Hodgkin–Huxley model
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[[Image:Hodgkin-Huxley.jpg|thumb|right|350px|Basic components of Hodgkin–Huxley-type models. Hodgkin–Huxley type models represent the biophysical characteristic of cell membranes. The lipid bilayer is represented as a capacitance (''C''<SUB>m</SUB>). Voltage-gated and leak ion channels are represented by nonlinear (''g''<SUB>n</SUB>) and linear (''g''<SUB>L</SUB>) conductances, respectively. The electrochemical gradients driving the flow of ions are represented by batteries (E), and ion pumps and exchangers are represented by current sources (''I''<SUB>p</SUB>).]]
The '''Hodgkin–Huxley model''' is a [[scientific model]] that describes how [[action potential]]s in [[neuron]]s are initiated and propagated.
It is a set of [[nonlinearity|nonlinear]] [[ordinary differential equation]]s that approximates the electrical characteristics of excitable cells such as neurons and [[cardiac muscle|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]].{{ref|HH}} They received the [[1963]] [[Nobel Prize in Physiology or Medicine]] for this work.
==Basic components==
The components of a typical Hodgkin–Huxley 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<SUB>m</SUB>). [[Voltage-gated ion channel]]s are represented by [[nonlinear]] [[electrical conductance]]s (''g''<SUB>''n''</SUB>, where ''n'' is the specific ion channel), meaning that the conductance is voltage and time-dependent. This was later shown to be mediated by voltage-gated cation channel proteins, each of which has an open probability that is voltage-dependent. [[Leak channel]]s are represented by linear conductances (''g''<SUB>''L''</SUB>). The [[electrochemical gradient]]s driving the flow of ions are represented by batteries (''E''<SUB>''n''</SUB> and ''E''<SUB>''L''</SUB>), the values of which are determined from the [[Reversal potential|Nernst potential]] of the ionic species of interest. Finally, [[Ion pump (biology)|ion pumps]] are represented by [[current sources]] (''I''<SUB>''p''</SUB>).
The time derivative of the potential across the membrane (<math>\dot{V}_m</math>) is proportional to the sum of the currents in the circuit. This is represented as follows:
: <math>\dot{V}_m= -\frac{1}{C_m} \left(\sum\limits ^{}_i I_i \right),</math>
where ''I''<sub>''i''</sub> denotes the individual ionic currents of the model.
==Ionic current characterization==
The current flowing through the ion channels is mathematically represented by the following equation:
: <math>I_i(V_m,t)= (V_m - E_i) {g_i}\;</math>
where <math>E_i</math> is the [[reversal potential]] of the ''i''-th ion channel.
In voltage-gated ion channels, the channel conductance ''g''<SUB>''i''</SUB> is a function of both time and voltage (''g''<SUB>''n''</SUB>(''t'', ''V'') in the figure), while in leak channels ''g''<SUB>''i''</SUB> is a constant (''g''<SUB>''L''</SUB> 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.
===Voltage-gated ion channels===
Under the Hodgkin–Huxley formulation, conductances for voltage-gated channels (''g''<SUB>''n''</SUB>(''t'', ''V'')) are expressed as:
: <math>{g}_n(V_m,t) = \bar{g}_n \varphi^\alpha \chi^\beta\,</math>
: <math>\dot{\varphi}(V_m,t) = \frac{1}{\tau_\varphi} (\varphi_\infty - \varphi) </math>
: <math>\dot{\chi}(V_m,t) = \frac{1}{\tau_\chi} (\chi_\infty - \chi),</math>
where <math>\varphi</math> and <math>\chi</math> are gating variables for activation and inactivation, respectively, representing the fraction of the maximum conductance available at any given time and voltage. <math>\bar{g}_n</math> is the maximal value of the conductance. <math>\alpha</math> and <math>\beta</math> are constants and <math>\tau_\varphi</math> and <math>\tau_{\chi}</math> are the time constants for activation and inactivation, respectively. <math>\varphi_\infty</math> and <math>\chi_\infty</math> are the steady state values for activation and inactivation, respectively, and are usually represented by Boltzmann equations as functions of <math>V_m</math>.
