Penelope J. Noble

Mathematical models of the electrical action potential of Purkinje fibre cells

Early development of ionic models for cardiac myocytes, from the pioneering modification of the Hodgkin–Huxley giant squid axon model by Noble to the iconic DiFrancesco–Noble model integrating voltage-gated ionic currents, ion pumps and exchangers, Ca2+ sequestration and Ca2+-induced Ca2+ release, provided a general description for a mammalian Purkinje fibre (PF) and the framework for modern cardiac models. In the past two decades, development has focused on tissue-specific models with an emphasis on the sino-atrial (SA) node, atria and ventricles, while the PFs have largely been neglected. However, achieving the ultimate goal of creating a virtual human heart will require detailed models of all distinctive regions of the cardiac conduction system, including the PFs, which play an important role in conducting cardiac excitation and ensuring the synchronized timing and sequencing of ventricular contraction. In this paper, we present details of our newly developed model for the human PF cell including validation against experimental data. Ionic mechanisms underlying the heterogeneity between the PF and ventricular action potentials in humans and other species are analysed. The newly developed PF cell model adds a new member to the family of human cardiac cell models developed previously for the SA node, atrial and ventricular cells, which can be incorporated into an anatomical model of the human heart with details of its electrophysiological heterogeneity and anatomical complexity.

How the Hodgkin–Huxley equations inspired the Cardiac Physiome Project

Abstract  Early modelling of cardiac cells (1960–1980) was based on extensions of the Hodgkin–Huxley nerve axon equations with additional channels incorporated, but after 1980 it became clear that processes other than ion channel gating were also critical in generating electrical activity. This article reviews the development of models representing almost all cell types in the heart, many different species, and the software tools that have been created to facilitate the cardiac Physiome Project.

Mechanical interaction of heterogeneous cardiac muscle segments in silico: Effects on Ca2+ handling and action potential

Effects of cardiac mechanical heterogeneity on the electrical function of the heart are difficult to assess experimentally, yet they pose a serious (patho-)physiological challenge. Here, we present an in silico study of the effects of mechanical heterogeneity on action potential duration (APD) in mechanically interacting muscle regions and consequent effects on the dispersion of repolarization, a well-established determinant of cardiac arrhythmogenesis. Using a novel mathematical description of ventricular electromechanical activity (virtual muscle), we first assessed how differences in intrinsic contractile properties affect the electrical behavior of cardiac muscle representations. In spite of identical electrophysiological model descriptions in virtual muscle samples, faster muscle models show shorter APD than their slower counterparts. This is a consequence of Ca2+-mediated feedback from mechanical to electrical activity in the individual muscle models. This mechano-electric feedback (MEF) is, of course, significantly more complex in native cardiac tissue, as the heterogeneous muscle elements interact both mechanically and electrically. Cardiac mechanical heterogeneity, in its most reduced form, can be represented by a duplex consisting of two mechanically interacting muscle segments. Our in silico model of heterogeneous myocardium therefore consists of two individual virtual muscles that are mechanically interconnected in-series to form a virtual heterogeneous duplex. During isometric contraction of the duplex (i.e. at constant external length), internal mechanical interactions affect Ca2+ handling and APD of muscle elements, resulting in an increased dispersion of repolarization beyond the intrinsic APD differences. Duplex electromechanical activity is strongly affected by the activation sequence of its elements. Late activation of the faster (subepicardial type) duplex element, postponed by time-lags that correspond to normal transmural activation delays, optimizes duplex contractility and smoothes out intrinsic APD differences, thereby reducing dispersion in repolarization. This smoothing effect is not observed upon delayed activation of the slower (subendocardial type) duplex element. In both settings, changes in repolarization timing follow a nonlinear dependence of APD on activation delay. Furthermore, asynchronous activation of identical elements in a homogeneous duplex causes an impairment of contractile function and increases dispersion of repolarization. This suggests that the normal electrical activation sequence in the heart requires matching mechanical and electrical heterogeneity for optimal cardiac performance. On the subcellular level, our results suggest that mechanical modulation of Ca2+ handling is a key mechanism of MEF in heterogeneous myocardium, which contributes to the matching of local mechanical and/or electrical activity to global hemodynamic demand.

