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Graph-facilitated resonant mode counting in stochastic interaction networks.

Michael F Adamer1, Thomas E Woolley2, Heather A Harrington3

  • 1Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX1 2JD, UK adamer@maths.ox.ac.uk.

Journal of the Royal Society, Interface
|December 8, 2017
PubMed
Summary

This study introduces an efficient framework for analyzing stochastic resonance in dynamical systems. It simplifies identifying resonant modes and parameter ranges using graph theory and root counting, crucial for complex networks.

Keywords:
chemical reaction networksgraph theoretic methodsquasi-cyclesresonant modessturm chains

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Area of Science:

  • Applied Mathematics
  • Dynamical Systems Theory
  • Stochastic Processes

Background:

  • Oscillations are common in deterministic dynamical systems.
  • Low copy number stochastic regimes can induce cyclic behavior in non-oscillatory systems.
  • Current methods for analyzing stochastic resonance are computationally intensive for large networks.

Purpose of the Study:

  • To develop a systematic and efficient framework for determining parameter ranges and the number of resonant modes in stochastic oscillations.
  • To establish stochastic resonance as a network property.

Main Methods:

  • Utilizing real root counting algorithms.
  • Applying graph theoretic methods for network analysis.
  • Analyzing the Jacobian matrix squared (J^2) to characterize resonant modes.

Main Results:

  • The proposed framework efficiently identifies parameter ranges and resonant modes for stochastic oscillations.
  • Stochastic resonance is demonstrated to be a network property, dependent on J^2.
  • Graph theory simplifies the analysis of stochastic behavior in large interaction networks.

Conclusions:

  • The developed framework offers an efficient approach to studying stochastic resonance.
  • The findings highlight the network-dependent nature of stochastic resonance.
  • This work facilitates the identification of multiple resonant modes in complex stochastic dynamical systems.