American Institute of Mathematical Sciences

Advanced
2013, 10(4): 1253-1264. doi: 10.3934/mbe.2013.10.1253

Heart rate variability as determinism with jump stochastic parameters

Jiongxuan Zheng 1, , Joseph D. Skufca 1, and  Erik M. Bollt 1,

1. 

8 Clarkson AVE. P.O.5817, Potsdam, NY 13676, United States, United States, United States

Received  May 2012 Revised  January 2013 Published  June 2013

We use measured heart rate information (RR intervals) to develop a one-dimensional nonlinear map that describes short term deterministic behavior in the data. Our study suggests that there is a stochastic parameter with persistence which causes the heart rate and rhythm system to wander about a bifurcation point. We propose a modified circle map with a jump process noise term as a model which can qualitatively capture such this behavior of low dimensional transient determinism with occasional (stochastically defined) jumps from one deterministic system to another within a one parameter family of deterministic systems.
Keywords: circle map, next angle map, electrocardiography, jump process., Heart rate variability.
Mathematics Subject Classification: Primary: 15A15, 15A09, 15A2.
Citation: Jiongxuan Zheng, Joseph D. Skufca, Erik M. Bollt. Heart rate variability as determinism with jump stochastic parameters. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1253-1264. doi: 10.3934/mbe.2013.10.1253
References:
[1]

Katrin Suder, Friedhelm R. Drepper, Michael Schiek and Hans-Henning Abel, One-dimensional, nonlinear determinism characterizes heart rate pattern during paced respiration, Am. J. Phiciol Heart Circ. Physiol, 275 (1998), 1092-1102. Google Scholar

[2]

N. B. Janson, A. G. Balanov, V. S. Anishchenko and P. V. E. McClintock, Modelling the dynamics of angles of human R-R intervals, Physiol. Meas., 22 (2001), 565-579. doi: 10.1088/0967-3334/22/3/313.  Google Scholar

[3]

Yuo-Hsien Shiau, Shu-Shya Hseu and Huey-Wen Yien, One-dimensional deterministic process extracted from noisy R-R intervals under spontaneous breathing conditions, Journal of Medical and Biological Engineering, 26 (2001), 121-124. Google Scholar

[4]

T. Schreiber, Extremely simple nonlinear noise-reduction method, Phys. Rev. E, 47 (1993), 2401-2404. doi: 10.1103/PhysRevE.47.2401.  Google Scholar

[5]

Claudia Lerma, Trine Krogh-Madsen, Micheal Guevara and Leon Glass, Stochastic aspects of cardiac arrhymias, Journal Statistical Physics, 128 (2007), 347-374. doi: 10.1007/s10955-006-9191-y.  Google Scholar

[6]

Tom Kuusela, Tony Shepherd and Jarmo Hietarinta, Stochastic model for heart-rate fluctuations, Phys. Rev. E, 67 (2003), 061904. doi: 10.1103/PhysRevE.67.061904.  Google Scholar

[7]

Task Force of the European Socirty of Cariology and the North American Scirty of Pacing and Electrophysiology, Heart rate variability: Standards of measurement, physiological interpretation and clinical use, European Heart Journal, 17 (1996), 354-381. Google Scholar

[8]

M. Pomeranz, R. J. B. Macaulay and M. A. Caudill, Assessment of autonomic function in humans by heart rate spectral analysis, Am. J. Physiol, 248 (1985), H151-3. Google Scholar

[9]

M. Pagani, F. Lombardi and S. Guzzetti, et al., Power spetral analysis of heart rate and arterial pressure variablities as a marker of sympatho-vagal interaction in man and conscious dog, Circ. Res., 59 (1986), 178-193. Google Scholar

[10]

C-K. Peng, J. E. Mietus, Y. Liu, G. Khalsa, P. S. Douglas, H. Benson and A. L. Goldberger, Exaggerated heart rate oscillations during two meditation techniques, International Journal of Cardiology, 70 (1999), 101-107. Google Scholar

[11]

Leon Glass, Cardiac arrhythmias and circles maps-A classical problem, Chaos: An Interdisciplinary Journal of Nonlinear Science, 1 (1991), 13-19. doi: 10.1063/1.165810.  Google Scholar

