生物神经元模型

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文件:Neuron3.png

Fig. 1. Neuron and myelinated axon, with signal flow from inputs at dendrites to outputs at axon terminals. The signal is a short electrical pulse called action potential or 'spike'.

文件:Spike HH 2.png

Fig 2. Time course of neuronal action potential ("spike"). Note that the amplitude and the exact shape of the action potential can vary according to the exact experimental technique used for acquiring the signal.

Biological neuron models, also known as a spiking neuron models, are mathematical descriptions of the properties of certain cells in the nervous system that generate sharp electrical potentials across their cell membrane, roughly one millisecond in duration, called action potentials or spikes (Fig. 2). Since spikes are transmitted along the axon and synapses from the sending neuron to many other neurons, spiking neurons are considered to be a major information processing unit of the nervous system. Spiking neuron models can be divided into different categories: the most detailed mathematical models are biophysical neuron models (also called Hodgkin-Huxley models) that describe the membrane voltage as a function of the input current and the activation of ion channels. Mathematically simpler are integrate-and-fire models that describe the membrane voltage as a function of the input current and predict the spike times without a description of the biophysical processes that shape the time course of an action potential. Even more abstract models only predict output spikes (but not membrane voltage) as a function of the stimulation where the stimulation can occur through sensory input or pharmacologically. This article provides a short overview of different spiking neuron models and links, whenever possible to experimental phenomena. It includes deterministic and probabilistic models.

生物神经元模型，也被称为尖峰神经元模型，是对神经系统中某些细胞特性的数学描述，这些细胞在细胞膜上产生尖峰电位，持续时间大约为1毫秒，由于尖峰信号是通过轴突和突触从发送神经元传递到其他神经元的，因此尖峰信号神经元被认为是神经系统中主要的信息处理单元。尖峰神经元模型可以分为不同的类别:最详细的数学模型是生物物理神经元模型(也称为霍奇金-赫胥黎模型)，它描述了膜电压作为输入电流和离子通道激活的函数。数学上更简单的是积分-点火模型，它将膜电压描述为输入电流的函数，并预测峰值时间，而没有描述形成动作电位的时间过程的生物物理过程。甚至更抽象的模型只预测输出峰值(而不是膜电压)作为刺激的函数，刺激可以通过感官输入或药理学发生。这篇文章提供了一个简短的概述，不同的尖峰神经元模型和链接，只要可能的实验现象。它包括确定性模型和概率模型。

thumb|right|400px|Fig. 1. Neuron and myelinated axon, with signal flow from inputs at dendrites to outputs at axon terminals. The signal is a short electrical pulse called action potential or 'spike'. thumb|Fig 2. Time course of neuronal action potential ("spike"). Note that the amplitude and the exact shape of the action potential can vary according to the exact experimental technique used for acquiring the signal. Biological neuron models, also known as a spiking neuron models, are mathematical descriptions of the properties of certain cells in the nervous system that generate sharp electrical potentials across their cell membrane, roughly one millisecond in duration, called action potentials or spikes (Fig. 2). Since spikes are transmitted along the axon and synapses from the sending neuron to many other neurons, spiking neurons are considered to be a major information processing unit of the nervous system. Spiking neuron models can be divided into different categories: the most detailed mathematical models are biophysical neuron models (also called Hodgkin-Huxley models) that describe the membrane voltage as a function of the input current and the activation of ion channels. Mathematically simpler are integrate-and-fire models that describe the membrane voltage as a function of the input current and predict the spike times without a description of the biophysical processes that shape the time course of an action potential. Even more abstract models only predict output spikes (but not membrane voltage) as a function of the stimulation where the stimulation can occur through sensory input or pharmacologically. This article provides a short overview of different spiking neuron models and links, whenever possible to experimental phenomena. It includes deterministic and probabilistic models.

thumb|right|400px|Fig.1.神经元和有髓神经元，信号从树突输入到轴突末梢输出。这个信号是一个叫做动作电位的短电脉冲。图2。神经元动作电位的时间进程。请注意，动作电位的幅度和精确形状可以根据获取信号的精确实验技术而改变。生物神经元模型，也被称为尖峰神经元模型，是对神经系统中某些细胞的特性的数学描述，这些细胞通过其细胞膜产生尖锐的电位，大约持续一毫秒，称为动作电位或尖峰。2).由于尖峰信号沿着轴突和突触从发送神经元传递到许多其他神经元，尖峰神经元被认为是神经系统的主要信息处理单元。尖峰神经元模型可以分为不同的类别: 最详细的数学模型是生物物理神经元模型(也称为 Hodgkin-Huxley 模型) ，它将膜电位描述为输入电流和离子通道激活的函数。数学上比较简单的是整合-着火模型，该模型将膜电位电位描述为输入电流的函数，并且在没有描述塑造动作电位时间过程的生物物理过程的情况下预测峰值时间。更抽象的模型只能预测刺激的输出峰值(而不是膜电位) ，刺激可以通过感觉输入或药理作用发生。这篇文章提供了一个简短的概述不同的脉冲神经元模型和链接，只要有可能的实验现象。它包括确定性模型和概率模型。

Introduction: Biological background, classification and aims of neuron models

Non-spiking cells, spiking cells, and their measurement

Non-spiking cells, spiking cells, and their measurement

= = 介绍: 神经元模型的生物学背景、分类和目的 = = 非尖峰细胞、尖峰细胞及其测量

Not all the cells of the nervous system produce the type of spike that define the scope of the spiking neuron models. For example, cochlear hair cells, retinal receptor cells, and retinal bipolar cells do not spike. Furthermore, many cells in the nervous system are not classified as neurons but instead are classified as glia.

并不是所有的神经系统细胞都能产生这种类型的尖刺，从而确定尖刺神经元模型的范围。例如，耳蜗毛细胞、视网膜感受器细胞和视网膜双极细胞不会突增。此外，神经系统中的许多细胞并没有被归类为神经元，而是被归类为神经胶质细胞。

Not all the cells of the nervous system produce the type of spike that define the scope of the spiking neuron models. For example, cochlear hair cells, retinal receptor cells, and retinal bipolar cells do not spike. Furthermore, many cells in the nervous system are not classified as neurons but instead are classified as glia.

并非所有神经系统的细胞都能产生这种类型的尖峰，从而界定了尖峰神经元模型的范围。例如，耳蜗毛细胞、视网膜受体细胞和视网膜双极细胞不会发生突起。此外，神经系统中的许多细胞并不归类为神经元，而是归类为神经胶质细胞。

Neuronal activity can be measured with different experimental techniques, such as the "Whole cell" measurement technique, which captures the spiking activity of a single neuron and produces full amplitude action potentials.

神经元活动可以用不同的实验技术来测量，比如“全细胞”测量技术，它捕捉单个神经元的尖峰活动，并产生全振幅动作电位。

Neuronal activity can be measured with different experimental techniques, such as the "Whole cell" measurement technique, which captures the spiking activity of a single neuron and produces full amplitude action potentials.

神经元的活动可以用不同的实验技术来测量，例如“全细胞”测量技术，它可以捕捉单个神经元的尖峰活动，并产生全振幅的动作电位。

With extracellular measurement techniques an electrode (or array of several electrodes) is located in the extracellular space. Spikes, often from several spiking sources, depending on the size of the electrode and its proximity to the sources, can be identified with signal processing techniques. Extracellular measurement has several advantages: 1) Is easier to obtain experimentally; 2) Is robust and lasts for a longer time; 3) Can reflect the dominant effect, especially when conducted in an anatomical region with many similar cells.

在细胞外测量技术中，一个电极(或多个电极阵列)位于细胞外空间。尖峰，通常来自几个尖峰源，取决于电极的大小和其接近的来源，可以识别与信号处理技术。细胞外测量有几个优点:1)更容易通过实验获得;2)坚固耐用，持续时间长;3)能反映显性效应，特别是在有许多相似细胞的解剖区域。

With extracellular measurement techniques an electrode (or array of several electrodes) is located in the extracellular space. Spikes, often from several spiking sources, depending on the size of the electrode and its proximity to the sources, can be identified with signal processing techniques. Extracellular measurement has several advantages: 1) Is easier to obtain experimentally; 2) Is robust and lasts for a longer time; 3) Can reflect the dominant effect, especially when conducted in an anatomical region with many similar cells.

利用细胞外测量技术，一个电极(或多个电极阵列)位于细胞外液。尖峰，通常来自几个尖峰源，取决于电极的大小和它的接近源，可以识别与信号处理技术。细胞外测量有以下几个优点: 1)更容易通过实验获得; 2)稳定性好，持续时间长; 3)能反映显性效应，尤其是在解剖区域有许多相似细胞时。

Overview of neuron models

Overview of neuron models

神经元模型综述

Neuron models can be divided into two categories according to the physical units of the interface of the model. Each category could be further divided according to the abstraction/detail level:

神经元模型根据其接口的物理单元可分为两类。每个类别还可以根据抽象/细节层次进一步划分:

Neuron models can be divided into two categories according to the physical units of the interface of the model. Each category could be further divided according to the abstraction/detail level:

神经元模型根据模型界面的物理单元可分为两大类。每个类别可以根据抽象/详细程度进一步划分:

1. Electrical input–output membrane voltage models – These models produce a prediction for membrane output voltage as a function of electrical stimulation given as current or voltage input. The various models in this category differ in the exact functional relationship between the input current and the output voltage and in the level of details. Some models in this category predict only the moment of occurrence of output spike (also known as "action potential"); other models are more detailed and account for sub-cellular processes. The models in this category can be either deterministic or probabilistic.电输入-输出膜电压模型-这些模型预测膜输出电压作为电流或电压输入电刺激的函数。这一类中的各种模型在输入电流和输出电压之间的精确函数关系和细节水平上有所不同。这类模型中的一些模型只预测输出峰值的发生时刻(也称为“动作电位”);其他的模型更详细，并考虑到亚细胞过程。这类模型可以是确定性的，也可以是概率的
2. Natural stimulus or pharmacological input neuron models – The models in this category connect between the input stimulus which can be either pharmacological or natural, to the probability of a spike event. The input stage of these models is not electrical, but rather has either pharmacological (chemical) concentration units, or physical units that characterize an external stimulus such as light, sound or other forms of physical pressure. Furthermore, the output stage represents the probability of a spike event and not an electrical voltage. 自然刺激或药物输入神经元模型-这类模型连接输入刺激，可以是药物或自然的，与脉冲事件的概率。这些模型的输入阶段不是电的，而是具有药理学(化学)浓度单位或物理单位来表征外部刺激，如光、声或其他形式的物理压力。此外，输出级代表了尖峰事件的概率，而不是电压。

1. Electrical input–output membrane voltage models – These models produce a prediction for membrane output voltage as a function of electrical stimulation given as current or voltage input. The various models in this category differ in the exact functional relationship between the input current and the output voltage and in the level of details. Some models in this category predict only the moment of occurrence of output spike (also known as "action potential"); other models are more detailed and account for sub-cellular processes. The models in this category can be either deterministic or probabilistic.
2. Natural stimulus or pharmacological input neuron models – The models in this category connect between the input stimulus which can be either pharmacological or natural, to the probability of a spike event. The input stage of these models is not electrical, but rather has either pharmacological (chemical) concentration units, or physical units that characterize an external stimulus such as light, sound or other forms of physical pressure. Furthermore, the output stage represents the probability of a spike event and not an electrical voltage.

1. 电力输入-输出/膜电位模型-这些模型预测膜输出电压作为电流或电压输入刺激的函数。这一类别中的各种模型在输入电流和输出电压之间的精确函数关系以及细节层次上都有所不同。这类模型中的一些模型只能预测输出尖峰发生的时刻(也被称为“动作电位”) ; 其他模型则更为详细，并考虑了亚细胞过程。这类模型可以是确定性的，也可以是概率性的。# 自然刺激或药理学输入神经元模型-这类模型连接的输入刺激，可以是药理或自然的，以概率的尖峰事件。这些模型的输入阶段不是电的，而是具有药理(化学)浓度单位，或表征外部刺激的物理单位，如光、声或其他形式的物理压力。此外，输出阶段代表的概率尖峰事件，而不是电压。

Although it is not unusual in science and engineering to have several descriptive models for different abstraction/detail levels, the number of different, sometimes contradicting, biological neuron models is exceptionally high. This situation is partly the result of the many different experimental settings, and the difficulty to separate the intrinsic properties of a single neuron from measurements effects and interactions of many cells (network effects). To accelerate the convergence to a unified theory, we list several models in each category, and where applicable, also references to supporting experiments.

尽管在科学和工程领域，对于不同的抽象/细节层次有不同的描述模型并不罕见，但不同的、有时相互矛盾的生物神经元模型的数量却异常高。这种情况在一定程度上是由于许多不同的实验设置，以及很难将单个神经元的固有特性与测量多个细胞的效应和相互作用(网络效应)区分开来。为了加快收敛到一个统一的理论，我们列出了每个类别的几个模型，在适用的情况下，还参考了支持实验。

Although it is not unusual in science and engineering to have several descriptive models for different abstraction/detail levels, the number of different, sometimes contradicting, biological neuron models is exceptionally high. This situation is partly the result of the many different experimental settings, and the difficulty to separate the intrinsic properties of a single neuron from measurements effects and interactions of many cells (network effects). To accelerate the convergence to a unified theory, we list several models in each category, and where applicable, also references to supporting experiments.

尽管在科学和工程学中，为不同的抽象/细节层次建立几个描述性模型并不罕见，但不同的、有时相互矛盾的生物神经元模型的数量特别多。这种情况在一定程度上是许多不同实验设置的结果，以及很难将单个神经元的内在属性从许多细胞的测量效应和相互作用(网络效应)中分离出来。为了加速收敛到一个统一的理论，我们列出了几个模型在每个范畴，并在适用的情况下，也参考了支持的实验。

Aims of neuron models

Aims of neuron models

神经元模型的目的

Ultimately, biological neuron models aim to explain the mechanisms underlying the operation of the nervous system. However several approaches can be distinguished from more realistic models (e.g., mechanistic models) to more pragmatic models (e.g., phenomenological models). Modeling helps to analyze experimental data and address questions such as: How are the spikes of a neuron related to sensory stimulation or motor activity such as arm movements? What is the neural code used by the nervous system? Models are also important in the context of restoring lost brain functionality through neuroprosthetic devices.

最终，生物神经元模型旨在解释神经系统运行的机制。然而，有几种方法可以从更现实的模型(如机械模型)和更实用的模型(如现象学模型)区分开来。建模有助于分析实验数据，并解决以下问题:神经元的峰值与感觉刺激或运动活动(如手臂运动)有何关联?神经系统使用的神经代码是什么?模型在通过神经修复装置恢复失去的大脑功能方面也很重要。

Ultimately, biological neuron models aim to explain the mechanisms underlying the operation of the nervous system. However several approaches can be distinguished from more realistic models (e.g., mechanistic models) to more pragmatic models (e.g., phenomenological models). Modeling helps to analyze experimental data and address questions such as: How are the spikes of a neuron related to sensory stimulation or motor activity such as arm movements? What is the neural code used by the nervous system? Models are also important in the context of restoring lost brain functionality through neuroprosthetic devices.

最终，生物神经元模型旨在解释神经系统运作的基本机制。然而，有几种方法可以从更加现实的模型(例如，机械模型)到更加实用的模型(例如，现象学模型)加以区分。建模有助于分析实验数据和解决问题，如: 神经元的尖峰是如何与感觉刺激或运动活动，如手臂运动？神经系统使用的神经代码是什么？模型在通过神经修复设备恢复大脑功能方面也很重要。

Electrical input–output membrane voltage models 模板:Anchor

The models in this category describe the relationship between neuronal membrane currents at the input stage, and membrane voltage at the output stage. This category includes (generalized) integrate-and-fire models and biophysical models inspired by the work of Hodgkin–Huxley in the early 1950s using an experimental setup that punctured the cell membrane and allowed to force a specific membrane voltage/current.

这类模型描述了输入阶段的神经元膜电流和输出阶段的膜电压之间的关系。这类模型包括(广义的)集成-点火模型和生物物理学模型，这些模型的灵感来自于20世纪50年代早期的霍奇金-赫胥黎(Hodgkin-Huxley)的工作，使用的是一种刺破细胞膜并强制施加特定膜电压/电流的实验装置。

The models in this category describe the relationship between neuronal membrane currents at the input stage, and membrane voltage at the output stage. This category includes (generalized) integrate-and-fire models and biophysical models inspired by the work of Hodgkin–Huxley in the early 1950s using an experimental setup that punctured the cell membrane and allowed to force a specific membrane voltage/current.

= = 电输入输出膜电位模型 = = 这类模型描述了输入阶段神经细胞膜电流和输出阶段神经细胞膜电流之间的膜电位关系。这一类包括(广义的)集成与火的模型和生物物理模型，这些模型的灵感来自于 Hodgkin-Huxley 在20世纪50年代早期的工作，他们使用一种实验装置刺穿细胞膜，使其产生特定的膜电位/电流。

Most modern electrical neural interfaces apply extra-cellular electrical stimulation to avoid membrane puncturing which can lead to cell death and tissue damage. Hence, it is not clear to what extent the electrical neuron models hold for extra-cellular stimulation (see e.g.).

大多数现代电神经接口应用细胞外电刺激，以避免膜穿刺可导致细胞死亡和组织损伤。因此，尚不清楚电神经元模型对细胞外刺激的支持程度(见例)。

Most modern electrical neural interfaces apply extra-cellular electrical stimulation to avoid membrane puncturing which can lead to cell death and tissue damage. Hence, it is not clear to what extent the electrical neuron models hold for extra-cellular stimulation (see e.g.).

大多数现代电神经界面应用细胞外电刺激来避免可能导致细胞死亡和组织损伤的膜穿刺。因此，电神经元模型在多大程度上适用于细胞外刺激尚不清楚。).

Hodgkin–Huxley

Experimental evidence supporting the model

Property of the H&H model	References
The shape of an individual spike
The identity of the ions involved
Spike speed across the axon

Main article: Hodgkin–Huxley model

The Hodgkin–Huxley model (H&H model) is a model of the relationship between the flow of ionic currents across the neuronal cell membrane and the membrane voltage of the cell. It consists of a set of nonlinear differential equations describing the behaviour of ion channels that permeate the cell membrane of the squid giant axon. Hodgkin and Huxley were awarded the 1963 Nobel Prize in Physiology or Medicine for this work.

霍奇金-赫胥黎模型(H&H模型)是一种描述离子电流通过神经元细胞膜与细胞膜电压之间关系的模型。它由一组非线性微分方程描述离子通道的行为，渗透细胞膜的鱿鱼巨型轴突。霍奇金和赫胥黎因这项工作被授予1963年诺贝尔生理学或医学奖。

Experimental evidence supporting the model

Property of the H&H model	References
The shape of an individual spike
The identity of the ions involved
Spike speed across the axon

The Hodgkin–Huxley model (H&H model) is a model of the relationship between the flow of ionic currents across the neuronal cell membrane and the membrane voltage of the cell. It consists of a set of nonlinear differential equations describing the behaviour of ion channels that permeate the cell membrane of the squid giant axon. Hodgkin and Huxley were awarded the 1963 Nobel Prize in Physiology or Medicine for this work.

Hodgkin–Huxley

Experimental evidence supporting the model

Property of the H&H model	References
The shape of an individual spike
The identity of the ions involved
Spike speed across the axon

The Hodgkin–Huxley model (H&H model) is a model of the relationship between the flow of ionic currents across the neuronal cell membrane and the membrane voltage of the cell.它由一组非线性微分方程组成，描述了渗入乌贼巨大神经轴细胞膜的离子通道的行为。霍奇金和赫胥黎因为这项工作获得了1963年的诺贝尔生理学或医学奖奖。

We note the voltage-current relationship, with multiple voltage-dependent currents charging the cell membrane of capacity C_m

我们注意到电压-电流关系，与多个电压依赖的电流充电的细胞膜的容量Cm

We note the voltage-current relationship, with multiple voltage-dependent currents charging the cell membrane of capacity

我们注意到电压-电流的关系，多个电压依赖的电流充电的细胞膜的容量

[math]\displaystyle{ C_\mathrm{m} \frac{d V(t)}{d t} = -\sum_i I_i (t, V). }[/math]

C_\mathrm{m} \frac{d V(t)}{d t} = -\sum_i I_i (t, V).

c _ mathrm { m } frac { d v (t)}{ d t } =-sum _ i _ i (t，v).

