更改

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>).

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>).

Hodgkin-Huxley 型模型的基本组成部分。Hodgkin-Huxley 型模型代表了细胞膜的生物物理特性。磷脂双分子层表示为一个电容项(''C<SUB>m</SUB>'')。电压门控离子通道和漏离子通道分别用非线性(''g<SUB>n</SUB>'')和线性电导(''g<SUB>L</SUB>'')表示。驱动离子流动的电化学梯度用电池(''E'')表示，离子泵和离子交换器用电流源表示(''I<SUB>p</SUB>'')。

Hodgkin-Huxley 类模型的基本组成部分。Hodgkin-Huxley 类模型代表了细胞膜的生物物理特性。磷脂双分子层表示为一个电容项（<math>C_m</math>）。电压门控离子通道和漏离子通道分别用非线性(<math>g_n</math>)和线性电导(<math>g_L</math>)表示。驱动离子流动的电化学梯度用电池(<math>E</math>)表示，离子泵和离子交换器用电流源表示(<math>I_p</math>)。

The '''Hodgkin–Huxley model''', or '''conductance-based model''', is a [[mathematical model]] that describes how [[action potential]]s in [[neuron]]s are initiated and propagated. It is a set of [[nonlinear]] [[differential equation]]s that approximates the electrical characteristics of excitable cells such as neurons and [[cardiac muscle|cardiac myocytes]]. It is a continuous-time [[dynamical system]].

The '''Hodgkin–Huxley model''', or '''conductance-based model''', is a [[mathematical model]] that describes how [[action potential]]s in [[neuron]]s are initiated and propagated. It is a set of [[nonlinear]] [[differential equation]]s that approximates the electrical characteristics of excitable cells such as neurons and [[cardiac muscle|cardiac myocytes]]. It is a continuous-time [[dynamical system]].

典型的 Hodgkin-Huxley 模型将可兴奋细胞的每个成分都当作电子元件来处理(如图所示)。

典型的 Hodgkin-Huxley 模型将可兴奋细胞的每个成分都当作电子元件来处理(如图所示)。

[[磷脂双分子层]]表示为[[电容]]。

[[磷脂双分子层]]表示为[[电容]]。

电压门控离子通道表示为电导(''g_n''，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(''g_l'')。

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(''e_n'')，电压源的电压取决于相关离子种类在细胞内和细胞外的浓度的比值。

驱使离子流动的电化学梯度表示为电压源(<math>e_n</math>)，电压源的电压取决于相关离子种类在细胞内和细胞外的浓度的比值。

最后，离子泵表示为电流源(''i_p'')。

最后，离子泵表示为电流源(<math>i_p</math>)。

[[膜电位]]表示为''v_m''。

[[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 ''v_i'' 是 ''i''-th 离子通道的反转电位。

其中 <math>V_i</math> 是 <math>i</math>-th 离子通道的反转电位。

因此，对于具有钠和钾离子通道的细胞，通过细胞膜的总电流为：

因此，对于具有钠和钾离子通道的细胞，通过细胞膜的总电流为：

where ''I'' is the total membrane current per unit area, ''C''_''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 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，<math>C_m</math>为单位面积的膜电容，<math>g_K</math>和<math>g_{Na}</math>分别为单位面积的钾和钠的电导，<math>V_K</math>和<math>V_{Na}</math>分别为钾和钠的反转电位，<math>g_l</math>和<math>V_l</math>分别为单位面积的漏电导和漏反转电位。

这个方程中的时间依赖项为 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<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。

==Ionic current characterization==

==Ionic current characterization==

= 离子电流的描述 =

==离子电流的描述==

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.

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

bar{ g } _ n 是电导的最大值。

<math>bar{g}_n</math>是电导的最大值。

N、 m 和 h 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。

N、 m 和 h 是0和1之间的无量纲量，分别与钾通道激活、钠通道激活和钠通道失活有关。

对于 p = (n，m，h) ，alpha _ p 和 beta _ p 的形式是

对于 p = (n，m，h) ，<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>

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 <math>V_m(t), n(t), m(t)</math>， 和 <math>h(t)</math>的微分方程系统，它们随着时间<math>t</math>变化。

这个系统很难研究，因为它是一个非线性系统，无法用解析法求解。

这个系统很难研究，因为它是一个非线性系统，无法用解析法求解。

然而，有许多数值方法可用于分析该系统。可以证明某些性质和一般行为（如极限环）是存在的。

然而，有许多数值方法可用于分析该系统。可以证明某些性质和一般行为（如极限环）是存在的。

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''v_m(t)''和钾门控变量'' n (t) ''，这样就能想象出极限环。

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。

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。

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 <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 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发放率。

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。

霍普夫分岔的一个结果就是存在一个最低的发放率。

这意味着要么神经元根本没有发放(对应于零频率) ，要么以最低发放率发放。

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”。

==Improvements and alternative models==

==Improvements and alternative models==