Periodically forced Fokker-Planck equation and stochastic resonance

Effects of Stochastic Noises on Limit-Cycle Oscillations and Power Losses in Fusion Plasmas and Information Geometry

We investigate the effects of different stochastic noises on the dynamics of the edge-localised modes (ELMs) in magnetically confined fusion plasmas by using a time-dependent PDF method, path-dependent information geometry (information rate, information length), and entropy-related measures (entropy production, mutual information). The oscillation quenching occurs due to either stochastic particle or magnetic perturbations, although particle perturbation is more effective in this amplitude diminishment compared with magnetic perturbations. On the other hand, magnetic perturbations are more effective at altering the oscillation period; the stochastic noise acts to increase the frequency of explosive oscillations (large ELMs) while decreasing the frequency of more regular oscillations (small ELMs). These stochastic noises significantly reduce power and energy losses caused by ELMs and play a key role in reproducing the observed experimental scaling relation of the ELM power loss with the input power. Furthermore, the maximum power loss is closely linked to the maximum entropy production rate, involving irreversible energy dissipation in non-equilibrium. Notably, over one ELM cycle, the information rate appears to keep almost a constant value, indicative of a geodesic. The information rate is also shown to be useful for characterising the statistical properties of ELMs, such as distinguishing between explosive and regular oscillations and the regulation between the pressure gradient and magnetic fluctuations.

Analytical solution of stochastic resonance in the nonadiabatic regime

We generalize stochastic resonance to the nonadiabatic limit by treating the double-well potential using two quadratic potentials. We use a singular perturbation method to determine an approximate analytical solution for the probability density function that asymptotically connects local solutions in boundary layers near the two minima with those in the region of the maximum that separates them. The validity of the analytical solution is confirmed numerically. Free from the constraints of the adiabatic limit, the approach allows us to predict the escape rate from one stable basin to another for systems experiencing a more complex periodic forcing.

Multiscale noise suppression and feature frequency extraction in SSVEP based on underdamped second-order stochastic resonance

OBJECTIVE

As one of the commonly used control signals of brain-computer interface (BCI), steady-state visual evoked potential (SSVEP) exhibits advantages of stability, periodicity and minimal training requirements. However, SSVEP retains the non-linear, non-stationary and low signal-to-noise ratio (SNR) characteristics of EEG. The traditional SSVEP extraction methods regard noise as harmful information and highlight the useful signal by suppressing the noise. In the collected EEG, noise and SSVEP are usually coupled together, the useful signal is inevitably attenuated while the noise is suppressed. Also, an additional band-pass filter is needed to eliminate the multi-scale noise, which causes the edge effect.

APPROACH

To address this issue, a novel method based on underdamped second-order stochastic resonance (USSR) is proposed in this paper for SSVEP extraction.

MAIN RESULTS

A synergistic effect produced by noise, useful signal and the nonlinear system can force the energy of noise to be transferred into SSVEP, and hence amplifying the useful signal while suppressing multi-scale noise. The recognition performances of detection are compared with the widely-used canonical coefficient analysis (CCA) and multivariate synchronization index (MSI).

SIGNIFICANCE

The comparison results indicate that USSR exhibits increased accuracy and faster processing speed, which effectively improves the information transmission rate (ITR) of SSVEP-based BCI.

Bodily motion fluctuation improves reaching success rate in a neurophysical agent via geometric-stochastic resonance