The above equation is the time derivative of the law of capacitance, Q = CV where the change of the total charge must be explained as the sum over the currents. Each current is given by

上面的方程是电容定律的时间导数，Q = CV，其中总电荷的变化必须解释为对电流的和。每个电流由

The above equation is the time derivative of the law of capacitance, where the change of the total charge must be explained as the sum over the currents. Each current is given by

上面的方程是电容定律的时间导数，其中总电荷的变化必须解释为电流之和。每个电流由

[math]\displaystyle{ I(t,V) = g(t,V)\cdot(V-V_\mathrm{eq}) }[/math]

I(t,V) = g(t,V)\cdot(V-V_\mathrm{eq})

i (t，v) = g (t，v) cdot (V-V _ mathrm { eq })

where g(t,V) is the conductance, or inverse resistance, which can be expanded in terms of its maximal conductance ḡ and the activation and inactivation fractions m and h, respectively, that determine how many ions can flow through available membrane channels. This expansion is given by

其中g(t,V)是电导，或逆电阻，它可以根据其最大电导ḡ和分别决定有多少离子可以通过可用膜通道的活化和失活组分m和h来扩大。这个展开式由

where is the conductance, or inverse resistance, which can be expanded in terms of its maximal conductance and the activation and inactivation fractions and , respectively, that determine how many ions can flow through available membrane channels. This expansion is given by

哪里是电导，或逆电阻，它可以根据它的最大电导，活化和失活分数，分别决定有多少离子可以通过可用的膜通道。这个展开式是由

[math]\displaystyle{ g(t,V)=\bar{g}\cdot m(t,V)^p \cdot h(t,V)^q }[/math]

g(t,V)=\bar{g}\cdot m(t,V)^p \cdot h(t,V)^q

g (t，v) = bar { g } cdot m (t，v) ^ p cdot h (t，v) ^ q

and our fractions follow the first-order kinetics

我们的分数遵循一级动力学

and our fractions follow the first-order kinetics

我们的分数遵循一级动力学

[math]\displaystyle{ \frac{d m(t,V)}{d t} = \frac{m_\infty(V)-m(t,V)}{\tau_\mathrm{m} (V)} = \alpha_\mathrm{m} (V)\cdot(1-m) - \beta_\mathrm{m} (V)\cdot m }[/math]

\frac{d m(t,V)}{d t} = \frac{m_\infty(V)-m(t,V)}{\tau_\mathrm{m} (V)} = \alpha_\mathrm{m} (V)\cdot(1-m) - \beta_\mathrm{m} (V)\cdot m

frac { dm (t，v)}{ dt } = frac { m _ infty (v)-m (t，v)}{ tau _ mathrm { m }(v)} = alpha _ mathrm { m }(v) cdot (1-m)-beta _ mathrm { m }(v) cdot

with similar dynamics for h, where we can use either τ and m_∞ or α and β to define our gate fractions.

对于h类似的动力学，我们可以使用τ和m∞或α和β来定义我们的门分数。

with similar dynamics for , where we can use either and or and to define our gate fractions.

类似的动力学，在这里我们可以使用和或来定义我们的门分数。

The Hodgkin–Huxley model may be extended to include additional ionic currents. Typically, these include inward Ca²⁺ and Na⁺ input currents, as well as several varieties of K⁺ outward currents, including a "leak" current.

霍奇金-赫胥黎模型可以扩展到包括额外的离子电流。通常，这些电流包括内向Ca2+和Na+输入电流，以及几种不同的K+输出电流，包括“泄漏”电流。

The Hodgkin–Huxley model may be extended to include additional ionic currents. Typically, these include inward Ca2+ and Na+ input currents, as well as several varieties of K+ outward currents, including a "leak" current.

Hodgkin-Huxley 模型可以扩展到包括更多的离子电流。通常，这些包括内向的 Ca2 + 和 Na + 输入电流，以及几种不同的 k + 外向电流，包括“泄漏”电流。

The end result can be at the small end 20 parameters which one must estimate or measure for an accurate model. In a model of a complex systems of neurons, numerical integration of the equations are computationally expensive. Careful simplifications of the Hodgkin–Huxley model are therefore needed.

最终结果可以是在小的20个参数，一个人必须估计或测量一个准确的模型。在一个复杂的神经元系统的模型中，方程的数值积分是计算昂贵的。因此，我们需要对霍奇金-赫胥黎模型进行仔细的简化。

The end result can be at the small end 20 parameters which one must estimate or measure for an accurate model. In a model of a complex systems of neurons, numerical integration of the equations are computationally expensive. Careful simplifications of the Hodgkin–Huxley model are therefore needed.

最终的结果可以是20个小的参数，这些参数必须估计或测量才能得到一个精确的模型。在一个复杂神经元系统的模型中，方程的数值积分是昂贵的。因此，我们需要对 Hodgkin-Huxley 模式进行谨慎的简化。

The model can be reduced to two dimensions thanks to the dynamic relations which can be established between the gating variables. it is also possible to extend it to take into account the evolution of the concentrations (considered fixed in the original model).

通过建立门控变量之间的动态关系，可以将模型简化为二维。也可以将其扩展到考虑浓度的演化(在原始模型中被认为是固定的)。

The model can be reduced to two dimensions thanks to the dynamic relations which can be established between the gating variables. it is also possible to extend it to take into account the evolution of the concentrations (considered fixed in the original model).

由于门控变量之间可以建立动态关系，该模型可以降低到二维。考虑到浓度的演变(在原始模型中被认为是固定的) ，也可以对其进行扩展。

Perfect Integrate-and-fire

One of the earliest models of a neuron is the perfect integrate-and-fire model (also called non-leaky integrate-and-fire), first investigated in 1907 by Louis Lapicque. A neuron is represented by its membrane voltage V which evolves in time during stimulation with an input current I(t) according

神经元最早的模型之一是完美的积分-点火模型(也称为无漏积分-点火模型)，由Louis Lapicque于1907年首次研究。神经元由其膜电压V表示，其在刺激过程中随时间变化，输入电流I(t)为

One of the earliest models of a neuron is the perfect integrate-and-fire model (also called non-leaky integrate-and-fire), first investigated in 1907 by Louis Lapicque. A neuron is represented by its membrane voltage which evolves in time during stimulation with an input current according

= = = 完美积分-着火 = = = 神经元最早的模型之一是完美积分-着火模型(也称为非泄漏积分-着火模型) ，路易斯 · 拉皮克于1907年首次对其进行研究。一个神经元由它的膜电位表示，它在刺激时随着输入电流的变化而进化

[math]\displaystyle{ I(t)=C \frac{d V(t)}{d t} }[/math]

I(t)=C \frac{d V(t)}{d t}

i (t) = c frac { d v (t)}{ d t }

which is just the time derivative of the law of capacitance, Q = CV. When an input current is applied, the membrane voltage increases with time until it reaches a constant threshold V_th, at which point a delta function spike occurs and the voltage is reset to its resting potential, after which the model continues to run. The firing frequency of the model thus increases linearly without bound as input current increases.

也就是电容定律的时间导数Q = CV。当输入电流施加时，膜电压随时间增加，直到达到一个恒定的阈值Vth，此时出现一个delta函数尖峰，电压复位到其静止电位，之后模型继续运行。因此，该模型的触发频率随着输入电流的增加而线性无界地增加。

which is just the time derivative of the law of capacitance, . When an input current is applied, the membrane voltage increases with time until it reaches a constant threshold , at which point a delta function spike occurs and the voltage is reset to its resting potential, after which the model continues to run. The firing frequency of the model thus increases linearly without bound as input current increases.

这是电容定律的时间导数。当输入电流被加入时，膜电位随时间增加，直到达到一个恒定的阈值，在这一点上出现一个增量函数尖峰，电压被复位到它的静息电位，之后模型继续运行。因此，模型的触发频率随着输入电流的增加而线性增加，没有界限。

The model can be made more accurate by introducing a refractory period t_ref that limits the firing frequency of a neuron by preventing it from firing during that period. For constant input I(t)=I the threshold voltage is reached after an integration time t_int=CV_thr/I after start from zero. After a reset, the refractory period introduces a dead time so that the total time until the next firing is t_ref+t_int . The firing frequency is the inverse of the total inter-spike interval (including dead time). The firing frequency as a function of a constant input current is therefore

通过引入不应期tref，可以通过阻止神经元在不应期放电来限制神经元的放电频率，从而使模型更加精确。对于恒定输入I(t)=I，从零开始后经过积分时间tint=CVthr/I达到阈值电压。复位后，不应期引入死时间，使总时间直到下次点火是tref+色调。发射频率是脉冲间总间隔(包括死区时间)的倒数。因此，作为恒定输入电流的函数的触发频率

The model can be made more accurate by introducing a refractory period that limits the firing frequency of a neuron by preventing it from firing during that period. For constant input the threshold voltage is reached after an integration time after start from zero. After a reset, the refractory period introduces a dead time so that the total time until the next firing is . The firing frequency is the inverse of the total inter-spike interval (including dead time). The firing frequency as a function of a constant input current is therefore

这个模型可以通过引入一个不应期(性)来限制神经元的放电频率，防止它在放电期间放电，从而使其更加精确。对于常量输入，从零开始积分一段时间后达到阈值电压。在复位之后，不应期(性)引入死时间，这样直到下一次发射的总时间是。放电频率与总的穗间期(包括死亡时间)成反比。因此，发射频率作为恒定输入电流的函数是

[math]\displaystyle{ \,\! f(I)= \frac{I} {C_\mathrm{} V_\mathrm{th} + t_\mathrm{ref} I}. }[/math]

\,\! f(I)= \frac{I} {C_\mathrm{} V_\mathrm{th} + t_\mathrm{ref} I}.

\,\!F (i) = frac { i }{ c _ mathrm {} v _ mathrm { th } + t _ mathrm { ref } i }.

A shortcoming of this model is that it describes neither adaptation nor leakage. If the model receives a below-threshold short current pulse at some time, it will retain that voltage boost forever - until another input later makes it fire. This characteristic is clearly not in line with observed neuronal behavior. The following extensions make the integrate-and-fire model more plausible from a biological point of view.

该模型的一个缺点是既不能描述适应性也不能描述泄漏。如果模型在某一时刻接收到低于阈值的短电流脉冲，它将永远保持电压升压-直到稍后另一个输入使其着火。这一特征显然与观察到的神经元行为不一致。下面的扩展使集成-点火模型从生物学的角度来看更加可信。

A shortcoming of this model is that it describes neither adaptation nor leakage. If the model receives a below-threshold short current pulse at some time, it will retain that voltage boost forever - until another input later makes it fire. This characteristic is clearly not in line with observed neuronal behavior. The following extensions make the integrate-and-fire model more plausible from a biological point of view.

该模型的一个缺点是既没有描述自适应性，也没有描述泄漏性。如果模型在某一时刻接收到低于阈值的短电流脉冲，它将永远保持那个电压升压——直到后来的另一个输入使它触发。这种特性显然不符合观察到的神经元行为。从生物学的角度来看，下面的扩展使得整合-开火模型更加合理。

Leaky integrate-and-fire

The leaky integrate-and-fire model which can be traced back to Louis Lapicque, contains, compared to the non-leaky integrate-and-fire model a "leak" term in the membrane potential equation, reflecting the diffusion of ions through the membrane. The model equation looks like

可追溯到Louis Lapicque的漏积-火模型，与非漏积-火模型相比，在膜势方程中包含了一个“漏”项，反映了离子通过膜的扩散。模型方程是这样的

The leaky integrate-and-fire model which can be traced back to Louis Lapicque, contains, compared to the non-leaky integrate-and-fire model a "leak" term in the membrane potential equation, reflecting the diffusion of ions through the membrane. The model equation looks like

= = = = 渗漏的积分与着火 = = = 渗漏的积分与着火模型可以追溯到 Louis Lapicque，与非渗漏的积分与着火模型相比，在膜电位方程中包含了一个“渗漏”项，反映了离子通过薄膜的扩散。模型方程看起来像

[math]\displaystyle{ C_\mathrm{m} \frac{d V_\mathrm{m} (t)}{d t}= I(t)-\frac{V_\mathrm{m} (t)}{R_\mathrm{m}} }[/math]

文件:Leaky Integrate-and-Fire model neuron (schematic).jpg

A neuron is represented by an RC circuit with a threshold. Each input pulse (e.g. caused by a spike from a different neuron) causes a short current pulse. Voltage decays exponentially. If the threshold is reached an output spike is generated and the voltage is reset.

where V_m is the voltage across the cell membrane and R_m is the membrane resistance. (The non-leaky integrate-and-fire model is retrieved in the limit R_m to infinity, i.e. if the membrane is a perfect insulator). The model equation is valid for arbitrary time-dependent input until a threshold V_th is reached; thereafter the membrane potential is reset.

其中Vm是跨细胞膜的电压，Rm是膜电阻。(在Rm为无穷大的极限下，即膜是完美绝缘体时，可以得到无泄漏的积分-点火模型)。模型方程对任意时变输入都是有效的，直到达到阈值Vth;此后膜电位被重置。

C_\mathrm{m} \frac{d V_\mathrm{m} (t)}{d t}= I(t)-\frac{V_\mathrm{m} (t)}{R_\mathrm{m}}

thumb|A neuron is represented by an RC circuit with a threshold. Each input pulse (e.g. caused by a spike from a different neuron) causes a short current pulse. Voltage decays exponentially. If the threshold is reached an output spike is generated and the voltage is reset. where is the voltage across the cell membrane and is the membrane resistance. (The non-leaky integrate-and-fire model is retrieved in the limit to infinity, i.e. if the membrane is a perfect insulator). The model equation is valid for arbitrary time-dependent input until a threshold is reached; thereafter the membrane potential is reset.

c _ mathrm { m } frac { d _ mathrm { m }(t)}{ d } = i (t)-frac { v _ mathrm { m }(t)}{ r _ mathrm { m } thumb | a 神经元用一个带阈值的 RC 电路表示。每个输入脉冲(例如:。由另一个神经元的尖峰引起)导致短电流脉冲。电压呈指数衰减。如果阈值达到一个输出尖峰产生和电压是复位。细胞膜上的电压和膜电阻。(非泄漏的集成与火灾模型在无穷远极限下进行反演，即在无穷远极限下进行反演。如果薄膜是完美的绝缘体)。在达到阈值之前，该模型方程对于任意时变输入都是有效的; 之后重置膜电位。

For constant input, the minimum input to reach the threshold is I_th = V_th / R_m. Assuming a reset to zero, the firing frequency thus looks like

对于恒定输入，达到阈值的最小输入为Ith = Vth / Rm。假设重置为零，那么触发频率看起来是这样的

For constant input, the minimum input to reach the threshold is . Assuming a reset to zero, the firing frequency thus looks like

对于常量输入，达到阈值的最小输入是。假设重置为零，发射频率因此看起来像

[math]\displaystyle{ f(I) = \begin{cases} 0, & I \le I_\mathrm{th} \\ \left[ t_\mathrm{ref}-R_\mathrm{m} C_\mathrm{m} \log\left(1-\tfrac{V_\mathrm{th}}{I R_\mathrm{m}}\right) \right]^{-1}, & I \gt I_\mathrm{th} \end{cases} }[/math]

f(I) =

\begin{cases}

 0,  & I \le I_\mathrm{th} \\
 \left[ t_\mathrm{ref}-R_\mathrm{m} C_\mathrm{m} \log\left(1-\tfrac{V_\mathrm{th}}{I R_\mathrm{m}}\right) \right]^{-1}, & I > I_\mathrm{th} 

\end{cases}

F (i) = begin { cases }0，& i le i _ mathrm { th } left [ t _ mathrm { ref }-r _ mathrm { m } log left (1-tfrac { v _ mathrm { th }{ th }{ i r _ mathrm { m }} right ] ^ {1} ，& i > i _ mathrm { th } end { cases }

which converges for large input currents to the previous leak-free model with refractory period. The model can also be used for inhibitory neurons.

当输入电流较大时，该模型收敛于无漏模型。该模型也可用于抑制性神经元。

which converges for large input currents to the previous leak-free model with refractory period. The model can also be used for inhibitory neurons.

该模型可以收敛于大输入电流到先前的带有不应期(性)的无泄漏模型。该模型也可用于抑制性神经元。

The biggest disadvantage of the Leaky integrate-and-fire neuron is that it does not contain neuronal adaptation so that it cannot describe an experimentally measured spike train in response to constant input current. This disadvantage is removed in generalized integrate-and-fire models that also contain one or several adaptation-variables and are able to predict spike times of cortical neurons under current injection to a high degree of accuracy.

Leaky积分-点火神经元的最大缺点是它不包含神经元适应，因此它不能描述一个在恒定输入电流下响应的实验测量的尖峰序列。在包含一个或多个适应变量的广义积分-点火模型中，该缺点被消除了，并且能够以较高的精确度预测皮层神经元在电流注入下的尖峰时间。

The biggest disadvantage of the Leaky integrate-and-fire neuron is that it does not contain neuronal adaptation so that it cannot describe an experimentally measured spike train in response to constant input current. This disadvantage is removed in generalized integrate-and-fire models that also contain one or several adaptation-variables and are able to predict spike times of cortical neurons under current injection to a high degree of accuracy.

漏点积分-触发神经元的最大缺点是它不包含神经元适应，因此它不能描述实验测量的对恒定输入电流的反应的尖峰列车。这种缺点在包含一个或多个适应变量的广义积分-开火模型中被消除，并且能够高度准确地预测皮层神经元在电流注射下的尖峰时间。

Adaptive integrate-and-fire

Experimental evidence supporting the model

Adaptive integrate-and-fire model model	References
Sub-threshold voltage for time-dependent input current
Firing times for time-dependent input current
Firing Patterns in response to step current input

Experimental evidence supporting the model

Adaptive integrate-and-fire model model	References
Sub-threshold voltage for time-dependent input current
Firing times for time-dependent input current
Firing Patterns in response to step current input

= = = = = = = = { | class = “ wikable”style = “ float: right; margin-left: 15px”| + 支持该模型的实验证据！自适应集成火灾模型！参考文献 |-| 依赖时间的输入电流的亚阈值电压 | |-| 依赖时间的输入电流的触发时间 | |-| 响应步进电流输入的触发模式 | | }

Neuronal adaptation refers to the fact that even in the presence of a constant current injection into the soma, the intervals between output spikes increase. An adaptive integrate-and-fire neuron model combines the leaky integration of voltage V with one or several adaptation variables w_k (see Chapter 6.1. in the textbook Neuronal Dynamics)

神经元适应指的是这样一个事实，即即使在向躯体注入恒定电流的情况下，输出峰值之间的间隔也会增加。自适应积分-点火神经元模型结合了电压V的漏积分和一个或几个自适应变量wk(见第6.1章)。在《神经元动力学》教科书中)

Neuronal adaptation refers to the fact that even in the presence of a constant current injection into the soma, the intervals between output spikes increase. An adaptive integrate-and-fire neuron model combines the leaky integration of voltage with one or several adaptation variables (see Chapter 6.1. in the textbook Neuronal Dynamics)

神经元适应是指即使在体细胞内有恒定的电流注入，输出尖峰之间的间隔也会增加。自适应积分-火神经元模型将电压的漏综合与一个或多个自适应变量相结合(见第6.1章)。在《神经动力学》一书中

[math]\displaystyle{ \tau_\mathrm{m} \frac{d V_\mathrm{m} (t)}{d t} = R I(t)- [V_\mathrm{m} (t) - E_\mathrm{m} ]- R \sum_k w_k }[/math]

\tau_\mathrm{m} \frac{d V_\mathrm{m} (t)}{d t} = R I(t)- [V_\mathrm{m} (t) - E_\mathrm{m} ]- R \sum_k w_k

tau _ mathrm { m } frac { d _ mathrm { m }(t)}{ d } = r i (t)-[ v _ mathrm { m }(t)-e _ mathrm { m }]-r sum _ k w _ k

[math]\displaystyle{ \tau_k \frac{d w_k (t)}{d t} = - a_k [V_\mathrm{m} (t) - E_\mathrm{m} ]- w_k + b_k \tau_k \sum_f \delta (t-t^f) }[/math]

\tau_k \frac{d w_k (t)}{d t} = - a_k [V_\mathrm{m} (t) - E_\mathrm{m} ]- w_k + b_k \tau_k \sum_f \delta (t-t^f)

tau _ k frac { d _ k (t)}{ d t } =-a _ k [ v _ mathrm { m }(t)-e _ mathrm { m }]-w _ k + b _ k tau _ k sum _ f delta (t-t ^ f)

where [math]\displaystyle{ \tau_m }[/math] is the membrane time constant , w_k is the adaptation current number, with index k, [math]\displaystyle{ \tau_k }[/math] is the time constant of adaptation current w_k, E_m is the resting potential and t^f is the firing time of the neuron and the Greek delta denotes the Dirac delta function. Whenever the voltage reaches the firing threshold the voltage is reset to a value V_r below the firing threshold. The reset value is one of the important parameters of the model. The simplest model of adaptation has only a single adaptation variable w and the sum over k is removed.

