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November  2013, 18(9): 2377-2396. doi: 10.3934/dcdsb.2013.18.2377

Nonlinear dynamics of a mathematical model on action potential duration and calcium transient in paced cardiac cells

Jiying Ma 1, and  Dongmei Xiao 2,

1. 

Department of Mathematics, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China
2. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240

Received  July 2012 Revised  August 2013 Published  September 2013

In the paper we focus on the dynamics of a two-dimensional discrete-time mathematical model, which describes the interaction between the action potential duration (APD) and calcium transient in paced cardiac cells. By qualitative and bifurcation analysis, we prove that this model can undergo period-doubling bifurcation and Neimark-Sacker bifurcation as parameters vary, respectively. These results provide theoretical support on some experimental observations, such as the alternans of APD and calcium transient, and quasi-periodic oscillations between APD and calcium transient in paced cardiac cells. The rich and complicated bifurcation phenomena indicate that the dynamics of this model are very sensitive to some parameters, which might have important implications for the control of cardiovascular disease.
Keywords: period-doubling bifurcation, Discrete-time mathematical model, APD restitution, Neimark-Sacker bifurcation., calcium transient.
Mathematics Subject Classification: Primary: 37E05, 37G10; Secondary: 92B0.
Citation: Jiying Ma, Dongmei Xiao. Nonlinear dynamics of a mathematical model on action potential duration and calcium transient in paced cardiac cells. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2377-2396. doi: 10.3934/dcdsb.2013.18.2377
References:
[1]

R. Aguilar-Lpez, R. Martnez-Guerra, H. Puebla and R. Hernndez-Surez, High order sliding-mode dynamic control for chaotic intracellular calcium oscillations, Nonlinear Analysis : Real World Appl., 11 (2010), 217-231. doi: 10.1016/j.nonrwa.2008.10.054.  Google Scholar

[2]

J. W. M. Bassani, W. Yuan and D. M. Bers, Fractional SR Ca release is regulated by trigger Ca and SR Ca content in cardiac myocytes, Am. J. Physiol., 268 (1995), C1313-C1319. Google Scholar

[3]

D. M. Bers, Cardiac excitation-contraction coupling, Nature, 415 (2002), 198-205. doi: 10.1038/415198a.  Google Scholar

[4]

H. Bien, L. Yin and E. Entcheva, Calcium instabilities in mammalian cardiomyocyte networks, Biophysical Journal, 90 (2006), 2628-2640. doi: 10.1529/biophysj.105.063321.  Google Scholar

[5]

E. Chudin, J. Goldhaber, A. Garfinkel, J. Weiss and B. Kogan, Intracellular Ca(2+) dynamics and the stability of ventricular tachycardia, Biophysics Journal, 77 (1999), 2930-2941. doi: 10.1016/S0006-3495(99)77126-2.  Google Scholar

[6]

D. E. Euler, Cardiac alternans: mechanisms and pathophysiological significance, Cardiovascular Research, 42 (1999), 583-590. doi: 10.1016/S0008-6363(99)00011-5.  Google Scholar

[7]

J. I. Goldhaber, L. H. Xie, T. Duong, C. Motter, K. Khuu and J. N. Weiss, Action potential duration restitution and alternans in rabbit ventricular myocytes: the key role of intracellular calcium cycling, Circulation Research, 96 (2005), 459-466. doi: 10.1161/01.RES.0000156891.66893.83.  Google Scholar

[8]

M. R. Guevara, L. Glass and A. Shrier, Phase locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells, Science, 214 (1981), 1350-1353. doi: 10.1126/science.7313693.  Google Scholar

[9]

G. M. Hall, S. Bahar and D. J. Gauthier, Prevalence of rate-dependent behaviors in cardiac muscle, Phys. Rev. Lett., 82 (1999), 2995-2998. doi: 10.1103/PhysRevLett.82.2995.  Google Scholar

[10]

A. Karma, Electrical alternans and spiral wave breakup in cardiac tissue, Chaos, 4 (1994), 461-472. doi: 10.1063/1.166024.  Google Scholar

[11]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Applied Mathematical Sciences, 3rd edition, Springer-Verlag, New York, LLC 2004.  Google Scholar

[12]

C. H. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes, Circulation Research, 74 (1994), 1071-1096. doi: 10.1161/01.RES.74.6.1071.  Google Scholar

[13]

R. Mehra, Global public health problem of sudden cardiac death, Journal of Electrocardiology, 40 (2007), S118-S122. doi: 10.1016/j.jelectrocard.2007.06.023.  Google Scholar

[14]

J. B. Nolasco and R. W. Dahlen, A graphic method for the study of alternation in cardiac action potentials, J. Appl. Physiol., 25 (1968), 191-196. Google Scholar

[15]

N. F. Otani and R. F. Gilmour, Memory models for the electrical properties of local cardiac systems, Journal of Theoretical Biology, 187 (1997), 409-436. doi: 10.1006/jtbi.1997.0447.  Google Scholar

