Abstract
A cable-type model for simulating the action potential propagation along a myelinated axon is studied. Compared with the previous works done in [4,8,11], spectral method is used rather than finite difference method and provides better numerical stability; Furthermore, an interpretation of the modeling on Ranvier nodes first given by Fitzhugh in [8] is suggested and the effect of the corresponding parameter on Ranvier node length is examined, which was ignored in [5], therein the author considered the nodes being modeled as points without length. Moreover, the simulation results are firstly obtained to fit the experimental data on frog in [12], and secondly tested by adjusting the anatomical parameters of the axon and found to be in accordance with a recent research on auditory brainstem axons [7], where the authors used a more complicated two-cable distributed-parameter model to include the more detailed anatomical parameters. Thirdly, the model is used to see the application on Multiple Sclerosis disease. Since the cable model can be derived from the classical Poisson-Nernst-Planck (PNP) equation [1], we may consider a recently modified PNP model with steric effect [22], and formally derive a new PNP-type model using Energetic Variational Approach with volume exclusion effect under strong repulsion assumption between ions. This modified PNP model includes ion-ion interaction, which is lacked in the classical PNP model and may be the continuum model to be compared with the classical multi-ion barrier model which is discrete. Thus it is promising to use this continuum model to study the selective permeability and the phenomena of saturation and bindings in the ion channel. Among all the fruitful results in Neuroscience up to now, where discrete models had much been studied and applied [30,31], the main contribution of this work is on the cable-type model as the continuum model in different scales including axon-level and ion-channel level. While there also exists works using spectral methods on the cable-type model, our work is on the myelinated axon, which is different to the works either on branched dendrites [32] or on unmyelinated axon [31]. That is, we extend the study on cable-type model using spectral methods to a myelinated axon situation.