Hodgkin–Huxley model 2991090 216774874 2008-06-03T03:54:34Z DumZiBoT 6085301 robot Adding: [[es:Modelo de Hodgkin y Huxley]] [[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'',&nbsp;''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'',&nbsp;''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]] [[de:Hodgkin-Huxley-Modell]] [[es:Modelo de Hodgkin y Huxley]] [[it:Modello di Hodgkin-Huxley]] [[he:מודל הודג'קין-הקסלי]]