Facilitation of the L-type calcium current in rabbit sino-atrial cells: effect on cardiac automaticity

OBJECTIVE The L-type Ca(2+) current (I(Ca,L)) contributes to the generation and modulation of the pacemaker action potential (AP). We investigated facilitation of I(Ca,L) in sino-atrial cells. METHODS Facilitation was studied in regularly-beating cells isolated enzymatically from young albino rabbits (0.8-1 kg). We used the whole-cell patch-clamp technique to vary the frequency of the test depolarizations evoked at -10 mV or the conditioning diastolic membrane potential prior to the test pulse. RESULTS High frequencies (range 0.2-3.5 Hz) slowed the decay kinetics of I(Ca,L) evoked from a holding potential (HP) of -80 mV in 68% of cells resulting in a larger Ca(2+) influx during the test pulse. The amount of facilitation increased progressively between 0.2 and 3.0 Hz. When the frequency was changed from 0.1 to 1 Hz, the averaged increase in the time integral of I(Ca,L) was 27+/-7% (n=22). Application of conditioning voltages between -80 and -50 mV induced similar facilitation of I(Ca,L) in 73% of cells. The maximal increase of Ca(2+) entry occurred between -60 and -50 mV, and was on average 38+/-14% for conditioning prepulses of 5 s in duration (n=15). Numerical simulations of the pacemaker activity showed that facilitation of I(Ca,L) promotes stability of sino-atrial rate by enhancing Ca(2+) entry, thus establishing a negative feedback control against excessive heart rate slowing. CONCLUSION Facilitation of I(Ca,L) is present in rabbit sino-atrial cells. The underlying mechanism reflects modulation of I(Ca,L) decay kinetics by diastolic membrane potential and frequency of depolarization. This phenomenon may provide an important regulatory mechanism of sino-atrial automaticity.

Competing Oscillators in Cardiac Pacemaking: Historical Background

Interaction between a membrane oscillator generated by voltage-dependent ion channels and an intracellular calcium signal oscillator was present in the earliest models (1984 to 1985) using representations of the sarcoplasmic reticulum. Oscillatory release of calcium is inherent in the calcium-induced calcium release process. Those historical results fully support the synthesis proposed in the articles in this review series. The oscillator mechanisms do not primarily compete with each; they entrain each other. However, there is some asymmetry: the membrane oscillator can continue indefinitely in the absence of the calcium oscillator. The reverse seems to be true only in pathological conditions. Studies from tissue-level work and on the development of the heart also provide valuable insights into the integrative action of the cardiac pacemaker.

Ca2+-induced delayed afterdepolarizations are triggered by dyadic subspace Ca2+ affirming that increasing SERCA reduces aftercontractions

Ca(2+)-induced delayed afterdepolarizations (DADs) are depolarizations that occur after full repolarization. They have been observed across multiple species and cell types. Experimental results have indicated that the main cause of DADs is Ca(2+) overload. The main hypothesis as to their initiation has been Ca(2+) overflow from the overloaded sarcoplasmic reticulum (SR). Our results using 37 previously published mathematical models provide evidence that Ca(2+)-induced DADs are initiated by the same mechanism as Ca(2+)-induced Ca(2+) release, i.e., the modulation of the opening of ryanodine receptors (RyR) by Ca(2+) in the dyadic subspace; an SR overflow mechanism was not necessary for the induction of DADs in any of the models. The SR Ca(2+) level is better viewed as a modulator of the appearance of DADs and the magnitude of Ca(2+) release. The threshold for the total Ca(2+) level within the cell (not only the SR) at which Ca(2+) oscillations arise in the models is close to their baseline level (∼1- to 3-fold). It is most sensitive to changes in the maximum sarco(endo)plasmic reticulum Ca(2+)-ATPase (SERCA) pump rate (directly proportional), the opening probability of RyRs, and the Ca(2+) diffusion rate from the dyadic subspace into the cytosol (both indirectly proportional), indicating that the appearance of DADs is multifactorial. This shift in emphasis away from SR overload as the trigger for DADs toward a multifactorial analysis could explain why SERCA overexpression has been shown to suppress DADs (while increasing contractility) and why DADs appear during heart failure (at low SR Ca(2+) levels).

Comparative study of rabbit sino-atrial node cell models

Abstract This paper compares recent mathematical models of rabbit sino-atrial node pacemaker cell activity [Cellular and Neuronal Oscillators, Dekker, New York, 1989, p. 59; Am. J. Physiol. 266 (1994) C832; J. Theor. Biol. 181 (1996) 245; Am. J. Physiol. 279 (2000) H397] and evaluates them with the perspective of developing detailed multicellular models of the right atrium. (i) All evaluated models reproduce control action potential shapes, which have been recorded experimentally (although one of them (Dokos et al., loc. cit.) shows an unusually long spontaneous diastolic depolarisation phase, probably more compatible with room-temperature rather than body-temperature conditions). This is achieved on the basis of implementing sarcolemmal ion fluxes as a function of (computed) internal and (computed/fixed) external ion concentrations. Also, all models address, to some extent, intracellular calcium handling processes. (ii) Application of the various models to simulated experimental interventions (such as block of selected ion currents) reveals a wide range of responses (partially outside patho-physiologically plausible ranges) and inconsistencies between simulated and experimental data, thus defining the need for further model improvement. (iii) The heterogeneity of cell parameters within the sino-atrial node is addressed only by one of the models (Zhang et al., loc. cit.). (iv) Computation time differs greatly between the various models, with a ratio of 1:6 between the slowest and the fastest models. We conclude that, out of the currently available set, the Zhang et al. (loc. cit.) model is best suited for application to multicellular modelling of the right atrium.

