 2001AcademicPress
202 P.OG STBORNE„A‚.
junctions (De Haan, 1982; Jalife, 1984; Delmar interior. Leakage reduces the "ring frequency
et al., 1986). Cai et al. (1993) have studied the since diastolic depolarization becomes slower. It
complex dynamics arising in a pair of model SN isalsoassumedinthesimulationsthattheborder
cells with di!erent natural frequencies when this between the SN and the atrium is sharp, and
coupling is reduced. Examples of Wenckebach speci"cally that the impulse leaves the SN at
rhythms and slow aperiodic rhythms in weakly discrete exit sites. The validity of these assump-
coupled model SN cells have been obtained pre- tions is discussed in Section 4.
viously by Michaels et al. (1986).
We studied the activity of poorly coupled SN
2. Methods
cells in an atrium with both simple and more
complexmodels,inordertotestthegeneralityof TheIrisawa&Noma(1982)modelwasusedto
the observed phenomena, and to gain an under- simulateSNelements.ThisisaHodgkin&Hux-
standingofthedynamics.Thepaperisorganized ley (1952)-type membrane model based on volt-
along a line of progressive simpli"cation. First, ageclampmeasurementsonaggregatesofrabbit
we studied a network model of the vicinity of an SN cells. Five currents (i , i , i , i and i )
C! K N! h b
impulse exit site to the atrium. This model in- contribute to the total current I #owing across
cluded some ultrastructural qualities suggested the cell membrane. To save computation time,
by experiments. We investigated the e!ects both the Na-channel was removed, and the Ca-
of a uniform gap junction conductance decrease, channel activation variable d was set to its
and of the introduction of electrically insulating steady-statevalued . These operationsare justi-
=
connective tissue strands. The same types of ar- "ed since the Na-current stays very small
rhythmiaswereproducedineithercase.Thenwe throughout the limit cycle, and d adjusts very
investigateda more schematic network model in quickly to d .
=
whichthein#uenceoftheatriumwasreplacedby To produce a frequency variation among
a constant leakage current. The leakage arises elements, the membrane densities of Ca- and
becausethe maximumdiastolic potential(MDP) K-current channels were altered, both with the
of SN cells is less negative than the atrial cell same factor Q compared to their normal values.
resting potential. The same arrhythmias were Frequency increased with Q, and the waveform
seeninthismodel.TheSNelementsoftensplitted changed in a way similar to that recorded by
into two frequency domains, one normally "ring Boyettet al. (1999)whenthey studieddi!erences
interior domain, and one slow peripheral. This inspontaneousactivitybetweencentralandperi-
allowed the use of only two SN elements, one of pheral parts of the rabbit SN. The model used is
whichsu!eredfromastrongleakagecurrent.The speci"ed in Appendix A.
bifurcation diagram resulting from changing Non-automatic (atrial) elements were simu-
coupling strength was studied in detail. Again, lated with the membrane model developed by
the same arrhythmias were seen. Owing to their Wohlfart&OhleHn(1999).Theequationsde"ning
seeming generality, in an accompanying paper, this model are also given in Appendix A.
we attempt to explain the rhythms and bifurca- The gap junction current j from cell k to
kl
tions of this system using circle maps based on a neighbour cell l was described as j "
kl
phase response curves. Model systems including g (<!<), with a constant conductance g .
kl k l kl
two SN exit sites showed the same type of dy- Here< isthemembranepotentialofcellk.Thus,
k
namics as those with a single exit site, but also < followed d</dt"!(I #+ j )/C , where
k k k l kl k
a kind of re-entry including both SN and atrial I is the current #owing across the cell
k
tissue. This might be one mechanism behind the membrane and C is the membrane capacitance.
k
brady-tachy syndrome. Cells were treated as two-dimensional squares,
Severe sinus arrhythmia, as simulated in this and were coupled to neighbours l having a
study, can only appear in vivo if the leakage common edge.
current to the atrium makes it impossible for Since a large amount of data was to be col-
theSN cellscloseto the SAjunctionto drivethe lected, we aimed to use the quickest possible
atrium properly without help from the SN integrationmethodineachcase.Tointegratethe
POORCOUPLINGINTHESINUSNODE 203
systems in Figs 1 and 4}8, the forward Euler dynamicswas recorded.After that, the dynamics
method was used. For the two-element system always seemed independent of initial conditions.
