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HodgkinHuxley model
The HodgkinHuxley Model is a scientific model that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear ordinary differential equations that approximates the electrical characteristics of excitable cells such as neurons and cardiac myocytes. Alan Lloyd 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. Additional recommended knowledge
Basic ComponentsThe components of a typical HodgkinHuxley model are shown in the figure. Each component of an excitable cell has a biophysical analog. The lipid bilayer is represented as a capacitance (C_{m}). Voltagegated ion channels are represented by nonlinear electrical conductances (g_{n}, where n is the specific ion channel), meaning that the conductance is voltage and timedependent. This was later shown to be mediated by voltagegated cation channel proteins, each of which has an open probability that is voltagedependent. Leak channels are represented by linear conductances (g_{L}). The electrochemical gradients driving the flow of ions are represented by batteries (E_{n} and E_{L}), the values of which are determined from the Nernst potential of the ionic species of interest. Finally, ion pumps are represented by current sources (I_{p}). The time derivative of the potential across the membrane () is proportional to the sum of the currents in the circuit. This is represented as follows:
where I_{i} denotes the individual ionic currents of the model. Ionic Current CharacterizationThe current flowing through the ion channels is mathematically represented by the following equation:
where E_{i} is the reversal potential of the ith ion channel. In voltagegated ion channels, the channel conductance g_{i} is a function of both time and voltage (g_{n}(t,V) in the figure), while in leak channels g_{i} is a constant (g_{L} 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. VoltageGated Ion ChannelsUnder the HodgkinHuxley formulation, conductances for voltagegated channels (g_{n}(t,V)) are expressed as:
, where φ and χ are gating variables for activation and inactivation, respectively, representing the fraction of the maximum conductance available at any given time and voltage. is the maximal value of the conductance. α and β are constants and τ_{φ} and τ_{χ} are the time constants for activation and inactivation, respectively. and are the steady state values for activation and inactivation, respectively, and are usually represented by Boltzmann equations as functions of V_{m}. In order to characterize voltagegated channels, the equations will be fit to voltageclamp data. For a derivation of the HodgkinHuxley equations under voltageclamp see. Briefly, when the membrane potential is held at a constant value (i.e., voltageclamp), for each value of the membrane potential the nonlinear gating equations reduce to linear differential equations of the form:
Thus, for every value of membrane potential, V_{m}, the following equation can be fit to the current curve: . The LevenbergMarquardt algorithm, a modified GaussNewton algorithm, is often used to fit these equations to voltageclamp data.Leak ChannelsLeak channels account for the natural permeability of the membrane to ions and take the form of the equation for voltagegated channels, where the conductance g_{i} is a constant. Pumps and ExchangersThe 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 sodiumpotassium and sodiumcalcium exchangers are the best known of these. Some of the basic properties of the Na/Ca exchanger have already been wellestablished: the stoichiometry of exchange is 3 Na^{+}:1 Ca^{2+} and the exchanger is electrogenic and voltagesensitive. The Na/K exchanger has also been described in detail. Improvements and Alternative ModelsThe HodgkinHuxley model is widely regarded as one of the great achievements of 20thcentury biophysics. Nevertheless, modern HodgkinHuxleytype models have been extended in several important ways:
Several simplified neuronal models have also been developed, facilitating efficient largescale simulation of groups of neurons, as well as mathematical insight into dynamics of action potential generation. See also
References
Categories: Electrophysiology  Ion channels  Computational neuroscience 

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "HodgkinHuxley_model". A list of authors is available in Wikipedia. 