TY - JOUR

T1 - Derivation of cable equation by multiscale analysis for a model of myelinated axons

AU - Jerez-Hanckes, Carlos

AU - Pettersson, Irina

AU - Rybalko, Volodymyr

N1 - Funding Information:
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.
Funding Information:
2010 Mathematics Subject Classification. Primary: 35B27; Secondary: 35Q92. Key words and phrases. Hodgkin-Huxley model, nonlinear cable equation, cellular electrophysiology, multiscale modeling, homogenization. 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.
Publisher Copyright:
© 2020 American Institute of Mathematical Sciences. All rights reserved.

PY - 2020

Y1 - 2020

N2 - 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 ε, a small positive parameter. Assuming a nonzero conductivity of the myelin sheath, we find a critical scaling with respect to ε 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.

AB - 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 ε, a small positive parameter. Assuming a nonzero conductivity of the myelin sheath, we find a critical scaling with respect to ε 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.

KW - Cellular electrophysiology

KW - Hodgkin-Huxley model

KW - Homogenization

KW - Multiscale modeling

KW - Nonlinear cable equation

UR - http://www.scopus.com/inward/record.url?scp=85076437746&partnerID=8YFLogxK

U2 - 10.3934/dcdsb.2019191

DO - 10.3934/dcdsb.2019191

M3 - Article

AN - SCOPUS:85076437746

VL - 25

SP - 815

EP - 839

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 3

ER -