The development of mathematical models of the heart

Chaos

Pergamon

Solitonr

& Fractolr Vol. 5, Nos 314, pp. 321-333, 1995 Copyright 0 1995 Elsevier Soence Ltd Printed in Great Bntam. All rights reserved 09@l-0779/95$9.50 + .oo

MO-0779(93)EOO25-7

The Development of Mathematical DEN’IS University

Laboratory

of Physiology,

Models of the Heart

NOBLE

University

of Oxford,

Parks Road, Oxford OX1 3PT, UK

Abstract-Mathematical modelling of cardiac cell activity has developed in roughly ten-year cycles over the last 30 years, with each major phase of modelling being focused around particular experimental questions. The first phase (around 1962) was based on the dissection of the K-currents in heart cells into the inward rectifier and delayed current components. The second phase (1975) was based on identifvina seuarate slow current mechanisms in the olateau and oacemaker ranges of potential and on* th‘k dkcovery of the calcium current in cardiac muscle. The most recent-phase (starting with the 1985 DiFrancesco-Noble model) was based on the identification of the hvperpolar&ion-activated pacemaker current and on the electrogemcity of sodium-calcium exchange. Although each of these developments has depended on advances in experiment method, it is also true that each has also needed to theorize ahead of the experiment work. There is, therefore, a bi-directional interaction between theory and experiment. Sometimes experimental work leads, sometimes the theoretical work does so. A major use of such models in the case of cardiac cells is their incorporation into integrative studies of how large networks of cardiac cells interact to produce normal and abnormal rhythms. This work has received a major boost from the introduction of massively parallel computers that provide the required speed and capacity. Already, models of networks of sinus node and atria1 cells have been constructed.

INTRODUCTION

The use of mathematical models in the study of cardiac excitation and rhythm generation has a long history starting with the use of the van der Pal equation for representing cardiac rhythm as a form of relaxation oscillator [l]. Although it is possible to relate the parameters of the van der Pol equation to real cardiac excitation parameters (see ref. [2]) and even to obtain some useful insights into conditions for instability (see ref. [3]), the equations were not developed, of course, with these applications in mind. Modelling of cardiac electrical activity using equations based on actual measurements of ionic conductances began 30 years ago with the Noble model of the cardiac Purkinje fibre [4]. Since then, modelling has followed periodic fashion s almost every decade, with peak modelling activity following on two or three years after each major advance in technique. Thus, the first full voltage-clamp experiments of the 1960s and early 1970s led to the development of the McAllister et al. Purkinje fibre model [5] and the Beeler and Reuter ventricular model [6]. The work on ion accumulation/depletion and the discovery of ir led to the DiFrancesco-Noble Purkinje model [7], the sinus node model of Noble and Noble [8] and the Hilgemann-Noble atria1 model [9]. More recently, the single-cell and patch-clamp data have led to a very wide range of single-cell models (single atria1 cell: Earm and Noble [lo]; single sinus cell: Noble et al. [lj.]; Wilders et al. [12]; guinea-pig ventricular cell: Noble et al. [13]), some of which are now being incorporated into modelling of massive cell networks using parallel computers (Winslow et al. [14]). There are many different motives for modelling work. At one extreme, simplified and tractable mathematical models are required for analytical work and for incorporation into more complex multicellular systems. At the other extreme, very detailed models are 321

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required to test the interactions of many different cellular components. It is always a matter of judgement what to incorporate and what to leave out of such work. That judgement depends in part on what the models are to be used for. It is not, for example, worthwhile exploring sodium-overload arrhythmias in models that do not include the relevant sodium- and calcium-dependent mechanisms inside the cell, while study of mechano-electric feedback will require more extensive modelling of length-dependent processes than has so far proved possible, though they are now being developed. We end up therefore with a vast range of possible models each with their own range of application. At the cellular level, they vary from equations with as few as 2 or 3 dependent variables (like the van der Pol equation) to systems of equations with more than 50, as in some of the recent cellular models incorporating some of the cell biochemistry. My own experience has been that the best modelling work is achieved around a pivotal problem that is also of great experimental interest. In this paper I will illustrate this point with some of the models I have been involved in developing. ANALYSIS OF CARDIAC-K CONDUCTANCE CHANGES

