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The statistical and dynamical properties of ions in the selectivity filter of the KcsA ion channel are considered on the basis of molecular dynamics (MD) simulations of the KcsA protein embedded in a lipid membrane surrounded by an ionic solution. A new approach to the derivation of a Brownian dynamics (BD) model of ion permeation through the filter is discussed, based on unbiased MD simulations. It is shown that depending on additional assumptions, ion's dynamics can be described either by under-damped Langevin equation with constant damping and white noise or by Langevin equation with a fractional memory kernel. A comparison of the potential of the mean force derived from unbiased MD simulations with the potential produced by the umbrella sampling method demonstrates significant differences in these potentials. The origin of these differences is an open question that requires further clarifications., to appear in Eur. Phys. J. Spec. Top

Resonant activation, rectification of fluctuations, and current reversal, are investigated numerically in an asymmetric periodic potential. The results are compared to recent predictions based on the theory of the logarithmic susceptibility. Possible applications are discussed.

For pt.I see ibid., vol.46, no.9, pp.1205-14 (1999). Stochastic resonance (SR), in which a periodic signal in a nonlinear system can be amplified by added noise, is discussed. The application of circuit modeling techniques to the conventional form of SR, which occurs in static bistable potentials, was considered in a companion paper. Here, the investigation of nonconventional forms of SR in part using similar electronic techniques is described. In the small-signal limit, the results are well described in terms of linear response theory. Some other phenomena of topical interest, closely related to SR, are also treated.

Thermally activated escape in the presence of a periodic external field is investigated theoretically and through analogue experiments and digital simulations. The observed variation of the activation energy for escape with driving force parameters is accurately described by the logarithmic susceptibility (LS). The frequency dispersion of the LS is shown to differ markedly from the standard linear susceptibility. Experimental data on the dispersion are in quantitative agreement with the theory. Switching between different branches of the activation energy is demonstrated for a nonsinusoidal (biharmonic) force.

The design of analogue electronic experiments to investigate phenomena in nonlinear dynamics, especially stochastic phenomena, is described in practical terms. The advantages and disadvantages of this approach, in comparison to more conventional digital methods, are discussed. It is pointed out that analogue simulation provides a simple, inexpensive, technique that is easily applied in any laboratory to facilitate the design and implementation of complicated and expensive experimental projects; and that there are some important problems for which analogue methods have so far provided the only experimental approach. Applications to several topical problems are reviewed. Large rare fluctuations are studied through measurements of the prehistory probability distribution, thereby testing for the first time some fundamental tenets of fluctuation theory. It has thus been shown for example that, whereas the fluctuations of equilibrium systems obey time-reversal symmetry, those under non-equilibrium conditions are temporally asymmetric. Stochastic resonance, in which the signal-to-noise ratio for a weak periodic signal in a nonlinear system can be enhanced by added noise, has been widely studied by analogue methods, and the main results are reviewed; the closely related phenomena of noise-enhanced heterodyning and noise-induced linearization are also described. Selected examples of the use of analogue methods for the study of transient phenomena in time-evolving systems are reviewed. Analogue experiments with quasimonochromatic noise, whose power spectral density is peaked at some characteristic frequency, have led to the discovery of a range of interesting and often counter-intuitive effects. These are reviewed and related to large fluctuation phenomena. Analogue studies of two examples of deterministic nonlinear effects, modulation-induced negative differential resistance (MINDR) and zero-dispersion nonlinear resonance (ZDNR) are described. Finally, some speculative remarks about possible future directions and applications of analogue experiments are discussed.

Zero-dispersion stochastic resonance (ZDSR) in underdamped SQUIDs (superconducting quantum interference devices) has been investigated theoretically and by analogue electronic experiment. It is shown that, in common with the standard form of stochastic resonance (SR) that occurs at low frequencies in overdamped bistable systems, ZDSR can result in large noise-induced enhancements of the signal/noise ratio for a weak periodic signal. Unlike standard SR, however, ZDSR is to be anticipated over a vastly wider and tunable range of high frequencies in highly underdamped SQUIDs that need not necessarily be bistable.

