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1952年，[[Alan Hodgkin]] 和 [[Andrew Huxley]] 描述了这个模型，来解释[[乌贼巨大轴突]]中动作电位的产生和传导的离子机制。<ref name="HH" />他们因为这项工作获得了1963年的[[诺贝尔生理学或医学奖]]。
1952年，[[Alan Hodgkin]] 和 [[Andrew Huxley]] 描述了这个模型，来解释[[乌贼巨大轴突]]中动作电位的产生和传导的离子机制。<ref name="HH" />他们因为这项工作获得了1963年的[[诺贝尔生理学或医学奖]]。
==基本成分==
==基本组成部分==
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>''.
典型的 Hodgkin-Huxley 模型将可兴奋细胞的每个成分都当作电子元件来处理(如图所示)。
典型的 Hodgkin-Huxley 模型将可兴奋细胞的每个成分都当作电子元件来处理（如图所示）。
[[磷脂双分子层]]表示为[[电容]]。
[[磷脂双分子层]]表示为[[电容]]。
[[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) ，它同时依赖于电压和时间。[[漏通道]]表示为线性电导(<math>g_l</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），它同时依赖于电压和时间。[[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。
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>)。
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>。
[[膜电位]]表示为<math>V_m</math>。
Thus, for a cell with sodium and potassium channels, the total current through the membrane is given by:
Thus, for a cell with sodium and potassium channels, the total current through the membrane is given by:
其中 <math>V_i</math> 是 <math>i</math>-th 离子通道的反转电位。
其中 <math>V_i</math> 是第<math>i</math>个离子通道的[[反转电位]]。
因此，对于具有钠和钾离子通道的细胞，通过细胞膜的总电流为：
因此，对于具有钠和钾离子通道的细胞，通过细胞膜的总电流为：
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.
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.
其中 i 为单位面积的总膜电流，<math>C_m</math>为单位面积的膜电容，<math>g_K</math>和<math>g_{Na}</math>分别为单位面积的钾和钠的电导，<math>V_K</math>和<math>V_{Na}</math>分别为钾和钠的反转电位，<math>g_l</math>和<math>V_l</math>分别为单位面积的漏电导和漏反转电位。
其中<math>i</math>为单位面积的总膜电流，<math>C_m</math>为单位面积的膜电容，<math>g_K</math>和<math>g_{Na}</math>分别为单位面积的钾和钠的电导，<math>V_K</math>和<math>V_{Na}</math>分别为钾和钠的反转电位，<math>g_l</math>和<math>V_l</math>分别为单位面积的漏电导和漏反转电位。
这个方程中的时间依赖项为<math>V_m</math>、<math>g_{Na}</math>和<math>g_K</math>，其中最后两个电导项也明确地取决于电压。
这个方程中的时间依赖项为<math>V_m</math>、<math>g_{Na}</math>和<math>g_K</math>，其中最后两个电导项也明确地取决于电压。
In voltage-gated ion channels, the channel conductance <math>g_l</math> is a function of both time and voltage (<math>g_n(t,V)</math> in the figure), while in leak channels <math>g_l</math> is a constant (<math>g_L</math> 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.
In voltage-gated ion channels, the channel conductance <math>g_l</math> is a function of both time and voltage (<math>g_n(t,V)</math> in the figure), while in leak channels <math>g_l</math> is a constant (<math>g_L</math> 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.
