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[[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.

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>{{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>他们因为这项工作获得了1963年的[[诺贝尔生理学或医学奖]]。

==基本组成部分==

==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>). 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>。

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。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===

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:

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。<ref name="HH"/>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>

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。<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

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>\alpha_p(V_m) = p_\infty(V_m)/\tau_p</math>

: <math>\alpha_p(V_m) = p_\infty(V_m)/\tau_p</math>

where <math> V = V_{rest} - V_m </math> denotes the negative depolarization in mV.

where <math> V = V_{rest} - V_m </math> denotes the negative depolarization in mV.

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

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

\end{array} </math>

\end{array} </math>

 

其中<math> V = V_{rest} - V_m </math> <font color = "#32CD32">表示负去极化</font>，单位为mV。

其中<math> V = V_{rest} - V_m </math> <nowiki><font color = "32CD32">表示负去极化</font></nowiki>，单位为mV。

 

While in many current software programs,<ref>Nelson ME (2005) [http://nelson.beckman.illinois.edu/courses/physl317/part1/Lec3_HHsection.pdf Electrophysiological Models In: Databasing the Brain: From Data to Knowledge.] (S. Koslow and S. Subramaniam, eds.) Wiley, New York, pp. 285–301</ref> Hodgkin–Huxley type models generalize <math> \alpha </math> and <math> \beta </math> to

While in many current software programs,<ref>Nelson ME (2005) [http://nelson.beckman.illinois.edu/courses/physl317/part1/Lec3_HHsection.pdf Electrophysiological Models In: Databasing the Brain: From Data to Knowledge.] (S. Koslow and S. Subramaniam, eds.) Wiley, New York, pp. 285–301</ref> Hodgkin–Huxley type models generalize <math> \alpha </math> and <math> \beta </math> to

在当前的软件程序中，霍奇金-赫胥黎类模型将<nowiki><math>\alpha</math></nowiki>和<nowiki><math>\beta</math></nowiki>归纳为

在当前的软件程序中<ref>Nelson ME (2005) [http://nelson.beckman.illinois.edu/courses/physl317/part1/Lec3_HHsection.pdf Electrophysiological Models In: Databasing the Brain: From Data to Knowledge.] (S. Koslow and S. Subramaniam, eds.) Wiley, New York, pp. 285–301</ref>，霍奇金-赫胥黎类模型将<math>\alpha</math>和><math>\beta</math>归纳为

:<math> \frac{A_p(V_m-B_p)}{\exp\big(\frac{V_m-B_p}{C_p}\big)-D_p} </math>

:<math> \frac{A_p(V_m-B_p)}{\exp\big(\frac{V_m-B_p}{C_p}\big)-D_p} </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 | 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:

为了刻画电压门控通道，'''<font color="#32CD32">该方程拟合自电压钳数据</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：

: <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>

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

为了得到传导的动作电位的完全解，<font color="#32CD32"> 必须将第一个微分方程左侧的电流项 <math>I</math>写为<math>V</math>的形式</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 = \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。

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。

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>

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.

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 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>.

漏通道解释了膜对离子的天然渗透性，其形式为电压门控通道方程，其中电导<math>g_{leak}</math>为常数。

漏通道解释了膜对离子的天然渗透性，其形式为电压门控通道方程，其中电导<math>g_{leak}</math>为常数。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===

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>

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 3 Na<SUP>+</SUP>: 1 Ca<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 3 Na<SUP>+</SUP>: 2 K<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==

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。<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>  

* 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。<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>

* 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>