Organisms generate a variety of noise types, including neural noise, sensory noise, and noise resulting from fluctuations associated with movement. Sensory and neural noises are known to induce stochastic resonance (SR), which improves information transfer to the subjects control systems, including the brain. As a consequence, sensory and neural noise provide behavioral benefits, such as stabilization of posture and enhancement of feeding efficiency. In contrast, the benefits of fluctuations in the movements of a biological system remain largely unclear. Here, we describe a novel type of noise-induced order (NIO) that is realized by actively exploiting the motion fluctuations of an embodied system. In particular, we describe the theoretical analysis of a feedback-controlled embodied agent system that has a geometric end-effector. Furthermore, through several numerical simulations we demonstrate that the ratio of successful reaches to goal positions and capture of moving targets are improved by the exploitation of motion fluctuations. We report that reaching success rate improvement (RSRI) is based on the interaction of the geometric size of an end-effector, the agents motion fluctuations, and the desired motion frequency. Therefore, RSRI is a geometrically induced SR-like phenomenon. We also report an interesting result obtained through numerical simulations indicating that the agents neural and motion noise must be optimized to match the prey's motion noise in order to maximize the capture rate. Our study provides a new understanding of body motion fluctuations, as they were found to be the active noise sources for a behavioral NIO.

Multi-scale modeling in biology: how to bridge the gaps between scales?

Human physiological functions are regulated across many orders of magnitude in space and time. Integrating the information and dynamics from one scale to another is critical for the understanding of human physiology and the treatment of diseases. Multi-scale modeling, as a computational approach, has been widely adopted by researchers in computational and systems biology. A key unsolved issue is how to represent appropriately the dynamical behaviors of a high-dimensional model of a lower scale by a low-dimensional model of a higher scale, so that it can be used to investigate complex dynamical behaviors at even higher scales of integration. In the article, we first review the widely-used different modeling methodologies and their applications at different scales. We then discuss the gaps between different modeling methodologies and between scales, and discuss potential methods for bridging the gaps between scales.

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.

What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology

Stochastic resonance is said to be observed when increases in levels of unpredictable fluctuations--e.g., random noise--cause an increase in a metric of the quality of signal transmission or detection performance, rather than a decrease. This counterintuitive effect relies on system nonlinearities and on some parameter ranges being "suboptimal". Stochastic resonance has been observed, quantified, and described in a plethora of physical and biological systems, including neurons. Being a topic of widespread multidisciplinary interest, the definition of stochastic resonance has evolved significantly over the last decade or so, leading to a number of debates, misunderstandings, and controversies. Perhaps the most important debate is whether the brain has evolved to utilize random noise in vivo, as part of the "neural code". Surprisingly, this debate has been for the most part ignored by neuroscientists, despite much indirect evidence of a positive role for noise in the brain. We explore some of the reasons for this and argue why it would be more surprising if the brain did not exploit randomness provided by noise--via stochastic resonance or otherwise--than if it did. We also challenge neuroscientists and biologists, both computational and experimental, to embrace a very broad definition of stochastic resonance in terms of signal-processing "noise benefits", and to devise experiments aimed at verifying that random variability can play a functional role in the brain, nervous system, or other areas of biology.

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.

Probability distribution of work done on a two-level system during a nonequilibrium isothermal process

We present an exact calculation of the probability density for the work done by an external agent on a two-level system. Due to the external drive, both the transition rates between the two states and their energies depend on time. Within this setting we calculate the probability of every possible sample path of the system evolution and also the work done along any such path. The general procedure yields an evolution equation for the characteristic function of the work. Assuming that the energies change with constant rates, the properties of the work distribution are controlled by a single parameter representing the ratio of the time scales of the driving protocol, and of the internal dynamics, respectively. We calculate the mean work and characterize those sample paths which are not in agreement with the second law. In the slow driving limit, the probability density for the work collapses to a delta function localized at the reversible work. In the strongly nonequilibrium regime, the most probable work is smaller and the mean work is bigger than the reversible work.

Probability densities of periodically driven noisy systems: an approximation scheme incorporating linear-response and adiabatic theory

We consider periodically driven noisy systems in the limit of long times. To deduce their asymptotic time-periodic probability distributions, two approaches are commonly used: adiabatic theory, valid if driving is very slow, and linear-response theory, applicable when driving is weak. We introduce an approximation scheme that combines these two approaches to yield the driven probability distribution even when driving is strong and moderately fast, so that both linear-response and adiabatic approximations break down. The high accuracy of this scheme is demonstrated on a driven overdamped noisy oscillator in a bistable quartic potential.