(Figs 5 and 6) the time step was set to 1ms, However, this was not systematically investi-
otherwise 0.5ms was used. For the systems in gated.
Figs 2 and 3, we used the integration method of
Victorri et al. (1985). In this method, voltage
3. Results
changes are controlled in each step. Such
a method had to be used because the large leak- 3.1. FREQUENCYDOMAINS
age currents across the SA junction contributed Figure 1 shows results of simulations with
tofastvoltagechangesofthesmallperipheralSN a network of 20]20 SN elements. In panel (a)
elements.ThismadetheEuleralgorithmdiverge, these were not coupled, so that the mean "ring
unless very small time steps were used. In our intervals were the same as the natural intervals.
version of the Victorri method, if the voltage These were distributed according to a gradient
change DD<D of any element exceeded 0.5mV, with a superposed smaller random variation.
the time step Dt was halved and calculations Values of Q (Section 2) were assigned according
wereredone.IfDD<D(0.5mVforallelements,Dt toQ"0.8#((k!1)/19)3#r,wherekistherow
wasdoubledinthenextiteration.Dtwasinitially number, and r is a uniform random variable
set to 0.5ms, and was not allowed to exceedthis between 0 and 0.2. This produced natural inter-
value. vals between 369 and 288ms. This design was
In all simulations, measurements were started inspiredbytheexperimentsbyOpthofetal.(1987)
after su$ciently long time so that no transient and Boyett et al. (1999), which demonstrated
FIG.1. Simulationsof20]20SNelementscoupledhomogenouslywithconductanceg.Naturalintervals(a)werevaried
throughadistributionofQ(Section2).(b)Formationofsixfrequencydomainsforweakcoupling.Theelementmarkedwith
anarrowisadomainofitsown,sinceithasnonearestneighbourswiththesamefrequency.Adi!erentscaleisusedinpanels
(b)}(d),sothatthedomainsareeasiertodistinguish.(c)Frequencyentrainmentforhighercoupling.(d)Aninteractionblock
( )causedsplittingintotwodomainsofanentrainednetwork.(e)Standarddeviationsofthemeanintervalsforthesystem
in(d)dividedbythemeanintervals.(f)Potentialtracesoftwoelementsmarkedin(e).Theelementfarfromthedomainborder
"redmore regularlythan theelementclosetoit.
204 P.OG STBORNE„A‚.
a gradient with longer natural intervals in
3.2. NETWORKMODELOFTHEVICINITYOFASINUS
the rabbit SN centre (400}500 ms) than in the
NODEIMPULSEEXITSITETOTHEATRIUM
periphery towards the crista terminal is (CT)
(&300ms). Figure2(a) showsthe systemunderconsidera-
The introduction of a uniform coupling con- tion. It is intended to incorporate some ultra-
ductance g, caused frequency entrainment for structural qualities suggested by experiments.
g’4.5S/F [panel (c)]. For lower values of g, Arial cells are bigger than SN cells (James, 1977;
frequencydomainsappeared.Thesewerede"ned Denyer&Brown,1990;Verheijcketal.,1998).To
asgroupsofneighbourelementshavingidentical obtain a simple two-dimensional geometry, the
mean frequencies. In panel (b), six domains non-automatic element capcitance (area) was set
can be recognized for a uniform conductance 4 times higher than the SN element capacitance.
g"4S/F. The thick lines represent electric insulation
Thenaturalfrequencygradientwasapproxim- between nodal and atrial tissue. This design is
atelylinear.Nevertheless,thefastestdomainwas motivatedbythestudybytenVeldeetal.(1995),
substantially larger than the slower ones. This whichindicatedthattheatriumformsnarrowing
was seen in other simulations too (not shown), strands penetrating into the SN, and that SN
and thus seemed to be a general quality, at least cells are coupledto atrial cells only at the tips of
of the present SN element model. these strands. The coupling between the shaded
Panel (d) shows how an interaction block SN elements was set 10 times higher than the
madeanentrainedsystemsplitintotwodomains coupling gSS between the other SN elements.
