Anomalous Transport and Nonlinear Reactions in Spiny Dendrites

TL;DR

This paper develops a mesoscopic nonlinear model for particle transport and reactions in spiny dendrites, revealing memory effects and anomalous convection phenomena that differ from traditional Markovian models.

Contribution

It extends previous linear models to include nonlinear reactions and memory effects, deriving fractional equations for particle densities in dendritic structures.

Findings

Flux depends on chemical reactions in spines.

Particle mean position exhibits sublinear growth, indicating anomalous convection.

Derived fractional convection-diffusion equation for particle density.

Abstract

We present a \textit{mesoscopic}description of the anomalous transport and reactions of particles in spiny dendrites. As a starting point we use two-state Markovian model with the transition probabilities depending on residence time variable. The main assumption is that the longer a particle survives inside spine, the smaller becomes the transition probability from spine to dendrite. We extend a linear model presented in [PRL, \textbf{101}, 218102 (2008)] and derive the nonlinear Master equations for the average densities of particles inside spines and parent dendrite by eliminating residence time variable. We show that the flux of particles between spines and parent dendrite is not local in time and space. In particular the average flux of particles from a population of spines through spines necks into parent dendrite depends on chemical reactions in spines. This memory effect means that one can not separate the exchange flux of particles and the chemical reactions inside spines. This phenomenon does not exist in the Markovian case. The flux of particles from dendrite to spines is found to depend on the transport process inside dendrite. We show that if the particles inside a dendrite have constant velocity, the mean particle's position $<x(t)> $ increases as $t^{\mu}$ with $\mu <1$ (anomalous convection). We derive a fractional convection-diffusion equation for the total density of particles.