Michael R. Guevara et al- Alternans in periodically stimulated isolated ventricular myocytes:...

8/3/2019 Michael R. Guevara et al- Alternans in periodically stimulated isolated ventricular myocytes: Experiment and model

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Alternans in periodically stimulated isolatedventricular myocytes: Experiment and model

MICHAEL R. GUEVARA,a FRANCISCO ALONSO,aDOM INIQUE JEANDUPEUXa AN D ANTON1 C. G. VANG I N N E K E N ~BDepartment of Physiology and C entre for Nonlinear Dynamicsin Physiology and Medicine, McGill University, 3655 DrummondStreet, Montreal, Quebec, Canada H3G 1Y6bFysiologisch Laboratorium, Universiteit van Amsterdam,Acaden~ischMedisch Centrum, Meibergdreef 15, 1105A ZAmsterdam, The Netherlands

The alter nt~tin g ontraction o r he ventricle is not a simple pathological pl~eno nie-not1 but rather a general pl iysiological function, that makes its appcnr;tl lce in manycircutnstances, .. .We have, i n fact, t o deal wit11 a capacity of the cardiac musclewl l ic l~ t~: tblcs t l ~ e entr icle to go o n w i th r hy t l~ n i ica l ontrac t ions even underabnor t~ ia l ondi tions. Musketis (1907--08)I N T R O D U C T I O NUc a t t o bea t a l t e rna t i on in t he i n tens i t y o f l i e pe r i phe ra l a r te r i a l pu ls e wasl i r s t des c ri bed us i ng g raph i c a l rec o r d i ng tec hn iques mor e han a c en tu ry ago .'Th is a l t c r t i a t i oa i s us ua l ly a t t r i bu ted t o an a l t e rna t i on in t he fo rc e ofc o n t r a c t io r i o f t h e l e f t v e r ~ t r i c l eof t l i e hea r t . P e r l ~ aps he s i n l p les t wa y inw l ~ i c l ~l t cr n :l t io n c o u l d o c c u r i s f t l ~ e r c o u l d b e a s i n ~ u l t a n e o u s o n c o r d a n ta l t e rna t i on of l i e l b r c e of c o n t r a c t i o n i n a l l c e ll s of l i e l e f t v ent r i c le . I n 1882G a s k c l l p u t f o r t l i a n a l t e r n a t iv e h y po t h e s is s t a t i n g t h a t a b e a t t o b e a tt ~ l t c n i a t i o n n t he i n tens i t y of t l ~ c i e a r t b e a t c o u l d b e d u e t o s p a t i ali n l i o tnogene i t i es in c o n t r a c t il e a c t i v i t y i n d i l re ren t a reas of he ventr ic le : 'T heexplanat ion, therefore, o f t h i s a l t e rna t i on in t h e f o r c e o f t h e c o n t r a c ti o n sn i u s t b e s o u g h t f o r i n t l ie m u s c u l a r t i ss ue i t s el f, a n d i t s ee ms t o m e t h a t t h eCcll lo Ccll Sigllrlling: From CopyrigM C 1 9x 9 Aude m t Frcu Limirrdl i r ~ ~ i ~ ~ i c n ~ r, 'I corc~ical dcls An righu of tqmduclion inl S N N 0-12-287960-0 , 551 ram ,-


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55 2 M. R . GUE V ARA E T AL .

~ l l o s t~ j rob ;~ \ ) l cx l ) l : r l ~a t i o r~s I l l a t a l a rger a l l l ou l l t o f t i ssue co t it r i1c ts when the bea ts :Ire I:~ rg c tl1;111 WIICI~ tI1at tl lcrclbrc, ill al l l ~ e y r e s ~ n a l l , IIICI ~ j r o h : ~ \ j i l it y . c r t ; ~ i ~ ~ r e s p o ~ l d n l y t o e ve r y s e c o ~ l d o r t i o ns o f t h e v e ~ ~ t r i c l e' i ~ ~ ~ p u l s e , r c s p o ~ ~ dl l i l c o t l l e r p o r t i o ~ ~ s to every i 111l )u lse' . A t l~ i r d oss ib il i t y, tI1:1t t l lere co r~ ld e ;I2 : 1 r e sp o us e i n t w o d i l l' e re n t s u b p o p u l a t i o ~ ~ s , it11 the c o ~ ~ t l u c t e t l a l t c r ~ l a t e e at s in he tw o subpopu la t i ons , was ea t occur r i ng Os r~gges tc t l y Min es (1913).

W i t l i t he a d v e ~ ~ t l ' e lec t r ica l record ing tec l~n iqnes , l ec t r i ca l a l te rnans was o b s er v e d i n w l ~ o l el car ts us in g ex t race ll u la r ( I - l e ll e rs te in a nd L iebow, 1950) a ~ l d n t ra c c ll u la r ( D o w n ; ~ r et tr l . , 1977) record ing tec l i n iques . I n the la t te r c as e, eve11 1 11 ou gl 1 l l e re c o rr l s t h e t r a ~ l s ~ l ~ c ~ l ~ b r ; ~ n e o f a s ingle cel l. i t o t e ~ ~ t i a l i s i ~ ~ ~ l ) v s s i b l eo say WIIC~I IC~r 1101 t he b e i ~ t o b e at a l t e r n a t i o ~ ~ n t l ie I IIO~-pllology o f l ~ e~ c t i o ~ ~ see11 i n l ~ a t t o t h e c e ll o r i s o t c ~ ~ t i a l c ell is i ~ ~ t r i r ~ s i c d u e t o :III c l ec t ro t o ~l ic ~ ~ j c c t i o ~ ~ c el ls t h at a re f c ur re nt f ro m ~ ~ e i g l ~ b o u r i ~ ~ g c o u l j l c d t o t h e c e l l i n c l u c s t i o ~ ~ g a p - j u n c t i o ~ ~ a l 111y l o w - r c s i s t ; ~ ~ ~ c e p i ~ t l ~ w a y s . the I ;~ t te r ase . t l ~ c ~ l t e r n i ~ r ~ s b e a t le as t p : ~ r t lydue to t l i e fac t tha t the s t i ~n u l u s l ~ r r e l ~ t i ne l iv e re d t o t h e c e ll i s i t s e lf a l tc r n ; ~ t i~ l g . o r e x a ~ ~ ~ p l e , v i rt u a ll y a l l t yp es o f c a rd i ac t is su e, a l t e r ~ ~ a t i o ~ ~ p o t e ~ ~ t i a l f a c t i o ~ ~ ~ ~ ~ o r p l ~ o -l o g y c a l l b e see11 i n t h e r e g i o n o f b l o c k w l i e ~ ~: 1 block occurs (see Guevara, 1984, I b r re fe reaccs ). Mod e l l i n g wo rk i l i d i ca tes tha t i n one such ins tance a l t e r ~ ~ : ~ t i o ~ ~ cu rr en ts i l ld e e d s e co n tl a ry t o t h e f a c t t h a t t h e l l o w o f s t i ~ l l u l u s i n t o t l lc r e g i o n o f b l o c k f r o m r e g io u s d is t a l t o t h e s it e o f b l o c k i s i t s el f a l t e r ~ ~ ; ~ t i ~ ~ g : h e a l t e r n ; ~ l i o n s t h u s n o t i l l t ri l l s ic t o t h e c e ll s ill ques t ion (Cineva ri~. 1988, l igu re 3D ). In l i gh t o f he :~bove onsi t l era t i ons, i t seemed t i~n e l yor us t o i nves tiga te : ( i ) w l ~ e t l l e ro r ~ i o t s in g le , i s ol a te d , q u ie s c en t v e ~ ~ t r i c u l a re l l c o u l d b e m a d e t o d i s p l : ~ y a l t e r ~ l a n s; ( i i ) w l l e t l le r o r n o t a l t e r n a ~ ~ s s ee no u l d b e ill a ~ l ~ a t l l e ~ n a t i c a l m e m b r a n e . 1 10 de l o f a p a t c h o l ' i s o p o t e ~ ~ t i a l

METHODS

E x p e r i m e n t a lS ing le ve~ ~t r i c u l a rel l s a re i so la ted f rom ra bb i t hear ts us ing s tandardt e c l ~ n i q u e sG i le s a n d v a n G i n ~ l e k e n , 9 85 ; G i le s a n d I n ~ a i z u ~ n i ,988). The l ~ e a r ts r e ~ n o v e dro111 h e a ~ ~ i n ~ a l , nd the coro l l a ry a r te r ies per fused w i th a heated (37C) . oxyg e~~ ate d, f l ow in g t l l roug h the aor ta u l l e r e d s a lt s o l u t i o ~ i i n l i e r e t r og r a d e d ir e c t i o n . ' l' l ~ e e a r t i s li r s t p e r fu s c d u s i n g a s o l u t i o ~ i i t h a p l ~ y s io l o g i ca l a2.* c o ~ l c e ~ ~ t r a t i o ~ i t o w a sh o u t t h e b l o o d f r o m th e 2. 2 mt.1) c o r o l l a r y b e d, t l ~ e n it 11 a l o w - C a 2 + so l u ti o n , a n d l i ~ l a l l y i t1 1 a l o w - C a 2 + s o l u t i o t ~ o ~ ~ t a i ~ ~ i ~ ~ g nixl lure o f the e l lzy lnes co l lagenase (S ignla , t ype IA :5UU U/IIII) i1 11 d r Y P s i ~ < ( ~ i g ~ ~ ~ a . l ' l l c l o w - C a 2 + s o l u l i o ~ ~y p e 1 1: 24 9 U/III~). is ~ n a d e y a d d i ng I 5 pln C a 2 + t o a n o ~ n i n a l l yCa2+- f ree so lu ti on . The

ALTERNANS IN ISOLATED VENTRICULAR MYOCYTES 55 3

c o ~ ~ i l ) o s i t i o ~ ~ o f t il e ~ ~ o ~ l i i n a l l y 121, K C 1HIM) zero [Ca 2+] sol ut iot ~s: N;ICI5.0, MgCI, 1.0, N a ll C O , 15, Na,ll l 'O, l .U, Na-C11,COOI 1 2.8, glucose 5.5.A l l solut ion s :I re gasset l wi th carbogen (95"/n 0,-5% CO,). St1 ps of issue aret11e11 u t 0111 o f l ~ cigh t vent r i c le . I ' l l ese s t r i ps a re cu t i n to c l l nnks , wh ic h arep laced i l l l o :I h e a te d ( 3 7 C ) b a t h c o ~ i t a i n i n go w - Ca 2 + s o l ~ ~ t i o ni t t i n g o n agyr :r t ing tab le ro ta t i ng a t 70 r.p.nI.. T h i s c o m bi n e d m e c l ~ a ~ ~ i c a ln d c l i e n ~ i c a ld i g e s t i o ~ ~ u p, y i e ld i n g a s u s p en s io ~ i f s i ~ ~ g l er e ak s t h e c l l u ~ ~ k s c el ls .The ce l ls a re then p laced i l l to a chamb er s i t t i ng a t roo111 te lnpera ture (24-27C) O the s tage o f an nver ted mic roscope. l ' he above i so la t i on procedureproduc es quiescent, rod-shaped cel ls wl i i ch are 'calc ium-tolerant ' , i .e. whic l id o 1101 d ie w l l en the exper imenta l c l i amber i s per fused w i t h so lu t i on a t ap l ~ y s i o l o g i c a lC a 2 +c o ~ ~ c e n t r a t i o nf 2.2 m M . A g la ss s u c ti o n ~ i ~ i c r o e l e c t r o d ei s t l ~ e nl o s i t io n e d u s i n g a n i i c r c h i a n i p u l a t o r t o c o m e i n t o c o n t a c t w it 11 t l i em e n i b r a ~ ~ e s ea l o f e s is ta n ce 2-f a ce ll , w l~er eup on n e lect rode-niembraneI 5 GlIl i s f o r ~ n c r l p o n a p p l i c a t i on o f s l ig h t s u ct i on . A s u c t i o n p u ls e t h end is r up ts th e p alc h o f ~ l ~ e ~ ~ i b r a ~ ~ ee al ed o l r b y t h e t i p o f t h e n ~ i c r o e l e ct r o d e .S i nc e t h e e le c t r ol y t ic s o l u t i o n w i t l ~ i nh e p i p e t t e ( c o m p o s i t i o n (111t.t):K C 1 1 40 ,ECi l 'A 5. CaCI, 1.54, Mg CI, I,Na,A?'P 5, HE PE S 5, bulPered to p H = 7.0w i t l i K O H ) i s then cont iguous w it11 the in t race l l u la r f l u id , the po ten t ia lt l i l l e re~ lce e tween the n i i c roe lec t rode an d a second e lec t rode p laced in theb a t l ~ i ~ ~ gl u i d s u r r o u ~ l d i n g h e c e l l c a n t h e n b e m e a su r ed . l ' l i i s p o t e n t i a ld i l l e r c ~ l c cs t he t r a t ~ s ~ i ~ e n ~ b r a n e A p e r io d i c t r a i n o f current pulseso t e ~ ~ t i a l .o f c o ~ ~ s t a ~ ~ t c a n t h e n b e d e li ve re d t o th e c el l t h r o u g li t li e s am e~ ~ ~ p l i t u d en~ic roe lec t ro de o s t i n lu la te the ce l l .

N u m e r i c a lW e k n o w o f n o p u b l is h ed i o n i c m o d el s o f h e r a b b i t v e n t r i c i ~ l a r e a rt c el l.F or th i s reason, we inves t iga te the mod e l o f Bee ler and Reuter (1977) w h ichdescr ibes ungu la te vent r i cu la r l i bres . There are f i ve i on i c cur ren ts in thism o d el : th e fa st i n w a r d s o d iu n i c u r re n t ( ) t h e s l ow i n w a r d c al c iu m c u r re n t(I,), t l i e t ime-dependent ou t war d po tass ium p la teau cur ren t (I,,),n d t h et ime- i l l dependent po tass ium and sod ium leakage cur ren ts (I,,,,,,).u-m e r i c a l i n t e g r a ti o n i s c a r r i e d o u t in s ing le p rec i s ion (app rox imate l y sevens i g ~ i i f i c a n td e c i m a l d ig i t s) o n a H e w l e t t - P a c k a r d m i n i c o m p u t e r ( m o d e lIUUUF) us ing an e l l i c i e~ i t a r i ab le t ime-s tep Eu ler a lg or i l h ~n V i c t o r r i et al.,1985) p rev ious ly used b y us in a m o d e l l i n g s t u d y o f p h a s e -r e s et t in g inPurk in je f i b re (Guevara and Shr ier , 1987) . l ' he ma x im um change in th et rans me~n branepotent ia l AV a l l o w e d in i t e r a t i n g f r o m t i m e t t o t i m e t + At is0 .4 m V . W l i e n a v a l u e o f AV l a rger than th i s up per l i m i t resu l ts , thei ~ ~ t e g r a t i o ~ ~in ie s tep At is successively halved and the calculat ions redoneu n t i l A V i s le ss i l l a n 0 . 4 m V . W l i e n A V is less tha n 0.2 nlV, At i s d o u b l e d f o rthe fo l l ow in g i te ra t i on . Un der these cond i ti ons , the va lue o f At rangesbetween Ips and 8 .192 ms, a nd the vo l tage wave form i s very c lose to th a t
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55 4 M. R. GUEVARA E T A L . ALTERNANS IN ISOLATED VENTRICULAR MYOCYTES 555: ~ p p c a r i~ ~ g o f he ~ ~ ~ o d c lll t he o r i g i na l de s c r i p t i o ~~ ( I l ee le r and Ren te r . 1977) .I n a d v a n c i ~ ~ gr o m t i ~ n e t o t i ni e r + A r, t he c o n t r i b u t i o n o f t h e l r a n s ni e l n-b r :~~ :ec a r re~ l t o A V s c a l c ~ ~ l a t e d i n f o o tn o t e 2s i n g t h e f o r ~ i i u l a~ p p e a r i n go f V i c t o r r i e r ul . (1985). Since the t ime s tep is var iab le, i t IIIUS~ be ad jus tedw11e11 curr ent pu lses are del ivered so tha t th e curr ent s tar ts an d s tops ate x a c tl y t h e r i g h t t im e s . T h e t i n ~ e o n s ta n t s a n d a s y ~ i i p t o t i c a l ue s o f t h ev a r io u s : ~ c t i v a t i o n n d in a c t i v a t io n v a r ia b le s ( /I, ,/, [ I , / ; .yI) a re s to red i n at a bl e a t s t e p s o f 0 . 2 I ~ I V 111d li n e a r i ~ ~ t e r p o l a t i o ns us ed to ex t rac t ou t t he i rv a l r ~ e sa t a l l y g i v en v o l t :~ge V . L ' H b p i l a l ' s r u l e n i u s t b e a p p li e d w h e nn ec es sa ry i n c a l c u l a t i ~ ~ g I n i t i a lh e r a t e c o ~ l s t a l l tu,,, a n d t h e c u r r e n t I,!.c o ~ ~ d i t i o ~ ~ s a n d n a c t i va t i o n v a ri ab le s w e re t ho se a p p r op r i at e t on a c t i v a t i o ~ ~th e n i e ~ ~ ~ b r : ~ n ees t i ng a t V = - the4 n i V f o r a n i nf in it el y lo n g tim e, a ~ ~ di ~ ~ i t i a l o n [ C a 2 + j w as [ C al +] = O .l p ~ .h i s v o l t a g e o fo ~ ~ t l i t io n i ~ i t e r ~ ~ a l- 84 I ~ IV s c l os e to the res t i ng po te ll t i t11 n the n i ode l .R E S U L T SE x p e r i r i i e n t a lI To r he pulse a ~ ~ ~ p l i t u d c o ~ ~ l yA ) s u l l ic i e n tl y l o w , o n e o b t : ~ i ~ i s a s u b t l ~ r e s l ~ o l dr cs l) cm sc t o c i ~ c l ~l i n ~ r~ lu sf h e ~ ~ ~ l s cl ' l ~ i s : 0 r e sp o ll sc c a l l b e s c c ~ ~~:I~II. a t

