Michael R. Guevara and Habo J. Jongsma- Three ways of abolishing automaticity in sinoatrial node: ionic modeling and nonlinear dynamics

8/3/2019 Michael R. Guevara and Habo J. Jongsma- Three ways of abolishing automaticity in sinoatrial node: ionic modeling

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Three ways of abolishing automaticity in sinoatrial node:ionic modeling and nonlinear dynamicsMICHAEL R. GUEVARA AND HABO J. JONGSMADepartment of Physiology, McG ill University, Montreal, Quebec H3G 1 Y6, Canada;and Physiolog ical Laboratory, University of Amsterdam, 1105 AZ Amsterdam, The Netherlands

Guevara, Michael R., and Habo J. Jongsm a. Three way sof abolishing automatic ity in sinoatrial node: ionic modelingand nonlinear dynam ics. Am. J. Physiol. 262 (Heart Circ.Physiol. 31): H1268-H1286,1992.-A review of the experimen-tal literature reveals that there are esse ntially three qualita-t ively different way s in which spontaneous activ ity in thesinoatr ial node can be abolished. We show that these threeway s also occur in an ionic model of space-clamped nodalmembrane. In one of these three way s, injection of a currentpulse abolishes (annihilates) spontaneous action potentialgeneration. In the other two way s, as some parameter ischanged, one sees a sequence of qualitative changes in thebehavior of the membrane as it is brought to quiescence. In oneof these two way s there are incrementing prepotentials inter-mixed with action potentials, with a maintained small-ampli-tude subthreshold oscillation being the limit ing case of suchbehavior. Thus both experimental and modeling work indicatethat the number of way s in which spontaneous activ ity can beabolished , or initiated, in the sinoatrial node is limited. Theclassif ication into three ways is based on ideas drawn from thequalitative theory of differential equatio ns, which are intro-duced. The classif ication schem e can be extended to encomp assbehaviors seen in other cardiac oscillators.annihilation; single-pulse triggering; afterpo tentials; delayedafterdepolarizations; subthreshold oscillations; oscillatory pre-potentials; bifurcations; chaosONE CAUSE of the potentia lly life-threatening cardiacarrhythmia called sinoatrial arrest is the cessation ofspontaneous action potential generation in the sinoatrialnode (SAN), the princ ipal natural pacemaker of themammalian heart. In a search of the literature we haveturned up many traces of the transmembrane potentialshowing cessation of spontaneous activity in the SAN asthe result of some experimental intervention. Perusal ofthese recordings has led us to class ify these tracings intothree groups, showing that there are essentially threequalitatively different ways in which the normal spon-taneous act ivity of the node can be abolished. In the firstway, there is a gradual progressive decline in the ampli-tude of the action potential until quiescence occurs . Inthe second way, injection of a brief stimulus pulse anni-hilates spontaneous act ivity , which can then be restartedor triggered by injecting a second stimulus pulse. In thethird way, before the membrane becomes quiescent asthe result of some intervention that gradually takes hold,one sees skipped-beat runs in which spontaneously oc-curring action potentials are preceded by one or moresmall-amplitude subthreshold oscillatory prepotentials.Because all the traces mentioned come from experi-

ments carried out on isolated right at ria1 preparations oron small pieces of tissue isolated from the SAN, theextent to which the behaviors seen might be accountedfor solely by membrane properties of SAN cells is uncer-tain. For example, subthreshold deflections resemblingpre- or afterpotentials recorded in one ce ll might actua llybe electrotonic potentials reflecting occurrence of blockof propagation into that area of the SAN (10, 44). Toavoid this complication in the interpretation of the re-sults, we carried out simulations using an ionic model ofan isopotential patch of membrane, where spatial factorsof this sort cannot occur. In addition, numerical inves-tigation of an ionic model allows one to probe the ionicbasis underlying the particular behavior observed.The model of isopotentia l SAN membrane studied hereis that of Irisawa and Noma (36). Because it is a Hodgkin-Huxley-type model, it is formulated as a system of non-linear ordinary differential equations. A major point ofthis paper is that a branch of nonlinear mathematics,bifurcation theory, can be used to obtain significantinsights into qualitative aspects of the various behaviorsdisplayed by this class of model. Indeed, it is our claimthat one cannot fully appreciate the results of the nu-merical simulations presented below and the correspond-ing experimental findings without also at least a passingacquaintance with concepts stemming from the qualita-tive theory of differential equations. We therefore inter-weave presentation of the resu lts of numerical simulat ionof the model with interpreta tions of those simula tions interms of the qualitative dynamics of the system. Forreaders wishing to obtain further details about bifurca-tion theory, the qualitative theory of differential equa-tions, and related aspects of nonlinear mathematics,several textbooks that are readable by someone with abiological sciences background are now available (1, 25,71, 74, 84).METHODS

We investigated the effect of changing, one at a t ime, manydifferent parameters in the Ir isawa-Noma (36) ionic model ofisopotential SAN membrane. In several instances, when aparameter was altered suff ic iently from its normal value, ter-mination of spontaneous activ ity resulted. The Ir isawa-Nomamodel is based on voltage-clamp data recorded from smallpieces of t issue taken from the rabbit SAN and provides aquantitative description of f ive currents: the fast inward sodiumcurrent (&), the slow inward calcium current ( Is), the delayedrectif ier potassium current ( IK), the hyperpolar ization-acti-

HE68 0363-6135/92 $2.00 Copyright 0 1992 the American Physiological Society


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BIFURCATIONS AND SINUS NODE AUTOMATICITY H1269vated pacemaker current (Ih; commonly termed If), and a time-independent leakage current (11).We used two different meth-ods of investigation, direct numerical integration of the equa-tions and bifurcation analysis.Numerical integration. We numerically integrated the Iri-sawa-Noma equations in a manner identical to that employedin a recent study that showed that this model accounts verywell for experimentally observed phase-resetting phenomenol-ogy (30). We use a variable time-step algorithm that is muchmore efficient than fixed time-step algorithms, yielding equiv-alent accuracy with much less computation (79). In addition,the convergence of the algorithm for equations of the Hodgkin-Huxley type used in this model can be mathematically proven(79). By adjusting the integration time step At at any time t tobe 1 of the 10 values 2N(0.016) ms with 0 5 N 5 9, the changein the transmembrane potential AV in iterating from time t totime t + At can be kept ~0.4 mV in the simulations shownbelow (with one exception, see Fig. 15). When AV is >0.4 mV,the time step is successive ly halved and the calculation redoneuntil a value of A V of


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H1270 BIFURCATIONS AND SINUS NODE AUTOMATICITYsite ly directed. The changes in 1k, 1Na, and 1h are muchsmaller, with 1k changing considerably more than eitherlNa or 1h. In fact, lNa and 1h contribute little to the overallactivity: removing them both from the model causessmall changes in the waveform of the action potentialand increases the spontaneous interbeat interval byN20%.In the Irisawa-Noma model, the state of the membraneat any given point in time is specified completely by thevalues of V, the activation variables m/, d, p, and Q (of1Na, &, 1k, and 1h, respectively), and the inactivationvariables h and f (of 1Na and I,, respectively) at that pointin time. These seven variables define a seven-dimen-sional state point (V, m, h, d, f, p, q) in the seven-dimensional state space of the system. Activity thencorresponds to the movement of this state point in thestate space, generating a curve called a trajectory. Start-ing out at time t = 0 from almost any initial condition(i.e., particular combination of V, m, h, d, f, p, q), thetrajectory asymptotically (i.e., as t --) 00) approaches aclosed curve in the seven-dimensional state space. Thisclosed curve is called a limit cycle or periodic orbit; thespontaneous periodic generation of action potentialsshown in Fig. lA (top trace) corresponds to the projectiononto the V-axis of the periodic movement of the statepoint along this limit cycle. The l imit cycle is said to bestable, since any trajectory starting out from an initialcondition sufficiently close to the limit cycle is asymp-totically attracted onto it. For example, injection of acurrent pulse produces a phase-resetting response inwhich recovery from the perturbation occurs as the tra-jectory .asymptotically regains the limit cycle (30). Incontrast, trajectories originating from initial conditionsin the neighborhood of an unstable limi t cycle would berepelled from the limit cycle.The steady-state current-voltage (I-V) curve of theIrisawa-Noma model crosses the current axis at only onepoint, at V = -34.1266 mV. Clamping the membrane tothis potential therefore asymptotically results in zerocurrent f low through the membrane. The other variables(i.e., m, h, d, f, p, Q) approach the asymptotic valuesappropriate to that potential [e.g., p + pm = ol&, +P) -l, where aP and & are the rate constants for p, thea&ivation variable for 1k, evaluated at V = -34.1266mV]. The set of initial conditions ( V, moo, hco, d,, fW, pm,qm) for V = -34.1266 mV corresponds to an equilibriumpoint or steady state in the phase space of the system,since releasing the clamp should theoretically result inzero current flow and so quiescence. The fact that com-putationally the membrane does not rest at this equilib-rium point when computations are started from init ialconditions very close to it (see below) shows that thisequilibrium point is unstable. Thus only if one were tostart off at the exact set of initial conditions (specifiedto an infinite number of decimal places) and use infi -nitely precise computation would the membrane poten-tial remain at V = -34.1266 mV. There is also a set ofinitial conditions, apart from the equilibrium point,starting from any member of which the trajectory wil lasymptotically approach the equilibrium point. This setof points forms the stable manifold of the equilibriumpoint. When the equilibrium point is stable, the dimen-sionality of this set will be equal to the dimensionality

