|
|
| Line 1: |
Line 1: |
| [[Image:Hodgkin-Huxley.svg|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 person who wrote the article is called Jayson Hirano and he completely digs that title. My day occupation is a travel agent. Ohio is where her house is. The preferred pastime for him and his kids is to play lacross and he'll be beginning something else alongside with it.<br><br>Visit my web blog ... [http://www.sirudang.com/siroo_Notice/2110 real psychics] |
| | |
| The '''Hodgkin–Huxley model''' (or "conductance-based model") is a [[mathematical model]] (a type of [[scientific model]]) that describes how [[action potential]]s in [[neuron]]s are initiated and propagated.
| |
| It is a set of [[nonlinearity|nonlinear]] [[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 name=HH>{{cite pmid|12991237}}</ref> They received the 1963 [[Nobel Prize in Physiology or Medicine]] for this work.
| |
| | |
| ==Basic components==
| |
| | |
| The typical Hodgkin–Huxley model treats each component of an excitable cell as an electrical element (as shown in the figure). The [[lipid bilayer]] is represented as a [[capacitance]] (C<SUB>m</SUB>). [[Voltage-gated ion channel]]s are represented by [[electrical conductance]]s (''g''<SUB>''n''</SUB>, where ''n'' is the specific ion channel) that depend on both voltage and time. [[Leak channel]]s are represented by linear conductances (''g''<SUB>''L''</SUB>). The [[electrochemical gradient]]s driving the flow of ions are represented by [[voltage source]]s (''E''<SUB>''n''</SUB>) whose [[voltage]]s are determined by the ratio of the intra- and extracellular concentrations of the ionic species of interest. Finally, [[Ion pump (biology)|ion pumps]] are represented by [[current sources]] (''I''<SUB>''p''</SUB>). The difference between the [[membrane potential]] and the [[resting potential]] is typically called "V<SUB>m</SUB>".
| |
| | |
| Mathematically, the current flowing through the lipid bilayer is written as
| |
| | |
| : <math>I_c = C_m\frac{{\mathrm d} V_m}{{\mathrm d} t} </math>
| |
| | |
| and the current through a given ion channel is
| |
| | |
| : <math>I_i = {g_i}(V_m - V_i) \;</math>
| |
| | |
| where <math>V_i</math> is the [[reversal potential]] of the ''i''-th ion channel.
| |
| Thus, for a cell with sodium and potassium channels, the total current through the membrane is given by:
| |
| | |
| : <math>I = C_m\frac{{\mathrm d} V_m}{{\mathrm d} t} + g_K(V_m - V_K) + g_{Na}(V_m - V_{Na}) + g_l(V_m - V_l),</math>
| |
|
| |
| where ''I'' is the total membrane current per unit area, ''C''<sub>''m''</sub> is the membrane capacitance per unit area, ''g''<sub>''K''</sub> and ''g''<sub>''Na''</sub> are the potassium and sodium conductances per unit area, respectively, ''V''<sub>''K'' </sub> and ''V''<sub>''Na''</sub> are the potassium and sodium reversal potentials, respectively, and ''g''<sub>''l''</sub> and ''V''<sub>''l''</sub> are the leak conductance per unit area and leak reversal potential, respectively. The time dependent elements of this equation are ''V''<sub>''m''</sub>, ''g''<sub>''Na''</sub>, and ''g''<sub>''K''</sub>, where the last two conductances depend explicitly on voltage as well.
| |
| | |
| ==Ionic current characterization==
| |
| | |
| 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===
| |
| 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 equation]]s.<ref>Hodgkin, A.L., and Huxley, A.F., "A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (1952) 177, 500-544</ref> Together with the equation for the total current mentioned above, these are:
| |
| | |
| : <math>I = C_m\frac{{\mathrm d} V_m}{{\mathrm d} t} + \bar{g}_Kn^4(V_m - V_K) + \bar{g}_{Na}m^3h(V_m - V_{Na}) + \bar{g}_l(V_m - V_l),</math>
| |
| | |
| : <math>\frac{dn}{dt} = \alpha_n(1 - n) - \beta_n n</math>
| |
| | |
| : <math>\frac{dm}{dt} = \alpha_m(1 - m) - \beta_m m</math>
| |
| | |
| : <math>\frac{dh}{dt} = \alpha_h(1 - h) - \beta_h h</math>
| |
| | |
| where ''I'' is the current per unit area, and <math>\alpha_i </math> and <math>\beta_i </math> are rate constants for the ''i''-th ion channel, which depend on voltage but not time. <math>\bar{g}_n</math> is the maximal value of the conductance. ''n'', ''m'', and ''h'' are dimensionless quantities between 0 and 1 that are associated with potassium channel activation, sodium channel activation, and sodium channel inactivation, respectively. For <math> p = (n, m, h)</math>, <math> \alpha_p </math> and <math> \beta_p </math> take the form
| |
| | |
| : <math>\alpha_p = p_\infty/\tau_p</math>
| |
| | |
| : <math> \beta_p = (1 - p_\infty)/\tau_p</math>.
