6,281 research outputs found
Analytical approximations for the amplitude and period of a relaxation oscillator
<p>Abstract</p> <p>Background</p> <p>Analysis and design of complex systems benefit from mathematically tractable models, which are often derived by approximating a nonlinear system with an effective equivalent linear system. Biological oscillators with coupled positive and negative feedback loops, termed hysteresis or relaxation oscillators, are an important class of nonlinear systems and have been the subject of comprehensive computational studies. Analytical approximations have identified criteria for sustained oscillations, but have not linked the observed period and phase to compact formulas involving underlying molecular parameters.</p> <p>Results</p> <p>We present, to our knowledge, the first analytical expressions for the period and amplitude of a classic model for the animal circadian clock oscillator. These compact expressions are in good agreement with numerical solutions of corresponding continuous ODEs and for stochastic simulations executed at literature parameter values. The formulas are shown to be useful by permitting quick comparisons relative to a negative-feedback represillator oscillator for noise (10× less sensitive to protein decay rates), efficiency (2× more efficient), and dynamic range (30 to 60 decibel increase). The dynamic range is enhanced at its lower end by a new concentration scale defined by the crossing point of the activator and repressor, rather than from a steady-state expression level.</p> <p>Conclusion</p> <p>Analytical expressions for oscillator dynamics provide a physical understanding for the observations from numerical simulations and suggest additional properties not readily apparent or as yet unexplored. The methods described here may be applied to other nonlinear oscillator designs and biological circuits.</p
Probabilistic description of extreme events in intermittently unstable systems excited by correlated stochastic processes
In this work, we consider systems that are subjected to intermittent
instabilities due to external stochastic excitation. These intermittent
instabilities, though rare, have a large impact on the probabilistic response
of the system and give rise to heavy-tailed probability distributions. By
making appropriate assumptions on the form of these instabilities, which are
valid for a broad range of systems, we formulate a method for the analytical
approximation of the probability distribution function (pdf) of the system
response (both the main probability mass and the heavy-tail structure). In
particular, this method relies on conditioning the probability density of the
response on the occurrence of an instability and the separate analysis of the
two states of the system, the unstable and stable state. In the stable regime
we employ steady state assumptions, which lead to the derivation of the
conditional response pdf using standard methods for random dynamical systems.
The unstable regime is inherently transient and in order to analyze this regime
we characterize the statistics under the assumption of an exponential growth
phase and a subsequent decay phase until the system is brought back to the
stable attractor. The method we present allows us to capture the statistics
associated with the dynamics that give rise to heavy-tails in the system
response and the analytical approximations compare favorably with direct Monte
Carlo simulations, which we illustrate for two prototype intermittent systems:
an intermittently unstable mechanical oscillator excited by correlated
multiplicative noise and a complex mode in a turbulent signal with fixed
frequency, where multiplicative stochastic damping and additive noise model
interactions between various modes.Comment: 29 pages, 15 figure
First Passage Time Densities in Resonate-and-Fire Models
Motivated by the dynamics of resonant neurons we discuss the properties of
the first passage time (FPT) densities for nonmarkovian differentiable random
processes. We start from an exact expression for the FPT density in terms of an
infinite series of integrals over joint densities of level crossings, and
consider different approximations based on truncation or on approximate
summation of this series. Thus, the first few terms of the series give good
approximations for the FPT density on short times. For rapidly decaying
correlations the decoupling approximations perform well in the whole time
domain.
As an example we consider resonate-and-fire neurons representing stochastic
underdamped or moderately damped harmonic oscillators driven by white Gaussian
or by Ornstein-Uhlenbeck noise. We show, that approximations reproduce all
qualitatively different structures of the FPT densities: from monomodal to
multimodal densities with decaying peaks. The approximations work for the
systems of whatever dimension and are especially effective for the processes
with narrow spectral density, exactly when markovian approximations fail.Comment: 11 pages, 8 figure
Quantum dynamics in strong fluctuating fields
A large number of multifaceted quantum transport processes in molecular
systems and physical nanosystems can be treated in terms of quantum relaxation
processes which couple to one or several fluctuating environments. A thermal
equilibrium environment can conveniently be modelled by a thermal bath of
harmonic oscillators. An archetype situation provides a two-state dissipative
quantum dynamics, commonly known under the label of a spin-boson dynamics. An
interesting and nontrivial physical situation emerges, however, when the
quantum dynamics evolves far away from thermal equilibrium. This occurs, for
example, when a charge transferring medium possesses nonequilibrium degrees of
freedom, or when a strong time-dependent control field is applied externally.
Accordingly, certain parameters of underlying quantum subsystem acquire
stochastic character. Herein, we review the general theoretical framework which
is based on the method of projector operators, yielding the quantum master
equations for systems that are exposed to strong external fields. This allows
one to investigate on a common basis the influence of nonequilibrium
fluctuations and periodic electrical fields on quantum transport processes.
Most importantly, such strong fluctuating fields induce a whole variety of
nonlinear and nonequilibrium phenomena. A characteristic feature of such
dynamics is the absence of thermal (quantum) detailed balance.Comment: review article, Advances in Physics (2005), in pres
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