Jesús Casado-Pascual

Two-State Theory of Nonlinear Stochastic Resonance

An amenable, analytical two-state description of the nonlinear population dynamics of a noisy bistable system driven by a rectangular subthreshold signal is put forward. Explicit expressions for the driven population dynamics, the correlation function (its coherent and incoherent parts), the signal-to-noise ratio (SNR), and the stochastic resonance (SR) gain are obtained. Within a suitably chosen range of parameter values this reduced description yields anomalous SR gains exceeding unity and, simultaneously, gives rise to a nonmonotonic behavior of the SNR vs the noise strength. The analytical results agree well with those obtained from numerical solutions of the Langevin equation.

Gain in stochastic resonance: precise numerics versus linear response theory beyond the two-mode approximation.

In the context of the phenomenon of stochastic resonance (SR), we study the correlation function, the signal-to-noise ratio (SNR), and the ratio of output over input SNR, i.e., the gain, which is associated to the nonlinear response of a bistable system driven by time-periodic forces and white Gaussian noise. These quantifiers for SR are evaluated using the techniques of linear response theory (LRT) beyond the usually employed two-mode approximation scheme. We analytically demonstrate within such an extended LRT description that the gain can indeed not exceed unity. We implement an efficient algorithm, based on work by Greenside and Helfand (detailed in the Appendix), to integrate the driven Langevin equation over a wide range of parameter values. The predictions of LRT are carefully tested against the results obtained from numerical solutions of the corresponding Langevin equation over a wide range of parameter values. We further present an accurate procedure to evaluate the distinct contributions of the coherent and incoherent parts of the correlation function to the SNR and the gain. As a main result we show for subthreshold driving that both the correlation function and the SNR can deviate substantially from the predictions of LRT and yet the gain can be either larger or smaller than unity. In particular, we find that the gain can exceed unity in the strongly nonlinear regime which is characterized by weak noise and very slow multifrequency subthreshold input signals with a small duty cycle. This latter result is in agreement with recent analog simulation results by Gingl et al. [ICNF 2001, edited by G. Bosman (World Scientific, Singapore, 2002), pp. 545-548; Fluct. Noise Lett. 1, L181 (2001)].

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.

Stochastic resonance: Theory and numerics

We address the phenomenon of stochastic resonance in a noisy bistable system driven by a time-dependent periodic force (not necessarily sinusoidal) and in its two-state approximation. Even for driving forces with subthreshold amplitudes, the behavior of the system response might require a nonlinear description. We introduce analytical and numerical tools to analyze the power spectral amplification and the signal-to-noise ratio in a nonlinear regime. Our analysis shows the importance of the effects of the driving force on the system fluctuations in a nonlinear regime. These effects can be usefully exploited to achieve high quality output signals with gains larger than unity, which is impossible within a linear regime.

Effects of additive noise on vibrational resonance in a bistable system.

We study the overdamped motion of a particle in a bistable potential subject to the action of a bichromatic force and additive noise, within the context of the vibrational resonance phenomenon. Under appropriate conditions, we obtain analytical expressions for the relevant observables which quantifies this phenomenon. The theoretical results are compared with those obtained by the numerical solution of the stochastic differential equation which describes the dynamics of the system. The limits of validity of the theoretical approach are also discussed.

Rocking bistable systems: Use and abuse of linear response theory

The response of a nonlinear stochastic system driven by an external sinusoidal time-dependent force is studied by a variety of numerical and analytical approximations. The validity of linear response theory is put to a critical test by comparing its predictions with numerical solutions over an extended parameter regime of driving amplitudes and frequencies. The relevance of the driving frequency for the applicability of linear response theory is explored.

High-frequency effects in the Fitzhugh-Nagumo neuron model

The effect of a high-frequency signal on the FitzHugh-Nagumo excitable model is analyzed. We show that the firing rate is diminished as the ratio of the high-frequency amplitude to its frequency is increased. Moreover, it is demonstrated that the excitable character of the system, and consequently the firing activity, is suppressed for ratios above a given threshold value. In addition, we show that the vibrational resonance phenomenon turns up for sufficiently large noise strength values.

Quantum stochastic synchronization.

We study, within the spin-boson dynamics, the synchronization of a quantum tunneling system with an external, time-periodic driving signal. As a main result, we find that at a sufficiently large system-bath coupling strength (i.e., for a friction strength alpha > 1) the thermal noise plays a constructive role in yielding forced synchronization. This noise-induced synchronization can occur when the driving frequency is larger than the zero-temperature tunneling rate. As an application evidencing the effect, we consider the charge transfer dynamics in molecular complexes.

The role of different reorganization energies within the Zusman theory of electron transfer

We consider the kinetics of electron transfer reactions in condensed media with different reorganization energies for the forward and backward processes. The starting point of our analysis is an extension of the well-known Zusman equations to the case of parabolic diabatic curves with different curvatures. A generalized master equation for the populations as well as formal expressions for their long-time limit is derived. We discuss the conditions under which the time evolution of the populations of reactants and products can be described at all times by a single exponential law. In the limit of very small tunnel splitting, a novel rate formula for the nonadiabatic transitions is obtained. It generalizes previous results derived within the contact approximation. For larger values of the tunnel splitting, we make use of the consecutive step approximation leading to a rate formula that bridges between the nonadiabatic and solvent-controlled adiabatic regimes. Finally, the analytical predictions for the long...

Theory of frequency and phase synchronization in a rocked bistable stochastic system.

We investigate the role of noise in the phenomenon of stochastic synchronization of switching events in a rocked, overdamped bistable potential driven by white Gaussian noise, the archetype description of stochastic resonance. We present an approach to the stochastic counting process of noise-induced switching events: starting from the Markovian dynamics of the nonstationary, continuous particle dynamics, one finds upon contraction onto two states a non-Markovian renewal dynamics. A proper definition of an output discrete phase is given, and the time rate of change of its noise average determines the corresponding output frequency. The phenomenon of noise-assisted phase synchronization is investigated in terms of an effective, instantaneous phase diffusion. The theory is applied to rectangular-shaped rocking signals versus increasing input-noise strengths. In this case, for an appropriate choice of the parameter values, the system exhibits a noise-induced frequency locking accompanied by a very pronounced suppression of the phase diffusion of the output signal. Precise numerical simulations corroborate very favorably our analytical results. The novel theoretical findings are also compared with prior ones.