Spontaneous activity of electrically excitable pacemaker cells
We study a mathematical model of spontaneous activity of a cardiac pacemaker cell. The model includes five ionic membrane which are responsible for the spontaneous generation of an action potential of the pacemaker cell. These are the potassium, sodium and calcium ionic channels, the sodium-potassiu...
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| Datum: | 2001 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine
2001
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Institution
Surface| _version_ | 1869291155503448064 |
|---|---|
| author | Pinchuk, A. O. Grechko, L. G. Chepelevska, N. V. |
| author_facet | Pinchuk, A. O. Grechko, L. G. Chepelevska, N. V. |
| author_institution_txt_mv | [
{
"author": "A. O. Pinchuk",
"institution": "Інститут хімії поверхні НАН України"
},
{
"author": "L. G. Grechko",
"institution": "Інститут хімії поверхні НАН України"
},
{
"author": "N. V. Chepelevska",
"institution": "Інститут хімії поверхні НАН України"
}
] |
| author_sort | Pinchuk, A. O. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-11-27T09:42:39Z |
| description | We study a mathematical model of spontaneous activity of a cardiac pacemaker cell. The model includes five ionic membrane which are responsible for the spontaneous generation of an action potential of the pacemaker cell. These are the potassium, sodium and calcium ionic channels, the sodium-potassium pump and the sodium-calcium exchangers. The dynamics of intracellular concentrations of the main cations, responsible for the spontaneous activity of the pacemaker cell, was obtained. The computer simulation of the membrane potential and ionic currents was performed. The limit cycles oscillations were analyzed using the faze diagrams method. The pathology pacemaker activity during arrhythmia was simulated. |
| first_indexed | 2025-09-24T17:25:26Z |
| format | Article |
| fulltext |
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SPoNTA]\ЕoUs AСTIVIТY oF ЕLЕCTRIсALLY
ЕхCITABLЕ PACЕMAKЕR CЕLLS
A. o. Pinсhulq L. G. Grесhko, and N. V. Cheрelеvskа
Institutе o| Suфcе Сhеmistry, Nаtionаl Асаdemу of Sсiencе
Gеn. Nqumov str. ]7, Kуiv-tб1, 03б80, L\KRАINЕ, e-mаil,'"u.ф,,1,пеmJrееnеt.kiеv,uа
Abstraсt
We study a math.еmatiсal modеl of spontanеous aсtivity of a сaгdiac paсеmakег сell.
Thе model inсludes fivе ioniс mеmbгanе whiсh are геsponsiblе ior the spontanеous genегation
of an aсtion potеntial of the paсеmakеr оеll' Thеsе aгe ihе potassium, ,odiu' and сaйium ioniс
сhannеls, thе sodium-potassium pumр and thе sodium.сalсium ехсhangeгs. The dynamiсs of
intraсеllular сonсentгations of thе main сations, rеsponsible foг thе spontaneous aсtivity of the
paсеmakеr сеll, was obtainеd. Тhе сomputeг simulation of thе mеmbгane potеntial and ioniс
сuггеnts was pегГoгmеd. Thе limit сyсIеs osсillations wеге analyzеd using thе faze diagгams
mеthod. Thе pathology paсеmakег activity duгing aггhythmia was simulatеd.
Introduсtion
Тhе devеlopmеnt of a mathеmatiсal modе1 foг еlесtгoсhemiсal osсillatoгs, suсh as
сardiaс paсеmakег сеlls, is an impoгtant task oГ modeгn biophysiсs and еlесtгo-phyiiology of
ехсitablе mеmbгanеs, Тhis pгoblеm has bееn attгaсtеd muсh attеntion duгing tasi jесaaй
Jt.el
Dеnis Noblе was thе fiгst who modifiеd thе сlassiсal Hodgkin-Huxlеy еquations foг thе
desсription of spontanеous aсtivity of сaгdiaс paсеmakеr се1ls [Б]. Although thЬ main results of
thе Noblе's modеl arе in good agгeеmеnt with eхpeгimental findings, this model is based on
modifiеd empiгiсal еquations, whiсh usе a lot of parametегs and ]hus thе рhysiсs.оhеmiсal
intегpгеtation of thе spontanеous aсtivitу of a paсеmakег оеll is muсh. сompliсated.
Fuгthеrmoге' some of thе rеsults of thе Noblе's model (an amрlitudе of the aсtion poiеntia1, a
peгiod of thе limit сyсle osсillations) aгe poоrlу сoinсidе with еxpегimеntal data [3] Sеvегal
modеls havе bееn dеvеloped |aier |2-4J, whiсh havе had moге vaгiеd dеsсгiption oГ singlе
сhanлеls [3] and anothег сhoiсe of paсеmaker сuггents that dгivе mеmbгane dеpolarizatйn
during diastole[2-5].
