Fractional Brownian motion with a reflecting wall

Wada, Alexander H. O.; Vojta, Thomas

doi:10.1103/physreve.97.020102

Cited by 50 publications

(69 citation statements)

References 41 publications

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“…6). The density distribution patterns varied dramatically across the three diffusion regimes, as anticipated (Wada and Vojta, 2018). However, they were robust within the superdiffusion regime, suggesting that the results can be safely generalized to other H values, beyond the one that was used in the simulations (0.8).…”

Section: Resultssupporting

confidence: 61%

“…A particularly important problem for biological sciences is the behavior of FBM in bounded domains (e.g., in two-or three-dimensional shapes). Reflected BM is well understood (Ito and McKean, 1965), but it is not until very recently that the first description of the properties of reflected FBM (rFBM) have become available, in onedimensional domains (Wada and Vojta, 2018;Guggenberger et al, 2019;Wada et al, 2019). The present study is the first application of this theoretical framework to serotonergic fiber distributions, on the whole-brain scale.…”

Section: Introductionmentioning

confidence: 92%

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Serotonergic Axons as Fractional Brownian Motion Paths: Insights into the Self-organization of Regional Densities

Janušonis

Detering

Metzler

et al. 2019

Preprint

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All vertebrate brains contain a dense matrix of thin fibers that release serotonin (5hydroxytryptamine), a neurotransmitter that modulates a wide range of neural, glial, and vascular processes. Perturbations in the density of this matrix have been associated with a number of mental disorders, including autism and depression, but its self-organization and plasticity remain poorly understood. We introduce a model based on reflected Fractional Brownian Motion (FBM), a rigorously defined stochastic process, and show that it recapitulates some key features of regional serotonergic fiber densities. Specifically, we use supercomputing simulations to model fibers as FBM-paths in two-dimensional brain-like domains and demonstrate that the resultant steady state distributions approximate the fiber distributions in physical brain sections immunostained for the serotonin transporter (a marker for serotonergic axons in the adult brain). We suggest that this framework can support predictive descriptions and manipulations of the serotonergic matrix and that it can be further extended to incorporate the detailed physical properties of the fibers and their environment.

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Section: Resultssupporting

confidence: 61%

Section: Introductionmentioning

confidence: 92%

Serotonergic Axons as Fractional Brownian Motion Paths: Insights into the Self-organization of Regional Densities

Janušonis

Detering

Metzler

et al. 2019

Preprint

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“…Motivated by a recent study of FBM in a semi-infinite interval with a reflecting boundary at the origin [46], we here investigate by extensive simulations FBM confined to a finite interval with reflecting boundary conditions. The central result is that the naively expected constant amplitude 1/L in an interval of length L in the stationary limit for Brownian motion is replaced by a solution for FBM in which the amplitude closer to the boundaries is decreased or increased for subdiffusive and superdiffusive FBM.…”

Section: Introductionmentioning

confidence: 99%

Fractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries

et al. 2019

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Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) á ñ a  ( ) X t t 2 with 1<α<2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0<α<1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.

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“…This is consistent with the toy model results, suggesting that the simple fact that a particle observed near a boundary necessarily has a biased history will introduce these depletion effects. Furthermore, it suggests that super-diffusion, where motion increments are positively correlated, would be expected to introduce a commensurate boundary enrichment, as observed in [23] for FBM.…”

Section: 2mentioning

confidence: 83%

“…They have again however focused primarily on the statistical analysis of particle paths to investigate, for example, the effects of confinement on ergodicity. A more recent investigation [23] has shown confinement of particles undergoing FBM can have significant consequences on their spatial distribution. This study however focused on transient rather than steady state dynamics and studied those dynamics on an unbounded domain (half line).…”

Section: Introductionmentioning

confidence: 99%

Sub-Diffusive Dynamics Lead to Depleted Particle Densities Near Cellular Borders

Holmes

2018

Preprint

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It has long been known that the complex cellular environment leads to anomalous motion of intracellular particles. At a gross level, this is characterized by mean squared displacements that deviate from the standard linear profile. Statistical analysis of particle trajectories has helped further elucidate how different characteristics of the cellular environment can introduce different types of anomalousness. A significant majority of this literature has however focused on characterizing the properties of trajectories that do not interact with cell borders (e.g. cell membrane or nucleus). Numerous biological processes ranging from protein activation to exocytosis however require particles to be near a membrane. This study investigates the consequences of a canonical type of sub-diffusive motion, Fractional Brownian Motion (FBM), and its physical analogue Generalized Langevin Equation (GLE) Dynamics, on the spatial localization of particles near reflecting boundaries. Results show that this type of sub-diffusive motion leads to the formation of significant zones of depleted particle density near boundaries, and that this effect is independent of the specific model details encoding those dynamics. Rather these depletion layers are a natural and robust consequence of the anti-correlated nature of motion increments that is at the core of FBM / GLE dynamics. If such depletion zones are present, it would be of profound importance given the wide array of signaling and transport processes that occur near membranes. If not, that would suggest our understanding of this type of anomalous motion may be flawed. Either way, this result points to the need to further investigate the consequences of anomalous particle motions near cell borders from both theoretical and experimental perspectives.

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Fractional Brownian motion with a reflecting wall

Cited by 50 publications

References 41 publications

Serotonergic Axons as Fractional Brownian Motion Paths: Insights into the Self-organization of Regional Densities

Serotonergic Axons as Fractional Brownian Motion Paths: Insights into the Self-organization of Regional Densities

Fractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries

Sub-Diffusive Dynamics Lead to Depleted Particle Densities Near Cellular Borders

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