Robust Asymptotic Stability of Desynchronization in Impulse-Coupled Oscillators

The property of desynchronization in an all-to-all network of homogeneous impulse-coupled oscillators is studied. Each impulse-coupled oscillator is modeled as a hybrid system with a single timer state that self-resets to zero when it reaches a threshold, at which event all other impulse-coupled oscillators adjust their timers following a common reset law. In this setting, desynchronization is considered as each impulse-coupled oscillator's timer having equal separation between successive resets. We show that for the considered model, desynchronization is an asymptotically stable property. For this purpose, we recast desynchronization as a set stabilization problem and employ Lyapunov stability tools for hybrid systems. Furthermore, several perturbations are considered showing that desynchronization is a robust property. Perturbations on the continuous and discrete dynamics are considered. Numerical results are presented to illustrate the main contributions.

[1]  Rodolphe Sepulchre,et al.  Clustering behaviors in networks of integrate-and-fire oscillators. , 2008, Chaos.

[2]  R. Sanfelice,et al.  Hybrid dynamical systems , 2009, IEEE Control Systems.

[3]  Ricardo G. Sanfelice,et al.  Hybrid Dynamical Systems: Modeling, Stability, and Robustness , 2012 .

[4]  Abbott,et al.  Asynchronous states in networks of pulse-coupled oscillators. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[6]  F. L. D. Silva,et al.  Event-related EEG/MEG synchronization and desynchronization: basic principles , 1999, Clinical Neurophysiology.

[7]  P.A. Tass,et al.  Desynchronization in networks of globally coupled neurons: effects of inertia , 2004, 2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No.04CH37541).

[8]  S. Strogatz,et al.  Synchronization of pulse-coupled biological oscillators , 1990 .

[9]  Radhika Nagpal,et al.  Desynchronization: The Theory of Self-Organizing Algorithms for Round-Robin Scheduling , 2007, First International Conference on Self-Adaptive and Self-Organizing Systems (SASO 2007).

[10]  Ali Nabi,et al.  Nonlinear hybrid control of phase models for coupled oscillators , 2010, Proceedings of the 2010 American Control Conference.

[11]  Charles S. Peskin,et al.  Mathematical aspects of heart physiology , 1975 .

[12]  Ricardo G. Sanfelice,et al.  A toolbox for simulation of hybrid systems in matlab/simulink: hybrid equations (HyEQ) toolbox , 2013, HSCC '13.

[13]  G. Laurent,et al.  Impaired odour discrimination on desynchronization of odour-encoding neural assemblies , 1997, Nature.

[14]  Leonard Maler,et al.  A Synchronization-Desynchronization Code for Natural Communication Signals , 2006, Neuron.

[15]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[16]  Jian Pei,et al.  A dynamic clustering and scheduling approach to energy saving in data collection from wireless sensor networks , 2005, 2005 Second Annual IEEE Communications Society Conference on Sensor and Ad Hoc Communications and Networks, 2005. IEEE SECON 2005..

[17]  From Clocks to Chaos: The Rhythms of Life , 1988 .

[18]  C.-C. Jay Kuo,et al.  Cooperative Communications and Networking , 2010 .

[19]  Bastian Goldlücke,et al.  Variational Analysis , 2014, Computer Vision, A Reference Guide.

[20]  Wulfram Gerstner,et al.  SPIKING NEURON MODELS Single Neurons , Populations , Plasticity , 2002 .