在[数学]\ displaystyle {\ tau_m}[/数学]是膜时间常数,工作是适应当前数字,与指数k,[数学]\ displaystyle {\ tau_k}[/数学]是适应当前工作的时间常数,Em是静态电位和tf是神经元的点火时间和希腊δ表示狄拉克δ函数。当电压达到触发阈值时，电压被重置为低于触发阈值的Vr值。复位值是模型的重要参数之一。最简单的适应模型只有一个适应变量w，对k的和被移除了。

文件:Spike Time Prediction with Generalized Integrate-and-Fire model.jpg

Spike times and subthreshold voltage of cortical neuron models can be predicted by generalized integrate-and-fire models such as the adaptive integrate-and-fire model, the adaptive exponential integrate-and-fire model, or the spike response model. In the example here, adaptation is implemented by a dynamic threshold which increases after each spike.

Integrate-and-fire neurons with one or several adaptation variables can account for a variety of neuronal firing patterns in response to constant stimulation, including adaptation, bursting and initial bursting. Moreover, adaptive integrate-and-fire neurons with several adaptation variables are able to predict spike times of cortical neurons under time-dependent current injection into the soma.

具有一个或多个适应变量的整合-点火神经元可以解释对持续刺激的各种神经元点火模式，包括适应、爆发和初始爆发。此外，具有多种适应变量的自适应整合-点火神经元能够预测脑皮层神经元在时间依赖电流注入躯体后的峰突时间。

where \tau_m is the membrane time constant , is the adaptation current number, with index k, \tau_k is the time constant of adaptation current , is the resting potential and is the firing time of the neuron and the Greek delta denotes the Dirac delta function. Whenever the voltage reaches the firing threshold the voltage is reset to a value below the firing threshold. The reset value is one of the important parameters of the model. The simplest model of adaptation has only a single adaptation variable and the sum over k is removed. thumb|Spike times and subthreshold voltage of cortical neuron models can be predicted by generalized integrate-and-fire models such as the adaptive integrate-and-fire model, the adaptive exponential integrate-and-fire model, or the spike response model. In the example here, adaptation is implemented by a dynamic threshold which increases after each spike. Integrate-and-fire neurons with one or several adaptation variables can account for a variety of neuronal firing patterns in response to constant stimulation, including adaptation, bursting and initial bursting. Moreover, adaptive integrate-and-fire neurons with several adaptation variables are able to predict spike times of cortical neurons under time-dependent current injection into the soma.

其中 tau _ m 是膜时间常数，是适应电流数，指数 k，tau _ k 是适应电流的时间常数，是静息电位，是神经元的放电时间，希腊的 δ 表示狄拉克δ函数。当电压达到触发阈值时，电压复位到低于触发阈值的值。重置值是模型的重要参数之一。最简单的自适应模型只有一个单一的自适应变量，除去了 k 上的和。大脑皮层神经元模型的尖峰时间和阈下电压可以通过广义的积分-着火模型来预测，如自适应积分-着火模型、自适应指数积分-着火模型或尖峰反应模型。在这个例子中，自适应是通过一个动态阈值来实现的，该阈值在每次峰值之后都会增加。具有一个或多个适应变量的整合-放电神经元可以解释多种神经元放电模式对持续刺激的反应，包括适应、爆发和初次爆发。此外，具有多个适应变量的自适应整合-放电神经元能够预测时间依赖性电流注入体细胞时皮层神经元的放电峰时间。

Fractional-order leaky integrate-and-fire

Recent advances in computational and theoretical fractional calculus lead to a new form of model, called Fractional-order leaky integrate-and-fire. An advantage of this model is that it can capture adaptation effects with a single variable. The model has the following form

[math]\displaystyle{ I(t)-\frac{V_\mathrm{m} (t)}{R_\mathrm{m}} = C_\mathrm{m} \frac{d^{\alpha} V_\mathrm{m} (t)}{d^{\alpha} t} }[/math]

Once the voltage hits the threshold it is reset. Fractional integration has been used to account for neuronal adaptation in experimental data.

Recent advances in computational and theoretical fractional calculus lead to a new form of model, called Fractional-order leaky integrate-and-fire. An advantage of this model is that it can capture adaptation effects with a single variable. The model has the following form

一旦电压达到阈值就会被重置。分数整合已经被用来解释实验数据中的神经元适应。

随着分数阶微积分计算和理论的发展，提出了一种新的分数阶泄漏积分-点火模型。这个模型的一个优点是它可以用一个变量来捕捉适应效应。模型有如下形式

I(t)-\frac{V_\mathrm{m} (t)}{R_\mathrm{m}} = C_\mathrm{m} \frac{d^{\alpha} V_\mathrm{m} (t)}{d^{\alpha} t}

Once the voltage hits the threshold it is reset. Fractional integration has been used to account for neuronal adaptation in experimental data.

= = = = 分数阶泄漏积分与火焰 = = = 计算与理论分数微积分的最新进展导致了一种新的模型形式，称为分数阶泄漏积分与火焰。这个模型的一个优点是它可以用一个单一的变量来捕捉适应效果。该模型具有以下形式: i (t)-frac { v _ mathrm { m }(t)}{ r _ mathrm { m }} = c _ mathrm { m } frac { d ^ { alpha } v _ mathrm { m }(t)}{ d ^ { alpha } t }一旦电压达到阈值就被重置。分数整合已经被用来解释实验数据中的神经元适应。

'Exponential integrate-and-fire' and 'adaptive exponential integrate-and-fire'

Main article: Exponential integrate-and-fire

Experimental evidence supporting the model

Adaptive exponential integrate-and-fire	References
The sub-threshold current-voltage relation
Firing patterns in response to step current input
Refractoriness and adaptation

Experimental evidence supporting the model

Adaptive exponential integrate-and-fire	References
The sub-threshold current-voltage relation
Firing patterns in response to step current input
Refractoriness and adaptation

{ | class = “ wikitable”style = “ float: right; margin-left: 15px”| + 支持该模型的实验证据！自适应指数集成与开火！参考文献 |-| 亚阈值电流-电压关系 | |-| 触发模式响应步进电流输入 | |-| 耐火度和自适应 | | }

In the exponential integrate-and-fire model, spike generation is exponential, following the equation:

In the exponential integrate-and-fire model, spike generation is exponential, following the equation:

在指数积分-火力模型中，穗的产生是指数的，下面是一个等式:

[math]\displaystyle{ \frac{dV}{dt} - \frac{R} {\tau_m} I(t)= \frac{1} {\tau_m} \left[ E_m-V+\Delta_T \exp \left( \frac{V - V_T} {\Delta_T} \right) \right]. }[/math]

\frac{dV}{dt} - \frac{R} {\tau_m} I(t)= \frac{1} {\tau_m} \left[ E_m-V+\Delta_T \exp \left( \frac{V - V_T} {\Delta_T} \right) \right].

frac { dV }{ dt }-frac { r }{ tau _ m } i (t) = frac {1}{ tau _ m }左[ e _ m-v + Delta _ t exp 左(frac { v-v _ t }{ Delta _ t }右)]。

where [math]\displaystyle{ V }[/math] is the membrane potential, [math]\displaystyle{ V_T }[/math] is the intrinsic membrane potential threshold, [math]\displaystyle{ \tau_m }[/math] is the membrane time constant, [math]\displaystyle{ E_m }[/math]is the resting potential, and [math]\displaystyle{ \Delta_T }[/math] is the sharpness of action potential initiation, usually around 1 mV for cortical pyramidal neurons. Once the membrane potential crosses [math]\displaystyle{ V_T }[/math], it diverges to infinity in finite time. In numerical simulation the integration is stopped if the membrane potential hits an arbitrary threshold (much larger than [math]\displaystyle{ V_T }[/math]) at which the membrane potential is reset to a value V_r . The voltage reset value V_r is one of the important parameters of the model. Importantly, the right-hand side of the above equation contains a nonlinearity that can be directly extracted from experimental data. In this sense the exponential nonlinearity is strongly supported by experimental evidence.

where V is the membrane potential, V_T is the intrinsic membrane potential threshold, \tau_m is the membrane time constant, E_mis the resting potential, and \Delta_T is the sharpness of action potential initiation, usually around 1 mV for cortical pyramidal neurons. Once the membrane potential crosses V_T, it diverges to infinity in finite time. In numerical simulation the integration is stopped if the membrane potential hits an arbitrary threshold (much larger than V_T) at which the membrane potential is reset to a value . The voltage reset value is one of the important parameters of the model. Importantly, the right-hand side of the above equation contains a nonlinearity that can be directly extracted from experimental data. In this sense the exponential nonlinearity is strongly supported by experimental evidence.

其中 v 代表膜电位，v t 代表内在膜电位阈值，tau m 代表膜时间常数，e m 代表静息电位，Delta t 代表动作电位启动的敏锐度，通常在大脑皮层锥体神经元1 mV 左右。一旦膜电位与 v t 相交，它在有限时间内就会发散到无穷远处。在数值模拟中，如果膜电位达到一个任意的阈值(比 v _ t 大很多) ，在这个阈值上膜电位被重置为一个值，积分就停止了。电压复位值是模型的重要参数之一。重要的是，上述方程的右边含有非线性项，可以直接从实验数据中提取出来。在这个意义上，指数非线性是强烈支持的实验证据。

In the adaptive exponential integrate-and-fire neuron the above exponential nonlinearity of the voltage equation is combined with an adaptation variabe w

在自适应指数积分-点火神经元中，电压方程的上述指数非线性与自适应变量w相结合

In the adaptive exponential integrate-and-fire neuron the above exponential nonlinearity of the voltage equation is combined with an adaptation variabe w

在自适应指数积分-火神经元中，将电压方程的上述指数非线性特性与自适应变量相结合，构造了一个自适应指数积分-火神经元模型

[math]\displaystyle{ \tau_m \frac{dV}{dt} = R I(t) + \left[ E_m-V+\Delta_T \exp \left( \frac{V - V_T} {\Delta_T} \right) \right] - R w }[/math]

\tau_m \frac{dV}{dt} = R I(t) + \left[ E_m-V+\Delta_T \exp \left( \frac{V - V_T} {\Delta_T} \right) \right] - R w

τ _ m frac { dV }{ dt } = r i (t) + 左[ e _ m-v + Delta _ t exp left (frac { v-v _ t }{ Delta _ t }右)]-r w

[math]\displaystyle{ \tau \frac{d w (t)}{d t} = - a [V_\mathrm{m} (t) - E_\mathrm{m} ]- w + b \tau \delta (t-t^f) }[/math]

文件:Initial bursting AdEx model.png

Firing pattern of initial bursting in response to a step current input generated with the Adaptive exponential integrate-and-fire model. Other Firing patterns can also be generated.

where w denotes the adaptation current with time scale [math]\displaystyle{ \tau }[/math]. Important model parameters are the voltage reset value V_r, the intrinsic threshold [math]\displaystyle{ V_T }[/math], the time constants [math]\displaystyle{ \tau }[/math] and [math]\displaystyle{ \tau_m }[/math] as well as the coupling parameters a and b. The adaptive exponential integrate-and-fire model inherits the experimentally derived voltage nonlinearity of the exponential integrate-and-fire model. But going beyond this model, it can also account for a variety of neuronal firing patterns in response to constant stimulation, including adaptation, bursting and initial bursting. However, since the adaptation is in the form of a current, aberrant hyperpolarization may appear. This problem was solved by expressing it as a conductance.

\tau \frac{d w (t)}{d t} = - a [V_\mathrm{m} (t) - E_\mathrm{m} ]- w + b \tau \delta (t-t^f)

thumb|Firing pattern of initial bursting in response to a step current input generated with the Adaptive exponential integrate-and-fire model. Other Firing patterns can also be generated. where denotes the adaptation current with time scale \tau. Important model parameters are the voltage reset value , the intrinsic threshold V_T, the time constants \tau and \tau_m as well as the coupling parameters and . The adaptive exponential integrate-and-fire model inherits the experimentally derived voltage nonlinearity of the exponential integrate-and-fire model. But going beyond this model, it can also account for a variety of neuronal firing patterns in response to constant stimulation, including adaptation, bursting and initial bursting. However, since the adaptation is in the form of a current, aberrant hyperpolarization may appear. This problem was solved by expressing it as a conductance.

tau frac { d (t)}{ dt } =-a [ v _ mathrm { m }(t)-e _ mathrm { m }]-w + b tau delta (t-t ^ f) thumb | 初始爆炸模式，响应用自适应指数积分-点火模型产生的阶跃电流输入。还可以生成其他触发模式。其中表示适应电流与时间尺度 tau。重要的模型参数包括电压复位值、固有阈值 v _ t、时间常数 tau 和 tau _ m 以及耦合参数和电流密度。自适应指数积分-着火模型继承了指数积分-着火模型的实验导出的电压非线性。但是，超越这个模型，它也可以解释各种神经元的放电模式，以响应不断的刺激，包括适应，爆发和初始爆发。然而，由于这种适应是以电流的形式出现的，可能会出现异常的超极化。把它表示为电导就解决了这个问题。

Stochastic models of membrane voltage and spike timing

The models in this category are generalized integrate-and-fire models that include a certain level of stochasticity. Cortical neurons in experiments are found to respond reliably to time-dependent input, albeit with a small degree of variations between one trial and the next if the same stimulus is repeated. Stochasticity in neurons has two important sources. First, even in a very controlled experiment where input current is injected directly into the soma, ion channels open and close stochastically and this channel noise leads to a small amount of variability in the exact value of the membrane potential and the exact timing of output spikes. Second, for a neuron embedded in a cortical network, it is hard to control the exact input because most inputs come from unobserved neurons somewhere else in the brain.

这类模型是广义的集成-触发模型，包含一定程度的随机性。实验发现，皮层神经元对依赖时间的输入有可靠的反应，尽管如果重复相同的刺激，在一次和下一次试验之间会有小程度的变化。神经元中的随机性有两个重要来源。首先，即使是在一个非常可控的实验中，直接将输入电流注入躯体，离子通道随机打开和关闭，这种通道噪声导致膜电位的精确值和输出峰值的准确时间有少量的变化。其次，对于嵌入皮层网络的神经元来说，很难控制准确的输入，因为大多数输入来自大脑中其他未被观察到的神经元。

The models in this category are generalized integrate-and-fire models that include a certain level of stochasticity. Cortical neurons in experiments are found to respond reliably to time-dependent input, albeit with a small degree of variations between one trial and the next if the same stimulus is repeated. Stochasticity in neurons has two important sources. First, even in a very controlled experiment where input current is injected directly into the soma, ion channels open and close stochastically and this channel noise leads to a small amount of variability in the exact value of the membrane potential and the exact timing of output spikes. Second, for a neuron embedded in a cortical network, it is hard to control the exact input because most inputs come from unobserved neurons somewhere else in the brain.

= = = 随机模型的膜电位和脉冲时间 = = 这类模型是广义积分和火模型，包括一定水平的随机性。实验中发现，皮层神经元对时间依赖性输入的反应是可靠的，尽管在同一刺激重复的情况下，两个实验之间会有一定程度的差异。神经元的随机性有两个重要来源。首先，即使是在一个非常可控的实验中，输入电流直接被随机地注入到胞体中，离子通道的开启和关闭，这种通道噪声导致膜电位的精确值和输出尖峰的精确时间有少量的变化。其次，对于嵌入皮层网络中的神经元来说，很难控制精确的输入，因为大多数输入来自大脑中其他地方未被观察到的神经元。

Stochasticity has been introduces into spiking neuron models in two fundamentally different forms: either (i) a noisy input current is added to the differential equation of the neuron model; or (ii) the process of spike generation is noisy. In both cases, the mathematical theory can be developed for continuous time, which is then, if desired for the use in computer simulations, transformed into a discrete-time model.

随机特性以两种根本不同的形式引入到尖峰神经元模型中:一种是(i)在神经元模型的微分方程中加入一个噪声输入电流;或者(2)脉冲产生的过程是有噪声的。在这两种情况下，数学理论都可以发展为连续时间，然后，如果需要在计算机模拟中使用，转换成离散时间模型。

Stochasticity has been introduces into spiking neuron models in two fundamentally different forms: either (i) a noisy input current is added to the differential equation of the neuron model; or (ii) the process of spike generation is noisy. In both cases, the mathematical theory can be developed for continuous time, which is then, if desired for the use in computer simulations, transformed into a discrete-time model.

随机性以两种截然不同的形式被引入到尖峰神经元模型中: 要么(i)在神经元模型的微分方程中加入噪声输入电流; 要么(ii)尖峰产生的过程是噪声的。在这两种情况下，数学理论可以发展为连续时间，然后，如果希望在计算机模拟中使用，转换成一个离散时间模型。

The relation of noise in neuron models to variability of spike trains and neural codes is discussed in Neural Coding and in Chapter 7 of the textbook Neuronal Dynamics.

神经元模型中的噪声与脉冲序列和神经编码变异性的关系在神经编码和《神经动力学》教材的第7章中进行了讨论。

The relation of noise in neuron models to variability of spike trains and neural codes is discussed in Neural Coding and in Chapter 7 of the textbook Neuronal Dynamics.

在《神经编码》和《神经动力学》教科书第7章中讨论了神经元模型中噪声与神经元序列和编码变异性的关系。

Noisy input model (diffusive noise) 模板:Anchor

A neuron embedded in a network receives spike input from other neurons. Since the spike arrival times are not controlled by an experimentalist they can be considered as stochastic. Thus a (potentially nonlinear) integrate-and-fire model with nonlinearity f(v) receives two inputs: an input [math]\displaystyle{ I(t) }[/math] controlled by the experimentalists and a noisy input current [math]\displaystyle{ I^{\rm noise}(t) }[/math] that describes the uncontrolled background input.

网络中的一个神经元接收来自其他神经元的脉冲输入。由于峰值到达时间不是由实验主义者控制的，因此可以认为它们是随机的。因此，具有非线性f(v)的(潜在非线性)积分-触发模型接收到两个输入:由实验人员控制的输入[math]\displaystyle{I(t)}[/math]和描述非受控背景输入的有噪声的输入电流[math]\displaystyle{I^{\rm噪声}(t)}[/math]。

A neuron embedded in a network receives spike input from other neurons. Since the spike arrival times are not controlled by an experimentalist they can be considered as stochastic. Thus a (potentially nonlinear) integrate-and-fire model with nonlinearity f(v) receives two inputs: an input I(t) controlled by the experimentalists and a noisy input current I^{\rm noise}(t) that describes the uncontrolled background input.

= = = = 噪声输入模型(扩散噪声) = = = 嵌入在网络中的神经元接收来自其他神经元的尖峰输入。由于尖峰到达时间不受实验者控制，因此可以认为尖峰到达时间是随机的。因此，一个具有非线性的(潜在的非线性)积分-着火模型接受两个输入: 由实验者控制的输入 i (t)和描述非控制背景输入的噪声输入电流 i ^ { rm 噪声}(t)。

[math]\displaystyle{ \tau_m \frac{dV}{dt} = f(V) + R I(t) + R I^\text{noise}(t) }[/math]

\tau_m \frac{dV}{dt} = f(V) + R I(t) + R I^\text{noise}(t)

{ dV }{ dt } = f (v) + r i (t) + r i ^ text { noise }(t)

Stein's model is the special case of a leaky integrate-and-fire neuron and a stationary white noise current [math]\displaystyle{ I^{\rm noise}(t) = \xi(t) Stein's model is the special case of a leaky integrate-and-fire neuron and a stationary white noise current \lt math\gt I^{\rm noise}(t) = \xi(t) 斯坦因模型是一个漏泄的积分-激发神经元和一个静止的白噪声电流 \lt math \gt i ^ { rm 噪声}(t) = xi (t)的特例 }[/math] with mean zero and unit variance. In the subthreshold regime, these assumptions yield the equation of the Ornstein–Uhlenbeck process

</math> with mean zero and unit variance. In the subthreshold regime, these assumptions yield the equation of the Ornstein–Uhlenbeck process

</math > 均值为零，方差为单位。在亚阈值情况下，这些假设得到 Ornstein-Uhlenbeck 过程方程

[math]\displaystyle{ \tau_m \frac{dV}{dt} = [E_m-V] + R I(t) + R \xi(t) }[/math]

\tau_m \frac{dV}{dt} = [E_m-V] + R I(t) + R \xi(t)

\tau_m \frac{dV}{dt} = [E_m-V] + R I(t) + R \xi(t)

However, in contrast to the standard Ornstein–Uhlenbeck process, the membrane voltage is reset whenever V hits the firing threshold V_th . Calculating the interval distribution of the Ornstein–Uhlenbeck model for constant input with threshold leads to a first-passage time problem. Stein's neuron model and variants thereof have been used to fit interspike interval distributions of spike trains from real neurons under constant input current.

