The Discrete Nature of Biological Membrane Conductance, Channel Interaction Through Electrolyte Layers and the Cable Equation

An initial value problem for membrane potential is formulated and solved to allow for the discrete (non-continuous) nature of biological membrane conductance. The discrete model solutions are discussed and compared with those resulting from the cable equation. It is shown that the solutions of the discrete model approach the latter when the channel distance becomes smaller than the space constant of the system. This convergence is interpreted as a consequence of "channel interaction" through electrolyte layers surrounding the membrane. In the case of an axon (channel distance ⪡ space constant) the effect of the discreteness on both the potential profile and the kinetics of the membrane response may be regarded as negligible, the physical reason for this behavior being the "mean-field like" properties of the system.