The original 1962 model was based on experiments showing a remarkable property of the K-current in heart cells. Hall et al. [15] (see also [16]) and Carmeliet [17] showed that the 50

A r

Fig. 1. The 1962 model of the cardiac Purkinje fibre. This model was based on experimental evidence showing that there are two K conductances, one (i& showing rapid inward-going rectification, the other (iK-called ix2 in ref. [4]) showing slow delayed rectification. Together with a sodium conductance and a background inward current, these were sufficient to generate the required action potential and pacemaker waveforms. Although the later models are much more complex, including ia, if, iNaCaamongst other components not known in 1962, the overall behaviour of the K-conductance is correctly represented by this model.

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immediate effect of membrane depolarization is greatly to reduce the potassium conductance of the cardiac cell membrane, a phenomenon called inward-going rectification. This effect was in total contrast to the situation in the squid nerve axon, which shows only a delayed increase in K-conductance. In cardiac cells, this delayed K-current change also exists, but is much slower and very much smaller. This work led to the first realization that there is more than one type of K-channel in heart cells. The early experimental work was analysed using two components, the inward-rectifier iK1 and the delayed rectifier iK. At that time, the voltage-dependence of iK1 was thought to be instantaneous. More recent work by Carmeliet and others has shown that it is actually time-dependent, but the relaxation time is rapid enough for it to be regarded as close to instantaneous during the cardiac action potential. The 1962 model was based on using these two K-conductances to see whether they were sufficient, together with a sodium conductance, to generate the Purkinje fibre-action potentials and pacemaker potentials. The model succeeded in doing this. This successwas both its strength and its weakness. Its strength lay in the fact that, even now, with many more K-channels identified, the underlying pattern during the normal action potential is very much that identified in the 1962 model. The weakness lay in the fact that its very successwith so few channel mechanisms may have delayed understanding of the complexity of cardiac cell behaviour. There was no hint in the 1962 model of the involvement of a calcium channel. That was to come later with Reuter’s work in 1967 [18]. Nor was it noticed that Weidmann’s experiments [19] already implied that the slow conductance changes must be more complex. The fact that there are two distinct voltage ranges in which slow conductance changes occur came with the experimental work of Noble and Tsien in 1968 and 1969 [20-221. They showed that the slow conductance change in the pacemaker range of potentials is quite distinct from that in the plateau range. This discovery, together with that of the calcium current by Reuter [18], led to the development of the McAllister, Noble and Tsien model

[51.

The central pivot in this case was the need to explore the consequences of the dissociation in the Purkinje fibre of the processes controlling the plateau from those involved in controlling the pacemaker depolarization. One of the successesof the model was the reconstruction of Weidmann’s experiment [19] on the effect of brief current pulses applied during the pacemaker depolarization. This experiment showed that current pulses have the opposite effect to that expected. Brief hyperpolarizing pulses are followed by an acceleration of the pacemaker depolarization, while brief depolarization pulses are followed by a slowing. This result was in fact first reconstructed by Hauswirth [23], using the McAllister et al. model [5]. These features are also extremely well reproduced by the DiFrancesco--Noble model, illustrated in Fig. 2. The explanation is simple: there must be a membrane channel gating mechanism in the pacemaker range itself, so that small voltage changes in this range strongly influence the channel properties. The 1962 model could not cope with this phenomenon. The fact that both the 1975 and 1985 models succeed in reproducing it so well (there is no significant difference between the two models in this reconstruction) shows that Weidmann’s result, while showing that there is voltage gating in this range, could not distinguish between very different conductance mechanisms. THE DISCOVERY

OF THE HYPERPOLARIZATION-ACTIVATED

CURRENT

The 1975 model was successful in dealing with this kind of result, amongst many others, yet it was precisely in this mechanism that the model contained the seeds of its most spectacular failure. This was that the mechanism used in the pacemaker range was a highly

