# American Institute of Mathematical Sciences

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

## Heart rate variability as determinism with jump stochastic parameters

 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.
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. [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. [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. [4] T. Schreiber, Extremely simple nonlinear noise-reduction method, Phys. Rev. E, 47 (1993), 2401-2404. doi: 10.1103/PhysRevE.47.2401. [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. [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. [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. [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. [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. [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. [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. [12] Edward Ott, "Chaos in Dynamical Systems," First ed., Cambridge University Press., 1993. [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. [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. [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. [16] T. Schreiber and A. Schmitz, Improved surrogate data for nonlinearity tests, Phys. Rev. Lett., 77 (1996), 635-638. [17] Holger Kantz and Thomas Schreiber, "Nonlinear Time Series Analysis," Second ed., Cambridge Press., 2004.

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##### 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. [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. [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. [4] T. Schreiber, Extremely simple nonlinear noise-reduction method, Phys. Rev. E, 47 (1993), 2401-2404. doi: 10.1103/PhysRevE.47.2401. [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. [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. [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. [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. [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. [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. [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. [12] Edward Ott, "Chaos in Dynamical Systems," First ed., Cambridge University Press., 1993. [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. [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. [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. [16] T. Schreiber and A. Schmitz, Improved surrogate data for nonlinearity tests, Phys. Rev. Lett., 77 (1996), 635-638. [17] Holger Kantz and Thomas Schreiber, "Nonlinear Time Series Analysis," Second ed., Cambridge Press., 2004.
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