We study stochastic dynamics of an ensemble of N globally coupled excitable elements. Each element is modeled by a FitzHugh-Nagumo oscillator and is disturbed by independent Gaussian noise. In simulations of the Langevin dynamics we characterize the collective behavior of the ensemble in terms of its mean field and show that with the increase of noise the mean field displays a transition from a steady equilibrium to global oscillations and then, for sufficiently large noise, back to another equilibrium. Diverse regimes of collective dynamics ranging from periodic subthreshold oscillations to large-amplitude oscillations and chaos are observed in the course of this transition. In order to understand details and mechanisms of noise-induced dynamics we consider a thermodynamic limit N → ∞ of the ensemble, and derive the cumulant expansion describing temporal evolution of the mean field fluctuations. In the Gaussian approximation this allows us to perform the bifurcation analysis; its results are in good agreement with dynamical scenarios observed in the stochastic simulations of large ensembles.
The dynamics of a spatially extended system of two competing species in the presence of two noise sources is studied. A correlated dichotomous noise acts on the interaction parameter and a multiplicative white noise affects directly the dynamics of the two species. To describe the spatial distribution of the species we use a model based on Lotka-Volterra (LV) equations. By writing them in a mean field form, the corresponding moment equations for the species concentrations are obtained in Gaussian approximation. In this formalism the system dynamics is analyzed for different values of the multiplicative noise intensity. Finally by comparing these results with those obtained by direct simulations of the time discrete version of LV equations, that is coupled map lattice (CML) model, we conclude that the anticorrelated oscillations of the species densities are strictly related to non-overlapping spatial patterns. PACS numbers: 05.40.-a, 05.45.-a, 87.23.Cc
Bistability generated via a pure noise-induced phase transition is reexamined from the view of bifurcations in macroscopic cumulant dynamics. It allows an analytical study of the phase diagram in more general cases than previous methods. In addition, using this approach we investigate spatially extended systems with two degrees of freedom per site. For this system, the analytic solution of the stationary Fokker-Planck equation is not available and a standard mean field approach cannot be used to find noise-induced phase transitions. A different approach based on cumulant dynamics predicts a noise-induced phase transition through a Hopf bifurcation leading to a macroscopic limit cycle motion, which is confirmed by numerical simulation.
We investigate Turing pattern formation in the presence of additive dichotomous fluctuations in the context of an extended system with diffusive coupling and FitzHugh-Nagumo kinetics. The fluctuations vary in space and/or time. Depending on the realization of the dichotomous switching the system is, at a given time (for spatial disorder at a given position) in one of two possible excitable dynamical regimes. Each of the two excitable dynamics for itself does not support pattern formation. With proper dichotomous fluctuations, however, the homogeneous steady state is destabilized via a Turing instability. We investigate the influence of different switching rates (different correlation length of the spatial disorder) on pattern formation. We find three distinct mechanisms: For slow switching existing boundaries become unstable, for high rates the system exhibits "effective bistability" which allows for a Turing instability. For medium rates the fluctuations create spatial structures via a new mechanism where the influence of the fluctuations is twofold. First they produce local inhomogeneities, which then grow (again caused by fluctuations) until the whole space is covered. Utilizing a nonlinear map approach we show bistability of a period-one and a period-two orbit being associated with the steady homogeneous and the Turing pattern state, respectively. Finally, for purely static dichotomous disorder we find destabilization of homogeneous steady states for finite nonzero correlation length of the disorder resulting again in Turing patterns.
We study numerically the stationary solutions of the Fokker-Planck equation for the FitzHugh-Nagumo model with additive noise. In the parameter regimes where the deterministic model is excitable we find various sets of maxima, minima, and saddle points of the stationary probability distribution depending on the noise intensity and separation of the time scales between activator and inhibitor.
Bistability generated via a noise-induced phase transition is reexamined from the view of macroscopic dynamical systems, which clarifies the role of fluctuation better than the conventional Fokker-Plank or Langevin equation approach. Using this approach, we investigated the spatially-extended systems with two degrees of freedom per site. The model systems undergo a noise-induced phase transition through a Hopf bifurcation, leading to a macroscopic limit cycle motion similar to the deterministic relaxation oscillation.
We examine an ensemble of globally coupled FitzHugh–Nagumo systems with Gaussian, white noise. In the case of spatially uncorrelated noise, at high and at low noise levels the mean of the ensemble is steady. In between it exhibits a complex behavior. Depending on the noise intensity we find small scale oscillations, period doubling, chaos, and spiking. We derive equations of motion for the cumulants of the ensemble distribution. The results of the cumulant dynamics analysis are in good qualitative agreement with results from the Langevin dynamics. Once we additionally apply correlated noise but keep the sum of both noise intensities constant, the mean starts spiking where it did not spike with uncorrelated noise. For increasing correlation strength a minimum of the coefficient of variation appears.
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