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Fig. 2. Reconstruction of Weidmann’s pacemaker pulse experiment [19]. The top recordings are from computations done with the DiFrancesco-Noble model [7]. The bottom recordings are the original experimental traces. This experiment was in fact first reconstructed by Hauswirth (1971) using the McAllister, Noble and Tsien (1975) model [5]. Both reconstructions are highly successful, which shows that, although the experiment requires a conductance mechanism to be voltage-gated in the pacemaker range of potentials, it does not reveal what the conductance is nor the direction in which its voltage activation occurs.

specific K-channel, called iK2. The experimental evidence for this mechanism seemed watertight. The current recorded in the pacemaker range of voltages not only displayed a reversal potential at very negative voltages, this reversal potential shifted with external [K] in a Nernstian fashion, with the correct slope of 60 mV per tenfold change in [K],. The hypothesis that a specific K-conductance declined during the pacemaker depolarization also provided a natural explanation for Weidmann’s resistance measurements [ 191.

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Yet, on this particular feature, the model was completely wrong! The story of this spectacular failure has been fully documented elsewhere [24]. It is sometimes thought that this is a prime example of some awkward experimental facts destroying a beautiful hypothesis. It is indeed the case that it was the identification of the hyperpolarizationactivated current if that finally nailed the coffin of the I‘K2 hypothesis, but it is also true to say that this hypothesis itself was already proving internally inconsistent. If there had been a specific K-conductance with the properties of i K2 it would have been impossible to record it in the multicellular Purkinje fibre preparation, since the accumulation and depletion of potassium ions in the extracellular space would have obscured its properties [25,26]. This fact was understood even before the identification of ir in the Purkinje fibre. That explains why the if model gained such rapid acceptance. Even in the same year (1980) as the if mechanism was analysed in Purkinje fibres [27,28], the modelling work showing its consistency with all the experimental data previously thought to support the iK2 model was done. There was no lengthy war between the two models, since even the architects of the old model were involved in showing its internal inconsistency and the total compatibility of the new model with the experimental basis of the old one. The two models were fully mapped onto each other in a detailed way by DiFrancesco and Noble [29]. CALCIUM

BALANCE

IN THE HEART

The advances reviewed so far concentrated on the analysis of the K-conductances and of the pacemaker current, ir. Although the if model was the cornerstone of the development of the series of models owing their origin to the 1985 DiFrancesco-Noble model [7], the pivot around which these models were eventually to develop was quite different. The central question that has controlled their more recent development is that of calcium balance. This was in fact a natural development initially required by the work on ir and the processes of K-ion accumulation and depletion. For to represent these processes it was necessary to include the activity of the Na-K exchange pump, together with representation of its intracellular activator, sodium. With sodium concentrations included and with calcium as one of the main ions carrying charge during the excitation process, it was then necessary to include the process that is largely responsible for linking calcium movements and sodium movements, i .e . the sodium-calcium exchange. DiFrancesco and I were then faced with a dilemma typical of modelling work in this field. We were forced to take the modelling way ahead of its experimental basis. In doing this we knew the risks we were taking. After all, the model we were working on was itself prompted by a dramatic failure of its predecessor. We were either lucky or prescient enough to be very successful. We opted not only for the right stoichiometry for the exchange process but also for the correct voltage-dependence. Was it luck or good judgement? That can best be assessedby briefly reviewing what we knew and how the story then developed. At that time (l980) the exchange was considered by many to be electrically neutral, exchanging two monovalent sodium ions for each divalent calcium ion, the original stoichiometry favoured by Reuter and Seitz [30], who first discovered the exchange mechanism in the heart. Mullins [31,32] had outlined the alternative view based on an electrogenic mechanism in which more than two sodium ions move in exchange for each calcium ion. This, of course, generates an electric current and the question inevitably arose: to what does this current correspond? Mullins proposed a number of possibilities. Probably he proposed too many, for the effect was to make it easy to refute most of his proposals and so to dismiss the central idea itself. I was myself also initially highly sceptical, and when we first incorporated the Na-Ca

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n GATEDANDEXCHANGECURRRNTS

-50

BACKGROUNDANDPUMPC”RRENTS

6.N.

