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Stochastic resonance (SR) provides a glaring example of a noise-induced transition in a nonlinear system driven by an information signal and noise simultaneously. In the regime of SR some characteristics of the information signal (amplification factor, signal-to-noise ratio, the degrees of coherence and of order, etc.) at the output of the system are significantly improved at a certain optimal noise level. SR is realized only in nonlinear systems for which a noise-intensity-controlled characteristic time becomes available. In the present review the physical mechanism and methods of theoretical description of SR are briefly discussed. SR features determined by the structure of the information signal, noise statistics and properties of particular systems with SR are studied. A nontrivial phenomenon of stochastic synchronization defined as locking of the instantaneous phase and switching frequency of a bistable system by external periodic force is analyzed in detail. Stochastic synchronization is explored in single and coupled bistable oscillators, including ensembles. The effects of SR and stochastic synchronization of ensembles of stochastic resonators are studied both with and without coupling between the elements. SR is considered in dynamical and nondynamical (threshold) systems. The SR effect is analyzed from the viewpoint of information and entropy characteristics of the signal, which determine the degree of order or self-organization in the system. Applications of the SR concept to explaining the results of a series of biological experiments are discussed.

We study the nonlinear response of the Hodgkin-Huxley model without external periodic signal to the noisy synaptic current near the saddle-node bifurcation of limit cycles. The coherence of the system, estimated from the interspike interval histogram and from the power spectra of membrane potentials and spike trains, is maximal at a certain noise intensity, so that the coherence resonance occurs. The mechanism of this phenomenon is found to be different from previously studied models of coherence resonance and explained in terms of rigid excitations of periodic oscillations, and the combined effect of amplitude and phase fluctuations.

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A general mechanism of coherence resonance that occurs in noisy dynamical systems close to the onset of bifurcation is demonstrated through examples of period-doubling and torus-birth bifurcations. Near the bifurcation of a periodic orbit, noise produces the characteristic peaks of ''noisy precursors'' in the power spectrum. The signal-to-noise ratio evaluated at these peaks is maximal for a certain optimal noise intensity in a manner that resembles a stochastic resonance. ͓S1063-651X͑97͒06307-1͔ PACS number͑s͒: 05.40.ϩj, 05.20.Ϫy Nonlinear systems perturbed by noise have the potential to display a wide range of complex responses including, somewhat paradoxically, an enhancement of net order and coherence as noise levels increase. A distinguished example of this phenomenon is stochastic resonance ͑SR͒ ͓1͔ which has attracted considerable attention over the last decade ͑see for references the reviews ͓2͔͒. Conventional SR occurs in noisy dynamical systems when perturbed by a weak external periodic signal. For such systems, significant amplification of the weak periodic signal may occur solely by increasing the level of the noise intensity. The signal-to-noise ratio ͑SNR͒, and other appropriate measures of signal coherence, pass through a maximum at an optimal noise strength when the noise-controlled time scale of the system matches the period of the external signal.A similar effect of noise-induced coherence may also be observed in systems which lack an external signal, but whose intrinsic dynamics are controlled by noise intensity. In earlier studies ͓3,4͔ the noise-induced enhancement of coherence in underdamped nonlinear oscillators has been found. The noise-induced peak at zero frequency appeared in the vicinity of a pitchfork bifurcation ͓3͔, whereas the decrease of the width of a fluctuating peak in the power spectrum is shown for an underdamped oscillator, whose eigenfrequency possesses an extreme in energy, in ͓4͔. Recently, a noiseinduced coherent motion has been observed for autonomous systems in ͓5͔, where the effect of noise on a nonuniform limit cycle has been studied, and in ͓6͔, where a coherence resonance in a noise-driven excitable system has been reported. This group of phenomena can be called coherence resonance or ''internal'' SR, which underlines the fact that one can observe SR-like phenomena without an external periodic signal.In the present paper we study the response of nonlinear dynamical systems to noise excitation near the onset of dynamical instabilities of periodic orbits. Our starting point is the key paper of Wiesenfeld ͓7͔, which carefully elaborates the way in which noise controls the qualitative structure of the power spectrum. In brief, Wiesenfeld demonstrates that the power spectrum of a system observed after a bifurcation point can, nevertheless, be visible even before the bifurcation actually occurs if there is noise present. We thus observe a noisy precursor of the bifurcation.To follow this line of thought further, let us suppose that noise induces a peak of height H at ...

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.

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We report experimental observation of phase synchronization in an array of nonidentical noncoupled noisy neuronal oscillators, due to stimulation with external noise. The synchronization derives from a noise-induced qualitative change in the firing pattern of single neurons, which changes from a quasiperiodic to a bursting mode. We show that at a certain noise intensity the onsets of bursts in different neurons become synchronized, even though the number of spikes inside the bursts may vary for different neurons. We demonstrate this effect both experimentally for the electroreceptor afferents of paddlefish, and numerically for a canonical phase model, and characterize it in terms of stochastic synchronization.