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Probabilistic convergence guarantees for type-II pulse-coupled oscillators
Joel Nishimura1, Eric J Friedman
1Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
Summary
This study demonstrates that pulse-coupled oscillators reliably synchronize from random states across various networks with time delays. The findings offer new methods for analyzing oscillator networks and designing synchronization systems.
Area of Science:
- Dynamical Systems
- Network Science
- Computational Neuroscience
Background:
- Pulse-coupled oscillators are fundamental in modeling biological systems and engineered networks.
- Understanding synchronization dynamics, especially with time delays, is crucial for system stability and function.
- Previous research established local convergence but lacked comprehensive network-wide analysis.
Purpose of the Study:
- To demonstrate the high-probability convergence of a large class of pulse-coupled oscillators.
- To analyze convergence on diverse network structures incorporating time delays.
- To establish rigorous bounds for convergence probabilities based on network properties.
Main Methods:
- Combining local convergence results with probabilistic network analysis.
- Utilizing a classification scheme for type-II phase response curves.
- Developing rigorous lower bounds for convergence probabilities tied to network density.
Main Results:
- A large class of pulse-coupled oscillators exhibit high-probability convergence from random initial conditions.
- Convergence is demonstrated across a broad range of network topologies with time delays.
- Network density is shown to be a key factor in determining convergence probabilities.
Conclusions:
- The study provides robust analytical methods for pulse-coupled oscillator networks.
- Insights are offered into the excitation-inhibition balance in biological systems.
- The findings support the design of decentralized clock synchronization in sensor networks.
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