Abstract
Abnormal cardiac rhythms (cardiac arrhythmias) often display complex changes over time that can have a random or haphazard appearance. Mathematically, these changes can on occasion be identified with bifurcations in difference or differential equation models of the arrhythmias. One source for the variability of these rhythms is the fluctuating environment. However, in the neighborhood of bifurcation points, the fluctuations induced by the stochastic opening and closing of individual ion channels in the cell membrane, which results in membrane noise, may lead to randomness in the observed dynamics. To illustrate this, we consider the effects of stochastic properties of ion channels on the resetting of pacemaker oscillations and on the generation of early afterdepolarizations. The comparison of the statistical properties of long records showing arrhythmias with the predictions from theoretical models should help in the identification of different mechanisms underlying cardiac arrhythmias.
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Y. Asano, J. M. Davidenko, W. T. Baxter, R. A. Gray and J. Jalife, Optical mapping of drug-induced polymorphic arrhythmias and torsade de pointes in the isolated rabbit heart, J. Am. Coll. Cardiol. 29: 831–842 (1997).
M.R. Boyett, H. Honjo and I. Kodama, The sinoatrial node, a heterogeneous pacemaker structure, Cardiovasc. Res. 47: 658–687 (2000).
D. Cai, R. L. Winslow and D. Noble, Effects of gap junction conductance on dynamics of sinoatrial node cells: two-cell and large-scale network models, IEEE Trans. Biomed. Eng. 41: 217–231 (1994).
T. R. Chay and Y. S. Lee, Impulse responses of automaticity in the Purkinje fiber, Biophys. J. 45: 841–849 (1984).
T. R. Chay and Y. S. Lee, Phase resetting and bifurcation in the ventricular myocardium, Biophys. J. 47: 641–651 (1985).
C. E. Clancy and Y. Rudy, Na+ channel mutation that causes both Brugada and long-QT syndrome phenotypes: a simulation study of mechanism, Circulation 105: 1208–1213 (2002).
J. R. Clay, Monte Carlo simulation of membrane noise: an analysis of fluctuations in graded excitation of nerve membrane, J. Theor. Biol. 64: 671–680 (1977).
J. R. Clay and L. J. DeFelice, Relationship between membrane excitability and single channel open-close kinetics, Biophys. J. 42: 151–157 (1983).
J. R. Clay and R. L. DeHaan, Fluctuations in interbeat interval in rhythmic heart-cell clusters. Role of membrane voltage noise. Biophys. J. 28: 377–389 (1979).
A. Coulombe, E. Coraboeuf and E. Deroubaix, Computer simulation of acidosis-induced abnormal repolarization and repetitive activity in dog Purkinje fibers, J. Physiol. (Paris) 76: 107–112 (1980).
M. Courtemanche, L. Glass, M. D. Rosengarten and A. L. Goldberger, Beyond pure parasystole: promises and problems in modelling complex arrhythmias, Am. J. Physiol. 257: H693–H706 (1989).
P. F. Cranefield and R. S. Aronson, Cardiac Arrhythmias: The Role of Triggered Activity and Other Mechanisms (Futura Publishing Co., Mount Kisco, NY, 1988).
E. J. Doedel, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations (Department of Computer Science, Concordia University, Montreal, Canada, 1997). http://cmvl.cs.concordia.ca/auto
D. S. Echt, P. R. Liebson, L. B. Mitchell, et al., Mortality and morbidity in patients receiving encainide, flecainide, or placebo. The Cardiac Arrhythmia Suppression Trial, N. Engl. J. Med. 324: 781–788 (1991).
N. El-Sherif and G. Turitto, Torsade de pointes, Curr. Opin. Cardiol. 18: 6–13 (2003).
R. Fischmeister and G. Vassort, The electrogenic Na-Ca exchange and the cardiac electrical activity. I-Simulation on Purkinje fibre action potential, J. Physiol. (Paris) 77: 705–709 (1981).
R. FitzHugh, Thresholds and plateaus in the Hodgkin–Huxley nerve equations, J. Gen. Physiol. 43: 867–896 (1960).
T. Gedeon and L. Glass, Continuity of resetting curves for FitzHugh-Nagumo equations on the circle. In Fields Institute Communications: Differential Equations with Applications Biology, R. Shigui, G. S. K. Wolkowicz and J. Wu (eds.), pp. 225–236 (1999).
W. J. Gibb, M. B. Wagner and M. D. Lesh, Effects of simulated potassium blockade on the dynamics of triggered cardiac activity, J. Theor. Biol. 168: 245–257 (1994).
D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81: 2340–2361 (1977).
