27,939 research outputs found
Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation
We present computer-assisted methods for analyzing stochastic models of gene
regulatory networks. The main idea that underlies this equation-free analysis
is the design and execution of appropriately-initialized short bursts of
stochastic simulations; the results of these are processed to estimate
coarse-grained quantities of interest, such as mesoscopic transport
coefficients. In particular, using a simple model of a genetic toggle switch,
we illustrate the computation of an effective free energy and of a
state-dependent effective diffusion coefficient that characterize an
unavailable effective Fokker-Planck equation. Additionally we illustrate the
linking of equation-free techniques with continuation methods for performing a
form of stochastic "bifurcation analysis"; estimation of mean switching times
in the case of a bistable switch is also implemented in this equation-free
context. The accuracy of our methods is tested by direct comparison with
long-time stochastic simulations. This type of equation-free analysis appears
to be a promising approach to computing features of the long-time,
coarse-grained behavior of certain classes of complex stochastic models of gene
regulatory networks, circumventing the need for long Monte Carlo simulations.Comment: 33 pages, submitted to The Journal of Chemical Physic
Variable-free exploration of stochastic models: a gene regulatory network example
Finding coarse-grained, low-dimensional descriptions is an important task in
the analysis of complex, stochastic models of gene regulatory networks. This
task involves (a) identifying observables that best describe the state of these
complex systems and (b) characterizing the dynamics of the observables. In a
previous paper [13], we assumed that good observables were known a priori, and
presented an equation-free approach to approximate coarse-grained quantities
(i.e, effective drift and diffusion coefficients) that characterize the
long-time behavior of the observables. Here we use diffusion maps [9] to
extract appropriate observables ("reduction coordinates") in an automated
fashion; these involve the leading eigenvectors of a weighted Laplacian on a
graph constructed from network simulation data. We present lifting and
restriction procedures for translating between physical variables and these
data-based observables. These procedures allow us to perform equation-free
coarse-grained, computations characterizing the long-term dynamics through the
design and processing of short bursts of stochastic simulation initialized at
appropriate values of the data-based observables.Comment: 26 pages, 9 figure
Stochastic Differential Equations for Quantum Dynamics of Spin-Boson Networks
The quantum dynamics of open many-body systems poses a challenge for
computational approaches. Here we develop a stochastic scheme based on the
positive P phase-space representation to study the nonequilibrium dynamics of
coupled spin-boson networks that are driven and dissipative. Such problems are
at the forefront of experimental research in cavity and solid state
realizations of quantum optics, as well as cold atom physics, trapped ions and
superconducting circuits. We demonstrate and test our method on a driven,
dissipative two-site system, each site involving a spin coupled to a photonic
mode, with photons hopping between the sites, where we find good agreement with
Monte Carlo Wavefunction simulations. In addition to numerically reproducing
features recently observed in an experiment [Phys. Rev. X 4, 031043 (2014)], we
also predict a novel steady state quantum dynamical phase transition for an
asymmetric configuration of drive and dissipation.Comment: 15 pages, 8 figure
Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience
This essay is presented with two principal objectives in mind: first, to
document the prevalence of fractals at all levels of the nervous system, giving
credence to the notion of their functional relevance; and second, to draw
attention to the as yet still unresolved issues of the detailed relationships
among power law scaling, self-similarity, and self-organized criticality. As
regards criticality, I will document that it has become a pivotal reference
point in Neurodynamics. Furthermore, I will emphasize the not yet fully
appreciated significance of allometric control processes. For dynamic fractals,
I will assemble reasons for attributing to them the capacity to adapt task
execution to contextual changes across a range of scales. The final Section
consists of general reflections on the implications of the reviewed data, and
identifies what appear to be issues of fundamental importance for future
research in the rapidly evolving topic of this review
Fluctuation effects in metapopulation models: percolation and pandemic threshold
Metapopulation models provide the theoretical framework for describing
disease spread between different populations connected by a network. In
particular, these models are at the basis of most simulations of pandemic
spread. They are usually studied at the mean-field level by neglecting
fluctuations. Here we include fluctuations in the models by adopting fully
stochastic descriptions of the corresponding processes. This level of
description allows to address analytically, in the SIS and SIR cases, problems
such as the existence and the calculation of an effective threshold for the
spread of a disease at a global level. We show that the possibility of the
spread at the global level is described in terms of (bond) percolation on the
network. This mapping enables us to give an estimate (lower bound) for the
pandemic threshold in the SIR case for all values of the model parameters and
for all possible networks.Comment: 14 pages, 13 figures, final versio
Why are probabilistic laws governing quantum mechanics and neurobiology?
We address the question: Why are dynamical laws governing in quantum
mechanics and in neuroscience of probabilistic nature instead of being
deterministic? We discuss some ideas showing that the probabilistic option
offers advantages over the deterministic one.Comment: 40 pages, 8 fig
Equation-free analysis of a dynamically evolving multigraph
In order to illustrate the adaptation of traditional continuum numerical
techniques to the study of complex network systems, we use the equation-free
framework to analyze a dynamically evolving multigraph. This approach is based
on coupling short intervals of direct dynamic network simulation with
appropriately-defined lifting and restriction operators, mapping the detailed
network description to suitable macroscopic (coarse-grained) variables and
back. This enables the acceleration of direct simulations through Coarse
Projective Integration (CPI), as well as the identification of coarse
stationary states via a Newton-GMRES method. We also demonstrate the use of
data-mining, both linear (principal component analysis, PCA) and nonlinear
(diffusion maps, DMAPS) to determine good macroscopic variables (observables)
through which one can coarse-grain the model. These results suggest methods for
decreasing simulation times of dynamic real-world systems such as
epidemiological network models. Additionally, the data-mining techniques could
be applied to a diverse class of problems to search for a succint,
low-dimensional description of the system in a small number of variables
Data-driven modelling of biological multi-scale processes
Biological processes involve a variety of spatial and temporal scales. A
holistic understanding of many biological processes therefore requires
multi-scale models which capture the relevant properties on all these scales.
In this manuscript we review mathematical modelling approaches used to describe
the individual spatial scales and how they are integrated into holistic models.
We discuss the relation between spatial and temporal scales and the implication
of that on multi-scale modelling. Based upon this overview over
state-of-the-art modelling approaches, we formulate key challenges in
mathematical and computational modelling of biological multi-scale and
multi-physics processes. In particular, we considered the availability of
analysis tools for multi-scale models and model-based multi-scale data
integration. We provide a compact review of methods for model-based data
integration and model-based hypothesis testing. Furthermore, novel approaches
and recent trends are discussed, including computation time reduction using
reduced order and surrogate models, which contribute to the solution of
inference problems. We conclude the manuscript by providing a few ideas for the
development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and
Multiscale Dynamics (American Scientific Publishers
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