We investigate globally coupled stochastic three-state oscillators, which we
consider as general models of stochastic excitable systems. We compare two
situations:in the first case the transitions between the three states of each
unit 1->2->3->1 are determined by Poissonian waiting time distributions. In the
second case only transition 1->2 is Poissonian whereas the others are
deterministic with a fixed delay. When coupled the second system shows coherent
oscillations whereas the first remains in a stable stationary state. We show
that the coherent oscillations are due to a Hopf-bifurcation in the dynamics of
the occupation probabilities of the discrete states and discuss the bifurcation
diagram.Comment: 10 pages, 4 figures, submitted to Physica
The control of coherence and spectral properties of noise-induced oscillations by time-delayed feedback is studied in a FitzHugh–Nagumo system which serves as a paradigmatic model of excitable systems. A semianalytical approach based on a discrete model with waiting time densities is developed, which allows one to predict quantitatively the increase of coherence measured by the correlation time, and the modulation of the main frequencies of the stochastic dynamics in dependence on the delay time. The analytical mean-field approximation is in good agreement with numerical results for the full nonlinear model.
We consider stochastic excitable units with three discrete states. Each state is characterized by a waiting time density function. This approach allows for a non-Markovian description of the dynamics of separate excitable units and of ensembles of such units. We discuss the emergence of oscillations in a globally coupled ensemble with excitatory coupling. In the limit of a large ensemble we derive the non-Markovian mean-field equations: nonlinear integral equations for the populations of the three states. We analyze the stability of their steady solutions. Collective oscillations are shown to persist in a large parameter region beyond supercritical and subcritical Hopf bifurcations. We compare the results with simulations of discrete units as well as of coupled FitzHugh-Nagumo systems.
We develop a theory to calculate the effective phase diffusion coefficient and the mean phase velocity in periodically driven stochastic models with two discrete states. This theory is applied to a dichotomically driven Markovian two-state system. Explicit expressions for the mean phase velocity, the effective phase diffusion coefficient, and the Pe clet number are analytically calculated. The latter indicates as a measure of phase-coherence forced synchronization of the stochastic system with respect to the periodic driving and exhibits a "bona fide" resonance. In a second step, the theory is applied to a non-Markovian two-state system modeling excitable systems. The results prove again stochastic synchronization to the periodic driving and are in good agreement with simulations of a stochastic FitzHugh-Nagumo system.
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