In the 1950's Hodgkin and Huxley (1952) introduced the first continuum mathematical model designed to reproduce
membrane action potentials. In their model, they assumed that when the membrane potential was not equal to the equilibrium potential,
there would be a net flow of ions proportional to the difference between the membrane potential and the equilibrium potential.
Thus for an ion species labeled y with a corresponding current , Ohm's law gave the
equation , where is the equilibrium potential, is the
resistance to the flow of ions, which can be expressed in terms of a membrane conductance .
The ionic conductance is, in general, a nonlinear function of the membrane voltage and can be represented
by several channels. In cells, the ion channels are either open or closed, and in the case of voltage gating,
the percentage of channels that are open out of the total number of channels in the cell is a function of the
membrane voltage. Therefore, the probability function y (known as a gate variable) can be constructed to define
what fraction of the channels are open as a function of voltage and time. Since gate variables range between 0
(all ion channels closed) and 1 (all ion channels open), the total cell or tissue conductance gion due to a
given ion is given by , where g is the maximum possible conductance obtained when all channels are open.
Hodgkin and Huxley found that squid axon sodium and potassium conductances, obtained from voltage-clamp
techniques, could be reproduced by using gate variables obeying simple first order equations of the form: