Journal
JOURNAL OF MATHEMATICAL BIOLOGY
Volume 59, Issue 6, Pages 761-808Publisher
SPRINGER
DOI: 10.1007/s00285-009-0251-1
Keywords
Dendrite; Cable equation; Anomalous diffusion; Fractional derivative
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Funding
- NIDCD NIH HHS [DC05669] Funding Source: Medline
- NIMH NIH HHS [MH071818] Funding Source: Medline
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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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