A modified van der Pol equation with delay in a description of the heart action

• Autorzy: Beata Zduniak , Marek Bodnar , Urszula Foryś
• Języki publikacji: EN

Abstrakty

EN

In this paper, a modified van der Pol equation is considered as a description of the heart action. This model has a number of interesting properties allowing reconstruction of phenomena observed in physiological experiments as well as in Holter electrocardiographic recordings. Our aim is to study periodic solutions of the modified van der Pol equation and take into consideration the influence of feedback and delay which occur in the normal heart action mode as well as in pathological modes. Usage of certain values for feedback and delay parameters allows simulating the heart action when an accessory conducting pathway is present (Wolff-Parkinson-White syndrome).

Słowa kluczowe

EN

system with feedback; stability; action potential

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2014
• Tom: 24
• Numer: 4
• Strony: 853-863

Daty

wydano

2014

otrzymano

2013-10-02

poprawiono

2014-01-08

Twórcy

autor

Beata Zduniak

• Faculty of Applied Informatics and Mathematics, Warsaw University of Life Sciences, Nowoursynowska 159, 02-776 Warsaw, Poland

autor

Marek Bodnar

• Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

autor

Urszula Foryś

• Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

Bibliografia

• Atay, F. (1998). Van der Pol's oscillator under delayed feedback, Journal of Sound and Vibration 218(2): 333-339.
• Bielczyk, N., Bodnar, M. and Foryś, U. (2012). Delay can stabilize: Love affairs dynamics, Applied Mathematics and Computation 219(2): 3923-3937.
• Cooke, K. and van den Driessche, P. (1986). On zeroes of some transcendental equations, Funkcialaj Ekvacioj 29(2): 77-90.
• Erneux, T. and Grasman, J. (2008). Limit cycle oscillators subject to a delayed feedback, Physical Review E 78(2): 026209-1-8.
• Foryś, U. (2004). Biological delay systems and the Mikhailov criterion of stability, Journal of Biological Systems 12(1): 45-60.
• Giacomin, H. and Neukirch, S. (1997). Number of limit cycles of the Lienard equation, Physical Review E 56(3809): 3809-3813.
• Grudziński, K. (2007). Modeling the Electrical Activity of the Heart Conduction, Ph.D. thesis, Warsaw University of Technology, Warsaw.
• Hale, J. and Lunel, S. (1993). Introduction to Functional Differential Equations, Springer-Verlag, New York, NY.
• Jiang, W. and Wei, J. (2008). Bifurcation analysis in van der Pol's oscillator with delayed feedback, Journal of Computational and Applied Mathematics 213(2): 604-615.
• Johnson, L. (1997). Essential Medical Physiology, Lippincott Williams and Wilkins, London.
• Kaplan, D. and Glass, L. (1995). Understanding Nonlinear Dynamics, Springer, New York, NY.
• Liu, Y., Yang, R., Lu. J. and Cai, X. (2013). Stability analysis of high-order Hopfield-type neural networks based on a new impulsive differential inequality, International Journal of Applied Mathematics and Computer Science 23(1): 201-211, DOI: 10.2478/amcs-2013-0016.
• Maccari, A. (2001). The response of a parametrically excited van der Pol oscillator to a time delay state feedback, Nonlinear Dynamics 26(2): 105-119.
• Palit, A. and Datta, D.P. (2010). On a finite number of limit cycles in a Lienard system, International Journal of Pure and Applied Mathematics 59: 469-488.
• Reddy, R.D., Sen, A. and Johnston, G. (2000). Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators, Physical Review Letters 85(3381): 3381-3384.
• Xu, J. and Chung, K. (2003). Effects of time delayed position feedback on a van der Pol-Duffing oscillator, Physica D 180(1): 17-39.
• Yu, W. and Cao, J. (2005). Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delay, Nonlinear Analysis 62: 141-165.
• Żebrowski, J., Kuklik, P. and Baranowski, R. (2008). Relaxation oscillations in the atrium-a model, Proceedings of the 5th Conference of the European Study Group on Cardiovascular Oscillations, Parma, Italy, pp. 04:16-04:19.
• Zhou, X., Jiang, M. and Cai, X. (2011). Hopf bifurcation analysis for the van der Pol equation with discrete and distributed delays, Discrete Dynamics in Nature and Society, 2011: 1-8, Article ID: 569141.

Identyfikatory

DOI

10.2478/amcs-2014-0063