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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]].

Hodgkin-Huxley 模型，或者说基于电导的模型，是一个描述神经元中动作电位如何产生和传导的数学模型。它是一组非线性微分方程，用于近似可兴奋细胞（如神经元和心肌细胞）的电学特性。它是一个时间连续的动力系统。

'''Hodgkin-Huxley 模型'''，或者说'''基于电导的模型'''，是一个描述[[神经元]]中[[动作电位]]如何产生和传导的[[数学模型]]。它是一组[[非线性微分方程]]，用于近似可兴奋细胞（如神经元和[[心肌细胞]]）的电学特性。它是一个时间连续的[[动力系统]]。

[[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" />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==

==基本成分==

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(<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>)。

[[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。

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>)。

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>。

[[膜电位]]表示为<math>V_m</math>。

: <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 第<math>i</math>个 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:

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。

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==

==离子电流的刻画==

 

==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.

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:

: <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 <math>i</math>-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>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。

<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 p = (n，m，h) ，<math>alpha_p</math>和<math>beta_p</math>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>

<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 Boltzmann 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[[Boltzmann 方程]]表示为 <math>V_m</math>的函数。

在 Hodgkin 和 Huxley 的原始论文中，alpha 和 beta函数如下给出<math> \begin{array}{lll}

在 Hodgkin 和 Huxley 的原始论文中，alpha 和 beta函数如下给出

<math> \begin{array}{lll}

  \alpha_n(V_m) = \frac{0.01(10-V)}{\exp\big(\frac{10-V}{10}\big)-1} & \alpha_m(V_m) = \frac{0.1(25-V)}{\exp\big(\frac{25-V}{10}\big)-1} & \alpha_h(V_m) = 0.07\exp\bigg(-\frac{V}{20}\bigg)\\

  \alpha_n(V_m) = \frac{0.01(10-V)}{\exp\big(\frac{10-V}{10}\big)-1} & \alpha_m(V_m) = \frac{0.1(25-V)}{\exp\big(\frac{25-V}{10}\big)-1} & \alpha_h(V_m) = 0.07\exp\bigg(-\frac{V}{20}\bigg)\\

\beta_n(V_m) = 0.125\exp\bigg(-\frac{V}{80}\bigg) &  \beta_m(V_m) = 4\exp\bigg(-\frac{V}{18}\bigg) & \beta_h(V_m) = \frac{1}{\exp\big(\frac{30-V}{10}\big) + 1}

\beta_n(V_m) = 0.125\exp\bigg(-\frac{V}{80}\bigg) &  \beta_m(V_m) = 4\exp\bigg(-\frac{V}{18}\bigg) & \beta_h(V_m) = \frac{1}{\exp\big(\frac{30-V}{10}\big) + 1}

\end{array} </math>其中<math> V = V_{rest} - V_m </math> 表示mV 中的负去极化（？？？）

\end{array} </math>

其中<math> V = V_{rest} - V_m </math> 表示mV 中的负去极化（？？？）

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。

I 和 v 之间的关系可以从电缆理论中推导出来，并给出了

I 和 v 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 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，<math>R</math> 是[[轴浆]]的[[比阻力]]（？？？），x 是沿着神经纤维的位置。

用这个表达式代替 i，将原来的方程组转变为一组偏微分方程，因为电压变成了 x 和 t 的函数。

用这个表达式代替 i，将原来的方程组转变为一组偏微分方程，因为电压变成了 x 和 t 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 算法为这些方程对电压钳数据进行拟合。

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 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。

[[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]].|链接=Special:FilePath/Hodgkin_Huxley_Limit_Cycle.png]]

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.

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。

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。

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.

Hodgkin-Huxley 模型有两个负的特征值和两个具有轻微取正的实部的复特征值。

Hodgkin-Huxley 模型有两个负的特征值和两个具有轻微取正的实部的复特征值。

随着时间 t 的增加，与两个负特征值相关的特征向量将减少到零。

随着时间 t 的增加，与两个负特征值相关的特征向量将减少到零。

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。

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。

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。

==Improvements and alternative models==

==Improvements and alternative models==

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。

* 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。

* 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。

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。

==See also==

==See also==