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

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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  • 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.

    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)

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

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

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