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第15行： − 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. 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，来解释乌贼巨大神经轴突中动作电位的产生和传导的离子机制。他们因为这项工作获得了1963年的诺贝尔生理学或医学奖。+ − ==Basic components== − 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 channels are represented by electrical conductances (g<SUB>n</SUB>, where n is the specific ion channel) that depend on both voltage and time. Leak channels are represented by linear conductances (g<SUB>L</SUB>). The electrochemical gradients driving the flow of ions are represented by voltage sources (E<SUB>n</SUB>) whose voltages are determined by the ratio of the intra- and extracellular concentrations of the ionic species of interest. Finally, ion pumps are represented by current sources (I<SUB>p</SUB>). The membrane potential is denoted by V<SUB>m</SUB>. − − = = 基本成分 = = 第31行：
第27行： − − Mathematically, the current flowing through the lipid bilayer is written as 第43行：
第37行： − − and the current through a given ion channel is the product 第57行：
第49行： − Thus, for a cell with sodium and potassium channels, the total current through the membrane is given by: − − where V_i is the reversal potential of the i-th ion channel. 第70行：
第59行： − − where I is the total membrane current per unit area, Cm is the membrane capacitance per unit area, gK and gNa are the potassium and sodium conductances per unit area, respectively, VK and VNa are the potassium and sodium reversal potentials, respectively, and gl and Vl are the leak conductance per unit area and leak reversal potential, respectively. The time dependent elements of this equation are Vm, gNa, and gK, where the last two conductances depend explicitly on voltage as well. − − ==Ionic current characterization== 第92行：
第77行： − − 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 equations. Together with the equation for the total current mentioned above, these are: 第122行：
第105行： − − where I is the current per unit area, and \alpha_i and \beta_i are rate constants for the i-th ion channel, which depend on voltage but not time. \bar{g}_n 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 p = (n, m, h), \alpha_p and \beta_p take the form 第176行：
第157行： − − 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. 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: 第201行：
第180行： − − Thus, for every value of membrane potential V_{m} the sodium and potassium currents can be described by 第219行：
第196行： − − 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 第232行：
第207行： − − where a is the radius of the axon, R is the 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 equations, because the voltage becomes a function of both x and t. 第245行：
第218行： − − 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 channels. 第252行：
第223行： − − 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 g_{leak} is a constant. Thus, the leak current due to passive leak ion channels in the Hodgkin-Huxley formalism is I_l=g_{leak}(V-V_{leak}). + − − 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. 第270行：
第238行： − − The Hodgkin–Huxley model can be thought of as a differential equation system with four state variables, V_m(t), n(t), m(t), and h(t), that change with respect to time t. The system is difficult to study because it is a 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 cycles, can be proven to exist. 第279行：
第245行： − + − − ===Center manifold=== − − − − Because there are four state variables, visualizing the path in phase space can be difficult. Usually two variables are chosen, voltage V_m(t) and the potassium gating variable n(t), 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. 第310行：
第270行： − − If the injected current I were used as a 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 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 canard. 第330行：
第288行： − − The Hodgkin–Huxley model is regarded as one of the great achievements of 20th-century biophysics. Nevertheless, modern Hodgkin–Huxley-type models have been extended in several important ways: − *Additional ion channel populations have been incorporated based on experimental data. − *The Hodgkin–Huxley model has been modified to incorporate transition state theory and produce thermodynamic Hodgkin–Huxley models. − *Models often incorporate highly complex geometries of dendrites and axons, often based on microscopy data. − *Stochastic models of ion-channel behavior, leading to stochastic hybrid systems. − *The Poisson–Nernst–Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. 第346行：
第297行： − − 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.
无编辑摘要
[[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，来解释乌贼巨大神经轴突中动作电位的产生和传导的离子机制。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==
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。
最后，离子泵表示为电流源(i < sub > p )。
最后，离子泵表示为电流源(i < sub > p )。
膜电位表示 v < sub > m 。
膜电位表示 v < sub > m 。
Mathematically, the current flowing through the lipid bilayer is written as
Mathematically, the current flowing through the lipid bilayer is written as
: i c = c m frac { mathrm d } v m }{ mathrm d } t
: i c = c m frac { mathrm d } v m }{ mathrm d } t
and the current through a given ion channel is the product
and the current through a given ion channel is the product
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.
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:
: 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_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)
: 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)
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。
==Ionic current characterization==
==Ionic current characterization==
===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:
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 i 是单位面积的电流，而 alpha _ i 和 beta _ i 是 i-th 离子通道的速率常数，它取决于电压而非时间。
其中 i 是单位面积的电流，而 alpha _ i 和 beta _ i 是 i-th 离子通道的速率常数，它取决于电压而非时间。
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。
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 v { m }，钠电流和钾电流可以如下描述
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 i（？？？），使方程成为单独的电压方程。
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，r 是轴浆的比阻力（？？？），x 是沿着神经纤维的位置。
其中 a 是轴突的半径，r 是轴浆的比阻力（？？？），x 是沿着神经纤维的位置。
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g {漏}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g {漏}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 i _ l = g _ { leak }(V-V _ { leak })。
因此，在 Hodgkin-Huxley 公式中，被动漏离子通道引起的漏电流为 i _ l = g _ { leak }(V-V _ { leak })。
===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。
==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 v _ m (t) ，n (t) ，m (t)和 h (t)的微分方程系统，它们随着时间 t 变化。
可以认为Hodgkin-Huxley 模型是一个具有4个状态变量 v _ m (t) ，n (t) ，m (t)和 h (t)的微分方程系统，它们随着时间 t 变化。
[[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]].]]
===Center manifold===
===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。
===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 i 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放电频率。霍普夫分岔的一个结果就是有一个最低的开火率。这意味着要么神经元根本没有放电(对应于零频率) ，要么以最低放电速率放电。由于“全有或全无”原理，动作电位振幅的增加不是平稳的，而是突然的“跳跃”。由此产生的转变被称为谣言。
如果将注入电流 i 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放电频率。霍普夫分岔的一个结果就是有一个最低的开火率。这意味着要么神经元根本没有放电(对应于零频率) ，要么以最低放电速率放电。由于“全有或全无”原理，动作电位振幅的增加不是平稳的，而是突然的“跳跃”。由此产生的转变被称为谣言。
*[[Stochastic]] models of ion-channel behavior, leading to stochastic hybrid systems.<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>
*[[Stochastic]] models of ion-channel behavior, leading to stochastic hybrid systems.<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>
*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:
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。
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。