Detecting and quantifying temporal correlations in stochastic resonance via information theory measures

We show that Information Theory quantifiers are suitable tools for detecting and for quantifying noise-induced temporal correlations in stochastic resonance phenomena. We use the Bandt & Pompe (BP) method [Phys. Rev. Lett. 88, 174102 (2002)] to define a probability distribution, P, that fully characterizes temporal correlations. The BP method is based on a comparison of neighboring values, and here is applied to the temporal sequence of residence-time intervals generated by the paradigmatic model of a Brownian particle in a sinusoidally modulated bistable potential. The probability distribution P generated via the BP method has associated a normalized Shannon entropy, H[P], and a statistical complexity measure, C[P], which is defined as proposed by Rosso et al. [Phys. Rev. Lett. 99, 154102 (2007)]. The statistical complexity quantifies not only randomness but also the presence of correlational structures, the two extreme circumstances of maximum knowledge (“perfect order") and maximum ignorance (“complete randomness") being regarded an “trivial", and in consequence, having complexity C = 0. We show that both, H and C, display resonant features as a function of the noise intensity, i.e., for an optimal level of noise the entropy displays a minimum and the complexity, a maximum. This resonant behavior indicates noise-enhanced temporal correlations in the sequence of residence-time intervals. The methodology proposed here has great potential for the precise detection of subtle signatures of noise-induced temporal correlations in real-world complex signals.     

Escape of Brownian particles in an asymmetric bistable sawtooth potential driven by cross-correlated noises

The relative escape rate (RER) for Brownian particles in an asymmetric bistable sawtooth potential driven by cross correlations between multiplicative white noise and additive white noise is studied. A new expression of the mean first-passage time is derived under the condition of piecewise linear potentials and discontinuous diffusion function. Based on the results of RER numerically calculated, it is found that (i) under positively correlated noises action (i.e. λ > 0, and λ is the correlation strength), the escape rate exhibits the suppression platform as the intensity of multiplicative noise varies. The effect of suppression becomes more pronounced with the growth of height of the deterministic potential barrier for transition, and with the increase of λ. However, for the case of uncorrelated noises (λ = 0) and of negatively correlated noises (λ < 0), the suppression platform disappears. (ii) The positive correlation between noises amplifies the change of the escape rate with the height of barrier for transition, while the negative correlation between noises suppresses this change. Received 20 November 2002 / Received in final form 19 October 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: kmdcmei@public.km.yn.cn     

Stochastic resonance: influence of a f-κ noise spectrum

With the aim of studying stochastic resonance (SR) in a double-well potential when the noise source has a spectral density of the form f^-κ (with varying κ), we have extended a procedure introduced by Kaulakys et al. (Phys. Rev. E 70, 020101 (2004)). In order to achieve an analytical understanding of the results, we have obtained an effective Markovian approximation that allows us to make a systematic study of the effect of such noise on the SR phenomenon. A comparison of the numerical and analytical results shows an excellent qualitative agreement indicating that the effective Markovian approximation is able to correctly describe the general trends.     

Power, Lévy, exponential and Gaussian-like regimes in autocatalytic financial systems

We study by theoretical analysis and by direct numerical simulation the dynamics of a wide class of asynchronous stochastic systems composed of many autocatalytic degrees of freedom. We describe the generic emergence of truncated power laws in the size distribution of their individual elements. The exponents α of these power laws are time independent and depend only on the way the elements with very small values are treated. These truncated power laws determine the collective time evolution of the system. In particular the global stochastic fluctuations of the system differ from the normal Gaussian noise according to the time and size scales at which these fluctuations are considered. We describe the ranges in which these fluctuations are parameterized respectively by: the Lévy regime α < 2, the power law decay with large exponent ( α > 2), and the exponential decay. Finally we relate these results to the large exponent power laws found in the actual behavior of the stock markets and to the exponential cut-off detected in certain recent measurement. Received 29 July 2000 and Received in final form 25 September 2000     

Associated relaxation time and intensity correlation function of a bistable system driven by cross-correlation additive and multiplicative coloured noise sources