of the system (7 in the case of the Irisawa-Noma model)so that starting out from any init ial condition in a(perhaps relatively small) seven-dimensional neighbor-hood of the equilibrium point will result in a trajectorythat is asymptotically attracted onto the equil ibriumpoint. When the equilibrium point is unstable, the di-mensionality of the stable manifold will be less thanseven.Because it is difficult to visualize dynamics occurringin a seven-dimensional state space, let us consider thesimple schematic two-variable or two-dimensional limit-cycle oscillator shown in Fig. 1B to illustrate the aboveconcepts. The closed curve is the limit cycle, with thearrow indicating the direction in which the state pointmoves along the cycle as time progresses. The limit cycleis stable, since starting from initial conditions, such asthe points labeled a, b, or c, results in an asymptoticapproach of the corresponding trajectory to the limi tcycle. The equilibrium point is indicated by the symbolx and is unstable, since starting from an initial conditionvery close to the equilibrium point (e.g., point c) results

in a repulsion of the trajectory away from the equilibriumpoint. Thus there are two structures of interest in thephase space of the two-variable system shown in Fig. 1B(and in the 7-dimensional phase space of the Irisawa-Noma model), an unstable equilibrium point and a stablelimit cycle. Figure 1B (and other 2-dimensional sketchesin Figs. 3, 4, 7, 8, 10, 12, and 15) is not to be regarded asa formal reduction of the full seven-dimensional statespace of the Irisawa-Noma equations to a two-dimen-sional state space; instead it represents an intrins icallytwo-dimensional system used to illustrate concepts weintroduce from nonlinear dynamics.Firs t way: gradual decline in action potential amplitude.

In the first way of abolishing activity as some interven-tion takes hold, the action potential gradually and con-tinuously decreases in amplitude until quiescence occurs.Figure 2 shows an example where the maximal conduct-ance of I, is decreased gradually in the model. Activi tyvery similar in appearance can be seen in experimentsin which 1s is progressively diminished using one of avariety of calcium-channel blocking agents (10, 40). InFig. 2, the frequency of beating slowly decreases to aboutone-half of the initial frequency, and the amplitude ofthe action potential fal ls in a gradual smooth manneruntil spontaneous activity is extinguished, which agreeswith the experimental result (e.g., see Refs. 10, 40). Thefirst way is also seen experimentally in the SAN whenmany interventions other than blockade of IS are carriedout, e.g., application of Ba2+ (63) or injection of a con-stant depolarizing bias current (53, 55). In these twocases, analogous tracings can also be seen in the Irisawa-

-100 0 t (set) 40Fig. 2. V as a function of t as maximal conductance of I, is graduallyreduced to 0 in a linear fashion over 40 s.


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BIFURCATIONS AND SINUS NODE AUTOMATICITY H1271Noma model when parameters appropriate to the partic-ular experimental intervention are changed.Figure 3 shows the behavior when the maximal con-ductance of & is reduced to increasingly smaller fixedlevels and allowed to stay at each of those levels. At anygiven level of block in Fig. 3, B-E, spontaneous activityis seen, with the amplitude of the action potential as wellas the frequency of beating declining as 1s s increasinglyblocked. Finally, for the maximal conductance set to one-fifth of its normal value, spontaneous activity ceases(Fig. 3F).During the normal spontaneous activity shown in Fig.3A, an unstable equilibrium point and a stable limit cycleare present. As Is is increasingly blocked (Fig. 3, A-F),the limit cycle contracts in size until it collapses ontothe equilibrium point and disappears. Again, because itis difficult to visualize this in a seven-dimensional statespace, Fig. 3, G-I, shows the analogous case in our simpletwo-dimensional limit-cycle oscillator. As a parameter ischanged, the original limit cycle (Fig. 3G) shrinks in size(Fig. 3H), maintaining the same topology, i.e., a stablelimit cycle and an unstable equilibrium point. Eventu-ally, the limit cycle contracts down onto the equilibrium

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Fig, 3, A-F: Hopf bifurcation in ionic m odel. V as a function of t.Maxima l conductance of I, is reduced from its normal value (A) to 0.8(B), 0.6 (C), 0.4 (D), 0.3 (E), and 0.2 (F) of that v alue. G-I: H opfbifurcation in simple 2-dimensional sys tem . As some parameter ischanged, l imit cycle (G) shrinks in size (H), e ventually disappearingand reversing stability of equilibrium point (I). See tex t for furtherdescription.

point and disappears (Fig. 31)) simultaneously convertingthe unstable equilibrium point in Fig. 3H into a stableequil ibrium point in Fig. 31. Thus at a value of theparameter somewhere between those used in Fig. 3, Hand I, a bifurcation, a qualitative change in the dynamics,has occurred. The exact value of the bifurcation param-eter (maximal conductance of 1s in the case of Fig. 3, A-F) at which the bifurcation occurs is termed the bifur-cation value. The particular type of bifurcation occurringhere in which the stability of an equilibrium point isreversed with the concurrent appearance or disappear-ance of a limit cycle is a Hopf bifurcation (1, 25, 71, 74).Second way: annihilation. To the best of our knowl-edge, the second way of abolishing spontaneous activityin the SAN has been described only once, in experimentscarried out on strips of kitten atrium containing nodaltissue placed in a sucrose gap apparatus (37). Injectionof a subthreshold current pulse of the correct duration,amplitude, and timing could then abolish (annihilate)spontaneous activity. Once activity was so annihilated,it could be restarted by injecting a second, suprathresholdcurrent pulse. Although we know of only one report ofannihilation in the SAN, the phenomenon has beendescribed in several other cardiac preparations (20, 24,38, 68, 72) and in other biological oscillators (35; seeother refs. in Ref. 84).These findings indicate the coexistence of two stablestructures in the phase space of the system, a stable l imitcycle and a stable equilibrium point, with the formercorresponding to spontaneous activity and the latter toquiescence. In Hodgkin-Huxley-like systems, such asthat studied here, it is relatively easy to locate equilib-rium points. One simply plots the steady-state I-V curveand looks for zero-current crossings as described above.The voltage at which such a crossing occurs gives the V-coordinate of the equilibrium point, and the other coor-dinates (m, h, etc.) are obtained from the asymptoticvalues of those variables (i.e., mco,h,, etc.) appropriateto that potential. This procedure yields all equilibriumpoints present in the system, since dV/dt = 0 when thetotal current is zero for a space-clamped system, and therates of change of all activation and inactivation vari-ables are zero, since they are set to their asymptotic orsteady-state values. The Irisawa-Noma model admitsonly one equilibrium point, since, as previously men-tioned, there is only one zero-current crossing of itssteady-state I-V curve (see Fig. 6 of Ref. 36). Thisequilibrium point is unstable, since starting computationfrom initial conditions very near to it results in a re-sumption of spontaneous activity (Fig. 4A). Thus anni-hilation cannot occur in the standard Irisawa-Nomamodel, since the only equilibrium point present is unsta-ble. This prediction of the model agrees with the corre-sponding experimental finding in the rabbit SAN whereclamping the membrane potential of a small, effectivelyisopotential piece of nodal tissue to the voltage corre-sponding to the zero-current point for some time andthen releasing the clamp results not in quiescence but inthe resumption of spontaneous beating (56).However, the standard Irisawa-Noma model can bemodified to allow annihilation. The first step in thisprocess is to remove 1h. Removing 1h has only a smalleffect on the spontaneous frequency, as might be ex-