Wе havе сonsidеrеd the rесеntly pгopоsеd modеl [2] of spontanеous aсtivity of a paсеmakеr
сardiaс сеII of sinoatгial nodе This modеl involvеs less variеd dеsсгiption of ioniс сhannels
than most of eхistеnt modеls and usеs more detail physiоs-сhеmiсal dеsсгiption of the
pгoсеssеs whiсh aге responsiblе foг thе auto-гh}thm osсillations of оaгdiас paсЬmakег сells.
Тhus, thе modеl is rathег useful for thе simulation of intегaсtions of biologiсally aсtivе agеnts
with thе ехсitablе mеmЬтanе of a сеll and thеir influеnсе on the auto гhythm osсillatiois of
Iiving сaгdiaс сеlls.
Gating meсhanisms of ioniс сhаnnеls
Lеt us сonsideг a kinеtiс tгansition sсhеmе betwееn thе opеn (o) and сlоsеd (C) statеs
of an ioniс сhannеl, whiсh is dеsспbеd by a Мarkov Dгoсеss
2',75
, (t)
whеге с and B"* .n],"?.j",.*.s' whiсh depеnd оn thе tтansmеmbranе potеntial. Thе
ioniс сhannеl is еithег еntirelу op.еn "':]:':d
and randomiу fluсtuatеs bеtwеen thеsе two
statеs' Lеt avaгiablе х dеnoiеsih. uu.,ug. ft;.;;; oiop"nЪhаnnеls. Thе dуnamiсs of ioniс
fluх through un "n,.'ьt"Ъ?;;;;;";".r,
i, а.,.,iЬ"а bу ihе fotlo*ing diffеrеntial еquation
(2)- dх _-а(|_х)_ft=-
dt
'.=з1 ,, =:n (3)
а * F
. а+ Р .. ^:-^ I .+ tЬo An.гfr,
whеrех-isthеstеadystatегraсtionofopеnсhannеls,тisthегеlaxationtimе.Lеtthееneгgу
diffеrеnсе bеtwееn thе opеn and сlosеd statеs ofthе сhannеl is givеn by
AЕ = Еo -Е, = q\p, _ p) (4)
whеrеgisagatingсhшgе'qpisthесhangеinelесtгiсalpotепtialеnеrgуduеtothе
геdistribution of сhargеs during thе transition, q"p, i* tь" diffеrеnс.е.in сonfoгmational еnеrgу
bеtwееn thе сlosеd *o
",.",,.,еs
of the ioni.Ъьu*., 1ljl]''o'\um
dхf dt =O and thе
,,"o.oun, for a singlе сhannеl ю bе щеn is given bу thе еxprеsston
(s)
т-= p
Тhis probabilitу is also givеn by thе Boltzmann distгibution
х. r ^Е) (6)
еЬo\ -ттJi-x- '\
Thus,fromthеexprеssions(a)-(6)withq=+4ewеobtatn
i ( ц,\,p' - ?})l '
,*=11-*p[-- *7-tjL'
Thе simplеst сhoiсе for аaldpts
( z'('P' - q)'\
с = 2ехРf_ _- kT )
( z,l1'P' - e)\
Р = 2eх01_17-
)
whеге2is thе сonstant. Thе rеlaxati *'.u^"
,::,|':1jii1-.
potеntial is givеn bу
т =|zlсoshl -.-;г-J]L\
loniс flow and Nernst еquilibriuiч":Т*llг
"nu,*.o
paгtiсlеs. Thе fiгst onе is a
.l*;s;# il]:,:;:ж';i:l.ffi iЁ.{i.91.ь;Бiс^^"11*1.,"ж-;:]i.]'"1,]lilii
onе is a motion oг сtrшe;j рuni.r., in an еlесtгiс fiеld, in our сasе tnls
(7)
(8)
(e)
(10)
!io
(1)
:оtеntial. Thе
эзn thеsе two
:зlliсs ofioniс
::l еquation
(2)
(3)
.
