The Hodgkin-Huxley equations.
Hodgkin and Huxley published a series of (now classical) papers in 1952 in which they investigated membrane currents through the membrane of the squid giant axon. A mathematical analysis of this work was published in a review paper
Hodgkin A.L. and Huxley, A.F. (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond.) 117;500-544
Current flowing across the giant axon membrane may be represented by the sum of conductive components (What we now identify as ion channels) and capacitance (the cell membrane). The currents are described in the following circuit diagram.
Circuit diagram summarizing the currents across the cell membrane of a squid giant axon used to construct the Hodgkin-Huxley model.
In this page we consider an example of how one of these components may be modeled. Here we demonstrate the method that Hodgkin and Huxley use to describe Na conductances.
In the Hodgkin-Huxley model conductive components are a function of potential difference across the cell membrane (Vm) and the equilibrium potentials (E) of the ions. The equilibrium potentials may be derived from the Nernst-Einstein relationship. The currents through the conductive elements may then be expressed as:
(1)
Hodgkin’s and Huxley’s results demonstrated that gNa and gK are a function of time as well as voltage, but the conductances of the other ions (and probably some gK) are constant. Depolarization of the axonal membrane causes a transient increase in Na conductance and a slower non-inactivating increase in K conductance. The time dependence of this conductance may be represented by an activation coefficient x which (if we consider that the conductance is represented by the opening of many individual channels) represents the probability of a gate in the channel being open. The conductance for a time dependent channel can thus be written in terms of its activation coefficient x (0 <= x <= 1) and a maximum conductance gion,max:
(2)
x may be described by the following equation:
(3)
ax and bx are rate coefficients which are non-linear functions of voltage (units are 1/time).
The activation coefficient can be raised to a higher power if one hypothesizes that a larger number of gates are present in one ion channel. Additionally, each channel may have multiple activation coefficients to describe activation followed by, for example, voltage-dependent inactivation. Current may then be represented by equation with the following general form:
(4)
In this equation x is the activation coefficient and y is the inactivation coefficient. The Na conductance responsible for the depolarization during the action potential takes this form because it inactivates without repolarization.
Hodgkin and Huxley observed that the Na conductance recorded in the squid giant axon opened in response to depolarization but closed without repolarization. They concluded from this that there must be two control gates. One of these gates opens on depolarization and a second closes with different kinetic parameters (necessarily the second gate must close more slowly than the first opens. Thus the rate coefficients for the inactivation variable ay and by are necessarily much slower than those for the activation variable.
Consequently, the model they chose has two control gates. One is activated when a threshold depolarization is achieved, the second closes more slowly to subsequently block flow of Na ions. These are represented by two variables (m is the activation coefficient (substituted for x above), h the inactivation coefficient (substituted for y above)) each described by a differential equation as .
(5)
(6)
where a and b are rate constants that are functions of voltage but not of time as for equation (4).
The Na conductance may then be written:
(7)
where gNa,max is the maximum Na conductance.
Expressions for rate constants were found by fitting curves to experimental data from squid giant axons.
The Na current is described by:
(8)
where ENa is the Na reversal potential.
The potassium conductance may be described similarly although, instead of voltage operated activation and inactivation gates, one need only include an activation gate.