We consider noise-driven exit from a domain of attraction in a two-dimensional bistable system lacking detailed balance. Through analog and digital stochastic simulations, we find a theoretically predicted bifurcation of the most probable exit path as the parameters of the system are changed, and a corresponding nonanalyticity of the activation energy. We also investigate the extent to which the bifurcation is related to the local breaking of time-reversal invariance., 4 pages, 5 figures, REVTEX

Brownian dynamics simulations provide evidence for a remote knock-on mechanism facilitating the permeation of a biological ion channel by an ion that is initially trapped at the selectivity filter (SF). Unlike the case of conventional direct knock-on, the second ion that instigates permeation does not need to enter the channel. Nor does it necessarily take the place of the permeating ion at the SF, and it can even be of a different ionic species. The study is based on the simultaneous, self-consistent, solution of the coupled Poisson and Langevin equations for a simple generic model, taking account of all the charges present. The new permeation mechanism involves electrostatic amplification attributable to the permittivity mismatch between water and protein: the arrival of the instigating ion at the channel entrance reduces the exit barrier for the ion trapped at the SF, facilitating escape.

The ionic permeation of a biological ion channel is a multi-particle, non-equilibrium, stochastic process. Brownian dynamics simulations for a simple electrostatic model of the calcium channel reveal regular structure in the conductance and selectivity as functions of the negative fixed charge Qf on the protein wall at the selectivity filter. This structure consists of distinct high conductance regions (conduction bands) separated by regions of near non-conductance (stop-bands). We report self-consistent electrostatic calculations of single-file, double-ion, stochastic optimal trajectories, and of the energy profiles along these trajectories, for different Qf. We show that the energy difference ΔE along the optimal path exhibits a pronounced minimum near Qf = 3e corresponding to an almost barrier-less (ΔE ~ kBT) resonance-like form of conduction. We demonstrate explicitly that the sharply-defined conduction/selectivity peak of the L-type calcium channel is attributable to the barrier-less knock-on motion of a pair of calcium ions that can occur when their mutual electrostatic repulsion balances their electrostatic attraction to the charge at the selectivity filter. The electrostatics calculations agree well with the results of Brownian dynamics simulations. These results clarify the longstanding puzzle of how the L-type calcium channel exhibits, simultaneously, both high calcium selectivity and conduction at almost the rate of free diffusion.

We consider the stochastic dynamics of escape in an excitable system, the FitzHugh-Nagumo (FHN) neuronal model, for different classes of excitability. We discuss, first, the threshold structure of the FHN model as an example of a system without a saddle state. We then develop a nonlinear (nonlocal) stability approach based on the theory of large fluctuations, including a finite-noise correction, to describe noise-induced escape in the excitable regime. We show that the threshold structure is revealed via patterns of most probable (optimal) fluctuational paths. The approach allows us to estimate the escape rate and the exit location distribution. We compare the responses of a monostable resonator and monostable integrator to stochastic input signals and to a mixture of periodic and stochastic stimuli. Unlike the commonly used local analysis of the stable state, our nonlocal approach based on optimal paths yields results that are in good agreement with direct numerical simulations of the Langevin equation. DOI: 10.1103/PhysRevE.87.032116

The effect of fluctuations on the nonlinear response of an underdamped oscillator to an external periodic field at a subharmonic frequency has been investigated theoretically, numerically, and with an analog electronic circuit model. The system studied has often been analyzed in nonlinear optics in the context of two-photon absorption and second-harmonic generation. We consider its nonlinear spectroscopy. Its resonant nonlinear response is described over a broad range of the fluctuation intensities. It is shown that the fluctuation intensity can be used to "tune" the oscillator so as to maximize the nonlinear response. The dependence of the absorption cross section on the fluctuation intensity displays a clearly resolved maximum. If the eigenfrequency of the oscillator is a nonmonotonic function of its energy, the signal at the second harmonic displays a resonant peak at one of two different frequencies, depending on the noise intensity.

Statistical distributions, power spectra and susceptibilities of an all-optical bistable system subject to noise are considered. In the presence of an additional small periodic signal, a stochastic-resonance phenomenon for additive noise is found to occur, and has been investigated theoretically and experimentally. A new form of optical heterodyning related to stochastic resonance, in which two high-frequency signals (an input signal and a reference one) are applied to an all-optical bistable system, is reported. A noise-induced enhancement of the heterodyne signal has been investigated theoretically, by means of analogue electronic simulation and in experiments in which a optical system was driven by two modulated laser beams at different wavelengths.