:t thе еnетgy
(4)
-:'. duе to thе
::::эnal еnelgy
;.= 0 and thе
a solution upon the aсtion of an еlесtгiс fiеld. A simplе diffi:sion is desсribеd by the еmpiriсalFiсk's law
П = -uk,tYN (11)
( l6)
(t'7)
(18)
whегeй is thе ioniс florv, Nandи aгe the соnсеntгation and mobility oг tь" paпi"t.,t. ,imotion of ions in an еlесtгiс flrcld is desсгibеd by well known ohm's law
П= _zеul\rYrp
U2)whеге z is thе valеnсе, e is thе еlеmеntary сhaгgе, p is the еlесtriс potеntial'
Combining thе ехpгessions (1l) and (12) wе оbtain the еquation, whiсh dеsсribes thе totalioniс fluх upon thе aсtion of thе еlесtгic fiеld and thе difr:sion
Й = _цkT eхo( -':g)o| ^, ^ -( ,,v\1..'.. -..н\
kT )'LN.-p[ rr)] (13)
Wе сan find an еquilibгium Nегnst potеntialbу puшing in (lз) fr = 0 and intеgгating fгom thе
еxtегnal (с) to intегnal (l) sidе of thе mеmbranе
'!,!, o, *,(- #)"1- "*(ff)]0,0, = o
Integгating (i4) wе оbtain thе follоwing еxpгession
N' | ,r(p'-a"\1
lv"=.*рL_ * 1 (l5)
Solving (l5) with геspесt to (Р'_ v")and introduсing thе notations p, =kTZ =
RЧF, whеге
,Еis thе Faгaday сonstant, wе obtain the equilibгium Nегnst potеntials for thе main сations of a
сell
t. = t-lnNf
ll'K
,," =)пЦЬ
Qьlo = Qтl"*
(l 4)
(s)
(б)
(7)
(8)
(e)
(10)
Iоniс сhannel сurrеnts
Lеt us сonsidег thе сurгеnt through a singlе ioniс channеl with a lеngth d and a сгoss
sесtion arеa S(х). Lеt x =_% dеnotеs аn intеrnal mеmbrane sidе and '=ц an eхteгna1
sidе. In сasе oГa stationary flow thе сuггеnt tгough all сross sесtions must be thе samе. and thе
fluх is invеrsеly propoпional to thе аrеa
Using (l9) for the fluх and ,","*;{TjJt1i]i1,.
"o',"i ( .u^\ Г-, (,,p\1
т= -ukTeхpt _#Jv| мехp[ +] lzeJ '\ kt / L .\k7-./1
Wе multiplу thе геsulting еquation bу eхp(zе(9 - ,Й l kT) and obtain
(1e)
(20).: onе is a
. зs sесond
:. of ions in
i "-,[=и-_Р) = - z е ukT *Г, "-'[-'-#)l
(21)
whеrе thе сonstant po is сhosеnin suсh away, that ,p(-alz)=сpo+9f2, ,p\dl2)=9o_Q12.
in.qu*ion (21) p=,p(*),y=ш(x), S=,S(х) and allthеоthеrsvaluеs donot dеpеnd on х,
Wеintеgratе(21)fгomtheinsidrsidеofthеmеmbranех=-d.f2totheoutsideх=dl2and
obtain
t .. ' ( w,) -,, ^...( рn.)\ t,,,,
i. I = -zеukT{N" ехг[_,kT) _ N eхp\2kT)
|
,i
eхp1zе(,p -,po), kT)'.
I-|-щt- |J.\
'1t
(2з)
Wе ехtraсt a faсtor ^tNN" and wгitе thе ratio of thе сonсentгations in thе Nеrnst potсntlals
ffi = ехP(- zeplzkT\ and obtain thе fo'lowing ехprеssion foг thе ioniо сuгrеnt through
thесhannеl \ / / ,\\''," ",'"i'-
**I Е ",^(
цg\ - .Е *,(:'' )\ -_zzечкт ^17, r, . nn|
,e\p - q')
] p+)
i=:}_.Jм'N.t\jN, "*P\zkT)-1ru,.''"\Я7)I=- 1 '.'.' '..-.\ zkT )
Thе intеgral (23) dеpеnds on thе potеntial p = q(x) as wеll as on thе сross sесtion of thе
