Analytical solutions of the Frankenhaeuser-Huxley equations I: minimal model for backpropagation of action potentials in sparsely excitable dendrites

Journal of Integrative Neuroscience
R R Poznanski

Abstract

Hodgkin and Huxley's ionic theory of the nerve impulse embodies principles, applicable also to the impulses in vertebrate nerve fibers, as demonstrated by Bernhard Frankenhaeuser and Andrew Huxley 40 years ago. Frankenhaeuser and Huxley reformulated the classical Hodgkin-Huxley equations, in terms of electrodiffusion theory, and computed action potentials specifically for saltatory conduction in myelinated axons. In this paper, we obtain analytical solutions to the most difficult nonlinear partial differential equations in classical neurophysiology. We solve analytically the Frankenhaeuser-Huxley equations pertaining to a model of sparsely excitable, nonlinear dendrites with clusters of transiently activating, TTX-sensitive Na(+) channels, discretely distributed as point sources of inward current along a continuous (non-segmented) leaky cable structure. Each cluster or hot-spot, corresponding to a mesoscopic level description of Na(+) ion channels, includes known cumulative inactivation kinetics observed at the microscopic level. In such a third-order system, the 'recovery' variable is an electrogenic sodium-pump imbedded in the passive membrane, and the system is stabilized by the presence of a large leak conductance mediated ...Continue Reading

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Citations

Aug 26, 2014·PloS One·Nicolangelo IannellaShigeru Tanaka
Jun 20, 2006·Journal of Integrative Neuroscience·Nicolangelo Iannella, Shigeru Tanaka
Jun 20, 2006·Journal of Integrative Neuroscience·Roman R Poznanski, Jorge J Riera
Jul 1, 2005·Journal of Integrative Neuroscience·M D Goldfinger
Apr 7, 2010·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Roman R Poznanski
Oct 18, 2019·PLoS Computational Biology·Helmut Schmidt, Thomas R Knösche
Jan 30, 2020·Journal of Mathematical Neuroscience·Lu Yihe, Yulia Timofeeva

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