Pattern selection in a lattice of pulse-coupled oscillators

Area of Science:

• Dynamical systems
• Computational neuroscience
• Network science

Background:

• Investigating spatio-temporal pattern formation in systems of coupled oscillators is crucial for understanding complex phenomena.
• Oscillator networks with inhibitory pulselike interactions exhibit diverse dynamic behaviors, including limit cycles where each oscillator fires once.

Purpose of the Study:

• To analyze pattern formation in a ring of N oscillators with unidirectional inhibitory pulselike interactions.
• To introduce and quantify "pattern degeneracy," accounting for multiple limit cycles leading to the same spatio-temporal pattern.
• To estimate the basin of attraction for each pattern and analyze its dependence on coupling strength.

Main Methods:

• Analysis of limit cycle attractors in a network of N coupled oscillators.
• Introduction of the concept of pattern degeneracy to classify distinct dynamic outcomes.
• Application of probabilistic arguments and pattern degeneracy for qualitative estimation of basin of attraction volumes.
• Comparison of theoretical estimations with numerical simulations.

Main Results:

• Identified limit cycles where each oscillator fires exactly once as attractors.
• Quantified pattern degeneracy to account for multiple dynamic states resulting in identical spatio-temporal patterns.
• Developed a method to qualitatively estimate the volume of the basin of attraction for each pattern.
• Demonstrated that the estimative formula accurately predicts simulation results in the limit of small coupling strength.

Conclusions:

• The study provides insights into the complex dynamics of coupled oscillator networks.
• Pattern degeneracy is a key concept for understanding the stability and predictability of spatio-temporal patterns.
• The developed estimation method offers a valuable tool for analyzing network dynamics, particularly under weak coupling conditions.