сhannеl S = S(х) ' Тo dеtеrminе thе pгofilе of thе рotеntial insidе thе сhannel wе havе to solvе
Poisson,sеquationfoгthеpotеntial,.takingintoaосountthегеalсhargеdistгibutioninsidеthе
сhannеl, whiсh is not avaiiablе at the momеni д .o*onlу usеd apprоximation is that the
Dotеntial inside thе.r'."".ii' ri"еaг' that,is thе еleсtгiс fiеlj insidе thе сhannеl is a сonstant
,:;;i;
=. ,",',. With thr assumptiоn .S(x) = 5o wе havе
г-..=.sosinh(ze(и
_e,)lz*т) (25)
1 =(zе)'u'lN'N"
_
d;il\zе,Еd-
Тhеequation(25)isknownasGoldman,sconstantfiеldapprоximation.Inthеlimitсasе
N,=ly'nitisrеduсеdtothеsimplеohm,slaw,sinсetharlсp,=0.Amorerealistiоsituatton
whеnthеintеgraltzзlсanstirrbесalсulatеd,isthatofanion.сhannеlwithaсonstantсross
sесtion,ехсеptforashortandnaгrowporеwitharadius3Aandlеngth5A,whiсhisa
typiсal assumption гo,jo"i.;;;;' [l]. If wе will assume that thе field is a сonstant, а poге
length is d , thanth. "^;;;;;.lii
j ut", iш"gration willtravе thе following form
zkTd| | , '(zсp) . [-l l l^,-"1, аеa\\ Q6)
'
= "#
\ *sinhlйJ
- [',
_
цJ''""[-zrr ,'t I
In thе limit сase 9 _+ 0
ll " " l сd
',=4?.r" i=;L "0 -РJ
Assuming I = Constwе obtain thе eхprеssion for thе сurтеnt
(21)
2't8
(21)
-|=Рo-Q//
Jеpеnd on x '
: t=dl2 and
(э1\
\--l
(23)
:-.st potеntials
:.::еnt thгough
, - r ltl-"
| {.211
.t- )
..'.:ion of thе
c - зi е to solvе
; .:. insidе thе
:.: .s that thе
: : a сonstant
(2s)
t. l. : iimit сasе
rl .' : situation
|: .l:ant сross
!"r' '''hiсh is a
ш .:лt, a porе
tu*
i (26)
, = д" .,n6[i*(q
_,,)]
- t2kr)
Wеrе /r, is indеpеndеnt fгom p
k, = 2,,ukТ'{ifr So
еd Q9)
fllff.Tjr;;gh
thе potassiшn сhannеls is givеп by (2s), than thе intеgгalmеmbranе сuгrent
l. =}*rsinh(ч*)
(28)
(30)
# = *.o,n ('_+){ l,] l * tunь( 9:l-']
] _'}
whеге r* = |l22,,,n.;;"' J'J;l1] |н ; ;'\ *:,l"::"е paгamеtег, whiсh isgivеn bу thе еxpгеssion (29)
Тhе sodium and сalсium сhanлеls havе an inaсtion and aсtivation mесhanisms. Тhеsе twomeсhanisms саn bе .::'19:'d u'
''uo
iia.i.nа.nt
.Мl..kou proсеsses. As thе aсtivationmесhanism is a vеry'u::..y,.1':1nтеduсе the Jуstеm of.diffеrеntial еquations. Thе maximumtlmе сonstant for thе inaсtivation pгoсеss of сaiсium and sodium
"ь.nn.t,
have the samе ordег(sevегal hundгеds of milliseсondо, tь"'iь"Ъ"i.еnts oгthes.
"ьunn.i*
ьuve the following viеw
iro = kr".f d*r'"nf 4tr.Jl
\ kr / (31)
o. = :1, * runn2'(P ..P)
I.L кI j
i,,, = k,o ь ,,-',inь(
,(p - p.""))
t2kr)
^- = Il, -,^'n(Цk-е))\'t \ /(1 )J
i=;"o'.L-*r-Jt;L'_'^'n|-i'11_oу (з6)
whегr &"o and'k-" aгe the paгamеtег of thе сondu.tiuiiy oiсalсium and sodium сhannеlsгеspесtivеly, po and Q. are thе half.aсtivation
рotеntiаls.