It is argued, on the basis of linear response theory (LRT), that new types of stochastic resonance (SR) are to be anticipated in diverse systems, quite different from the one most commonly studied to date, which has a static double-well potential and is driven by a net force equal to the sum of periodic and stochastic terms. On this basis, three new nonconventional forms of SR are predicted, sought, found, and investigated both theoretically and by analogue electronic experiment: (a) in monostable systems; (b) in bistable systems with periodically modulated noise; and (c) in a system with coexisting periodic attractors. In each case, it is shown that LRT can provide a good quantitative description of the experimental results for sufficiently weak driving fields. It is concluded that SR is a much more general phenomenon than has hitherto been appreciated.

A different form of stochastic resonance, in which the weak periodic force is applied multiplicatively (rather than additively) in the noise, has been investigated for a Brownian particle moving in a double-well potential. A regular periodic signal whose amplitude increases sharply with increasing noise intensity is shown to arise when the potential is asymmetric. The experimental measurements are in good agreement with a theoretical analysis.

Population fluctuations in a predator-prey system are analyzed for the case where the number of prey could be determined, subject to measurement noise, but the number of predators was unknown. The problem of how to infer the unmeasured predator dynamics, as well as the model parameters, is addressed. Two solutions are suggested. In the first of these, measurement noise and the dynamical noise in the equation for predator population are neglected; the problem is reduced to a one-dimensional case, and a Bayesian dynamical inference algorithm is employed to reconstruct the model parameters. In the second solution a full-scale Markov Chain Monte Carlo simulation is used to infer both the unknown predator trajectory, and also the model parameters, using the one-dimensional solution as an initial guess.

The problem of how to reconstruct the parameters of a stochastic nonlinear dynamical system when these are time-varying is considered in the context of online decoding of physiological information from neuron signaling activity. To model the spiking of neurons, a set of FitzHugh-Nagumo (FHN) oscillators is used. It is assumed that only a fast dynamical variable can be detected for each neuron, and that the monitored signals are mixed by an unknown measurement matrix. The Bayesian framework introduced in Paper I (Phys. Rev. E 77, 06110500 (2008)) is applied both for reconstruction of the model parameters and elements of the measurement matrix, and for inference of the time-varying parameters in the non-stationary system. It is shown that the proposed approach is able to reconstruct unmeasured (hidden) slow variables of the FHN oscillators, to learn to model each individual neuron, and to track continuous, random and step-wise variations of the control parameter for each neuron in real time.

Recent progress towards an understanding of fluctuational escape from chaotic attractors (CAs) is reviewed and discussed in the contexts of both continuous systems and maps. It is shown that, like the simpler case of escape from a regular attractor, a unique most probable escape path (MPEP) is followed from a CA to the boundary of its basin of attraction. This remains true even where the boundary structure is fractal. The importance of the boundary conditions on the attractor is emphasized. It seems that a generic feature of the escape path is that it passes via certain unstable periodic orbits. The problems still remaining to be solved are identified and considered.

A novel conceptual model is introduced in which ion permeation is coupled to the protein wall vibration and the later in turn modulates exponentially strongly the permeation via radial oscillations of the potential of mean force. In the framework of this model of ion-wall-water interaction we discuss problems of selectivity between alike ions and coupling of ion permeation to gating.

A direct comparison between continuous and discrete forms of analysis of control and stability is investigated theoretically and numerically. We demonstrate that the continuous method provides a more energy-efficient means of controlling the switching of a periodically-driven class-B laser between its stable and unstable pulsing regimes. We provide insight into this result using the close correspondence that exists between the problems of energy-optimal control and the stability of a steady state.

We consider the following general problem of applied stochastic nonlinear dynamics. We observe a time series of signals y(t) = y(t0+hn) corrupted by noise. The actual state and the nonlinear vector field of the dynamical system is not known. The question is how and with what accuracy can we determine x(t) and functional form of f(x). In this talk we discuss a novel approach to the solution of this problem based on the application of the path-integral approach to the full Bayesian inference. We demonstrate a reconstruction of a dynamical state of a system from corrupted by noise measurements. Next we reconstruct the corresponding nonlinear vector field. The emphasis are on the theoretical analysis. The results are compared with the results of earlier research.© (2005) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

A new technique is introduced to reconstruct a nonlinear stochastic model of the cardiorespiratory interaction. Its inferential framework uses a set of polynomial basis functions representing the nonlinear force governing the system oscillations. The strength and direction of coupling, and the noise intensity are simultaneously inferred from a univariate blood pressure signal, monitored in a clinical environment. The technique does not require extensive global optimization and it is applicable to a wide range of complex dynamical systems subject to noise., Comment: 5 pages, 4 figures, 1 table. accepted for publication in Phys. Rev. Lett. 2005

Ionic motion through an open ion channel is analyzed within the framework of self‐consistent Brownian dynamics. A novel conceptual model in which the ions motion is coupled to the vibrations of the pore walls is introduced. The model allows to include into simulations an important additional mechanism of energy dissipation and the effects of self‐induced strong modulation of the channel conductivity.

