Non-equilibrium distributions at finite noise intensities

We analyse the non-equilibrium distribution in dissipative dynamical systems at finite noise intensities. The effect of finite noise is described in terms of topological changes in the pattern of optimal paths. Theoretical predictions are in good agreement with the results of numerical solution of the Fokker-Planck equation and Monte Carlo simulations.Comment: 4 pages, 3 figure

Fast Monte Carlo simulations and singularities in the probability distributions of non-equilibrium systems

A numerical technique is introduced that reduces exponentially the time required for Monte Carlo simulations of non-equilibrium systems. Results for the quasi-stationary probability distribution in two model systems are compared with the asymptotically exact theory in the limit of extremely small noise intensity. Singularities of the non-equilibrium distributions are revealed by the simulations.Comment: 4 pages, 4 figure

Exponentially fast Monte Carlo simulations for non equilibrium systems.

A new numerical technique is demonstrated and shown to reduce exponentially the time required for Monte Carlo simulations of non-equilibrium systems. The quasi stationary probability dis- tribution is computed for two model systems, and the results are compared with the asymptotically exact theory in the limit of extremely small noise intensity. Singularities of the non-equilibrium distributions are revealed by the simulations

Long-term cyclic persistence in an experimental predator–prey system

Predator–prey cycles rank among the most fundamental concepts in ecology, are predicted by the simplest ecological models and enable, theoretically, the indefinite persistence of predator and prey1,2,3,4. However, it remains an open question for how long cyclic dynamics can be self-sustained in real communities. Field observations have been restricted to a few cycle periods5,6,7,8 and experimental studies indicate that oscillations may be short-lived without external stabilizing factors9,10,11,12,13,14,15,16,17,18,19. Here we performed microcosm experiments with a planktonic predator–prey system and repeatedly observed oscillatory time series of unprecedented length that persisted for up to around 50 cycles or approximately 300 predator generations. The dominant type of dynamics was characterized by regular, coherent oscillations with a nearly constant predator–prey phase difference. Despite constant experimental conditions, we also observed shorter episodes of irregular, non-coherent oscillations without any significant phase relationship. However, the predator–prey system showed a strong tendency to return to the dominant dynamical regime with a defined phase relationship. A mathematical model suggests that stochasticity is probably responsible for the reversible shift from coherent to non-coherent oscillations, a notion that was supported by experiments with external forcing by pulsed nutrient supply. Our findings empirically demonstrate the potential for infinite persistence of predator and prey populations in a cyclic dynamic regime that shows resilience in the presence of stochastic events

Cardiovascular dynamics - multiple time scales, oscillations and noise

Modelling the cardiovascular system (CVS) presents a challenging and important problem. The CVS is a complex dynamical system that is vital to the function of the human organism, and it reflects numerous different states of health and disease. Its complexity lies in a combination of oscillatory modes spanning a wide frequency scale that can synchronize for short episodes of time, coupled with a strong stochastic contribution. Motivated by these properties, we discuss the problem of characterising dynamics when there is a combination of oscillatory components in the presence of strong noise and, in particular, where the characteristic frequencies and corresponding amplitudes vary in time. We show that, where there are several noisy oscillatory modes, the slower modes are difficult to characterise because the length of the recorded time series is inevitably limited in real measurements. We argue that, in the case of strong noise combined with a limited observation time, such oscillatory dynamics with several modes may appear to manifest as a 1/f‐like behaviour. We also show that methods of time‐frequency analysis can provide a basis for characterising noisy oscillations, but that a straightforward characterisation of multi‐scale oscillatory dynamics in the presence of strong noise still remains an unsolved problem

Simple approximation of the singular probability distribution in a nonadiabatically driven system.

Singular behavior and the formation of plateaus in the probability distribution in a nonadiabatically driven system are investigated numerically in the weak noise limit. A simple extension of the recently introduced logarithmic susceptibility theory is proposed to construct an approximation of the nonequilibrium potential that is valid throughout whole of the phase space

Noise-induced shift of singularities in the pattern of optimal paths

We analyse the non-equilibrium distribution in dissipative dynamical systems at finite noise intensities. The effect of finite noise is described in terms of topological changes in the pattern of optimal paths. Theoretical predictions are in good agreement with the numerical results

Inference of systems with delay and applications to cardiovascular dynamics

A Bayesian inference technique, able to encompass stochastic nonlinear systems, is described. It is applicable to differential equations with delay and enables values of model parameters, delay, and noise intensity to be inferred from measured time series. The procedure is demonstrated on a very simple one-dimensional model system, and then applied to inference of parameters in the Mackey-Glass model of the respiratory control system based on measurements of ventilation in a healthy subject. It is concluded that the technique offers a promising tool for investigating cardiovascular interactions

Stochastic dynamics of anaesthesia

Recent developments in the analysis of synchronization and directionality of couplings for noisy nonlinear oscillators are being applied to study the complex interactions between cardiac and respiratory oscillations, and brain waves (especially delta and gamma), during anæsthesia. It is found that marked changes occur in the inter‐oscillator interactions during anæsthesia in both rats and humans. These could form a new basis for measurement of depth of anæsthesia. The new EC programme BRACCIA will explore and quantify causal relationships between the oscillatory processes for the different stages of anæsthesia and consciousness