L. Glass, A. L. Goldberger, M. Courtemanche and A. Shrier, Nonlinear dynamics, chaos and complex cardiac arrhythmias, Proc. Roy. Soc. (London) A 413, 9–26 (1987).
L. Glass, M. R. Guevara, J. Belair and A. Shrier, Global bifurcations of a periodically forced biological oscillator, Phys. Rev. A 29: 1348–1357 (1984).
J. Guckenheimer, Isochrons and phaseless sets, J. Math. Biol. 1: 259–273 (1975).
M. R. Guevara and T. J. Lewis, A minimal single-channel model for the regularity of beating in the sinoatrial node, Chaos 5: 174–183 (1995).
M. R. Guevara, L. Glass and A. Shrier, Phase locking, period-doubling bifurcations and irregular dynamics in periodically stimulated cardiac cells, Science 214: 1350–1353 (1981).
M. R. Guevara, A. Shrier and L. Glass, Phase resetting of spontaneously beating embryonic ventricular heart cell aggregates, Am. J. Physiol. 251: H1298–H1305 (1986).
J. Han and G. K. Moe, Nonuniform recovery of excitability in ventricular muscle, Circ. Res. 14: 44–60 (1964).
H. Henry and W.-J. Rappel, The role of M cells and the long QT syndrome in cardiac arrhythmias: simulation studies of reentrant excitations using a detailed electrophysiological model, Chaos 14: 172–182 (2004).
A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117: 500–544 (1952).
D. J. Huelsing, K. W. Spitzer and A. E. Pollard, Electrotonic suppression of early afterdepolarizations in isolated rabbit Purkinje myocytes, Am. J. Physiol. 279: H250–H259 (2000).
R. Huffaker, S. T. Lamp, J. N. Weiss and B. Kogan, Intracellular calcium cycling, early afterdepolarizations and reentry in simulated long QT syndrome, Heart Rhythm 4: 441–448 (2004).
R. L. Hughson, A. Maillet, G. Dureau, Y. Yamamoto and C. Gharib, Spectral analysis of blood pressure variability in heart transplant patients, Hypertension 25: 643–650 (1995).
H. V. Huikuri, A. Castellanos and R. J. Myerburg, Sudden death due to cardiac arrhythmias, N. Eng. J. Med. 345: 1473–1482 (2001).
P. Ch. Ivanov, L. A. N. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, H. E. Stanley and Z. R. Struzik, From 1/f noise to multifractal cascades in heartbeat dynamics, Chaos 11: 641–652 (2001).
J. Jalife and G. K. Moe, A biologic model of parasystole, Am. J. Cardiol. 43: 761–772 (1979).
J. Jalife, M. Delmar, J. M. Davidenko and J. M. B. Anumonwo, Basic Cardiac Electrophysiology for the Clinician (Futura, Armonk, NY, 1999).
R. E. Kleiger, J. P. Miller, J. T. Bigger Jr. and A. J. Moss, Decreased heart rate variability and its association with increased mortality after acute myocardial infarction, Am. J. Cardiol. 59: 256–262 (1987).
K. Kotani, Z. R. Struzik, K. Takamasu, et al., Model for complex heart rate dynamics in health and diseases, Phys. Rev. E 72: 041904 (2005).
T. Krogh-Madsen, L. Glass, E. J. Doedel and M. R. Guevara, Apparent discontinuities in the phase-resetting response of cardiac pacemakers, J. Theor. Biol. 230: 499–519 (2004).
T. Krogh-Madsen, P. Schaffer, A. D. Skriver, L. K. Taylor, B. Pelzmann, B. Koidl and M. R. Guevara, An ionic model for rhythmic activity in small clusters of embryonic chick ventricular cells, Am. J. Physiol. 289: H398–H413 (2005).
T. Kuusela, T. Shepherd and J. Hietarinta, Stochastic model for heart-rate fluctuations, Phys. Rev. E 67: 061904 (2003).
R. Langendorf, A. Pick and M. Winternitz, Mechanisms of intermittent ventricular bigeminy. I. Appearance of ectopic beats dependent upon length of the ventricular cycle, the “rule of bigeminy,” Circulation 11: 422–430 (1955).
K. R. Laurita, S. D. Girouard and D. S. Rosenbaum, Modulation of ventricular repolarization by a premature stimulus. Role of epicardial dispersion of repolarization kinetics demonstrated by optical mapping of the intact guinea pig heart, Circ. Res. 79: 493–503 (1996).
H. Lecar and R. Nossal, Theory of threshold fluctuations in nerves. I. Relationships between electrical noise and fluctuations in axon firing, Biophys. J. 11: 1048–1067 (1971).