Fluctuational transitions between two co-existing chaotic attractors, separated by a fractal basin boundary, are studied in a discrete dynamical system. It is shown that the mechanism for such transitions is determined by a hierarchy of homoclinic points. The most probable escape path from the chaotic attractor to the fractal boundary is found using both statistical analyses of fluctuational trajectories and the Hamiltonian theory of fluctuations., 4 pages; 3 figures

In this chapter the authors discuss the application of simulation techniques to the study of fluctuational escape and related phenomena in nonliner optical systems: that is, situations where a large deviation of the system from an equilibrium state occurs under the influence of relatively weak noise present in the system. The authors are interested primarily in the analysis of situations where large deviations lead to new nontrivial behaviour or to a transition to a different state.

Fluctuational transitions between two coexisting attractors are investigated. Two different systems are considered: the periodically driven nonlinear oscillator and the two‐dimensional map introduced by Holmes. These two systems have smooth and fractal boundaries, respectively, separating their coexisting attractors. It is shown that, starting from a cycle embedded in the chaotic attractor, the periodically‐driven oscillator escapes to a saddle cycle at the boundary of the basin of attraction, and does so through sequential transitions between saddles cycles embedded in the attractor. In the case of discrete dynamics with locally disconnected fractal boundaries, it is shown that escape from an attractor always seems to occur through an accessible boundary orbit and further through the specific homoclinic points forming a fractal structure of the boundary. It is shown that analysis of fluctuational transitions between attractors can be used to solve a problem of the energy‐optimal migration of a chaotic system. The deterministic optimal control functions are identified with the corresponding optimal fluctuational forces in the limit of small noise intensity. We discuss possible applications and related unsolved problems of stochastic dynamics.

Fluctuational transitions between the stationary states of periodically-driven nonlinear oscillators are investigated by means of numerical simulations and analogue experiments for over-damped, weakly damped and chaotic motion. It is shown that transitions take place along distinct most probable escape paths (MPEPs) in all three cases. The transition probabilities are compared with predictions based on the theory of the logarithmic susceptibility (LS). It is found that, in agreement with theoretical predictions, the log of the transition probability displays an exponentially sharp dependence on the field frequency for the case of over-damped motion, and resonant behaviour in the case of weak damping, and that it is linear in the field amplitude in both cases. Particular attention is paid to the analysis of noise-induced escape from the quasi-attractor of a periodically-driven underdamped oscillator: it is demonstrated that analysis of the escape process can be reduced to the analysis of transitions between a few saddle limit cycles of low period, thus raising the possibility of an analytic description based on an extension of the LS theory.

The energy-optimal entraining of the dynamics of a periodically driven oscillator, moving it from a chaotic attractor to a coexisting stable limit cycle, is investigated via analysis of fluctuational transitions between the two states. The deterministic optimal control function is identified with the corresponding optimal fluctuational force, which is found by numerical and analog simulations.

We generalize Chirikov’s resonance-overlap criterion for the onset of global chaos in Hamiltonian systems to describe the onset of chaotic attractors in weakly dissipative systems.

Noise-induced transitions between coexisting stable states of a periodically driven nonlinear oscillator have been investigated by means of analog experiments and numerical simulations in the non-adiabatic limit for a wide range of oscillator parameters. It is shown that, for over-damped motion, the field-induced corrections to the activation energy can be described quantitatively in terms of the logarithmic susceptibility (LS) and that the measured frequency dispersion of the corresponding corrections for a weakly damped nonlinear oscillator is in qualitative agreement with the theoretical prediction. Resonantly directed diffusion is observed in numerical simulations of a weakly damped oscillator. The possibility of extending the LS approach to encompass escape from the basin of attraction of a quasi-attractor is discussed.

The problem of activated escape in a far-from-equilibrium system is analyzed theoretically, numerically and by analogue electronic experiments. The process is shown to occur via optimal fluctuations that are well described, under diverse conditions, in terms of a new physical quantity, the logarithmic susceptibility (LS).

