American Institute of Mathematical Sciences

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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 and 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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

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