In order to characterize voltage-gated channels, the equations will be fit to voltage-clamp data. For a derivation of the Hodgkin–Huxley equations under voltage-clamp see.{{ref|JohnstonAndWu}} Briefly, when the membrane potential is held at a constant value (i.e., voltage-clamp), for each value of the membrane potential the nonlinear gating equations reduce to linear differential equations of the form:
: <math>\varphi(t) = \varphi_{0} - [ (\varphi_{0}-\varphi_{\infty})(1 - e^{-t/\tau_\varphi})]\, </math>
: <math>\chi(t) = \chi_{0} - [ (\chi_{0}-\chi_{\infty})(1 - e^{-t/\tau_\chi})].</math>
Thus, for every value of membrane potential, <math>V_{m}</math>, the following equation can be fit to the current curve:
: <math>I_n(t)=\bar{g}_n \varphi^\alpha \chi^\beta (V_m-E_n).</math>
The [[Levenberg–Marquardt algorithm]],{{ref|Marquardt}}{{ref|Levenberg}} a modified [[Gauss–Newton algorithm]], is often used to fit these equations to voltage-clamp data.
===Leak channels===
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 <math>g_i</math> is a constant.
===Pumps and exchangers===
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<SUP>+</SUP>:1 Ca<SUP>2+</SUP> and the exchanger is electrogenic and voltage-sensitive. The Na/K exchanger has also been described in detail.{{ref|Hille}}
==Improvements and alternative models==
{{main|Biological neuron models}}
The Hodgkin–Huxley model is widely 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.
*Models often incorporate highly complex geometries of [[dendrites]] and [[axons]], often based on microscopy data.
Several simplified neuronal models have also been developed, facilitating efficient large-scale simulation of groups of neurons, as well as mathematical insight into dynamics of action potential generation.
==See also==
*[[Fitzhugh-Nagumo model]]
*[[Soliton model]]
*[[Action potential]]
*[[Biological neural network]]
==References==
#{{note|HH}}Hodgkin, A., and Huxley, A. (1952): A quantitative description of membrane current and its application to conduction and excitation in nerve. ''J. Physiol.'' '''117''':500–544. PMID 12991237
#{{note|Marquardt}}Marquardt, D. (1963): An algorithm for the least-squares estimation of nonlinear parameters. ''SIAM J. Appl. Math.'' '''11''' (2):431–441.
#{{note|Levenberg}}Levenberg, K. (1944): A method for the solution of certain non-linear problems in least-squares. ''Q. Appl. Math.'' '''2''' (2):164–168.
#{{note|JohnstonAndWu}}Johnston, D., and Wu, S. (1997): Foundations of Cellular Neurophysiology, chapter 6. MIT Press, Cambridge, MA. ISBN 0-262-10053-3
#{{note|Hille}}Hille, B. (2001): Ionic Channels of Excitable Membranes (3rd ed.). Sinauer Associates, Inc., Sunderland, MA. ISBN 0-87893-321-2
==External links==
*[http://thevirtualheart.org/HHindex.html Interactive Java applet of the HH model ] Parameters of the model can be changed as well as excitation parameters and phase space plottings of all the variables is possible.
*[http://comp.uark.edu/~jostmey/Hodgkin-Huxley%20Equations/Hodgkin-Huxley%20Applet.html Java applet of the HH Equations] Numerically solves the Hodgkin-Huxley Equations. Parameters may be varied, and allows for user to select from any arbitrary current.
[[Category:Non-linear systems]]
[[Category:Electrophysiology]]
[[Category:Ion channels]]
[[Category:Computational neuroscience]]
[[Category:Excitable Membranes]]
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