Noise-induced escape from a quasi-attractor, and from the Lorenz attractor with non-fractal boundaries, are compared through measurements of optimal paths. It has been found that, for both types of attractor, there exists a most probable (optimal) escape trajectory, the prehistory of the escape being defined by the structure of the chaotic attractor. For a quasi-attractor the escape process is realized via several steps, which include transitions between low-period saddle cycles co-existing in the system phase space. The prehistory of escape from the Lorenz attractor is defined by stable and unstable manifolds of the saddle center point, and the escape itself consists of crossing the saddle cycle surrounding one of the stable point-attractors.

Escape from a metastable potential is considered on time-scales less than are needed for the creation of quasi-equilibrium within the well. It is shown that the escape flux may then depend exponentially strongly, and in a complicated way, on friction and time.

Noise-induced escape from a metastable potential is considered on time-scales preceding the formation of quasi-equilibrium within the metastable part of the potential. It is shown that the escape flux may then depend exponentially strongly, and in a complicated manner, on time and friction.

The first measurements of an exit location distribution are reported for an overdamped nonconservative system perturbed by weak white noise, modeled both numerically and by an analog electronic circuit. In the weak-noise limit the distribution is increasingly concentrated near a saddle point of the dynamics and is increasingly well approximated by a Weibull distribution, in agreement with theoretical predictions. A physical explanation for this behavior is given, which should facilitate the computation of corrections to the limiting form.

Fluctuations in a periodically driven overdamped oscillator are studied theoretically and experimentally in the limit of low noise intensity by investigation of their prehistory. It is shown that, for small noise intensity, fluctuations to points in coordinate space that are remote from the stable states occur along paths that form narrow tubes. The tubes are centered on the optimal paths corresponding to trajectories of an auxiliary Hamiltonian system. The optimal paths themselves, and the tubes of paths around them, are visualized through measurements of the prehistory probability distribution for an electronic model. Some general features of fluctuations in nonequilibrium systems, such as singularities in the pattern of optimal paths, the corresponding nondifferentiability of the generalized nonequilibrium potential, and the feasibility of their experimental investigation, are discussed.

Fluctuations around some average or equilibrium state arise universally in physical systems. Large fluctuations — fluctuations that are much larger than average — occur only rarely but are responsible for many physical processes, such as nucleation in phase transitions, chemical reactions, mutations in DNA sequences, protein transport in cells and failure of electronic devices. They lie at the heart of many discussions1,2,3,4,5 of how the irreversible thermodynamic behaviour of bulk matter relates to the reversible (classical or quantum-mechanical) laws describing the constituent atoms and molecules. Large fluctuations can be described theoretically using hamiltonian6,7 and equivalent path-integral8,9,10,11,12 formulations, but these approaches remain largely untested experimentally, mainly because such fluctuations are rare and also because only recently was an appropriate statistical distribution function formulated11. It was shown recently, however, that experiments on fluctuations using analogue electronic circuits allow the phase-space trajectories of fluctuations in a dynamical system to be observed directly12. Here we show that this approach can be used to identify a fundamental distinction between two types of random motion: fluctuational motion, which takes the system away from a stable state, and relaxational motion back towards this state. We suggest that macroscopic irreversibility is related to temporal asymmetry of these two types of motion, which in turn implies a lack of detailed balance and corresponds to non-differentiability of the generalized nonequilibrium potential in which the motion takes place.

Underdamped oscillators that possess a maximum or minimum in their dependence of eigenfrequency on energy have recently been shown to exhibit a range of unusual phenomena. Because they are associated with the presence of an extremum in whose vicinity the eigenfrequency is almost energy-independent, they have been named zero-dispersion phenomena. They manifest themselves both in the deterministic dynamics and in the presence of noise. When the oscillator is driven by a weak periodic force at a frequency close to that of the extremum, a novel type of nonlinear resonance, zero-dispersion nonlinear resonance (ZDNR) can occur. A giant response then arises even in the absence of resonance between the drive frequency and any eigenoscillation of the system. The properties of ZDNR, the nature of the transition from ZDNR to conventional nonlinear resonance as relevant parameters are varied, the occurrence of dynamical chaos associated with ZDNR, and the influence of noise, are analysed and discussed for both Hamiltonian and dissipative systems.

