Kick synchronization versus diffusive synchronization

Proceedings of the IEEE Conference on Decision and Control

Volumn , Issue , 2012, Pages 7171-7183

  Mauroy, Alexandre ^a   Sacre, Pierre ^b   Sepulchre, Rodolphe ^b  

^a University of California   (United States)

^b UNIVERSITY OF LIÈGE   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

CONTINUOUS TIME SYSTEMS;

CONTINUOUS-TIME; DIFFUSIVE MODEL; FUNDAMENTAL MODELS; MATHEMATICAL LITERATURE; WEAK COUPLINGS;

SYNCHRONIZATION;

EID: 84874232831     PISSN: 07431546     EISSN: 25762370     Source Type: Conference Proceeding    
DOI: 10.1109/CDC.2012.6425821     Document Type: Conference Paper

References (85)

1. L. F. Abbott, "Lapicque's introduction of the integrate-and-fire model neuron (1907)," Brain Res. Bull., vol. 50, no. 5-6, pp. 303-304, Nov. 1999.
2. L. F. Abbott and C. van Vreeswijk, "Asynchronous states in networks of pulse-coupled oscillators," Phys. Rev. E, vol. 48, no. 2, pp. 1483-1490, Aug. 1993.
3. D. Angeli, "A Lyapunov approach to incremental stability properties," IEEE Trans. Autom. Control, vol. 47, no. 3, pp. 410-421, Mar. 2002.
4. M. Arcak, "Passivity as a design tool for group coordination," IEEE Trans. Autom. Control, vol. 52, no. 8, pp. 1380-1390, Aug. 2007.
5. M. R. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, "Huygens's clocks," Proc. R. Soc. Lond. A, vol. 458, pp. 563-579, Mar. 2002.
6. I. I. Blekhman, Synchronization in Science and Technology. American Society of Mechanical Engineers, 1988.
7. S. Bottani, "Synchronization of integrate and fire oscillators with global coupling," Phys. Rev. E, vol. 54, no. 3, pp. 2334-2350, Sep. 1996.
8. P. C. Bressloff and S. Coombes, "A dynamical theory of spike train transitions in networks of integrate-and-fire oscillators," SIAM J. Appl. Math., vol. 60, no. 3, pp. 820-841, 2000.
9. E. T. Brown, P. Holmes, and J. Moehlis, "Globally coupled oscillator networks," in Perspectives and Problems in Nonlinear Science: a Celebratory Volume in Honor of Larry Sirovich, E. Kaplan, J. E. Marsden, and K. R. Sreenivasan, Eds. New York, NY: Springer, 2003, pp. 183-215.
10. E. T. Brown, J. Moehlis, and P. Holmes, "On the phase reduction and response dynamics of neural oscillator populations," Neural Comput., vol. 16, no. 4, pp. 673-715, Apr. 2004.
11. J. B. Buck, "Synchronous rhythmic flashing of fireflies. II." Q. Rev. Biol., vol. 63, no. 3, pp. 265-289, Sep. 1988.
12. Y.-C. Chang and J. Juang, "Stable synchrony in globally coupled integrate-and-fire oscillators," SIAM J. Appl. Dyn. Syst., vol. 7, no. 4, pp. 1445-1476, 2008.
13. N. Chopra, "Output synchronization on strongly connected graphs," IEEE Trans. Autom. Control, vol. PP, no. 99, pp. 1-1, Apr. 2012.
14. H. Daido, "Order function and macroscopic mutual entrainment in uniformly coupled limit-cycle oscillators," Prog. Theor. Phys., vol. 88, no. 6, pp. 1213-1218, Dec. 1992.
15. F. De Smet and D. Aeyels, "Coexistence of stable stationary behavior and partial synchrony in an all-to-all coupled spiking neural network," Phys. Rev. E, vol. 82, no. 6 Pt 2, p. 066208, Dec. 2010.
16. M. Denker, M. Timme, M. Diesmann, F. Wolf, and T. Geisel, "Breaking synchrony by heterogeneity in complex networks," Phys. Rev. Lett., vol. 92, no. 7, p. 074103, Feb. 2004.
17. A. Díaz-Guilera, A. Arenas, A. Corral, and C. J. Pérez, "Stability of spatio-temporal structures in a lattice model of pulse-coupled oscillators," Physica D, vol. 103, no. 1-4, pp. 419-429, Apr. 1997.
