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Approximate analytical solution of a (V,m,h) reduced system for backpropagating action potentials in sparsely excitable dendrites

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Nicolangelo L. Iannella and Roman R.  Poznanski  (2023). Approximate analytical solution of a (V,m,h) reduced system for backpropagating action potentials in sparsely excitable dendrites. Journal of Multiscale Neuroscience 2(2), 350-372     


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We derive an approximate analytical solution of a nonlinear cable equation describing the backpropagation of action potentials in sparsely excitable dendrites with clusters of transiently activating, TTX-sensitive Na+ channels of low density, discretely distributed as point sources of transmembrane current along a continuous (non-segmented) passive cable structure. Each cluster or hotspot, corresponding to a mesoscopic level description of Na+ ion channels, included known cumulative inactivation kinetics observed at the microscopic level. In such a reduced third-order system, the ‘recovery’ variable is an electrogenic sodium-pump and/or a Na+-Ca2+ exchanger imbedded in the passive membrane, and a high leakage conductance stabilizes the system. A nonlinear cable equation was used to investigate back-propagation and repetitive activity of action potentials, exhibiting characteristics of the modified Hodgkin-Huxley kinetics (in the presence of suprathreshold input). In particular, a time-dependent analytical solution was obtained through a perturbation expansion of the membrane potential (V) for all voltage dependent terms including the voltage dependent Na+ activation (m) and state-dependent inactivation (h) gating variables and then solving the resulting system of integral equations. It was shown that back-propagating action potentials attenuate in amplitude are dependent on the discrete and low-density distributions of transient Na+ channels along the cable structure. A major significance of integrative modelling is the provision of a continuous description of the non-dimensional membrane potential (Φ) as a function of position.

Keywords:  Backpropagation, ionic cable theory, dendritic action potentials, frequency, repetitive discharge, analytical modeling, Hodgkin-Huxley kinetics

Conflict of Interest

The authors declare that they were an editorial board member of JMN, at the time of submission. This had no impact on the peer review process and the final decision.

This article belongs to the Special Issue                

Dynamic Multiscaling in Neuroscience

       Lead Editor:  Dr. Nicolangelo Iannella

               University of Oslo, Norway

Copyright: © 2023 The Author(s). Published by Neural Press.

This is an open access article distributed under the terms and conditions of the CC BY 4.0 license.

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that made by its manufacturer, is not guaranteed or endorsed by the publisher.

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