Й glw LrlU rrdlr.al;livallon рaгametегs, P7 and Qh are thе half-aсtivation
Sodium . рotassium рumр
Thе геaсtion of Na* ,K* puпlp is fully dеsсгibеd by the folloйng еquation
(з2)
z4e-e,)\1
1_- *-]rf r (з3)
/J
)
(з4)
#=|"*{фP)ЕГ,_'-'[
(3s)
(27)
279
АTP -ЗI,/а, +2K" <э АDP +3|'Iаi +2Ki + P,".р (37)
rvhеге А,tP,АDP,P,. aге adеnosinе tгiphosphatе, adеnоsinе diрhosphatе and inorgantс
phosphatе rеspесtivеlу. Thе еnегgу, whiсh is rеquirеd fог thе aсtive trаnsрoп of 3 сations .A/a-
und z
"ations
Kt against thе еlесtroсhеmiсal gradient
AG'o = _зе\v_Фu"\ (38)
ЬG* - +zе\,l - Ф) (-з9)
Тhе сhangе in Gibbs еnеrgу- is dеsсribеd bу l . \ z l лl
^G
= ЬGo,, + ЬG,'+ AG* = e\цn,, +З9,,,.-zqr _,p) (4Ul
whеrе A(i,n." is thе еnеrgу, assoсiatеd with thе brеakdown of АTP, 9юp= ЬGn,,f е'Тhe
sum оffonvard and baсkward ratеs сaп bе еqualеd to a сonstant
q+P=i $1)
At еquilibnum the foгward and baсkwaгd rеaсtions must oссur at thе samе fгеquеnсу, that is
" =.*o[_Щ. _) Ф2)
P '\ KI J
Solving thе еquations (41) and (42) foг а and B wе obtain
z ехр(_ #)
сI = ---7--|;\
l + ехp\_ йJ
7
Д=
'n.-р9
Thе diffеrеnсе
а- lJ=,
*p!_f l-*4tl=,ttunь[-*], ц_ *,,.i* ;p(#) -
,".-...^\ 2kT)
(43)
(44)
(4 s.i
(4б)
givеs thе total сuгrent of Nс",K- pump
l
",oк
= Ме\а - Р) = k**'"n(фP)
\ LYт
whеге A^'.* = ]у4е7 .
Sodium-сalсium ехсhangеr
Thе rеaсtion wьiсir dеiсribеs thе sodium-сalcium ехсhangеr has thе follойng viеw
Саi- +змаi },Cа!' +3Nаi (47)
Theеnегgy,whiсhisnеcessarytoеxtraсtonесatiоnСа2*againstthееlесtroсhemiсal
gгаd i еnt, i, p,o аu".а d".й ; ;
" {";:: : l:llu ! [
" "\*'
thе el е сt гo сhеmiс al *'.o'
"".'o
*,
ЬG,.o = -2Nа(rp _ сp,")
Thе total work is givеn by
AG = AGш" + ЬG',. = ,(v _ Зсp ," + Zrp,') (49)
ThегatioаtoBhasthеsamеfoгmasin(42),butasoppositеtosodiцm-pоtassiumpumpthе
saтuгatiоn еffесt is not еxpесtеd. and thus
280
(37)
and inогgantс
' j сations Nсt
(3 8)
(з9)
(40)
=3rj^",,fе.TЬe
(41)
:-::rсу, that is
(42)
Intеgrating (55) wе obtain
o = л"*o(-:\у-l'*:зd\'\ zkT )
ГУ tp- ^ INL
L
+ N'n,") = д
(.5О)
в'hеге 2isaоonstant
I na:,,
whегe Ano.o = 2ne)"
'
Thе membrane potеntial
Supposе that thе еlесtгOсhепiсal аоtivity of a cardiaс сell is dеfinеd bу thгеe
K- 'Nа- ,Сq2- сhannеls' .^/a-К- pump and |,/а* Саz* ехсhangег, and all thе othег сuгrеnts
сan bе neglесtеd' Тhе standard diffегеntial еquation foг thе mеmbгane potеntial and thе
сonsеrvation laws foг thе intгaсеllulаг ioniс сonсrntгations of thе main сations havе thе
followins viеw
rirp I I
ff = -i\,*+rN, + ir + i** + i*"",)
dN,к
=2iNoК
- iK
dt F'tr,
dN Ь"
=
2i ],,oс,. - i co
dt zFV
d|l,*''
-:
l.'ц _ зi NoKo - 3i voсo
whеге С is the mеmbгanе сapaсitanсе , Л is thе Faгaday сonstant. Solйng thе еquation systеm
(52) wе obtain the ехpгеssions Гor i*,i"o,i,o
. n,d t,tК = _I'у
-
1у L, 1 llNaK
сl
i,' = _FL, *N,N. _зiN"K - 3i"o,o
dt
Substituting obtainеd ехpгеssions into thе еquation foг thе potеntial wе havе
dсo F|,' d '-; = . ;\N r +2N'r"+ruN"J=0' а| ( dI'
Тhis еquation сan be written as
d( FV,
d,|e_ С (Ni +2|'I','