18. R. O. Dror, C. C. Canavier, R. J. Butera, J. W. Clark, and J. H. Byrne, "A mathematical criterion based on phase response curves for stability in a ring of coupled oscillators," Biol. Cybern., vol. 80, no. 1, pp. 11-23, 1999.
19. G. B. Ermentrout and N. Kopell, "Parabolic bursting in an excitable system coupled with a slow oscillation," SIAM J. Appl. Math., vol. 46, no. 2, pp. 233-253, 1986.
20. U. Ernst, K. Pawelzik, and T. Geisel, "Delay-induced multistable synchronization of biological oscillators," Phys. Rev. E, vol. 57, no. 2, pp. 2150-2162, 1998.
21. M. Farkas, Periodic Motions, ser. Applied Mathematical Sciences. New York, NY: Springer-Verlag, 1994, vol. 104.
22. R. Fitzhugh, "Impulses and physiological states in theoretical models of nerve membrane," Biophys. J., vol. 1, no. 6, pp. 445-466, Jul. 1961.
23. W. Gerstner and W. M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge, England: Cambridge University Press, Aug. 2002.
24. L. Glass and M. C. Mackey, From Clocks to Chaos: the Rhythms of Life. Princeton, NJ: Princeton University Press, 1988.
25. P. Goel and G. B. Ermentrout, "Synchrony, stability, and firing patterns in pulse-coupled oscillators," Physica D, vol. 163, no. 3-4, pp. 191-216, 2002.
26. A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: the Molecular Bases of Periodic and Chaotic Behaviour. Cambridge, United Kingdom: Cambridge University Press, 1996.
27. D. Golomb, D. Hansel, B. Shraiman, and H. Sompolinsky, "Clustering in globally coupled phase oscillators," Phys. Rev. A, vol. 45, no. 6, pp. 3516-3530, Mar. 1992.
28. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 2nd ed., ser. Applied Mathematical Sciences. New York, NY: Springer, 1983, vol. 42.
29. A. Hamadeh, G.-B. Stan, R. Sepulchre, and J. Gonçalves, "Global state synchronization in networks of cyclic feedback systems," IEEE Trans. Autom. Control, vol. 57, no. 2, pp. 478-483, Feb. 2012.
30. A. L. Hodgkin and A. F. Huxley, "A quantitative description of membrane current and its application to conduction and excitation in nerve," J. Physiol., vol. 117, no. 4, pp. 500-544, Aug. 1952.
31. Y.-W. Hong and A. Scaglione, "A scalable synchronization protocol for large scale sensor networks and its applications," IEEE J. Sel. Areas Commun., vol. 23, no. 5, pp. 1085-1099, 2005.
32. F. C. Hoppensteadt and E. M. Izhikevich, Weakly Connected Neural Networks, ser. Applied Mathematical Sciences. New York, NY: Springer-Verlag, 1997, vol. 126.
33. C. Huygens, Horologium Oscillatorium. Parisiis, France: Apud F. Muguet, 1673.
34. - Oeuvres Complètes de Christiaan Huygens. The Hague, The Netherlands: Martinus Nijhoff, 1932, vol. 17.
35. E. M. Izhikevich, Dynamical Systems in Neuroscience: the Geometry of Excitability and Bursting. Cambridge, MA: The MIT Press, 2007.
36. H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2002.
37. B. W. Knight, "Dynamics of encoding in a population of neurons," The Journal of General Physiology, vol. 59, no. 6, pp. 734-766, Jun. 1972.
38. Y. Kuramoto, "Self-entrainment of a population of coupled non-linear oscillators," in International Symposium on Mathematical Problems in Theoretical Physics. Berlin/Heidelberg, Germany: Springer-Verlag, 1975, pp. 420-422.
39. - Chemical Oscillations, Waves, and Turbulence, 1st ed., ser. Springer Series in Synergetics. Berlin/Heidelberg, Germany: Springer-Verlag, 1984, vol. 19.
40. - "Cooperative dynamics of oscillator community: a study based on lattice of rings," Prog. Theor. Phys. Suppl., vol. Suppl. 79, pp. 223-240, 1984.
41. - "Collective synchronization of pulse-coupled oscillators and excitable units," Physica D, vol. 50, no. 1, pp. 15-30, May 1991.
