Mathematical models of threshold phenomena in the nerve membrane

The types of mathematical model which have been used to represent all-or-none behavior in the nerve membrane may be classified as follows: (1) thediscontinuous threshold phenomenon, in which differential equations with discontinuous functions provide both a discontinuity of response as a function of stimulus intensity at threshold and a finite maximum latency, (2) thesingular-point threshold phenomenon which exists in a phase space having analytic functions in its differential equations and having a singular point with one characteristic root positive and the rest with negative real parts, the latency being unbounded, and (3) thequasi threshold phenomenon, which has a finite maximum latency and continuous functions, but neither a true discontinuity in response nor an exact threshold. Several models of the nerve membrane in the literature are classified accordingly, and the applicability of the different types of threshold phenomena to the membrane is discussed, including an extension to a stochastic model.