Conditions of appearance of local response and action potential in a simple mathematical model of nerve

A simple mathematical model of a nerve fibre is proposed. According to the model the current-voltage relation for the peak Na-current is represented as a broken line. It is shown that within the potential range, where the differential (slope) Na-conductance isnegative but its absolutevalue is smaller than the membrane conductance, the fibre gives a subthreshold local response. Within the range, where differential Na-conductance is negative and has absolute value larger than the membrane conductance, the action potential arises, the rise being independent on the strength of the stimulus and conditioned only by the membrane parameters. A simple formula is obtained which describes the relation between the strength of the threshold stimulus, I, and its duration, t₀. There are two parameters in this formula, which can be measured independently: the time constant of the membrane and the lag-period in the Na-current. Insertion of these parameters into the formula allows one to construct a theoretical I-t₀ curve in good agreement with experiment.