H. Lecar and R. Nossal, Theory of threshold fluctuations in nerves. II. Analysis of various sources of membrane noise, Biophys. J. 11: 1068–1084 (1971).
C. Lerma, C. F. Lee, L. Glass and A. L. Goldberger, The rule of bigeminy revisited: analysis in sudden cardiac death syndrome, J. Electrocardiol., accepted (2006).
M. D. Lesh, M. Pring and J. F. Spear, Cellular uncoupling can unmask dispersion of action potential duration in ventricular myocardium. A computer modeling study, Circ. Res. 65: 1426–1440 (1989).
G.-R. Li, J. Feng, L. Yue and M. Carrier, Transmural heterogeneity of action potentials and I1 in myocytes isolated from the human right ventricle, Am. J. Physiol. 275: H369–H377 (1998).
T. Liu, B.-R. Choi, M.-D. Drici and G. Salama, Sex modulates the arrhythmogenic substrate in prepubertal rabbit hearts with long QT 2, J. Cardiovasc. Electrophysiol. 16: 516–524 (2005).
C.-H. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential. II. Afterdepolarizations, triggered activity, and potentiation, Circ. Res. 74: 1097–1113 (1994).
E. M. MacLachlan, An analysis of the release of acetylcholine from preganglionic nerve terminals, J. Physiol. 245: 447–466 (1975).
M. Monserrat, J. Saiz, J. M. Ferrero Jr., J. M. Ferrero and N. V. Thakor, Ectopic activity in ventricular cells induced by early afterdepolarizations developed in Purkinje cells, Ann. Biomed. Eng. 28: 1343–1351 (2000).
C. Napolitano, S. G. Priori, P. J. Schwartz, et al., Genetic testing in the long QT syndrome: development and validation of an efficient approach to genotyping in clinical practice, J. Am. Med. Assoc. 294: 2975–2980 (2005).
S. Nattel, New ideas about atrial fibrillation 50 years on, Nature 415: 219–226 (2002).
D. Noble, A modification of the Hodgkin–Huxley equations applicable to Purkinje fibre action and pace-maker potentials, J. Physiol. 160: 317–352 (1962).
C. Nordin and Z. Ming, Computer model of current-induced early afterdepolarizations in guinea pig ventricular myocytes, Am. J. Physiol. 268: H2440–H2459 (1995).
O. Odemuyiwa, M. Malik, T. Farrell, Y. Bashir, J. Poloniecki and J. Camm, Comparison of the predictive characteristics of heart rate variability index and left ventricular ejection fraction for all-cause mortality, arrhythmic events and sudden death after acute myocardial infarction, Am. J. Cardiol. 68: 434–439 (1991).
H. I. Oei, A. C. G. van Ginneken, H. J. Jongsma and L. N. Bouman, Mechanisms of impulse generation in isolated cells from the rabbit sinoatrial node, J. Mol. Cell. Cardiol. 21: 1137–1149 (1989).
J. M. Pastore, S. D. Girouard, K. R. Laurita, F. G. Akar and D. S. Rosenbaum, Mechanism linking T-wave alternans to the genesis of cardiac fibrillation, Circulation 99: 1385–1394 (1999).
M. Rosenblum and J. Kurths, A model of neural control of the heart rate, Physica A 215: 439–450 (1995).
J. Saiz, J. M. Ferrero Jr., M. Monserrat, J. M. Ferrero and N. V. Thakor, Influence of electrical coupling on early afterdepolarizations in ventricular myocytes, IEEE Trans. Biomed. Eng. 46: 138–147 (1999).
K. J. Sampson and C. S. Henriquez, Electrotonic influences on action potential duration dispersion in small hearts: a simulation study, Am. J. Physiol. 289: H350–H360 (2005).
I. Schafferhofer-Steltzer, E. Hofer, D. J. Huelsing, S. P. Bishop, and A. E. Pollard, Contributions of Purkinje-myocardial coupling to suppression and facilitation of early afterdepolarization-induced triggered activity, IEEE Trans. Biomed. Eng. 52: 1522–1531 (2005).
V. Schulte-Frohlinde, Y. Ashkenazy, P. Ch. Ivanov, L. Glass, A. L. Goldberger and H. E. Stanley, Noise effects on the complex patterns of abnormal heartbeats, Phys. Rev. Lett. 87: 068104 (2001).
V. Schulte-Frohlinde, Y. Ashkenazy, A. L. Goldberger, P. Ch. Ivanov, M. Costa, A. Morley-Davies, H. E. Stanley and L. Glass, Complex patterns of abnormal heartbeats, Phys. Rev. E 66: 031901 (2002).