在电压门控离子通道中，通道电导<math>g_l</math>是时间和电压（图中为 <math>g_n(t,V)</math>）的函数，而在漏通道中，<math>g_l</math>是常数（图中为<math>g_L</math>）。
在电压门控离子通道中，通道电导<math>g_l</math>是时间和电压(图中为 <math>g_n(t,V)</math>)的函数，而在漏通道中，<math>g_l</math>是常数(图中为<math>g_L</math>)。
由离子泵产生的电流取决于离子泵特定的离子种类。
由离子泵产生的电流取决于离子泵特定的离子种类。
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。
<math>bar{g}_n</math>是电导的最大值。
<math>bar{g}_n</math>是电导的最大值。
<math>N</math>、 <math>m</math> 和 <math>h</math> 是0和1之间的无量纲量，分别与钾通道激活、钠通道激活和钠通道失活有关。
<math>N</math>、 <math>m</math> 和 <math>h</math> 是0和1之间的无量纲量，分别与钾通道激活、钠通道激活和钠通道失活有关。
对于 p = (n，m，h) ，<math>\alpha_p</math>和<math>\beta_p</math>的形式是
对于<math>p = (n，m，h)</math>，<math>\alpha_p</math>和<math>\beta_p</math>的形式是
: <math>\alpha_p(V_m) = p_\infty(V_m)/\tau_p</math>
: <math>\alpha_p(V_m) = p_\infty(V_m)/\tau_p</math>
<math>p_\infty</math> 和 <math>(1-p_\infty)</math>分别是激活和失活的稳态值，通常用[[Boltzmann 方程]]表示为 <math>V_m</math>的函数。
<math>p_\infty</math> 和 <math>(1-p_\infty)</math>分别是激活和失活的稳态值，通常用[[玻尔兹曼方程]]表示为<math>V_m</math>的函数。
在 Hodgkin 和 Huxley 的原始论文中，alpha 和 beta函数如下给出
在 Hodgkin 和 Huxley 的原始论文中，<math>\alpha</math>和<math>\beta</math>函数如下给出
<math> \begin{array}{lll}
<math> \begin{array}{lll}
\alpha_n(V_m) = \frac{0.01(10-V)}{\exp\big(\frac{10-V}{10}\big)-1} & \alpha_m(V_m) = \frac{0.1(25-V)}{\exp\big(\frac{25-V}{10}\big)-1} & \alpha_h(V_m) = 0.07\exp\bigg(-\frac{V}{20}\bigg)\\
\beta_n(V_m) = 0.125\exp\bigg(-\frac{V}{80}\bigg) & \beta_m(V_m) = 4\exp\bigg(-\frac{V}{18}\bigg) & \beta_h(V_m) = \frac{1}{\exp\big(\frac{30-V}{10}\big) + 1}
\beta_n(V_m) = 0.125\exp\bigg(-\frac{V}{80}\bigg) & \beta_m(V_m) = 4\exp\bigg(-\frac{V}{18}\bigg) & \beta_h(V_m) = \frac{1}{\exp\big(\frac{30-V}{10}\big) + 1}
\end{array} </math>
\end{array} </math>
其中<math> V = V_{rest} - V_m </math> 表示负去极化，单位为mV（？？？）
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 | name-list-style = vanc |title=Foundations of cellular neurophysiology|year=1997|publisher=MIT Press|location=Cambridge, Massachusetts [u.a.]|isbn=978-0-262-10053-3|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:
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 | name-list-style = vanc |title=Foundations of cellular neurophysiology|year=1997|publisher=MIT Press|location=Cambridge, Massachusetts [u.a.]|isbn=978-0-262-10053-3|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:
为了表征电压门控通道，该方程拟合电压钳数据。关于电压钳下 Hodgkin-Huxley 方程的推导，请参阅。
简单地说，当膜电位保持为一个恒定值(例如，电压钳)时，对于膜电位的每个值，非线性门控方程可以归结为以下形式的方程:
简单地说，当膜电位保持为一个恒定值(例如，电压钳)时，对于膜电位的每个值，非线性门控方程可以归结为以下形式的方程:
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
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
为了得到传导的动作电位的完整解，必须在第一个微分方程的左侧写上电流项 i（？？？），使方程成为单独的电压方程。
为了得到传导的动作电位的完整解，必须在第一个微分方程的左侧写上电流项 <math>i</math>（？？？），使方程成为单独的电压方程。
I 和 v 之间的关系可以从[[电缆理论]]中推导出来，并给出了
<math>I</math>和<math>V</math>之间的关系可以从[[电缆理论]]中推导出来，并给出了
: <math>I = \frac{a}{2R}\frac{\partial^2V}{\partial x^2}, </math>
: <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''.
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''.
其中 a 是[[轴突]]的半径，<math>R</math> 是[[轴浆]]的[[比阻力]]（？？？），x 是沿着神经纤维的位置。
其中<math>a</math>是[[轴突]]的半径，<math>R</math> 是[[轴浆]]的[[比阻力]]（？？？），<math>x</math>是沿着神经纤维的位置。
用这个表达式代替 i，将原来的方程组转变为一组偏微分方程，因为电压变成了 x 和 t 的函数。
用这个表达式代替<math>i</math>，将原来的方程组转变为一组[[偏微分方程]]，因为电压变成了<math>x</math>和<math>t</math>的函数。
The [[Levenberg–Marquardt algorithm]] is often used to fit these equations to voltage-clamp data.<ref>{{cite book |title=New Ecoinformatics Tools in Environmental Science : Applications and Decision-making |first1=Vladimir F. |last1=Krapivin |first2=Costas A. |last2=Varotsos |first3=Vladimir Yu. |last3=Soldatov |year=2015 |pages=37–38 |publisher=Springer |isbn=9783319139784 |url=https://www.google.com/books/edition/New_Ecoinformatics_Tools_in_Environmenta/bWpnBgAAQBAJ?hl=en&gbpv=1&pg=PA37 }}</ref>
The [[Levenberg–Marquardt algorithm]] is often used to fit these equations to voltage-clamp data.<ref>{{cite book |title=New Ecoinformatics Tools in Environmental Science : Applications and Decision-making |first1=Vladimir F. |last1=Krapivin |first2=Costas A. |last2=Varotsos |first3=Vladimir Yu. |last3=Soldatov |year=2015 |pages=37–38 |publisher=Springer |isbn=9783319139784 |url=https://www.google.com/books/edition/New_Ecoinformatics_Tools_in_Environmenta/bWpnBgAAQBAJ?hl=en&gbpv=1&pg=PA37 }}</ref>
维持这些浓度梯度需要这几种离子的主动运输。
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。
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经得到公认: 交换的化学计量比为 3 Na<SUP>+</SUP>: 1 Ca<SUP>2+</SUP>，交换器具有产电性和电压敏感性。
钠钙交换器的一些基本性质已经得到公认: 交换的化学计量比为3Na<SUP>+</SUP>:1Ca<SUP>2+</SUP>，且交换器具有产电性和电压敏感性。
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。
==数学性质==
==数学性质==
The Hodgkin–Huxley model can be thought of as a [[differential equation]] system with four [[state variable]]s, <math>V_m(t), n(t), m(t)</math>, and <math>h(t)</math>, that change with respect to time <math>t</math>. The system is difficult to study because it is a [[nonlinear|nonlinear system]] and cannot be solved analytically. However, there are many numerical methods available to analyze the system. Certain properties and general behaviors, such as [[limit cycle]]s, can be proven to exist.
