The Hodgkin and Huxley equations

In the fourth paper of their classic 1952 series, Hodgkin and Huxley concatenated all of their analytic expressions into the succinct set of equations shown here. This image has been scanned in from the original paper.

The upper line (their equation 26) describes the total membrane current flowing through the four electrical paths through the membrane. These four elements are shown in the diagram of the equivalent electrical circuit for the membrane, where they are arranged from left to right in the same order that they appear in the equation.

The upper equation (26) states that the total membrane current equals the sum of four currents through the membrane:

1. The capacitive current: IC = Cm * dV/dt.

2. The K current: IK = gKbar * n^4 (V - EK)
   where:
   • gKbar in mho(S)/cm^2 is the maximum K conductance possible (all K channels are open)
   • n^4 provides the kinetic pattern of the K conductance
   • (V - EK) is the electrical driving force on K: the difference between the membrane voltage at any time and the equilibrium potential at which the net flow of K ions is zero

3. The Na current: INa = gNabar * m^3 * h (V - ENa)
   where:
   • gNabar in mho(S)/cm^2 is the maximum Na conductance possible
   • m^3 and h are dynamic variables providing the kinetic pattern of the Na conductance
   • (V - ENa) is the electrical driving force on Na: the difference between the membrane voltage at any time and the equilibrium potential at which the net flow of Na ions is zero.

4. the leakage current: Ileak = gl * (V - El)
   where
   • gl is the leak conductance
   • (V - El) is the electrical dirving force on the leakage channel.

The lower set of equations (7, 15,16) describe the kinetics of the variables, n, m, and h.

These variables are probabilities allowing one to calculate the conductance at any point in time as follows:

1. The K conductance: gK= gKbar * n^4.

2. The Na conductance: gNa= gNabar * m^3 * h.

The Hodgkin/Huxley equations have so successfully encapsulated the details of the mechanisms underlying impulse generation and propagation in squid giant axon that they are still the de facto standard for simulations of axons.They serve also as a framework by which equations for other excitable membranes are described.