基于电导的模型

参考http://www.scholarpedia.org/article/Conductance-based_models

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Conductance-based models are the simplest possible biophysical representation of an excitable cell, such as a neuron, in which its protein molecule ion channels are represented by conductances and its lipid bilayer by a capacitor.

‎基于电导的模型conductance-based model是‎‎可兴奋‎‎细胞（如‎‎神经元‎‎）最简单的‎生物物理‎表示，其中它的蛋白质分子‎‎离子通道‎‎用电导表示，它的磷脂双分子层用电容表示。‎

理论基础

Conductance-based models are based on an equivalent circuit representation of a cell membrane as first put forth by Hodgkin and Huxley (1952).

‎基于电导的模型建立在‎‎Hodgkin和Huxley‎‎（1952）首次提出的细胞膜的一个等效电路表示的基础之上。

These models represent a minimal biophysical interpretation for an excitable cell in which current flow across the membrane is due to charging of the membrane capacitance,[math]\displaystyle{ I_C }[/math],and movement of ions across ion channels.

这些模型代表了可兴奋细胞的最小生物物理解释，其中电流能够跨膜流动的原因是膜电容的充电，[math]\displaystyle{ I_C }[/math]以及离子在离子通道上的运动。

In its simplest version, a conductance-based model represents a neuron by a single isopotential electrical compartment, neglects ion movements between subcellular compartments, and represents only ion movements between the inside and outside of the cell.

在其最简单的版本中，基于电导的模型通过单个等势电房室表示‎‎神经元‎‎，忽略了亚细胞房室之间的离子运动，并且仅表示细胞内部和外部之间的离子运动。

Ion channels are selective for particular ionic species, such as sodium[math]\displaystyle{ Na }[/math] or potassium[math]\displaystyle{ K }[/math], giving rise to currents[math]\displaystyle{ I_{Na} }[/math] \) or([math]\displaystyle{ I_K }[/math]\ ,\) respectively.

离子通道对特定的离子种类具有选择性，例如对钠（[math]\displaystyle{ Na^{+} }[/math]）或钾（[math]\displaystyle{ K^{+} }[/math]），分别产生电流[math]\displaystyle{ I_{Na} }[/math]或[math]\displaystyle{ I_{K} }[/math]。

Thus, the total membrane current, \( [math]\displaystyle{ I_m(t) }[/math] \ ,\) is the sum of the capacitive current and the ionic current,

因此，总膜电流[math]\displaystyle{ I_m(t) }[/math]为电容电流和离子电流之和，‎

[math]\displaystyle{ I_m(t) = I_C + I_{ionic} }[/math]，其中

[math]\displaystyle{ I_C = C_m dV(t)/dt }[/math]

In the Hodgkin-Huxley model, the original conductance-based model,

在‎‎霍奇金-赫胥黎模型‎‎中，基于电导的原始模型，‎

[math]\displaystyle{ I_{ionic} = I_{Na} + I_K + I_L }[/math]

The leak current, \( [math]\displaystyle{ I_L }[/math] \ ,\) approximates the passive properties of the cell.

‎漏电流 [math]\displaystyle{ I_L }[/math] 用于近似神经元的被动属性。

Each ionic current is associated with a conductance (inverse of resistance) and a driving force (battery) which is due to the different concentrations of ions in the intracellular and extracellular media of the cell.

每项离子电流都与电导（电阻的倒数）和驱动力（电池）相关，这是细胞内与细胞外的介质中离子的不同浓度所导致的。

Thus,

因此

[math]\displaystyle{ I_{ionic} = g_{Na}(V) [V(t) - V_{Na}] + g_K(V) [V(t) - V_K] + g_L [V(t) - V_L] . }[/math]

This is illustrated in Figure 1.

这在图‎‎1‎‎的原理图中用电阻上的箭头表示。

The voltage dependence or non-constant nature of the conductance, \( [math]\displaystyle{ g }[/math] \) (1/resistance) of ion channels is captured using "activation" and "inactivation" gating variables which are described using first-order kinetics.

离子通道的电压依赖性或电导的非恒定性[math]\displaystyle{ g }[/math]（1/电阻）用"激活"和"失活"‎‎的门控变量来‎‎体现，这些变量用一阶动力学方程来描述。

This is represented with an arrow across the resistor in the schematic representation of Figure 1.

