Fractional cable equation models for anomalous electrodiffusion in nerve cells: finite domain solutions

Langlands, T. A. M. and Henry, B. I. and Wearne, S. L. (2011) Fractional cable equation models for anomalous electrodiffusion in nerve cells: finite domain solutions. SIAM Journal on Applied Mathematics, 71 (4). pp. 1168-1203. ISSN 0036-1399

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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.

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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: Published version deposited in accordance with the copyright policy of the publisher (SIAM) '[The author(s) retain] The right to post an electronic version of the final SIAM file of the work on the author's current institutionalInternet server for limited noncommercial distribution, provided that proper notice of SIAM's copyright is included and that no separate or additional fees are collected for access to or distribution of the work apart from Internet access fees which may be paid to an Internet access provider.'
Faculty / Department / School: Historic - Faculty of Sciences - Department of Maths and Computing
Date Deposited: 28 Dec 2011 10:34
Last Modified: 26 Aug 2014 03:49
Uncontrolled Keywords: dendrite; cable equation; anomalous diffusion; fractional derivative; finite domain solution
Fields of Research : 01 Mathematical Sciences > 0104 Statistics > 010406 Stochastic Analysis and Modelling
01 Mathematical Sciences > 0101 Pure Mathematics > 010110 Partial Differential Equations
01 Mathematical Sciences > 0102 Applied Mathematics > 010202 Biological Mathematics
Socio-Economic Objective: C Society > 92 Health > 9201 Clinical Health (Organs, Diseases and Abnormal Conditions) > 920112 Neurodegenerative Disorders Related to Ageing
E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences
Identification Number or DOI: doi: 10.1137/090775920

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