(s l)
ngeг wе haveum ехсha
2,Р""))
-)
(43)
(44)
(4s)
(48)
(4e)
oump thе
(s2)
(46)
+ш;)) = o
\Yhеге thе сonstant l is found flгom thе сondition, that thе mеmbranе potеntial is еqual to zегo,
whеn thе insidе and оutsidе сation сonсеntrations aге еqual. Thus' the еquation Гor thе
mеmbгanе Dotеntial has thе following viеrv
o=|W;_Ni + 2(NL"_1/Ё")*N; -^-^1")
Тhе parametеrs usеd during соmрuter simulаtiоns
Wе have usеd thе following paramеteгs to simulаtе numегiсally thе paсеmakег aсtivitу
of a sinoatrial nodе сеl1 [2]: thе сonduсtanсе paramеtеrs k," =26.2 pА' ku."=О'01645 pA'
l," =] 12 1 pА-, k"=З29 PЦ k"',.=1400 pА' й'*=1l a6 pА' thе ехtгaсеllulaг ioniс
соnсеntrations Ni:5 a mМ, N,i,:l40 mМ, NZ":2 mМ, thе tempeгature Z= 310K, thе
сapaсitancе of a mеmbгanе С:1O pF, thе volumе of a оеll V =7О|mf , thе геlaxation timе
Т = TK = ТNo = Tсo = 200 ms, thе half-aсtivation potеntia|s g' _ -25.1 mV, Qa = -6.6 mУ,
Q1=-25 mv, 9n|= -4l .4mУ, 9ь
_ -9l mV, Q,qтp = -450 mV
Numеriсal rеsults and disсussion
Thе еleсtro-phуsiologiсal spontanеous aсtivity of real paсеmakeг сеlls is rеasonablу
dеsсribеd by abovе dеsспbеd thеоry' In paпiсular, wе havе simulatеd thе spontanеous aсtivity
of a rabbit sinoatгial nodе сеll using this modеl. Thе simulatеd mеmbгanе aсtion potеntial of
thе paсеmakег се1l is shown in Fig' 1.
1,0
Timе |s]
(s7)
> -10
.Е -zo
U
.Si^
;:
-ЭU
Fig. 1 Simulatеd mеmbranе aсtion potеntial of a paсеmakеr сеll
Тhе wavеfoгm and amplitudе of thе simulatеd aсtion potеntial rеasonably сoinсidе with
еxpегimеntal rеsults [6]. Using this thеory wе сan prеdiоt ехpегimеntallу оbsегvablе iоniс
.u,1..nt,, rvhiсh undеrliе thе spontanеous aсtivity of alivе paсеmakег сеlls, Fuгthеrmorе, wе
Fio_'5
282
..]i to zегo,
:-.n for thе
(s7)
::.3г aсtivitу
. ;645 PA.
е ,'llaг ioniс
= ] 10K, thе
.'.;tion timе
-.б.6 mV,
:еasonablY
.ls activity
-.tеntiai of
I
сan gеt somе physiсs-molесular mеaning of thе pгoсеssеs whiсhosсillations in thе paсеmakеr сеll' Thе dynamiсs of thе ioniс сuггents K'inFigs2-4
dгive auto-гhythmiс
,Nа*,Cа2' is shown
80
70
OU
50
40
30
20
10
0
J'9 U,о 1,5
t,imе [s]
Fig. 2 Thе outward dеlayеd гесtiфing potassium сurrеnt of a pacеmakег сеll
0
-40
-ou
-80
-1 00
,с
г
-:idе with
. зblе ioniс
::п1orе, wе
-1 60
- t80
-200
T'inlе [sJ
Fig. 3 Тhe inward сalсium сurгent оf a paсеmakег сеlI
28з
Fig. 4 Thе inward sodium сuггеnt of a pacеmakег сеll.
Thе long-timе simulation of thе mеmbranе potеntials with thе еqual staгting
intraсеllulaг and ехtгaсеllulaг ioniс сonсеntrations is shown in Fig.5.
As сan bе sееn fгom thе long-term simulation thе Nегnst еquilibrium potеntials arе
геaсhеd гoughly aftег 750 о (12 5 minutеs) Thеsе long.timе simulations сonfrгm ahуpothеsis
on thе limit-сусlе osсillations in thе simulatеd sinoatгial nodе сеll' Thе impoпanсе of thе
|,.,еnt"d mojеi as wеll as othеrs rеasonable modеls liеs in thе possibilitу of applуing of thе
Ьodel foг invеstigation of somr pathologirs' whiсh havе wеll definеd сliniсal manifеstation.