[12]

Edward Ott, "Chaos in Dynamical Systems," First ed., Cambridge University Press., 1993.  Google Scholar

[13]

Steven H. Strogatz, "Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry and Engineering," First ed., Westview Press., 1995. doi: 10.1063/1.2807947.  Google Scholar

[14]

Otakar Fojt and Jiri Holcik, Applying nonlinear dynamics to ECG signal processing, IEEE Engineering in Medicine and Biology., (1998). doi: 10.1109/51.664037.  Google Scholar

[15]

Joseph D. Skufca and Erik M. Bollt, A concept of homeomorphic defect for defining mostly conjugate dynamical systems, Chaos., 18 (2008), 013008. doi: 10.1063/1.2837397.  Google Scholar

[16]

T. Schreiber and A. Schmitz, Improved surrogate data for nonlinearity tests, Phys. Rev. Lett., 77 (1996), 635-638. Google Scholar

[17]

Holger Kantz and Thomas Schreiber, "Nonlinear Time Series Analysis," Second ed., Cambridge Press., 2004.  Google Scholar

show all references

References:
[1]

Katrin Suder, Friedhelm R. Drepper, Michael Schiek and Hans-Henning Abel, One-dimensional, nonlinear determinism characterizes heart rate pattern during paced respiration, Am. J. Phiciol Heart Circ. Physiol, 275 (1998), 1092-1102. Google Scholar

[2]

N. B. Janson, A. G. Balanov, V. S. Anishchenko and P. V. E. McClintock, Modelling the dynamics of angles of human R-R intervals, Physiol. Meas., 22 (2001), 565-579. doi: 10.1088/0967-3334/22/3/313.  Google Scholar

[3]

Yuo-Hsien Shiau, Shu-Shya Hseu and Huey-Wen Yien, One-dimensional deterministic process extracted from noisy R-R intervals under spontaneous breathing conditions, Journal of Medical and Biological Engineering, 26 (2001), 121-124. Google Scholar

[4]

T. Schreiber, Extremely simple nonlinear noise-reduction method, Phys. Rev. E, 47 (1993), 2401-2404. doi: 10.1103/PhysRevE.47.2401.  Google Scholar

[5]

Claudia Lerma, Trine Krogh-Madsen, Micheal Guevara and Leon Glass, Stochastic aspects of cardiac arrhymias, Journal Statistical Physics, 128 (2007), 347-374. doi: 10.1007/s10955-006-9191-y.  Google Scholar

[6]

Tom Kuusela, Tony Shepherd and Jarmo Hietarinta, Stochastic model for heart-rate fluctuations, Phys. Rev. E, 67 (2003), 061904. doi: 10.1103/PhysRevE.67.061904.  Google Scholar

[7]

Task Force of the European Socirty of Cariology and the North American Scirty of Pacing and Electrophysiology, Heart rate variability: Standards of measurement, physiological interpretation and clinical use, European Heart Journal, 17 (1996), 354-381. Google Scholar

[8]

M. Pomeranz, R. J. B. Macaulay and M. A. Caudill, Assessment of autonomic function in humans by heart rate spectral analysis, Am. J. Physiol, 248 (1985), H151-3. Google Scholar

[9]

M. Pagani, F. Lombardi and S. Guzzetti, et al., Power spetral analysis of heart rate and arterial pressure variablities as a marker of sympatho-vagal interaction in man and conscious dog, Circ. Res., 59 (1986), 178-193. Google Scholar

[10]

C-K. Peng, J. E. Mietus, Y. Liu, G. Khalsa, P. S. Douglas, H. Benson and A. L. Goldberger, Exaggerated heart rate oscillations during two meditation techniques, International Journal of Cardiology, 70 (1999), 101-107. Google Scholar

[11]

Leon Glass, Cardiac arrhythmias and circles maps-A classical problem, Chaos: An Interdisciplinary Journal of Nonlinear Science, 1 (1991), 13-19. doi: 10.1063/1.165810.  Google Scholar

[12]