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H1272 BIFURCATIONS AND SINUS NODE AUTOMATICITY

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Fig. 4. A-C: saddle-node bifurc ation in ionic mod el. A: integration isstarted from initial conditions appropriate to equilibrium point at V =-34.1266 mV in unmodified ( i.e., Ih # 0) model. T his simulationcorresponds to clamping membrane potential for infinitely long timeand then releasing clamp at t = 0. B: steady-state current-voltage (I-V) characteristic curve of model with Ih removed and bias current ( Ibias)= 0.0 (curve a) or 0.392 PA/cm2 (curve b). C: expanded view of curve bin B near equilibrium point at VI = -57.85 mV with Ibias = 0.392 PA /cm2 . D-F: saddle-node bifurcation in 2-dimensional sys tem . As someparam eter is change d, a saddle-node equilibrium point (indicated by x)suddenly appears (E) in a region of phase space previously containingno equilibr ium points (D). As parameter is changed further, saddle-node breaks up into 2 equilibrium points (F): an unstable node (left)and a saddle point (right).petted from its smal l contribution to the total cu .rrentduring the phase of diastolic depolarization (Fig. 14 .This small contribution of Ih to generation of the pace-maker potential is seen in certa .in regions of the SAN,since blocking &, pharmacologically sometimes producesonly slight changes in the beat rate (12, 41, 57) andperforming voltage-clamp experiments reveals that 1h isoften present only in relatively small amounts (11, 12,59). The I-V curve with 1h thus removed is shown incurue a of Fig. 4B. Note that because total current (I) isnot a variable of the system, the I-V plane of Fig. 4B isnot the state space, nor is it some projection of thatspace onto two dimensions. As in the unmodified model,there is only one zero-current crossing in curue a of Fig.4B, showing that there is stil l only one equilibrium pointin the model. Numerical simulation reveals that thisequilibrium point is unstable, with a trace similar to thatshown in Fig. 4A, which is from the unmodified model,being produced i f one starts computation with initia lconditions close to the equilibrium point. Injection of aconstant hyperpolarizing bias current (Ibias) to slow fur-ther the spontaneous rate causesof Fig. 4B to move upward. Eve the I-Vntually, curve (curue a)because of theN-shape of the I- V curve, two new zero-current crossingsof the I-V curve are created as the peak of the I- V curvemoves through the voltage axis (curve b of Fig. 4B;expanded view in Fig. 4C), producing two new equil ib-rium points in the phase space of the system. There isthus, once again, a qualitative change in the dynamics,since two new equilibrium points (at potentials Vl and

Vz in Fig. 4C) are created. This de novo creation of twonew equilibrium points is called a saddle-node bifurca-tion of equilibrium points, with one of the points beinga node and the other a saddle (1, 25, 74).Figure 4, D-F, illustrates a saddle-node bifurcation ina two-dimensional system. Before the bifurcation takesplace, there are no equilibrium points present in the partof the phase space shown in Fig. 40, which also showsfive representative trajectories. At the bifurcation point(Fig. 4E), there is the creation of a saddle-node, a specialkind of equilibrium point. Note that this point exists atone and only one value of the bifurcation parameter,which is again called the bifurcation value. In Fig. 4B,the bifurcation parameter is Zbias, and the bifurcationvalue is the value of 1biasat which the peak of the N-shaped I- V curve just touches the V-axis at one point as1bias is increased. At that exact value, a saddle-nodeequilibrium point is born via a saddle-node bifurcation.As the bifurcation parameter is pushed just beyond thebifurcation value (Fig. 4F), the saddle-node breaks upinto two equilibrium points, one a saddle point (equilib-rium point to the right in Fig. 4F) and the other a node(the equilibrium point to the left in Fig. 48). The nodeis unstable in this case, since trajectories starting fromanywhere very close to it leave its immediate vicinity.The saddle point is also unstable, since trajectories start-ing out from initial conditions almost anywhere in asmall neighborhood around it leave its vicinity . However,there exist two sets of initial conditions that are attractedto the saddle asymptotically: starting off with an initialcondition anywhere on the trajectories labeled a and bresults in an asymptotic (i.e., as t + 00) approach to thesaddle point. These two trajectories thus form the stablemanifold of the equilibrium point, whereas the pair la-beled c and d form the unstable manifold of the equil ib-rium point, since they asymptotically approach the equi-librium point should the direction of time be reversed.Each of these four trajectories (a-d) is termed a separa-trix, since they separate the phase space in the neigh-borhood of the saddle point into four distinct regions. Atrajectory cannot cross over from any one of these fourregions into another. Thus a saddle point can confer onthe membrane true all-or-none threshold behavior (17).The equilibrium point created in Fig. 4B for the I-Vcurve labeled b and associated with the more depolarizedpotential, labeled V2 in the exploded view of Fig. 4C, is asaddle; the other newly created equilibrium point, a node,is associated with the more hyperpolarized potential,labeled Vl in Fig. 4C. A saddle point, as mentioned above,is always unstable. However, a node can be stable orunstable. Indeed, on being born at the bifurcation point,the equilibrium point associated with the potential Vl isunstable. Thus annihilation is once again impossible.However, as I bias s increased, there CO m eS a point (at Ibias= 0.391 pA/cm2) where this equilibrium point reversesits stability: for example, by the point where the highervalue of Ibias employed in Fig. 4c is reached, this equilib-rium point has indeed become stable. Thus startingintegration from an initial condition appropriate to thatpoint results in quiescence (Fig. 5A, trace a); in contrast,if one starts from, for example, our standard init ialcondition (see METHODS) , one obtains spontaneous ac-tivity (Fig. 5A, trace b). In this situation, activity can


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BIFURCATIONS AND SINUS NODE AUTOMATICITY H1273therefore be triggered (Fig. 5B) or annihilated (Fig. 5C)by injecting a single current pulse. To trigger activity,the pulse must be large enough in amplitude; otherwise,only a damped subthreshold response, similar to thatseen following annihilation in Fig. 5C, occurs. To anni-hilate activity, the current pulse amplitude, duration,and timing must all lie within certain critical ranges fora given polarity of the pulse. For the pulse polarity,amplitude, and duration used in Fig. 5C, annihilation isseen over a range of coupling intervals that is -60 mswide.We have not been able to find annihilation in thestandard unmodified (i.e., 1h not set to zero) model when1biae s applied and changed in increments of 0.001 PA/cm2, despite the fact that there is also in that case arange Of Ibias over which three equilibrium points exist.The equilibrium point with V = Vl in Fig. 4C is stable,since starting out with an initial condition appropriateto that point results in quiescence (Fig. 5A, trace a). Inthis case, the system is six-dimensional, since 1h and itsassociated activation variable 4 have been removed.There is thus a full six-dimensional null space (6) sur-rounding the equilibrium point such that trajectoriesstarting at an initial condition anywhere within this six-dimensional volume, which forms part of the stable man-ifold of the equilibrium point (the set of all initial con-ditions from which trajectories asymptotically approachthe equilibrium point), are asymptotically attracted tothe stable equilibrium point. Because periodic actionpotential generation can also be seen (Fig. 5A, trace b),a stable limit cycle producing t hat activity must also bepresent in the phase space of the system. Thus in thephase space of the system of Fig. 5A there are present atleast four 0 bjects of topological interest: three equili .b-rium points, only one of which is stable, and one stablelimit cycle. Because there is a full six-dimensional nullspace, there must be a five-dimensional surface (theseparatrix surface) that divides the phase space into tworegions. The presence of such a repelling hypersurface isnecessary to produce a separation of trajectories, withtrajectories starting from the set of initial conditions tothe inside of the separatrix hypersurface, within thenull space (which forms part of the basin of attrac tion

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-202>-80Fig. 5 . Annihilation and single-pulse triggering in ionic model. A: Ih =0 and Ibias= 0.392 pA/cm 2: integration is started from initial condit ionsappropriate to equilibrium point asso ciated with potential Vl in Fig.4C in truce a and from our standard init ial condit ions (see METHODS)in trace b. B: injection of depolarizing current pulse (duration = 20 ms ,amplitude = -2.0 pA/cm2) at t = 5.85 s tr iggers activ ity ( init ialcondit ions corresponding to point VI). C: injection of depolariz ingcurrent pulse (duration = 20 ms , amplitude = -0.2 pA/cm 2) at t = 5.85s annihilates spontaneous activ ity (standard init ial condit ions).