[16]

Z. Qu, Y. Shiferaw and J. N. Weiss, Nonlinear dynamics of cardiac excitation-contraction coupling: an iterated map study, Phys. Rev. E, 75 (2007), 011927. doi: 10.1103/PhysRevE.75.011927.  Google Scholar

[17]

T. R. Shannon, K. S. Ginsburg and D. M. Bers, Potentiation of fractional sarcoplasmic reticulum calcium release by total and free intra-sarcoplasmic reticulum calcium concentration, Biophysical Journal, 78 (2000), 334-343. doi: 10.1016/S0006-3495(00)76596-9.  Google Scholar

[18]

Y. Shiferaw and A. Karma, Turing instability mediated by voltage and calcium diffusion in paced cardiac cells, Proc. Natl. Acad. Sci., 103 (2006), 5670-5675. doi: 10.1073/pnas.0511061103.  Google Scholar

[19]

Y. Shiferaw, Z. Qu, A. Garfinkel, A. Karma and J. N. Weiss, Nonlinear dynamics of paced cardiac cells, Annals of the New York Academy of Sciences, 1080 (2006), 376-394. doi: 10.1196/annals.1380.028.x.  Google Scholar

[20]

Y. Shiferaw, D. Sato and A. Karma, Coupled dynamics of voltage and calcium in paced cardiac cells, Phys. Rev. E, 71 (2005), 021903. doi: 10.1103/PhysRevE.71.021903.  Google Scholar

[21]

Y. Shiferaw, M. A. Watanabe, A. Garfinkel, J. N. Weiss and A. Karma, Model of intracellular calcium cycling in ventricular myocytes, Biophysical Journal, 85 (2003), 3666-3686. doi: 10.1016/S0006-3495(03)74784-5.  Google Scholar

[22]

M. D. Stubna, R. H. Rand and R. F. Gilmour, Analysis of a non-linear partial difference equation, and its application to cardiac dynamics, Journal of Difference Equations and Applications, 8 (2002), 1147-1169. doi: 10.1080/1023619021000054006.  Google Scholar

[23]

R. Thul and S. Coombes, Understanding cardiac alternans: A piecewise linear modeling framework, Chaos, 20 (2010), 045102. doi: 10.1063/1.3518362.  Google Scholar

[24]

E. G. Tolkacheva, M. M. Romeo and D. J. Gauthier, Control of cardiac alternans in a mapping model with memory, Physica D: Nonlinear Phenomena, 194 (2004), 385-391. doi: 10.1016/j.physd.2004.03.008.  Google Scholar

[25]

M. L. Walker and D. S. Rosenbaum, Repolarization alternans: implications for the mechanism and prevention of sudden cardiac death, Cardiovascular Research, 57 (2003), 599-614. doi: 10.1016/S0008-6363(02)00737-X.  Google Scholar

[26]

G. S. B. Williams, G. D. Smith, E. A. Sobie and M. S. Jafri, Models of cardiac excitation-contraction coupling in ventricular myocytes, Mathematical Biosciences, 226 (2010), 1-15. doi: 10.1016/j.mbs.2010.03.005.  Google Scholar

show all references

References:
[1]

R. Aguilar-Lpez, R. Martnez-Guerra, H. Puebla and R. Hernndez-Surez, High order sliding-mode dynamic control for chaotic intracellular calcium oscillations, Nonlinear Analysis : Real World Appl., 11 (2010), 217-231. doi: 10.1016/j.nonrwa.2008.10.054.  Google Scholar

[2]

J. W. M. Bassani, W. Yuan and D. M. Bers, Fractional SR Ca release is regulated by trigger Ca and SR Ca content in cardiac myocytes, Am. J. Physiol., 268 (1995), C1313-C1319. Google Scholar

[3]

D. M. Bers, Cardiac excitation-contraction coupling, Nature, 415 (2002), 198-205. doi: 10.1038/415198a.  Google Scholar

[4]

H. Bien, L. Yin and E. Entcheva, Calcium instabilities in mammalian cardiomyocyte networks, Biophysical Journal, 90 (2006), 2628-2640. doi: 10.1529/biophysj.105.063321.  Google Scholar

[5]

E. Chudin, J. Goldhaber, A. Garfinkel, J. Weiss and B. Kogan, Intracellular Ca(2+) dynamics and the stability of ventricular tachycardia, Biophysics Journal, 77 (1999), 2930-2941. doi: 10.1016/S0006-3495(99)77126-2.  Google Scholar

[6]

D. E. Euler, Cardiac alternans: mechanisms and pathophysiological significance, Cardiovascular Research, 42 (1999), 583-590. doi: 10.1016/S0008-6363(99)00011-5.  Google Scholar

[7]