V. Schulte-Frohlinde, Y. Ashkenazy, A. L. Goldberger, P. Ch. Ivanov, M. Costa, A. Morley-Davies, H. E. Stanley, L. Glass, Heartprints: a Dynamical Portrait of Cardiac Arrhythmia. http://www.physionet.org/physiotools/heartprints/.
E. Skaugen, Firing behaviour in stochastic nerve membrane models with different pore densities, Acta Physiol. Scand. 108: 49–60 (1980).
E. Skaugen and L. Walløe, Firing behaviour in a stochastic nerve membrane model based upon the Hodgkin–Huxley equations, Acta Physiol. Scand. 107: 343–363 (1979).
R. Soma, D. Nozaki, S. Kwak and Y. Yamamoto, 1/f noise outperforms white noise in sensitizing baroreflex function in the human brain, Phys. Rev. Lett. 91: 078101 (2003).
W. G. Stevenson, Catheter ablation of monomorphic ventricular tachycardia, Curr. Opin. Cardiol. 20: 42–47 (2005).
A. J. Tanskanen, J. L. Greenstein, B. O'Rourke and R. L. Winslow, The role of stochastic and modal gating of cardiac L-type Ca2+ channels on early after-depolarizations, Biophys. J. 88: 85–95 (2005).
The Sudden Cardiac Death Holter Database. http://www.physionet.org/physiobank/database/sddb/.
A. A. Verveen and H. E. Derksen, Fluctuation phenomena in nerve membrane, Proc. IEEE 56: 906–916 (1968).
A. Vinet and F. A. Roberge, A model study of stability and oscillations in the myocardial cell membrane, J. Theor. Biol. 147: 377–412 (1990).
P. C. Viswanathan and Y. Rudy, Pause induced early afterdepolarizations in the long QT syndrome: a simulation study, Cardiovasc. Res. 42: 530–542 (1999).
P. C. Viswanathan and Y. Rudy, Cellular arrhythmogenic effects of congenital and acquired long-QT syndrome in the heterogeneous myocardium, Circulation 101: 1192–1198 (2000).
P. G. A. Volders, K. R. Sipido, M. A. Vos, A. Kulcsár, S. C. Verduyn and H. J. J. Wellens, Cellular basis of biventricular hypertrophy and arrhythmogenesis in dogs with chronic complete atrioventricular block and acquired torsade de pointes, Circulation 98: 1136–1147 (1998).
M. B. Wagner, W. J. Gibb and M. D. Lesh, A model study of propagation of early afterdepolarizations, IEEE Trans. Biomed. Eng. 42: 991–998 (1995).
T. Watanabe, L. M. Delbridge, J. O. Bustamante and T. F. McDonald, Heterogeneity of the action potential in isolated rat ventricular myocytes and tissue, Circ. Res. 52: 280–290 (1983).
T. Watanabe, P. M. Rautaharju and T. F. McDonald, Ventricular action potentials, ventricular extracellular potentials and the ECG of guinea pig, Circ. Res. 57: 362–373 (1985).
H. J. J. Wellens, A. Vermeulen and D. Durrer, Ventricular fibrillation occurring on arousal from sleep by auditory stimuli, Circulation 46: 661–665 (1972).
R. Wilders. From Single Channel Kinetics to Regular Beating. A Model Study of Cardiac Pacemaker Activity. Doctoral Thesis (University of Amsterdam, Amsterdam, 1993).
R. Wilders and H. J. Jongsma, Beating irregularity of single pacemaker cells isolated from the rabbit sinoatrial node, Biophys. J. 65: 2601–2613 (1993).
A. T. Winfree, When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias (Princeton University Press, Princeton, 1987).
A. T. Winfree, The Geometry of Biological Time, 2nd ed. (Springer-Verlag, New York, 2001).
Y. Yamamoto, Y. Nakamura, H. Sato, M. Yamamoto, K. Kato and R. L. Hughson, On the fractal nature of heart rate variability in humans: effects of vagal blockade, Am. J. Physiol. 269: R830–R837 (1995).
M. Zaniboni, A. E. Pollard, L. Yang and K. W. Spitzer, Beat-to-beat repolarization variability in ventricular myocytes and its suppression by electrical coupling, Am. J. Physiol. 278: H677–H687 (2000).
J. Zeng and Y. Rudy, Early afterdepolarizations in cardiac myocytes: mechanism and rate dependence, Biophys. J. 68: 949–964 (1995).
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Lerma, C., Krogh-Madsen, T., Guevara, M. et al. Stochastic Aspects of Cardiac Arrhythmias. J Stat Phys 128, 347–374 (2007). https://doi.org/10.1007/s10955-006-9191-y
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DOI: https://doi.org/10.1007/s10955-006-9191-y