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{{short description|Describes how neurons transmit electric signals}}

{{short description|Describes how neurons transmit electric signals}}

[[Image:Hodgkin-Huxley.svg|thumb|right|350px|Hodgkin-Huxley 类模型的基本组成部分。Hodgkin-Huxley 类模型代表了细胞膜的生物物理特性。磷脂双分子层表示为一个电容项（<math>C_m</math>）。电压门控离子通道和漏离子通道分别用非线性（<math>g_n</math>）和线性电导（<math>g_L</math>）表示。驱动离子流动的电化学梯度用<font color = "32CD32">电池</font>（<math>E</math>）表示，离子泵和离子交换器用电流源表示(<math>I_p</math>)。|链接=Special:FilePath/Hodgkin-Huxley.svg]]

[[Image:Hodgkin-Huxley.svg|thumb|right|350px|Hodgkin-Huxley 类模型的基本组成部分。Hodgkin-Huxley 类模型代表了细胞膜的生物物理特性。磷脂双分子层表示为一个电容项（<math>C_m</math>）。电压门控离子通道和漏离子通道分别用非线性(<math>g_n</math>)和线性电导(<math>g_L</math>)表示。驱动离子流动的电化学梯度用电池(<math>E</math>)表示，离子泵和离子交换器用电流源表示(<math>I_p</math>)。|链接=Special:FilePath/Hodgkin-Huxley.svg]]

The '''Hodgkin–Huxley model''', or '''conductance-based model''', is a [[mathematical model]] that describes how [[action potential]]s in [[neuron]]s are initiated and propagated. It is a set of [[nonlinear]] [[differential equation]]s that approximates the electrical characteristics of excitable cells such as neurons and [[cardiac muscle|cardiac myocytes]]. It is a continuous-time [[dynamical system]].

The '''Hodgkin–Huxley model''', or '''conductance-based model''', is a [[mathematical model]] that describes how [[action potential]]s in [[neuron]]s are initiated and propagated. It is a set of [[nonlinear]] [[differential equation]]s that approximates the electrical characteristics of excitable cells such as neurons and [[cardiac muscle|cardiac myocytes]]. It is a continuous-time [[dynamical system]].

'''Hodgkin-Huxley 模型'''，或者说'''基于电导的模型'''，是一个描述[[神经元]]中[[动作电位]]如何产生和传导的[[数学模型]]。它是一组[[非线性微分方程]]，用于近似可兴奋细胞（如神经元和[[心肌细胞]]）的电学特性。它是一个时间连续的[[动力系统]]。

'''霍奇金-赫胥黎模型Hodgkin-Huxley model'''，或者说'''基于电导的模型conductance-based model'''，是一个描述[[神经元]]中[[动作电位]]如何产生和传导的数学模型。它是一组[[非线性微分方程]]，用于近似可兴奋细胞（如神经元和[[心肌细胞]]）的电学特性。它是一个时间连续的[[动力系统]]。

[[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]].<ref name=HH>{{cite journal | vauthors = Hodgkin AL, Huxley AF | title = A quantitative description of membrane current and its application to conduction and excitation in nerve | journal = The Journal of Physiology | volume = 117 | issue = 4 | pages = 500–44 | date = August 1952 | pmid = 12991237 | pmc = 1392413 | doi = 10.1113/jphysiol.1952.sp004764 }}</ref>  They received the 1963 [[Nobel Prize in Physiology or Medicine]] for this work.

[[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]].<ref name=HH>{{cite journal | vauthors = Hodgkin AL, Huxley AF | title = A quantitative description of membrane current and its application to conduction and excitation in nerve | journal = The Journal of Physiology | volume = 117 | issue = 4 | pages = 500–44 | date = August 1952 | pmid = 12991237 | pmc = 1392413 | doi = 10.1113/jphysiol.1952.sp004764 }}</ref>  They received the 1963 [[Nobel Prize in Physiology or Medicine]] for this work.