b e tw e e n s t i t ~ ~ u l i . a t a g i ve n rn y v al ue o fr ,, t he t i t ~ ~ e F o r A s u ll ic ie n tl y l ~ i g l ~ F i g . 1. Allernans in an isolated rab bit ventricular cell . The transrnernbranet l ~ es t i ~ n r ~ l r ~ s a n d s o o ne c a ll o b t ai n a c t i o ~ ~ potent ia l is s l~ ow n s a funct ion of t i rne. (A ) 1: l rhy thm. t ,=700ms. (B ) 2: 2s s u p r i ~ t l ~ r e s l ~ o l t l , p o t e n ti a l s. I nt l ~ :~ t: lse, I 'or r, su l l ic ient ly large, one sees a i n w l i ic l~ ach:I r h y l l ~ ~ n , rhyth m, t,=430 rns. (C) 2: l rhytl lni, t,=350 rns. Pulse amp litud e=l. O nA, pulses t i ~ ~ ~ ~ ~ l r ~ s (1-ig . I A) . ( A n N :A l r l ~ y l l ~ n i durat io n=lOm s. Ternperature=26"C. S uf f ic ient t irne has been a l lowed to le tp ro t l r l c c s a l l : ~c l i on ~ ~ o l e n t i a l is ap c ri o tl ic ~ > i l t t c r ~ ~o ~ ~ s i s t i ~ ~ gf '~.cpe :~t ingN : M cycles e:icl~ OII~:I~II~II~ N s t i n ~ u l i transients pass. Sliniulus artefact retouched.: ~ n d 4 : ~ c t i o ~ ~ o l ' t l ~eo t en l ia l s. I ' h c p c r i o d o f r c p c t i t i o ~ ~ wave lor^^^ is thu s Nr,.' l ' l ~ c t a c t i o n ~ > o t e ~ ~ l i : ~ l s 11:ivc M di l l c rc nt 111ol -pholopies .)i l l ill g c ~ ~ c r a l

