Multiplicative Gaussian White Noise Research Articles

Analytic expressions for rate and CV of a type I neuron driven by white gaussian noise.

We study the one-dimensional normal form of a saddle-node system under the influence of additive gaussian white noise and a static "bias current" input parameter, a model that can be looked upon as the simplest version of a type I neuron with stochastic input. This is in contrast with the numerous studies devoted to the noise-driven leaky integrate-and-fire neuron. We focus on the firing rate and coefficient of variation (CV) of the interspike interval density, for which scaling relations with respect to the input parameter and noise intensity are derived. Quadrature formulas for rate and CV are numerically evaluated and compared to numerical simulations of the system and to various approximation formulas obtained in different limiting cases of the model. We also show that caution must be used to extend these results to the Theta neuron model with multiplicative gaussian white noise. The correspondence between the first passage time statistics for the saddle-node model and the Theta neuron model is obtained only in the Stratonovich interpretation of the stochastic Theta neuron model, while previous results have focused only on the Ito interpretation. The correct Stratonovich interpretation yields CVs that are still relatively high, although smaller than in the Ito interpretation; it also produces certain qualitative differences, especially at larger noise intensities. Our analysis provides useful relations for assessing the distance to threshold and the level of synaptic noise in real type I neurons from their firing statistics. We also briefly discuss the effect of finite boundaries (finite values of threshold and reset) on the firing statistics.

Reconstruction of noisy patterns by bistable gradient neural-like network

An attractor bistable gradient neural-like network (BGN) described in Chinarov and Menzinger (BioSystems 55 (2000) 137) is applied to restoration of unknown patterns, which have been highly corrupted by multiplicative and additive Gaussian white noise. This becomes possible due to competitive advantage of BGN that derives from it: nice generalization capabilities, existence of the unique attractor with the lowest energy that is worked out when several patterns are stored by the network, and fast guaranteed convergence to this attractor.

Approximate solution of the Fokker–Planck–Kolmogorov equation

The aim of this paper is to present a thorough investigation of approximate techniques for estimating the stationary and non-stationary probability density function (PDF) of the response of nonlinear systems subjected to (additive and/or multiplicative) Gaussian white noise excitations. Attention is focused on the general scheme of weighted residuals for the approximate solution of the Fokker–Planck–Kolmogorov (FPK) equation. It is shown that the main drawbacks of closure schemes, such as negative values of the PDF in some regions, may be overcome by rewriting the FPK equation in terms of log-probability density function (log-PDF). The criteria for selecting the set of weighting functions in order to obtain improved estimates of the response PDF are discussed in detail. Finally, a simple and very effective iterative solution procedure is proposed.

Critical behavior of nonequilibrium phase transitions to magnetically ordered states.

We describe nonequilibrium phase transitions in arrays of dynamical systems with cubic nonlinearity driven by multiplicative Gaussian white noise. Depending on the sign of the spatial coupling we observe transitions to ferromagnetic or antiferromagnetic ordered states. We discuss the phase diagram, the order of the transitions, and the critical behavior. For global coupling we show analytically that the critical exponent of the magnetization exhibits a transition from the value 1/2 to a nonuniversal behavior depending on the ratio of noise strength to the magnitude of the spatial coupling.

Flux in the Case of Gaussian White Noises**The project supported by National Natural Science Foundation of China, the National Basic Research Project “Nonlinear Science”, the Youth Foundation of Beijing Normal University and the Educational Committee of the State Council through the Foundation of Doctoral Training

The transport of a spatially periodic system driven by additive and multiplicative Gaussian white noises (between which there is a correlation function) is investigated. The probability current (flux) shows .that the correlation function between the additive and multiplicative noises, and the spatial asymmetry are ingredients for the flux of particles. It is a new phenomenon that the correlation function between the additive and multiplicative noises can cause transport.

Transport of particles caused by correlation between additive and multiplicative noise

The transport of a spatially periodic system driven by additive and multiplicative Gaussian white noises (between which there is a correlation) is investigated in the presence of spatial symmetry. The probability current shows that the correlation between additive and multiplicative noise is an ingredient for the flux of particles. Using the formulas obtained by us we investigate the transport in the superconducting junction and the transport of motor proteins in the case of environmental perturbation. For the motor proteins we find that the multiplicative noise can also induce the flux of the moleculae, even if the correlation between additive and multiplicative noise is zero. The results of superconducting junction and motor proteins provide a theoretical foundation for further study.

The influence of noise on critical collapse in the nonlinear Schrödinger equation

Multiplicative Gaussian white noise is included in the nonlinear Schrödinger equation of one and two dimensions in the critical case, in which collapse occurs. The collective coordinate method is used to derive analytical results concerning the effects of the noise on the collapse process. This approach makes it possible to distinguish between the one- and two-dimensional case. The analytical results are in good agreement with numerical results obtained directly from the partial differential equation.

White noise in the two-dimensional nonlinear schrödinger equation

Multiplicative Gaussian white noise is included in the two-dimensional nonlinear Schrodinger equation. The collective coordinate method is used to derive an effective stochastic equation for the width of the wave function. From this equation the coherence time of the ground state solitary wave solution, tcoh , is found to be proportional to D-1/3 (D is the variance of the noise). Direct numerical simulations indicate tcohαD-0.4

Escape rates for noisy maps.