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>).{{Clarify|reason=where to find in the mathematical model below?|date=June 2014}} The [[membrane potential]] is denoted by ''V<SUB>m</SUB>''.

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>).{{Clarify|reason=where to find in the mathematical model below?|date=June 2014}} The [[membrane potential]] is denoted by ''V<SUB>m</SUB>''.

经典的 Hodgkin-Huxley 模型将可兴奋细胞的每个部分都当作电路元件来处理（如图所示）。[[磷脂双分子层]]表示为[[电容]]（<math>C_m</math>）。[[电压门控离子通道]]表示为[[电导]]（<math>g_n</math>，其中 n 是特定的离子通道），它依赖于电压和时间。[[漏通道]]表示为线性电导（<math>g_L</math>）。驱使离子流动的[[电化学梯度]]表示为[[电压源]]（<math>E_n</math>），电压源的[[电压]]取决于相关离子种类在细胞内和细胞外的浓度的比值。最后，[[离子泵]]表示为[[电流源]]（<math>i_p</math>）。[[膜电位]]表示为<math>V_m</math>。

经典的 Hodgkin-Huxley 模型将可兴奋细胞的每个部分都当作电路元件来处理（如图所示）。[[磷脂双分子层]]表示为电容（<math>C_m</math>）。[[电压门控离子通道]]表示为[[电导]]（<math>g_n</math>，其中 n 是特定的离子通道），它依赖于电压和时间。[[漏通道]]表示为线性电导（<math>g_L</math>）。驱使离子流动的[[电化学梯度]]表示为[[电压源]]（<math>E_n</math>），电压源的[[电压]]取决于相关离子种类在细胞内和细胞外的浓度的比值。最后，[[离子泵]]表示为[[电流源]]（<math>i_p</math>）。[[膜电位]]表示为<math>V_m</math>。

Mathematically, the current flowing through the lipid bilayer is written as

Mathematically, the current flowing through the lipid bilayer is written as

以下各节将更详细地描述这些公式。

以下各节将更详细地描述这些公式。

===电压门控离子通道 voltage-gated ion channel===

===电压门控离子通道===

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 name="HH"/> Together with the equation for the total current mentioned above, these are:

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 name="HH"/> Together with the equation for the total current mentioned above, these are:

虽然最初的实验只处理钠和钾通道，Hodgkin-Huxley 模型也可以扩展到其他种类的离子通道。

虽然最初的实验只处理钠和钾通道，Hodgkin-Huxley 模型也可以扩展到其他种类的离子通道。

===漏通道 leak channel===

===漏通道===

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_{leak}</math> is a constant. Thus, the leak current due to passive leak ion channels in the Hodgkin-Huxley formalism is <math>I_l=g_{leak}(V-V_{leak})</math>.

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_{leak}</math> is a constant. Thus, the leak current due to passive leak ion channels in the Hodgkin-Huxley formalism is <math>I_l=g_{leak}(V-V_{leak})</math>.

因此，在 Hodgkin-Huxley 公式中，被动漏离子通道引起的漏电流为<math>I_l=g_{leak}(V-V_{leak})</math>。

因此，在 Hodgkin-Huxley 公式中，被动漏离子通道引起的漏电流为<math>I_l=g_{leak}(V-V_{leak})</math>。

===泵和交换器 pump and exchanger===

===泵和交换器===

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, with a 3 Na<SUP>+</SUP>: 2 K<SUP>+</SUP> stoichiometry.<ref name="Rakowski_1989">{{cite journal | vauthors = Rakowski RF, Gadsby DC, De Weer P | title = Stoichiometry and voltage dependence of the sodium pump in voltage-clamped, internally dialyzed squid giant axon | journal = The Journal of General Physiology | volume = 93 | issue = 5 | pages = 903–41 | date = May 1989 | pmid = 2544655 | doi = 10.1085/jgp.93.5.903 | pmc=2216238}}</ref><ref name="Hille">{{cite book|last=Hille|first=Bertil | name-list-style = vanc | title=Ion channels of excitable membranes|year=2001|publisher=Sinauer|location=Sunderland, Massachusetts|isbn=978-0-87893-321-1 | edition = 3rd }}</ref>