| |
| | |
| <math>n_\infty</math> and <math>m_\infty</math>, and <math>h_\infty</math> are the steady state values for activation and inactivation, respectively, and are usually represented by [[Boltzmann equation]]s as functions of <math>V_m</math>.
| |
| | |
| In order to characterize voltage-gated channels, the equations are fit to voltage clamp data. For a derivation of the Hodgkin–Huxley equations under voltage-clamp, see.<ref name=JohnstonAndWu>{{cite book|last1=Gray|first1=Daniel Johnston|first2=Samuel Miao-Sin|last2=Wu|title=Foundations of cellular neurophysiology|year=1997|publisher=MIT Press|location=Cambridge, Mass. [u.a.]|isbn=9780262100533|edition=3rd.}}</ref> 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 equations of the form:
| |
| | |
| : <math>m(t) = m_{0} - [ (m_{0}-m_{\infty})(1 - e^{-t/\tau_m})]\, </math>
| |
| | |
| : <math>h(t) = h_{0} - [ (h_{0}-h_{\infty})(1 - e^{-t/\tau_h})]\, </math>
| |
| | |
| : <math>n(t) = n_{0} - [ (n_{0}-n_{\infty})(1 - e^{-t/\tau_n})]\, </math>
| |
| | |
| Thus, for every value of membrane potential <math>V_{m}</math> the sodium and potassium currents can be described by
| |
| | |
| : <math>I_{Na}(t)=\bar{g}_{Na} m(V_m)^3h(V_m)(V_m-E_{Na}),</math>
| |
| | |
| : <math>I_K(t)=\bar{g}_K n(V_m)^4(V_m-E_K).</math>
| |
| | |
| 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
| |
| | |
| : <math>I = \frac{a}{2R}\frac{\partial^2V}{\partial x^2} </math>,
| |
| | |
| where ''a'' is the radius of the [[axon]], ''R'' is the [[Resistivity|specific resistance]] of the [[axoplasm]], and ''x'' is the position along the nerve fiber. Substitution of this expression for ''I'' transforms the original set of equations into a set of [[partial differential equation]]s, because the voltage becomes a function of both ''x'' and ''t''. | |
| | |
| The [[Levenberg–Marquardt algorithm]],<ref name=Marquardt>{{cite doi|10.1137/0111030}}</ref><ref name=Levenberg>{{cite journal|last=Levenberg|first=K|title=A method for the solution of certain non-linear problems in least squares|journal=Qu. App. Maths.|year=1944|volume=2|pages=164}}</ref> a modified [[Gauss–Newton algorithm]], is often used to fit these equations to voltage-clamp data.{{Citation needed|date=August 2010}} | |
| | |
| While the original experiments treated only sodium and potassium channels, the Hodgkin Huxley model can also be extended to account for other species of [[ion channel]]s.
| |
| | |
| ===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 name=Hille>{{cite book|last=Hille|first=Bertil|title=Ion channels of excitable membranes|year=2001|publisher=Sinauer|location=Sunderland, Mass.|isbn=9780878933211|edition=3. ed.}}</ref>
| |
| | |
| ==Mathematical properties==
| |
| The Hodgkin-Huxley model can be thought of as a [[differential equation]] with four [[state variable|state variables]], '' v(t)'', ''m(t)'', ''n(t)'', and ''h(t)'', that change with respect to time ''t''. The system is difficult to study because it is a [[nonlinear|nonlinear system]] and cannot be solved analytically. However, there are many numeric methods available to analyze the system. Certain properties and general behaviors, such as [[limit cycles]], can be proven to exist.
| |
| | |
| [[File:Hodgkin Huxley Limit Cycle.png|thumb|left|A simulation of the Hodgkin-Huxley model in phase space, in terms of voltage v(t) and potassium gating variable n(t). The closed curve is known as a [[limit cycle]].]]
