Emergence of Fractional Kinetics in Spiny Dendrites

TL;DR

This paper demonstrates that fractional kinetics in spiny dendrites can be naturally derived from continuous time random walks and generalized Brownian motion models, explaining anomalous diffusion observed in neural structures.

Contribution

It shows that the time-fractional cable equation emerges naturally from CTRW models and ggBm-like constructions, linking geometry and stochastic processes in dendritic diffusion.

Findings

Fractional cable equations relate to CTRW and ggBm models.

Anomalous diffusion in dendrites can be explained by superimposing Markovian processes.

The fractional model arises naturally from stochastic process considerations.

Abstract

Fractional extensions of the cable equation have been proposed in the literature to describe transmembrane potential in spiny dendrites. The anomalous behavior has been related in the literature to the geometrical properties of the system, in particular, the density of spines, by experiments, computer simulations, and in comb-like models.~The same PDE can be related to more than one stochastic process leading to anomalous diffusion behavior. The time-fractional diffusion equation can be associated to a continuous time random walk (CTRW) with power-law waiting time probability or to a special case of the Erd\'ely-Kober fractional diffusion, described by the ggBm. In this work, we show that time fractional generalization of the cable equation arises naturally in the CTRW by considering a superposition of Markovian processes and in a {\it ggBm-like} construction of the random variable.