然而，与标准的奥恩斯坦-乌伦贝克过程相反，当V达到触发阈值Vth时，膜电压被重置。计算具有阈值的常数输入的Ornstein-Uhlenbeck模型的区间分布会导致首通过时间问题。Stein的神经元模型及其变体已被用于拟合恒定输入电流下真实神经元脉冲序列的脉冲间区间分布。

However, in contrast to the standard Ornstein–Uhlenbeck process, the membrane voltage is reset whenever V hits the firing threshold . Calculating the interval distribution of the Ornstein–Uhlenbeck model for constant input with threshold leads to a first-passage time problem. Stein's neuron model and variants thereof have been used to fit interspike interval distributions of spike trains from real neurons under constant input current.

然而，与标准的 Ornstein-Uhlenbeck 过程不同的是，每当 v 达到起爆阈值时，膜电位就会被重置。计算 Ornstein-Uhlenbeck 模型中带阈值的常数输入的区间分布导致了一个首次通过时间问题。本文利用 Stein 的神经元模型及其变体，拟合了恒定输入电流下实际神经元电刺激序列的峰间期分布。

In the mathematical literature, the above equation of the Ornstein–Uhlenbeck process is written in the form

In the mathematical literature, the above equation of the Ornstein–Uhlenbeck process is written in the form

在数学文献中，奥恩斯坦-乌伦贝克过程的上述方程是以这种形式写成的

[math]\displaystyle{ dV = [E_m-V + R I(t)] \frac{dt}{\tau_m} + \sigma \, dW }[/math]

dV = [E_m-V + R I(t)] \frac{dt}{\tau_m} + \sigma \, dW

dV = [E_m-V + R I(t)] \frac{dt}{\tau_m} + \sigma \, dW

where [math]\displaystyle{ \sigma where \lt math\gt \sigma 在哪里 }[/math] is the amplitude of the noise input and dW are increments of a Wiener process. For discrete-time implementations with time step dt the voltage updates are

</math> is the amplitude of the noise input and dW are increments of a Wiener process. For discrete-time implementations with time step dt the voltage updates are

</math > 是噪声输入的振幅，dW 是维纳过程的增量。对于具有时间步长 dt 的离散时间实现，电压更新为

[math]\displaystyle{ \Delta V = [E_m-V + R I(t)] \frac{\Delta t}{\tau_m} + \sigma \sqrt{\tau_m}y }[/math]

\Delta V = [E_m-V + R I(t)] \frac{\Delta t}{\tau_m} + \sigma \sqrt{\tau_m}y

Delta v = [ e_m-v + r i (t)] frac { Delta t }{ tau _ m } + sigma sqrt { tau _ m } y

where y is drawn from a Gaussian distribution with zero mean unit variance. The voltage is reset when it hits the firing threshold V_th .

其中y来自于单位均值方差为零的高斯分布。当电压达到触发阈值Vth时，电压被重置。

where y is drawn from a Gaussian distribution with zero mean unit variance. The voltage is reset when it hits the firing threshold .

其中 y 是从均值单位方差为零的正态分布中得出的。当电压达到触发阈值时，电压被复位。

The noisy input model can also be used in generalized integrate-and-fire models. For example, the exponential integrate-and-fire model with noisy input reads

噪声输入模型也可用于广义的积分-火灾模型。例如，带噪声输入的指数积分-点火模型

The noisy input model can also be used in generalized integrate-and-fire models. For example, the exponential integrate-and-fire model with noisy input reads

噪声输入模型也可用于广义积分-火灾模型。例如，带有噪声输入读数的指数积分-点火模型

[math]\displaystyle{ \tau_m \frac{dV}{dt} =E_m-V+\Delta_T \exp \left( \frac{V - V_T} {\Delta_T} \right) + R I(t) + R\xi(t) }[/math]

\tau_m \frac{dV}{dt} =E_m-V+\Delta_T \exp \left( \frac{V - V_T} {\Delta_T} \right) + R I(t) + R\xi(t)

τ _ m frac { dV }{ dt } = e _ m-v + Delta _ t exp left (frac { v-v _ t }{ Delta _ t }右) + r i (t) + r xi (t)

For constant deterministic input [math]\displaystyle{ I(t)=I_0 }[/math] it is possible to calculate the mean firing rate as a function of [math]\displaystyle{ I_0 }[/math]. This is important because the frequency-current relation (f-I-curve) is often used by experimentalists to characterize a neuron. It is also the transfer function in

对于恒定的确定性输入[math]\displaystyle{I(t)=I_0}[/math]，可以将平均发射率作为[math]\displaystyle{I_0}[/math]的函数来计算。这是很重要的，因为频率-电流关系(f- i曲线)经常被实验人员用来描述一个神经元。这也是传递函数

For constant deterministic input I(t)=I_0 it is possible to calculate the mean firing rate as a function of I_0. This is important because the frequency-current relation (f-I-curve) is often used by experimentalists to characterize a neuron. It is also the transfer function in

对于常数确定性输入 i (t) = i _ 0，可以计算平均激发率作为 i _ 0的函数。这一点很重要，因为频率-电流关系(f-i 曲线)经常被实验人员用来描述一个神经元。它也是传递函数

The leaky integrate-and-fire with noisy input has been widely used in the analysis of networks of spiking neurons. Noisy input is also called 'diffusive noise' because it leads to a diffusion of the subthreshold membrane potential around the noise-free trajectory (Johannesma, The theory of spiking neurons with noisy input is reviewed in Chapter 8.2 of the textbook Neuronal Dynamics.

带噪声输入的泄漏积分-点火算法被广泛应用于尖峰神经元网络的分析中。噪声输入也被称为“扩散噪声”，因为它导致阈下膜电位在无噪声轨道周围的扩散(Johannesma，带噪声输入的尖峰神经元理论在《神经动力学》课本第8.2章中进行了综述。

The leaky integrate-and-fire with noisy input has been widely used in the analysis of networks of spiking neurons. Noisy input is also called 'diffusive noise' because it leads to a diffusion of the subthreshold membrane potential around the noise-free trajectory (Johannesma, The theory of spiking neurons with noisy input is reviewed in Chapter 8.2 of the textbook Neuronal Dynamics.

在脉冲神经元网络的分析中，有噪声输入的漏泄积分-着火算法得到了广泛的应用。噪声输入也被称为“扩散噪声”，因为它导致亚阈值膜电位在无噪声轨迹周围扩散(约翰内斯马，在神经元动力学教科书第8.2章中回顾了噪声输入刺激神经元的理论。

Noisy output model (escape noise)模板:Anchor

In deterministic integrate-and-fire models, a spike is generated if the membrane potential V(t) hits the threshold [math]\displaystyle{ V_{th} }[/math]. In noisy output models the strict threshold is replaced by a noisy one as follows. At each moment in time t, a spike is generated stochastically with instantaneous stochastic intensity or 'escape rate'

在确定性积分-点火模型中，如果膜电位V(t)达到阈值[math]\displaystyle{V_{th}}[/math]，就会产生一个脉冲。在有噪声的输出模型中，严格阈值被一个有噪声的阈值代替，如下所示。在t时刻的每一时刻，以瞬时随机强度或“逃逸率”随机产生一个脉冲。

In deterministic integrate-and-fire models, a spike is generated if the membrane potential hits the threshold V_{th}. In noisy output models the strict threshold is replaced by a noisy one as follows. At each moment in time t, a spike is generated stochastically with instantaneous stochastic intensity or 'escape rate'

在确定性积分-火灾模型中，如果膜电位达到阈值 v { th } ，就会产生尖峰。在噪声输出模型中，严格的阈值被一个噪声阈值代替，如下所示。在时间 t 的每个时刻，随机产生一个峰值，即瞬时随机强度或“逃逸率”

[math]\displaystyle{ \rho(t) = f(V(t)-V_{th}) }[/math]

\rho(t) = f(V(t)-V_{th})

rho (t) = f (v (t)-v _ { th })

that depends on the momentary difference between the membrane voltage V(t) and the threshold [math]\displaystyle{ V_{th} }[/math]. A common choice for the 'escape rate' [math]\displaystyle{ f }[/math] (that is consistent with biological data) is

that depends on the momentary difference between the membrane voltage and the threshold V_{th}. A common choice for the 'escape rate' f (that is consistent with biological data) is

这取决于膜电位和阈值 v { th }之间的瞬间差异。逃逸率 f (与生物学数据一致)的一个常见选择是

[math]\displaystyle{ f(V-V_{th}) = \frac{1}{\tau_0} \exp[\beta(V-V_{th}] }[/math]

文件:Spike Response Model with escape noise (noisy output).jpg

Stochastic spike generation (noisy output) depends on the momentary difference between the membrane potential V(t) and the threshold. The membrane potential V of the spike response model (SRM) has two contributions. First, input current I is filtered by a first filter k. Second the sequence of output spikes S(t) is filtered by a second filter η and fed back. The resulting membrane V(t) potential is used to generate output spikes by a stochastic process ρ(t) with an intensity that depends on the distance between membrane potential and threshold. The spike response model (SRM) is closely related to the Generalized Linear Model (GLM).

where [math]\displaystyle{ \tau_0 }[/math]is a time constant that describes how quickly a spike is fired once the membrane potential reaches the threshold and [math]\displaystyle{ \beta }[/math] is a sharpness parameter. For [math]\displaystyle{ \beta\to\infty }[/math] the threshold becomes sharp and spike firing occurs deterministically at the moment when the membrane potential hits the threshold from below. The sharpness value found in experiments is [math]\displaystyle{ 1/\beta\approx 4mV }[/math] which means that neuronal firing becomes non-negligible as soon the membrane potential is a few mV below the formal firing threshold.

f(V-V_{th}) = \frac{1}{\tau_0} \exp[\beta(V-V_{th}]

thumb|Stochastic spike generation (noisy output) depends on the momentary difference between the membrane potential V(t) and the threshold. The membrane potential V of the spike response model (SRM) has two contributions. First, input current I is filtered by a first filter k. Second the sequence of output spikes S(t) is filtered by a second filter η and fed back. The resulting membrane V(t) potential is used to generate output spikes by a stochastic process ρ(t) with an intensity that depends on the distance between membrane potential and threshold. The spike response model (SRM) is closely related to the Generalized Linear Model (GLM). where \tau_0is a time constant that describes how quickly a spike is fired once the membrane potential reaches the threshold and \beta is a sharpness parameter. For \beta\to\infty the threshold becomes sharp and spike firing occurs deterministically at the moment when the membrane potential hits the threshold from below. The sharpness value found in experiments is 1/\beta\approx 4mV which means that neuronal firing becomes non-negligible as soon the membrane potential is a few mV below the formal firing threshold.

f (V-V { th }) = frac {1}{ tau _ 0} exp [ beta (V-V { th }] thumb | 随机脉冲产生(噪声输出)取决于膜电位 v (t)和阈值之间的瞬时差。尖峰反应模型(SRM)的膜电位 v 有两个贡献。首先，输入电流 i 用第一个滤波器滤波，然后输出尖峰序列 s (t)用第二个滤波器 η 滤波并反馈。由此产生的膜电位 v (t)被用来产生一个随机过程 ρ (t)的输出尖峰，其强度取决于膜电位和阈值之间的距离。尖峰反应模型(SRM)与广义线性模式反应模型(GLM)密切相关。其中 tau _ 0是一个时间常数，用来描述当膜电位达到阈值时尖峰发射的速度，而 beta 是一个锐度参数。对于贝塔系数来说，阈值变得尖锐，当膜电位从下面击中阈值时，尖峰放电就确定地发生了。在实验中发现的清晰度值是1/beta 大约4mv，这意味着神经元的放电变得不可忽视，只要膜电位低于正式放电阈值几 mV。

The escape rate process via a soft threshold is reviewed in Chapter 9 of the textbook Neuronal Dynamics.

通过软阈值的逃逸率过程在教科书《神经动力学》的第9章中进行了回顾。

The escape rate process via a soft threshold is reviewed in Chapter 9 of the textbook Neuronal Dynamics.

通过软阈值的逃逸速率过程在神经元动力学教科书的第9章中进行了回顾。

For models in discrete time, a spike is generated with probability

对于离散时间模型，有可能产生一个脉冲

For models in discrete time, a spike is generated with probability

对于离散时间模型，用概率生成尖峰

[math]\displaystyle{ P_F(t_n) = F[V(t_n)-V_{th}] }[/math]

P_F(t_n) = F[V(t_n)-V_{th}]

p _ f (t _ n) = f [ v (t _ n)-v _ { th }]

that depends on the momentary difference between the membrane voltage V at time [math]\displaystyle{ t_n }[/math] and the threshold [math]\displaystyle{ V_{th} }[/math]. The function F is often taken as a standard sigmoidal [math]\displaystyle{ F(x) = 0.5[1 + \tanh(\gamma x)] }[/math] with steepness parameter [math]\displaystyle{ \gamma }[/math], similar to the update dynamics in artificial neural networks. But the functional form of F can also be derived from the stochastic intensity [math]\displaystyle{ f }[/math] in continuous time introduced above as [math]\displaystyle{ F(y_n)\approx 1 - \exp[y_n\Delta t] }[/math] where [math]\displaystyle{ y_n = V(t_n)-V_{th} }[/math] is the distance to threshold.

that depends on the momentary difference between the membrane voltage at time t_n and the threshold V_{th}. The function F is often taken as a standard sigmoidal F(x) = 0.5[1 + \tanh(\gamma x)] with steepness parameter \gamma, similar to the update dynamics in artificial neural networks. But the functional form of F can also be derived from the stochastic intensity f in continuous time introduced above as F(y_n)\approx 1 - \exp[y_n\Delta t] where y_n = V(t_n)-V_{th} is the distance to threshold.

这取决于时间 t n 和阈值 v { th }之间膜电位的瞬间差异。函数 f 通常被看作是具有陡度参数 γ 的标准 sigmoidal 函数 f (x) = 0.5[1 + tanh (gamma x)] ，类似于人工神经网络的更新动力学。但是 f 的函数形式也可以从 f (y _ n)近1-exp [ y _ n △ t ]中引入的连续时间内的随机强度 f 推导出来，其中 y _ n = v (t _ n)-v _ (th)为到阈值的距离。

Integrate-and-fire models with output noise can be used to predict the PSTH of real neurons under arbitrary time-dependent input. For non-adaptive integrate-and-fire neurons, the interval distribution under constant stimulation can be calculated from stationary renewal theory.

带输出噪声的积分-点火模型可用于预测任意时变输入下真实神经元的PSTH。对于非自适应的积分-点火神经元，利用平稳更新理论可以计算出其在恒定刺激下的区间分布。

Integrate-and-fire models with output noise can be used to predict the PSTH of real neurons under arbitrary time-dependent input. For non-adaptive integrate-and-fire neurons, the interval distribution under constant stimulation can be calculated from stationary renewal theory.

利用输出噪声的积分-着火模型可以预测任意时变输入下实际神经元的 PSTH。对于非自适应整合-放电神经元，可以利用平稳更新理论计算恒定刺激下的区间分布。

= = 尖峰反应模型(SRM) = =

Experimental evidence supporting the model

Spike response model	References
Sub-threshold voltage for time-dependent input current
Firing times for time-dependent input current
Firing Patterns in response to step current input
Interspike interval distribution
Spike-afterpotential
refractoriness and dynamic firing threshold

Experimental evidence supporting the model

Spike response model	References
Sub-threshold voltage for time-dependent input current
Firing times for time-dependent input current
Firing Patterns in response to step current input
Interspike interval distribution
Spike-afterpotential
refractoriness and dynamic firing threshold

{ | class = “ wikitable”style = “ float: right; margin-left: 15px”| + 支持该模型的实验证据！尖峰反应模型！参考文献 |-| 依赖时间的输入电流的亚阈值电压 | |-| 依赖时间的输入电流的触发时间 | |-| 触发模式响应步进电流的输入 | | | | |-| Interspike 间隔分布 | | | |-| 尖峰-后电位 | |-| 耐火度和动态触发阈值 | | | }

main article: Spike response model

main article: Spike response model

主要文章: 穗反应模型

The spike response model (SRM) is a general linear model for the subthreshold membrane voltage combined with a nonlinear output noise process for spike generation. The membrane voltage V(t) at time t is

脉冲响应模型(SRM)是阈下膜电压与非线性输出噪声过程相结合的一般线性模型。t时刻的膜电压V(t)为

The spike response model (SRM) is a general linear model for the subthreshold membrane voltage combined with a nonlinear output noise process for spike generation. The membrane voltage at time t is

尖峰响应模型(SRM)是亚阈值噪声与非线性输出噪声过程相结合产生尖峰的一般线性模型膜电位。时间 t 的膜电位

[math]\displaystyle{ V(t)= \sum_f \eta(t-t^f) + \int_0^\infty \kappa(s) I(t-s)\,ds + V_\mathrm{rest} }[/math]

V(t)= \sum_f \eta(t-t^f) + \int_0^\infty \kappa(s) I(t-s)\,ds + V_\mathrm{rest}

v (t) = sum _ f eta (t-t ^ f) + int _ 0 ^ infty kappa (s) i (t-s) ，ds + v _ mathrm { rest }

where t^f is the firing time of spike number f of the neuron, V_rest is the resting voltage in the absence of input, I(t-s) is the input current at time t-s and [math]\displaystyle{ \kappa(s) }[/math] is a linear filter (also called kernel) that describes the contribution of an input current pulse at time t-s to the voltage at time t. The contributions to the voltage caused by a spike at time [math]\displaystyle{ t^f }[/math] are described by the refractory kernel [math]\displaystyle{ \eta(t-t^f) }[/math]. In particular, [math]\displaystyle{ \eta(t-t^f) }[/math] describes the reset after the spike and the time course of the spike-afterpotential following a spike. It therefore expresses the consequences of refractoriness and adaptation. The voltage V(t) can be interpreted as the result of an integration of the differential equation of a leaky integrate-and-fire model coupled to an arbitrary number of spike-triggered adaptation variables.

where is the firing time of spike number f of the neuron, is the resting voltage in the absence of input, is the input current at time t-s and \kappa(s) is a linear filter (also called kernel) that describes the contribution of an input current pulse at time t-s to the voltage at time t. The contributions to the voltage caused by a spike at time t^f are described by the refractory kernel \eta(t-t^f). In particular, \eta(t-t^f) describes the reset after the spike and the time course of the spike-afterpotential following a spike. It therefore expresses the consequences of refractoriness and adaptation. The voltage V(t) can be interpreted as the result of an integration of the differential equation of a leaky integrate-and-fire model coupled to an arbitrary number of spike-triggered adaptation variables.

其中神经元尖峰数 f 的点火时间，是没有输入时的静止电压，是时间 t-s 处的输入电流，kappa (s)是一个线性滤波器(也称为核) ，它描述了时间 t-s 处的输入电流脉冲对时间 t 处电压的贡献。难熔核 eta (t-t ^ f)描述了 t ^ f 时尖峰对电压的贡献。特别地，eta (t-t ^ f)描述了尖峰后的复位和尖峰后电位的时间过程。因此，它表达了难耐性和适应性的后果。电压 v (t)可以被解释为一个渗漏的积分-着火模型的微分方程与一个任意数目的尖峰触发适应变量相结合的结果。

Spike firing is stochastic and happens with a time-dependent stochastic intensity (instantaneous rate)

Spike firing is stochastic and happens with a time-dependent stochastic intensity (instantaneous rate)

脉冲放电是随机的，并且具有随时间变化的随机强度(瞬时速率)

[math]\displaystyle{ f(V-\vartheta(t)) = \frac{1}{\tau_0} \exp[\beta(V-\vartheta(t)] }[/math]

f(V-\vartheta(t)) = \frac{1}{\tau_0} \exp[\beta(V-\vartheta(t)]

f (v-vartheta (t)) = frac {1}{ tau _ 0} exp [ beta (v-vartheta (t)]

with parameters [math]\displaystyle{ \tau_0 }[/math] and [math]\displaystyle{ \beta }[/math] and a dynamic threshold [math]\displaystyle{ \vartheta(t) }[/math] given by

with parameters \tau_0 and \beta and a dynamic threshold \vartheta(t) given by

参数 tau _ 0和 beta 和动态阈值 vartheta (t)由

[math]\displaystyle{ \vartheta(t)= \vartheta_0 + \sum_f \theta_1(t-t^f) }[/math]

\vartheta(t)= \vartheta_0 + \sum_f \theta_1(t-t^f)

\vartheta(t)= \vartheta_0 + \sum_f \theta_1(t-t^f)

Here [math]\displaystyle{ \vartheta_0 }[/math] is the firing threshold of an inactive neuron and [math]\displaystyle{ \theta_1(t-t^f) }[/math] describes the increase of the threshold after a spike at time [math]\displaystyle{ t^f }[/math]. In case of a fixed threshold, one sets [math]\displaystyle{ \theta_1(t-t^f) }[/math]=0. For [math]\displaystyle{ \beta \to \infty }[/math] the threshold process is deterministic.

