Fractional cable equation models for anomalous electrodiffusion in nerve cells: finite domain solutions
Article
Article Title | Fractional cable equation models for anomalous electrodiffusion in nerve cells: finite domain solutions |
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ERA Journal ID | 393 |
Article Category | Article |
Authors | Langlands, T. A. M. (Author), Henry, B. I. (Author) and Wearne, S. L. (Author) |
Journal Title | SIAM Journal on Applied Mathematics |
Journal Citation | 71 (4), pp. 1168-1203 |
Number of Pages | 36 |
Year | 2011 |
Place of Publication | United States |
ISSN | 0036-1399 |
1095-712X | |
Digital Object Identifier (DOI) | https://doi.org/10.1137/090775920 |
Web Address (URL) | https://epubs.siam.org/doi/10.1137/090775920 |
Abstract | In recent work we introduced fractional Nernst–Planck equations and related fractional cable equations to model electrodiffusion of ions in nerve cells with anomalous subdiffusion along and across the nerve cells. This work was motivated by many computational and experimental studies showing that anomalous diffusion is ubiquitous in biological systems with binding, crowding, or trapping. For example, recent experiments have shown that anomalous subdiffusion occurs along the axial direction in spiny dendrites due to trapping by the spines. We modeled the subdiffusion in two ways leading to two fractional cable equations and presented fundamental solutions on infinite and semi-infinite domains. Here we present solutions on finite domains for mixed Robin boundary conditions. The finite domain solutions model passive electrotonic properties of spiny dendritic branch segments with ends that are voltage clamped, sealed, or killed. The behavior of the finite domain solutions is similar for both fractional cable equations. With uniform subdiffusion along and across the nerve cells, the solution approaches the standard nonzero steady state, but the approach is slowed by the anomalous subdiffusion. If the subdiffusion is more anomalous along the axial direction, then (boundary conditions permitting) the solution converges to a zero steady state, whereas if the subdiffusion is less anomalous along the axial direction, then the solution approaches a spatially linear steady state. These solutions could be compared with realistic electrophysiological experiments on actual dendrites. |
Keywords | dendrite; cable equation; anomalous diffusion; fractional derivative; finite domain solution |
Contains Sensitive Content | Does not contain sensitive content |
ANZSRC Field of Research 2020 | 490510. Stochastic analysis and modelling |
490410. Partial differential equations | |
490102. Biological mathematics | |
Public Notes | File reproduced in accordance with the copyright policy of the publisher/author. |
Byline Affiliations | Department of Mathematics and Computing |
University of New South Wales | |
Icahn School of Medicine at Mount Sinai, United States | |
Institution of Origin | University of Southern Queensland |
https://research.usq.edu.au/item/q0z74/fractional-cable-equation-models-for-anomalous-electrodiffusion-in-nerve-cells-finite-domain-solutions
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