Analysis of stochastic resonances

We investigate the one-dimensional diffusion of a particle in a piecewise linear potential superimposed with a step of a harmonically modulated height. Employing the matching conditions, we solve the corresponding Fokker-Planck equation and we analyze nonlinear features of the particle's mean position as a function of time. We present detailed results in two physically relevant cases. First, we take the unperturbed potential as a symmetrical up-oriented tip, which is placed between two reflecting boundaries and we add the jump at the tip coordinate. The setting yields resonance-like behavior of the stationary-response amplitude. Second, if the discontinuity at origin is combined with the constant force in the symmetrical region between the boundaries, the stationary response displays a time-independent shift against the potential slope. The driving-induced force exhibits a resonance-like behavior both with respect to the diffusion constant and the slope of the unperturbed potential.

Observing stochastic resonance in an underdamped bistable Duffing oscillator by the method of moments

The method of moments is applied to an underdamped bistable oscillator driven by Gaussian white noise and a weak periodic force for the observations of stochastic resonance and the resulting resonant structures are compared with those from Langevin simulation. The physical mechanisms of the stochastic resonance are explained based on the evolution of the intrawell frequency peak and the above-barrier frequency peak via the noise intensity and the fluctuation-dissipation theorem, and the three possible sources of stochastic resonance in the system are confirmed. Additionally, with the noise intensity fixed, the stochastic resonant structures are also observed by adjusting the nonlinear parameter.

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.

Coherent and incoherent trapping of a diffusion-assisted system in the presence of an external periodic field

Field induced trapping on a line segment of a diffusion-driven system is studied with an aim to gain an insight into the occurrence of coherent stochastic resonance, which is thoroughly explored as synchronized mean-free passages to the traps. Synchronization (coherence) between the external bias, the noise in the system and the temporal trapping events is found to attain an optimum value by increasing the forcing frequency towards the relevant resonant frequency, revealing a minimum in the nonmonotonic mean-free-passage time (MFPT) to trapping. The MFPT at a given forcing frequency, is also nonmonotonic when considered as a function of the diffusion coefficient of the medium, and reveals a maximum exhibiting the least synchronization effect (incoherent trapping).

Effect of asymmetry on stochastic resonance and stochastic resonance induced by multiplicative noise and by mean-field coupling

In the paper, we investigate the effect of asymmetry of the potential on stochastic resonance (SR) for a model with an asymmetric bistable potential and driven by additive noise, the signal-to-noise ratio (SNR) for a model with a monostable potential and driven by additive and multiplicative noises, and the SNR for a mean-field coupled model with infinite globally coupling oscillators driven by additive noises. It is shown that for the first model,the asymmetry of the potential can weaken the phenomenon of SR; for the second and third models, a SR induced by multiplicative noise and a different one caused by mean-field coupling are found.

Robust stochastic resonance: signal detection and adaptation in impulsive noise

Stochastic resonance (SR) occurs when noise improves a system performance measure such as a spectral signal-to-noise ratio or a cross-correlation measure. All SR studies have assumed that the forcing noise has finite variance. Most have further assumed that the noise is Gaussian. We show that SR still occurs for the more general case of impulsive or infinite-variance noise. The SR effect fades as the noise grows more impulsive. We study this fading effect on the family of symmetric alpha-stable bell curves that includes the Gaussian bell curve as a special case. These bell curves have thicker tails as the parameter alpha falls from 2 (the Gaussian case) to 1 (the Cauchy case) to even lower values. Thicker tails create more frequent and more violent noise impulses. The main feedback and feedforward models in the SR literature show this fading SR effect for periodic forcing signals when we plot either the signal-to-noise ratio or a signal correlation measure against the dispersion of the alpha-stable noise. Linear regression shows that an exponential law gamma(opt)(alpha)=cA(alpha) describes this relation between the impulsive index alpha and the SR-optimal noise dispersion gamma(opt). The results show that SR is robust against noise "outliers." So SR may be more widespread in nature than previously believed. Such robustness also favors the use of SR in engineering systems. We further show that an adaptive system can learn the optimal noise dispersion for two standard SR models (the quartic bistable model and the FitzHugh-Nagumo neuron model) for the signal-to-noise ratio performance measure. This also favors practical applications of SR and suggests that evolution may have tuned the noise-sensitive parameters of biological systems.