Here \vartheta_0 is the firing threshold of an inactive neuron and \theta_1(t-t^f) describes the increase of the threshold after a spike at time t^f. In case of a fixed threshold, one sets \theta_1(t-t^f)=0. For \beta \to \infty the threshold process is deterministic.

在这里，vartheta _ 0是非活动神经元的放电阈值，theta _ 1(t-t ^ f)描述了 t ^ f 时放电后阈值的增加。在固定阈值的情况下，设置 theta _ 1(t-t ^ f) = 0。对于贝塔来说，阈值过程是确定的。

The time course of the filters [math]\displaystyle{ \eta,\kappa,\theta_1 }[/math] that characterize the spike response model can be directly extracted from experimental data. With optimized parameters the SRM describes the time course of the subthreshold membrane voltage for time-dependent input with a precision of 2mV and can predict the timing of most output spikes with a precision of 4ms. The SRM is closely related to linear-nonlinear-Poisson cascade models (also called Generalized Linear Model). The estimation of parameters of probabilistic neuron models such as the SRM using methods developed for Generalized Linear Models is discussed in Chapter 10 of the textbook Neuronal Dynamics.

文件:Spike Response Model (schematic).jpg

Spike arrival causes postsynaptic potentials (red lines) which are summed. If the total voltage V reaches a threshold (dashed blue line) a spike is initiated (green) which also includes a spike-afterpotential. The threshold increases after each spike. Postsynaptic potentials are the response to incoming spikes while the spike-afterpotential is the response to outgoing spikes.

The name spike response model arises because in a network, the input current for neuron i is generated by the spikes of other neurons so that in the case of a network the voltage equation becomes

The time course of the filters \eta,\kappa,\theta_1 that characterize the spike response model can be directly extracted from experimental data. With optimized parameters the SRM describes the time course of the subthreshold membrane voltage for time-dependent input with a precision of 2mV and can predict the timing of most output spikes with a precision of 4ms. The SRM is closely related to linear-nonlinear-Poisson cascade models (also called Generalized Linear Model). The estimation of parameters of probabilistic neuron models such as the SRM using methods developed for Generalized Linear Models is discussed in Chapter 10 of the textbook Neuronal Dynamics. thumb|Spike arrival causes postsynaptic potentials (red lines) which are summed. If the total voltage V reaches a threshold (dashed blue line) a spike is initiated (green) which also includes a spike-afterpotential. The threshold increases after each spike. Postsynaptic potentials are the response to incoming spikes while the spike-afterpotential is the response to outgoing spikes. The name spike response model arises because in a network, the input current for neuron i is generated by the spikes of other neurons so that in the case of a network the voltage equation becomes

可以直接从实验数据中提取出表征尖峰反应模型的滤波器 eta，kappa，theta _ 1的时间进程。通过优化参数，SRM 以2 mv 的精度描述了亚阈值膜电位对依赖时间的输入的时间过程，并以4 ms 的精度预测了大多数输出峰值的时间。SRM 与线性-非线性-泊松级联模型(也称为广义线性模式模型)密切相关。在神经元动力学教科书第10章中，讨论了使用为广义线性模型开发的方法估计概率神经元模型(如 SRM)的参数。拇指 | 脉冲到达引起突触后电位(红线)。如果总电压 v 达到一个阈值(虚线蓝线)一个尖峰被启动(绿色) ，这也包括一个尖峰后电位。阈值增加后，每个峰值。突触后电位是对传入尖峰的反应，而尖峰后电位是对传出尖峰的反应。由于神经元 i 的输入电流是由其他神经元的尖峰产生的，因此在网络的情况下，电压方程就变成了

[math]\displaystyle{ V_i(t)= \sum_f \eta_i(t-t_i^f) + \sum_{j=1}^N w_{ij} \sum_{f'}\varepsilon_{ij}(t-t_j^{f'}) + V_\mathrm{rest} }[/math]

V_i(t)= \sum_f \eta_i(t-t_i^f) + \sum_{j=1}^N w_{ij} \sum_{f'}\varepsilon_{ij}(t-t_j^{f'}) + V_\mathrm{rest}

v _ i (t) = sum _ f eta _ i (t-t _ i ^ f) + sum _ { j = 1} n w { ij } sum _ { f’} varepsilon _ { ij }(t-t _ j ^ { f’}) + v _ mathrm { rest }

where [math]\displaystyle{ t_j^{f'} }[/math] are the firing times of neuron j (i.e., its spike train) , and [math]\displaystyle{ \eta_i(t-t^f_i) }[/math] describes the time course of the spike and the spike after-potential for neuron i, [math]\displaystyle{ w_{ij} }[/math] and [math]\displaystyle{ \varepsilon_{ij}(t-t_j^{f'}) }[/math] describe the amplitude and time course of an excitatory or inhibitory postsynaptic potential (PSP) caused by the spike [math]\displaystyle{ t_j^{f'} }[/math] of the presynaptic neuron j. The time course [math]\displaystyle{ \varepsilon_{ij}(s) }[/math] of the PSP results from the convolution of the postsynaptic current [math]\displaystyle{ I(t) }[/math] caused by the arrival of a presynaptic spike from neuron j with the membrane filter [math]\displaystyle{ \kappa(s) }[/math].

where t_j^{f'} are the firing times of neuron j (i.e., its spike train) , and \eta_i(t-t^f_i) describes the time course of the spike and the spike after-potential for neuron i, w_{ij} and \varepsilon_{ij}(t-t_j^{f'}) describe the amplitude and time course of an excitatory or inhibitory postsynaptic potential (PSP) caused by the spike t_j^{f'} of the presynaptic neuron j. The time course \varepsilon_{ij}(s) of the PSP results from the convolution of the postsynaptic current I(t) caused by the arrival of a presynaptic spike from neuron j with the membrane filter \kappa(s).

其中 t _ j ^ { f’}是神经元 j (即其尖峰列车)的放电次数，而 eta _ i (t-t ^ f)描述了神经元 i 的尖峰和尖峰后电位的时间过程，w _ { ij }和 varepsilon _ { ij }(t-t _ j ^ { f’})描述了由突触前神经元 j _ j ^ { f’}引起的兴奋性或抑制性突触(PSP)的振幅和时间过程。PSP 的时间进程变化是突触后电流 i (t)与膜滤器 kappa (s)的卷积作用的结果。

SRM0 模板:Anchor

The SRM0 is a stochastic neuron model related to time-dependent nonlinear renewal theory and a simplification of the Spike Renose Model (SRM). The main difference to the voltage equation of the SRM introduced above is that in the term containing the refractory kernel [math]\displaystyle{ \eta(s) }[/math] there is no summation sign over past spikes: only the most recent spike (denoted as the time [math]\displaystyle{ \hat{t} }[/math]) matters. Another difference is that the threshold is constant. The model SRM0 can be formulated in discrete or continuous time. For example, in continuous time, the single-neuron equation is

SRM0是一种与时间依赖非线性更新理论相关的随机神经元模型，是Spike Renose模型(SRM)的简化。与上面介绍的SRM的电压方程的主要区别是，在包含不可解内核[math]\displaystyle{\eta(s)}[/math]的项中，对过去的尖峰没有求和符号:只有最近的尖峰(记作时间[math]\displaystyle{\hat{t}}[/math])起作用。另一个区别是，阈值是恒定的。模型SRM0可以用离散时间或连续时间来表示。例如，在连续时间下，单神经元方程为

The SRM0 is a stochastic neuron model related to time-dependent nonlinear renewal theory and a simplification of the Spike Renose Model (SRM). The main difference to the voltage equation of the SRM introduced above is that in the term containing the refractory kernel \eta(s) there is no summation sign over past spikes: only the most recent spike (denoted as the time \hat{t}) matters. Another difference is that the threshold is constant. The model SRM0 can be formulated in discrete or continuous time. For example, in continuous time, the single-neuron equation is

Srm0是一个与时变非线性更新理论有关的随机神经元模型，是 Spike Renose 模型的简化。上面介绍的 SRM 电压方程的主要区别在于，在包含难熔的内核 eta (s)的项中，没有过去尖峰的求和符号: 只有最近的尖峰(表示为时间帽{ t })起作用。另一个区别是阈值是恒定的。Srm0模型可以在离散时间或连续时间内建立。例如，在连续时间中，单神经元方程是

[math]\displaystyle{ V(t)= \eta(t-\hat{t}) + \int_0^\infty \kappa(s) I(t-s) \, ds + V_\mathrm{rest} }[/math]

V(t)= \eta(t-\hat{t}) + \int_0^\infty \kappa(s) I(t-s) \, ds + V_\mathrm{rest}

v (t) = eta (t-hat { t }) + int _ 0 ^ infty kappa (s) i (t-s) ，ds + v _ mathrum { rest }

and the network equations of the SRM0 are

SRM0的网络方程为

and the network equations of the SRM0 are

网络方程为 SRM0

[math]\displaystyle{ V_i(t\mid\hat{t}_i) = \eta_i(t-\hat{t}_i) + \sum_j w_{ij} \sum_f \varepsilon_{ij}(t-\hat{t}_i,t-t^f) + V_\mathrm{rest} }[/math]

V_i(t\mid\hat{t}_i) = \eta_i(t-\hat{t}_i) + \sum_j w_{ij} \sum_f \varepsilon_{ij}(t-\hat{t}_i,t-t^f) + V_\mathrm{rest}

v _ i (mid hat { t } _ i) = eta _ i (t-hat { t } _ i) + sum _ j w _ j { ij } sum _ f varepsilon _ { ij }(t-hat { t } _ i，t-t ^ f) + v _ mathrm { rest }

where [math]\displaystyle{ \hat{t}_i }[/math] is the last firing time neuron i. Note that the time course of the postsynaptic potential [math]\displaystyle{ \varepsilon_{ij} }[/math] is also allowed to depend on the time since the last spike of neuron i so as to describe a change in membrane conductance during refractoriness. The instantaneous firing rate (stochastic intensity) is

where \hat{t}_i is the last firing time neuron i. Note that the time course of the postsynaptic potential \varepsilon_{ij} is also allowed to depend on the time since the last spike of neuron i so as to describe a change in membrane conductance during refractoriness. The instantaneous firing rate (stochastic intensity) is

其中 i 是最后一个触发时间神经元 i。注意，突触后电位 varepsilon { ij }的时间过程也被允许依赖于自神经元 i 最后一个尖峰以来的时间，以此来描述在无效期间膜电导的变化。瞬时放电率(随机强度)为

[math]\displaystyle{ f(V-\vartheta) = \frac{1}{\tau_0} \exp[\beta(V-V_{th})] }[/math]

f(V-\vartheta) = \frac{1}{\tau_0} \exp[\beta(V-V_{th})]

F (v-vartheta) = frac {1}{ tau _ 0} exp [ beta (V-V _ { th })]

where [math]\displaystyle{ V_{th} }[/math] is a fixed firing threshold. Thus spike firing of neuron i depends only on its input and the time since neuron i has fired its last spike.

where V_{th} is a fixed firing threshold. Thus spike firing of neuron i depends only on its input and the time since neuron i has fired its last spike.

其中 v _ { th }是一个固定的触发阈值。因此神经元 i 的尖峰放电只取决于其输入和自神经元 i 发射最后一个尖峰以来的时间。

With the SRM0, the interspike-interval distribution for constant input can be mathematically linked to the shape of the refractory kernel [math]\displaystyle{ \eta }[/math] . Moreover the stationary frequency-current relation can be calculated from the escape rate in combination with the refractory kernel [math]\displaystyle{ \eta }[/math]. With an appropriate choice of the kernels, the SRM0 approximates the dynamics of the Hodgkin-Huxley model to a high degree of accuracy. Moreover, the PSTH response to arbitrary time-dependent input can be predicted.

使用SRM0，恒定输入下的尖峰区间分布可以在数学上与难解内核[math]\displaystyle{\eta}[/math]的形状联系起来。此外，结合难解核[math]\displaystyle{\eta}[/math]，可由逃逸率计算出稳态频率-电流关系。通过适当的kernel选择，SRM0将Hodgkin-Huxley模型的动力学近似到一个很高的精度。此外，可以预测PSTH对任意时变输入的响应。

With the SRM0, the interspike-interval distribution for constant input can be mathematically linked to the shape of the refractory kernel \eta . Moreover the stationary frequency-current relation can be calculated from the escape rate in combination with the refractory kernel \eta. With an appropriate choice of the kernels, the SRM0 approximates the dynamics of the Hodgkin-Huxley model to a high degree of accuracy. Moreover, the PSTH response to arbitrary time-dependent input can be predicted.

使用 SRM0，常数输入的穗间隔分布可以在数学上与难处理的籽粒 eta 的形状相联系。此外，还可以结合难熔粒径和逃逸率计算出稳态频率-电流关系。通过对内核的适当选择，srm0可以高度精确地逼近 Hodgkin-Huxley 模型的动态性。此外，还可以预测 PSTH 对任意时变输入的响应。

Galves–Löcherbach model

文件:GalvesLocherbach - Low resolution.gif

3D visualization of the Galves–Löcherbach model for biological neural nets. This visualization is set for 4,000 neurons (4 layers with one population of inhibitory neurons and one population of excitatory neurons each) at 180 intervals of time.

Main article: Galves–Löcherbach model

The Galves–Löcherbach model is a stochastic neuron model closely related to the spike response model SRM0 and to the leaky integrate-and-fire model. It is inherently stochastic and, just like the SRM0 linked to time-dependent nonlinear renewal theory. Given the model specifications, the probability that a given neuron [math]\displaystyle{ i }[/math] spikes in a time period [math]\displaystyle{ t }[/math] may be described by

Galves-Löcherbach模型是一个与脉冲响应模型SRM0和泄漏的积分-点火模型密切相关的随机神经元模型。它本质上是随机的，就像与依赖时间的非线性更新理论相关联的SRM0。给定模型规格，一个给定神经元[math]\displaystyle{i}[/math]在一个时间段[math]\displaystyle{t}[/math]出现峰值的概率可以描述为

thumb|3D visualization of the Galves–Löcherbach model for biological neural nets. This visualization is set for 4,000 neurons (4 layers with one population of inhibitory neurons and one population of excitatory neurons each) at 180 intervals of time.

The Galves–Löcherbach model is a stochastic neuron model closely related to the spike response model SRM0 and to the leaky integrate-and-fire model. It is inherently stochastic and, just like the SRM0 linked to time-dependent nonlinear renewal theory. Given the model specifications, the probability that a given neuron i spikes in a time period t may be described by

= = = Galves-Löcherbach 模型 = = = 拇指 | 生物神经网络 Galves-Löcherbach 模型的三维可视化。这种可视化设置为4000个神经元(4层，每层有一个抑制神经元群和一个兴奋神经元群)在180个时间间隔。Galves-Löcherbach 模型是一个与神经元模型 srm0和泄漏积分-着火模型密切相关的随机神经元模型。它具有固有的随机性，就像 srm0与时间相关的非线性更新理论一样。给定模型规范，给定的神经元 i 在一个时间周期 t 达到峰值的概率可以用

[math]\displaystyle{ \mathop{\mathrm{Prob}}(X_{t}(i) = 1\mid \mathcal{F}_{t-1}) = \varphi_i \Biggl( \sum_{j\in I} W_{j \rightarrow i} \sum_{s=L_t^i}^{t-1} g_j(t-s) X_s(j),~~~ t-L_t^i \Biggl), }[/math]

\mathop{\mathrm{Prob}}(X_{t}(i) = 1\mid \mathcal{F}_{t-1}) = \varphi_i \Biggl( \sum_{j\in I} W_{j \rightarrow i} \sum_{s=L_t^i}^{t-1} g_j(t-s) X_s(j),Moonscar（讨论） t-L_t^i \Biggl),

1 mid mathcal { f }{ t-1}) = varphi i Biggl (sum { j in i } w { j right tarrow i } sum { s = l _ t ^ i } ^ { t-1} g _ j (t-s) x _ s (j) ，~ ~ t-L _ t ^ i) ,

where [math]\displaystyle{ W_{j \rightarrow i} }[/math] is a synaptic weight, describing the influence of neuron [math]\displaystyle{ j }[/math] on neuron [math]\displaystyle{ i }[/math], [math]\displaystyle{ g_j }[/math] expresses the leak, and [math]\displaystyle{ L_t^i }[/math] provides the spiking history of neuron [math]\displaystyle{ i }[/math] before [math]\displaystyle{ t }[/math], according to

where W_{j \rightarrow i} is a synaptic weight, describing the influence of neuron j on neuron i, g_j expresses the leak, and L_t^i provides the spiking history of neuron i before t, according to

其中 w { j right tarrow i }是一个突触权重，描述了神经元 j 对神经元 i 的影响，g 表示漏，l _ t ^ i 提供了神经元 i 在 t 之前的尖峰历史，根据

[math]\displaystyle{ L_t^i =\sup\{s\lt t:X_s(i)=1\}. }[/math]

L_t^i =\sup\{s<t:X_s(i)=1\}.

l _ t ^ i = sup { s < t: x _ s (i) = 1}.

Importantly, the spike probability of neuron i depends only on its spike input (filtered with a kernel [math]\displaystyle{ g_{j} }[/math] and weighted with a factor [math]\displaystyle{ W_{j\to i} }[/math]) and the timing of its most recent output spike (summarized by [math]\displaystyle{ t-L_t^i }[/math]).

Importantly, the spike probability of neuron i depends only on its spike input (filtered with a kernel g_{j} and weighted with a factor W_{j\to i}) and the timing of its most recent output spike (summarized by t-L_t^i).

重要的是，神经元 i 的尖峰概率仅取决于它的尖峰输入(用核函数 g _ { j }过滤，用因子 w _ { j to i }加权)和它最近的输出尖峰时间(t-L _ t ^ i)。

Didactic toy models of membrane voltage

The models in this category are highly simplified toy models that qualitatively describe the membrane voltage as a function of input. They are mainly used for didactic reasons in teaching but are not considered valid neuron models for large-scale simulations or data fitting.

这类模型是高度简化的玩具模型，定性地将膜电压描述为输入的函数。它们主要用于教学中的教学原因，但并不被认为是大规模模拟或数据拟合的有效神经元模型。

The models in this category are highly simplified toy models that qualitatively describe the membrane voltage as a function of input. They are mainly used for didactic reasons in teaching but are not considered valid neuron models for large-scale simulations or data fitting.