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, with a 3 Na<SUP>+</SUP>: 2 K<SUP>+</SUP> stoichiometry.<ref name="Rakowski_1989">{{cite journal | vauthors = Rakowski RF, Gadsby DC, De Weer P | title = Stoichiometry and voltage dependence of the sodium pump in voltage-clamped, internally dialyzed squid giant axon | journal = The Journal of General Physiology | volume = 93 | issue = 5 | pages = 903–41 | date = May 1989 | pmid = 2544655 | doi = 10.1085/jgp.93.5.903 | pmc=2216238}}</ref><ref name="Hille">{{cite book|last=Hille|first=Bertil | name-list-style = vanc | title=Ion channels of excitable membranes|year=2001|publisher=Sinauer|location=Sunderland, Massachusetts|isbn=978-0-87893-321-1 | edition = 3rd }}</ref>

膜电位取决于其跨膜离子浓度梯度的保持。维持这些浓度梯度需要这几种离子的主动运输。其中钠钾交换器和钠钙交换器最为著名。钠钙交换器的一些基本性质已得到公认: 交换的化学计量比为3Na<SUP>+</SUP>:1Ca<SUP>2+</SUP>，且具有生电性和电压敏感性。文献中还详细描述了Na/K交换器，它有着3Na<SUP>+</SUP>:2K<SUP>+</SUP>的化学计量比。<ref name="Rakowski_1989">{{cite journal | vauthors = Rakowski RF, Gadsby DC, De Weer P | title = Stoichiometry and voltage dependence of the sodium pump in voltage-clamped, internally dialyzed squid giant axon | journal = The Journal of General Physiology | volume = 93 | issue = 5 | pages = 903–41 | date = May 1989 | pmid = 2544655 | doi = 10.1085/jgp.93.5.903 | pmc=2216238}}</ref><ref name="Hille">{{cite book|last=Hille|first=Bertil | name-list-style = vanc | title=Ion channels of excitable membranes|year=2001|publisher=Sinauer|location=Sunderland, Massachusetts|isbn=978-0-87893-321-1 | edition = 3rd }}</ref>

膜电位取决于其跨膜离子浓度梯度的保持。维持这些浓度梯度需要这几种离子的主动运输。其中钠钾交换器和钠钙交换器最为著名。钠钙交换器的一些基本性质已得到公认: 交换的化学计量比为3Na<SUP>+</SUP>:1Ca<SUP>2+</SUP>，且具有生电性和电压敏感性。文献中还详细描述了Na/K交换器，它具有3Na<SUP>+</SUP>:2K<SUP>+</SUP>的化学计量比。<ref name="Rakowski_1989">{{cite journal | vauthors = Rakowski RF, Gadsby DC, De Weer P | title = Stoichiometry and voltage dependence of the sodium pump in voltage-clamped, internally dialyzed squid giant axon | journal = The Journal of General Physiology | volume = 93 | issue = 5 | pages = 903–41 | date = May 1989 | pmid = 2544655 | doi = 10.1085/jgp.93.5.903 | pmc=2216238}}</ref><ref name="Hille">{{cite book|last=Hille|first=Bertil | name-list-style = vanc | title=Ion channels of excitable membranes|year=2001|publisher=Sinauer|location=Sunderland, Massachusetts|isbn=978-0-87893-321-1 | edition = 3rd }}</ref>

==数学性质==

==数学性质==

可以认为Hodgkin-Huxley 模型是一个具有4个[[状态变量]]<math>V_m(t)，n(t)，m(t)</math>，和<math>h(t)</math>的[[微分方程]]系统，它们随着时间<math>t</math>变化。这个系统很难研究，因为它是一个[[非线性系统]]，无法用解析法求解。然而，可以用许多数值方法分析该系统。可以证明某些性质和一般行为（如[[极限环]]）是存在的。