-50 0

0.50

1.5 1.0 seconds

2.0

, 2.5

Fig. 3. The DiFrancesco-Noble (1985) model [7] of the cardiac Purkinje fibre. In addition to the sodium and two K-conductances represented in the 1962 model, this model includes the calcium current iCa, hyperpolarizationactivated current if, sodium-calcium exchange current &ca and representation of the internal and external concentrations of sodium, potassium and calcium ions. This model is the generic one from which models for the SA node, atrium and ventricle have been subsequently developed.

exchange into our Purkinje fibre model we tried a neutral mechanism. We found, quite simply, that it does not work. There is, then, insufficient energy in the sodium gradient to keep internal free calcium sufficiently low. We found that a stoichiometry of 3:l (already suggested by some workers-see review by Chapman and Noble [33]) was sufficient and that 4:l (actually favoured by Mullins) was too strong. This discovery was both disappointing and awkward. It was disappointing, because a neutral mechanism would have been so much easier to deal with. It was awkward, because it forced us to run against the prevailing fashion and because, like Mullins, we had to answer the question how the electrogenicity of the exchange mechanism manifests itself during normal electrical activity. At least some of

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the ionic current attributed to other mechanisms had to be reassigned to the sodiumcalcium exchange. There are only two factors that control this exchange that change rapidly enough to induce substantial changes during normal electrical activity: membrane voltage and intracellular calcium. At constant voltage only internal calcium remains. The search therefore centred on current changes that reflect changes in internal calcium. One such current variation was already known to exist: the oscillatory transient inward current iTI identified by Lederer and Tsien in 1976 as the current underlying glycosideinduced arrhythmias. This is an inward current change in response to each rise in internal calcium as calcium oscillates in conditions of sodium-overload. Moreover, Tsien and his colleagues had already recently proposed Na-Ca exchange as one of several possible charge carriers [34]. The problem was that this was thought to be a pathological phenomenon, not a mechanism operating during normal beating. Yet internal calcium also rises during normal beating. There seemed to me to be no obvious reason why it should activate sodium-calcium exchange in one case but not in the other. If it does activate the exchange during normal beating then we would be looking for an inward current change that follows the calcium current, once that current had triggered internal calcium release. The problem with looking for such a current was also clear: it would be likely to be relatively small compared to the peak calcium current, and might therefore appear indistinguishable from a slower phase of calcium current inactivation. Although this meant that testing the hypothesis was going to be difficult, it was nevertheless a reassuring conclusion, for it readily explained why the exchange current had not so far been detected: it had probably always been confused with calcium current inactivation. We therefore started looking for ways in which it might nevertheless be possible to separate out the two processes. Fashion in science can lead to results being suppressed. The irony in this case is that my own laboratory had already been doing exactly that! We had, for some time, been recording the voltage dependence of calcium current availability in work on small sinus node preparations and had found that, at low levels of calcium current, the initial calcium current peak is sometimes followed by a second much slower peak, exactly as the model required. We had simply stored the records in a bottom drawer, convinced that they must, in some way, be artefactual. Once we had seen what their interpretation might be, they were resurrected and eventually published [35,36]. Soon after, Per Arlock showed that a similar phenomenon could be recorded in ferret ventricular strips [37]. He also made the crucial observation that while the first, rapid current peak was abolished by calcium current blockers such as Mn ions, the slower phase was initially resistant for so long as the ventricular strip remained contractile. The interpretation was obvious: the second, slower peak was calcium-activated and was probably sodium-calcium exchange. These conclusions were later strongly reinforced by work on isolated ferret ventricular cells [38; see also 391. At the same time, Trevor Powell and his collaborators had taken a different route to the same conclusion. They were working on the rat ventricular cell, which has an action potential shape resembling that of the atrium in most species: a rapid initial phase of repolarization is followed by a long low plateau at around -40 mV. They showed that the inward current maintaining the late plateau had all the characteristics of sodium-calcium exchange: activated by internal calcium and requiring external sodium [40,41]. Even lithium could not substitute for sodium; a sure sign that the exchange was involved since all sodium channels can also conduct lithium. This work set the scene for the final and conclusive test of the hypothesis. If the later phase of inward current was generated by sodium-calcium exchange, then it should be accompanied by a net efflux of calcium, whereas when the calcium current is activated