A one-dimensional bistable map in the presence of multiplicative Gaussian white noise is considered. An exact expression for the escape rate for asymptotically small noise strengths is derived consisting of an exponentially leading Arrhenius factor and a pre-exponential factor that shows a nontrivial dependence on the noise strength. The basic ingredients are a WKB ansatz for the invariant density and a discrete-time version of Kramers's flux-over-population method for the determination of the rate. In the particular case of a piecewise linear map with additive noise, the general rate formula takes a simple closed form and compared very well with numerical results.

A new approximate solution technique for randomly excited non-linear oscillators—II

The method of weighted residuals is applied to the reduced Fokker-Planck equation associated with a non-linear oscillator, which is subjected to both additive and multiplicative Gaussian white noise excitations. A set of constraints are deduced for obtaining an approximate stationary probability density for the system response. One of the constraints coincides with the previously proposed criterion of dissipation energy balancing, and the others are useful for calculating the equivalent conservative force. It is shown that these constraints imply certain relationships among certain statistical moments; their imposition guarantees that such moments computed from the approximate probability density satisfy the corresponding exact equations derived from the original equation of motion. Moreover, the well-known procedure of stochastic linearization and its improved version of partial linearization are shown to be special cases of this scheme, and they are less accurate since the approximations are not chosen from the entire set of the solution pool of generalized stationary potential. Applications of the scheme are illustrated by examples, and its accuracy is substantiated by Monte Carlo simulation results.

On the effect of interference of additive and multiplicative noises

A bistable kinetic model under simultaneous influence of additive and multiplicative Gaussian white noises is considered. In the case of independent noises, the stationary properties of the system are dominated by the multiplicative noise, the role of the additive one being rather small. The presence of correlation between both noises changes this picture. Especially, the transition between unimodal and bimodal stationary distribution (i.e. the transition between stationary states of the system) is strongly influenced by the correlations between both noises. It is suggested that (i) the dependence of the critical value a c of the steering parameter on other deterministic parameters of the model can serve as an indicator of appearance of such correlations, (ii) stochastic parameters characterizing the correlated noises can be determined from measurements of a c and of the stationary values of the state variable.

Transient solution to the diffusion equation for nonlinear stochastically perturbed systems

We study the time evolution of nonlinear physical systems subject to the influence of random perturbations, which are modelled by additive and multiplicative stationary Gaussian white noises. The transient behaviour of the probability density related to the states of the system which are far from the stationary asymptotic condition is determined by means of a mixed numerical-analytic technique based on the differential quadrature method. Some applications in nonlinear physics follow, and the results obtained are analysed and discussed.

Delayed bifurcations in the presence of noise

We study how an additive or multiplicative gaussian white noise reduces the delay caused by a sweeping of the control parameter through a steady bifurcation point. In particular an analytic solution can be provided when the solution which becomes unstable is the trivial zero solution.

Renormalized perturbation theory for a multiplicative noise system

We study an anharmonic overdamped oscillator driven by both additive and multiplicative Gaussian white noise. Within the Martin-Siggia-Rose formalism we employ renormalized perturbation theory to calculate the stationary correlation function and the response function of the oscillator variable. Due to the absence of a simple fluctuation-dissipation theorem renormalization cannot make use of exact stationary moments, but is perormed with the help of renormalization conditions.

Carleman imbedding of multiplicative stochastic processes

A class of exactly solvable nonlinear stochastic models with multiplicative Gaussian white noise is here solved by a method of linear imbedding. The models describe a continuous instability with fluctuations of the bifurcation parameter. The method of solution is a generalization of an idea, originally introduced by Carleman for the solution of deterministic rate equations. Our application of this method to the stochastic models studied here provides further insight into the applicability of the method. The solutions we find are compared with results, which have been obtained earlier by Fokker-Planck methods. Complete agreement with the Fokker-Planck results is found, contrary to recent claims by Brenig and Banai and also in disagreement with recent results of Suzuki et al. The method presented here also allows us, for the first time, to obtain solutions in a domain of parameter space where the Fokker-Planck equation has not been solved as yet.

Forward Hopf bifurcation with multiplicative gaussian white noise: Exact Fokker-Planck solution

Abstract The general Fokker-Planck equation for forward Hopf bifurcation of an oscillatory mode under the influence of multiplicative gaussian white noise is solved exactly.

Spectrum and dynamic-response function of transmitted light in the absorptive optical bistability

We consider the dynamical behavior of the (nonlinear) amplitude fluctuations in the transmitted radiation of the optical bistability in the transition region. A Fokker-Planck model with nonlinear diffusion coefficient (multiplicative Gaussian white noise) is used to describe the system in the "good-quality" cavity limit. Choosing a parameter set corresponding to a numerically small but tractable tunneling rate between the stable and metastable states we study, as functions of the external coherent field and system size, the spectrum and the response to a small additional injected coherent signal. The numerical calculations are facilitated by the construction of complex-valued continued-fraction representations for the spectrum of amplitude fluctuations and for the response function. This representation allows us to discuss the memory effects caused by deviations from a simple Gaussian-Markov behavior around the transition region. In addition, such a representation enables the construction of "a posteriori" error bounds on the power spectrum.