Periodically time-modulated bistable systems: Nonstationary statistical properties

Autonomous stochastic resonance driven by colored noise

A one-dimensional linear autonomous system coupled to a generic stationary nonequilibrium fluctuating bath can exhibit resonant response when its damped oscillation period matches some characteristic bath's relaxation time. This condition justifies invoking the stochastic resonance paradigm, even if it can be achieved more easily by tuning the system to the bath and not vice versa, as is usually the case. The simple nature of the mechanism numerically investigated here suggests a number of interesting applications for instance in the context of (1) energy harvesting from random ambient vibrations, (2) activated barrier crossing through a saddle point, or (3) an unstable limit cycle.

Dissipative Double-Well Potential for Cold Atoms: Kramers Rate and Stochastic Resonance

We experimentally study particle exchange in a dissipative double-well potential using laser-cooled atoms in a hybrid trap. We measure the particle hopping rate as a function of barrier height, temperature, and atom number. Single-particle resolution allows us to measure rates over more than 4 orders of magnitude and distinguish the effects of loss and hopping. Deviations from the Arrhenius-law scaling at high barrier heights occur due to cold collisions between atoms within a well. By driving the system periodically, we characterize the phenomenon of stochastic resonance in the system response.

Analytic formula for leading-order nonlinear coherent response in stochastic resonance

The response of an overdamped bistable system driven by a Gaussian white noise and perturbed by a weak monochromatic signal is studied analytically. The perturbation theory is employed to calculate the nonlinear coherent response in the leading order of the amplitude of the weak signal. Simple analytic formulas for the linear and the nonlinear responses have been derived in low noise and low-frequency regime and the results based on the derived formulas are compared with the numerical results.

Double stochastic resonance over an asymmetric barrier

In a recent experiment [Müller, Phys. Rev. A 79, 031804(R) (2009)] reported a splitting of the stochastic resonance peak, which they attributed to the asymmetry of an effective double-well restoring potential in their optomechanical read-out device. We show here that such an effect, though smaller than reported, is indeed consistent with a characterization of stochastic resonance as a synchronization phenomenon, while it proves elusive in terms of spectral quantifiers.

Diffusion process with two reflecting barriers in a time-dependent potential

We consider a Brownian particle which is driven by a harmonically oscillating force, the motion of which is restricted by two reflecting boundaries. We solve the Fokker-Planck equation using the finite-element method and focus on the dynamics of the mean position of the particle in the time-asymptotic regime. As a function of the strength of the external force, the response of the system, i.e., the amplitude of the mean position and the dynamical shift, in the stationary limit shows a resonancelike behavior as a function of the diffusion coefficient for certain parameter regimes. We explain these numerical results heuristically and give some qualitative estimates.

Stochastic synchronization via noise recycling

The nonequilibrium escape dynamics in a bistable system under the influence of two Gaussian noises, one obtained by recycling, the other with a constant delay time, is shown by numerical simulation to exhibit stochastic synchronization; i.e., under stationary conditions, an appreciable fraction of renewal trajectories gets locked to the noise-recycling delay time. The conditions for optimal synchronization, reminiscent of stochastic resonance, are interpreted at any order within the framework of Kramers' theory.

Asymmetric probability densities in symmetrically modulated bistable devices

A Brownian particle hopping in a symmetric double-well potential can be statistically confined into a single well by the simultaneous action of (a) two periodic input signals, one tilting the minima and the other one modulating the barrier height, and (b) an additive and a purely multiplicative random signal, generated by a unique source and thus preserving a certain degree of statistical correlation. The underlying gating mechanism is quite robust when compared, for instance, with biharmonic rocking. In view of technological implementation, asymmetric confinement through gating can be conveniently maximized by tuning the input signal parameters (correlation time, phase-time lag, amplitudes), thus revealing a resonant localization mechanism of general applicability.

Subthreshold stochastic resonance: rectangular signals can cause anomalous large gains

The main objective of this work is to explore aspects of stochastic resonance (SR) in noisy bistable, symmetric systems driven by subthreshold periodic rectangular external signals possessing a large duty cycle of unity. Using a precise numerical solution of the Langevin equation, we carry out a detailed analysis of the behavior of the first two cumulant averages, the correlation function, and its coherent and incoherent parts. We also depict the nonmonotonic behavior versus the noise strength of several SR quantifiers such as the average output amplitude, i.e., the spectral amplification, the signal-to-noise ratio, and the SR gain. In particular, we find that with subthreshold amplitudes and for an appropriate duration of the pulses of the driving force, the phenomenon of stochastic resonance is accompanied by SR gains exceeding unity. This analysis thus sheds light on the interplay between nonlinearity and the nonlinear response, which in turn yields nontrivial unexpected SR gains above unity.

