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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]]

[[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]]

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]].

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===

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

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:

通过改变细胞外钠离子和钾离子的浓度，进行一系列的[[电压钳]]实验，Hodgkin 和 Huxley 建立了一个由四个[[常微分方程]]描述可兴奋细胞特性的模型。加上上述总电流的方程，这些方程为:

通过改变细胞外钠离子和钾离子的浓度，进行一系列的[[电压钳]]实验，Hodgkin 和 Huxley 建立了一个由四个[[常微分方程]]描述的可兴奋细胞特性的模型。加上上述总电流的方程，这些方程为:

: <math>I = C_m\frac{{\mathrm d} V_m}{{\mathrm d} t}  + \bar{g}_\text{K}n^4(V_m - V_K) + \bar{g}_\text{Na}m^3h(V_m - V_{Na}) + \bar{g}_l(V_m - V_l),</math>

: <math>I = C_m\frac{{\mathrm d} V_m}{{\mathrm d} t}  + \bar{g}_\text{K}n^4(V_m - V_K) + \bar{g}_\text{Na}m^3h(V_m - V_{Na}) + \bar{g}_l(V_m - V_l),</math>

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===

===漏通道 leak channel===

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===

===泵和交换器 pump and exchanger===

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>

[[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]].‎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]]

===中心流形===

===中心流形 center manifold===

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.

此外，极限环包含在中心流形上。

此外，极限环包含在中心流形上。

[[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]]

[[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]]

===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].

*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.

{{DEFAULTSORT:Hodgkin-Huxley Model}}

{{DEFAULTSORT:Hodgkin-Huxley Model}}