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

TL;DR

This paper develops a multiscale model of myelinated axons, deriving a nonlinear cable equation that accounts for imperfect insulation of myelin and microstructural details, providing insights into potential propagation.

Contribution

It introduces a novel derivation of a nonlinear cable equation incorporating microstructural effects and imperfect myelin insulation through multiscale analysis.

Findings

Derived a nonlinear cable equation for myelinated axons.

Identified the impact of imperfect myelin insulation on potential propagation.

Connected microstructure geometry to effective potential in the model.

Abstract

The paper concerns the multiscale modeling of a myelinated axon. Taking into account the microstructure with alternating myelinated parts and nodes Ranvier, we derive a nonlinear cable equation describing the potential propagation along the axon. We assume that the myelin is not a perfect insulator, and assign a low (asymptotically vanishing) conductivity in the myelin. Compared with the case when myelin is assumed to have zero conductivity, an additional potential arises in the limit equation. The coefficient in front of the effective potential contains information about the geometry of the myelinated parts.