Asymptotic distributions of periodically driven stochastic systems

We study the large-time behavior of Brownian particles moving through a viscous medium in a confined potential, and which are further subjected to position-dependent driving forces that are periodic in time. We focus on the case where these driving forces are rapidly oscillating with an amplitude that is not necessarily small. We develop a perturbative method for the high-frequency regime to find the large-time behavior of periodically driven stochastic systems. The asymptotic distribution of Brownian particles is then determined to second order. To first order, these particles are found to execute small-amplitude oscillations around an effective static potential that can have interesting forms.

Spatiotemporal stochastic resonance and its consequences in neural model systems

The realization of spatiotemporal stochastic resonance is studied in a two-dimensional FitzHugh-Nagumo system, and in a one-dimensional system of integrate-and-fire neurons. We show that spatiotemporal stochastic resonance occurs in these neural model systems, independent of the method of modeling. Moreover, the ways of realization are analogous in the two model systems. The biological implications and open questions are discussed. (c) 2001 American Institute of Physics.

Effects of noise coherence on stochastic resonance enhancement in a bithreshold system

We identify a method for optimizing the stochastic resonance (SR) in a symmetric bithreshold device: by varying the coherence of the added noise series. To show SR enhancement via this method, we compare the performance of the system using noise sources with different coherence at normalized amplitude. The normalization of the noise amplitude is based on the mean threshold crossing rate of the Gaussian white noise, which is considered as the standard noise in SR studies, at optimal variance. The amplitude for optimal performance of the Gaussian white noise is determined using a signal-to-noise ratio (Q). The Q measure is also used to compare and examine the system performance for different noise cases. This measure is used because it is particularly sensitive to the effects of coherence on the quality of the output power spectrum.

Nonlinear stochastic resonance: the saga of anomalous output-input gain

We reconsider stochastic resonance (SR) for an overdamped bistable dynamics driven by a harmonic force and Gaussian noise from the viewpoint of the gain behavior, i.e., the signal-to-noise ratio (SNR) at the output divided by that at the input. The primary issue addressed in this work is whether a gain exceeding unity can occur for this archetypal SR model, for subthreshold signals that are beyond the regime of validity of linear response theory: in contrast to nondynamical threshold systems, we find that the nonlinear gain in this conventional SR system exceeds unity only for suprathreshold signals, where SR for the spectral amplification and/or the SNR no longer occurs. Moreover, the gain assumes, at weak to moderate noise strengths, rather small (minimal) values for near-threshold signal amplitudes. The SNR gain generically exhibits a distinctive nonmonotonic behavior versus both the signal amplitude at fixed noise intensity and the noise intensity at fixed signal amplitude. We also test the validity of linear response theory; this approximation is strongly violated for weak noise. At strong noise, however, its validity regime extends well into the large driving regime above threshold. The prominent role of physically realistic noise color is studied for exponentially correlated Gaussian noise of constant intensity scaling and also for constant variance scaling; the latter produces a characteristic, resonancelike gain behavior. The gain for this typical SR setup is further contrasted with the gain behavior for a "soft" potential model.

Stochastic resonance in a single neuron model: theory and analog simulation

Here, we consider a noisy, bistable, single neuron model in the presence of periodic external modulation. The modulation induces a correlated switching between states driven by the noise. The information flow through the system, from the modulation, or signal, to the output switching events, leads to a succession of strong peaks in the power spectrum. The signal-to-noise ratio (SNR) obtained from this power spectrum is a measure of the information content in the neuron response. With increasing noise intensity, the SNR passes through a maximum: an effect which has been called stochastic resonance, and which was first advanced as a possible explanation of the observed periodicity in the recurrences of the Earth's ice ages. We treat the problem within the framework of a recently developed approximate theory, valid in the limits of weak noise intensity, weak periodic forcing and low forcing frequency, for both additive and multiplicative noise. Moreover, we have constructed an analog simulator of the neuron which demonstrates the stochastic resonance effect, and with which we have measured the SNRs for comparison with the theoretical results. Our model should be of interest in situations where a single inherently noisy neuron is the receptor of a periodic signal, which is itself noisy, either from the network or from an external source.