Fo r A c hos en c o r rec t ly , :S r, is t lccrc :~sct l . l ~ e :I r l ~ y t l ~ n i s een a t a p :~ r t i c u l :~ r n~ p l i t ud ea t f i x ed r, depends o n whe the r A was increasedw i l l b c convertedd i re ct ly i l lt o a 2:2 r l ~ y t l ~ ~ ~ ~ . o r decreas ed to a r r i v e a t t ha t v a l ue . If a we l l - t i l l l ed ex t ra s t i ~ i i u l u s u ls e i sw i t l ~ o u t n y o t l ~ e r at te r1 1 b e in g se en . F i g u r e I Ds l ~ o w sa n c x ;~ r i~ p le s l ~ o w i ~ ~ g i n t he ~ n o r p l i o lo g y added to the bas i c c u r ren t p u l s e tra i n , i t s pos sib l e l o c onv e r t a 2 : l r hy th r no f a 2:2 r l i y l l ~ ~ ~ ~ , a ll a l t c r ~ ~ a ~ i so f t he act in^^ p o t e n ti a l . W l ~ i l c l l e re i s a l l a l t e r n a t io n ill I n a n y o f t l ie i n t o a 2:2 o r a 1 :1 r l i y t t i ~ i i . n j e c t in g a n e x t ra s t i mu l u s o r d r o p p i n g o n eo f l i e p e r i o d i c d r i v e t r a i n c a n a l so c o n v e r t a I:I o r 2 :2 r h y t l i r n i n t o aI ) :t r a~ l lc t cr s l ~ i ~ t p o t e n ti a l , t h e a l t e r n a t io n i s s t i ~ n r ~ l u sc a l l b e u s c tl t o de s c ri b e a n a c t i o ~ ~p r o b a b l y ~ i i o s t t r i k in g ill ac t i o n po ten t i a l an i p l i t ude and du ra t i on . As r, i s 2 : l rh y t l ~ n i . h i s i ll u s t ra tes tha t b i s tab i l i t y i s p res ent , w i th [ l i e c e ll be i ngd ec rc as cd f u rt li e r, th e a l t c r ~ ~ a ~ ~ s ' c a p ab l e o f s u p p o r t i n g t w o d i f k r e n t p e r i o d i c r h y t h m s a t a f i x e d c o n i b i n a t i o nb e c o n ~ e s n o r e n i ar k e d , w i t h t l ie a ~ n p l i t u d ea n d t l u r a t i o ~ ~ E v e n t u al l y, t l i e 2:2 o f A a nd r,.o f h e s ~ n a l l e r ~ c l i o n o t e n t i a l d e c r e a s i ~ ~ g .r h y t l ~ n ~ i l l t o a 2 :l l on ge ro ~ ~ v e r t s r l ~ y t h n i ,w i t11 ev e ry s ec ond s l i ~ i i u l us IOp r o d u c i n g a n a c t i o n p o t e n t i a l ( F i g . IC). M o d e l l i n gI ' h c r c i s b o t h I ~ y s t e r e s is ~ n d i s t a b i li t y p r e s en t ill t h e b c h a v i o u r s h o w n i nF i g . I. If r, is f i rs t dec rc i ~se d nd t11e11 l lc reased so tha t t l ie bord er between F i g u r e 2 A s h o w s t ha t, f o r r a n d A suf l ic ier i l l y large, 1:1 s y nc i i ron i z a t i ont w o p a t t c r r ~ ss t r :~versed wice, the v :~ lueo f t , a t w h i c h t l i e t r a n s i ti o n o c c u rs s results . A s r, is decreased, t l ie po in t is arr iv ed at w here one begins t o see and i l l e rc ~ ~ t,l c p c ~ ~ d i ~ ~ g o f c l ~ a ~ ~ g e l 'l ii s ' f req ue nc y n l t er n a t io n o f a c t i o ~ i o t e n t i al m o r p h o l o g y . F i g u r e 2 B s h ow s a n e x a mp l e o fp 0 11 t he d i r e c t i o ~ ~ o f r,.hys teres is ' has a counterpart ill 'amplitude hysteresis', in w h i c h t l ie p a t t e r n s u c l ~a 2:2 r l i y t l i ~ n . s i n l i e c as e o f he ex pe r imen ta l wo rk , t he a l t e rna t i on i s


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55 6 M. R. GUEVARA ETAL. ALTERNANS IN ISOLATED VENTRICULAR MYOCYTES 55 7pcrh:tps triost obvious i n ac t ion po tcn t i : t l :~~~~p l i tudend durat iot~,andb e c o ~ ~ ~ e snore n~arkecl s I , is decreased. Eventually, as in the experimentalwork. :I 2: 1 r l ~ y t l ~ ~ ~ ~csi~lts Fig. 2C).- A s s u n ~ i ~ ~ gty pic :~ l ~ pp ar cI tt ~ ~ c ~ n b r a ~ t eurfi~ cc rea of 5200 lt1n2 and anapp:trcnt specilic c:tpacitiince of 1.4 IIF /C II I~or a single rabbit ven tricular cell(Giles and Imaiz u~ni. 988). one can calculate that the cur rent of I nA used inFig. I corresponds to ;I t ruc current dcns ity of :tbout 14 ~ A /C I I I ~ ,rovidedtI1:tt thc true spccilic cap:~citnnce s I ~ r l : / c n ~ ~n tlte absence of i~~lbldingsnthe ccll membrane. Fo r a 20 ms dura tio ~l ulse, an equivalent ell'ect would beproduced at a current density of a bout 7 pA/cm2. wl~ich s quite close to the3 pA/cm2 actually used in the model in Fig. 2. I'he cu rrent density ernployedil l the tiiodel also conlpares well with experi~~ientalesults in isopotentialspontaneou sly beating (Guevar a, 1984) and quiescent (Guevara er a/., 1984)

Fig. 2. Alternarls in the Beeler-Reuter model. The transmembrane potential (V )is shown as a function of time ( r ) . (A ) 1 :l rhythm, tS=375 ms. (B) 2:2 rhythm,t,=355 ms. (C ) 2 :l rhythm. t,=250 rns. Pulse amplitude=3 pA/cm2. pulse dura-tion=20 ms. The first ftimulus pulse is injected at t=0, and 5 s are allowed to lettransients pass.

aggregates of e~nbry oni c hick ventricular cells, where a current density ofabout 6pA /cm2 is needed to produce alternans when current pulses ofduration 20ms are used (aggregate diameter2:100pm, total surfacearea 2: .5 x 10-' cm 2, pulse amplitud e 2.16 nA).