| |
| | |
| ===Center manifold===
| |
| | |
| Because there are four state variables, visualizing the path in [[phase space]] can be difficult. Usually two variables are chosen, voltage ''v(t)'' and the potassium gating variable ''n(t)'', 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 (mathematics)| 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|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.
| |
| | |
| [[File:Hodgkins Huxley Plot.gif|thumb|right|360px|The voltage '' v(t)'' (in millivolts) of the Hodgkin-Huxley model, graphed over 50 milliseconds. The injected current varies from -5 nanoamps to 12 nanoamps. The graph passes through 3 stages: an equilibrium stage, a single-spike stage, and a limit cycle stage. ]]
| |
| | |
| ===Bifurcations===
| |
| If we use the injected current <math> I </math> as a [[bifurcation|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 law |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 [[classical canard phenomenon]], or simply a canard.
| |
| | |
| ==Improvements and alternative models==
| |
| {{Main|Biological neuron models}}
| |
| | |
| 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.
| |
| *Models often incorporate highly complex geometries of [[dendrites]] and [[axons]], often based on microscopy data.
| |
| *[[Stochastic]] models of ion-channel behavior, leading to stochastic hybrid systems<ref name=stochastic>{{cite journal|last=Pakdaman|first=K|title=Fluid limit theorems for stochastic hybrid systems with applications to neuron models|year=2010|journal=Adv.Appl.Proba|volume=43.}}</ref>
| |
| | |
| 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. Another new model, the [[Soliton model in neuroscience|Soliton model]], explains why an action potential traveling along an axon results in a slight local thickening and outward displacement of the membrane. It also accounts for a slight increase in temperature, followed by a decrease in temperature, during an action potential.
| |
| | |
| ==See also==
| |
| {{Div col}}
| |
| *[[Action potential]]
| |
| *[[Anode break excitation]]
| |
| *[[Autowave]]
| |
| *[[Biological neuron model]]
| |
| *[[Biological neural network]]
| |
| *[[FitzHugh-Nagumo model]]
| |
| *[[GHK current equation]]
| |
| *[[Goldman equation]]
| |
| *[[Memristor]]
| |
| *[[Neural accommodation]]
| |
| *[[Reaction-diffusion]]
| |
| *[[Theta model]]
| |
| {{Div col end}}
| |
| | |
| ==References==
| |
| <references/>
| |
| | |
| ==External links==
| |
| *[http://myselph.de/hodgkinHuxley.html Interactive Javascript simulation of the HH model ] Runs in any HTML5 - capable browser. Allows for changing the parameters of the model and current injection.
| |
| *[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://www.ebi.ac.uk/biomodels-main/BIOMD0000000020 Direct link to Hodgkin-Huxley model] and a [http://www.ebi.ac.uk/biomodels-main/static-pages.do?page=ModelMonth%2F2006-09 Description] in [[BioModels Database]]
| |
| *[http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=1392219&blobtype=pdf Direct link to Hodgkin-Huxley paper #1] via PubMedCentral
| |
| *[http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=1392213&blobtype=pdf Direct link to Hodgkin-Huxley paper #2] via PubMedCentral
| |
| *[http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=1392209&blobtype=pdf Direct link to Hodgkin-Huxley paper #3] via PubMedCentral
| |
| *[http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=1392212&blobtype=pdf Direct link to Hodgkin-Huxley paper #4] via PubMedCentral
| |
| *[http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=1392413&blobtype=pdf Direct link to Hodgkin-Huxley paper #5] via PubMedCentral
| |
| *[http://www.demonstrations.wolfram.com/NeuralImpulsesTheActionPotentialInAction/ Neural Impulses: The Action Potential In Action] by Garrett Neske, [[The Wolfram Demonstrations Project]]
| |
| *[http://demonstrations.wolfram.com/HodgkinHuxleyActionPotentialModel/ Interactive Hodgkin-Huxley model] by Shimon Marom, [[The Wolfram Demonstrations Project]]
| |
| *[http://senselab.med.yale.edu/modeldb ModelDB] A computational neuroscience source code database containing 4 versions (in different simulators) of the original Hodgkin–Huxley model and hundreds of models that apply the Hodgkin–Huxley model to other channels in many electrically excitable cell types.
| |
| *Several [http://sites.google.com/site/gwainrib/papers articles] about the stochastic version of the model and its link with the original one.
| |
| | |
| {{DEFAULTSORT:Hodgkin-Huxley Model}}
| |
| [[Category:Nonlinear systems]]
| |
| [[Category:Electrophysiology]]
| |
| [[Category:Ion channels]]
| |
| [[Category:Computational neuroscience]]
| |