The associated relaxation time and the intensity correlation function of a bistable system driven by an additive and a multiplicative coloured noise with coloured cross-correlation are investigated. Using the Novikov theorem and the projection operator method, the analytic expressions of the stationary probability distribution P_st(x), the relaxation time T_c, and the normalized correlation function C(s) of the system are obtained. The effects of the noise intensity, the cross-correlation strength λ and the cross-correlation time τ are discussed. By numerical computation, it is found that the cross-correlation strength |λ| and the quantum noise intensity D decrease the relaxation of the system from unstable points. The cross-correlation time τ delays relaxation of the system from unstable points. The cross-correlation strength λ and the cross-correlation time τ can alter the effects of the pump noise intensity Q. Thus, the relaxation time T_c is a stochastic resonant phenomenon, and distribution curves exhibit a single-maximum structure.     

Quantum-mechanical tunneling in associative neural networks

We investigate the quantum-mechanical tunneling between the “patterns" of the, so-called, associative neural networks. Being the relatively stable minima of the “configuration-energy" space of the networks, the “patterns" represent the macroscopically distinguishable states of the neural nets. Therefore, the tunneling represents a macroscopic quantum effect, but with some special characteristics. Particularly, we investigate the tunneling between the minima of approximately equal depth, thus requiring no energy exchange. If there are at least a few such minima, the tunneling represents a sort of the “random walk" process, which implies the quantum fluctuations in the system, and therefore “malfunctioning" in the information processing of the nets. Due to the finite number of the minima, the “random walk" reduces to a dynamics modeled by the, so-called, Pauli master equation. With some plausible assumptions, the set(s) of the Pauli master equations can be analytically solved. This way comes the main result of this paper: the quantum fluctuations due to the quantum-mechanical tunneling can be “minimized" if the “pattern"-formation is such that there are mutually “distant" groups of the “patterns", thus providing the “zone" structure of the “pattern" formation. This qualitative result can be considered as a basis of the efficient deterministic functioning of the associative neural nets. Received 15 July 1999     

Modelling fluctuations of financial time series: from cascade process to stochastic volatility model

In this paper, we provide a simple, “generic” interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that in this context 1/f power spectra, as recently observed in reference [23], naturally emerge. We then propose a simple solvable “stochastic volatility” model for return fluctuations. This model is able to reproduce most of recent empirical findings concerning financial time series: no correlation between price variations, long-range volatility correlations and multifractal statistics. Moreover, its extension to a multivariate context, in order to model portfolio behavior, is very natural. Comparisons to real data and other models proposed elsewhere are provided. Received 22 May 2000     

Impact of network activity on noise delayed spiking for a Hodgkin-Huxley model

In a Hodgkin-Huxley neuron model driven just above threshold, external noise can increase both jitter and latency of the first spike, an effect called “noise delayed decay” (NDD). This phenomenon is important when considering how neuronal information is represented, thus by the precise timing of spikes or by their rate. We examine how NDD can be affected by network activity by varying the model's membrane time constant, τ_m. We show that NDD is significant for small τ_m or high network activity, and decreases for large τ_m, or low network activity. Our results suggest that for inputs just above threshold, the activity of the network constrains the neuronal coding strategy due to, at least in part, the NDD effect.     

Stability of a nonlinear oscillator with random damping

A noisy damping parameter in the equation of motion of a nonlinear oscillator renders the fixed point of the system unstable when the amplitude of the noise is sufficiently large. However, the stability diagram of the system can not be predicted from the analysis of the moments of the linearized equation. In the case of a white noise, an exact formula for the Lyapunov exponent of the system is derived. We then calculate the critical damping for which the nonlinear system becomes unstable. We also characterize the intermittent structure of the bifurcated state above threshold and address the effect of temporal correlations of the noise by considering an Ornstein-Uhlenbeck noise.     

