# 基于电导的模型

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.

## 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,$\displaystyle{ I_C }$,and movement of ions across ion channels.

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.

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

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

$\displaystyle{ I_m(t) = I_C + I_{ionic} }$，其中

$\displaystyle{ I_C = C_m dV(t)/dt }$

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

$\displaystyle{ I_{ionic} = I_{Na} + I_K + I_L }$

The leak current, $$$\displaystyle{ I_L }$ \ ,$$ approximates the passive properties of the cell.

‎漏电流 $\displaystyle{ I_L }$ 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.

Thus,

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

This is illustrated in Figure 1.

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

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

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

## 公式、参数和假设

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

where

$\displaystyle{ g_j = \overline{g}_j a_j^x b_j^y }$ with

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

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

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.

$\displaystyle{ V_j }$是电流$\displaystyle{ j }$能斯特电位反转电位$\displaystyle{ (V - V_j) }$称为$\displaystyle{ j }$的驱动力，而$\displaystyle{ I_{ext} }$是可能存在的外部电流。

$\displaystyle{ a }$$\displaystyle{ b }$分别是提升为小整数幂$\displaystyle{ x }$$\displaystyle{ y }$的门控变量。

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

$\displaystyle{ \tau }$是时间常数，取决于电压。

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.

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.

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.

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).

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).

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.

## 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.

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).

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

## 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

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《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。