这一类的模型是高度简化的玩具模型，定性地描述了作为输入函数的膜电位膜电位。它们主要用于教学的教学原因，但不被认为是有效的神经元模型大规模模拟或数据拟合。

FitzHugh–Nagumo

Main article: FitzHugh–Nagumo model

Sweeping simplifications to Hodgkin–Huxley were introduced by FitzHugh and Nagumo in 1961 and 1962. Seeking to describe "regenerative self-excitation" by a nonlinear positive-feedback membrane voltage and recovery by a linear negative-feedback gate voltage, they developed the model described by

菲茨休和南云在1961年和1962年提出了对霍奇金-赫胥黎的全面简化。为了用非线性正反馈膜电压来描述“再生自激励”，用线性负反馈栅电压来描述恢复，他们开发了以下描述的模型 Sweeping simplifications to Hodgkin–Huxley were introduced by FitzHugh and Nagumo in 1961 and 1962. Seeking to describe "regenerative self-excitation" by a nonlinear positive-feedback membrane voltage and recovery by a linear negative-feedback gate voltage, they developed the model described by

= = = FitzHugh-Nagumo = = = 对 Hodgkin-Huxley 进行了彻底的简化，由 FitzHugh 和 Nagumo 于1961年和1962年引入。为了通过非线性正反馈膜电位和线性负反馈门电压恢复来描述“再生自励磁”，他们发展了以下模型:

[math]\displaystyle{ \begin{array}{rcl} \dfrac{d V}{d t} &=& V-V^3/3 - w + I_\mathrm{ext} \\ \tau \dfrac{d w}{d t} &=& V-a-b w \end{array} }[/math]

\begin{array}{rcl}

 \dfrac{d V}{d t} &=& V-V^3/3 - w + I_\mathrm{ext} \\
 \tau \dfrac{d w}{d t} &=& V-a-b w

\end{array}

开始{ array }{ rcl } dfrac { d v }{ d t } & = & v ^ 3/3-w + i _ mathrom { ext } tau dfrac { d }{ d }{ d }{ t } & = & V-a-b w end { array }

where we again have a membrane-like voltage and input current with a slower general gate voltage w and experimentally-determined parameters a = -0.7, b = 0.8, τ = 1/0.08. Although not clearly derivable from biology, the model allows for a simplified, immediately available dynamic, without being a trivial simplification. The experimental support is weak, but the model is useful as a didactic tool to introduce dynamics of spike generation through phase plane analysis. See Chapter 7 in the textbook Methods of Neuronal Modeling

在这里，我们再次有膜状电压和输入电流，较慢的一般栅极电压w和实验确定的参数a = -0.7, b = 0.8， τ = 1/0.08。虽然不能从生物学中明确推导出来，但该模型允许一种简化的、立即可用的动态，而不是简单的简化。虽然该模型的实验支持较弱，但可以作为一种教学工具，通过相平面分析来介绍脉冲产生的动力学过程。参见《神经元建模方法》教材第7章

where we again have a membrane-like voltage and input current with a slower general gate voltage and experimentally-determined parameters . Although not clearly derivable from biology, the model allows for a simplified, immediately available dynamic, without being a trivial simplification. The experimental support is weak, but the model is useful as a didactic tool to introduce dynamics of spike generation through phase plane analysis. See Chapter 7 in the textbook Methods of Neuronal Modeling

这里我们再次有一个薄膜样的电压和输入电流与一个较慢的一般栅极电压和实验确定的参数。尽管不能清楚地从生物学推导出来，但该模型允许一个简化的、立即可用的动态，而不是一个简单的简化。虽然实验支持较弱，但该模型可以作为通过相平面分析引入尖峰发生动力学的有用工具。见教科书《神经元建模方法》第7章

Morris–Lecar

Main article: Morris–Lecar model

In 1981 Morris and Lecar combined the Hodgkin–Huxley and FitzHugh–Nagumo models into a voltage-gated calcium channel model with a delayed-rectifier potassium channel, represented by

1981年，Morris和Lecar将Hodgkin-Huxley和FitzHugh-Nagumo模型结合成一个电压门控钙通道模型，其中延迟整流钾通道为

In 1981 Morris and Lecar combined the Hodgkin–Huxley and FitzHugh–Nagumo models into a voltage-gated calcium channel model with a delayed-rectifier potassium channel, represented by

= = = Morris-Lecar = = = 1981年 Morris 和 Lecar 将 Hodgkin-Huxley 模型和 FitzHugh-Nagumo 模型结合成一个具有延迟整流钾离子通道的电压门控钙离子通道模型

[math]\displaystyle{ \begin{align} & C\frac{d V}{d t} &=& -I_\mathrm{ion}(V,w) + I \\[6pt] & \frac{d w}{d t} &=& \varphi \cdot \frac{w_\infty - w}{\tau_{w}} \end{align} }[/math]

\begin{align}

& C\frac{d V}{d t} &=& -I_\mathrm{ion}(V,w) + I \\[6pt] & \frac{d w}{d t} &=& \varphi \cdot \frac{w_\infty - w}{\tau_{w}} \end{align}

\begin{align}

& C\frac{d V}{d t} &=& -I_\mathrm{ion}(V,w) + I \\[6pt] & \frac{d w}{d t} &=& \varphi \cdot \frac{w_\infty - w}{\tau_{w}} \end{align}

where [math]\displaystyle{ I_\mathrm{ion}(V,w) = \bar{g}_\mathrm{Ca} m_\infty \cdot(V-V_\mathrm{Ca}) + \bar{g}_\mathrm{K} w\cdot(V-V_\mathrm{K}) + \bar{g}_\mathrm{L}\cdot(V-V_\mathrm{L}) }[/math]. The experimental support of the model is weak, but the model is useful as a didactic tool to introduce dynamics of spike generation through phase plane analysis. See Chapter 7 in the textbook Methods of Neuronal Modeling.

where I_\mathrm{ion}(V,w) = \bar{g}_\mathrm{Ca} m_\infty \cdot(V-V_\mathrm{Ca}) + \bar{g}_\mathrm{K} w\cdot(V-V_\mathrm{K}) + \bar{g}_\mathrm{L}\cdot(V-V_\mathrm{L}). The experimental support of the model is weak, but the model is useful as a didactic tool to introduce dynamics of spike generation through phase plane analysis. See Chapter 7 in the textbook Methods of Neuronal Modeling.

其中 i _ mathrm { ion }(v，w) = bar { g } _ mathrm { Ca } _ infty cdot (v _ mathrm { Ca }) + bar { g } _ mathrm { k }(v _ mathrm { k }) + bar { g } _ mathrm { l } cdot (v _ mathrm { l })。该模型的实验支持较弱，但可作为通过相平面分析引入尖峰发生动力学的教学工具。见教科书《神经元建模方法》第7章。

A two-dimensional neuron model very similar to the Morris-Lecar model can be derived step-by-step starting from the Hodgkin-Huxley model. See Chapter 4.2 in the textbook Neuronal Dynamics.

A two-dimensional neuron model very similar to the Morris-Lecar model can be derived step-by-step starting from the Hodgkin-Huxley model. See Chapter 4.2 in the textbook Neuronal Dynamics.

一个非常类似于 Morris-Lecar 模型的二维神经元模型可以从 Hodgkin-Huxley 模型开始逐步推导出来。见教科书《神经动力学》中的第4.2章。

Hindmarsh–Rose

Main article: Hindmarsh–Rose model

Building upon the FitzHugh–Nagumo model, Hindmarsh and Rose proposed in 1984 a model of neuronal activity described by three coupled first-order differential equations:

1984年，Hindmarsh和Rose在FitzHugh-Nagumo模型的基础上，提出了一个由三个耦合一阶微分方程描述的神经元活动模型:

Building upon the FitzHugh–Nagumo model, Hindmarsh and Rose proposed in 1984 a model of neuronal activity described by three coupled first-order differential equations:

= = = Hindmarsh-Rose = = = 在 FitzHugh-Nagumo 模型的基础上，Hindmarsh 和 Rose 在1984年提出了一个由三个一阶耦合微分方程描述的神经元活动模型:

[math]\displaystyle{ \begin{align} & \frac{d x}{d t} &=& y+3x^2-x^3-z+I \\[6pt] & \frac{d y}{d t} &=& 1-5x^2-y \\[6pt] & \frac{d z}{d t} &=& r\cdot (4(x + \tfrac{8}{5})-z) \end{align} }[/math]

\begin{align}

& \frac{d x}{d t} &=& y+3x^2-x^3-z+I \\[6pt] & \frac{d y}{d t} &=& 1-5x^2-y \\[6pt] & \frac{d z}{d t} &=& r\cdot (4(x + \tfrac{8}{5})-z) \end{align}

\begin{align}

& \frac{d x}{d t} &=& y+3x^2-x^3-z+I \\[6pt] & \frac{d y}{d t} &=& 1-5x^2-y \\[6pt] & \frac{d z}{d t} &=& r\cdot (4(x + \tfrac{8}{5})-z) \end{align}

with r² = x² + y² + z², and r ≈ 10⁻² so that the z variable only changes very slowly. This extra mathematical complexity allows a great variety of dynamic behaviors for the membrane potential, described by the x variable of the model, which include chaotic dynamics. This makes the Hindmarsh–Rose neuron model very useful, because being still simple, allows a good qualitative description of the many different firing patterns of the action potential, in particular bursting, observed in experiments. Nevertheless, it remains a toy model and has not been fitted to experimental data. It is widely used as a reference model for bursting dynamics.

with , and so that the variable only changes very slowly. This extra mathematical complexity allows a great variety of dynamic behaviors for the membrane potential, described by the variable of the model, which include chaotic dynamics. This makes the Hindmarsh–Rose neuron model very useful, because being still simple, allows a good qualitative description of the many different firing patterns of the action potential, in particular bursting, observed in experiments. Nevertheless, it remains a toy model and has not been fitted to experimental data. It is widely used as a reference model for bursting dynamics.

因此变量的变化非常缓慢。这种额外的数学复杂性使得膜电位的动态行为多种多样，由模型的变量描述，其中包括混沌动力学。这使得 Hindmarsh-罗斯神经元模型非常有用，因为它仍然简单，可以很好地定性描述动作电位的许多不同的放电模式，特别是在实验中观察到的放电模式。然而，它仍然是一个玩具模型，并没有适合的实验数据。它被广泛用作爆破动力学的参考模型。

= Theta 模型和二次积分-火。==

The theta model, or Ermentrout–Kopell canonical Type I model, is mathematically equivalent to the quadratic integrate-and-fire model which in turn is an approximation to the exponential integrate-and-fire model and the Hodgkin-Huxley model. It is called a canonical model because it is one of the generic models for constant input close to the bifurcation point, which means close to the transition from silent to repetitive firing.

θ模型，或Ermentrout-Kopell典型I型模型，在数学上等同于二次积分-点火模型，而二次积分-点火模型又近似于指数积分-点火模型和Hodgkin-Huxley模型。它被称为规范模型，因为它是一个一般的模型，用于接近分岔点的恒定输入，这意味着接近从沉默到重复触发的过渡。

The theta model, or Ermentrout–Kopell canonical Type I model, is mathematically equivalent to the quadratic integrate-and-fire model which in turn is an approximation to the exponential integrate-and-fire model and the Hodgkin-Huxley model. It is called a canonical model because it is one of the generic models for constant input close to the bifurcation point, which means close to the transition from silent to repetitive firing.

Theta 模型，或 Ermentrout-Kopell 典型 i 型模型，在数学上等同于二次积分-着火模型，而二次积分-着火模型又是指数积分-着火模型和 Hodgkin-Huxley 模型的近似。它被称为正则模型，因为它是一个通用的模型，用来描述接近分叉点的常数输入，这意味着接近从无声到重复激发的转变。

The standard formulation of the theta model is

模型的标准公式为

The standard formulation of the theta model is

Θ 模型的标准公式是

[math]\displaystyle{ \frac{d\theta(t)}{d t} = (I-I_0) [1+ \cos(\theta)] + [1- \cos(\theta)] }[/math]

\frac{d\theta(t)}{d t} = (I-I_0) [1+ \cos(\theta)] + [1- \cos(\theta)]

frac { d theta (t)}{ d t } = (i _ 0)[1 + cos (theta)] + [1-cos (theta)]

The equation for the quadratic integrate-and-fire model is (see Chapter 5.3 in the textbook Neuronal Dynamics ))

二次积分-点火模型的方程为(见《神经元动力学》教材第5.3章)

The equation for the quadratic integrate-and-fire model is (see Chapter 5.3 in the textbook Neuronal Dynamics ))

二次积分-着火模型的方程式是(见《神经元动力学》一书第5.3章)

[math]\displaystyle{ \tau_\mathrm{m} \frac{d V_\mathrm{m} (t)}{d t} = (I-I_0) R + [V_\mathrm{m} (t) - E_\mathrm{m} ][V_\mathrm{m} (t) - V_\mathrm{T} ] }[/math]

\tau_\mathrm{m} \frac{d V_\mathrm{m} (t)}{d t} = (I-I_0) R + [V_\mathrm{m} (t) - E_\mathrm{m} ][V_\mathrm{m} (t) - V_\mathrm{T} ]

tau _ mathrm { m } frac { d _ mathrm { m }(t)}{ d } = (i _ 0) r + [ v _ mathrm { m }(t)-e _ mathrm { m }][ v _ mathrm { m }(t)-v _ mathrm { t }]

The equivalence of theta model and quadratic integrate-and-fire is for example reviewed in Chapter 4.1.2.2 of spiking neuron models.

例如，在关于尖峰神经元模型的4.1.2.2章中，我们回顾了θ模型和二次积分-点火模型的等价性。

The equivalence of theta model and quadratic integrate-and-fire is for example reviewed in Chapter 4.1.2.2 of spiking neuron models.

例如，在脉冲神经元模型的第4.1.2.2章中回顾了 theta 模型和二次积分-着火模型的等价性。

For input I(t) that changes over time or is far away from the bifurcation point, it is preferable to work with the exponential integrate-and-fire model (if one wants the stay in the class of one-dimensional neuron models), because real neurons exhibit the nonlinearity of the exponential integrate-and-fire model.

输入我(t)随时间的变化或远离分歧点,最好使用指数integrate-and-fire模型(如果一个人想要留在类神经元的一维模型),因为真正的神经元表现出的非线性指数integrate-and-fire模型。

For input I(t) that changes over time or is far away from the bifurcation point, it is preferable to work with the exponential integrate-and-fire model (if one wants the stay in the class of one-dimensional neuron models), because real neurons exhibit the nonlinearity of the exponential integrate-and-fire model.

对于随时间变化或远离分岔点的输入 i (t) ，最好采用指数积分-着火模型(如果想保留在一维神经元模型中) ，因为实际神经元具有指数积分-着火模型的非线性。

= 感觉输入-刺激编码神经元模型 =

The models in this category were derived following experiments involving natural stimulation such as light, sound, touch, or odor. In these experiments, the spike pattern resulting from each stimulus presentation varies from trial to trial, but the averaged response from several trials often converges to a clear pattern. Consequently, the models in this category generate a probabilistic relationship between the input stimulus to spike occurrences. Importantly, the recorded neurons are often located several processing steps after the sensory neurons, so that these models summarize the effects of the sequence of processing steps in a compact form

这类模型是根据自然刺激(如光、声音、触摸或气味)的实验推导出来的。在这些实验中，每个刺激呈现的峰值模式因试验而异，但多次试验的平均反应往往趋同于一个明确的模式。因此，这类模型在输入刺激和峰值出现之间产生了一种概率关系。重要的是，被记录的神经元通常位于感觉神经元之后的几个处理步骤，因此这些模型以紧凑的形式总结了处理步骤序列的影响

The models in this category were derived following experiments involving natural stimulation such as light, sound, touch, or odor. In these experiments, the spike pattern resulting from each stimulus presentation varies from trial to trial, but the averaged response from several trials often converges to a clear pattern. Consequently, the models in this category generate a probabilistic relationship between the input stimulus to spike occurrences. Importantly, the recorded neurons are often located several processing steps after the sensory neurons, so that these models summarize the effects of the sequence of processing steps in a compact form

这一类型的模型是在包括自然刺激(如光、声音、触摸或气味)的实验之后得出的。在这些实验中，每次刺激呈现所产生的刺激模式因试验而异，但是几次试验的平均反应通常会收敛到一个清晰的模式。因此，这类模型在输入刺激和脉冲发生之间产生一种概率关系。重要的是，记录下来的神经元通常位于感觉神经元之后的几个处理步骤，因此这些模型以紧凑的形式总结了处理步骤序列的影响

The non-homogeneous Poisson process model (Siebert)

Siebert modeled the neuron spike firing pattern using a non-homogeneous Poisson process model, following experiments involving the auditory system. According to Siebert, the probability of a spiking event at the time interval [math]\displaystyle{ [t, t+\Delta_t] }[/math] is proportional to a non negative function [math]\displaystyle{ g[s(t)] }[/math], where [math]\displaystyle{ s(t) }[/math] is the raw stimulus.:

在听觉系统的实验之后，Siebert用非齐次泊松过程模型模拟了神经元脉冲放电模式。根据Siebert，在时间间隔[math]\displaystyle{[t, t+\Delta_t]}[/math]发生峰值事件的概率与非负函数[math]\displaystyle{g[s(t)]}[/math]成正比，其中[math]\displaystyle{s(t)}[/math]是原始刺激:

Siebert modeled the neuron spike firing pattern using a non-homogeneous Poisson process model, following experiments involving the auditory system. According to Siebert, the probability of a spiking event at the time interval [t, t+\Delta_t] is proportional to a non negative function g[s(t)], where s(t) is the raw stimulus.:

= = = 非齐次泊松过程模型(Siebert) = = Siebert 用非齐次泊松过程模型模拟了神经元放电模式，接着进行了涉及听觉系统的实验。根据希伯特的观点，在时间间隔[ t，t + Delta _ t ]处出现尖峰事件的概率与非负函数 g [ s (t)]成正比，其中 s (t)是原始刺激。:

[math]\displaystyle{ P_\text{spike}(t\in[t',t'+\Delta_t])=\Delta_t \cdot g[s(t)] }[/math]

P_\text{spike}(t\in[t',t'+\Delta_t])=\Delta_t \cdot g[s(t)]

p _ text { spike }(t in [ t’，t’+ Delta _ t ]) = Delta _ t cdot g [ s (t)]

Siebert considered several functions as [math]\displaystyle{ g[s(t)] }[/math], including [math]\displaystyle{ g[s(t)] \propto s^2(t) }[/math] for low stimulus intensities.

Siebert considered several functions as g[s(t)], including g[s(t)] \propto s^2(t) for low stimulus intensities.

希伯特认为有几个函数是 g [ s (t)] ，包括 g [ s (t)] propto s ^ 2(t)。

The main advantage of Siebert's model is its simplicity. The shortcomings of the model is its inability to reflect properly the following phenomena:

• The transient enhancement of the neuronal firing activity in response to a step stimulus.
• The saturation of the firing rate.
• The values of inter-spike-interval-histogram at short intervals values (close to zero).

These shortcoming are addressed by the age-dependent point process model and the two-state Markov Model.

Siebert模型的主要优点是简单。这个模式的缺点是它不能适当地反映下列现象:

阶跃刺激引起的神经元放电活动的短暂增强。

射速的饱和度。

在短时间间隔的直方图值(接近于零)。

The main advantage of Siebert's model is its simplicity. The shortcomings of the model is its inability to reflect properly the following phenomena:

• The transient enhancement of the neuronal firing activity in response to a step stimulus.
• The saturation of the firing rate.
• The values of inter-spike-interval-histogram at short intervals values (close to zero).

These shortcoming are addressed by the age-dependent point process model and the two-state Markov Model.

Siebert 模型的主要优点是它的简单性。该模型的缺点是不能很好地反映以下现象:

• 神经元的放电活动对阶跃刺激的瞬时性增强。
• 燃烧率的饱和度。
• 短时间区间直方图值(接近于零)。针对这些不足，提出了年龄相关点过程模型和两状态马尔可夫模型。

Refractoriness and age-dependent point process model模板:Anchor

Berry and Meister studied neuronal refractoriness using a stochastic model that predicts spikes as a product of two terms, a function f(s(t)) that depends on the time-dependent stimulus s(t) and one a recovery function [math]\displaystyle{ w(t-\hat{t}) }[/math] that depends on the time since the last spike

Berry和Meister使用一种随机模型来研究神经元的难解性，该模型将尖峰预测为两个项的乘积，一个函数f(s(t))依赖于依赖于时间的刺激s(t)，另一个是恢复函数[math]\displaystyle{w(t-\hat{t})}[/math]依赖于上一次尖峰后的时间

Berry and Meister studied neuronal refractoriness using a stochastic model that predicts spikes as a product of two terms, a function f(s(t)) that depends on the time-dependent stimulus s(t) and one a recovery function w(t-\hat{t}) that depends on the time since the last spike

= = = = 折射率和年龄相关点过程模型 = = = Berry 和 Meister 用一个随机模型研究了神经元的折射率，该模型预测尖峰是两个项的乘积，一个函数 f (s (t)依赖于时间相关的刺激 s (t) ，另一个函数 w (t-hat { t }依赖于上次尖峰后的时间

[math]\displaystyle{ \rho(t) = f(s(t))w(t-\hat{t}) }[/math]

\rho(t) = f(s(t))w(t-\hat{t})

rho (t) = f (s (t)) w (t-hat { t })

The model is also called an inhomogeneous Markov interval (IMI) process. Similar models have been used for many years in auditory neuroscience. Since the model keeps memory of the last spike time it is non-Poisson and falls in the class of time-dependent renewal models. It is closely related to the model SRM0 with exponential escape rate. Importantly, it is possible to fit parameters of the age-dependent point process model so as to describe not just the PSTH response, but also the interspike-interval statistics.

该模型也称为非齐次马尔可夫区间过程。类似的模型已经在听觉神经科学中使用了很多年。由于该模型保持了对上一个峰值时间的记忆，它是非泊松的，属于时间依赖更新模型类。它与模型SRM0密切相关，具有指数逃逸率。重要的是，可以拟合年龄相关的点过程模型的参数，从而不仅可以描述PSTH反应，还可以描述峰值间期统计量。

The model is also called an inhomogeneous Markov interval (IMI) process. Similar models have been used for many years in auditory neuroscience. Since the model keeps memory of the last spike time it is non-Poisson and falls in the class of time-dependent renewal models. It is closely related to the model SRM0 with exponential escape rate. Importantly, it is possible to fit parameters of the age-dependent point process model so as to describe not just the PSTH response, but also the interspike-interval statistics.