onе of suсh hеaгt disеasе is an isсhaейia[l,7] Тhе simulatеd aсtion potеntial at diffеrеnt
еxtrасеllular сonсеntrations of K-, as it takеs plaсе during isсhaеmia, is shown in Fig.б
As it сan bе seеn from Fig.6 thе arrhутhmia pathology takеs plaсе with inсгеasing of
ехtraсеllulaг potasslum сonсеntra1ion Nthough somе еvidеnt- suссеss in dеsсгibing of thе
paсеmakег activitу during isсhaеmiapathоlogy, the rnodеl falls to prеdiсt all thе possiblе
ехtraсеllu1aг K- сonсеntгations, that сan takе plaсе duгing this disеasе [7-9]. Thе problеm сan
liе paгtlу in that faоt, that thе modеl involvеs onlу thrее main сations K- ,Nа* ,Са,* ' and does
not inсludе for instanсе Сi - anions. This is a subjесt of thе follоwing invеstigations, as it сan
bе mentionеd from еlесtrо-nеutralitу rеquirеments of thе intraсеllulaг and еxtгaсеllulaг salt
aquеous solutions, whiсh havе not bееn сonsidегеd in thе modеl'
It is known, that thе spontanеous osсillations in a paоеmakег сеlls havе thе limit сусlе
сhaгaсtег, To analyzе this phеnomеnon wе havе simulatеd thе phase diagгam of thе
spontanеous oscillations usinj the modеl. Тhе phasе diagгam of the limit сусlе osсillations is
shorvn in Fig 7.
284
1 000 2000
Timе [sJ
3000 4000 5000
5000
100
60
40
ZO
n
20
40
60'
80'
40
а
I
,|
\
\
с̂r
б
\
сi
\,<
2000 3000
Time [sJ
Fig. 5. Long-timе simulation of the mеmbгanе potеntials оf a paсеmakеr сеll
10
0
.JU
1 000 4000
:-:- staгtlng
n::зntials arе
i -\Dothesis
:::,се of thе
г ..rg of thе
г-":-lЭstation.
ц .: diffеrеnt
-- -o.
. ::еasing of
;-:::.g of thе
-. possiblе
в..-.эlеmсan
- . and doеs
ш.-s. as it сan
r: ::llulaг salt
-:.: .imit сусlе
ц-.,rl of thе
r I ,:rllations is
ь
':
^o
б
U
t(
-40
-ou
-70
Fig. б Aл influenсе of an ехtгaсеllulaг potasslum сonсеntration on thе aпh1thmia of
a paсеmaker сеll
285
f----
\
0006
l
\
ooo4
1
I
o,oo2 I
t
\
O2
Fig. ? Limit сус1е osсillatiоns of а paсепlakеr сеll
lfthеstaгtingiпtraсеllu]alсonсеntгaticnofmainсationsisсhansedinthеlimitсусlе
rangе, thе tгajеctory oriь. pt]u,. diagгam -'',;.ln;;йiу uрf,ou"ь"' tье timit сусlе, whilе at
thе сonсеntrations K- > 5 44 rгм thе trаjесtог1 ot.n. liпut суо1е fоllоws to infinitу'
Rеfеrеnсеs
1.HillеBIoшсСhannеlsofЕхсitablе}{еmbranеs'-2nс1еd'-Iv{assaсhusеtts.Sinauеr
, i:::::ТL'}., fj,i-
,;:,l.,
,
..:9 )]iin:'l;l; ii}Тfl.';: ;ъ:,ъ;.branе
potеntia' оГ
living се1ls ul вu,opеan'вiopьу',i.ul Jоuгna.l - 2000 - \, ' 29 , - P. 90- 1 03
3. Doсos S'' Сеllет в.l"Jl.,'.irш , !о1
Сu'.,.nt, UndеriуingNоdе.Paоеmakеr
Aсtivitу'
ANеw Singlе Cеll fi;;;;..tilooеt, ;;;;ilJo,Jti.ul Biologу. - l996. . V 18l -
P.245-21?
4.NoblеDТhеinitrationofthеhеaпbеatoхfoтd'oxfordUnivеrsitуPгеss,1979.-412p
5. Noblе D , Noblе S-Amodеl of sino.atгial noj" .t"",,i"al aсtility basеd on a modifiсation of
thе DiFranсе'"o.N;;';;;,.',
p.o. к"io..;;"; B. _ 19s4 -У 222. -P'295-з04.
6. DiFranсеsko о pa-йuк..
'..ьonisms
in сardiaс tissuе // Aлnual Rеv. of Physiologу _
"
issз - v 5. _P 455-4T2
18,732
1a 1tO
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Сh'еn F.F , Vaughan-JonT R P ' Сlarkе K', Noblе D' Modеling myoсardial isсhaеmia ant.irеpегlтsion // Pгogrеss in Biophуsiсs anсt Мolесular Biology. _ йs;] ] v. оs - P 515.-538Lеi М., Brorvn H., Noblе D' Whаt rоlе do T-tуpe .ul.iu' сhaпnеIs play in сardiaс
!ffi::Llъ:.Шy),1,..
