Нейрон қабылдауларының электр сигналдарын тасымалдау моделі

Describes how neurons transmit electric signals

The Hodgkin–Huxley model, or conductance based model, is a mathematical model that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear differential equations that approximates the electrical engineering characteristics of excitable cells such as neurons and muscle cells. It is a continuous time dynamical system. 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. They received the 1963 Nobel Prize in Physiology or Medicine for this work.

In voltage gated ion channels, the channel conductance is a function of both time and voltage ( in the figure), while in leak channels, , it is a constant ( in the figure). The current generated by ion pumps is dependent on the ionic species specific to that pump. The following sections will describe these formulations in more detail.

Leak channels account for the natural permeability of the membrane to ions and take the form of the equation for voltage gated channels, where the conductance is a constant. Thus, the leak current due to passive leak ion channels in the Hodgkin Huxley formalism is .

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. The sodium potassium and sodium calcium exchangers are the best known of these. Some of the basic properties of the Na/Ca exchanger have already been well established: the stoichiometry of exchange is 3 Na+: 1 Ca2+ and the exchanger is electrogenic and voltage sensitive. The Na/K exchanger has also been described in detail, with a 3 Na+: 2 K+ stoichiometry.

The Hodgkin–Huxley model can be thought of as a differential equation system with four state variables, , and , that change with respect to time The system is difficult to study because it is a nonlinear system, cannot be solved analytically, and therefore has no closed form solution. However, there are many numerical methods available to analyze the system. 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. Usually two variables are chosen, voltage and the potassium gating variable , allowing one to visualize the limit cycle. However, one must be careful because this is an ad hoc method of visualizing the 4 dimensional system. This does not prove the existence of the limit cycle. 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 Hodgkin–Huxley model has two negative eigenvalues and two complex eigenvalues with slightly positive real parts. 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. In other words, the 4 dimensional system collapses onto a 2 dimensional plane. Any solution starting off the center manifold will decay towards the center manifold. Furthermore, the limit cycle is contained on the center manifold.

If the injected current were used as a bifurcation parameter, then the Hodgkin–Huxley model undergoes a Hopf bifurcation. As with most neuronal models, increasing the injected current will increase the firing rate of the neuron. One consequence of the Hopf bifurcation is that there is a minimum firing rate. This means that either the neuron is not firing at all (corresponding to zero frequency), or firing at the minimum firing rate. Because of the all or none principle, there is no smooth increase in action potential amplitude, but rather there is a sudden "jump" in amplitude. The resulting transition is known as a canard.

The Hodgkin–Huxley model is regarded as one of the great achievements of 20th century biophysics. Nevertheless, modern Hodgkin–Huxley type models have been extended in several important ways:
Additional ion channel populations have been incorporated based on experimental data. The Hodgkin–Huxley model has been modified to incorporate transition state theory and produce thermodynamic Hodgkin–Huxley models. Models often incorporate highly complex geometries of dendrites and axons, often based on microscopy data. Conductance based models similar to Hodgkin–Huxley model incorporate the knowledge about cell types defined by single cell transcriptomics. Stochastic models of ion channel behavior, leading to stochastic hybrid systems. The Poisson–Nernst–Planck (PNP) model is based on a mean field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. 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.