该模型也称为非齐次马尔可夫区间过程。类似的模型已经在听觉神经科学领域应用了很多年。由于该模型保持了上次峰值时间的记忆性，所以它是非泊松的，属于一类依赖时间的更新模型。它与指数逃逸率的 srm0模型密切相关。重要的是，可以拟合年龄相关点过程模型的参数，从而不仅描述 PSTH 响应，而且可以描述峰间期统计量。

Linear-nonlinear Poisson cascade model and GLM

模板:Main articles The linear-nonlinear-Poisson cascade model is a cascade of a linear filtering process followed by a nonlinear spike generation step. In the case that output spikes feed back, via a linear filtering process, we arrive at a model that is known in the neurosciences as Generalized Linear Model (GLM). The GLM is mathematically equivalent to the spike response model SRM) with escape noise; but whereas in the SRM the internal variables are interpreted as the membrane potential and the firing threshold, in the GLM the internal variables are abstract quantities that summarizes the net effect of input (and recent output spikes) before spikes are generated in the final step.

线性非线性泊松级联模型是一个线性滤波过程的级联，后面跟着一个非线性脉冲产生步骤。在输出峰值反馈的情况下，通过线性滤波过程，我们得到了一个在神经科学中称为广义线性模型(GLM)的模型。该模型在数学上等价于具有逃逸噪声的脉冲响应模型SRM;但在SRM中，内部变量被解释为膜电位和触发阈值，而在GLM中，内部变量是在最后一步产生尖峰之前总结输入(和最近的输出尖峰)净效应的抽象量。 The linear-nonlinear-Poisson cascade model is a cascade of a linear filtering process followed by a nonlinear spike generation step. In the case that output spikes feed back, via a linear filtering process, we arrive at a model that is known in the neurosciences as Generalized Linear Model (GLM). The GLM is mathematically equivalent to the spike response model SRM) with escape noise; but whereas in the SRM the internal variables are interpreted as the membrane potential and the firing threshold, in the GLM the internal variables are abstract quantities that summarizes the net effect of input (and recent output spikes) before spikes are generated in the final step.

= = = = 线性-非线性泊松级联模型和 GLM = = = 线性-非线性-泊松级联模型是一个线性滤波过程和一个非线性尖峰产生步骤的级联。在输出尖峰反馈的情况下，通过一个线性过滤过程，我们得到了一个模型，被称为神经科学的广义线性模式。在数学上，GLM 等价于存在逃逸噪声的尖峰响应模型 SRM; 但是在 SRM 中，内部变量被解释为膜电位和发射阈值，而在 GLM 中，内部变量是抽象量，在最后一步产生尖峰之前，总结了输入(和最近的输出尖峰)的净效应。

= = 两态马尔可夫模型(Nossenson & Messer) =

The spiking neuron model by Nossenson & Messer produces the probability of the neuron to fire a spike as a function of either an external or pharmacological stimulus. The model consists of a cascade of a receptor layer model and a spiking neuron model, as shown in Fig 4. The connection between the external stimulus to the spiking probability is made in two steps: First, a receptor cell model translates the raw external stimulus to neurotransmitter concentration, then, a spiking neuron model connects between neurotransmitter concentration to the firing rate (spiking probability). Thus, the spiking neuron model by itself depends on neurotransmitter concentration at the input stage.

由Nossenson & Messer所建立的脉冲神经元模型产生了神经元在外部或药理学刺激作用下产生脉冲的概率。该模型由受体层级联模型和尖峰神经元模型组成，如图4所示。外部刺激与脉冲概率之间的联系分为两个步骤:首先，一个受体细胞模型将外部刺激转化为神经递质浓度，然后，一个脉冲神经元模型将神经递质浓度与放电率(脉冲概率)联系起来。因此，脉冲神经元模型本身依赖于输入阶段的神经递质浓度。

文件:Nerve model 2wiki.png

Fig 4: High level block diagram of the receptor layer and neuron model by Nossenson & Messer.

文件:Firing rate improved.png

Fig 5. The prediction for the firing rate in response to a pulse stimulus as given by the model by Nossenson & Messer.

An important feature of this model is the prediction for neurons firing rate pattern which captures, using a low number of free parameters, the characteristic edge emphasized response of neurons to a stimulus pulse, as shown in Fig. 5. The firing rate is identified both as a normalized probability for neural spike firing, and as a quantity proportional to the current of neurotransmitters released by the cell. The expression for the firing rate takes the following form:

该模型的一个重要特征是预测神经元的放电率模式，利用较少的自由参数捕捉特征边缘强调神经元对刺激脉冲的响应，如图5所示。放电率被确定为神经脉冲放电的归一化概率，以及与细胞释放的神经递质电流成比例的数量。射击率的表达式采用以下形式:

The spiking neuron model by Nossenson & Messer produces the probability of the neuron to fire a spike as a function of either an external or pharmacological stimulus. The model consists of a cascade of a receptor layer model and a spiking neuron model, as shown in Fig 4. The connection between the external stimulus to the spiking probability is made in two steps: First, a receptor cell model translates the raw external stimulus to neurotransmitter concentration, then, a spiking neuron model connects between neurotransmitter concentration to the firing rate (spiking probability). Thus, the spiking neuron model by itself depends on neurotransmitter concentration at the input stage. thumb|400px|Fig 4: High level block diagram of the receptor layer and neuron model by Nossenson & Messer. thumb|400px|Fig 5. The prediction for the firing rate in response to a pulse stimulus as given by the model by Nossenson & Messer. An important feature of this model is the prediction for neurons firing rate pattern which captures, using a low number of free parameters, the characteristic edge emphasized response of neurons to a stimulus pulse, as shown in Fig. 5. The firing rate is identified both as a normalized probability for neural spike firing, and as a quantity proportional to the current of neurotransmitters released by the cell. The expression for the firing rate takes the following form:

诺森森和梅塞尔的尖峰神经元模型产生了神经元作为外部刺激或药理刺激的函数发出尖峰的概率。该模型包括级联的受体层模型和脉冲神经元模型，如图4所示。将外部刺激与放电概率之间的联系分为两个步骤: 首先，受体细胞模型将原始外部刺激转化为神经递质浓度，然后，将神经递质浓度与放电概率联系起来(放电概率)。因此，刺激神经元模型本身取决于输入阶段神经递质的浓度。拇指 | 400px | 图4: Nossenson & Messer 设计的受体层和神经元模型的高级框图。400px | Fig 5.由 Nossenson 和 Messer 建立的模型预测脉冲刺激下的放电频率。该模型的一个重要特征是对神经元放电频率模式的预测，该模式通过使用少量的自由参数，捕捉到神经元对刺激脉冲的特征边缘强调反应，如图所示。5.神经元放电频率既是神经元放电的归一化概率，又是与细胞释放的神经递质电流成正比的量。发射率的表达式如下:

[math]\displaystyle{ R_\text{fire}(t)=\frac{P_\text{spike}(t;\Delta_t)}{\Delta_t}=[y(t)+R_0] \cdot P_0(t) }[/math]

R_\text{fire}(t)=\frac{P_\text{spike}(t;\Delta_t)}{\Delta_t}=[y(t)+R_0] \cdot P_0(t)

r _ text { fire }(t) = frac { p _ text { spike }(t; Delta _ t)}{ Delta _ t } = [ y (t) + r _ 0] cdot p _ 0(t)

where,

• P0 is the probability of the neuron to be "armed" and ready to fire. It is given by the following differential equation:

where,

• P0 is the probability of the neuron to be "armed" and ready to fire. It is given by the following differential equation:

其中，

• p0是神经元“武装”并准备触发的概率。它是由以下微分方程提供的:

[math]\displaystyle{ \dot{P}_0=-[y(t)+R_0+R_1] \cdot P_0(t) +R_1 }[/math]

\dot{P}_0=-[y(t)+R_0+R_1] \cdot P_0(t) +R_1

点{ p } _ 0 =-[ y (t) + r _ 0 + r _ 1] cdot p _ 0(t) + r _ 1

P0 could be generally calculated recursively using Euler method, but in the case of a pulse of stimulus it yields a simple closed form expression.

• y(t) is the input of the model and is interpreted as the neurotransmitter concentration on the cell surrounding (in most cases glutamate). For an external stimulus it can be estimated through the receptor layer model:

P0 could be generally calculated recursively using Euler method, but in the case of a pulse of stimulus it yields a simple closed form expression.

• y(t) is the input of the model and is interpreted as the neurotransmitter concentration on the cell surrounding (in most cases glutamate). For an external stimulus it can be estimated through the receptor layer model:

通常可以用欧拉方法递归地计算 P0，但是对于一个脉冲的刺激，它会产生一个简单的封闭式表达式。

• y (t)是模型的输入，被解释为周围细胞(大多数情况下是谷氨酸)的神经递质浓度。对于外部刺激，可以通过受体层模型进行估计:

[math]\displaystyle{ y(t) \simeq g_\text{gain} \cdot \langle s^2(t)\rangle, }[/math]

y(t) \simeq g_\text{gain} \cdot \langle s^2(t)\rangle,

y (t) simeq g _ text { gain } cdot langle s ^ 2(t) rangle,

with [math]\displaystyle{ \langle s^2(t)\rangle }[/math] being short temporal average of stimulus power (given in Watt or other energy per time unit).

• R₀ corresponds to the intrinsic spontaneous firing rate of the neuron.
• R₁ is the recovery rate of the neuron from the refractory state.

with \langle s^2(t)\rangle being short temporal average of stimulus power (given in Watt or other energy per time unit).

• R0 corresponds to the intrinsic spontaneous firing rate of the neuron.
• R1 is the recovery rate of the neuron from the refractory state.

当 langle s ^ 2(t) rangle 为短时平均刺激功率(以瓦特或其他时间单位能量表示)。

• r0与神经元内在自发放电率相对应。
• r1是神经元从不应状态的恢复速率。

Other predictions by this model include:

Other predictions by this model include:

这个模型的其他预测包括:

1) The averaged evoked response potential (ERP) due to the population of many neurons in unfiltered measurements resembles the firing rate.

1) The averaged evoked response potential (ERP) due to the population of many neurons in unfiltered measurements resembles the firing rate.

1)未经滤波的测量结果中，由于大量神经元的聚集，所产生的平均诱发反应电位(ERP)与放电频率相似。

2) The voltage variance of activity due to multiple neuron activity resembles the firing rate (also known as Multi-Unit-Activity power or MUA).

2) The voltage variance of activity due to multiple neuron activity resembles the firing rate (also known as Multi-Unit-Activity power or MUA).

2)由于多个神经元活动引起的电压变化与放电率相似(也称为多单元活动功率或 MUA)。

3) The inter-spike-interval probability distribution takes the form a gamma-distribution like function.

Experimental evidence supporting the model by Nossenson & Messer

Property of the Model by Nossenson & Messer	References	Description of experimental evidence
The shape of the firing rate in response to an auditory stimulus pulse		The Firing Rate has the same shape of Fig 5.
The shape of the firing rate in response to a visual stimulus pulse		The Firing Rate has the same shape of Fig 5.
The shape of the firing rate in response to an olfactory stimulus pulse		The Firing Rate has the same shape of Fig 5.
The shape of the firing rate in response to a somato-sensory stimulus		The Firing Rate has the same shape of Fig 5.
The change in firing rate in response to neurotransmitter application (mostly glutamate)		Firing Rate change in response to neurotransmitter application (Glutamate)
Square dependence between an auditory stimulus pressure and the firing rate		Square Dependence between Auditory Stimulus pressure and the Firing Rate (- Linear dependence in pressure square (power)).
Square dependence between visual stimulus electric field (volts) and the firing rate		Square dependence between visual stimulus electric field (volts) - Linear Dependence between Visual Stimulus Power and the Firing Rate.
The shape of the Inter-Spike-Interval Statistics (ISI)		ISI shape resembles the gamma-function-like
The ERP resembles the firing rate in unfiltered measurements		The shape of the averaged evoked response potential in response to stimulus resembles the firing rate (Fig. 5).
MUA power resembles the firing rate		The shape of the empirical variance of extra-cellular measurements in response to stimulus pulse resembles the firing rate (Fig. 5).

3) The inter-spike-interval probability distribution takes the form a gamma-distribution like function.

Experimental evidence supporting the model by Nossenson & Messer

Property of the Model by Nossenson & Messer	References	Description of experimental evidence
The shape of the firing rate in response to an auditory stimulus pulse		The Firing Rate has the same shape of Fig 5.
The shape of the firing rate in response to a visual stimulus pulse		The Firing Rate has the same shape of Fig 5.
The shape of the firing rate in response to an olfactory stimulus pulse		The Firing Rate has the same shape of Fig 5.
The shape of the firing rate in response to a somato-sensory stimulus		The Firing Rate has the same shape of Fig 5.
The change in firing rate in response to neurotransmitter application (mostly glutamate)		Firing Rate change in response to neurotransmitter application (Glutamate)
Square dependence between an auditory stimulus pressure and the firing rate		Square Dependence between Auditory Stimulus pressure and the Firing Rate (- Linear dependence in pressure square (power)).
Square dependence between visual stimulus electric field (volts) and the firing rate		Square dependence between visual stimulus electric field (volts) - Linear Dependence between Visual Stimulus Power and the Firing Rate.
The shape of the Inter-Spike-Interval Statistics (ISI)		ISI shape resembles the gamma-function-like
The ERP resembles the firing rate in unfiltered measurements		The shape of the averaged evoked response potential in response to stimulus resembles the firing rate (Fig. 5).
MUA power resembles the firing rate		The shape of the empirical variance of extra-cellular measurements in response to stimulus pulse resembles the firing rate (Fig. 5).

3)穗间期概率分布呈类 γ 分布函数形式。{ | class = “ wikitable sortable”| + 实验证据支持 Nossenson & Messer 的模型！模型的属性作者: Nossenson & Messer！参考！描述实验证据 |-| 听觉刺激脉冲的放电率的形状 | | 放电率的形状与图5相同。视觉刺激脉冲作用下放电频率的形状与图5相同。嗅觉刺激脉冲引起的放电频率的形状与图5的形状相同。体感刺激引起的放电频率的形状与图5的形状相同。神经递质应用(主要是谷氨酸)引起的放电频率变化 | 神经递质应用引起的放电频率变化 | 听觉刺激压力和放电频率之间的平方相关 | 听觉刺激压力和放电频率之间的平方相关(- 压力平方的线性相关(幂))。视觉刺激电场(伏特)和射频率之间的平方相关性视觉刺激电场(伏特)之间的平方相关性视觉刺激电源和射频率之间的线性相关性。刺激诱发反应电位的平均形状与刺激引起的放电频率相似(图)。5).|-| MUA 功率类似于放电率 | | 细胞外测量的经验方差的形状在对刺激脉冲的反应类似于放电率(图。5).|}

模板:Anchor Pharmacological input stimulus neuron models

The models in this category produce predictions for experiments involving pharmacological stimulation.

The models in this category produce predictions for experiments involving pharmacological stimulation.

= = 药物输入刺激神经元模型 = = 这类模型为涉及药物刺激的实验提供预测。

Synaptic transmission (Koch & Segev)

See also: Neurotransmission

According to the model by Koch and Segev, the response of a neuron to individual neurotransmitters can be modeled as an extension of the classical Hodgkin–Huxley model with both standard and nonstandard kinetic currents. Four neurotransmitters primarily have influence in the CNS. AMPA/kainate receptors are fast excitatory mediators while NMDA receptors mediate considerably slower currents. Fast inhibitory currents go through GABA_A receptors, while GABA_B receptors mediate by secondary G-protein-activated potassium channels. This range of mediation produces the following current dynamics:

根据Koch和Segev的模型，神经元对单个神经递质的反应可以被建模为经典的霍奇金-赫胥黎模型的扩展，包括标准和非标准动能电流。四种神经递质主要影响中枢神经系统。AMPA/凯恩酸受体是快速兴奋性介质，而NMDA受体介导相当慢的电流。快速抑制电流通过GABAA受体，GABAB受体通过二级g蛋白激活钾通道介导。这种中介范围产生了以下当前动态:

According to the model by Koch and Segev, the response of a neuron to individual neurotransmitters can be modeled as an extension of the classical Hodgkin–Huxley model with both standard and nonstandard kinetic currents. Four neurotransmitters primarily have influence in the CNS. AMPA/kainate receptors are fast excitatory mediators while NMDA receptors mediate considerably slower currents. Fast inhibitory currents go through GABA_A receptors, while GABA_B receptors mediate by secondary G-protein-activated potassium channels. This range of mediation produces the following current dynamics:

根据 Koch 和 Segev 的模型，神经元对单个神经递质的反应可以用标准动力电流和非标准动力电流来描述，这是 Hodgkin-Huxley 模型的一个扩展。四种神经递质主要影响中枢神经系统。AMPA/kainate 受体是快速兴奋性介质，NMDA 受体介导的电流明显减慢。快速抑制电流通过 GABA < sub > a 受体，而 GABA < sub > b 受体通过次级 g 蛋白激活的钾通道介导。这一范围的调解产生了以下目前的动态:

• [math]\displaystyle{ I_\mathrm{AMPA}(t,V) = \bar{g}_\mathrm{AMPA} \cdot [O] \cdot (V(t)-E_\mathrm{AMPA}) }[/math]
• [math]\displaystyle{ I_\mathrm{NMDA}(t,V) = \bar{g}_\mathrm{NMDA} \cdot B(V) \cdot [O] \cdot (V(t)-E_\mathrm{NMDA}) }[/math]
• [math]\displaystyle{ I_\mathrm{GABA_A}(t,V) = \bar{g}_\mathrm{GABA_A} \cdot ([O_1]+[O_2]) \cdot (V(t)-E_\mathrm{Cl}) }[/math]
• [math]\displaystyle{ I_\mathrm{GABA_B}(t,V) = \bar{g}_\mathrm{GABA_B} \cdot \tfrac{[G]^n}{[G]^n+K_\mathrm{d}} \cdot (V(t)-E_\mathrm{K}) }[/math]

• I_\mathrm{AMPA}(t,V) = \bar{g}_\mathrm{AMPA} \cdot [O] \cdot (V(t)-E_\mathrm{AMPA})
• I_\mathrm{NMDA}(t,V) = \bar{g}_\mathrm{NMDA} \cdot B(V) \cdot [O] \cdot (V(t)-E_\mathrm{NMDA})
• I_\mathrm{GABA_A}(t,V) = \bar{g}_\mathrm{GABA_A} \cdot ([O_1]+[O_2]) \cdot (V(t)-E_\mathrm{Cl})
• I_\mathrm{GABA_B}(t,V) = \bar{g}_\mathrm{GABA_B} \cdot \tfrac{[G]^n}{[G]^n+K_\mathrm{d}} \cdot (V(t)-E_\mathrm{K})

• I_\mathrm{AMPA}(t,V) = \bar{g}_\mathrm{AMPA} \cdot [O] \cdot (V(t)-E_\mathrm{AMPA})

• I_\mathrm{NMDA}(t,V) = \bar{g}_\mathrm{NMDA} \cdot B(V) \cdot [O] \cdot (V(t)-E_\mathrm{NMDA})

• I_\mathrm{GABA_A}(t,V) = \bar{g}_\mathrm{GABA_A} \cdot ([O_1]+[O_2]) \cdot (V(t)-E_\mathrm{Cl})

• I_\mathrm{GABA_B}(t,V) = \bar{g}_\mathrm{GABA_B} \cdot \tfrac{[G]^n}{[G]^n+K_\mathrm{d}} \cdot (V(t)-E_\mathrm{K})

where ḡ is the maximal conductance (around 1S) and E is the equilibrium potential of the given ion or transmitter (AMDA, NMDA, Cl, or K), while [O] describes the fraction of receptors that are open. For NMDA, there is a significant effect of magnesium block that depends sigmoidally on the concentration of intracellular magnesium by B(V). For GABA_B, [G] is the concentration of the G-protein, and K_d describes the dissociation of G in binding to the potassium gates.

where is the maximal conductance (around 1S) and is the equilibrium potential of the given ion or transmitter (AMDA, NMDA, Cl, or K), while describes the fraction of receptors that are open. For NMDA, there is a significant effect of magnesium block that depends sigmoidally on the concentration of intracellular magnesium by . For GABAB, is the concentration of the G-protein, and describes the dissociation of G in binding to the potassium gates.

其中是最大电导(约1s) ，是给定离子或递质(AMDA，NMDA，Cl，或 k)的平衡电位，而描述了开放的受体的比例。对于 NMDA，镁阻滞有显著的作用，这主要取决于细胞内镁的浓度。对 GABAB 来说，是 g 蛋白的浓度，并描述了 g 与钾门结合时的解离。

The dynamics of this more complicated model have been well-studied experimentally and produce important results in terms of very quick synaptic potentiation and depression, that is, fast, short-term learning.