D. NOBLE

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0 b E ‘E Y -60

2

I

-10 0

I

1 60

I 160

I4

time/ms Fig. 4. This figure shows the Hilgemann-Noble (1987) atria1 model [9] and its success in reconstructing the net calcium fluxes during the action potential. Trace (i) shows the computed action potential, (ii) shows free internal calcium, (iii) shows contraction and (iv) shows the external calcium transient which falls when calcium enters the cell through the calcium channel and rises when calcium levels the cell via sodium-calcium exchange. The insert on the top right shows experimental traces of the action potential, contraction and external calcium.

there should be a net influx. The experimental test of this prediction was provided by Hilgemann [42] who showed that in rabbit atria1 muscle the late plateau is indeed a period of net calcium efflux. This work led to the development of the Hilgemann-Noble atria1 model, which accurately reproduced the net flux measurements (see Fig. 4). That work was the watershed, and was rapidly followed by the development of a single atria1 cell model [lo] based on voltage-clamp recordings of the exchange current in this cell [43]. Egan et al. [44] recorded exchange current tails in guinea-pig ventricular cells (these tails had previously been attributed to calcium current deactivation!) and this work led to the development of a single guinea-pig ventricular cell model [13]. Modelling the sodium-calcium exchange required not only a choice of stoichiometry; it was also necessary to describe its voltage dependance. This was also unknown at the beginning of the modelling work. We did have some clues though. The slow current changes that might be attributable to sodium-calcium exchange were typically recorded at fairly negative potentials. The slow plateau in atria1 cells, for example, starts around -35 mV. Moreover, inward current tails attributable to the exchange were easier to record at fairly negative potentials (see also [45]). Since one of the formulations proposed by Mullins gave an exponential dependance on voltage of the inward mode of the exchange, it seemed sensible to opt for this formulation. It was therefore very exciting to see the beautiful results of Kimura et al. [46] showing that, to a first approximation, the voltage dependence is exponential. I have not been able here to refer to all the work on sodium-calcium exchange that is relevant to cardiac modelling, and I have certainly not done justice to the priorities in relation to establishing the stoichiometry and kinetic properties of the exchanger. Readers

Development of mathematical models of the heart

are referred to the proceedings of the two international exchange [47,48] for further references.

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meetings on sodium-calcium

SUCCESSES AND FAILURES

The standard test of a theory in the physical sciences is to ask what it can predict beyond what was used to formulate the theory in the first place. This is not necessarily the best test of an integrative theory in physiological work. It might be, for example, that the purpose of the theory is to build it into further integrative work at a higher functional level. Even if it is incapable of further predictions at the level at which it was constructed, it would still be a valuable model provided that all the relevant features are successfully reproduced.

90% pump

block

- immediate

effect

block

- lon6-term

effect

( .IWJ

-100-1 0

90% pump

loomilliseconds

200

300

SW-

700

millisctonds600

900

Fig. 5. Reconstruction of the actions of cardiac glycosides using the Earm-Noble (1990) mode1 [lo] of the single rabbit atria1 cell. Top left: immediate effect of 90% sodium pump block is a small depolarization of the resting potential and a moderate lengthening of repolarization. Bottom left: computed internal sodium and contractions (fraction of cross-bridge formations, CR) over a time scale of 600 seconds following 90% sodium pump block. Note the massive positive inotropic effect, which turns into a negative inotropic effect as the resting contraction starts to appear. Top right: action potentials computed at the beginning (0 set continuous line), 100 set (dotted line), 2OOsec (dashed line), 250sec (dot-dashed line) and 500 set (long dashed line). Bottom right: corresponding computed contractions. Note that at 250 set there is oscillation of contraction (and hence of internal calcium). This is the transient inward current mechanism illustrated in later figures in this paper. (See ref. (511.)

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Nevertheless, it is very satisfying when a physiological model can be tested using the standard physical theory paradigm. There are, in fact, quite a large number of such tests that the present generation of cardiac cell models have been through, starting with the ability to map totally the experimental results on which iK2 model was based [29]. Glenna 501 A

Ectopic beats in atrial cell

0.

z -50.