42. - "Phase- and center-manifold reductions for large populations of coupled oscillators with application to non-locally coupled systems," Int. J. Bifurcat. Chaos, vol. 7, no. 4, pp. 789-805, 1997.
43. W. Lohmiller and J.-J. E. Slotine, "On contraction analysis for nonlinear systems," Automatica, vol. 34, no. 6, pp. 683-696, Jun. 1998.
44. A. Mauroy, "On the dichotomic collective behaviors of large populations of pulse-coupled firing oscillators," Ph.D. dissertation, University of Liège, Liège, Belgium, Oct. 2011.
45. A. Mauroy, J. M. Hendrickx, A. Megretski, and R. Sepulchre, "Global analysis of firing maps," in Proc. 19th Int. Symp. Mathematical Theory Networks and Systems, Budapest, Hungary, Jul. 2010, pp. 1775-1782.
46. A. Mauroy and R. Sepulchre, "Clustering behaviors in networks of integrate-and-fire oscillators," Chaos, vol. 18, no. 3, p. 037122, 2008.
47. - "Global analysis of a continuum model for monotone pulsecoupled oscillators," arXiv, vol. math.AP, Feb. 2011.
48. - "Contraction of monotone phase-coupled oscillators," arXiv, vol. math.DS, May 2012.
49. R. E. Mirollo and S. H. Strogatz, "Synchronization of pulse-coupled biological oscillators," SIAM J. Appl. Math., vol. 50, no. 6, pp. 1645-1662, 1990.
50. L. Moreau, "Stability of multiagent systems with time-dependent communication links," IEEE Trans. Autom. Control, vol. 50, no. 2, pp. 169-182, Feb. 2005.
51. J. Nagumo, S. Arimoto, and S. Yoshizawa, "An active pulse transmission line simulating nerve axon," Proc. IRE, vol. 50, no. 10, pp. 2061-2070, 1962.
52. H. Nijmeijer and I. Mareels, "An observer looks at synchronization," IEEE Trans. Circuits Syst. I, vol. 44, no. 10, pp. 882-890, Oct. 1997.
53. H. Nijmeijer and A. Rodríguez-Angeles, Synchronization of Mechanical Systems. World Scientific, 2003.
54. F. Nunez, Y. Wang, A. R. Teel, and F. J. Doyle III, "Bio-inspired synchronization of non-identical pulse-coupled oscillators subject to a global cue and local interactions," in Proc. 4th IFAC Conf. Analysis and Design of Hybrid Systems, Jun. 2012, pp. 115-120.
55. K. Okuda, "Variety and generality of clustering in globally coupled oscillators," Physica D, vol. 63, no. 3-4, pp. 424-436, Mar. 1993.
56. Z. Olami, H. J. S. Feder, and K. Christensen, "Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes." Phys. Rev. Lett., vol. 68, no. 8, pp. 1244-1247, Feb. 1992.
57. G. Orosz, J. Moehlis, and P. Ashwin, "Designing the dynamics of globally coupled oscillators," Prog. Theor. Phys., vol. 122, no. 3, pp. 611-630, Sep. 2009.
58. P. Östborn, "Phase transition to frequency entrainment in a long chain of pulse-coupled oscillators." Phys. Rev. E, vol. 66, no. 1 Pt 2, p. 016105, Jul. 2002.
59. A. V. Pavlov, "The output regulation problem: a convergent dynamics approach," Ph.D. dissertation, Technische Universiteit Eindhoven, 2004.
60. A. V. Pavlov, N. van de Wouw, and H. Nijmeijer, "Convergent systems: analysis and synthesis," in Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems. Berlin/Heidelberg, Germany: Springer-Verlag, 2005, pp. 131-146.
61. C. S. Peskin, Mathematical Aspects of Heart Physiology. New York, NY: Courant Institute of Mathematical Sciences, New York University, 1975.
62. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, ser. Cambridge Nonlinear Science Series. Cambridge, United Kingdom: Cambridge University Press, 2001, vol. 12.
63. M. B. H. Rhouma and H. Frigui, "Self-organization of pulse-coupled oscillators with application to clustering," IEEE Trans. Pattern Anal. Mach. Intell., vol. 23, no. 2, pp. 180-195, Feb. 2001.