这在图‎‎1‎‎的原理图中用电阻上的箭头表示。

A current due to ionic species \( [math]\displaystyle{ S }[/math] \) with an activation gating variable, \( [math]\displaystyle{ a }[/math] \ ,\) but no inactivation variable, would be given by \( [math]\displaystyle{ g_S = \overline{g}_S \times a }[/math] \ ,\) where \( [math]\displaystyle{ a }[/math] \) is described by first-order kinetics and \( [math]\displaystyle{ \overline{g}_S }[/math] \) represents the maximal conductance for the particular ion channel.

由具有激活门控变量但没有失活变量的离子种类引起的电流，由[math]\displaystyle{ g_S = \overline{g}_S \times a \ ， }[/math]给出，其中[math]\displaystyle{ a }[/math]由一阶动力学描述，[math]\displaystyle{ \overline{g}_S }[/math]表示特定离子通道的最大电导。‎

公式、参数和假设

From the theoretical basis described above, the standard formulation for a conductance-based model is given as [math]\displaystyle{ C_m dV/dt = \Sigma_j g_j (V_j - V) + I_{ext} }[/math]

从上述理论基础来看，基于电导的模型的标准公式给出为[math]\displaystyle{ C_m dV/dt = \Sigma_j g_j (V_j - V) + I_{ext} }[/math]

where

其中

[math]\displaystyle{ g_j = \overline{g}_j a_j^x b_j^y }[/math] with

[math]\displaystyle{ da/dt = [a_{\infty}(V) - a]/\tau_a(V) }[/math] 以及

[math]\displaystyle{ db/dt = [b_{\infty}(V) - b]/\tau_b(V) }[/math]

for each j. 对每个j。
\( V_j \) is the Nernst potential or reversal potential for current \( j \ ,\) \( (V - V_j) \) is called the driving force for \( j \ ,\) and \( I_{ext} \) is an external current that may be present. \( a, b \) are gating variables raised to small integer powers \( x, y \ ,\) respectively. \( a_{\infty}, b_{\infty} \) are the steady-state gating variable functions that are typically sigmoidal in shape. \( \tau \) is the time constant, which can be voltage-dependent. Further details and equation descriptions can be found in many texts such as Hille (2001) and Koch (1999). Thus, conductance-based models consist of a set of ordinary differential equations (ODEs), as derived from current flow in a circuit representation following Kirchoff's laws. The number of differential equations in the set of model equations depends on the number of different ion channel types being represented with their particular activation and inactivation gating variables. The conductances can depend not only on transmembrane potential \( V \ ,\) but also on concentrations of different ions, for example, the concentration of calcium.

[math]\displaystyle{ V_j }[/math]是电流[math]\displaystyle{ j }[/math]的能斯特电位或反转电位，[math]\displaystyle{ (V - V_j) }[/math]称为[math]\displaystyle{ j }[/math]的驱动力，而[math]\displaystyle{ I_{ext} }[/math]是可能存在的外部电流。

[math]\displaystyle{ a }[/math]，[math]\displaystyle{ b }[/math]分别是提升为小整数幂[math]\displaystyle{ x }[/math]，[math]\displaystyle{ y }[/math]的门控变量。

[math]\displaystyle{ a_{\infty} }[/math]， [math]\displaystyle{ b_{\infty} }[/math]是稳态门控变量函数，其形状通常为 sigmoid。

[math]\displaystyle{ \tau }[/math]是时间常数，取决于电压。

进一步的细节和方程描述可以在许多文本中找到，如Hille（2001）和Koch（1999）。

因此，基于电导的模型由一组常微分方程（ODE）组成，根据基尔霍夫定律从电路表示的电流推导得出。

模型方程组中微分方程的数量取决于不同离子通道类型的数量，这些离子通道类型用其特定的激活和失活门控变量表示。

电导不仅取决于跨膜电位[math]\displaystyle{ V }[/math]，还取决于不同离子的浓度，例如钙离子的浓度。

The parameters in conductance-based models are determined from empirical fits to voltage-clamp experimental data (e.g., see Willms 2002), assuming that the different currents can be adequately separated using pharmacological manipulations and voltage-clamp protocols. As shown in the model formulation, the activation and inactivation variables can be raised to a non-unity integer power, and this is dictated by empirical fits to the data.