[cf. with panel (c)]. The block could be inter- Microscope studies by Bleeker et al. (1980) in-
preted as a strand of connective tissue, since dicated that intercellular coupling is greater in
these seem to be electrically insulating the peripheral parts of the SN close to the CT.
(DeMazie‘reetal.,1992).Frequencyentrainment Kwong et al. (1998) found that SN cells that
was regained at g"5.7S/F. stained for the gap junction protein Cx43 were
The dynamics of the systems in panels (b) and arrangedinbundles.Insomesections,thesewere
(d) was aperiodic. This was typically the case seen to abut atrial cells. Coppen et al. (1999)
when global frequency entrainment was absent. found that Cx43 was present in the rabbit SN
However,itcouldhappenthatthemeanfrequen- only in a restricted zone near the border to the
cies of all domains became rationally related, CT, which is the preferential pathway out of the
leading to periodic dynamics. This was seen SN (Cx40 and Cx45 were present throughout
mainly when there were only two domains. the node). These results suggest that SN cells
Panel (e) shows the standard deviations of the close to outlets to the atrium indeed are more
"ring intervals for the system in panel (d), in stronglycoupled.ThenaturalintervalsoftheSN
percentofthemeanintervals.Theslowerdomain elements were made to vary between 336 and
showed larger interval variations. The interval 303ms, due to a uniform random probability
variations were greatest close to the border be- distribution of Q between 1 and 1.4. Since the
tween the domains, and tended to decrease with system was only intended to model an SN tissue
the distance from this border. This was seen in around an exit site, we did not include the full
other simulations too (not shown), and thus frequency gradient between the SN centre and
seemed to be a general quality. Panel (f) shows periphery (Opthof et al., 1987; Verheijck et al.,
potentialtracesoftheelementsaandbmarkedin 1998),neverthelesssomerandomcell-to-cellvari-
panel(e).The"ringsofaweremuchmoreregular ations are bound to exist. The coupling gSA
than those of b. This "nding suggests that if the betweenSNandnon-automaticelementswasset
splitting into domains is to have any clinical to 100S/F, and the coupling gAA between non-
signi"cance,it has to occur close to the SA junc- automatic elements to 500S/F (siemens per unit
tion. Otherwise, atrial intervals would probably SN element capacitance in both cases). We refer
be quite regular. This is one reason why we to Section 4 for a discussion of these values.
focusedourattentiononthedynamicsofSNcells Potential traces of the marked elements a, b,
close to the SA border. and c are shown in panels (b)}(e). The leakage
POORCOUPLINGINTHESINUSNODE 205
FIG.2. Dynamics of a system modelling the neighbourhood of an SN exit site. (a) System architecture. The thick lines
representinteractionblocks.TheshadedSNelementsare10timesmorestronglycoupledthantheotherSNcells.Potential
tracesofthemarkedelementsareshowninpanels(b)}(e).Forzerocoupling(gSS)betweenSNelements,thesixjunctionalSN
elementsdidnot"reasaresultoftheleakage(b).ForgSS’21S/F,wehad1:1conduction(e).Forweakercoupling,exit
blocksofvariousdegreesappeared(candd).Generally,the12shadedelementsformedaslowdomain,whereastheotherSN
elements"rednormally.Theleakagecurrent#owingfromelementbtotheatriumisshowninpanels(b)}(e).(----)a;(**)b;
())))))c.
current shown is the current #owing from ele- Figure 3 illustrates the e!ects of non-homo-
ment b to the non-automatic element in front. geneous coupling decrease in the SN part of the
Panel (b) illustrates that for gSS"0, the six SA system in Fig. 2. Interaction blocks of various
junctional SN elements were quiescent (no bars sizes and positions were introduced. These were
are shown). For gSS"10S/F, the SN tissue intended to model strands of connective tissue,
splittedintotwodomains,wherethe meaninter- and are representedby thick lines.gSA and gAA
vals of the 12 well-coupled junctional elements were the same as in Figure 2. gSS was set to
were twice that of the other elements [panel (c)]. 30S/F in all panels, which led to frequency en-
A2:1advancedexitblockwaspresent.Thenota- trainmentintheabsenceoftheblocks[Fig.2(e)].
