On the Geometry of Interacting Axon Tracts in Relation to Their Functional Properties

It is well known that demyelination associated with multiple sclerosis alters coupling between axons in a diseased nerve tract due to exposure of ion channels. In this work we ask the question whether the presence of electric fields as a coupling mechanism between nerve axons, has any impact on the expected conduction properties of the nerve under focal demyelination. Using an approximate geometry for a bundle of three axons, we allow electric fields generated at a node of Ranvier to influence other nodes of Ranvier on different axons. In order to introduce focal demyelination, we allow a new type of structure, intermediate between a node and an internodal segment. We find that the nerve's conduction profiles are different with and without field-based coupling. This is important because it opens up the possibility for an experimental investigation to study whether field-based coupling predominates over current-based coupling. Fields and currents may also be measured using different methods and so there are implications for bioinstrumentation. From a systems perspective, the study is based on a novel equation which allows electric fields to act as sources of current at axons' nodes of Ranvier, thereby generating new behaviors from the set of coupled partial differential equations that model the electrodynamics of the nerve. In addition to our simulation study of focal demyelination, we present a justification for a certain geometric W-matrix used in our phenomenological 2019 model. We also provide the first simulations of a tract of tortuous axons, showing the impact of tract tortuosity on conduction velocity. The tortuous model is analyzed geometrically to illustrate the mechanism of synchronization. Finally, this same model is shown to be castable as a learning machine - a formulation that has neuroscience implications. ### Competing Interest Statement The authors have declared no competing interest.