of the equilibrium point) ending up at the equilibriumpoint, whereas those originating from most initial con-ditions outside the separatrix surface enclosing the nullspace (the basin of attraction of the limit cycle) asymp-totically approach the stable limit cycle. Lying withinthis surface is an unstable limit cycle. How did thisunstable limit cycle originate? We mentioned that thenode created at the saddle-node bifurcation was unstablewhen born (at 1ias H 0.390 pA/cm2), but yet this equilib-rium point was stable at the slightly higher value of 1bias(0.392 pA/cm2) used in Fig. 5. Thus somewhere betweenI bias = 0.390 and 0.392 pA/cm2, there is a qualitativechange in the dynamics that came about via a bifurca-tion, with the unstable equilibrium point reversing itsstability, thereby becoming stable, and simultaneouslyspawning an unstable limit cycle in its immediate vicin-ity. The bifurcation is thus again a Hopf bifurcation.However, unlike the Hopf bifurcation shown earlier inFig. 3, G-I, which is termed supercritical, since stableobjects exist on both sides of the bifurcation, an unstablelimit cycle is born rather than a stable limit cycle dying,which is termed a subcritical Hopf bifurcation, sinceunstable objects exist on both sides of the bifurcation.However, in both cases, the unstable equil ibrium pointbecomes stable as the bifurcation parameter is changed.The saddle-node and Hopf bifurcations describedabove are summarized nicely in a bifurcation diagram inwhich the value of one of the variables in a system isplotted as a function of the bifurcation parameter. Inthis case (Fig. 6A) we plot transmembrane potential vs.I bias.The branch abcde of the bifurcation diagram of Fig.6A is a plot of the voltage of the equilibrium point(s) asIbias s changed. A solid curve indicates that the equilib-rium point is stable, whereas a dashed curve indicatesthat it is unstable. As the hyperpolarizing bias current isincreased, at point d in Fig. 6A, one has the saddle-nodebifurcation depicted in Fig. 4B (curue b), when the peakof the N-shaped I- V curve intersects the voltage axis. Atpoint c in Fig. 6A one has a reverse saddle-node bifurca-tion that results when the valley in the I-V curve of Fig.4B is pushed up with further increase in Ibias, eventuallyintersecting the voltage axis and resulting in the coales-cence and disappearance of two equilibrium points. Thusbetween the values of I biascorresponding to the points dand c in Fig. 6A, the system has three equilibrium points,with the point lying along the branch cd being the saddlepoint with transmembrane potential V2 in Fig. 4C.Also shown on Fig. 6A are two curves bf and bg, whichform the branch of the bifurcation diagram representingthe stable periodic orbit responsible for the usual spon-taneous generation of action potentials shown in Fig. 5,with curue bf giving the overshoot potential and curue bggiving the maximum diastolic potential at a particularvalue Of Ibias. As with the gradual decrease of the con-ductance of Is shown in Figs. 2 and 3, application of anincreasingly large depolarizing bias current (i.e., movingfrom right to left in Fig. 6A from Ibias = 0) results in agradual fall in the amplitude of this orbit, with bothovershoot potential and maximum diastolic potentialdeclining. Although we know of no corresponding sys-tematic experiment with respect to injection of a depo-larizing bias current, the partial experimental resultsnow available (55) are consistent with this modeling


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H1274 BIFURCATIONS AND SINUS NODE AUTOMATICITYV (mv)

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4000 t3000 b :I2000

11/,* a1000

t0: 1I I0.389 0.392IBIAS (pA/cm2)Fig. 6. Bifurcation diagrams generated with AUTO for ionic modelwith Ih = 0 (see text for further description). Exac t values of 1biasatwhich phenomena (e.g., unstable limit cycle in B) occur are slightlyshifted with respect to computations of Figs. 4 and 5, s ince doubleprecision arithmetic and a- different method of integration are used. C:period of stable limit cycle (7) is given as a function of the bifurcationparameter ( Ibias)*D: periods of both stable (curve a) and unstable (curveb) l imit cycles are given; the 2 orbits approach homoclinic ity within0.0005 PA/cm2 of each other (see also Refs. 16, 80).result. At point b, a supercritical Hopf bifurcation occurs,with a stable limit cycle and unstable equilibrium pointcoalescing, being replaced by a stable equilibrium point(branch ab).Figure 6B is an expanded view of Fig. 6A in a neigh-borhood of the saddle-node bifurcation (also termedknee, turning point, limit point, or fold) labeled d in Fig.6A. The equilibrium point at the most negative potentialin Fig. 6, A and B ( VI in Fig. 4C), is unstable for Ibiassufficiently low (i.e., along curue dh in Fig. 6B) but isstable for higher Ibias (i.e., to the right of point h). Thisreversal of stability is caused by a subcritical Hopf bifur-cation at point h, which produces a low-amplitude unsta-ble limit cycle (curues hi and hj give the overshoot andmaximum diastolic potentials of this orbit, respectively).There is thus only an extremely narrow range of Ibias(CO.001 pA/cm2) over which the stable equilibrium point(branch to right of h in Fig. 6B) coexists with the small-amplitude unstable limit cycle of Fig. 6B (branch hi-hj)and the large-amplitude stable limit cycle of Fig. 6A(branch bf-bg), thereby allowing single-pulse triggeringand annihilation to occur.Once again, because it is difficult to exercise onesimagination in six dimensions, to illustrate the topolog-ical concepts underlying annihilation and single-pulsetriggering we consider the simpler two-dimensional caseshown in Fig. 7A. This is the simplest possible configu-ration in which i t is possible to obtain annihilation andsingle-pulse triggering. Note that there is only one equi-librium point present (indicated by the x) that is stable,since starting from initial conditions sufficiently close to

Fig. 7. Single-pulse triggering (B) and annihilation (C) in 2-dimen-sional system (A) possessing stable limit cycle (outer solid curve),unstable limit cycle ( inner dashed curve), and stable equilibr ium point(x). See text for further description.that point (e.g., point c ) results in an attraction of thetrajectory asymptotical .ly onto the equilibrium point.This point is analogous to the stable equilibrium pointexisting just to the right of point h in Fig. 6B. There aretwo limit cycles present, one lying inside the other. Theouter limit cycle, the solid curve, is stable, since startingout at initial conditions at points a or b sufficiently closeto itthat results in trajectorieslimit cycle (Fig. 7A ). that asymptotically approachMovement of the state pointalong this limit-cycle trajectory corresponds to sponta-neous generation of action potentials in the Irisawa-Noma model (branch bf-bg in Fig. 6A). The inner closeddashed curve in Fig. 7A is an unstable limit cycle analo-gous to that existing in the Irisawa-Noma model (branchhi-hj in Fig. 6B). It is unstable, since trajectories startingout from initial conditions quite close to it (e.g., points band c) are repelled away from i t, ev entually en.ding up ateither one or the other of the two attracting structuresin the phase space,stable limit cycle. the stable equilibrium point or theFigure 7B illustrates the process of single-pulse trig-gering (Fig. 5B) in this configuration. Because the sys-tem is initia lly quiescent, this corresponds to the statepoint of the system sitting at the stable equilibrium pointindicated by the x. In the absence of external perturba-tions, the state point will sit there indefinitely. Injectionof a stimulus pulse can carry the state point away fromthe equilibrium point along the trajectory illustrated inFig. 7B, with the state point ending up at point d at theend of the stimul .us pu lse. The trajectory wil l then windout to the stable limit cycle as indicated. Thus sponta-neous activity can be triggered by injecting a singlestimulus pulse. It is apparent that the stimulus must besufficiently large so that point d lies in the region exteriorto the orbit of the unstable limit cycle; otherwise, thestate point will return to the stable equilibrium point,following a trajectory similar to that shown in Fig. 7Astarting from point c, giving a damped oscillatory sub-threshold response, and triggering wil l not occur. In thecase of Fig. 5B, the stimulus transports the state pointthrough the five-dimensional separatrix hypersurfaceand thus out of the six-dimensional null space surround-ing the equilibrium point.Figure 7C illustrates an example of annihilation in thetwo-variable model. Starting during spontaneous activ-ity, i.e., while the state point is moving along the outerlimit cycle, which is stable, a stimulus is injected whenthe state point lies at point e on the stable limit cycle.The stimulus causes the state point to move along thetrajectory indicated to point f, at which time the stimulus


