Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory

Area of Science:

• Computational Neuroscience
• Applied Mathematics
• Biophysics

Background:

• Myelinated nerve axon conduction involves potential changes jumping between nodes.
• Existing models often simplify or omit explicit treatment of forward and backward deviating arguments.

Purpose of the Study:

• To numerically solve a nonlinear functional differential equation modeling myelinated nerve axon conduction.
• To develop and validate a novel computational scheme that explicitly incorporates advance and delay terms.

Main Methods:

• A difference scheme was employed to approximate the functional differential equation on a finite interval.
• Asymptotic representation, cubic interpolation, iterative techniques, and a continuation method were utilized.
• The numerical scheme was validated against analytically solvable problems for accuracy and stability.

Main Results:

• The numerical scheme accurately solves the functional differential equation for myelinated nerve axon conduction.
• The study provides the first numerical analysis explicitly treating advance and delay terms in this context.
• The dependence of the solution on various physically relevant model parameters was investigated.

Conclusions:

• The developed numerical method is accurate and stable for simulating myelinated nerve conduction.
• This approach offers a more comprehensive understanding of impulse propagation by explicitly modeling delays.
• The findings contribute to the computational neuroscience of nerve signal transmission.