Characterizing an ensemble of interacting oscillators: the mean-field variability index

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Neuroscience

Background:

• Characterizing ensembles of interacting oscillators is crucial in various scientific fields.
• Existing methods may not fully capture the impact of inter-oscillator interactions on overall signal variability.
• Understanding synchronization dynamics is key to analyzing complex systems.

Purpose of the Study:

• To introduce a novel parameter, the mean-field variability index (κ), for characterizing interacting oscillator ensembles.
• To investigate how mutual interactions influence oscillator behavior, independent of oscillator number or spectral properties.
• To demonstrate the utility of κ in analyzing diverse oscillatory systems, including biological signals.

Main Methods:

• Defined the mean-field variability index (κ) as the ratio of the variance of the mean field (r) to the mean square of r.
• Analyzed the behavior of κ for noninteracting oscillators (purely random phasors), where it converges to 0.215.
• Calculated κ using numerically simulated data and electroencephalograph (EEG) signals from human subjects in different states (awake, anesthetized, epileptic).

Main Results:

• κ quantifies the impact of inter-oscillator interactions on signal variability.
• Interactions lower κ with increasing phase coherence, approaching zero for complete phase synchronization.
• Interactions increase κ for amplitude or intermittent synchronization patterns.

Conclusions:

• The mean-field variability index (κ) provides a robust measure of interaction effects in oscillator ensembles.
• κ successfully differentiates various synchronization states, including those observed in human brain activity.
• This parameter offers a valuable tool for analyzing complex oscillatory dynamics across different scientific domains.