-lOO-

0

aiw

loo0

Time (ms.1

SA node cell, INal, = 16 mM

Na-Ca exchange

current

Fig. 6. Top: computed ectopic rhythmic activity in artrial cell model during calcium overload following 90% sodium pump inhibition, as in the previous figure. Inhibition was maintained long enough for internal sodium to rise to 16.24 mM. This model is usually quiescent. The rhythmic activity computed here is therefore abnormal and is generated (as is the transient inward current and related ectopic beats in experimental situations) by osciUatory variations in internal calcium. The exact frequency and shape of action potential depend on the sensitivity assumed for the calcium-induced calcium release process, which is the variable that was changed between each of these computations. For the solid line a binding constant of 0.001 mM (i.e. 1 @I) was assumed. The short dotted line gives the result for 0.0008 mM, the dotted line (which shows only subthreshold oscillations after the initial beat) gives the result for 0.0006 mM, while the dot-dash line gives the result for 0.0015 mM. Bottom: oscillatory transient inward currents generated in the sinus node cell model following calcium overload generated by sodium pump inhibition.

Development of mathematical models of the heart

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Bett and I recently reviewed the successes and failures of the models [49]. Here, I will refer briefly to one of the major successesand to the area of greatest failure. The greatest successof the models must be the ability to reconstruct totally the inotropic and arrhythmic events following sodium pump block. Figures 5 and 6 show the relevant results using the atria1 cell model; 90% pump block was assumed. The immediate result of this is a prolongation of action potential duration as the outward pump current is reduced. This is followed by massive action potential shortening as the secondary consequences of a rise in intracellular sodium develop. As internal sodium increases the sodium gradient is reduced so weakening sodiumcalcium exchange. Less calcium is therefore moved out of the cell during each beat and more is stored for subsequent release. This produces the well-known positive inotropic effect. This effect develops with relatively little change in diastolic calcium or contraction. Eventually, the exchange is weakened to the point at which diastolic calcium rises substantially so inducing some calcium release even before excitation. This is the period during which a negative inotropic effect develops and internal calcium oscillations occur. These are then responsible for generating after-depolarizations which, if they are strong enough, trigger ectopic beating. It is impressive that all these features are so successfully reproduced since none of them was used in setting the models up. As so often happens, however, it is precisely in the area in which a model shows strength that it can be tested to reveal weakness. There are many other experimental results that one would like to reproduce that depend on successful reconstruction of the processes of calcium buffering, sequestration and release. In many of these, the present models fail and must be reformulated. Glenna Bett’s paper in this volume [50] deals with this issue. Acknowledgements-The work described in this paper is supported by the British Heart Foundation and the Medical Research Council. The computations in Figs 2-6 were done using the computer program OXSOFI’ HEART version 4.0. Information on the availability of this program can be obtained from OXSOFT Ltd, 49 Old Road, Oxford OX3 752, UK.

REFERENCES 1. B. van der Pol and J. van der Mark, The heartbeat considered as a relaxation oscillation and an electrical model of the heart. Phil. Mag. Suppl6, 763-775 (1928). 2. J. J. B. Jack. D. Noble and R. W. Tsien. Electrical Current Flow in Excitable Cells. Clarendon Press. Oxford (1975). 3. 0. Hauswirth, D. Noble and R. W. Tsien, The mechanism of oscillatory activity at low membrane potentials in cardiac Purkinie fibres. J. Physiol. 200. 255-265 (1969). 4. D. Noble, A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pacemaker potentials, J. Physiol. 160, 317-352 (1962). 5. R. E. McAllister, D. Noble and R. W. Tsien, Reconstruction of the electrical activity of cardiac Purkinie fibres, J. Physiol. 251, 1-59 (1975). 6. G. W. Beeler and H. Reuter, Reconstruction of the action potential of ventricular myocardial fibres, J. Physiol. 268, 177-210 (1977). 7. D. DiFrancesco and D.’ Noble, A model of cardiac electrical activity incorporating ionic pumps and concentration changes, Phil. Trans. R. Sot. Land. B3@7,353-398 (1985). 8. D. Noble and S. J. Noble, A model of S.A. node electrical activity using a modification of the DiFrancesco-Noble equations. Proc. Roy. Sot. B222, 295-304 (1984). 9. D. W. Hilgemann and D. Noble, Excitation-contraction coupling and extracellular calcium transients in rabbit atrium; reconstruction of basic cellular mechanisms. Proc. Roy. Sot. B230, 163-205 (1987). 10. Y. E. Earm and D. Noble, A model of the single atria1 cell: between calcium current and calcium release, Proc. Roy. Sot. 240, 83-96 (1990).

