On the role of fractional Brownian motion in models of chemotaxis and stochastic gradient ascent

TL;DR

This work investigates whether temporally correlated noise modeled by fractional Brownian motion with $H>1/2$ can enhance chemotactic search dynamics. By formulating chemotaxis as a stochastic gradient ascent driven by $doldsymbol{X}_t=oldsymbol{ abla}f(oldsymbol{X}_t)dt+oldsymbol{ Sigma}(oldsymbol{X}_t)doldsymbol{W}^H_t$ and extending to smooth manifolds, the authors analyze first hitting times to the global maximum of the chemoattractant field $f$, including scenarios with curvature and secondary cues. Across Euclidean and curved domains, and under both independent and interacting (swarming) dynamics, they show that memory effects from $H>1/2$ robustly accelerate reaching the global maximum, though long-range correlations can delay individual trajectories. These findings have broad implications for biological navigation in noisy environments and offer insights for designing optimization and sampling algorithms that leverage structured stochasticity.

Abstract

Cell migration often exhibits long-range temporal correlations and anomalous diffusion, even in the absence of external guidance cues such as chemical gradients or topographical constraints. These observations raise a fundamental question: do such correlations simply reflect internal cellular processes, or do they enhance a cell's ability to navigate complex environments? In this work, we explore how temporally correlated noise (modeled using fractional Brownian motion) influences chemotactic search dynamics. Through computational experiments, we show that superdiffusive motion, when combined with gradient-driven migration, enables robust exploration of the chemoattractant landscape. Cells reliably reach the global maximum of the concentration field, even in the presence of spatial noise, secondary cues, or irregular signal geometry. We quantify this behavior by analyzing the distribution of first hitting times under varying degrees of temporal correlation. Notably, our results are consistent across diverse conditions, including flat and curved substrates, and scenarios involving both primary and self-generated chemotactic signals. Beyond biological implications, these findings also offer insight into the design of optimization and sampling algorithms that benefit from structured stochasticity.

Figures (19)

• Figure 1: Examples of stationary chemoattractant distributions explored in Section \ref{['res:Euclid']}. The setting on the left is referred to as the "weakly trapping" experiment, whereas the one on the right corresponds to the "strongly trapping" experiment.
• Figure 2: Spatial positions of $100$ particles at $t=10$ for different parameter sets in a one-dimensional version of the weakly trapping experiment. An X-mark indicates the initial particle location, and the numbers in the red boxes denote the total number of particles that reached the global maximum within the observation window.
• Figure 3: Spatial positions of $100$ particles at $t=10$ for different parameter sets in a two-dimensional version of the weakly trapping experiment of Fig. \ref{['fig312']}. An X-mark indicates the initial particle location, and the numbers in the red boxes denote the total number of particles that reached the global maximum within the observation window.
• Figure 4: Spatial positions of $100$ particles at $t=10$ for different parameter sets in a one-dimensional version of the strongly trapping experiment. An X-mark indicates the initial particle location, and the numbers in the red boxes denote the total number of particles that reached the global maximum within the observation window.
• Figure 5: Spatial positions of $100$ particles at $t=10$ for different parameter sets in a two-dimensional version of the strongly trapping experiment of Fig. \ref{['fig314']}. An X-mark indicates the initial particle location, and the numbers in the red boxes denote the total number of particles that reached the global maximum within the observation window.