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Stochastic Hierarchical Systems: Excitable Dynamics

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Abstract

We present a discrete model of stochastic excitability by a low-dimensional set of delayed integral equations governing the probability in the rest state, the excited state, and the refractory state. The process is a random walk with discrete states and nonexponential waiting time distributions, which lead to the incorporation of memory kernels in the integral equations. We extend the equations of a single unit to the system of equations for an ensemble of globally coupled oscillators, derive the mean field equations, and investigate bifurcations of steady states. Conditions of destabilization are found, which imply oscillations of the mean fields in the stochastic ensemble. The relation between the mean field equations and the paradigmatic Kuramoto model is shown.

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Acknowledgements

We thank DFG-Sfb 555 for financial support. The authors thank our former coauthor Dr. T. Prager who substantially contributed to elaborate the presented three-state model for stochastic excitable systems.

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Authors and Affiliations

  1. Institute of Physics, Humboldt University at Berlin, Newtonstr. 15, D-12489, Berlin, Germany

    Helmar Leonhardt, Michael A. Zaks & Lutz Schimansky-Geier

  2. Mathematical Cell Physiology, Max Delbrück Centre for Molecular Medicine, Robert Rössle Str. 10, 13092, Berlin, Germany

    Martin Falcke

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  1. Helmar Leonhardt
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  2. Michael A. Zaks
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  3. Martin Falcke
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  4. Lutz Schimansky-Geier
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Correspondence to Lutz Schimansky-Geier.

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Leonhardt, H., Zaks, M.A., Falcke, M. et al. Stochastic Hierarchical Systems: Excitable Dynamics. J Biol Phys 34, 521–538 (2008). https://doi.org/10.1007/s10867-008-9112-1

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