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第2行：
第2行： − + 第33行：
第33行： − − : I_c = C_m\frac{{\mathrm d} V_m}{{\mathrm d} t} − − : i c = c m frac { mathrm d } v m }{ mathrm d } t 第43行：
第39行： − − : I_i = {g_i}(V_m - V_i) \; − − : i i = { g _ i }(v _ m-v _ i) ; 第58行：
第50行： − : 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) − − : 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 _ v _ l) 第81行：
第70行： − − : 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), − − : i = c _ m frac { mathrm d } v _ m }{{ mathrm d } t } + bar { g } text { k } n ^ 4(v _ m-v _ k) + bar { g }文本{ Na } m ^ 3 h (v _ m-v _ Na }) + bar { g } l (v _ m-v _ l) , − − : \frac{dn}{dt} = \alpha_n(V_m)(1 - n) - \beta_n(V_m) n − − : frac { dn }{ dt } = alpha _ n (v _ m)(1-n)-beta _ n (v _ m) n − − : \frac{dm}{dt} = \alpha_m(V_m)(1 - m) - \beta_m(V_m) m − − : frac { dm }{ dt } = alpha _ m (v _ m)(1-m)-beta _ m (v _ m) m − − : \frac{dh}{dt} = \alpha_h(V_m)(1 - h) - \beta_h(V_m) h − − : frac { dh }{ dt } = alpha _ h (v _ m)(1-h)-beta _ h (v _ m) h 第112行：
第85行： − − : \alpha_p(V_m) = p_\infty(V_m)/\tau_p − − : alpha _ p (v _ m) = p _ infty (v _ m)/tau _ p − − : \beta_p(V_m) = (1 - p_\infty(V_m))/\tau_p. − − : beta _ p (v _ m) = (1-p _ infty (v _ m))/tau _ p. 第152行：
第117行： − : \frac{A_p(V_m-B_p)}{\exp\big(\frac{V_m-B_p}{C_p}\big)-D_p} + − − : frac { a _ p (v _ m-b _ p)}{ exp big (frac { v _ m-b _ p }{ c _ p } big)-d _ p } − − 第162行：
第123行： − − : m(t) = m_{0} - [ (m_{0}-m_{\infty})(1 - e^{-t/\tau_m})]\, − − M (t) = m _ {0}-[(m _ {0}-m _ { infty })(1-e ^ {-t/tau _ m })] , − − : h(t) = h_{0} - [ (h_{0}-h_{\infty})(1 - e^{-t/\tau_h})]\, − − : h (t) = h _ {0}-[(h _ {0}-h _ { infty })(1-e ^ {-t/tau _ h })] , − − : n(t) = n_{0} - [ (n_{0}-n_{\infty})(1 - e^{-t/\tau_n})]\, − − n (t) = n _ {0}-[(n _ {0}-n _ { infty })(1-e ^ {-t/tau _ n })] , 第184行：
第133行： − − : I_\mathrm{Na}(t)=\bar{g}_\mathrm{Na} m(V_m)^3h(V_m)(V_m-E_\mathrm{Na}), − − : i _ mathrm { Na }(t) = bar { g } _ mathrm { Na } m (v _ m) ^ 3h (v _ m)(v _ m-e _ mathrm { Na }) , − − : I_\mathrm{K}(t)=\bar{g}_\mathrm{K} n(V_m)^4(V_m-E_\mathrm{K}). − − : i _ mathrm { k }(t) = bar { g } _ mathrm { k } n (v _ m) ^ 4(v _ m-e _ mathrm { k }). 第201行：
第142行： − − : I = \frac{a}{2R}\frac{\partial^2V}{\partial x^2}, − − 2 r } frac { partial ^ 2V }{ partial x ^ 2} , 第228行：
第165行： − + 第243行：
第180行： − + 第264行：
第201行： − +
无编辑摘要
{{short description|Describes how neurons transmit electric signals}}
{{short description|Describes how neurons transmit electric signals}}
[[Image:Hodgkin-Huxley.svg|thumb|right|350px|Basic components of Hodgkin–Huxley-type models. Hodgkin–Huxley type models represent the biophysical characteristic of cell membranes. The lipid bilayer is represented as a capacitance (''C''<SUB>m</SUB>). Voltage-gated and leak ion channels are represented by nonlinear (''g''<SUB>n</SUB>) and linear (''g''<SUB>L</SUB>) conductances, respectively. The electrochemical gradients driving the flow of ions are represented by batteries (E), and ion pumps and exchangers are represented by current sources (''I''<SUB>p</SUB>).]]