基于电导的模型中的参数是通过经验拟合和电压钳实验数据确定的（例如，参见Willms 2002），假设可以使用药理学操作和电压钳方案充分分离不同的电流。如模型公式所示，激活和失活变量可以提高到（非单位整数幂），这由对数据的经验拟合决定。

Since (i) it is rarely possible to obtain estimates of all parameter values in a conductance-based mathematical model from experimental data alone, and (ii) the model construct is necessarily a simplification of the biological cell, it is important to consider various optimization techniques to help constrain the problem for which the conductance-based model was developed to address.

由于（i）仅从实验数据中很难获得基于电导的数学模型中所有参数值的估计，并且（ii）模型构建必然是生物细胞的简化，因此重要的是要考虑各种优化技术，以帮助约束建立基于电导的模型来解决的问题。

In summary, the basic assumptions in conductance-based models are:

1. the different ion channels in the cell membrane are independent from each other,
2. activation and inactivation gating variables are voltage-dependent and independent of each other for a given ion channel type,
3. each gating variable follows first-order kinetics, and
4. the model cell compartment is isopotential.

总之，基于电导的模型中的基本假设为：

1. 细胞膜中的不同离子通道相互独立
2. 对于给定的离子通道类型，激活和失活门控变量依赖于电压且相互独立
3. 每个门控变量都遵循一阶动力学
4. 并且模型细胞房室是等电势的。

示例和变体

• Hodgkin-Huxley model (1952): Original conductance-based model based on the giant axon of the squid producing action potentials. There is a sodium current with activation and inactivation variables, a potassium current with only an activation variable, and a (passive) leak current.
• 霍奇金-赫胥黎模型（1952年）：原始的基于电导的模型，建立在产生动作电位的乌贼巨大轴突的基础上。模型中有具有激活和失活变量的钠电流，仅具有激活变量的钾电流，以及（被动）漏电流。
• Connor-Stevens model (1971): Extended action potential generating model using gastropod neuron somas. There is a sodium, potassium and leak current as in the Hodgkin-Huxley model, and in addition, another potassium current that is transient, the so-called A-current, is included. This current has an activation and an inactivation variable.
• Connor-Stevens模型（1971）：使用腹足纲神经元体建立的扩展动作电位生成模型。与霍奇金-赫胥黎模型一样，有钠离子项、钾离子项和漏电流项，此外，还包括另一个瞬态钾电流，即所谓的A电流。此电流项具有激活和失活变量。
• Morris-Lecar model (1981): Based on the barnacle muscle fiber. There is a calcium current with an instantaneous activation, a potassium current with an activation variable, and a (passive) leak current.

• Morris-Lecar模型（1981年）：基于藤壶肌纤维建立模型。模型中有具有瞬时激活的钙电流、具有激活变量的钾电流和（被动）漏电流。

Conductance-based models are the most common formulation used in neuronal models and can incorporate as many different ion channel types as are known for the particular cell being modeled.

A common extension found in many conductance-based models is the inclusion of an equation for calcium dynamics.

Ionic currents can be calcium-dependent in addition to voltage-dependent with calcium concentrations being controlled by calcium currents, pumps and exchangers.

For example, see section 6.2 in Dayan and Abbott (2001).

基于电导的模型是神经元模型中最常用的公式，它可以包含在要建模的特定细胞中已知的那么多的不同离子通道类型。

建立在许多基于电导的模型中的一个常见扩展为将钙动力学方程包含在内。

离子电流除了对钙离子依赖，还依赖于电压，由钙离子电流、离子泵和交换器控制的钙离子浓度。

例如，参见Dayan and Abbott （2001）中的第6.2节。

Furthermore, as details of various ion channels are determined, variants of conductance-based models have been developed to better match the experimental data.

For example, the standard conductance-based formalism derived from Hodgkin-Huxley models has been extended to account for state-dependent inactivation without voltage dependence in fast sodium and Kv3 potassium channels (Marom and Abbott 1994).

此外，随着各种离子通道的细节确定，已经建立了基于电导的模型的变体，以更好地适应实验数据。

例如，从霍奇金-赫胥黎模型派生的标准基于电导的形式体系，已经扩展到解释快速钠通道和Kv3钾通道中没有电压依赖性的状态依赖性失活（Marom和Abbott 1994）。

The simplest conductance-based model formulation from a spatial perspective consists of a single, isopotential compartment.

Ion movement is strictly between the inside and outside of the cell.

However, to incorporate spatial complexity of cells, several compartments can be connected to represent the cell's complex morphology (see compartmental model).

A conductance-based model formulation can then be used for each compartment with additional terms added to the equations to represent the connections.

That is, current flow occurs not only between the inside and outside of the cell, but also between different regions of the cell.