11. D. Noble, D. DiFrancesco and J. C. Denyer, Ionic mechanisms in normal and abnormal cardiac pacemaker activity, in Cellular and Neuronal Oscillators, edted by J. W. Jacklet, pp. 59-85, Dekker, New York (1989). 12. R. Wilders, H. Jongsma and A. C. G. van Ginneken, Pacemaker activity of the rabbit sinoatrial node: a comparison of mathematical models, Biophy. J. 60, 1202-1216 (1991).

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13. D. Noble, S. J. Noble, G. C. L. Bett, Y. E. Earm, W. K. Ho and I. S. So, The role of sodium-calcium exchange during the cardiac action potential Ann. NY Acnd. Sci. 639, 334-353 (1991). 14. R. L. Winslow, A. L. Kimball, A. Varghese and D. Noble, Simulating cardiac sinus and atria1 network dynamics on the Connection Machine, Physica D64, 281-298 (1993). 15. A. E. Hall, 0. F. Hutter and D. Noble, Current-voltage relations of Purkinje fibres in sodium-deficient solutions. J. Physiol. 166, 225-240 (1963). 16. 0. F. Hutter and D. Noble, Rectifying properties of heart muscle. Nature Land. 188, 495 (1960). 17. E. E. Carmeliet, Chloride ions and the membrane potential of Purkinje fibres. J. Physiol. 156, 375-388 (1961). 18. H. Reuter, The dependence of slow inward current in Purkinje fibres on the extracellular calcium concentration, J. Physiol. 192, 479-492 (1967). 19. S. Weidmann, Effects of current flow on the membrane potential of cardiac muscle, J. Physiol. 115, 227-236 (1951). 20. D. Noble and R. W. Tsien, The kinetics and rectifier properties of the slow potassium current in cardiac Purkinje fibres, J. Physiol. 195, 185-214 (1968). 21. D. Noble and R. W. Tsien, Outward membrane currents activated in the plateau range of potentials in cardiac Purkinje fibres, J. Physiol. 200, 205-231 (1969). 22. D. Noble, and R. W. Tsien, Reconstruction of the reoolarization urocess in cardiac Purkinie fibres based on voltage clamp measurements’of the membrane current: J. Physiol. iO0, 233-254 (1969). ” 23. 0. Hauswirth, Computer-Rekonstructionen der Effekte von Polarisation-stromen and Pharmaka auf Schrittmacher- und Aktionspotentiale von Herzmuskelfasern, Habilitltsionsschrift, University of Heidelberg (1971). 24. D. Noble, The surprising heart: a review of recent progress in cardiac electrophysiology. J. Physiol. 353, l-50 (1984). 25. D. E. Attwell, D. A. Eisner and I. Cohen, Voltage clamp and tracer flux data: effects of a restricted extracellular space, Q. Rev. Biophys 12, 213-261 (1979). 26. D. DiFrancesco and D. Noble, The time course of potassium current following potassium accumulation in frog atrium: analytical solutions using a linear approximation, J. Physiol. 306, 151-173 (1980). 27. D. DiFrancesco, A new interpretation of the pacemaker current, iK*, in Purkinje fibres, J. Physiol. 314, 359-376 (1981). 28. D. DiFrancesco, A study of the ionic nature of the pacemaker current in calf Purkinje fibres, 1. Physiol. 314, 377-393 (1981). 29. D. DiFrancesco and D. Noble, Implications of the re-interpretation of i ~2 for the modelling of the electrical activity of pacemaker tissues in the heart, in Cardiac Rate and Rhythm, edited by L. N. Bouman and H. J. Jongsma, pp. 93-128. Nijhoff, The Hague (1982). 30. H. Reuter and N. Seitz, The dependence of calcium efflux from cardiac muscle on temperature and external ion composition, J. Physiol. 195, 451-470 (1968). 31. L. J. Mullins, A mechanism for Na/Ca transport, J. Gen. Physiol. 70, 681-695 (1977). 32. L. J. Mullins, Zen Transport in the Heart. Raven, New York (1981). 33. R. A. Chapman and D. Noble, Sodium-calcium exchange in the heart, in Sodium-Calcium Exchange, edited by J. Allen, D. Noble and H. Reuter. Oxford University Press, Oxford (1989). 34. R. S. Kass, W. J. Lederer, R. W. Tsien and R. Weingart, Role of calcium ions in transient inward current and after contractions induced by strophanthidin in cardiac Purkinje fibres, J. Physiol. 281, 187-208 (1978). 35. H. F. Brown, J. Kimura, D. Noble, S. J. Noble and A. I. Taupignon, Mechanisms underlying the slow inward current, i,, , in the rabbit sino-atria1 node investigated by voltage clamp and computer simulation, Proc. Roy. Sot. B222, 305-328 (1984).