Economic returns of research: the Pareto law and its implications

At what level should government or companies support research? This complex multi-faceted question encompasses such qualitative bonus as satisfying natural human curiosity, the quest for knowledge and the impact on education and culture, but one of its most scrutinized component reduces to the assessment of economic performance and wealth creation derived from research. Many studies report evidences of positive economic benefits derived from basic research [#!Martin!#,#!NAS!#]. In certain areas such as biotechnology, semi-conductor physics, optical communications [#!Ehrenreich!#], the impact of basic research is direct while, in other disciplines, the path from discovery to applications is full of surprises. As a consequence, there are persistent uncertainties in the quantification of the exact economic returns of public expenditure on basic research. This gives little help to policy makers trying to determine what should be the level of funding. Here, we suggest that these uncertainties have a fundamental origin to be found in the interplay between the intrinsic “fat tail” power law nature of the distribution of economic returns, characterized by a mathematically diverging variance, and the stochastic character of discovery rates. In the regime where the cumulative economic wealth derived from research is expected to exhibit a long-term positive trend, we show that strong fluctuations blur out significantly the short-time scales: a few major unpredictable innovations may provide a finite fraction of the total creation of wealth. In such a scenario, any attempt to assess the economic impact of research over a finite time horizon encompassing only a small number of major discoveries is bound to be highly unreliable. New tools, developed in the theory of self-similar and complex systems [#!Dubrulleetal!#] to tackle similar extreme fluctuations in Nature [#!Mandelbrot!#], can be adapted to measure the economic benefits of research, which is intimately associated to this large variability. Received 26 October 1998 and Received in final form 27 October 1998     

A high-sensitivity laser-pumped Mx magnetometer

We discuss the design and performance of a laser-pumped cesium vapor magnetometer in the M_x configuration. The device will be employed in the control and stabilization of fluctuating magnetic fields and gradients in a new experiment searching for a permanent electric dipole moment of the neutron. We have determined the intrinsic sensitivity of the device to be 15 fT in a 1 Hz bandwidth, limited by technical laser noise. In the shot noise limit the magnetometer can reach a sensitivity of 10 fT in a 1 Hz bandwidth. We have used the device to study the fluctuations of a stable magnetic field in a multi-layer magnetic shield for integration times in the range of 2–100 seconds. The residual fluctuations for times up to a few minutes are traced back to the instability of the power supply used to generate the field.     

Shot noise suppression in avalanche photodiodes

We identify a new shot noise suppression mechanism in a thin (approximately 100 nm) heterostructure avalanche photodiode. In the low-gain regime the shot noise is suppressed due to temporal correlations within amplified current pulses. We demonstrate in a Monte Carlo simulation that the effective excess noise factors can be < 1, and reconcile the apparent conflict between theory and experiments. This shot noise suppression mechanism is independent of known mechanisms such as Coulomb interaction, or reflection at heterojunction interfaces.     

The maximal process of nonlinear shot noise

In the nonlinear shot noise system-model shots’ statistics are governed by general Poisson processes, and shots’ decay-dynamics are governed by general nonlinear differential equations. In this research we consider a nonlinear shot noise system and explore the process tracking, along time, the system’s maximal shot magnitude. This ‘maximal process’ is a stationary Markov process following a decay-surge evolution; it is highly robust, and it is capable of displaying both a wide spectrum of statistical behaviors and a rich variety of random decay-surge sample-path trajectories. A comprehensive analysis of the maximal process is conducted, including its Markovian structure, its decay-surge structure, and its correlation structure. All results are obtained analytically and in closed-form.     

Dynamical inference of hidden biological populations

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.     

Two different kinds of time delays in a stochastic system

The steady state properties of a noise-driven bistable system are investigated when there are two different kinds of time delays existed in the deterministic and fluctuating forces respectively. Using the approximation of the probability density approach, the delayed Fokker-Planck equation is obtained. The stationary probability distribution (SPD) and the variance of the system are derived. It is found that the time delay τ in the deterministic force can reduce the fluctuations while the time delay β in the fluctuating force can enhance the fluctuations. Numerical simulations are presented and are in good agreement with the approximate theoretical results.     

