Stochastic Hierarchical Systems: Excitable Dynamics |
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Authors: | Helmar Leonhardt Michael A Zaks Martin Falcke Lutz Schimansky-Geier |
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Institution: | 1.Institute of Physics,Humboldt University at Berlin,Berlin,Germany;2.Mathematical Cell Physiology,Max Delbrück Centre for Molecular Medicine,Berlin,Germany |
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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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Keywords: | Excitable systems Delayed integral equations Time-convoluted master equations |
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