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BIFURCATIONS AND SINUS NODE AUTOMATICITY H1275pulse is turned off. The trajectory will then asymptoti-cally approach the stable equilibrium point, which isattracting. Thus spontaneous activity is annihilated byinjecting a single stimulus. In the case of annihilationshown in Fig. 5C, the stimulus again takes the state pointto a point lying within the interior of the six-dimensionalnull space surrounding the equilibrium point, after pierc-ing the separatrix hypersurface. It is apparent from theconstruction shown in Fig. 7C that only stimuli withcertain combinations of strength and timing wil l leavethe state point within the two-dimensional null spaceforming the interior of the unstable limit cycle. Forexample, a stimulus with the same timing as shown inFig. 7C, but too small or too large in amplitude, mightput the state point into the annular region lying betweenthe two limit cycles (e.g., too large a stimulus deliveredat point e might take the state point to point b in Fig.7A) or even into the region outside the stable limit cycle.In both cases, one would have an eventual restoration ofspontaneous activity due to asymptotic return of thestate point to the outer, stable limit cycle. The region inthe (stimulus strength) - (stimulus timing) parameterspace at which annihilation wil l be seen has been termedthe black hole by Winfree (84). In general, as the sizeof the unstable cycle grows (e.g., with increasing 1bias nFig. 6B), one expects the size of this b lack hole to alsogrow.The area within the interior of the unstable limit cyclein Fig. 7 is thus the stable manifold of the equilibriumpoint, since any initial condition within this area isattracted asymptotically onto the equil ibrium point. Inthis simple two-dimensional system, the unstable limitcycle itself, which is a one-dimensional object, forms theseparatrix in the system, since trajectories starting atinitial conditions just to one side (inside) of this cycleasymptotically approach the equilibrium point, whereasthose starting from initial conditions just to the otherside (i.e., outside) go to the stable limit cycle. In contrast,in systems of dimension greater than two, the unstablecycle, being a one-dimensional object, cannot itself func-tion as a separatrix but is embedded in the higher di-mensional separatrix hypersurface.The stable and unstable limit cycles shown in Fig. 6,A and B, are both born in Hopf bifurcations (at points band h, respectively). They both gradually increase inamplitude (branch bf-bg in Fig. 6A, branch hi-hj in Fig.6B) and then abruptly disappear. Just before disappear-ing, there is a very steep growth in the period of bothoscillations (Fig. 6, C and 0). The bifurcation producingdestruction of both of these cycles is thus a homoclinicbifurcation (74) in which a periodic orbit col lides with asaddle point andOnce again, it disappears.is easier to visualize this bifurcation ina simple schematic two-dimensional system (Fig. 8). Weillustrate the homoclinic bifurcation analogous to thatinvolved in abolishing the small-amplitude unstable orbitof Fig. 6B (branch hi-hj). One starts off with the situationin Fig. 8A where there i s an unstable small-amplitudelimit cycle (dashed curve) around a stable equilibriumpoint (point a), analogous to the point associated withVI in Fig. 4C (and located on the branch to the right ofpoint h in Fig. 6B). There is also a saddle point (pointb) analogous to the saddle point associated with V2 in

Fig. 8. Homoc linic bifurcation in simple 2-dimensional model. As bi-furcation parameter is changed, unstable limit cycle init ially present(dashed curve in A) grows in amplitude until it coll ides with saddlepoint (point b in A), producing a hom oclinic orbit (dash ed curve in B).As parameter is changed further, homoclinic orbit disappears, leavinga heteroclinic connection (trajectory labeled g) between saddle pointand stable focus (point u in A). Reversing direction of all arrows ontrajector ies produces a homoclinic bifurcation in which a stable limitcycle is destroyed.Fig. 4C (and branch cd in Fig. 6A ). As the bifurcationparameter is changed, the unstable limi t cycle of Fig. 8Agrows in amplitude, with the state point increasinglyslowing its rate of progress along the part of the orbit inthe immediate vic ini ty of the saddle point. In fact, theperiod of the orbit becomes arbitrar ily large as the saddlepoint, where by definition rates of change of all variablesin the system are zero, is increasingly encroached on bythe orbit. Eventually, at one exact value of the bifurca-tion parameter (the bifurcation value), the orbit collideswith the saddle point and becomes of infinite period (Fig.8B). At this point, one has an intersection of the stableand unstable manifolds of the saddle point, with theseparatrix bc (Fig. 8A), which forms part of the unstablemanifold of the saddle point, uniting with the separatrixbf, which forms part of the stable manifold of that point,to produce a single trajectory that is biasymptotic (i.e.,as t + km) to the saddle point. This trajectory of infiniteperiod is called a homoclinic orbit. Pushing the bifurca-tion parameter further (Fig. 8C) results in the disap-pearance of the homoclinic orbit into the blue, hence onealternative name for this bifurcation, the blue-sky catas-trophe (74). The trajectory labeled g connecting thesaddle point to the stable equilibrium point in Fig. 8C istermed a heteroclinic connection or orbit.Even though we have chosen to illustrate the disap-pearance of a small-amplitude unstable limit cycle in Fig.8, a similar scenario holds for the disappearance of thelarge-amplitude stable limit cycle at point f-g in Fig. 6A.In fact, the approach to homoclinicity of this orbit isheralded by the slowing in beat rate caused by the dra-matic fal l in the rate of diastolic depolarization seen inFig. 5. This decrease is generated by the slow movementof the trajectory into a neighborhood of the saddle point,with subsequent rapid escape generating the upstroke ofthe action potential. The large-amplitude orbit charac-teristic of periodic action potential generation (branchbf-bg) thus disappears at large amplitude when hyper-polarizing 1bias s applied (at point f-g); in contrast, injec-tion of depolarizing 1biasproduces a gradual smooth de-cline in the amplitude of the action potential to zero (atpoint b).

Third way: skipped-beat runs. Figure 9 shows an ex-ample of the third way of abolishing spontaneous activ-ity. A hyperpolarizing current is again injected to slowthe rate of spontaneous activity but this time in the


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201E -501-8O- -60-zl~ 1::~~

0 t (sec) 40 0 t(sec1 40Fig. 9. Skipped-beat runs and subthreshold oscillation in ionic model.Spontaneous activ ity at 6 different values of 1biesn unmodified ( i.e., 1h# 0) model: 1biaa 0.78 (A), 0.79 B), 0.80 (C), 0.81 (D), 0.814 (E), and0.818 (F) pA/cm 2. Right: range of potentials from -60 to -50 mV onexpanded s cale. Note appearance of prepotentials in B-D, a maintainedsmall-amplitude oscillation in E, and a damped oscillation followed byquiescence in F. Standard init ial condit ions except for F where initialcondit ions corresponding to asym ptotic values appropriate to V = -55mV are chosen so as to make damped oscillation more evident.unmodified (i.e., 1h # 0) model. At first, one sees a gradualslowing in the frequency of action potential generationuntil the interbeat interval grows to -2 s (Fig. 9A). As1biaa is increased beyond this point, periodic patternscontaining both action potentials and subthreshold in-crementing oscillatory prepotentials (skipped-beatruns) are observed. These prepotentials are more clearlyseen in Fig. 9, right, which shows the pacemaker rangeof potential on an expanded scale. As 1bias ncreases, thefrequency of occurrence of prepotentials relative to ac-tion potentials also increases (Fig. 9, B-D). Eventuallya maintained subthreshold small-amplitude oscillation isseen (Fig. 9E). If 1biass increased sufficiently, quiescencefinally occurs (Fig. 98). Patterns of activity very similarto various members of the sequence illustrated in Fig. 9have been described in the SAN in response to a varietyof interventions (e.g., see Refs. 9, 20, 39, 54, 55, 58, 62,69, 76,81). In the one instance of these reports in whichit is straightforward to carry out the analogous simula-tion in the Irisawa-Noma model (adding acetylcholine),activity similar to that shown in Fig. 9 occurs in themodel. Indeed, the entire sequence of patterns shown inFig. 9 has been found in small pieces cut out of the SANin which the external potassium concentration is gradu-ally elevated and in the corresponding simulations on amodified version of the Noble-Noble (52) SAN model(M. R. Guevara, T. Opthof, and H. J. Jongsma, unpub-lished data). In that case, although excess K+ causes adepolarization of maximum diastolic potential and notthe slight hyperpolarization seen in Fig. 9, there is aprogressive slowing in the rate of spontaneous diastolicdepolarization similar to that shown in Fig. 9. Thisdecrease in the rate of rise of the pacemaker potential isa common feature in many of the reports cited above inwhich skipped-beat runs have been described in the SAN;it is also seen in the Irisawa-Noma model and other SANmodels when skipped-beat runs are produced bv a varietv

