Hodgkin–Huxley model

Alan Hodgkin and Andrew Huxley described the model in 1952 to explain the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon.

[1] They received the 1963 Nobel Prize in Physiology or Medicine for this work.

The typical Hodgkin–Huxley model treats each component of an excitable cell as an electrical element (as shown in the figure).

Leak channels are represented by linear conductances (gL).

The electrochemical gradients driving the flow of ions are represented by voltage sources (En) whose voltages are determined by the ratio of the intra- and extracellular concentrations of the ionic species of interest.

Finally, ion pumps are represented by current sources (Ip).

[clarification needed] The membrane potential is denoted by Vm.

Mathematically, the current flowing into the capacitance of the lipid bilayer is written as and the current through a given ion channel is the product of that channel's conductance and the driving potential for the specific ion where

Thus, for a cell with sodium and potassium channels, the total current through the membrane is given by: where I is the total membrane current per unit area, Cm is the membrane capacitance per unit area, gK and gNa are the potassium and sodium conductances per unit area, respectively, VK and VNa are the potassium and sodium reversal potentials, respectively, and gl and Vl are the leak conductance per unit area and leak reversal potential, respectively.

Using a series of voltage clamp experiments and by varying extracellular sodium and potassium concentrations, Hodgkin and Huxley developed a model in which the properties of an excitable cell are described by a set of four ordinary differential equations.

are rate constants for the i-th ion channel, which depend on voltage but not time.

For instance, given that potassium channels in squid giant axon are made up of four subunits which all need to be in the open state for the channel to allow the passage of potassium ions, the n needs to be raised to the fourth power.

are the steady state values for activation and inactivation, respectively, and are usually represented by Boltzmann equations as functions of

In many current software programs [2] Hodgkin–Huxley type models generalize

to In order to characterize voltage-gated channels, the equations can be fitted to voltage clamp data.

[3] Briefly, when the membrane potential is held at a constant value (i.e., with a voltage clamp), for each value of the membrane potential the nonlinear gating equations reduce to equations of the form: Thus, for every value of membrane potential

the sodium and potassium currents can be described by In order to arrive at the complete solution for a propagated action potential, one must write the current term I on the left-hand side of the first differential equation in terms of V, so that the equation becomes an equation for voltage alone.

The relation between I and V can be derived from cable theory and is given by where a is the radius of the axon, R is the specific resistance of the axoplasm, and x is the position along the nerve fiber.

The membrane potential depends upon the maintenance of ionic concentration gradients across it.

The maintenance of these concentration gradients requires active transport of ionic species.

[5][6] The Hodgkin–Huxley model can be thought of as a differential equation system with four state variables,

Certain properties and general behaviors, such as limit cycles, can be proven to exist.

Because there are four state variables, visualizing the path in phase space can be difficult.

A better projection can be constructed from a careful analysis of the Jacobian of the system, evaluated at the equilibrium point.

Specifically, the eigenvalues of the Jacobian are indicative of the center manifold's existence.

Likewise, the eigenvectors of the Jacobian reveal the center manifold's orientation.

The eigenvectors associated with the two negative eigenvalues will reduce to zero as time t increases.

The remaining two complex eigenvectors define the center manifold.

One consequence of the Hopf bifurcation is that there is a minimum firing rate.

Nevertheless, modern Hodgkin–Huxley-type models have been extended in several important ways: Several simplified neuronal models have also been developed (such as the FitzHugh–Nagumo model), facilitating efficient large-scale simulation of groups of neurons, as well as mathematical insight into dynamics of action potential generation.