D I S C U S S I O NA l t e r n a n s i n v e n t r i c u l a r t i s s u eThe trace shown in Fig. IB answers more than a century later a questionraised in the work ofGask ell(188 2). it demonstrates t hat it is indeed possibleto obtain intrinsic alternation in a single, quiescent, ventricular myocyte.However, it does not answer the question of what is happening in the leftventricle during electrical alternans. The re are relatively few recordings of thetran sn~e nibr ane otential in that circumstance, either in the intact ventricleor in isolated pieces of ventricular niuscle. In most such experiments, onlyone i~npale~rten ts made, and so on e docs not know if all cells are alternatingin a 2:2 rhythm. One experiment in which four simultaneous intracellulari~npalet t~entsere made during ischaemia indicates that a I:I pattern waspresent in the non-iscl~aernicmyocardium, while 1:0, 2:l and 2:2 patternswere present in the iscliaemic area (Do wnar er al., 1977, figure 10). In thatcase, the pattern of alternation that would be ascribed to the ventricle as awhole is due to the existence of an alternating rhythm in one subpopulationof cells as well as a 2:1 rhythm of block in yet anothe r subpop ulation . Theexistence of either one of these supopulations alone would tlieoretically besulliciertt to generate a n overall alternation in the ventricle. When both 2:2and 2:l rl~y thni s ccur together, it is i~npossib le o decide which ofth ese tworhyth ~ns ould be the more important in generating alternation of the QRS Tcomplex of the surface electrocardiogram or mechanical alterna ns, since thiswould depend both on the relative sizes of the two subpop ulatio ns and alsoon the degree of alternatiori present in the 2:2 response. Since mechanicalalternation is sometimes discordant with electrical alternation, with thelarger action potential producing the weaker contraction (Spear and Moore,1971). one might even imagine a situation in which an overall mechanicalalternans might not be seen, provided that the balance between the 2:2 and2:l sub populations was just right.In !lie case w hen 1: 1, 2:2 and 2: 1 rhythms are seen in di nr en t areas of theventriclc, i t is possible that the 2:l pattern is due to block of propagation ofthe cardiac impulse, with the 2:2 patte rn being seen in the region of block.The 2:2 pattern seen in this circunlstance is due to decremental conductionand can be seen when 2: 1 block occurs in tissue taken frorn virtually all areaso f the heart (see Gue vara, 1984, for references). The possibility then existsthat the 2:2 pattern in the region of block is partly due to electrotonic


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558 M. R. GUEVARA ETAL. ALTERNANS IN ISOLATED VENTRICULAR MYOCYTES 559i t i j e c t i o~ io f cur rent f rom the d is ta l 2 : l region, in a manner recent lydc ~ i i ons t r a ted n ~ r i ode l l i ti g o r k o t i a o ne - t l i t i i e~~s io~ ia lt rand o f Purk i t ijel ibre (Guevara, 1988). I ' l ie quest ion o f wl ie t lie r or n ot i t s poss ible to obta in. a ventricle wi tli a 2:2 p atte rn in a11 cells, or a 2:2 patt ern n one circumscribedarea and a I:I atte rn everywhere else. remains open. 'This que stion can beanswered onl y when techniques that al low rec ording of l ie t ranst iiembranepo t e~ i t i a l r o t n m any s it es s i ~ i i u l t a~ ieous l ye.g. us ing potential-s ensitiveopt ica l dyes) wo uld be applied.

Ionic b a s i s of a l t e r n a n s' I 'l ie ionic basis o f the response sl iown in Fig. I reniains undelerciiined.Howcvcr , t he e lcc trop l iysio log iceI or ig i i i o f he 2:2 rhyt l i t i l is essent ially dueto the fac t that, a t t he s t i ~ i~ u la t io t ia tes e~i i p loye dn F ig . I, he du r a ti on o f a nact ion potcciti : il decreases ~i ~o r i ot or ~i ca l l yi t l i :i ecrease i n recovery t iniesince the imniediately precedit ig act ion poten t ial (k lauswirth e l al., 1972;Boyerr ancl Jewell, 1978: Gueva ra cr rt l . . 1984). Since in niode l l ing wo rk onecan p lo t out r io t on ly t l ie t rans~i ier i ib r t~neotet i t ial , but also t l ie var iouscurrents , ac t iva t ion var iables ar id i~ ~a c t iv a t io ~iariables, it wil l be possible inf r ~ t u r e o r k t o d e t e r ~ i i i ~ i el ie ionic m ecliar i is~i is roduc ing the 2:2 responses hown in Fig. 2.

L o s s of 1:1 s y n c h r o n i z a t i o nWe l iave shown above that the sequence o f l i y t l ims {I:! 2:2+21) can beseer1 as I, s decreased (Fi g. I). owever, t l i is occurs onl y at a sunicient ly l i igl ist ir~ iulu s tnpl i tucle. We have also observed o n several occasions in solatedcel ls a direct t ransi t ion rro tn a I o a 2: 1 r hy t hm . w i t h n o evi dence o f l i e 2:2r l i y t l i ~ i i . l i e ~ l, s cli:uiged in steps o f 10 111s. (I 'erliaps tlie earliest de liniliv eexa~i ip le f a d irec t {I:I 2: I) t rans i tion is to be found in the work o f M ines( 1913). wh o studied the : i t ropinized ventr ic le o f he f rog.) In addi t ion t o th isdirect {I 4 2 : I) sequence. one can also see Wetickebacli-like rhythmsbetween the I: nd 2: 1 r l i y t l i rns a t i,s decreased, provid ed t hat t l ie st imulusat i ipl i tude is just suprat l i resl iold a nd r, is quite large (Gue vara e i al., 1989a).The existence o f these three quali tat ively di ll 'ereti t ways i n which I:!s y t ~c l i r o t ~ i z a t i o t ian be lost has been previously described i n per iodical lystiniul: itetl. spontan eously beating, erilbr yoriic cliick ventricu lar heart-cellaggregates wi th the { l:1+2:2) t ransit ion occurr ing at l i ig l i st iniulus ampli-lade. the {I:I2:1) t rat is i t ion at intermediate ampli tude, an d Wenckebachrhyth ms at low.amp li tude (Guevara, 1984; Guevara et al., 1989b).

