### Abstract

We present a 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 Fedotov [Phys. Rev. Lett. 101, 218102 (2008)10.1103/PhysRevLett.101.218102] 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 cannot 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 tμ with μ<1 (anomalous advection). We derive a fractional advection-diffusion equation for the total density of particles.

Original language | English |
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Article number | 041103 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 82 |

Issue number | 4 |

DOIs | |

Publication status | Published - Oct 6 2010 |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Statistics and Probability

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*82*(4), [041103]. https://doi.org/10.1103/PhysRevE.82.041103