tion m:n means that n out of m SN "rings reach In panels (a)}(d), a growing block, progressively
the atrium. For gSS"15S/F, the ratio of mean decoupledtheSNperipheryfromtheinterior.In
intervals between the junctional and interior SN panel(a),therewasa4:3Wenckebachrhythm,in
elementswas3/2,anda3:2Wenckebachrhythm panel(b)anadvanced2:1block,andinpanel(c)
was seen. All SN elements became frequency en- a5:2exitblock.Thelatterkindofrhythmwillbe
trained for gSS’21S/F, leading to perfect im- discussed below, and also in the accompanying
pulse transmission to the atrium. The dynamics paper.Finally,inpanel(d),a3:1blockisseen.If
was periodic in all panels. the12well-coupledSNelementswerecompletely
206 P.OG STBORNE„A‚.
developed. As in panel (a) a 4:3 block was pre-
sent. However, it was not of the Wenckebach
type.Rather,theblockcreateda&&miniatureSN’’
which was able to drive the atrium at a slightly
lower, but quite regular rate. Blocks placed
further inside the SN were not able to create
conduction disorders.
The dynamics was periodic in all panels.
3.3. SIMPLIFIEDNETWORKMODELOFSINUSNODE
TISSUEAROUNDANEXITSITE
The in#uence of the atrium was replaced by
a leakage current, and the geometry of the SN
tissuewassimpli"ed.Aconstantleakageof6A/F
was added to the four central elements in a net-
work of 10]10 elements [Fig. 4(b)]. These ele-
ments should be imagined to be placed around
adiscreteSNexitsite[Fig.4(a)].Foranisolated
element, this leakage abolished spontaneous ac-
tivity. As for the systems in Figs 2 and 3, each
element was randomly assigned a Q value be-
tween 1 and 1.4. The coupling between the 16
most central elements were set 10 times higher
than the coupling g between the other elements.
Figure4(b)}(f) shows mean"ring intervals for
"ve di!erent values of g. In panel (b) there were
three frequency domains. The relation between
their mean intervals was 6:3:2, giving a 3:1
block. In panel (c) there was a 2:1 block, and in
panel (d) a 3:2 Wenckebach rhythm. In these
panels,theelements"redperiodically.Beforeen-
trainment settled (f), there was a region around
g"23S/F,inwhichsomecellsadjacenttothe16
FIG.3. SimulationswiththesamesystemasinFig.2,but
central ones broke away from the large domain,
with interaction blocks (connective tissue strands) inserted
betweenSNelements.gSS"30S/Finallpanels.Successive and attained somewhat higher mean frequencies
extensionofasmallstrandproducedmoreandmoresevere (e). The dynamics was aperiodic during this
exit blocks. Potential traces from the same elements as in
breakdown of the two-domain situation. Tem-
Fig.2areshown.(a)4:3conduction,(b)2:1,(c)5:2,and(d)
poraryexitblocksappearedatirregularintervals,
3:1.AblockfurtherinsidetheSNproducedalarger,slow,
junctionaldomain(e),andgaveriseto4:3conduction. as seen in panel (g).
3.4. ASINUSNODEWITHONEEXITSITEMODELLED
decoupled from the rest of the SN tissue, a com- ASATWO-ELEMENTSYSTEM
plete SA block developed (not shown). In all Since states with only two domains occurred
panels (a)}(d), a two-domain situation was pre- often, the dynamics during weak intra-nodal
sent,withthe 12 junctionalelementsformingthe coupling was modelled with a two-element sys-
slower domain. tem. The similarity between the dynamics of the
In panel (e), the block was placed a bit further system in Figs 2 and 3 and that in Fig. 4, let us
away from the atrium. In this case, a larger model the in#uence of the atrium as a leakage
junctional domain consisting of 18 elements current. The SA blocks shown in Figs 2}4
POORCOUPLINGINTHESINUSNODE 207
FIG.5. Bifurcationdiagramforasimpli"edtwo-element
systemin which element 2 su!ered from a leakage current
1.4A/F. The occurring "ring intervals of element 2 are
shown for di!erent values of the coupling conductance g.