of interventions (Guevara, unpublished data).We now explore the bifurcation structure of the var-ious rhythms of Fig. 9. We stress that these rhythms areonly a sampling of those seen with 1biasbetween 0.780and 0.818 PA/cm? For theoretical reasons outlined be-low, one expects an infinity of different rhythms, bothperiodic and aperiodic, to exist within this range. Therhythm shown in Fig. 9D, which is periodic with a verylong period (10 s), suggests that the system is close topossessing a homoclinic orbit. This particular homoclinicorbit is not the same as that described earlier in Fig. 8in that it is associated not with a saddle point but witha saddle focus. Once again, it is difficult to visualizetrajectories in a seven-dimensional system: we thereforeschematically illustrate in Fig. 10 a saddle-focus equilib-rium point (indicated by the x) and its associated homo-clin ic orbit, which is the projection of an orbit in aninherently three-dimensional system down onto the twodimensions of a sheet of paper. Starting from an init ialcondition at a, which is a point on the homoclinic orbitclose to the saddle focus, the trajectory first spiralsoutwards roughly in the plane of the paper and thenmakes an excursion out of that plane into the thirddimension perpendicular to the plane of the paper (partof the trajectory labeled b). The orbit then reversesdirection at c, returning toward the plane of the paper(part of trajectory labeled d) . The speed of movement ofthe state point then slows, with the trajectory asymptot-ically (i.e., as t + 00) approaching the equilibrium point.Unlike the homoclinic orbit earlier illustrated in Fig. 8B,which involved a saddle point, the type of homoclinicorbit occurring in Figs. 9 and 10 cannot occur in a two-variable system. It requires a system of dimensionalityof at least three, since, should the trace shown in Fig. 10be regarded as being produced by a two-dimensionalsystem, the trajectory would cross itself, thus violatinguniqueness of solution.Starting with an initial condition closer to the equilib-rium point than the point a illustrated in Fig. 10 wouldresult in the trajectory making a larger number of spira lturns in the plane before being ejected out into the thirddimension. Starting with an initial condition infinitesi-mally close to the equilibrium point, an arbitrarily large

C

Fig. 10. Homoclinic orbit in 3-dimensional sys tem . x, Equilibr iumnoint, which is a saddle focus. See text for further description.


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BIFURCATIONS AND SINUS NODE AUTOMATICITY H1277numbe r of spiral turns of the trajectory would be pro-duced, whi ch would take an arbitrari ly long time. Ahomoclinic orbit is thus a closed trajectory of infinitelylong period, with the equi librium point bei w both thestarting and ending point of that orbit. Thus one wouldsee, in terms of membrane potential, an infinite numberof continually incrementing prepotentials (correspond-ing to the spiraling out in Fig. 10) that would take aninfinitely long time to produce a single action potential,following which the membrane would asymptotically andmonotonically return to its resting potential (corre-sponding to the approach to the saddle-focus point bythe part of the trajectory labeled d in Fig. 10). Thus thesimulations of Fig. 9 suggest that a homoclinic orbitinvolving a saddle-focus likely exists at one precise valueof 1biaa omewhere between 1bias= 0.810 PA/cm2 (Fig. 9D)and 1bias= 0.814 PA/cm2 (Fig. 9E).As 1bias s decreased away from the one exact value atwhich the homoclinic orbit involving the saddle focusexists, the homoclinic orbit disappears and one of twoscenarios generally then occurs (23, 26, 82). In the sim-pler alternative, which unfortunately does not occur here,as the homoclinic orbit is broken one sees a singleperiodic orbit of finite period, with the period of thatorbit smoothly decreasing as one moves the bifurcationparameter away from the bifurcation val ue at which thehomoclinic orbit existed (e.g., see Fig. 3.1(i) in Ref. 26).The homoclinic orbit can then be regarded as the limitingcase of this unique orbit as the bifurcation parameter ischanged in the other direction back to the value produc-ing homocl inicity. In the alternative, more complex, casethat occurs here, more than one periodic orbit is born asthe bifurcation parameter is moved through a range ofvalues close to the value producing the homoclinic orbitof Fig. 10. The number of such orbits can be finite (e.g.,see Figs. 3.8, 4.7 of Ref. 26) or infinite (e.g., see Fig. 3 ofRef. 23, Fig. 3.4 of Ref. 26, Fig. 3.2.41 of Ref. 82). In thesimulations of Fig. 9, a large number of stable periodicrhythms are produced over a rather small range of 1bias(-0.03 pA/cm2) adjacent to the value of 1biasat which thehomoclinic orbit exists. In Fig. 9, we show only a few ofthese traces; however, many other more complex periodicrhythms (not shown) are seen when 1bias s changed morefinely within the range 0.780 PA/cm2 c 1biasc 0.814 PA/c)cm.How aredie? Somebifurcation

these periodic orbits born, and how do theyof these orbits are born through saddle- nodes of periodic orbits, whereas others are bornvia period-doubling bifurcations (23, 26, 82). Figure 11,a partial bifurcation diagram computed using AUTO forthe simulations of Fig. 9, gives examples of both of thesebifurcations. As is the case when 1h is removed from themodel (Fig. 6A), injection of a depolarizing bias currentproduces a supercritical Hopf bifurcation at point b (Fig.1 lA ), and there is again a region of coexistence of threeequilibrium points in response to a hyperpolarizing biascurrent (Fig. 11B). However, there is not a subcriticalHopf bifurcation producing a small-amplitude unstablelimit cycle in this region of coexistence, as in Fig. 6B,but rather a supercritical Hopf bifurcation at point c (Fig.llB), which lies outside of this region, producing thesmall-amplitude stable oscillation of Fig. 9E. Thus, asdirect numerical integration has already shown (Fig. 4A ),

-50*0t \ -7o.ot f-75.0 -2.0 -1.0 0.0 1.0IBIAS (uA/cm2) IBIAS (uA/cm2)

w w W V)-50.0 C-60.0 F\/ I:~:~,~0 1000 0 1000 2000ww ww

Fig. 11. A and B: bifurcation diagrams generated by AUTO in unmod-if ied ( i.e., 1h # 0) ionic model. C : stable small-amplitude subthresholdoscillation at value of Ibias (-0.81102 pA/cm2) close to maxim um am-plitude of orbit (point g-h in B). S light decreas e in 1bias esults in aperiod-doubling bifurcation . D: stable period-doubled subthresh old os-cil lation at value of Ibias (-0.81085 pA/cm2) close to maxim um ampli-tude. Further slight decrease in 1bias roduces a second period-doublingbifurcation.the annihilation and single-pulse triggering of Fig. 5 (&,removed) cannot occur in the unmodified Irisawa-Nomamodel. Unlike the case in Fig. 6B, the small-amplitudeorbit of Fig. 11B (branch cg-ch) does not terminate andvanish in a homoclinic orbit involving the saddle point,which is relatively far removed. Rather a period-doublingbifurcation occurs in which the small-amplitude limitcycle, stable along branch cg-ch, grows in amplitude as1bias s decreased from its birth at point c until it becomesunstable at point g-h, spawning a stable limit cycle ofabout twice the period, but about the same amplitude, ofthe original but now destabilized limit cycle. This stableperiod-doubled orbit exists only over a very narrow rangeof Ibias, too small to be shown in Fig. 11B. It grows inamplitude as I bias s reduced and itself undergoes anotherperiod-doubling bifurcation. Figure 11C shows one cycleof the small-amplitude oscillation at a value of Ibias ustbefore it undergoes the first period doubling at point g-h, whereas Fig. 11D shows one cycle of the period-doubled oscillation just before it period doubles the sec-ond time. We return to discussion of period-doubledorbits at a later stage.How does the large-amplitude limit cycle representedby branch be@ in Fig. llA , corresponding to the spon-taneous action potential generation of Fig. 9A, disappearat point e-f? Again, the situation is different from thatshown in Fig. 6C (&, = 0): the limi t cycle does notdisappear abruptly in a homoclinic bifurcation as in Fig.6C; rather, it collides with an unstable limit cycle andboth orbits disappear at point e-f via a saddle-nodebifurcation of periodic orbits. Although, for reasons pre-viously mentioned, the traces shown in Fig. 9 can onlybe generated in a system having dimensionality of atleast three, a saddle-node bifurcation of periodic orbitscan occur in systems of dimensionality as low as two:Fig. 12 illustrates such a bifurcation in a simple two-dimensional system. In Fig. 12A, there are no periodicorbits present, only a stable equil ibrium point. As thebifurcation parameter is than d, a large-a .mplitude limit