64. G. Russo, M. di Bernardo, and E. D. Sontag, "Stability of networked systems: a multi-scale approach using contraction," in Proc. 49th IEEE Conf. Decision and Control, Atlanta, GA, Dec. 2010, pp. 6559-6564.
65. L. Scardovi, M. Arcak, and E. D. Sontag, "Synchronization of interconnected systems with applications to biochemical networks: an input-output approach," IEEE Trans. Autom. Control, vol. 55, no. 6, pp. 1367-1379, Jun. 2010.
66. W. Senn and R. Urbanczik, "Similar nonleaky integrate-and-fire neurons with instantaneous couplings always synchronize," SIAM J. Appl. Math., vol. 61, no. 4, pp. 1143-1155, 2000.
67. R. Sepulchre, "Oscillators as systems and synchrony as a design principle," in Current Trends in Nonlinear Systems and Control: In Honor of Petar Kokotović and Turi Nicosia. Boston, MA: Birkhäuser, 2006.
68. J.-J. E. Slotine and W. Wang, "A study of synchronization and group cooperation using partial contraction theory," in Cooperative Control. Berlin/Heidelberg, Germany: Springer-Verlag, 2005, pp. 207-228.
69. E. D. Sontag, "Contractive systems with inputs," in Perspectives in Mathematical System Theory, Control, and Signal Processing. Berlin/Heidelberg, Germany: Springer, 2010, pp. 217-228.
70. E. D. Sontag and M. Arcak, "Passivity-based stability of interconnection structures," in Recent Advances in Learning and Control. London, England: Springer-Verlag, 2008, pp. 195-204.
71. G.-B. Stan and R. Sepulchre, "Analysis of interconnected oscillators by dissipativity theory," IEEE Trans. Autom. Control, vol. 52, no. 2, pp. 256-270, Feb. 2007.
72. S. H. Strogatz, "From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators," Physica D, vol. 143, no. 1-4, pp. 1-20, Sep. 2000.
73. - Sync: the Emerging Science of Spontaneous Order. New York, NY: Hyperion, 2003.
74. M. Timme, F. Wolf, and T. Geisel, "Coexistence of regular and irregular dynamics in complex networks of pulse-coupled oscillators." Phys. Rev. Lett., vol. 89, no. 25, p. 258701, Dec. 2002.
75. - "Prevalence of unstable attractors in networks of pulse-coupled oscillators." Phys. Rev. Lett., vol. 89, no. 15, p. 154105, Oct. 2002.
76. - "Topological speed limits to network synchronization," Phys. Rev. Lett., vol. 92, no. 7, p. 074101, Feb. 2004.
77. B. van der Pol, "A theory of the amplitude of free and forced triode vibrations," Radio Rev., vol. 1, pp. 701-710, 1920.
78. C. van Vreeswijk, "Partial synchronization in populations of pulsecoupled oscillators," Phys. Rev. E, vol. 54, no. 5, pp. 5522-5537, Nov. 1996.
79. C. van Vreeswijk, L. F. Abbott, and G. B. Ermentrout, "When inhibition not excitation synchronizes neural firing," J. Comput. Neurosci., vol. 1, no. 4, pp. 313-321, 1994.
80. W. Wang and J.-J. E. Slotine, "On partial contraction analysis for coupled nonlinear oscillators," Biol. Cybern., vol. 92, no. 1, pp. 38-53, Dec. 2004.
81. S. Watanabe and S. H. Strogatz, "Constants of motion for superconducting Josephson arrays," Physica D, vol. 74, no. 3-4, pp. 197-253, Jul. 1994.
82. A. T. Winfree, "Biological rhythms and the behavior of populations of coupled oscillators," J. Theor. Biol., vol. 16, no. 1, pp. 15-42, Jul. 1967.
83. - The Geometry of Biological Time, 1st ed., ser. Biomathematics. New York, NY: Springer-Verlag, 1980, vol. 8.
84. YouTube. (2011, Mar.) Sync of metronomes. [Online]. Available: http://www.youtube.com/watch?v=gFnVmuU8-Lg
85. R. Zillmer, R. Livi, A. Politi, and A. Torcini, "Stability of the splay state in pulse-coupled networks," Phys. Rev. E, vol. 76, no. 4 Pt 2, p. 046102, Oct. 2007.