Detecting vorticity filaments using wavelet analysis: About the statistical contribution of vorticity filaments to intermittency in swirling turbulent flows

Swirling turbulent flows display intermittent pressure drops associated with intense vorticity filaments. Using the wavelet transform modulus maxima representation of pressure fluctuations, we propose a method of characterizing these pressure drop events from their time-scale properties. This method allows us to discriminate fluctuations induced by just formed (young) as well as by burst (old) filaments from background pressure fluctuations. The statistical characteristics of these filaments (core size, waiting time) are analyzed in details and compared with previously reported experimental and numerical findings. Their intermittent occurrence is found to be governed by a pure Poisson's law, the hallmark of independent events. Then we apply the wavelet transform modulus maxima (WTMM) method to the background pressure fluctuations. This study reveals that, once removed all the filaments, the “multifractal” nature of pressure fluctuations still persists. This is a clear indication that the statistical contribution of the filaments is not important enough to account for the intermittency phenomenon in turbulents flows. Received 27 July 1998 and Received in final form 23 November 1998     

Noise-controlled slow–fast oscillations in predator–prey models with the Beddington functional response

The influence of environmental fluctuations (modeled as a multiplicative dichotomous noise) on predator–prey interaction is studied using a metapopulation model with N prey-subpopulations. Investigating the role that predator interference plays in the dynamics of such trophic systems, the Beddington functional response is considered. In case the growth rates of prey and predator are widely different, we obtain analytic results by a dynamical mean-field approximation. In some regions of the system parameters, variations of noise amplitude or correlation time can cause transitions of the mean field from a globally stable equilibrium to the stable limit cycle as well as in the opposite direction. The conditions for the occurrence of such a phenomenon are found and illustrated by phase diagrams. Implications of the results on the colored-noise-induced extinction of a predator population are also discussed.     

Some first passage time problems for shot noise processes

It has recently been shown that the first passage time problem for a certain class of one-dimensional processes that includes shot noise can be formulated in terms of a set of integral equations. These are found by exact enumeration of all possible trajectories. We show that the equations can be found by more direct means for processes described by the evolution equation , wheren(t) is time-localized shot noise.     

Nonstationary stochastic resonance viewed through the lens of information theory

In biological systems, information is frequentlytransferred with Poisson like spike processes (shot noise) modulatedin time by information-carrying signals. How then to quantifyinformation transfer by such processes for nonstationary inputsignals of finite duration? Is there some minimal length of theinput signal duration versus its strength? Can such signals bebetter detected when immersed in noise stemming from thesurroundings by increasing the stochastic intensity? These are somebasic questions which we attempt to address within an analyticaltheory based on the Kullback-Leibler information concept applied torandom processes.     

The emergence of coordination in public good games

In physical models it is well understood that the aggregate behaviour of a system is not in one to one correspondence with the behaviour of the average individual element of that system. Yet, in many economic models the behaviour of aggregates is thought of as corresponding to that of an individual. A typical example is that of public goods experiments. A systematic feature of such experiments is that, with repetition, people contribute less to public goods. A typical explanation is that people “learn to play Nash” or something approaching it. To justify such an explanation, an individual learning model is tested on average or aggregate data. In this paper we will examine this idea by analysing average and individual behaviour in a series of public goods experiments. We analyse data from a series of games of contributions to public goods and as is usual, we test a learning model on the average data. We then look at individual data, examine the changes that this produces and see if some general model such as the EWA (Expected Weighted Attraction) with varying parameters can account for individual behaviour. We find that once we disaggregate data such models have poor explanatory power. Groups do not learn as supposed, their behaviour differs markedly from one group to another, and the behaviour of the individuals who make up the groups also varies within groups. The decline in aggregate contributions cannot be explained by resorting to a uniform model of individual behaviour. However, the Nash equilibrium of such a game is a total payment for all the individuals and there is some convergence of the group in this respect. Yet the individual contributions do not converge. How the individuals “self-organsise” to coordinate, even in this limited way remains to be explained.