33,018 research outputs found
A primer on noise-induced transitions in applied dynamical systems
Noise plays a fundamental role in a wide variety of physical and biological
dynamical systems. It can arise from an external forcing or due to random
dynamics internal to the system. It is well established that even weak noise
can result in large behavioral changes such as transitions between or escapes
from quasi-stable states. These transitions can correspond to critical events
such as failures or extinctions that make them essential phenomena to
understand and quantify, despite the fact that their occurrence is rare. This
article will provide an overview of the theory underlying the dynamics of rare
events for stochastic models along with some example applications
On the effect of heterogeneity in stochastic interacting-particle systems
We study stochastic particle systems made up of heterogeneous units. We
introduce a general framework suitable to analytically study this kind of
systems and apply it to two particular models of interest in economy and
epidemiology. We show that particle heterogeneity can enhance or decrease the
collective fluctuations depending on the system, and that it is possible to
infer the degree and the form of the heterogeneity distribution in the system
by measuring only global variables and their fluctuations
Effects of Noise on Ecological Invasion Processes: Bacteriophage-mediated Competition in Bacteria
Pathogen-mediated competition, through which an invasive species carrying and
transmitting a pathogen can be a superior competitor to a more vulnerable
resident species, is one of the principle driving forces influencing
biodiversity in nature. Using an experimental system of bacteriophage-mediated
competition in bacterial populations and a deterministic model, we have shown
in [Joo et al 2005] that the competitive advantage conferred by the phage
depends only on the relative phage pathology and is independent of the initial
phage concentration and other phage and host parameters such as the
infection-causing contact rate, the spontaneous and infection-induced lysis
rates, and the phage burst size. Here we investigate the effects of stochastic
fluctuations on bacterial invasion facilitated by bacteriophage, and examine
the validity of the deterministic approach. We use both numerical and
analytical methods of stochastic processes to identify the source of noise and
assess its magnitude. We show that the conclusions obtained from the
deterministic model are robust against stochastic fluctuations, yet deviations
become prominently large when the phage are more pathological to the invading
bacterial strain.Comment: 39 pages, 7 figure
Noise-induced synchronization and anti-resonance in excitable systems; Implications for information processing in Parkinson's Disease and Deep Brain Stimulation
We study the statistical physics of a surprising phenomenon arising in large
networks of excitable elements in response to noise: while at low noise,
solutions remain in the vicinity of the resting state and large-noise solutions
show asynchronous activity, the network displays orderly, perfectly
synchronized periodic responses at intermediate level of noise. We show that
this phenomenon is fundamentally stochastic and collective in nature. Indeed,
for noise and coupling within specific ranges, an asymmetry in the transition
rates between a resting and an excited regime progressively builds up, leading
to an increase in the fraction of excited neurons eventually triggering a chain
reaction associated with a macroscopic synchronized excursion and a collective
return to rest where this process starts afresh, thus yielding the observed
periodic synchronized oscillations. We further uncover a novel anti-resonance
phenomenon: noise-induced synchronized oscillations disappear when the system
is driven by periodic stimulation with frequency within a specific range. In
that anti-resonance regime, the system is optimal for measures of information
capacity. This observation provides a new hypothesis accounting for the
efficiency of Deep Brain Stimulation therapies in Parkinson's disease, a
neurodegenerative disease characterized by an increased synchronization of
brain motor circuits. We further discuss the universality of these phenomena in
the class of stochastic networks of excitable elements with confining coupling,
and illustrate this universality by analyzing various classical models of
neuronal networks. Altogether, these results uncover some universal mechanisms
supporting a regularizing impact of noise in excitable systems, reveal a novel
anti-resonance phenomenon in these systems, and propose a new hypothesis for
the efficiency of high-frequency stimulation in Parkinson's disease
Gaussian approximations for stochastic systems with delay: chemical Langevin equation and application to a Brusselator system
We present a heuristic derivation of Gaussian approximations for stochastic
chemical reaction systems with distributed delay. In particular we derive the
corresponding chemical Langevin equation. Due to the non-Markovian character of
the underlying dynamics these equations are integro-differential equations, and
the noise in the Gaussian approximation is coloured. Following on from the
chemical Langevin equation a further reduction leads to the linear-noise
approximation. We apply the formalism to a delay variant of the celebrated
Brusselator model, and show how it can be used to characterise noise-driven
quasi-cycles, as well as noise-triggered spiking. We find surprisingly
intricate dependence of the typical frequency of quasi-cycles on the delay
period.Comment: 14 pages, 9 figure
Stochastic processes with distributed delays: chemical Langevin equation and linear-noise approximation
We develop a systematic approach to the linear-noise approximation for
stochastic reaction systems with distributed delays. Unlike most existing work
our formalism does not rely on a master equation, instead it is based upon a
dynamical generating functional describing the probability measure over all
possible paths of the dynamics. We derive general expressions for the chemical
Langevin equation for a broad class of non-Markovian systems with distributed
delay. Exemplars of a model of gene regulation with delayed auto-inhibition and
a model of epidemic spread with delayed recovery provide evidence of the
applicability of our results.Comment: 21 pages, 7 figure
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