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Fig. 12. Saddle-node bifurcation of periodic orbits in &dimen sionalsys tem. As a parameter is changed to its bifurcation value, a large-amplitude semistable limit cycle (B) suddenly appears in a region ofphase space previously containing no periodic orbits (A). As parameteris changed further, semistable cycle splits up into 2 limit cycles (C):one stable (outer solid curve) and other unstable (inner dashed cu rve).See text for further description.cycle suddenly appears (Fig. 12B) via a saddle-nodebifurcation of periodic orbits. The limit cycle is semi-stable, being attracting from one side (trajectory withinitial condition a in Fig. l2B) but repelling from theother side (trajectory with initial condition b). Like thehomoclinic orbits of Figs. 8B and 10, this semistableorbit exists only at one precise value of the bifurcationparameter, the bifurcation value (the value of 1biascor-responding to the points e and f in Fig. 1lA). As thebifurcation parameter is pushed just beyond this point,the semistable limit cycle breaks up into two limit cycles,one stable and the other unstable (Fig. 12C). Note thatthe configuration shown in Fig. 12C is exactly that shownearlier in Fig. 7, which is the topology allowing annihi-lation and single-pulse triggering. Indeed, it is a saddle-node bifurcation generating stable and unstable limi tcycles that allows these phenomena to occur in theHodgkin-Huxley equations in response to injection of asteady bias current (see Fig. 9 of Ref. 35) instead of thehomoclinic bifurcation of Fig. 6B.Skipped-beat runs similar to some of those shown inFig. 9 can be seen in the response of the model to 1bias flNa or both lNa and 1h are removed. However, in the lattercase these rhythms take place over a range of 1biaswherethere are three equilibrium points in the system. Becausemore than one equilibrium point is present, it is possiblethat heteroclinic connections, cycles, or orbits can occur,with trajectories connecting different equilibrium points(1, 71, 74, 82). In addition, because one of the threeequilibrium points present is a saddle, it is possible thathomoclinic orbits of the type illustrated in Fig. 8 existabout that point. The existence of hetero- and homoclinictrajectories can result in various bifurcations that pro-duce or destroy limit cycles, e.g., the previously men-tioned homoclinic bifurcation, as well as the omega ex-plosion (74). This situation of coexistence of multipleequilibria when both lNa and 1h are removed is in contrastto the unmodified model (F ig. 11, A and B), or whenonly lNa is removed, or when 11 s increased in a reducedthree-dimensional model (33) where skipped-beatrhythms occur when there is only one equilibrium point(a saddle focus) present in the phase space of the system.In addition, the basic rhythm of Fig. 9A, as well as atleast some of the skipped-beat rhythms, can be annihi-lated and single-pulse triggered when both lNa and 1h areremoved, since the equilibrium point lying at the most

hyperpolarized potential is stable in that circumstance(as in Fig. 4C).Ionic mechanisms of skipped-beat runs. We now turnto investigate briefly the ionic mechanisms underlyingthe generation of the traces shown in Fig. 9. Figure 13displays the various ionic currents during the rhythmsshown in Fig. 9, B and E. Note that, as is the case duringspontaneous activity (Fig. l), the three currents 1Na, 1k,

and 1h contribute but lit tle to the evolution of the voltagewaveform during diastole, and once again the changes inIs and 11 are much larger, being almost exactly equal inmagnitude but opposite in sign. The three indispensiblecurrents involved in producing the skipped-beat rhythmsshown in Fig. 9 can be said to be IS, 1k, and 11, since, iflNa and 1h are both removed from the model, skippedbeat runs can be seen at 1bias= 0.348 pA/cm2. In addition,with lNa and 1h both so removed from the model, whichthen becomes four-dimensional (variables: V, d, f, p),annihilation and single-pulse triggering can also befound. In fact, should the activation of &, be made timeindependent (i.e., d = d, at all times), which is a reason-able approximation given the slow upstroke velocity ofthe action potential, the model becomes three-dimen-sional (variables: V, f, p) but is stil l capable of displayingannihilation and single-pulse triggering (33), as well asskipped-beat runs, both of which occur once again in arange Of Ibias where there are three equilibrium pointspresent in the system.

Afterpotentials and eigenvalues. In Fig. 9F, the mem-brane becomes quiescent in response to injection of asufficiently large hyperpolarizing bias current. A decre-menting oscillation of small amplitude is seen to precedethe eventual establishment of quiescence in the steadystate (Fig. 9F, right). If an action potential is provokedby a suprathreshold current pulse during this quiescence,one sees a train of decaying afterpotentials following theinduced action potential that is similar to the dampedoscillation shown in Fig. 9F but smaller in amplitude,indicating that the approach of the trajectory to theequilibrium point is modulated by currents activated

A B

0.25I 0.00 tlsec) tlsec)

Fig. 13. Ionic basis of skipped-beat rhythm of Fig. 9B (A) and ofsubthreshold oscillation of Fig. 9E (23). Transmembrane potential andcurrents as in Fig. 1. All currents drawn to same scale.


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BIFURCATIONS AND SINUS NODE AUTOMATICITY H1279during the action potentiaas soon as Ibias is j ust large 1e Indeed, this behavior is seennough to produce quiescence.In that circumstance, injection of a subthreshold stimu-lus pulse produces a ringing in the membrane (Fig.14B). As Ibias s increased, the damped oscillation startsoff with a lower amplitude of its first peak and dampsout more quickly (Fig. 14, B-E). Mathematically, theappearance of the damped oscil lation as Ibias s increasedis due to the conversion of the equilibrium point in thesystem of Fig. 14A, when a maintained subthresholdoscillation is present, from an unstable focus or spiralpoint to a stable focus via a Hopf bifurcation (at point c, in Fig. 11B).One can linearize the nonlinear Irisawa-Noma equa-tions around the equil ibrium point and calculate seveneigenvalues, numbers that characterize the behavior ofthe trajectories in an infinitesimally small neighborhoodof state space about the equilibrium point (1, 2, 14, 25,71, 74). When the oscillation in Fig. 14A exists, two ofthese eigenvalues are a pair of complex conjugate num-bers with positive real part; the other five eigenvaluesare negative real numbers. It is this positivity of the realpart of the complex eigenvalue pair that makes theequilibrium point unstable and that is responsible for thetrajectory spiraling away from i t as time proceeds (Fig.14A). As Ibias is increased, the real part of the complexpair decreases, and a Hopf bifurcation occurs, by defini-tion, when the eigenvalues become purely imaginary,with zero real part (at point c in Fig. 1lB). For Ibiasgreater than this bifurcation value, the real part of thepair of complex conjugate eigenvalues becomes negative,producing a stable equil ibrium point (branch to right ofpoint c in Fig. 11B). The fact that there is stil l a nonzeroimaginary part of the eigenvalues results in oscillatorybehavior, which is damped since the equilibrium point isstable (Fig. 14, B-E). As Ibias increases further, the realpart of the complex pair becomes increasingly negative,leading to the progressively increasing level of dampingshown in Fig. 14. In contrast, there is slower change inthe imaginary part, leading to much smaller changes in

F -54 cE>-591-541 D

-54 E

-595 0 t (set) 20Fig. 14. Subthreshold potentials during quiescence. 1bias= 0.814 (A),0.82 (B), 0.83 (C), 0.85 (D), and 0.90 (E) pA/cm 2. A m aintainedsubthreshold oscillation is present in A. Membrane is quiescent in B-E, with a hyperpolariz ing pulse of amplitude 0.05 PA/cm 2 and duration20 ms injected at t = 10 s. Init ial condit ions in each case are close toequilibr ium point, w hich is unstable in A but stable in B-E.

the frequency of the damped oscillatory response. Even-tually, at a sufficiently high value Of Ibias (-1.17 pA/cm2),the imaginary part of the complex pair of eigenvaluesgoes to zero, with the complex pair being replaced by twonegative real eigenvalues, removing the oscillatory com-ponent of the dynamics but maintaining the stability ofthe equilibrium point. The stable focus is thus replacedby a stable node (1, 14, 25, 71, 74), with a monotonic(i.e., nonoscillatory) approach of the state point to theequilibrium point following a perturbation. A similartransition from a stable focus to a stable node occurs asan increasingly large depolarizing Ibias is injected at Ibias= -4.55 PA/cm2 (to the left of point a in Fig. 1lA).Limit cycle s and Floquet multipliers. In the same waythat one can linearize a system of nonlinear differentialequations about an equilibrium point, one can also li-nearize a system about a limi t cycle. Instead of obtainingeigenvalues, one obtains Floquet multipliers, which char-acterize the nature of the trajectories in an infinitesimalneighborhood of the limit cycle (71, 74). One of thesemultipliers is always equal to unity, reflecting the motionof the state point along the direction of the limit cycletrajectory itself. The other multipliers, which are N - 1in number in an N-dimensional system, are generallycomplex numbers. The limit cycle is stable when all ofthose other N - 1 complex numbers lie within the unitcircle. A stable limit cycle thus destabilizes through abifurcation when one or two Floquet multipl iers moveaway from the origin and eventually cross the unit c ircleas some parameter is changed. This can happen in oneof three ways: a single real-valued Floquet multipliercrosses the unit circle by becoming more negative than-1, producing a period-doubling bifurcation; a singlereal-valued multiplier crosses through +l, producing asaddle-node bifurcation of periodic orbits; and a complexpair of multipliers intersects the unit circle, producing atorus bifurcation, after which orbits, which can beperiodic or aperiodic, move on a toroidal hypersurface.In the first and third cases, the original limit cyclepersists beyond the bifurcation point but is unstable.Thus it is numerical calculation of the Floquet multi-pliers using AUTO that allows us to determine the sta-bility of the limit cycles in Figs. 6 and 11.DISCUSSION