R e d u c t i o n t o a o n e - d im e n s i o n a l m a pWe l iave s l iown prev ious ly tl ia t dy r ia in ics s im i lar to that s l iown in F ig . 1 seenin per iodical ly st imulated, quiescent c l i ick ve ntr icular aggregates can bereduced to the study o f he i terat ion o f a one-dirnensiotial f ini te-direre nceequat ion or m ap (Guevara et al. . 1984). This m ap gives the i t h act ionpote l l r ia l durat ion as a func t ion o f the imnied ia te ly preced ing one and isobta ined f ro n i cons ideration o f the curve descr ibing ho w ac t ion potent ia ldura t ion is restored as a funct io n o f he recovery t ime since the immediatelypreceding act ion potent ial. I n l iat case, i t can be shown th at alternans occurswhen t l ie s lope o f the ma p at t l ie f ixed po int becomes more negat ive than - ,leading to a per iod-d oubling bifurca t ion (Guevara er a/.. 1984, f igure 4). Th esequelice o f r l iy t l im s predicted f rotn t l ie ma p is then {I:! 2:2+2:1), whichagrees wit h what is seen experinlental ly. A simila r analysis s l io uld be possiblein the case o f F igs I and 2. T he f irst analysis o f electr ical alternanslbrmu lated using an i lerat ive lecl inique seems to have been that o f Nolascoarid Da hlet i (1968). wl i i le mechanical alternans has been similar ly studied(Mnhler and Rogel, 1970).W hen t he r ec ove r y o f ac t i on po t en t ia l du r a t i on i s s u l l i c i e t i t l ~apid, al lpo in ts on the map have a s lope more pos i t ive than - I,and so a per iod-doubl in g b i furcat ion cannot occur . I n l ia t c i rcumstance, one obta ins a d i rec t{1:1+2: 1) t ransit ion. No te t l iat in l i is instance, t l ie t ransit ion is no t due to aperiod -doub ling bifurca t ion (cr. Chial vo and Jal ife. 1987). The 2: l rhy thmcor responds to a per iod - I orb i t o n the map, whi le the 2 :2 rhy thm seen i f hema p is suf l ic iently steep corresponds to a per iod-doubled pe r iod-2 orb it(Guevara el al., 1984, f igure 4). I 'he tran sition {2:2+2:1) seen i n a simplemode l o f a l imit -cycle osci l lator (Guevara and Glass, 1982) an d in experi-mental w ork on spontaneously beat ing heart -cel l aggregates (Guevara, 1984,l igure 5-16) is due to a change i n ro ta t io n number .

As n ient ioned prev ious ly , when s t imula t ing w i th la rge i, - 1 s) an d smallA, Wenckebach rhy t l i~ ns an be observed at values of t , intermediate to thosea t wh i c h I:I nd 2:1 r l i y t l ims occur . A t suc l i low s t i inu la t ion f requenc ies int l ie rabbit ventr ic le, the act ion potent ial durat ion shortens with increasingrecovery t ime (Saxon and Safronova, 1982; Giles and I~n aizu mi, 1988) - tdoes n o t p r o l ong as at the higher st imula t ion f requencies employed in Fig. 1.One thus obta ins a d i k r en t class o f maps which have two branches, each ofwhich has posit ive slope everywhere (Guevara et al., 1989a). Thus, a perio d-t loubl i ng bifu rcat ion can not occur. These maps predict the existence o fWenckebach rhythms, an d are quali tat ively s i milar t o a class o f maps der ivedin recent work o n Wenckebac l i rhy thms in he human at r iovent r icu lar node(Shrier er al., 1987). Wen ckebac li rhy thm s can also be seen in periodical lyst imulated, spontaneously beat ing, e mbryo nic chick ventr icular aggregates. (Guevara er al., 1988); in t l iat case, one c an somet imes reduce the dynamics


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562 M. R. GUEVARA ET AL . ALTERNANS IN ISOLATED VENTRICULAR MYOCYTES 563Gucvar:~. M. It. (1984) Chaotic cardiac dynalnics. P hD tticsis. Mc Gi l l University,Montrc i l l .Gucvara. M. R. (1988) Sp atiotc~ npora l atterns or block i n an ionic moclel or cardiacI'i l rkinjc l ibre. I n Ro rri Clirriricirl ro Biolo,qicnl Orgtn~iza liortcds M. Markus, S. C.Miil lcr and G . Nicolis), pp. 273-81. Springer. I3crl in.Gucv:lra, M. R. :uld Glnss, L. ( 1982) I'lrasc locking , pcriod dou bling bif itrca tions andc ll :tos in a n~at l icnrnt ica l no dc l Ta periodically drivcn oscil lator: a theory for thec ~ ~ t r a i t ~ n l c n tr biological oscil lators and the gcncration o r cardiac dysrhythmias.