m:1blocksoccupyagreatpartofthediagram.Wenckebach
rhythms were found in the region marked W . In the
1
W }W regions, other Wenckebach-like rhythms were
2 4
found(seetext).IntheC -andC -regionsthedynamicswas
1 2
FIG.4. Simulations of 10]10SN elements in a plane. chaotic. The insets show that these regions appeared after
The four central elements in the network su!ered from period-doubling cascades. (b) 4:3 Example of a Wencke-
a leakage current of 6A/F and should be imagined to be bachrhythmatg "11.5S/F.Thesolidlinecorrespondsto
1
placedaroundadiscreteSNexitsite.Thecouplingconduc- element 2. „"347ms, „ "344ms. (c) 7:3 rhythm from
I II
tancebetween the 16 centralelements was10 times higher theW regionatg "6.2S/F.(d)Chaoticrhythmfromthe
2 2
than the coupling g between the other elements. Natural C region at g "0.6S/F. Firing intervals are given in
2 3
frequencieswerevariedthroughadistributionofQ(Section seconds.
2). (a) Imagined position of the plane. (b)}(f) Mean "ring
intervalsforeachelementforasequenceofincreasingcon-
ductancesg.(b)Threedomainsand3:1exitblock.(c)Two
domains and 2:1 block. (d) Two domains and 3:2 appearedin suchasimpli"edsystemalso,and in
Wenckebachconduction.(e)Breakdownofthetwo-domain the same order of appearance as coupling was
situation, leading to aperiodic dynamics. (f) Perfect fre-
altered.
quency entrainment and 1:1 conduction. (g) Aperiodic
blockrhythmof thesystemin(e). We let element 2 su!er from the leakage
1.4A/F[Fig.5(a)].Thisvalueislowerthaninthe
208 P.OG STBORNE„A‚.
simpli"ed network simulations (Fig. 4) since the
elementwasintendedtocorrespondtotheentire
slow junctional domain. g is the coupling con-
ductance between the elements. Both elements
had Q"2 (Section 2). In Fig. 5(a), the occurring
"ring intervals for element 2 at a given g are
plotted against g. The "ring intervals of element
1 (corresponding to the SN interior) increased
only slightly from its uncoupled value 280ms as
g increased. The variance of its "ring inter-
vals was small irrespective of the dynamics of
element 2.
In Fig. 5(a), plateaux on which every m-th
"ring of the interior element triggered the SA
junctionaloneareseen.mincreasedinstepsofone
from 1 to 7 as g decreased. The widths of the
FIG.6. The ratio R between the number of "rings of
plateaux decreased with increasing m.
elements1and2asafunctionofgintheWenckebachregion
Between the 1:1 and 2:1 conduction areas, W in Fig. 5(a). R seems to form a Devil’s staircase, with
1
Wenckebach rhythms arose. The region is plateaux at all rationals between 1 and 2. The inset is an
enlargementaroundthe5:3region.
marked W . In Fig. 5(b), a 4:3 rhythm is shown.
1
The phase lag of the peripheral oscillator in-
creased (D(D (D ), and the "ring interval period-doubling cascade [see insets in Fig. 5(a)].
I II III
decreased („’„ ), before one excitation was After the "rst branching in the 7:1 (6:1) region,
I II
lost. These features are characteristic of Wen- every second "ring interval of element 2 was
ckebachconduction.In Fig.6, the ratioR"m/n alittlebitlonger.Afterthesecondbranchingfour
betweenthe numberof generatedand conducted slightly di!erent intervals occurred, then eight,
impulses in the W region is plotted against g. and so on. By calculating the mean rate of
1
R(g) seemed to form a complete Devil’s staircase exponential divergence of nearby trajectories
(Ott, 1993), in which there are steps at each ra- in phase space, the Lyapunov exponents for
tional valuebetween one and two. It is seen that g"1.25 (C ) and 0.6S/F (C ) were estimated to
1 2
theWenckebachratios(m#1):moccupiedmost be 0.32 and 0.45s~1, respectively. Chaos is de-
part of the parameter interval. The only other "ned to be present then there is a positive expo-
rhythm that occurred on an interval with ap- nent. In Fig. 5(d), a rhythm of element 2 for
preciable width was 5:3 conduction. In the en- g "0.6S/F is shown, with "ring intervals
3
larged picture, the more exotic ratios 8:5 and marked between the potential peaks. The most
11:7 are also seen. remarkablefeatureof thesedynamicalregimesis
TherhythmsinregionW [Fig.5(a)]werealso thatsome"ringintervalsweremuchlongerthanin
2
Wenckebach-like. Here, at regular intervals, an the regular 6:1 and 7:1 regions which preceded
additional "ring was lost in a basic 2:1 cycle. In them, respectively [Fig. 5(a)]. This is strange
Fig. 5(c), such a 7:3 rhythm is shown. A 5:2 sincethetriggeringimpulsesfromelement1were
rhythmofthistypeisseeninFig.3(c).Generally, stronger.Intheaccompanyingpaper,weattempt
R"(2m#1)/m for these rhythms. The charac- to explain this and other qualities of the bifurca-
teristic increasing phase lags and decreasing in- tion diagram using phase response curves.