Edward Ott, "Chaos in Dynamical Systems," First ed., Cambridge University Press., 1993.  Google Scholar

[13]

Steven H. Strogatz, "Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry and Engineering," First ed., Westview Press., 1995. doi: 10.1063/1.2807947.  Google Scholar

[14]

Otakar Fojt and Jiri Holcik, Applying nonlinear dynamics to ECG signal processing, IEEE Engineering in Medicine and Biology., (1998). doi: 10.1109/51.664037.  Google Scholar

[15]

Joseph D. Skufca and Erik M. Bollt, A concept of homeomorphic defect for defining mostly conjugate dynamical systems, Chaos., 18 (2008), 013008. doi: 10.1063/1.2837397.  Google Scholar

[16]

T. Schreiber and A. Schmitz, Improved surrogate data for nonlinearity tests, Phys. Rev. Lett., 77 (1996), 635-638. Google Scholar

[17]

Holger Kantz and Thomas Schreiber, "Nonlinear Time Series Analysis," Second ed., Cambridge Press., 2004.  Google Scholar

[1]

Sebastián Ferrer, Francisco Crespo. Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction. Journal of Geometric Mechanics, 2018, 10 (3) : 359-372. doi: 10.3934/jgm.2018013
[2]

Jan Rychtář, Dewey T. Taylor. Moran process and Wright-Fisher process favor low variability. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3491-3504. doi: 10.3934/dcdsb.2020242
[3]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006
[4]

Lluís Alsedà, Michał Misiurewicz. Semiconjugacy to a map of a constant slope. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3403-3413. doi: 10.3934/dcdsb.2015.20.3403
[5]

Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255
[6]

Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov. Quasiperiodic motion for the pentagram map. Electronic Research Announcements, 2009, 16: 1-8. doi: 10.3934/era.2009.16.1
[7]

John Erik Fornæss, Brendan Weickert. A quantized henon map. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 723-740. doi: 10.3934/dcds.2000.6.723
[8]

Zenonas Navickas, Rasa Smidtaite, Alfonsas Vainoras, Minvydas Ragulskis. The logistic map of matrices. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 927-944. doi: 10.3934/dcdsb.2011.16.927
[9]

Roberto De Leo, James A. Yorke. The graph of the logistic map is a tower. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5243-5269. doi: 10.3934/dcds.2021075
[10]

Jian Zhai, Jianping Fang, Lanjun Li. Wave map with potential and hypersurface flow. Conference Publications, 2005, 2005 (Special) : 940-946. doi: 10.3934/proc.2005.2005.940
[11]

Mila Nikolova. Model distortions in Bayesian MAP reconstruction. Inverse Problems & Imaging, 2007, 1 (2) : 399-422. doi: 10.3934/ipi.2007.1.399
[12]

Nalini Joshi, Pavlos Kassotakis. Re-factorising a QRT map. Journal of Computational Dynamics, 2019, 6 (2) : 325-343. doi: 10.3934/jcd.2019016
[13]

Juan Luis García Guirao, Marek Lampart. Transitivity of a Lotka-Volterra map. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 75-82. doi: 10.3934/dcdsb.2008.9.75
[14]

Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012
[15]

Luigi Ambrosio, Federico Glaudo, Dario Trevisan. On the optimal map in the $ 2 $-dimensional random matching problem. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7291-7308. doi: 10.3934/dcds.2019304
[16]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[17]

Miaohua Jiang, Qiang Zhang. A coupled map lattice model of tree dispersion. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 83-101. doi: 10.3934/dcdsb.2008.9.83
[18]

Anatoli F. Ivanov. On global dynamics in a multi-dimensional discrete map. Conference Publications, 2015, 2015 (special) : 652-659. doi: 10.3934/proc.2015.0652
[19]

Kokum R. De Silva, Shigetoshi Eda, Suzanne Lenhart. Modeling environmental transmission of MAP infection in dairy cows. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1001-1017. doi: 10.3934/mbe.2017052
[20]

Vladimír Špitalský. Transitive dendrite map with infinite decomposition ideal. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 771-792. doi: 10.3934/dcds.2015.35.771

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy