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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）。

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>。

[[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>）。

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

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> 是第<math>i</math>个离子通道的[[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：

: <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>  

: <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''_''m'' is the membrane capacitance per unit area, ''g''_''K'' and ''g''_''Na'' are the potassium and sodium conductances per unit area, respectively, ''V''_''K'' and ''V''_''Na'' are the potassium and sodium reversal potentials, respectively, and ''g''_''l'' and ''V''_''l'' are the leak conductance per unit area and leak reversal potential, respectively. The time dependent elements of this equation are ''V''_''m'', ''g''_''Na'', and ''g''_''K'', where the last two conductances depend explicitly on voltage as well.

where ''I'' is the total membrane current per unit area, ''C''_''m'' is the membrane capacitance per unit area, ''g''_''K'' and ''g''_''Na'' are the potassium and sodium conductances per unit area, respectively, ''V''_''K'' and ''V''_''Na'' are the potassium and sodium reversal potentials, respectively, and ''g''_''l'' and ''V''_''l'' are the leak conductance per unit area and leak reversal potential, respectively. The time dependent elements of this equation are ''V''_''m'', ''g''_''Na'', and ''g''_''K'', where the last two conductances depend explicitly on voltage as well.

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。

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。这个方程中对时间依赖的项为<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。

==离子电流的刻画==

==离子电流的刻画==

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

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

其中<math>I</math>是单位面积的电流，而<math>\alpha_i</math>和<math>\beta_i</math>是第<math>i</math>个离子通道的速率常数，它取决于电压而非时间。

其中<math>I</math>是单位面积的电流，而<math>\alpha_i</math>和<math>\beta_i</math>是第<math>i</math>个离子通道的速率常数，它取决于电压而非时间。<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。对于<math>p = (n，m，h)</math>，<math>\alpha_p</math>和<math>\beta_p</math>的形式是

<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。

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[[玻尔兹曼方程]]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，<math>\alpha</math>和<math>\beta</math>函数如下给出

在 Hodgkin 和 Huxley 的原始论文<ref name="HH"/>中，<math>\alpha</math>和<math>\beta</math>函数如下给出

其中<math> V = V_{rest} - V_m </math> 表示负去极化，单位为mV（？？？）

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。

为了刻画电压门控通道，'''<font color="#32CD32">该方程拟合电压钳数据</font>'''。关于电压钳下 Hodgkin-Huxley 方程的推导，请参阅<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>。简单来说，当膜电位保持为一个恒定值（即电压钳取值）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:

: <math>m(t) = m_{0} - [ (m_{0}-m_{\infty})(1 - e^{-t/\tau_m})]\, </math>

: <math>m(t) = m_{0} - [ (m_{0}-m_{\infty})(1 - e^{-t/\tau_m})]\, </math>

Thus, for every value of membrane potential <math>V_{m}</math> the sodium and potassium currents can be described by

Thus, for every value of membrane potential <math>V_{m}</math> the sodium and potassium currents can be described by

因此，对于膜电位的每个值 v { m }，钠电流和钾电流可以如下描述

因此，对于膜电位的每个值<math>V_m</math>，钠电流和钾电流可以描述为（次方的位置？？？）

: <math>I_\mathrm{Na}(t)=\bar{g}_\mathrm{Na} m(V_m)^3h(V_m)(V_m-E_\mathrm{Na}),</math>

: <math>I_\mathrm{Na}(t)=\bar{g}_\mathrm{Na} m(V_m)^3h(V_m)(V_m-E_\mathrm{Na}),</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

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 <math>i</math>（？？？），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全解，'''<font color="#32CD32"> 必须在第一个微分方程的左侧写上电流项 <math>I</math> </font>'''，使方程成为单独的电压方程。<math>I</math>和<math>V</math>之间的关系可以从[[电缆理论]]中推导出来，即

<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<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。

其中<math>a</math>是[[轴突]]的半径，<math>R</math> 是[[轴浆]]的[[比电阻]]，<math>x</math>是沿神经纤维的位置。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]]，因为电压变为<math>x</math>和<math>t</math>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>

通常采用 [[Levenberg-Marquardt 算法]]为这些方程对电压钳数据进行拟合。

通常采用 [[Levenberg-Marquardt 算法]]通过电压钳数据拟合这些方程。

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.

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.

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>

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。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。

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。

[[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]].|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===

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.

由于有四个状态变量，想象[[相空间]]中的路径会比较困难。

由于有四个状态变量，想象[[相空间]]中的路径会比较困难。通常选择两个变量，电压<math>V_m(t)</math>和钾门控变量<math>n(t)</math>，这样就能想象出[[极限环]]。但是，要注意这只是一个想象四维系统的特殊方法，并不能证明极限环的存在性。

通常选择两个变量，电压<math>V_m(t)</math>和钾门控变量<math>n(t)</math>，这样就能想象出[[极限环]]。

A better [[projection (mathematics)|projection]] can be constructed from a careful analysis of the [[Jacobian matrix and determinant|Jacobian]] of the system, evaluated at the [[equilibrium point]]. Specifically, the [[eigenvalues]] of the Jacobian are indicative of the [[center manifold]]'s existence. Likewise, the [[Eigenvalues and eigenvectors|eigenvectors]] of the Jacobian reveal the center manifold's [[Orientation (geometry)|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.

A better [[projection (mathematics)|projection]] can be constructed from a careful analysis of the [[Jacobian matrix and determinant|Jacobian]] of the system, evaluated at the [[equilibrium point]]. Specifically, the [[eigenvalues]] of the Jacobian are indicative of the [[center manifold]]'s existence. Likewise, the [[Eigenvalues and eigenvectors|eigenvectors]] of the Jacobian reveal the center manifold's [[Orientation (geometry)|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.

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.|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===

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]]。

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]]。

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==

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。<ref>{{cite journal |last=Forrest |first=M. D. |title=Can the Thermodynamic Hodgkin–Huxley Model of Voltage-Dependent Conductance Extrapolate for Temperature? |journal=Computation |volume=2 |issue=2 |pages=47–60 |date=May 2014 |doi=10.3390/computation2020047|url=http://wrap.warwick.ac.uk/60495/1/WRAP_computation-02-00047.pdf |doi-access=free }}</ref>

* 通常基于显微镜数据，模型通常包含[[树突]]和[[轴突]]的高度复杂的几何形状。

* 通常基于显微镜数据，模型通常包含[[树突]]和[[轴突]]的高度复杂的几何形状。

* 离子通道行为的[[随机]]模型，导致随机混合系统。

* 离子通道行为的[[随机]]模型，导致随机混合系统。<ref name=stochastic>{{cite journal |last1=Pakdaman |first1=K. |last2=Thieullen |first2=M. |first3=G. |last3=Wainrib  |title=Fluid limit theorems for stochastic hybrid systems with applications to neuron models |year=2010 |journal=Adv. Appl. Probab. |volume=42 |issue=3 |pages=761–794 |doi=10.1239/aap/1282924062 |arxiv=1001.2474 |bibcode=2010arXiv1001.2474P|s2cid=18894661 }}</ref>

* [[Poisson-Nernst-Planck]] 模型是基于离子相互作用的[[平均场近似]]以及浓度和静电势的连续描述建立的。

* [[Poisson-Nernst-Planck]] 模型是基于离子相互作用的[[平均场近似]]以及浓度和静电势的连续描述建立的。<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>

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.