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

Langlands, Trevor and Henry, B. I. and Wearne, S. L. (2009) Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions. Journal of Mathematical Biology, 59 (6). pp. 761-808. ISSN 1432-1416

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We introduce fractional Nernst-Planck equations and derive fractional cable equations as macroscopic models for electrodiffusion of ions in nerve cells when molecular diffusion is anomalous subdiffusion due to binding, crowding or trapping. The anomalous subdiffusion is modelled by replacing diffusion constants with time dependent operators parameterized by fractional order exponents. Solutions are obtained as functions of the scaling parameters for infinite cables and semi-infinite cables with instantaneous current injections. Voltage attenuation along dendrites in response to alpha function synaptic inputs is computed. Action potential firing rates
are also derived based on simple integrate and fire versions of the models. Our results show that electrotonic properties and firing rates of nerve cells are altered by anomalous subdiffusion in these models. We have suggested electrophysiological experiments to calibrate and validate the models.

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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: Author's version deposited in accordance with the copyright policy of the publisher (Springer Verlag)
Faculty / Department / School: Historic - Faculty of Sciences - Department of Maths and Computing
Date Deposited: 31 Oct 2009 06:06
Last Modified: 27 Apr 2017 06:16
Uncontrolled Keywords: dendrite, cable equation, anomalous diffusion, fractional derivative
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: 10.1007/s00285-009-0251-1

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