tervalsbeforethelost"ringwereseeninthiscase
also. Analogously, in regions W and W , addi-
3 4
tional "rings were lost at regular intervals in 3.5. TAKINGMULTIPLESINUSNODEEXITSITES
basic 3:1 and 4:1 rhythms. We had R" INTOACCOUNT
(3m#1)/m and R"(4m#1)/m, respectively. In order to "nd out whether the existence of
Thedynamicsin regions C and C seemed to twodiscreteoutletsfromtheSNmakesnewtypes
1 2
be chaotic. Both regions were entered via a of rhythmspossible,simulationsweremade with
POORCOUPLINGINTHESINUSNODE 209
a network of 5]10SN elements, in which two
elements at some distance from each other suf-
fered from leakage, simulating the exit sites. To
beabletostudythee!ectsofthesituationwhere
thetwo outletsweresurroundedby SNelements
with di!erent natural frequencies, a gradient of
Q values (Section 2) was introduced [Fig. 7(a)].
InFig.7(b),itisseenthatforg"3S/F,theSN
interior elements splitted into two domains, so
thatthetwoSAjunctionalelementswerefedwith
di!erent frequencies. Element b was locked in
a2:1conductionratiowithrespecttothe"rings
ofthesurroundingdomain,whereasnosuchrela-
tion existed for element a. The overall dynamics
was aperiodic. It seemed that all possible time
delaysbetweena "ringof elementaandthe next
"ring of element b occurred sometime (and vice
versa). The last two qualities were typical when
more than one interior domain was present.
For higher g, the interior elements formed
asingledomain[Fig.7(c)].IfnoQ-valuegradient
was used [Fig. 7(a)], this situation appeared at
averylowg(notshown).Inanycase,bothaand
bbecamefrequencylocked in arational conduc-
tion ratio with respect to the common feeding
frequency, and the dynamics became periodic.
Thereafter, each drained element followed the
sequence of bifurcations in the two-element sys-
tem [Fig. 5(a)]. For a given g, they could be at
di!erent positions in the bifurcation diagram,
dependingontheQvaluesofaandbandthoseof
the surrounding elements. In Fig. 7(c), a was in
a 2:1 state and b showed 5:3 conduction.
Naively, the uncorrelated "rings of a and b FIG.7. Meanfrequenciesinanetworkof5]10SNcells
in which cell a su!ered from a leakage of 1.1A/F and cell
at g"3S/F indicate that long atrial activation bof1.4A/F,simulatingtwodiscreteSNexitsites.Cellswere
intervals might alternate with very short ones. coupled with a common conductance g. A gradient of
However, we have not taken into account that Q values ranging from 1 to 2 with a superimposed minor
random variation created a gradient in natural periods.
junctional elements at di!erent outlets interact
Q(a)"1.5, Q(b)"1.7. (a) g"0S/F. Since a and b were
via the atrium. To this end, the system described quiescent,nobarsareshown.(b)g"3S/F.Fourfrequency
in Fig. 8(a) was studied. The two interior SN domains remained; two interior ones (2 and 3) and two
junctional (1 and 4), corresponding to cells a and b. Both
elements allow dynamics corresponding to the
aandb"redirregularly.awaslockedina2:1conduction
splitting of the SN interior into more than one
ratio with respect to the surrounding (irregularly "ring)
frequency domain [Fig. 7(b)]. domain.(c)g"4S/F.TheSNinteriorconstitutedasingle
Itwasfoundthatwheneverg washighenough domain. a was locked in a 2:1 state, and b showed 5:3
d
conduction.Periodicdynamics.
to allow impulse conduction though the non-
automatic elements, a "ring of SN element 4
immediately triggered element 1 via the atrium, the junctional domains in Fig. 7(b) and (c). The
and vice versa. They achieved a common mean junctional cell synchronization made impossible
frequency and could be said to form a single anyatrialactivationintervalshorterthannormal
domain. This corresponds to the uni"cation of due to uncorrelated "rings.
