From Cable Equation to Active and Passive Nerve Membrane Model

Action potential generation and propagation along a nerve fiber is an exciting phenomena and the process is described using a series of non-linear differential equations called as the telegraphers equation or the cable equation. A number of cable equation solutions are proposed so far to describe the propagation process considering different cable parameters under some specific boundary conditions. In the proposed work, the cable equation is extended into modeling equivalent electrical circuit and the solution of the propagation model for a passive fiber is computed based on the equivalent electrical circuit. Further, a similar solution is used in order to model the Hodgkin and Huxley type active membrane from the cable equation. The modeled differential equation is used to transform a neuronal fiber into equivalent electrical circuit that can represent the active and passive fiber characteristics. A combination of such active and passive membrane equivalent electrical circuits can be easily implemented to mimic more complex passive fiber dynamics having localized active channels with low computational complexity.