Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Synchronization of the Cardiac Pacemaker Model with Delayed Pulse-coupling

Discontinuity, Nonlinearity, and Complexity 3(1) (2014) 19--31 | DOI:10.5890/DNC.2014.03.002

M. Akhmet

Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, 06531 Ankara, Turkey

Download Full Text PDF



We reconsider the C. Peskin model of the cardiac pacemaker assuming that pulse-couplings are delayed. Sufficient conditions for synchronization of identical and non-identical oscillators are obtained. The results are demon- strated with numerical simulations.


The author wishes to express his sincere gratitude to the referee for the helpful criticism and valuable suggestions. This research was supported by a grant (111T320) from TUBITAK, the Scientific and Technological Research Council of Turkey.


  1. [1]  Brooks, C.M. and Lu, H.H. (1972), The sinoatrial pacemaker of the heart. Thomas, Springfield IL.
  2. [2]  Peskin, C.S.(1975),Mathematical aspects of heart physiology, Courant Institute of Mathematical sSciences, 268-278.
  3. [3]  Mirollo, R.E. and Strogatz, S.H.(1990), Synchronization of pulse-coupled biological oscillators, SIAM Journal on Applied Mathematics , 50, 1645-1662.
  4. [4]  Bottani, S.(1995), Pulse-coupled relaxation oscillators: From biological synchronization to self-organized driticality, Physical Review Letters, 74, 4189-4182.
  5. [5]  Ernst, U., Pawelzik, K. and Geisel, T.(1998),Delay-induced multistable synchronization of biological oscillators, Phys. Review E, 57, 2150-2162.
  6. [6]  Gerstner,W.(1996), Rapid phase locking in systems of pulse-coupled oscillators with delays, Physical Review Letters, 76, 1755-1758.
  7. [7]  Kuramoto, Y. (1991), Collective synchronization of pulse-coupled oscillators and excitable units, Physica D, 50, 15-30.
  8. [8]  Mathar, R. and Mattfeldt, J. (1996), Pulse-coupled decentral synchronization, SIAM Journal on Applied Mathematics , 56, 1094-1106.
  9. [9]  Senn, W. and Urbanzik, R. (2000), Similar non-leaky integrate-ans-fire neurons with instantaneous couplings always synchronize, SIAM Journal on Applied Mathematics , 61 , 1143-1155.
  10. [10]  Strogatz, S.(2003), Sync: The Emerging Science of Spontaneous Order, Hyperion, New York.
  11. [11]  Timme,M.,Wolf, F. and Geisel, T.(2002), Prevalence of Unstable Attractors in Networks of Pulse-Coupled Oscillators, Physical Review Letters, 89, 154105.
  12. [12]  Timme, M. and Wolf, F. (2008), The simplest problem in the collective dynamics of neural networks: is synchrony stable?, Nonlinearity, 21, 1579–1599.
  13. [13]  Akhmet, M.U. (2011) Nonlinear Hybrid Continuous/discrete-time Models, Atlantis Press, Amsterdam, Paris.
  14. [14]  Akhmet, M.U.(2011), Analysis of biological integrate-and-fire oscillators. Nonlinear Studies, 18 , 313-327.
  15. [15]  Akhmet,M.U.(2012), Self-synchronization of the integrate-and-fire pacemaker model with continuous couplings, Nonlinear Analysis: Hybrid Systems , 6 , 730-740.
  16. [16]  Vreeswijk, C. van (1996), Partial synchronization in populations of pulse-coupled oscillators, Physical Review E, 54 , 5522-5537.
  17. [17]  Akhmet, M.U. (2010), Principles Of Discontinuous Dynamical Systems, Springer, New York.
  18. [18]  Akhmet, M.U. (2005), Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis: TMA, 60, 163-178.
  19. [19]  Devi, J., Vasundara, and Vatsala, A. S.(2003), Generalized quasilinearization for an impulsive differential equation with variable moments of impulse,Dynamic Systems & Applications , 12, 369-382.
  20. [20]  Feckan, M.(1998), Bifurcation of periodic and chaotic solutions in discontinuous systems. Archiv der Mathematik (Brno), 34, 73-82.
  21. [21]  Frigon, M. and O’Regan, D. (1996), Impulsive differential equations with variable times, Nonlinear Analysis: TMA, 26, 1913-1922.
  22. [22]  Lakshmikantham, V. Bainov, D. D. and Simeonov, P. S. (1989), Theory of Impulsive Differential Equations, World Scientific, Singapore, NJ, London, Hong Kong.
  23. [23]  Lakshmikantham, V., Leela, S. and Kaul, S. (1994), Comparison principle for impulsive differential equations with variable times and stability theory, Nonlinear Analysis: TMA, 22, 49-503.
  24. [24]  Lakshmikantham, V. and Liu, X.(1989), On quasistability for impulsive differential equations, Nonlinear Analysis: TMA, 13, 819-828.
  25. [25]  Liu, X. and Pirapakaran, R. (1989), Global stability results for impulsive differential equations, Appl. Anal., 33, 87- 102.
  26. [26]  Luo, A. C. J. (2008), Global Transversality, Resonance and Chaotic Dynamics,World Scientific, Hackensack, NJ.
  27. [27]  Samoilenko, A. M. and Perestyuk, N. A.(1995), Impulsive Differential Equations,World Scientific, Singapore.
  28. [28]  Murray, J.D. (2002), Mathematical Biology: I. An Introduction, Springer, New-York.
  29. [29]  Buck, J.(1988), Synchronous Rhythmic Flashing of Fireflies, II.,The Quarterly Review of Biology , 63 , 265-290.
  30. [30]  Ko, T.-W. and Ermentrout,G.B. (2007), Effects of axonal time delay on synchronization and wave formation in sparsely coupled neuronal oscillators, Physical Review E , 76 , 1-8.
  31. [31]  Kuramoto (1984), Y. Chemical oscillators, Waves and Turbulence, Springer, Berlin.
  32. [32]  Herz, A.V.M. and Hopfield, J.J.(1995), Earthquake cycles and neural perturbations: collective oscillations in systems with pulse-coupled thresholds elements, Physical Review Letters, 75 , 1222-1225.
  33. [33]  Hopfield, J.J.(1994), Neurons, dynamics and computation, Physics Today, February, 40-46.
  34. [34]  Olami, Z., Feder, H.J.S. and Christensen, K.(1992), Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes, Physical Review Letters, 68 , 1244-1247.