'..ffliъun'u"'thе
IntегnationalЕleсtгophy,loiogy Мееting ( Basеl,
Нan Х,, Sеn L.,
'.'.n'I,'}T]]",..:|I1yl. сalсium сhannеls in гabbit atгiоyеntiсuiагnodе and сaгdiomуopathiс hamstег vеntгiсle ttiьiа - P. 110-122.
.'it n',пlo
:.. whilеat
гrj-
h
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. Sinauег
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|
| id | oai:ojs.pkp.sfu.ca:article-67 |
| institution | Surface |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-09-24T17:44:40Z |
| publishDate | 2001 |
| publisher | Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | surfacezbircomua/ea/0a105e011158c96467151b85278e5bea.pdf |
| spelling | oai:ojs.pkp.sfu.ca:article-672018-11-27T09:42:39Z Spontaneous activity of electrically excitable pacemaker cells Spontaneous activity of electrically excitable pacemaker cells Spontaneous activity of electrically excitable pacemaker cells Pinchuk, A. O. Grechko, L. G. Chepelevska, N. V. We study a mathematical model of spontaneous activity of a cardiac pacemaker cell. The model includes five ionic membrane which are responsible for the spontaneous generation of an action potential of the pacemaker cell. These are the potassium, sodium and calcium ionic channels, the sodium-potassium pump and the sodium-calcium exchangers. The dynamics of intracellular concentrations of the main cations, responsible for the spontaneous activity of the pacemaker cell, was obtained. The computer simulation of the membrane potential and ionic currents was performed. The limit cycles oscillations were analyzed using the faze diagrams method. The pathology pacemaker activity during arrhythmia was simulated. We study a mathematical model of spontaneous activity of a cardiac pacemaker cell. The model includes five ionic membrane which are responsible for the spontaneous generation of an action potential of the pacemaker cell. These are the potassium, sodium and calcium ionic channels, the sodium-potassium pump and the sodium-calcium exchangers. The dynamics of intracellular concentrations of the main cations, responsible for the spontaneous activity of the pacemaker cell, was obtained. The computer simulation of the membrane potential and ionic currents was performed. The limit cycles oscillations were analyzed using the faze diagrams method. The pathology pacemaker activity during arrhythmia was simulated. We study a mathematical model of spontaneous activity of a cardiac pacemaker cell. The model includes five ionic membrane which are responsible for the spontaneous generation of an action potential of the pacemaker cell. These are the potassium, sodium and calcium ionic channels, the sodium-potassium pump and the sodium-calcium exchangers. The dynamics of intracellular concentrations of the main cations, responsible for the spontaneous activity of the pacemaker cell, was obtained. The computer simulation of the membrane potential and ionic currents was performed. The limit cycles oscillations were analyzed using the faze diagrams method. The pathology pacemaker activity during arrhythmia was simulated. Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine 2001-06-10 Article Article application/pdf https://surfacezbir.com.ua/index.php/surface/article/view/67 Surface; No. 4-6 (2001): Chemistry, Physics and Technology of Surface; 275-289 Поверхность; № 4-6 (2001): Химия, физика и технология поверхности; 275-289 Поверхня; № 4-6 (2001): Хімія, фізика та технологія поверхні; 275-289 3154-8091 3154-8083 en https://surfacezbir.com.ua/index.php/surface/article/view/67/66 Авторське право (c) 2001 A.O. Pinchuk, L.G. Grechko, and N.V. Chepelevska |
| spellingShingle | Pinchuk, A. O. Grechko, L. G. Chepelevska, N. V. Spontaneous activity of electrically excitable pacemaker cells |
| title | Spontaneous activity of electrically excitable pacemaker cells |
| title_alt | Spontaneous activity of electrically excitable pacemaker cells Spontaneous activity of electrically excitable pacemaker cells |
| title_full | Spontaneous activity of electrically excitable pacemaker cells |
| title_fullStr | Spontaneous activity of electrically excitable pacemaker cells |
| title_full_unstemmed | Spontaneous activity of electrically excitable pacemaker cells |
| title_short | Spontaneous activity of electrically excitable pacemaker cells |
| title_sort | spontaneous activity of electrically excitable pacemaker cells |
| url | https://surfacezbir.com.ua/index.php/surface/article/view/67 |
| work_keys_str_mv | AT pinchukao spontaneousactivityofelectricallyexcitablepacemakercells AT grechkolg spontaneousactivityofelectricallyexcitablepacemakercells AT chepelevskanv spontaneousactivityofelectricallyexcitablepacemakercells |