Derivation of cable equation by multiscale analysis for a model of myelinated axons

Carlos Jerez-Hanckes; Irina Pettersson; Volodymyr Rybalko

doi:10.3934/dcdsb.2019191

Article Contents

2020, Volume 25, Issue 3: 815-839. Doi: 10.3934/dcdsb.2019191

This issue Previous Article Next Article Input-to-state stability of continuous-time systems via finite-time Lyapunov functions

Derivation of cable equation by multiscale analysis for a model of myelinated axons

1.
Faculty of Engineering and Sciences, Universidad Adolfo Ibáñez, Diagonal Las Torres 2700, Peñalolén, Santiago, Chile
2.
Department of Electrical Engineering, Mathematics and Science, University of Gävle, SE-801 76 Gävle, Sweden
3.
B. Verkin Institute for Low Temperature Physics and Engineering of NASU, 47 Nauky Ave., 61103 Kharkiv, Ukraine

Received: August 31, 2018

Revised: February 28, 2019

Early access: September 2019

Published: March 2020

This research was supported by the Swedish Foundation for International Cooperation in Research and Higher Education STINT (research grant IB 2017-7370) and Chile Fondecyt Regular 1171491

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Abstract

We derive a one-dimensional cable model for the electric potential propagation along an axon. Since the typical thickness of an axon is much smaller than its length, and the myelin sheath is distributed periodically along the neuron, we simplify the problem geometry to a thin cylinder with alternating myelinated and unmyelinated parts. Both the microstructure period and the cylinder thickness are assumed to be of order $ \varepsilon $, a small positive parameter. Assuming a nonzero conductivity of the myelin sheath, we find a critical scaling with respect to $ \varepsilon $ which leads to the appearance of an additional potential in the homogenized nonlinear cable equation. This potential contains information about the geometry of the myelin sheath in the original three-dimensional model.

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Mathematics Subject Classification: Primary: 35B27; Secondary: 35Q92.

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Figure 1. Simplified cross-section of a cylindrical myelinated axon (left) and the periodicity cell $ Y $ (right)

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Figure 2. 2D surface generating by revolution the periodicity cell in the neighborhood of a Ranvier node

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Figure 3. Overlapping cells $ \tilde Y_k $ covering $ \Omega_ \varepsilon $

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