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A universal description of stochastic oscillators.

Alberto Pérez-Cervera1, Boris Gutkin2, Peter J Thomas3

  • 1Department of Applied Mathematics, Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Madrid 28040, Spain.

Proceedings of the National Academy of Sciences of the United States of America
|July 11, 2023
PubMed
Summary

We developed a new mathematical method to unify the study of random oscillations across physics, chemistry, and biology. This approach simplifies analyzing spontaneous activity and coupled system dynamics.

Keywords:
cross-correlation of coupled oscillatorslinear responsenonlinear stochastic differential equationspower spectrum

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Area of Science:

  • Physics, chemistry, and biology
  • Nonlinear dynamics
  • Stochastic processes

Background:

  • Many natural systems exhibit oscillations with significant random components, arising from diverse mechanisms like noise-perturbed limit cycles or excitable systems.
  • Despite varied origins, these stochastic oscillations often share similar observable behaviors.
  • Existing methods struggle to unify the analysis of these diverse oscillatory systems.

Purpose of the Study:

  • To introduce a novel nonlinear transformation for unifying the mathematical description of stochastic oscillators.
  • To simplify the analysis of spontaneous activity, response to perturbations, and correlation statistics in weakly coupled oscillators.
  • To provide a universal framework for comparing and characterizing different types of random oscillations.

Main Methods:

  • Developed a complex-valued function, denoted as ψ(x), derived from stochastic oscillators.
  • Identified ψ(x) as the eigenfunction of the Kolmogorov backward operator with eigenvalue λ₁ = μ₁ + iω₁.
  • Utilized this transformation to derive exact formulas for power spectra, susceptibility, and cross-spectra.

Main Results:

  • The complex-valued function's power spectrum precisely follows a Lorentz profile with peak frequency ω₁ and half-width μ₁.
  • The system's susceptibility to weak external forcing is characterized by a simple one-pole filter centered at ω₁.
  • Cross-spectra of coupled oscillators are readily expressed using spontaneous power spectra and susceptibilities of individual systems.

Conclusions:

  • The introduced nonlinear transformation offers a unified and simplified mathematical framework for stochastic oscillators.
  • This method allows for direct comparison of qualitatively different stochastic oscillators and provides clear coherence characteristics.
  • The framework facilitates the description and analysis of weakly coupled oscillatory systems across scientific disciplines.