The traces presented above are not the first to showabolition of spontaneous activity in an SAN model assome intervention is made: for example, cessation ofactivity has been previously demonstrated as a result ofadding a bias current (86), decreasing 1s (12, 75), addingacetylcholine (75, 86), increasing the internal sodiumconcentration (52), or blocking either of the calciumcurrents IL or IT (51). However, in these previous studies,the parameter under consideration was changed in stepstoo coarse to allow precise determination of how spon-taneous activity would cease. In fact, the point of theseother simulations was simply that spontaneous activitywould disappear if the change made was sufficientlygreat. The present study is the first to probe finely thevarious routes leading to quiescence in an ionic model ofSAN isopotential membrane.First way. The first way of abolishing activity (Figs. 2


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HE80 BIFURCATIONS AND SINUS NODE AUTOMATICITYand 3) has been described as a result of several differentinterventions in the SAN (10,40,50,63, 70).A commonfeature of these interventions is a gradual progressiveprimary reduction in either I, or 1k, producing a continualsmooth fall in the size of the upstroke or repolarizationphase of the action potential, respectively. A fall in thesize of either of these two phases also leads to a fall inthe other: a primary decrease in 1s leads to a fall inovershoot potential, decreased activation of 1k, and amore depolarized maximum diastolic potential; a primarydecrease in 1k produces a more depolarized maximumdiastolic potential, increased resting inactivation of I,during diastole, and a fall in the overshoot potential.Thus, in both cases, the action potential amplitude (thedifference between overshoot and maximum diastolicpotentials) declines. As in Figs. 2 and 3, where Is isblocked, blocking 1k in the Irisawa-Noma model alsoproduces a gradual fall in action potential amplitude.Second way. Annihilation and single-pulse triggeringbecome possible when there is coexistence of a stableequilibrium point and a stable limit cycle (Fig. 7). Indeed,it was this theoretical concept that led Appleton and vander Pol (4 ) in 1922 to search for and find single-pulsetriggering in a simple electronic oscillator. Indeed, oscil-lators with the topology shown in Fig. 7A are referred toin the engineering literature as hard oscillators, sincethey must be excited in some way with a shock of finiteamplitude to be started up (in contrast to the softoscillator of Fig. 3G). Teorell (see Ref. 73 and referencesto earlier work contained therein) invoked the conceptof a hard oscillator to explain the single-pulse initiationand brief-pulse annihilation that he had observed inanalogue computer simulations of a physical model of amechanoreceptor. It was also theoretical work, based onconsideration of the topology of phase resetting and theimplied existence of a singular stimulus, that led Win-free (83) some years later to independently propose theconcept of annihilat ion. This pioneering work of Winfreedirectly led experimentalists to conduct the first system-atic searches for annihilation in several biological prep-arations, including cardiac tissue (35, 37, 38). Althoughseveral examples of a premature stimulus annihilatingspontaneous activity had appeared in the cardiac litera-ture before these systematic searches were made (seediscussion in chapt. VIII of Ref. 20), no theoreticalinterpretation of those isolated results was offered.Annihilation. Although there are many experimentalstudies showing at least some features of the first andthird routes to quiescence in the SAN, we know of onlyone report in which this second route (annihilation) hasbeen described in the SAN (37). As previously noted, theresults of another experimental study indicate that an-nihilation should not be possible (56). The reasons forthe diametrically opposing results in these two studiesare unknown; they might, for example, simply involvespecies differences [cat (37) vs. rabbit (56)] or differencesin the experimental preparation [sucrose-gap (37) vs.small piece (56)]. However, it must be kept in mind thatthe SAN is an inhomogeneous structure with, for exam-ple, action potentials of many different morphologiesbeing recorded in different areas of the node (7). Thusthe contrasting results of Jalife and Antzelevitch (37)and Noma and Irisawa (56) might be due to differences

in the sites from which the SAN specimens were taken.Recent work has shown that many of the electrophysio-logical differences in small pieces of tissue taken fromdifferent sites are largely intrinsic, being local propertiesof the cell membrane, and are not due to electrotonicinteractions (39,41,61). This conclusion is reinforced bymore recent work using single SAN cells (59).In some pieces of tissue isolated from the SAN, volt-age-clamp experiments indicate that 1h is present only inrelatively small amounts in the pacemaker range of po-tential, and so 1h would be expected to play a negligiblerole in generating spontaneous diastolic depolarizationin those pieces (11,12). More recent work on single cellsisolated from electrophysiologically unidentified areas ofthe SAN indicates that there is significant variabili ty inthe amount of 1h (pA/pF) present in different cells (59).In fact, blocking 1h with Cs2+ in some small-piece prep-arations produces only a slight decrease in rate (12, 41,57). In contast, in other pieces, a signif icant amount of1h is activated by a voltage-clamp step into the pacemakerrange of potentials (11, 12, 48), and relatively largechanges in beat rate can be produced when Cs2+ is applied(12, 41, 57). In the Irisawa-Noma model in which ahyperpolarizing bias current is applied, our work indi-cates that 1h must be absent (or sufficiently reduced) forannihilation to be seen. The exact degree of reduction isvery delicate, depending on the details of the model. Forexample, in the predecessor of the Irisawa-Noma model,which also has a small contribution of &, it is notnecessary to reduce 1h to see annihilation (43). In thisrespect, it is interesting to note that in both experimentaland modeling work on two other cardiac tissues in whichannihilation has been demonstrated [depolarized Pur-kinje fibre (15, 38, 68), depolarized ventricular muscle(1%24,68)1, ne would also expect a negligible contri-bution of 1b to the generation of diastolic depolarization.In addition, in all three of these tissues, lNa is substan-tially reduced as a consequence of inactivation secondaryto depolarization of the membrane, a feature shared withthe Irisawa-Noma model. In a very recent report using afourth preparation, the embryonic chick atria1 heart cellaggregate, which does not possess &, annihilation couldnot be produced unless I Napwhich is relatively large inthis preparation, was at least partially blocked withtetrodotoxin (72). Computer simulation using an ionicmodel of this preparation is also in agreement with thisexperimental finding (72). Once again, different areas ofthe SAN show evidence of intrins ically possessing radi-cal ly different amounts of INa (39, 41,61). Finally, in theone reported case of annihilation in the SAN, the inter-beat interval was quite long for a kitten C-390 ms in Fig.1 and -670 ms in Fig. 2 of Ref. 37 vs. a mean interval of375 t 41.4 (SD) ms in 12 isolated SAN preparationsfrom cats weighing 2.5-4.5 kg in Ref. 601. This is con-sistent with our finding in the model (Fig. 5C) thatannihilation is only seen when the interbeat interval isprolonged significantly by, for example, injecting a hy-perpolarizing bias current. The fact that annihilation isnot seen in the model with 1h present in sufficiently largeamounts might indicate that 1h is playing a protectiverole in the peripheral areas of the SAN, where it ispresent in considerable amounts, in that it preventsannihilation from occurring in the primary natural pace-


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BIFURCATIONS AND SINUS NODE AUTOMATICITY H1281maker of the mammalian heart in response to an other-wise suitably timed premature atria1 contraction. Thestudy with atria1 aggregates mentioned above might alsoindicate that lNa is playing a similar protective role inperipheral areas of the node where it is also present inlarge amounts.One other point that might have some bearing on thedifferences in the two studies is the fact that the equil ib-rium point that is created anew (at voltage VI in Fig. 4C)as 1bias s changed can be stable, thus allowing annihila-tion to be seen. For this point to be created, the steady-state I-V characterist ic curve must be N-shaped, havinga region of negative slope. However, most published I-Vcurves from the SAN have an N-shaped region that iseither very shallow or even entirely absent (48, 53, 57).Indeed, in the Irisawa-Noma model, either modified (byremoving 1h) or unmodified, the I- Vcurve is very shallow,with the region of negative slope conductance disappear-ing if, for example, the lNa window current is removed.Therefore three equilibria exist over only a very smallrange of the parameter being changed (0.3897 PA/cm2

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Michael R. Guevara and Habo J. Jongsma- Three ways of abolishing automaticity in sinoatrial node: ionic modeling and nonlinear dynamics