Vagal control of heart rate is modulated by extracellular potassium

Heart rate (HR) recovery from heavy exercise is associated with a shift in cardiac sympatho-vagal balance and a transient hypokalaemia. Since changes in extracellular potassium ([K+]0) affect membrane currents in the sino-atrial node, in particular the acetylcholine-activated potassium current (I(K,ACh)), the hyperpolarization-activated current (I(f)) and the L-type calcium current (I(Ca,L)), we investigated whether mimicking [K+]0 concentrations seen during and immediately after exercise could directly modulate the HR response to vagal nerve stimulation (VNS) in the isolated guinea-pig atria preparation pre-stimulated with noradrenaline (NA, 1 microM). Lowering [K+]0 from 4 to 3 mM significantly enhanced the HR response to VNS (5 Hz, 5 V, 30 s, deltaHR 84.5 +/- 14.1 bpm and 119.3 +/- 18.2 bpm, respectively). Increasing [K+]0 to 8 or 10 mM significantly decreased the drop in HR with VNS in comparison to the response to 3 mM K+ Tyrode (deltaHR 56.4 +/- 9.1 bpm and 52.1 +/- 8.7 bpm, respectively). These results could be simulated using the OXSOFT heart sino-atrial node computer model by activating I(K,ACh) during changes in [K+]0. However, changing [K+]0 in the model had no significant effect on the decrease in beating frequency brought about by decreasing I(f) or I(Ca,L). We conclude that the magnitude of the decrease in HR with VNS is enhanced in low [K +]0 and reduced in high [K+]0. The increased efficacy of cardiac vagal activation in low [K+]0 might therefore facilitate the drop in HR after heavy exercise where there is a transient hypokalaemia. Modelling suggests this result may be explained by the effects of changes in [K+]0 on the current-voltage relationship for I(K,ACh).

Resistance of cardiac cells to NCX knockout : A model study

Abstract:  The effects of NCX knockout were determined in a variety of cardiac cell models. Those of the mouse and rat ventricles, and of atrial cells in other species behave similarly to the experiments on mouse ventricle showing only small effects, and considerable tolerance of NCX knockout. Models of ventricular cells with high action potential plateaus, however, are more sensitive and require compensatory mechanisms to adjust other conductance parameters to enable the cells to resist NCX knockout. The effects can therefore be expected to be species‐specific, and it is not possible to extrapolate the mouse results to those that may occur in the Guinea pig or human.

Ascending aortic stenosis selectively increases action potential‐induced Ca2+ influx in epicardial myocytes of the rat left ventricle

A decrease of the transient outward potassium current (Ito) has been observed in cardiac hypertrophy and contributes to the altered shape of the action potential (AP) of hypertrophied ventricular myocytes. Since the shape and duration of the ventricular AP are important determinants of the Ca2+ influx during the AP (QCa), we investigated the effect of ascending aortic stenosis (AS) on QCa in endo‐ and epicardial myocytes of the left ventricular free wall using the AP voltage‐clamp technique. In sham‐operated animals, QCa was significantly larger in endocardial compared to epicardial myocytes (803 ± 65 fC pF−1, n= 27 vs. 167 ± 32 fC pF−1, n= 38, P < 0.001). Ascending aortic stenosis significantly increased QCa in epicardial myocytes (368 ± 54 fC pF−1, n= 42, P < 0.05), but did not alter QCa in endocardial myocytes (696 ± 65 fC pF−1, n= 26). Peak and current–voltage relation of the AP‐induced Ca2+ current were unaffected by AS. However, the time course of the current–voltage relation was significantly prolonged in epicardial myocytes of AS animals. Model calculations revealed that the increase in QCa can be ascribed to a prolonged opening of the activation gate, whereas an increase in inactivation prevents an excessive increase in QCa. In conclusion, AS significantly increased AP‐induced Ca2+ influx in epicardial but not in endocardial myocytes of the rat left ventricle.