Effects of colored noise on stochastic resonance in a bistable system subject to multiplicative and additive noise

The effects of colored noise on stochastic resonance (SR) in a bistable system driven by multiplicative colored noise and additive white noise and a periodic signal are studied by using the unified colored noise approximation and the theory of signal-to-noise ratio (SNR) in the adiabatic limit. In the case of no correlations between noises, there is an optimal noise intensities ratio R at which SNR is a maximum that identifies the characteristics of the SR when the correlation time tau of the multiplicative colored noise is small. However, when tau is increased, a second optimal value of R appears, and two peaks appear in the SNR simultaneously. In the case of correlations between noises, the SNR is not only dependent on the correlation time tau, but also on the intensity of correlations between noises. Moreover, the double peak phenomenon can also appear as tau is increased in certain situations.

Stochastic resonance on a circle without excitation: physical investigation and peak frequency formula

In this article the existence of stochastic resonance (SR) without external force in a simplified circular system for different values of the control parameter b is considered. The average power spectra are calculated as well as the signal-to-noise ratio as a measure for stochastic resonance. It is shown that in the monostable and semistable (b<1 and b=1) cases coherent oscillations occur and SR exists. For the case b>1, the system is oscillatory and noise plays only a destructive role; therefore no SR occurs. The rotation number of the system is calculated and compared to the peak frequency of the power spectrum. Although the coincidence in the noisy case is not as good as that in the deterministic case, we can derive an empirical formula between the peak frequency of the power spectrum and the rotation number of the system, which is in good agreement with results of numerical simulations.

Stochastic resonance in a bistable system subject to multiplicative and additive noise

The stochastic resonance (SR) phenomenon in a bistable system under the simultaneous action of multiplicative and additive noise and periodic signal is studied by using the theory of signal-to-noise ratio (SNR) in the adiabatic limit. Two cases have been considered: the case of no correlations between multiplicative and additive noise and the case of correlations between two noises. The expressions of the SNR for both cases are obtained. The effects of intensity of multiplicative and additive noise and the intensity of the correlations between noises on the SNR are discussed for both cases, respectively. It is found that the existence of a maximum in the SNR is the identifying characteristic of the SR phenomenon. In the case of no correlations between multiplicative and additive noise, the SNR is independent of the initial condition of the system. However, the SNR is not only dependent on the intensity of correlations between noises, but also on the initial condition of the system in the presence of correlations between two noises.

Stochastic resonance as a possible mechanism of amplification of weak electric signals in living cells

The most important but still unresolved problem in bioelectromagnetics is the interaction of weak electromagnetic fields (EMFs) with living cells. Thermal and other types of noise pose restrictions in cell detection of weak signals. As a consequence, some extant experimental results that indicate low-intensity field effects cannot be accounted for, and this renders the results themselves questionable. One way out of this dead end is to search for possible mechanisms of signal amplification. In this paper, we discuss a general mechanism in which a weak signal is amplified by system noise itself. This mechanism was discovered several years ago in physics and is known, in its simplest form, as a stochastic resonance. It was shown that signal amplification may exceed a factor of 1000, which renders existing estimations of EMF thresholds highly speculative. The applicability of the stochastic resonance concept to cells is discussed particularly with respect to the possible role of the cell membrane in the amplification process.