201 research outputs found
First passage time problem for biased continuous-time random walks
We study the first passage time (FPT) problem for biased continuous time
random walks. Using the recently formulated framework of fractional
Fokker-Planck equations, we obtain the Laplace transform of the FPT density
function when the bias is constant. When the bias depends linearly on the
position, the full FPT density function is derived in terms of Hermite
polynomials and generalized Mittag-Leffler functions.Comment: 12 page
First Passage Time Distribution for Anomalous Diffusion
We study the first passage time (FPT) problem in Levy type of anomalous
diffusion. Using the recently formulated fractional Fokker-Planck equation, we
obtain an analytic expression for the FPT distribution which, in the large
passage time limit, is characterized by a universal power law. Contrasting this
power law with the asymptotic FPT distribution from another type of anomalous
diffusion exemplified by the fractional Brownian motion, we show that the two
types of anomalous diffusions give rise to two distinct scaling behavior.Comment: 11 pages, 2 figure
Generalized Turing Patterns and Their Selective Realization in Spatiotemporal Systems
We consider the pattern formation problem in coupled identical systems after
the global synchronized state becomes unstable. Based on analytical results
relating the coupling strengths and the instability of each spatial mode
(pattern) we show that these spatial patterns can be selectively realized by
varying the coupling strengths along different paths in the parameter space.
Furthermore, we discuss the important role of the synchronized state (fixed
point versus chaotic attractor) in modulating the temporal dynamics of the
spatial patterns.Comment: 9 pages, 3 figure
Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data
It is often useful in multivariate time series analysis to determine
statistical causal relations between different time series. Granger causality
is a fundamental measure for this purpose. Yet the traditional pairwise
approach to Granger causality analysis may not clearly distinguish between
direct causal influences from one time series to another and indirect ones
acting through a third time series. In order to differentiate direct from
indirect Granger causality, a conditional Granger causality measure in the
frequency domain is derived based on a partition matrix technique. Simulations
and an application to neural field potential time series are demonstrated to
validate the method.Comment: 18 pages, 6 figures, Journal publishe
Uncovering interactions in the frequency domain
Oscillatory activity plays a critical role in regulating biological processes at levels ranging from subcellular, cellular, and
network to the whole organism, and often involves a large number of interacting elements. We shed light on this issue by
introducing a novel approach called partial Granger causality to reliably reveal interaction patterns in multivariate data with
exogenous inputs and latent variables in the frequency domain. The method is extensively tested with toy models, and
successfully applied to experimental datasets, including (1) gene microarray data of HeLa cell cycle; (2) in vivo multielectrode
array (MEA) local field potentials (LFPs) recorded from the inferotemporal cortex of a sheep; and (3) in vivo LFPs
recorded from distributed sites in the right hemisphere of a macaque monkey
Analyzing Stability of Equilibrium Points in Neural Networks: A General Approach
Networks of coupled neural systems represent an important class of models in
computational neuroscience. In some applications it is required that
equilibrium points in these networks remain stable under parameter variations.
Here we present a general methodology to yield explicit constraints on the
coupling strengths to ensure the stability of the equilibrium point. Two models
of coupled excitatory-inhibitory oscillators are used to illustrate the
approach.Comment: 20 pages, 4 figure
Mitigating the effects of measurement noise on Granger causality
Computing Granger causal relations among bivariate experimentally observed
time series has received increasing attention over the past few years. Such
causal relations, if correctly estimated, can yield significant insights into
the dynamical organization of the system being investigated. Since experimental
measurements are inevitably contaminated by noise, it is thus important to
understand the effects of such noise on Granger causality estimation. The first
goal of this paper is to provide an analytical and numerical analysis of this
problem. Specifically, we show that, due to noise contamination, (1) spurious
causality between two measured variables can arise and (2) true causality can
be suppressed. The second goal of the paper is to provide a denoising strategy
to mitigate this problem. Specifically, we propose a denoising algorithm based
on the combined use of the Kalman filter theory and the
Expectation-Maximization (EM) algorithm. Numerical examples are used to
demonstrate the effectiveness of the denoising approach.Comment: 16 pages, 7 figure
- …