Determination of time-constants in cables of finite length

Abstract

Current flow in cylindrical nerve and muscle fibre has been analysed in terms of a mathematical model leading to a linear partial differential equation for the voltage as a function of both position and time. In the case of a one-dimensional cable subject to a step input of current, the solution will consist of a steady-state behaviour preceded by an initial transient. The electrical properties of the fibre or cable itself determine a length-constant, λ, which can be determined experimentally from the steady-state response, and a time-constant, τ, which must be found from the initial transient.

When the cable is infinite and when there is a single input electrode, an exact solution can be produced which enables ready determination of the time-constant τ.

Two complications arise in experimental practice, however. In the first place, the fibre has finite length, and in the second, two spatially separated stimulation electrodes are often required. We thus analyse a more complicated and more general situation. The linearity of the membrane properties, however, allows the solution to the more general case to be built up by superposition of solutions from the simpler case (equivalent to the classical method of images). We also approximate the Hodgkin and Rushton solution by asymptotic formulae in order to allow more tractable expressions for the exact solution.

We are thus able to give a method for the ready evaluation of the time constant τ under more general conditions.