210 P.OG STBORNE„A‚.
a total of three domains. When elements 2 and
3 formed one domain*corresponding to the
situation where the SN as a whole contains
only two domains, one interior and one junc-
tional*the atrial rhythm again followed the bi-
furcation diagram in Fig. 5(a) as g }g increased.
a c
Thus, the two-element approximation of a
weakly coupled SN seems to be valid even if
several SN exit sites are considered.
However,ifconductionthroughoneoftheexit
siteswasperfect(1:1),butthatthroughtheother
was not, it was possible to induce re-entry. For
theparametervaluesgiveninthe"gurelegend,if
the ring was disconnected in the atrial part (b),
SN element 1 showed 4:3 Wenckebach conduc-
tion with respect to the frequency of elements
2 and 3, which formed a single domain. Their
mean "ring interval was 368ms. Element 4
showed 1:1 conduction. Panel (c) shows what
occurred in a simulation with an intact ring.
When one "ring was lost in the Wenckebach
cycle of element 1, the impulse transmitted
through the exit site to the right re-entered the
FIG.8. (a)Systemusedtostudyatrialrhythmwheninter-
SN through the left exit site, triggered SN ele-
actionsviatheatriumbetweenSAjunctionalcellsatdi!er-
entexitsitesaretakenintoaccount.FourSNelementswere ments 2}4, which sent away a new impulse, and
used[cf.Fig. 7(b)].The non-automaticelementswerecon- the cycle started over. The re-entrant rhythm
nectedtoaone-dimensionalrow.g wasthecouplingcon-
a persistedinde"nitely,withastablecyclelengthof
ductancebetweenSNelements1and2,g between2and3,
b
andg betweenelements3and4.Thesameconductanceg 222ms. It settled for seemingly arbitrary initial
c d
wasusedtocoupleoneSNandonenon-automaticelement, conditions. No extra stimulus was needed.
and two non-automatic elements. (b) and (c) Proposed
mechanism of brady-tachy rhythms. When the ring was
disconnectedintheatrialpart,therighthalf"rednormally
whereas the left half showed 4:3 Wenckebach conduction
4. Discussion
(b).Actionpotentialsoftwonon-automaticelementsaand
baredisplayedtoshowprogressionoftheimpulse.Whenan
4.1. FREQUENCYDOMAINSANDLEAKAGE
intact ring was used (c), the lost "ring in the Wenckebach
cycleenabledanimpulsefromtherighttopenetratetheSN Theformationoffrequencydomainsseemedto
retrogradelytotheleft,initiatingre-entranttachycardia.30 be such a generic phenomenon that if coupling
Non-automaticelementswereused.Allelementsinthering
had the same (unit) capacitance. Q "(2, 1.5, 2, 2) and ever becomes so weak that global frequency en-
1}4
g "(12, 20, 50, 100)S/F. (**) 4; (----) b; (})})})}) 1; trainment is lost, we expect that it arises no
a}d
()))))))a. matter how peculiar is the SN ultrastructure.
However, the splitting into domains seems to be
clinically signi"cant only if it occurs close to an
For the system in Fig. 8(a), consider the case SA junction and a severe leakage is assumed. If
where g and g were too low to allow 1:1 fre- thesplittingoccursfarfromanexitsite,thelarge
a c
quencylockingbetweenSNelements1and2and peripheral SN domain can drive the atrium at
between 3 and 4, respectively. Then, for g low areasonablerate(Fig.3).Theatrialintervalswill
b
enoughsothatSNelements2and3didnotform also be quite regular, due to the large distance
a single domain, an aperiodic slow atrial rhythm from the domain border (Fig. 1). If leakage is
resulted. This situation corresponds to that in weak, the frequencies of peripheral domains
Fig. 7(b), but with interactions between a and (smallorlarge)canonlybeslightlydecreasedand
b via the atrium taken into account, leading to nosevereexitblockscanappear.However,ifthe