J. Mnrlr. Biol. 14, 1-23.Gucvara, M. R. and Sl~r ic r . . (1987) Phase resetting in a model o f cardiac Purkinjel iber. Bioplrp. J. 52, 165-75.Gucvara, M. R., Glass, L. and Sliricr. A. (1981) Pllasc locking, period-doublingbirnrcations. a nd irregular dynaniics in periodically sti lnulated cardiac cells.Sci ~irc c 14, 1350-3.Gucvara. M. R., Ward. G., Shricr. A. and Glass. L. (1984) Elcctrical alternans andpcr io t l -doubl i t~gbir~trcations. n Coritprtrers ill c f ? r d i o / ~ ~ j ~ ,p. 167-70. IE E EConl putcr Socicly, Silvcr Spring, MD.Guevnra, M . R.. Shricr, A. and Glass. L. (1988) Phase-locked rllytlltns i n periodica llystimulated heart cell aggregates. Arrr. J. Pltjsiol. 254, H1-11 10.Gucvtlra. M. R.. Jcandupcux. D..Alonso. F.and Morissctte. N. l989a) Wenckehachr l ry t l l t~rs n iso la tcd vc t l t r ic r~ l :~ricart cells. I n Sir~ptlcrrBclmc.ioctr arrd No rrlir~ea rDj.rriirriics (cds St. Pn evlnatikos, T. Bountis and Sp. Pnevmatikos). W orl d Scien-tilic. Singapore (in press).Gncv:rr :~. M . 11. Sl~r ic r .A. and Gl:~ss. 1 (1989h) Cllaotic and coniplcx cardiac . .r l ly t l i~ l ls . n Corcl inc b l rc rro~ ~r l i !~s i~r Io~j :iorr t Ce ll o Beclsidc (eds D.P. Zipes andJ. J:~lir'c). S:~undcrs, I'lril:~dclpllia (i n press).H:luswirtl~, 0.. Noble. D. and Tsien. R. W. (1972) Tile dependence o r plateaucurrents in cardiac Purkinjc l ibres on the interval, between action potentials.J. I'/I,I-.rial.. Lorrd. 222, 27-49.I lc l lc rs tc i r~,II. . and Licbow, I.M . (1950) Elcctrical alternation in expcrinlcntalcoron:lry artery occlusion. Arrr. J. PItj.sio1. 160, 366-74.t - l o g : ~ ~ ~ c : i ~ r ~ p ..E. . Knrdcsch, M.. Danlorth. W. 1-1. nnd Ding, R. J. (1959) Trans-n ~ c n ~ b r a n clcctrical potentials in ventricular tachycardia and l ibri l lation. Anr.I l r t r r r J. 57, 214-22.Mall lcr. Y. and Rogcl. S. (1970) Irrtcrrclatio n bctwccn rcsl i tutio n time-constant andi ~ l t c r ~ ~ o t i r l g~ryocn rtIi:ll on tract ilily in dogs. Cliri. Sci. 39, 625-39.Mints.G. R. (1913) 01 1dyna n~ic qu i l ibr iun l in the heart. J.Pl~ysiol.,Lorltl. 46, 349-83.Muskcns. L. J. J. (1907-08) Gcncsis or the alternating pulse. J. Pliysiol., Lorrd. 36,104-12.Nolasco, J. 19. nllt l Dall lcn. R. B. (1968) A ~ r a p l l i c ic thod for thc s tudy ora l lernat ionin c :~rtl i :~c ction potcrlt ials. J. Al~p l . ' l i ~ r i o l .25, 191-6.Ri tzc t~hcrg. . I, . , A(I;II~I. D. 11. and Collcn, R. J. (1984) I 'criod multip lying-e vidc~~ ccf o r ~ l o n l i ~ i c i t rt l l : lv iour of t l ~ c anitic Ilcnrt. Norrtre 307. 159-61.S;ltlo, T.. Tsucl~illaslli.Il.r t~t l h in la~noto.T. (1958) Ventricular l ibri l lat ion studieshy tile tiiicroclcctrodc nicthod. Circitlorbrr Res. 6. 4 1-6.Saxon. M . E. and S arronova, V. G. (1982) The rest-dcpcntlcnt depression or ac t ionpo tc n l i i ~ l n r :~ t i on n rabbit tnyocard iutn nnd t l ~ c oss ible ro lc or t l rc transientoutward cnrrcnt : A pll:lr~nac ological analysis. J. Plrj~sic)l.,Pnr i - t78, 461-6.Sllricr. A,. 1)ubnrsky. M.. Roscng:lrten. M., Ciucvara. M. R., Nattcl, S. and Glass. I..(1987 ) I ' red ic t ion o~ c o ~r ~ p l c xtr iovet~tr icu lar ondl lc t ion rhy t l~rn sn hutnans w i thusc o f t l lc atriovcntrikular nodal recovery curve. Circrtlurior~ 6, 1196-205.

Spear. J. F. arid Moore. E. N. (1971) A con iparison o f alternation in myocardialaction polcnlials and co ntracti l i ty. Arrr. J. Pliy siol . 220, 1708-16.Tresser. C.. Coullet. P.and Arneodo. A. (1980) O n the existence o f hysteresis in atransition t o chaos after a single birurcation. J. PIi!.s. Ler r. (Pa ris) 41, L243 -L24 6.Victorri , D., Vinct. A., Roberge. F.A. and Drouhard, J. -P. (1985) Numericalintegration in the reconstruction of cardiac action potentials using Hodgkin-Huxle y-type m odels. Conrp. Biorired. Res. 18, 10-23.


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