可以认为Hodgkin-Huxley 模型是一个具有4个[[状态变量]]<math>V_m(t)，n(t)，m(t)</math>，和<math>h(t)</math>的[[微分方程]]系统，它们随着时间<math>t</math>变化。这个系统很难研究，因为它是一个[[非线性系统]]，无法用解析法求解。然而，可以用许多数值方法分析该系统。可以证明某些性质和一般行为（如[[极限环]]）是存在的。

[[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]].‎霍奇金-赫胥黎模型在相空间中对电压<math>v(t)</math>和钾门控变量<math>n(t)</math>的模拟。闭合曲线称为[[‎‎极限环]]‎‎。‎|链接=Special:FilePath/Hodgkin_Huxley_Limit_Cycle.png]]

[[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]].‎

===中心流形 center manifold===

 

===中心流形===

Because there are four state variables, visualizing the path in [[phase space]] can be difficult. Usually two variables are chosen, voltage <math>V_m(t)</math> and the potassium gating variable <math>n(t)</math>, 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.

Because there are four state variables, visualizing the path in [[phase space]] can be difficult. Usually two variables are chosen, voltage <math>V_m(t)</math> and the potassium gating variable <math>n(t)</math>, 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.

[[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 three stages: an equilibrium stage, a single-spike stage, and a limit cycle stage.

[[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 three stages: an equilibrium stage, a single-spike stage, and a limit cycle stage.

本模型的电压<math>v(t)</math>(mV)，图中超过50毫秒。注入电流从-5nA到12nA不等。该图经历了三个阶段：<font color="#32CD32">平衡阶段、单峰阶段和极限环阶段</font>。|链接=Special:FilePath/Hodgkins_Huxley_Plot.gif]]

本模型的电压<math>v(t)</math>(mV)，图中超过50毫秒。注入电流从-5nA到12nA不等。该图经历了三个阶段：<font color="#32CD32">平衡阶段、单峰阶段和极限环阶段</font>。|链接=Special:FilePath/Hodgkins_Huxley_Plot.gif]]

 

===分岔===

 

 

===分岔 bifurcation===

If the injected current <math>I</math> were used as a [[bifurcation theory|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 [http://www.scholarpedia.org/article/Canards canard].

If the injected current <math>I</math> were used as a [[bifurcation theory|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 [http://www.scholarpedia.org/article/Canards canard].

如果将注入电流 <math>I</math> 作为[[分岔参数]]，那么 Hodgkin-Huxley 模型将经历一个[[霍普夫分岔]]。和大多数神经元模型一样，增加注入电流会增加神经元的发放率。霍普夫分岔的一个结果就是存在一个最低的发放率。这意味着神经元要么根本没有发放(对应于零频率) ，要么以最低发放率放电。由于[[全或无原理]]，[[动作电位]]的幅度不存在平稳的增加，而是幅度上的突然“跳跃”。由此产生的转变被称为[[鸭解]]。

如果将注入电流 <math>I</math> 作为[[分岔参数]]，那么 Hodgkin-Huxley 模型将经历一个[[霍普夫分岔]]。和大多数神经元模型一样，增加注入电流会增加神经元的发放率。霍普夫分岔的一个结果就是存在一个最低的发放率。这意味着神经元要么根本没有发放（对应于零频率） ，要么以最低发放率放电。由于[[全或无原理]]，[[动作电位]]的幅度不存在平稳的增加，而是幅度上的突然“跳跃”。由此产生的转变被称为[[鸭解]]。

==改进与可替代模型==

==改进与可替代模型==

*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.

*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 articles about the stochastic version of the model and its link with the original one.

*Several articles about the stochastic version of the model and its link with the original one.

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{{DEFAULTSORT:Hodgkin-Huxley Model}}