9,584 research outputs found
Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics
Linearized catalytic reaction equations modeling e.g. the dynamics of genetic
regulatory networks under the constraint that expression levels, i.e. molecular
concentrations of nucleic material are positive, exhibit nontrivial dynamical
properties, which depend on the average connectivity of the reaction network.
In these systems the inflation of the edge of chaos and multi-stability have
been demonstrated to exist. The positivity constraint introduces a nonlinearity
which makes chaotic dynamics possible. Despite the simplicity of such minimally
nonlinear systems, their basic properties allow to understand fundamental
dynamical properties of complex biological reaction networks. We analyze the
Lyapunov spectrum, determine the probability to find stationary oscillating
solutions, demonstrate the effect of the nonlinearity on the effective in- and
out-degree of the active interaction network and study how the frequency
distributions of oscillatory modes of such system depend on the average
connectivity.Comment: 11 pages, 5 figure
A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients
In recent years deep artificial neural networks (DNNs) have been successfully
employed in numerical simulations for a multitude of computational problems
including, for example, object and face recognition, natural language
processing, fraud detection, computational advertisement, and numerical
approximations of partial differential equations (PDEs). These numerical
simulations indicate that DNNs seem to possess the fundamental flexibility to
overcome the curse of dimensionality in the sense that the number of real
parameters used to describe the DNN grows at most polynomially in both the
reciprocal of the prescribed approximation accuracy and the
dimension of the function which the DNN aims to approximate
in such computational problems. There is also a large number of rigorous
mathematical approximation results for artificial neural networks in the
scientific literature but there are only a few special situations where results
in the literature can rigorously justify the success of DNNs in
high-dimensional function approximation. The key contribution of this paper is
to reveal that DNNs do overcome the curse of dimensionality in the numerical
approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift
coefficients. We prove that the number of parameters used to describe the
employed DNN grows at most polynomially in both the PDE dimension and the reciprocal of the prescribed approximation accuracy . A crucial ingredient in our proof is the fact that the
artificial neural network used to approximate the solution of the PDE is indeed
a deep artificial neural network with a large number of hidden layers.Comment: 48 page
Wireless networks appear Poissonian due to strong shadowing
Geographic locations of cellular base stations sometimes can be well fitted
with spatial homogeneous Poisson point processes. In this paper we make a
complementary observation: In the presence of the log-normal shadowing of
sufficiently high variance, the statistics of the propagation loss of a single
user with respect to different network stations are invariant with respect to
their geographic positioning, whether regular or not, for a wide class of
empirically homogeneous networks. Even in perfectly hexagonal case they appear
as though they were realized in a Poisson network model, i.e., form an
inhomogeneous Poisson point process on the positive half-line with a power-law
density characterized by the path-loss exponent. At the same time, the
conditional distances to the corresponding base stations, given their observed
propagation losses, become independent and log-normally distributed, which can
be seen as a decoupling between the real and model geometry. The result applies
also to Suzuki (Rayleigh-log-normal) propagation model. We use
Kolmogorov-Smirnov test to empirically study the quality of the Poisson
approximation and use it to build a linear-regression method for the
statistical estimation of the value of the path-loss exponent
Theoretical connections between mathematical neuronal models corresponding to different expressions of noise
Identifying the right tools to express the stochastic aspects of neural
activity has proven to be one of the biggest challenges in computational
neuroscience. Even if there is no definitive answer to this issue, the most
common procedure to express this randomness is the use of stochastic models. In
accordance with the origin of variability, the sources of randomness are
classified as intrinsic or extrinsic and give rise to distinct mathematical
frameworks to track down the dynamics of the cell. While the external
variability is generally treated by the use of a Wiener process in models such
as the Integrate-and-Fire model, the internal variability is mostly expressed
via a random firing process. In this paper, we investigate how those distinct
expressions of variability can be related. To do so, we examine the probability
density functions to the corresponding stochastic models and investigate in
what way they can be mapped one to another via integral transforms. Our
theoretical findings offer a new insight view into the particular categories of
variability and it confirms that, despite their contrasting nature, the
mathematical formalization of internal and external variability are strikingly
similar
Spatio-Temporal Scaling of Solar Surface Flows
The Sun provides an excellent natural laboratory for nonlinear phenomena. We
use motions of magnetic bright points on the solar surface, at the smallest
scales yet observed, to study the small scale dynamics of the photospheric
plasma. The paths of the bright points are analyzed within a continuous time
random walk framework. Their spatial and temporal scaling suggest that the
observed motions are the walks of imperfectly correlated tracers on a turbulent
fluid flow in the lanes between granular convection cells.Comment: Now Accepted by Physical Review Letter
- âŠ