The Hodgkin–Huxley model can be thought of as a [[differential equation]] system with four [[state variable]]s, <math>V_m(t), n(t), m(t)</math>, and <math>h(t)</math>, that change with respect to time <math>t</math>. The system is difficult to study because it is a [[nonlinear|nonlinear system]] and cannot be solved analytically. However, there are many numerical methods available to analyze the system. Certain properties and general behaviors, such as [[limit cycle]]s, can be proven to exist.
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。
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。
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，有许多数值方法可用于分析该系统。可以证明某些性质和一般行为（如[[极限环]]）是存在的。
然而，有许多数值方法可用于分析该系统。可以证明某些性质和一般行为（如[[极限环]]）是存在的。
[[File: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.|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.|链接=Special:FilePath/Hodgkins_Huxley_Plot.gif]]
本模型的电压<math>v(t)</math>（毫伏），图中超过50毫秒。注入电流从 -5纳安到12纳安不等。该图经历了三个阶段: 平衡阶段、单峰阶段和极限环阶段。（？？？）
===分岔===
===分岔===
*The [[Nanofluidic circuitry#Ion transport|Poisson–Nernst–Planck]] (PNP) model is based on a [[mean-field theory|mean-field approximation]] of ion interactions and continuum descriptions of concentration and electrostatic potential.<ref>{{cite journal | last1=Zheng |first1=Q. |last2=Wei |first2=G. W. | title = Poisson-Boltzmann-Nernst-Planck model | journal = Journal of Chemical Physics | volume = 134 | issue = 19 | pages = 194101 | date = May 2011 | pmid = 21599038 | pmc = 3122111 | doi = 10.1063/1.3581031 | bibcode=2011JChPh.134s4101Z }}</ref>
*The [[Nanofluidic circuitry#Ion transport|Poisson–Nernst–Planck]] (PNP) model is based on a [[mean-field theory|mean-field approximation]] of ion interactions and continuum descriptions of concentration and electrostatic potential.<ref>{{cite journal | last1=Zheng |first1=Q. |last2=Wei |first2=G. W. | title = Poisson-Boltzmann-Nernst-Planck model | journal = Journal of Chemical Physics | volume = 134 | issue = 19 | pages = 194101 | date = May 2011 | pmid = 21599038 | pmc = 3122111 | doi = 10.1063/1.3581031 | bibcode=2011JChPh.134s4101Z }}</ref>
霍奇金-赫胥黎模型被认为是20世纪生物物理学的伟大成就之一。尽管如此，现代 Hodgkin-huxley 型模型已经在几个重要方面得到了扩展:
霍奇金-赫胥黎模型被认为是20世纪生物物理学的伟大成就之一。尽管如此，现代 Hodgkin-Huxley 型模型已经在几个重要方面得到了扩展:
* 根据实验数据引入了额外的离子通道群。
* 根据实验数据引入了额外的离子通道群。
* 对Hodgkin-Huxley 模型加以修正，加入了[[过渡态理论]]，并产生了[[热力学]] Hodgkin-Huxley 模型。
* 对Hodgkin-Huxley 模型加以修正，加入了[[过渡态理论]]，并产生了[[热力学]] Hodgkin-Huxley 模型。
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.
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.
一些简化的神经元模型（如 [[FitzHugh-Nagumo 模型]]）也发展了出来，它们有助于对神经元群进行高效的大规模模拟，以及对动作电位产生的动力学的数学洞察。
==参见==
==参见==