Regular patterns in dichotomically driven activator-inhibitor dynamics.

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.

Noise induced complexity: from subthreshold oscillations to spiking in coupled excitable systems.

We study the 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. In the course of this transition diverse regimes of collective dynamics ranging from periodic subthreshold oscillations to large-amplitude oscillations and chaos are observed. In order to understand the details and mechanisms of these noise-induced dynamics we consider the thermodynamic limit N-->infinity of the ensemble, and derive the cumulant expansion describing temporal evolution of the mean field fluctuations. In Gaussian approximation this allows us to perform the bifurcation analysis; its results are in good qualitative agreement with dynamical scenarios observed in the stochastic simulations of large ensembles.

Macroscopic limit cycle via pure noise-induced phase transitions.

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.