36. H. F. Brown, D. Noble, S. J. Noble and A. I. Taunpignon, Relationship between the transient inward current and slow inward current in the sino-atria1 node of the rabbit, J. Physiol. 370, 299-315 (1986). 37. P. Arlock and D. Noble, Two components of ‘second inward current’ in ferret papillary muscle. J. Physiol. 369, 88P (1985). 38. M. R. Boyett, M. S. Kirby and C. H. Orchard, Rapid regulation of the ‘second inward current’ by intracellular calcium in isolated rat and ferret ventricular myocytes, J. Physiol. 407, 77-102 (1988). 39. J. Simurda, M. Simurdova, P. Braveny and J. Sumbera, A contraction-related component of slow inward current in dog ventricular muscle and its relation to Na-Ca exchange, J. Physiol. 456, 49-70 (1992). 40. M. R. Mitchell, T. Powell, D. A. Terrar and V. W. Twist, The effects of ryanodine, EGTA and low-sodium on action potentials in rat and guinea-pig ventricular myocytes: evidence for two inward currents during the plateau. Brit. J. Pharmacol. 81, 543-550 (1984). 41. M. R. Mitchell, T. Powell, D. A. Terrar and V. W. Twist, Calcium-activated inward current and contraction in rat and guinea-pig ventricular myocytes. J. Physiol. 391, 545-560 (1987). 42. D. W. Hilgemann, Extracellular calcium transients at single excitations in rabbit atrium measured with tetramethylmurexide, J. Gen. Physiol. 87, 707-735 (1986). 43. Y. E. Earm. W. K. Ho and I. S. So, Inward current generated bv sodium-calcium exchanae durine the action potential in single atria1 cells of the rabbit, Proc. Roy- Sot. B240; 61-81 (1990). 44. T. M. Egan, D. Noble, S. J. Noble, T. Powell, A. J. Spindler and V. W. Twist, Sodium-calcium exchange during the action potential in guinea-pig ventricular cells. J. Physiof. 411, 639-661 (1989). 45. J. R. Hume and A. Uehara, ‘Creep currents’ in single frog atria1 cells may be generated by electrogenic Na/Ca exchange, J. Gen. Physiol. 87, 857-844 (1986).

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46. J. Kimura, S. Miyamae and A. Noma, Identification of sodium-calcium exchange current in single ventricular cells of guinea-pig, J. Pkysiol. 384, 199-222 (1987). 47. T. J. A. Allen, D. Noble and H. Reuter, Sodium-Calcium Exchange. Oxford University Press, Oxford (1989). 48. M. P. Blaustein, R. DiPolo and J. P. Reeves (editors), Proc. 2nd Int. Mg. on Sodium-Calcium Exchange, Ann. NY Acad. Sci. 639, (1991). 49. D. Noble and G. C. L. Bett, Reconstructing the heart: a challenge for integrative physiology, Cardiovas. Res. 27, 1701-1712 (1993). 50. G. C. L. Bett, Calcium handling in isolated guinea-pig ventricular myocytes, Chaos, Solitons & Fracfuls, 5(3/4), 335-346 (1995). 51. D. Noble, Ionic mechanisms determining the timing of ventricular repolarization: significance for cardiac arrhythmias. Ann. NY Acad. Sci. 644, l-22 (1991).