[[Image:Hodgkin-Huxley.svg|thumb|right|350px|Basic components of Hodgkin–Huxley-type models. Hodgkin–Huxley type models represent the biophysical characteristic of cell membranes. The lipid bilayer is represented as a capacitance (''C''<SUB>m</SUB>). Voltage-gated and leak ion channels are represented by nonlinear (''g''<SUB>n</SUB>) and linear (''g''<SUB>L</SUB>) conductances, respectively. The electrochemical gradients driving the flow of ions are represented by batteries (E), and ion pumps and exchangers are represented by current sources (''I''<SUB>p</SUB>).|链接=Special:FilePath/Hodgkin-Huxley.svg]]
: <math>I_c = C_m\frac{{\mathrm d} V_m}{{\mathrm d} t} </math>
: <math>I_c = C_m\frac{{\mathrm d} V_m}{{\mathrm d} t} </math>
and the current through a given ion channel is the product
and the current through a given ion channel is the product
: <math>I_i = {g_i}(V_m - V_i) \;</math>
: <math>I_i = {g_i}(V_m - V_i) \;</math>
where <math>V_i</math> is the [[reversal potential]] of the ''i''-th ion channel.
where <math>V_i</math> is the [[reversal potential]] of the ''i''-th ion channel.
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，Cm 为单位面积的膜电容，gK 和 gNa 分别为单位面积的钾和钠的电导，VK 和 VNa 分别为钾和钠的反转电位，gl 和 Vl 分别为单位面积的漏电导和漏反转电位。
其中 i 为单位面积的总膜电流，Cm 为单位面积的膜电容，gK 和 gNa 分别为单位面积的钾和钠的电导，VK 和 VNa 分别为钾和钠的反转电位，gl 和 Vl 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 Vm、 gNa 和 gK，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 Vm、 gNa 和 gK，其中最后两个电导项也明确地取决于电压。
: <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>
: <math>\frac{dn}{dt} = \alpha_n(V_m)(1 - n) - \beta_n(V_m) n</math>
: <math>\frac{dn}{dt} = \alpha_n(V_m)(1 - n) - \beta_n(V_m) n</math>
: <math>\frac{dm}{dt} = \alpha_m(V_m)(1 - m) - \beta_m(V_m) m</math>
: <math>\frac{dm}{dt} = \alpha_m(V_m)(1 - m) - \beta_m(V_m) m</math>
: <math>\frac{dh}{dt} = \alpha_h(V_m)(1 - h) - \beta_h(V_m) h</math>
: <math>\frac{dh}{dt} = \alpha_h(V_m)(1 - h) - \beta_h(V_m) h</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
: <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> \beta_p(V_m) = (1 - p_\infty(V_m))/\tau_p.</math>
: <math> \beta_p(V_m) = (1 - p_\infty(V_m))/\tau_p.</math>
<math>p_\infty</math> and <math>(1-p_\infty)</math> are the steady state values for activation and inactivation, respectively, and are usually represented by [[Boltzmann equation]]s as functions of <math>V_m</math>. In the original paper by Hodgkin and Huxley,<ref name="HH"/> the functions <math>\alpha</math> and <math>\beta</math> are given by
<math>p_\infty</math> and <math>(1-p_\infty)</math> are the steady state values for activation and inactivation, respectively, and are usually represented by [[Boltzmann equation]]s as functions of <math>V_m</math>. In the original paper by Hodgkin and Huxley,<ref name="HH"/> the functions <math>\alpha</math> and <math>\beta</math> are given by
:<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:
为了表征电压门控通道，该方程适合于电压钳位数据。关于电压箝下 Hodgkin-Huxley 方程的推导，请参阅。
为了表征电压门控通道，该方程适合于电压钳位数据。关于电压箝下 Hodgkin-Huxley 方程的推导，请参阅。
: <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>
: <math>h(t) = h_{0} - [ (h_{0}-h_{\infty})(1 - e^{-t/\tau_h})]\, </math>
: <math>h(t) = h_{0} - [ (h_{0}-h_{\infty})(1 - e^{-t/\tau_h})]\, </math>
: <math>n(t) = n_{0} - [ (n_{0}-n_{\infty})(1 - e^{-t/\tau_n})]\, </math>
: <math>n(t) = n_{0} - [ (n_{0}-n_{\infty})(1 - e^{-t/\tau_n})]\, </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
: <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>
: <math>I_\mathrm{K}(t)=\bar{g}_\mathrm{K} n(V_m)^4(V_m-E_\mathrm{K}).</math>
: <math>I_\mathrm{K}(t)=\bar{g}_\mathrm{K} n(V_m)^4(V_m-E_\mathrm{K}).</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
: <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''.
===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>
膜电位取决于其上的离子浓度梯度的保持。
膜电位取决于其上的离子浓度梯度的保持。
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: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]].]]
[[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]]
===中心流形===
===中心流形===
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.]]
[[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]]
模型的电压 v (t)(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。（？？？）
模型的电压 v (t)(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。（？？？）