Serotonergic Axons as Fractional Brownian Motion Paths: Insights Into the Self-Organization of Regional Densities

Abstract

All vertebrate brains contain a dense matrix of thin fibers that release serotonin (5-hydroxytryptamine), 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.

Probability density of the fractional Langevin equation with reflecting walls

“…‡ If the increments are positively correlated (persistent), the resulting motion is superdiffusive (α > 1) whereas anticorrelated (antipersistent) increments produce subdiffusive motion (α < 1). FBM has been employed to model a variety of system ranging from diffusion inside biological cells [14][15][16][17][18][19], the dynamics of polymers [20,21], electronic network traffic [22], and the geometry of serotonergic fibers in vertebrate brains [23], to fluctuations of financial markets [24,25]. FBM was put forward by Kolmogorov [26] as well as Mandelbrot and van Ness [27].…”

“…The deviations from the common master curve at the smallest x for each t stem from the time discretization, they occur for x of the order of the step width σ = 1. (b) Data for different α at the longest simulated times, ranging from t = 2 17 (for α = 1.6) to t = 223 (for α = 0.6). The solid lines are fits of the small-y behavior to the conjectured power law Y ∼ y κ with κ = 2/α − 1.…”

Tempered fractional Brownian motion on finite intervals

“…Finally, we point out that the tempering of the correlations provides a powerful tool in applications in which a stochastic process is used to model experimental data. For example, FBM was recently put forward as a model to explain the spatial distribution of serotonergic fibers in vertebrate brains [24,25]. Despite the limited "neurobiological input", the model captures important aspects of the highly nonuniform distributions of these fibers throughout the brain.…”

“…In the marginal case, α = 1, FBM is identical to normal Brownian motion with uncorrelated increments. FBM processes have been used to describe the motion inside biological cells [18][19][20][21][22][23], the patterns of serotonergic fibers in verebrate brains [24,25]. polymer dynamics [26,27], electronic network traffic [28], as well as fluctuations of financial markets [29,30].…”

Fractional Brownian motion in superharmonic potentials and non-Boltzmann stationary distributions

“…FBM-like correlations are particularly studied in modern financial market models to account for market "roughness" [57][58][59], and similar effects in network traffic [60]. FBM was also applied to describe observed density profiles of serotonergic brain fibres [61].…”