107 research outputs found
Smoluchowski-Kramers approximation in the case of variable friction
We consider the small mass asymptotics (Smoluchowski-Kramers approximation)
for the Langevin equation with a variable friction coefficient. The limit of
the solution in the classical sense does not exist in this case. We study a
modification of the Smoluchowski-Kramers approximation. Some applications of
the Smoluchowski-Kramers approximation to problems with fast oscillating or
discontinuous coefficients are considered.Comment: already publishe
Compositionality, stochasticity and cooperativity in dynamic models of gene regulation
We present an approach for constructing dynamic models for the simulation of
gene regulatory networks from simple computational elements. Each element is
called a ``gene gate'' and defines an input/output-relationship corresponding
to the binding and production of transcription factors. The proposed reaction
kinetics of the gene gates can be mapped onto stochastic processes and the
standard ode-description. While the ode-approach requires fixing the system's
topology before its correct implementation, expressing them in stochastic
pi-calculus leads to a fully compositional scheme: network elements become
autonomous and only the input/output relationships fix their wiring. The
modularity of our approach allows to pass easily from a basic first-level
description to refined models which capture more details of the biological
system. As an illustrative application we present the stochastic repressilator,
an artificial cellular clock, which oscillates readily without any cooperative
effects.Comment: 15 pages, 8 figures. Accepted by the HFSP journal (13/09/07
Nonconventional Large Deviations Theorems
We obtain large deviations theorems for nonconventional sums with underlying
process being a Markov process satisfying the Doeblin condition or a dynamical
system such as subshift of finite type or hyperbolic or expanding
transformation
Shaping bursting by electrical coupling and noise
Gap-junctional coupling is an important way of communication between neurons
and other excitable cells. Strong electrical coupling synchronizes activity
across cell ensembles. Surprisingly, in the presence of noise synchronous
oscillations generated by an electrically coupled network may differ
qualitatively from the oscillations produced by uncoupled individual cells
forming the network. A prominent example of such behavior is the synchronized
bursting in islets of Langerhans formed by pancreatic \beta-cells, which in
isolation are known to exhibit irregular spiking. At the heart of this
intriguing phenomenon lies denoising, a remarkable ability of electrical
coupling to diminish the effects of noise acting on individual cells.
In this paper, we derive quantitative estimates characterizing denoising in
electrically coupled networks of conductance-based models of square wave
bursting cells. Our analysis reveals the interplay of the intrinsic properties
of the individual cells and network topology and their respective contributions
to this important effect. In particular, we show that networks on graphs with
large algebraic connectivity or small total effective resistance are better
equipped for implementing denoising. As a by-product of the analysis of
denoising, we analytically estimate the rate with which trajectories converge
to the synchronization subspace and the stability of the latter to random
perturbations. These estimates reveal the role of the network topology in
synchronization. The analysis is complemented by numerical simulations of
electrically coupled conductance-based networks. Taken together, these results
explain the mechanisms underlying synchronization and denoising in an important
class of biological models
Large deviations for the macroscopic motion of an interface
We study the most probable way an interface moves on a macroscopic scale from an initial to a final position within a fixed time in the context of large deviations for a stochastic microscopic lattice system of Ising spins with Kac interaction evolving in time according to Glauber (non-conservative) dynamics. Such interfaces separate two stable phases of a ferromagnetic system and in the macroscopic scale are represented by sharp transitions. We derive quantitative estimates for the upper and the lower bound of the cost functional that penalizes all possible deviations and obtain explicit error terms which are valid also in the macroscopic scale. Furthermore, using the result of a companion paper about the minimizers of this cost functional for the macroscopic motion of the interface in a fixed time, we prove that the probability of such events can concentrate on nucleations should the transition happen fast enough
Global instability in the Ghil--Sellers model
The Ghil--Sellers model, a diffusive one-dimensional energy balance model of Earth's climate, features---for a considerable range of the parameter descriptive of the intensity of the incoming radiation---two stable climate states, where the bistability results from the celebrated ice-albedo feedback. The warm state is qualitatively similar to the present climate, while the cold state corresponds to snowball conditions. Additionally, in the region of bistability, one can find unstable climate states. We find such unstable states by applying for the first time in a geophysical context the so-called edge tracking method, which has been used for studying multiple coexisting states in shear flows. This method has a great potential for studying the global instabilities in multistable systems, and for providing crucial information on the possibility of transitions when forcing is present. We examine robustness, efficiency, and accuracy properties of the edge tracking algorithm. We find that the procedure is the most efficient when taking a single bisection per cycle. Due to the strong diffusivity of the system, the transient dynamics, is approximately confined to the heteroclininc trajectory, connecting the fixed unstable and stable states, after relatively short transient times. Such a constraint dictates a functional relationship between observables. We characterize such a relationship between the global average temperature and a descriptor of nonequilibrium thermodynamics, the large scale temperature gradient between low and high latitudes. We find that a maximum of the temperature gradient is realized at the same value of the average temperature, about 270 K, largely independent of the strength of incoming solar radiation. Due to this maximum, a transient increase and nonmonotonic evolution of the temperature gradient is possible and not untypical. We also examine the structural properties of the system defined by bifurcation diagrams describing the equilibria depending on a system parameter of interest, here the solar strength. We construct new bifurcation diagrams in terms of quantities relevant for describing thermodynamic properties such as the temperature gradient and the material entropy production due to heat transport. We compare our results for the energy balance model to results for the intermediate complexity general circulation model the Planet Simulator and find an interesting qualitative agreement
Time scales and exponential trends to equilibrium: Gaussian model problems
We review results on the exponential convergence of multi- dimensional Ornstein-Uhlenbeck processes and discuss related notions of characteristic timescales with concrete model systems. We focus, on the one hand, on exit time distributions and provide ecplicit expressions for the exponential rate of the distribution in the small noise limit. On the other hand, we consider relaxation timescales of the process to its equi- librium measured in terms of relative entropy and discuss the connection with exit probabilities. Along these lines, we study examples which il- lustrate specific properties of the relaxation and discuss the possibility of deriving a simulation-based, empirical definition of slow and fast de- grees of freedom which builds upon a partitioning of the relative entropy functional in conjuction with the observed relaxation behaviour
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