4,631 research outputs found
Characterization of Information Channels for Asymptotic Mean Stationarity and Stochastic Stability of Non-stationary/Unstable Linear Systems
Stabilization of non-stationary linear systems over noisy communication
channels is considered. Stochastically stable sources, and unstable but
noise-free or bounded-noise systems have been extensively studied in
information theory and control theory literature since 1970s, with a renewed
interest in the past decade. There have also been studies on non-causal and
causal coding of unstable/non-stationary linear Gaussian sources. In this
paper, tight necessary and sufficient conditions for stochastic stabilizability
of unstable (non-stationary) possibly multi-dimensional linear systems driven
by Gaussian noise over discrete channels (possibly with memory and feedback)
are presented. Stochastic stability notions include recurrence, asymptotic mean
stationarity and sample path ergodicity, and the existence of finite second
moments. Our constructive proof uses random-time state-dependent stochastic
drift criteria for stabilization of Markov chains. For asymptotic mean
stationarity (and thus sample path ergodicity), it is sufficient that the
capacity of a channel is (strictly) greater than the sum of the logarithms of
the unstable pole magnitudes for memoryless channels and a class of channels
with memory. This condition is also necessary under a mild technical condition.
Sufficient conditions for the existence of finite average second moments for
such systems driven by unbounded noise are provided.Comment: To appear in IEEE Transactions on Information Theor
Long-Run Accuracy of Variational Integrators in the Stochastic Context
This paper presents a Lie-Trotter splitting for inertial Langevin equations
(Geometric Langevin Algorithm) and analyzes its long-time statistical
properties. The splitting is defined as a composition of a variational
integrator with an Ornstein-Uhlenbeck flow. Assuming the exact solution and the
splitting are geometrically ergodic, the paper proves the discrete invariant
measure of the splitting approximates the invariant measure of inertial
Langevin to within the accuracy of the variational integrator in representing
the Hamiltonian. In particular, if the variational integrator admits no energy
error, then the method samples the invariant measure of inertial Langevin
without error. Numerical validation is provided using explicit variational
integrators with first, second, and fourth order accuracy.Comment: 30 page
Phenotypic switching of populations of cells in a stochastic environment
In biology phenotypic switching is a common bet-hedging strategy in the face
of uncertain environmental conditions. Existing mathematical models often focus
on periodically changing environments to determine the optimal phenotypic
response. We focus on the case in which the environment switches randomly
between discrete states. Starting from an individual-based model we derive
stochastic differential equations to describe the dynamics, and obtain
analytical expressions for the mean instantaneous growth rates based on the
theory of piecewise deterministic Markov processes. We show that optimal
phenotypic responses are non-trivial for slow and intermediate environmental
processes, and systematically compare the cases of periodic and random
environments. The best response to random switching is more likely to be
heterogeneity than in the case of deterministic periodic environments, net
growth rates tend to be higher under stochastic environmental dynamics. The
combined system of environment and population of cells can be interpreted as
host-pathogen interaction, in which the host tries to choose environmental
switching so as to minimise growth of the pathogen, and in which the pathogen
employs a phenotypic switching optimised to increase its growth rate. We
discuss the existence of Nash-like mutual best-response scenarios for such
host-pathogen games.Comment: 17 pages, 6 figure
Stochastic focusing coupled with negative feedback enables robust regulation in biochemical reaction networks
Nature presents multiple intriguing examples of processes which proceed at
high precision and regularity. This remarkable stability is frequently counter
to modelers' experience with the inherent stochasticity of chemical reactions
in the regime of low copy numbers. Moreover, the effects of noise and
nonlinearities can lead to "counter-intuitive" behavior, as demonstrated for a
basic enzymatic reaction scheme that can display stochastic focusing (SF).
Under the assumption of rapid signal fluctuations, SF has been shown to convert
a graded response into a threshold mechanism, thus attenuating the detrimental
effects of signal noise. However, when the rapid fluctuation assumption is
violated, this gain in sensitivity is generally obtained at the cost of very
large product variance, and this unpredictable behavior may be one possible
explanation of why, more than a decade after its introduction, SF has still not
been observed in real biochemical systems.
In this work we explore the noise properties of a simple enzymatic reaction
mechanism with a small and fluctuating number of active enzymes that behaves as
a high-gain, noisy amplifier due to SF caused by slow enzyme fluctuations. We
then show that the inclusion of a plausible negative feedback mechanism turns
the system from a noisy signal detector to a strong homeostatic mechanism by
exchanging high gain with strong attenuation in output noise and robustness to
parameter variations. Moreover, we observe that the discrepancy between
deterministic and stochastic descriptions of stochastically focused systems in
the evolution of the means almost completely disappears, despite very low
molecule counts and the additional nonlinearity due to feedback.
The reaction mechanism considered here can provide a possible resolution to
the apparent conflict between intrinsic noise and high precision in critical
intracellular processes
On the Performance of Short Block Codes over Finite-State Channels in the Rare-Transition Regime
As the mobile application landscape expands, wireless networks are tasked
with supporting different connection profiles, including real-time traffic and
delay-sensitive communications. Among many ensuing engineering challenges is
the need to better understand the fundamental limits of forward error
correction in non-asymptotic regimes. This article characterizes the
performance of random block codes over finite-state channels and evaluates
their queueing performance under maximum-likelihood decoding. In particular,
classical results from information theory are revisited in the context of
channels with rare transitions, and bounds on the probabilities of decoding
failure are derived for random codes. This creates an analysis framework where
channel dependencies within and across codewords are preserved. Such results
are subsequently integrated into a queueing problem formulation. For instance,
it is shown that, for random coding on the Gilbert-Elliott channel, the
performance analysis based on upper bounds on error probability provides very
good estimates of system performance and optimum code parameters. Overall, this
study offers new insights about the impact of channel correlation on the
performance of delay-aware, point-to-point communication links. It also
provides novel guidelines on how to select code rates and block lengths for
real-time traffic over wireless communication infrastructures
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