scholarly journals Entropic Approach to the Detection of Crucial Events

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 178 ◽  
Author(s):  
Garland Culbreth ◽  
Bruce West ◽  
Paolo Grigolini
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Anomalous Diffusion ◽  
Correlation Functions ◽  
Joint Action ◽  
Clear Distinction ◽  
Aging Effects ◽  
Recent Advances ◽  
The Brain

In this paper, we establish a clear distinction between two processes yielding anomalous diffusion and 1 / f noise. The first process is called Stationary Fractional Brownian Motion (SFBM) and is characterized by the use of stationary correlation functions. The second process rests on the action of crucial events generating ergodicity breakdown and aging effects. We refer to the latter as Aging Fractional Brownian Motion (AFBM). To settle the confusion between these different forms of Fractional Brownian Motion (FBM) we use an entropic approach properly updated to incorporate the recent advances of biology and psychology sciences on cognition. We show that although the joint action of crucial and non-crucial events may have the effect of making the crucial events virtually invisible, the entropic approach allows us to detect their action. The results of this paper lead us to the conclusion that the communication between the heart and the brain is accomplished by AFBM processes.

scholarly journals Generalized Fractional Master Equation for Self-Similar Stochastic Processes Modelling Anomalous Diffusion

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Gianni Pagnini ◽  
Antonio Mura ◽  
Francesco Mainardi
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Master Equation ◽  
Fractional Brownian Motion ◽  
Anomalous Diffusion ◽  
Natural Generalization ◽  
Fractional Diffusion ◽  
Gaussian Density ◽  
Master Equation Approach ◽  
Self Similar

The Master Equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdélyi-Kober fractional diffusion, that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes, and the standard Brownian motion. In this framework, the M-Wright function (known also as Mainardi function) emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.

A deep learning-based approach to model anomalous diffusion of membrane proteins: The case of the nicotinic acetylcholine receptor

2021 ◽  
Author(s):  
Héctor Buena Maizón ◽  
Francisco J. Barrantes
Keyword(s):  
Brownian Motion ◽  
Deep Learning ◽  
Fractional Brownian Motion ◽  
Acetylcholine Receptor ◽  
Nicotinic Acetylcholine Receptor ◽  
Single Molecule ◽  
Anomalous Diffusion ◽  
Translational Diffusion ◽  
Network System ◽  
Nicotinic Acetylcholine

AbstractWe present a concatenated deep-learning multiple neural network system for the analysis of single-molecule trajectories. We apply this machine learning-based analysis to characterize the translational diffusion of the nicotinic acetylcholine receptor at the plasma membrane, experimentally interrogated using superresolution optical microscopy. The receptor protein displays a heterogeneous diffusion behavior that goes beyond the ensemble level, with individual trajectories exhibiting more than one diffusive state, requiring the optimization of the neural networks through a hyperparameter analysis for different numbers of steps and durations, especially for short trajectories (<50 steps) where the accuracy of the models is most sensitive to localization errors. We next use the statistical models to test for Brownian, continuous-time random walk, and fractional Brownian motion, and introduce and implement an additional, two-state model combining Brownian walks and obstructed diffusion mechanisms, enabling us to partition the two-state trajectories into segments, each of which is independently subjected to multiple analysis. The concatenated multi-network system evaluates and selects those physical models that most accurately describe the receptor’s translational diffusion. We show that the two-state Brownian-obstructed diffusion model can account for the experimentally observed anomalous diffusion (mostly subdiffusive) of the population and the heterogeneous single-molecule behavior, accurately describing the majority (72.5% to 88.7% for α-bungarotoxin-labeled receptor and between 73.5% and 90.3% for antibody-labeled molecules) of the experimentally observed trajectories, with only ∼15% of the trajectories fitting to the fractional Brownian motion model.

scholarly journals Erdélyi-Kober fractional diffusion

2012 ◽  
Vol 15 (1) ◽  
Author(s):  
Gianni Pagnini
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
Anomalous Diffusion ◽  
Short Note ◽  
Natural Generalization ◽  
Fractional Diffusion ◽  
Gaussian Density ◽  
Fractional Integral Equation ◽  
Mainardi Function

AbstractThe aim of this Short Note is to highlight that the generalized grey Brownian motion (ggBm) is an anomalous diffusion process driven by a fractional integral equation in the sense of Erdélyi-Kober, and for this reason here it is proposed to call such family of diffusive processes as Erdélyi-Kober fractional diffusion. The ggBm is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion. This class is made up of self-similar processes with stationary increments and it depends on two real parameters: 0 < α ≤ 2 and 0 < β ≤ 1. It includes the fractional Brownian motion when 0 < α ≤ 2 and β = 1, the time-fractional diffusion stochastic processes when 0 < α = β < 1, and the standard Brownian motion when α = β = 1. In the ggBm framework, the Mainardi function emerges as a natural generalization of the Gaussian distribution recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bakka ◽  
S. Hajji ◽  
D. Kiouach
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter ◽  
Fixed Point Principle ◽  
Stochastic Functional

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

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