It is shown theoretically and by analogue electronic experiment that stochastic resonance (SR), in which a weak periodic signal can be optimally enhanced by the addition of noise of appropriate intensity, is to be anticipated in underdamped SQUIDs (superconducting quantum interference devices). It manifests under conditions quite unlike those needed for classical SR, which is restricted to low frequencies and confined to systems that are both overdamped and bistable. The zero-dispersion SR reported here can be expected over a vastly wider, tunable, range of high frequencies in highly underdamped SQUIDs that need not necessarily be bistable.

It is shown theoretically and by analog electronic experiment that, in dissipative oscillatory systems for which the frequency of eigenoscillation displays extremum as a function of energy, the dynamics of nonlinear resonance can differ markedly from the conventional case. Transitions between the conventional and novel types of nonlinear resonance, as parameters vary, correspond to changes in the topology of basins of attraction. With added noise, they can result in drastic changes in fluctuational transition rates between small- and large-amplitude regimes. {copyright} {ital 1996 The American Physical Society.}

We review some basic theoretical ideas and revealing experimental observations on stochastic resonance, a phenomenon in which the signal and/or signal-to-noise ratio in a nonlinear system increases with increasing intensity of noise. These ideas and results are related to those known from earlier work in condensed matter physics.

A new form of frequency‐selective all‐optical heterodyning at the difference frequency of two modulated laser beams, related to stochastic resonance, is reported. A noise‐induced enhancement of the heterodyne signal and of the signal‐to‐noise ratio has been predicted and observed in an all‐optical bistable system for the beams at different wavelengths.

We outline the historical development of stochastic resonance (SR), a phenomenon in which the signal and/or the signal-to-noise ratio in a nonlinear system increases with increasing intensity of noise. We discuss basic theoreticak ideas explaining and describing SR, and we review some revealing experimental data that place SR within the wider context of statistical physics. We emphasize the close relationship of SR to some effects that are well known in condensed-matter physics.

We present a nonequilibrium reaction rate model of the ionic transition through an open ion channel, taking account of the interaction between an ion at the entrance of the channel and an ion at the binding site in a self-consistent way. The electrostatic potential is calculated by solution of the Poisson equation for a channel modeled as a cylindrical tube. The transition rate, and the binding site occupancy as a function of the left bulk concentration are compared to 1D Brownian dynamics simulations. The analysis is performed for a single binding site of high-affinity, with the exit rate influenced by barrier fluctuations at the channel exit. The results are compared with experimental data for the permeation of the Na+ ion through the Gramicidin A channel, with which they are shown to be in good agreement.

It is found that the response of a nonlinear dynamical system can be linearised, and its frequency dispersion diminished, by the addition of external noise of sufficient intensity. Taking as an example an overdamped bistable system driven by a low-frequency periodic field, this noise-induced linearisation is investigated through analogue electronic experiments. The wider implications are considered.

A new form of heterodyning is reported in which a heterodyne signal at the difference frequency between an input signal and a reference signal can be enhanced by adding noise. The underlying mechanism is closely related to that of stochastic resonance. The dependences of the heterodyne signal and the signal-to-noise ratio on the amplitudes and frequencies of the input and reference signals have been investigated. Noise-induced enhancement of the heterodyning has been demonstrated both for white noise and for high-frequency noise (with the power spectrum centered at the frequency of the reference signal) added at the input.

Fluctuation‐induced transitions between coexisting periodic attractors in a periodically driven nonlinear oscillator have been investigated theoretically and by analogue electronic experiment. Calculations and measurements of the corresponding activation energies are in good agreement, and have enabled the position of the kinetic phase transition (KPT) line to be established over its full range.

The response of a bistable system to a weak periodic force is investigated using linear response theory (LRT) and by analogue electronic experiment. For quasithermal systems the response, and in particular its increase with increasing noise intensity D, are described by the fluctuation dissipation relations. For small D the low-frequency susceptibility of the system chi(omega) has been found in explicit form allowing for both forced oscillations about the states and periodic modulation of the probabilities of fluctuational transitions between the states. It is shown, both theoretically and experimentally, that a phase lag phi between the force and the response passes through a maximum when D is tuned through the range where stochastic resonance (SR) occurs. A giant nonlinearity of the response is shown to arise for small D and small frequencies of the driving force. It results in the signal induced by a sinusoidal force being nearly rectangular. The range of applicability of LRT is established.

A new form of heterodyning is reported, related to stochastic resonance, in which a heterodyne signal can be enhanced by adding noise.