# A distributed-parameter model of the myelinated nerve fiber

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 Title: A distributed-parameter model of the myelinated nerve fiber Halter, John Alan Clark, John W., Jr. This thesis is concerned with the development of a new distributed-parameter model of the myelinated nerve fiber that includes an adequate representation of the anatomical complexity present at the node of Ranvier, considers the peri-axonal conductance pathway and includes active ionic channels in the axolemmal membrane beneath the myelin sheath. A detailed review of the current anatomical and electrophysiological literature was performed and tables containing the biophysical properties of the amphibian and mammalian myelinated nerve fibers were compiled. Recent dynamics for the mammalian sodium channel were adopted and a non-linear parameter estimation method was applied to fit the dynamics of a fast and a slow amphibian potassium channel. Membrane action potentials were computed from the new space-clamped nodal and internodal membrane models. A multi-axial electrical equivalent circuit was developed with intra-axonal, peri-axonal and extra-axonal longitudinal conduction paths and independent transverse elements for the axolemmal membrane and the myelin sheath. Non-uniform spatial step sizes were used, enabling detailed representation of the nodal region while minimizing the number of segments necessary to represent the entire fiber. An implicit integration method was developed for the resulting multiple cross-coupled parabolic partial differential equations as represented in finite difference matrix form. The solution was implemented on a Sequent Systems S81 parallel processor, dividing the solution for the component ionic currents and the matrix operations across an arbitrary number of processors. The model allowed for the detailed examination of component potentials and currents. Physiological conduction velocities of 20.2 m/s at 20$\sp\circ$C for a 15$\mu$m diameter amphibian fiber and 57.6 m/s at 37$\sp\circ$C for a 17.5$\mu$m diameter mammalian fiber were achieved. An increase in conduction velocity of 32.3% was seen for a nodal constriction of 80%, contrary to previous modeling efforts. The peri-axonal space in the paranodal region was shown to have a strong influence on conduction velocity. Restriction of the extracellular volume resulted in slower conduction velocities for radii less than 25 $\mu$m. Paranodal ionic channels were activated during conduction and there was model evidence for the possible generation of slow (0.06 m/s) propagating action potentials beneath the myelin sheath. Halter, John Alan. (1989) "A distributed-parameter model of the myelinated nerve fiber." Doctoral Thesis, Rice University. http://hdl.handle.net/1911/16506. http://hdl.handle.net/1911/16506 1989