J. I. Goldhaber, L. H. Xie, T. Duong, C. Motter, K. Khuu and J. N. Weiss, Action potential duration restitution and alternans in rabbit ventricular myocytes: the key role of intracellular calcium cycling, Circulation Research, 96 (2005), 459-466. doi: 10.1161/01.RES.0000156891.66893.83.  Google Scholar

[8]

M. R. Guevara, L. Glass and A. Shrier, Phase locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells, Science, 214 (1981), 1350-1353. doi: 10.1126/science.7313693.  Google Scholar

[9]

G. M. Hall, S. Bahar and D. J. Gauthier, Prevalence of rate-dependent behaviors in cardiac muscle, Phys. Rev. Lett., 82 (1999), 2995-2998. doi: 10.1103/PhysRevLett.82.2995.  Google Scholar

[10]

A. Karma, Electrical alternans and spiral wave breakup in cardiac tissue, Chaos, 4 (1994), 461-472. doi: 10.1063/1.166024.  Google Scholar

[11]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Applied Mathematical Sciences, 3rd edition, Springer-Verlag, New York, LLC 2004.  Google Scholar

[12]

C. H. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes, Circulation Research, 74 (1994), 1071-1096. doi: 10.1161/01.RES.74.6.1071.  Google Scholar

[13]

R. Mehra, Global public health problem of sudden cardiac death, Journal of Electrocardiology, 40 (2007), S118-S122. doi: 10.1016/j.jelectrocard.2007.06.023.  Google Scholar

[14]

J. B. Nolasco and R. W. Dahlen, A graphic method for the study of alternation in cardiac action potentials, J. Appl. Physiol., 25 (1968), 191-196. Google Scholar

[15]

N. F. Otani and R. F. Gilmour, Memory models for the electrical properties of local cardiac systems, Journal of Theoretical Biology, 187 (1997), 409-436. doi: 10.1006/jtbi.1997.0447.  Google Scholar

[16]

Z. Qu, Y. Shiferaw and J. N. Weiss, Nonlinear dynamics of cardiac excitation-contraction coupling: an iterated map study, Phys. Rev. E, 75 (2007), 011927. doi: 10.1103/PhysRevE.75.011927.  Google Scholar

[17]

T. R. Shannon, K. S. Ginsburg and D. M. Bers, Potentiation of fractional sarcoplasmic reticulum calcium release by total and free intra-sarcoplasmic reticulum calcium concentration, Biophysical Journal, 78 (2000), 334-343. doi: 10.1016/S0006-3495(00)76596-9.  Google Scholar

[18]

Y. Shiferaw and A. Karma, Turing instability mediated by voltage and calcium diffusion in paced cardiac cells, Proc. Natl. Acad. Sci., 103 (2006), 5670-5675. doi: 10.1073/pnas.0511061103.  Google Scholar

[19]

Y. Shiferaw, Z. Qu, A. Garfinkel, A. Karma and J. N. Weiss, Nonlinear dynamics of paced cardiac cells, Annals of the New York Academy of Sciences, 1080 (2006), 376-394. doi: 10.1196/annals.1380.028.x.  Google Scholar

[20]

Y. Shiferaw, D. Sato and A. Karma, Coupled dynamics of voltage and calcium in paced cardiac cells, Phys. Rev. E, 71 (2005), 021903. doi: 10.1103/PhysRevE.71.021903.  Google Scholar

[21]

Y. Shiferaw, M. A. Watanabe, A. Garfinkel, J. N. Weiss and A. Karma, Model of intracellular calcium cycling in ventricular myocytes, Biophysical Journal, 85 (2003), 3666-3686. doi: 10.1016/S0006-3495(03)74784-5.  Google Scholar

[22]

M. D. Stubna, R. H. Rand and R. F. Gilmour, Analysis of a non-linear partial difference equation, and its application to cardiac dynamics, Journal of Difference Equations and Applications, 8 (2002), 1147-1169. doi: 10.1080/1023619021000054006.  Google Scholar

[23]

R. Thul and S. Coombes, Understanding cardiac alternans: A piecewise linear modeling framework, Chaos, 20 (2010), 045102. doi: 10.1063/1.3518362.  Google Scholar

[24]

E. G. Tolkacheva, M. M. Romeo and D. J. Gauthier, Control of cardiac alternans in a mapping model with memory, Physica D: Nonlinear Phenomena, 194 (2004), 385-391. doi: 10.1016/j.physd.2004.03.008.  Google Scholar

[25]

M. L. Walker and D. S. Rosenbaum, Repolarization alternans: implications for the mechanism and prevention of sudden cardiac death, Cardiovascular Research, 57 (2003), 599-614. doi: 10.1016/S0008-6363(02)00737-X.  Google Scholar

[26]

G. S. B. Williams, G. D. Smith, E. A. Sobie and M. S. Jafri, Models of cardiac excitation-contraction coupling in ventricular myocytes, Mathematical Biosciences, 226 (2010), 1-15. doi: 10.1016/j.mbs.2010.03.005.  Google Scholar

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