从空间角度来看，最简单的基于电导的模型公式由单个等电势房室表示。

离子运动严格地在细胞内和细胞外之间进行。

然而，为了结合细胞的空间复杂性，可以连接几个房室来表示细胞的复杂形态（参见区室模型）。

然后，基于电导的模型公式可以用在每个房室，并在方程中添加其他项以表示连接。

也就是说，电流的流动不仅发生在细胞的内部和外部之间，也发生在细胞的不同区域之间。

其他问题

Conductance-based models for excitable cells are developed to help understand underlying mechanisms that contribute to action potential generation, repetitive firing and bursting (i.e., oscillatory patterns) and so on.

In turn, these intrinsic characteristics affect behaviors in neuronal networks.

建立基于电导的可兴奋细胞模型，来帮助理解促进动作电位的产生、反复放电和簇放电（即振荡模式）等的潜在原理。

反过来，这些内在特征会影响神经元网络中的行为。

However, as the number of currents included in conductance-based models expands, it becomes more difficult to understand and predict the resulting model dynamics due to the increasing number of differential equations.

For example, the original Hodgkin-Huxley model is a 4th order system of ODEs.

Efforts have been made not only to capture the qualitative dynamics of conductance-based models (e.g., FitzHugh-Nagumo model) but also to reduce the complexity of the system (e.g., Kepler et al. 1992).

然而，随着基于电导的模型中包含的电流项数量的增加，由于微分方程数量的增加，对由此产生的模型的动力学行为的理解和预测变得更加困难。

例如，最初的霍奇金-赫胥黎模型是一个 4 阶ODE系统。

人们不仅努力建立基于电导的模型的定性动力学（例如，FitzHugh-Nagumo模型），而且还降低了系统的复杂性（例如，Kepler等人，1992年）。

Mathematical distinctions in conductance-based models using dynamical system and bifurcation analyses are available. Details are described in Izhikevich (2007).

在使用动力系统和分岔分析的基于电导的模型中，可以进行数学区分。更多细节在Izhikevich（2007）中有所描述。

参考文献

• Connor JA and Stevens CF. "Prediction of repetitive firing behaviour from voltage clamp data on an isolated neurone soma." J Physiol. 1971 Feb;213(1):31-53.
• Dayan P and Abbott LF. "Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems." The MIT Press, 2001.
• Hille B. "Ion Channels of Excitable Membranes". 3rd ed. Sinauer Associates Inc. Sunderland, MA, 2001.
• Hodgkin AL and Huxley AF. "A quantitative description of membrane current and its application to conduction and excitation in nerve." J Physiol. 1952 Aug;117(4):500-44.
• Izhikevich EM. "Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting." The MIT Press, 2007.
• Kepler TB, Abbott LF, Marder E. "Reduction of conductance-based neuron models." Biol Cybern. 1992, 66:381-387.
• Koch C. "Biophysics of Computation: Information processing in single neurons." Oxford University Press, New York, 1999.
• Marom S and Abbott LF. "Modeling state-dependent inactivation of membrane currents." Biophys J. 1994 Aug;67(2):515-20.
• Morris C and Lecar H. "Voltage oscillations in the barnacle giant muscle fiber." Biophys J. 1981 Jul;35(1):193-213.
• Willms AR. "NEUROFIT: software for fitting Hodgkin-Huxley models to voltage-clamp data." J Neurosci Meth. 2002, 121:139-150.

Internal references

• Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
• Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHugh-Nagumo model. Scholarpedia, 1(9):1349.

外部链接

• Frances K. Skinner's website

参见

Hodgkin-Huxley Model, Morris-Lecar Model, Hindmarsh-Rose Model, Dynamical Systems, Bifurcations, Excitability, NEURON, GENESIS, Neural Oscillators

Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Reviewed by: Prof. Leonid L. Rubchinsky, Indiana University Purdue University, Indianapolis, IN, and Indiana University School of Medicine, Indianapolis, IN, USA

Reviewed by: Astrid A. Prinz, Emory University, Atlanta, Georgia

Accepted on: 2006-11-22 15:04:01 GMT

Categories:

• Computational Neuroscience
• Neuroscience
• Dynamical Systems
• Models of Neurons

编者推荐

《神经科学的数学原理》（《Mathematical Foundations of Neuroscience》）G.Bard Ermentrout, David H.Terman著 吴莹，刘深泉译 1.5节详细推导、介绍了本词条内容。