这个更复杂模型的动力学已经在实验上得到了很好的研究，并在非常快速的突触电位增强和抑制方面产生了重要的结果，即快速的短期学习。

The dynamics of this more complicated model have been well-studied experimentally and produce important results in terms of very quick synaptic potentiation and depression, that is, fast, short-term learning.

这个更加复杂的模型的动力学已经在实验中得到了很好的研究，并在非常快的突触增强和抑制方面产生了重要的结果，即快速的短期学习。

The stochastic model by Nossenson and Messer translates neurotransmitter concentration at the input stage to the probability of releasing neurotransmitter at the output stage. For a more detailed description of this model, see the Two state Markov model section above.

Nossenson和Messer的随机模型将输入阶段的神经递质浓度转化为输出阶段释放神经递质的概率。有关此模型的更详细描述，请参阅上面的Two状态马尔可夫模型部分。

The stochastic model by Nossenson and Messer translates neurotransmitter concentration at the input stage to the probability of releasing neurotransmitter at the output stage. For a more detailed description of this model, see the Two state Markov model section above.

Nossenson 和 Messer 的随机模型将输入阶段的神经递质浓度转化为输出阶段神经递质释放的概率。有关此模型的更详细描述，请参阅上面的两状态马尔可夫模型部分。

HTM neuron model

The HTM neuron model was developed by Jeff Hawkins and researchers at Numenta and is based on a theory called Hierarchical Temporal Memory, originally described in the book On Intelligence. It is based on neuroscience and the physiology and interaction of pyramidal neurons in the neocortex of the human brain.

文件:Neuron comparison.png

Comparing the artificial neural network (A), the biological neuron (B), and the HTM neuron (C).

Artificial Neural Network (ANN)	Neocortical Pyramidal Neuron (Biological Neuron)	HTM Model Neuron
- Few synapses The HTM neuron model was developed by Jeff Hawkins and researchers at Numenta and is based on a theory called Hierarchical Temporal Memory, originally described in the book On Intelligence. It is based on neuroscience and the physiology and interaction of pyramidal neurons in the neocortex of the human brain. Artificial Neural Network (ANN) Neocortical Pyramidal Neuron (Biological Neuron) HTM Model Neuron - Few synapses 该模型是由 Jeff Hawkins 和 Numenta 的研究人员根据一个叫做分级暂存记忆的理论建立的，该理论最初在《论智力》一书中有所描述。它基于神经科学和人脑新皮层锥体神经元的生理学和相互作用。{ | class = “ wikable”| 人工神经网络 | 新皮层锥体神经元(生物神经元) | 模型神经元 | | |-少数突触 - No dendrites - No dendrites - 没有树突 - Sum input x weights - Sum input x weights - 总和输入 x 权数 - Learns by modifying weights of synapses - Thousands of synapses on the dendrites - Learns by modifying weights of synapses - Thousands of synapses on the dendrites - 通过调节突触的重量进行学习 |-树突上的数千个突触 - Active dendrites: cell recognizes hundreds of unique patterns - Active dendrites: cell recognizes hundreds of unique patterns - 活性树突: 细胞识别数百种独特的模式 - Co-activation of a set of synapses on a dendritic segment causes an NMDA spike and depolarization at the soma - Co-activation of a set of synapses on a dendritic segment causes an NMDA spike and depolarization at the soma - 树突上一系列突触的共激活引起突触体 NMDA 峰值和去极化 - Sources of input to the cell: Feedforward inputs which form synapses proximal to the soma and directly lead to action potentials NMDA spikes generated in the more distal basal Apical dendrites that depolarize the soma (usually not sufficient enough to generate a somatic action potential) - Sources of input to the cell: Feedforward inputs which form synapses proximal to the soma and directly lead to action potentials NMDA spikes generated in the more distal basal Apical dendrites that depolarize the soma (usually not sufficient enough to generate a somatic action potential) - 细胞输入来源: # 前馈输入，形成接近胞体的突触，并直接导致动作电位 # NMDA 尖峰产生于更远端的基底 # 树突，使胞体去极化(通常不足以产生体细胞动作电位) - Learns by growing new synapses - Inspired by the pyramidal cells in neocortex layers 2/3 and 5 - Learns by growing new synapses - Inspired by the pyramidal cells in neocortex layers 2/3 and 5 - 通过生长新的突触来学习-灵感来自新皮层2/3和5层的锥体细胞 - Thousands of synapses - Thousands of synapses - 成千上万的突触 - Active dendrites: cell recognizes hundreds of unique patterns - Active dendrites: cell recognizes hundreds of unique patterns - 活性树突: 细胞识别数百种独特的模式 - Models dendrites and NMDA spikes with each array of coincident detectors having a set of synapses - Models dendrites and NMDA spikes with each array of coincident detectors having a set of synapses - 模拟树突和 NMDA 峰值，每个重合探测器阵列有一组突触 - Learns by modeling growth of new synapses - Learns by modeling growth of new synapses

- 通过模拟新突触的生长来学习 | }

Applications

Main article: Brain–computer interface

Spiking Neuron Models are used in a variety of applications that need encoding into or decoding from neuronal spike trains in the context of neuroprosthesis and brain-computer interfaces such as retinal prosthesis: or artificial limb control and sensation. Applications are not part of this article; for more information on this topic please refer to the main article.

尖峰神经元模型被用于各种需要在神经假体和脑机接口(如视网膜假体或人工肢体控制和感觉)中对神经元尖峰序列进行编码或解码的应用中。应用程序不属于本文的一部分;有关此主题的更多信息，请参阅主要文章。

Spiking Neuron Models are used in a variety of applications that need encoding into or decoding from neuronal spike trains in the context of neuroprosthesis and brain-computer interfaces such as retinal prosthesis: or artificial limb control and sensation. Applications are not part of this article; for more information on this topic please refer to the main article.

在神经假体和大脑-计算机接口，如视网膜假体或人工假肢控制和感觉的背景下，神经元神经元模型被用于各种各样的应用，需要从神经元神经元序列中编码或解码。应用程序不是本文的一部分; 有关此主题的更多信息，请参阅主文章。

Relation between artificial and biological neuron models

The most basic model of a neuron consists of an input with some synaptic weight vector and an activation function or transfer function inside the neuron determining output. This is the basic structure used for artificial neurons, which in a neural network often looks like

神经元最基本的模型是由一个具有突触权向量的输入和一个神经元内部的激活函数或传递函数决定输出。这是人工神经元的基本结构，在神经网络中通常看起来是这样的

The most basic model of a neuron consists of an input with some synaptic weight vector and an activation function or transfer function inside the neuron determining output. This is the basic structure used for artificial neurons, which in a neural network often looks like

= = = 人工神经元模型与生物神经元模型之间的关系 = = 一个神经元的最基本模型由一个输入端和一些突触重量向量以及神经元输出端的激活函数或传递函数组成。这是人造神经元的基本结构，在神经网络中，神经元通常看起来像人造神经元

[math]\displaystyle{ y_i = \varphi\left( \sum_j w_{ij} x_j \right) }[/math]

y_i = \varphi\left( \sum_j w_{ij} x_j \right)

y _ i = varphi left (sum _ j w _ { ij } x _ j right)

where y_i is the output of the i th neuron, x_j is the jth input neuron signal, w_ij is the synaptic weight (or strength of connection) between the neurons i and j, and φ is the activation function. While this model has seen success in machine-learning applications, it is a poor model for real (biological) neurons, because it lacks time-dependence in input and output.

其中yi为第I个神经元的输出，xj为JTH输入神经元信号，wij为神经元I与j之间的突触权值(或连接强度)，φ为激活函数。虽然这个模型在机器学习应用中取得了成功，但对于真实的(生物)神经元来说，它是一个糟糕的模型，因为它在输入和输出方面缺乏时间依赖性。

where is the output of the th neuron, is the th input neuron signal, is the synaptic weight (or strength of connection) between the neurons and , and is the activation function. While this model has seen success in machine-learning applications, it is a poor model for real (biological) neurons, because it lacks time-dependence in input and output.

其中第个神经元的输出，是第个输入神经元信号，是神经元和神经元之间的突触重量(或连接强度) ，是激活函数。虽然这个模型在机器学习应用中取得了成功，但对于真正的(生物)神经元来说，它是一个糟糕的模型，因为它在输入和输出中缺乏时间依赖性。

When an input is switched on at a time t and kept constant thereafter, biological neurons emit a spike train. Importantly this spike train is not regular but exhibits a temporal structure characterized by adaptation, bursting, or initial bursting followed by regular spiking. Generalized integrate-and-fire model such as the Adaptive Exponential Integrate-and-Fire model, the spike response model, or the (linear) adaptive integrate-and-fire model are able to capture these neuronal firing patterns.

当一个输入在t时刻开启，并且此后保持不变时，生物神经元就会发出一个脉冲序列。重要的是，这种脉冲序列不是规则的，而是表现出一种时间结构特征，即适应、破裂或最初破裂后是规则的脉冲。广义的积分-点火模型，如自适应指数积分-点火模型、脉冲响应模型或(线性)自适应积分-点火模型能够捕捉这些神经元的点火模式。

When an input is switched on at a time t and kept constant thereafter, biological neurons emit a spike train. Importantly this spike train is not regular but exhibits a temporal structure characterized by adaptation, bursting, or initial bursting followed by regular spiking. Generalized integrate-and-fire model such as the Adaptive Exponential Integrate-and-Fire model, the spike response model, or the (linear) adaptive integrate-and-fire model are able to capture these neuronal firing patterns.

当一个输入在一个时刻 t 被打开，并且此后保持不变时，生物神经元就会发出一个尖峰信号。重要的是，这种脉冲序列并不规则，而是表现出一种时间结构---- 拥有属性适应性、突发性或者伴随着规则脉冲的初始突发性。广义的集成-着火模型，如自适应指数集成-着火模型，尖峰反应模型，或(线性)自适应集成-着火模型都能够捕捉这些神经元的放电模式。

Moreover, neuronal input in the brain is time-dependent. Time-dependent input is transformed by complex linear and nonlinear filters into a spike train in the output. Again, the spike response model or the adaptive integrate-and-fire model enable to predict the spike train in the output for arbitrary time-dependent input, whereas an artificial neuron or a simple leaky integrate-and-fire does not.

此外，大脑中的神经元输入是有时间依赖性的。时间相关的输入通过复杂的线性和非线性滤波器转换成输出中的脉冲序列。同样，脉冲响应模型或自适应积分-触发模型能够预测任意时间依赖输入的脉冲序列，而人工神经元或简单的泄漏积分-触发模型则不能。

Moreover, neuronal input in the brain is time-dependent. Time-dependent input is transformed by complex linear and nonlinear filters into a spike train in the output. Again, the spike response model or the adaptive integrate-and-fire model enable to predict the spike train in the output for arbitrary time-dependent input, whereas an artificial neuron or a simple leaky integrate-and-fire does not.

此外，大脑中的神经元输入是时间依赖性的。时变输入通过复杂的线性和非线性滤波器转化为输出中的尖峰列。同样，尖峰响应模型或自适应积分-着火模型能够预测任意时间依赖的输入下的输出尖峰列车，而人工神经元或简单的泄漏积分-着火模型则不能。

If we take the Hodkgin-Huxley model as a starting point, generalized integrate-and-fire models can be derived systematically in a step-by-step simplification procedure. This has been shown explicitly for the exponential integrate-and-fire model and the spike response model.

如果我们以hodgkin - huxley模型为出发点，可以通过一步一步的简化过程系统地推导出广义的集成-点火模型。这已经在指数积分-点火模型和脉冲响应模型中得到了明确的证明。

If we take the Hodkgin-Huxley model as a starting point, generalized integrate-and-fire models can be derived systematically in a step-by-step simplification procedure. This has been shown explicitly for the exponential integrate-and-fire model and the spike response model.

以 Hodkgin-Huxley 模型为出发点，可以通过逐步简化的方法系统地推导出广义积分与消防模型。这在指数积分-着火模型和尖峰响应模型中都得到了明确的证明。

In the case of modelling a biological neuron, physical analogues are used in place of abstractions such as "weight" and "transfer function". A neuron is filled and surrounded with water containing ions, which carry electric charge. The neuron is bound by an insulating cell membrane and can maintain a concentration of charged ions on either side that determines a capacitance C_m. The firing of a neuron involves the movement of ions into the cell that occurs when neurotransmitters cause ion channels on the cell membrane to open. We describe this by a physical time-dependent current I(t). With this comes a change in voltage, or the electrical potential energy difference between the cell and its surroundings, which is observed to sometimes result in a voltage spike called an action potential which travels the length of the cell and triggers the release of further neurotransmitters. The voltage, then, is the quantity of interest and is given by V_m(t).

在生物神经元建模的情况下，物理模拟被用来代替诸如“权重”和“传递函数”之类的抽象概念。神经元被含有离子的水充满和包围，这些离子携带电荷。神经元被一层绝缘的细胞膜所束缚，可以保持两侧带电离子的浓度，这决定了其电容为Cm。当神经递质导致细胞膜上的离子通道打开时，神经元的放电涉及离子进入细胞的运动。我们用物理时间相关的电流I(t)来描述它。伴随着电压的变化，也就是细胞和周围环境之间的电势能差，有时会导致被称为动作电位的电压尖峰，动作电位在细胞中传递，并触发进一步的神经递质的释放。电压为感兴趣的量，由Vm(t)给出。

In the case of modelling a biological neuron, physical analogues are used in place of abstractions such as "weight" and "transfer function". A neuron is filled and surrounded with water containing ions, which carry electric charge. The neuron is bound by an insulating cell membrane and can maintain a concentration of charged ions on either side that determines a capacitance . The firing of a neuron involves the movement of ions into the cell that occurs when neurotransmitters cause ion channels on the cell membrane to open. We describe this by a physical time-dependent current . With this comes a change in voltage, or the electrical potential energy difference between the cell and its surroundings, which is observed to sometimes result in a voltage spike called an action potential which travels the length of the cell and triggers the release of further neurotransmitters. The voltage, then, is the quantity of interest and is given by .

在建立生物神经元模型的情况下，使用物理类似物来代替”权重”和”传递函数”等抽象概念。神经元被带有电荷的含离子的水包围。神经元与绝缘细胞膜结合，并且能够保持两侧带电离子的浓度，从而确定电容。当神经递质使细胞膜上的离子通道打开时，神经元的放电就涉及到离子进入细胞的运动。我们用与物理时间有关的电流来描述这种现象。随之而来的是电压的变化，或者说是细胞与周围环境之间的电位能量差，这种差异被观察到有时会导致一种叫做动作电位的电压尖峰，动作电位沿着细胞的长度传递并触发进一步的神经递质的释放。那么，电压就是感兴趣的量，由。

If the input current is constant, most neurons emit after some time of adaptation or initial bursting a regular spike train. The frequency of regular firing in response to a constant current I is described by the frequency-current relation which corresponds to the transfer function [math]\displaystyle{ \varphi }[/math] of artificial neural networks. Similarly, for all spiking neuron models the transfer function [math]\displaystyle{ \varphi }[/math] can be calculated numerically (or analytically).

If the input current is constant, most neurons emit after some time of adaptation or initial bursting a regular spike train. The frequency of regular firing in response to a constant current is described by the frequency-current relation which corresponds to the transfer function \varphi of artificial neural networks. Similarly, for all spiking neuron models the transfer function \varphi can be calculated numerically (or analytically).

如果输入电流不变，大多数神经元在一段时间的适应或初始放电后，会发出一个规则的尖峰信号。用与人工神经网络传递函数曲线相对应的频率-电流关系来描述恒流作用下的规则放电频率。类似地，对于所有的尖峰神经元模型，传递函数可以用数值方法(或解析方法)计算。

= = 电缆理论和分室模型 =

See also: Cable theory

All of the above deterministic models are point-neuron models because they do not consider the spatial structure of a neuron. However, the dendrite contributes to transforming input into output. Point neuron models are valid description in three cases. (i) If input current is directly injected into the soma. (ii) If synaptic input arrives predominantly at or close to the soma (closeness is defined by a length scale [math]\displaystyle{ \lambda }[/math] introduced below. (iii) If synapse arrive anywhere on the dendrite, but the dendrite is completely linear. In the last case the cable acts as a linear filter; these linear filter properties can be included in the formulation of generalized integrate-and-fire models such as the spike response model.

以上所有确定性模型都是点神经元模型，因为它们不考虑神经元的空间结构。然而，树突有助于将输入转化为输出。在三种情况下，点神经元模型是有效的描述。(i)如果输入电流直接注入躯体。(ii)如果突触输入主要到达或接近体(接近度由下面介绍的长度刻度[math]\displaystyle{\lambda}[/math]定义。(iii)如果突触到达树突的任何位置，但树突是完全线性的。在最后一种情况下，电缆作为一个线性滤波器;这些线性滤波器的特性可以被包括在广义的积分-火灾模型中，例如尖峰响应模型。

All of the above deterministic models are point-neuron models because they do not consider the spatial structure of a neuron. However, the dendrite contributes to transforming input into output. Point neuron models are valid description in three cases. (i) If input current is directly injected into the soma. (ii) If synaptic input arrives predominantly at or close to the soma (closeness is defined by a length scale \lambda introduced below. (iii) If synapse arrive anywhere on the dendrite, but the dendrite is completely linear. In the last case the cable acts as a linear filter; these linear filter properties can be included in the formulation of generalized integrate-and-fire models such as the spike response model.

上述所有确定性模型都是点神经元模型，因为它们没有考虑神经元的空间结构。然而，枝晶有助于将输入转化为输出。点神经元模型在三种情况下都是有效的描述。(i)如果输入电流直接注入体细胞。如果突触输入主要到达或接近躯体(亲密度定义为一个长度尺度下面介绍的 lambda。如果突触到达树突的任何位置，但树突完全是线性的。在最后一种情况下，电缆充当线性滤波器，这些线性滤波特性可以包括在广义积分-火灾模型的表述中，例如尖峰响应模型。

The filter properties can be calculate from a cable equation.

滤波器的性质可以从缆索方程计算出来。

The filter properties can be calculate from a cable equation.

滤波器的性能可以从电缆方程式中计算出来。

Let us consider a cell membrane in the form a cylindrical cable. The position on the cable is denoted by x and the voltage across the cell membrane by V. The cable is characterized by a longitudinal resistance [math]\displaystyle{ r_l }[/math] per unit length and a membrane resistance [math]\displaystyle{ r_m }[/math] . If everything is linear, the voltage changes as a function of time

让我们设想一种圆柱形电缆形式的细胞膜。电缆上的位置用x表示，跨细胞膜的电压用v表示。电缆的特征是单位长度的纵向电阻[math]\displaystyle{r_l}[/math]和膜电阻[math]\displaystyle{r_m}[/math]。如果一切都是线性的，电压作为时间的函数变化

[math]\displaystyle{ \frac{r_m}{r_l} \frac{\partial ^2 V}{\partial x^2}=c_m r_m \frac{\partial V}{\partial t}+ V }[/math]

(19)

We introduce a length scale [math]\displaystyle{ \lambda^2 = {r_m}/{r_l} }[/math] on the left side and time constant [math]\displaystyle{ \tau = c_m r_m }[/math] on the right side. The cable equation can now be written in its perhaps best known form:

[math]\displaystyle{ \lambda^2 \frac{\partial ^2 V}{\partial x^2}=\tau \frac{\partial V}{\partial t}+ V }[/math]

(20)

The above cable equation is valid for a single cylindrical cable.

Let us consider a cell membrane in the form a cylindrical cable. The position on the cable is denoted by x and the voltage across the cell membrane by V. The cable is characterized by a longitudinal resistance r_l per unit length and a membrane resistance r_m . If everything is linear, the voltage changes as a function of timeWe introduce a length scale \lambda^2 = {r_m}/{r_l} on the left side and time constant \tau = c_m r_m on the right side. The cable equation can now be written in its perhaps best known form: The above cable equation is valid for a single cylindrical cable.

让我们考虑一下细胞膜上的圆柱形电缆。电缆上的位置用 x 表示，电压用 v 表示。电缆的纵向电阻为每单位长度1拥有属性，膜电阻为每单位长度1米。如果一切都是线性的，电压随时间的变化我们在左边引入一个长度尺度 λ ^ 2 = { r _ m }/{ r _ l } ，在右边引入时间常数 tau = c _ m r _ m。电缆方程式现在可以写成也许是最著名的形式: 上面的电缆方程式对于单根圆柱形电缆是有效的。

Linear cable theory describes the dendritic arbor of a neuron as a cylindrical structure undergoing a regular pattern of bifurcation, like branches in a tree. For a single cylinder or an entire tree, the static input conductance at the base (where the tree meets the cell body, or any such boundary) is defined as

Linear cable theory describes the dendritic arbor of a neuron as a cylindrical structure undergoing a regular pattern of bifurcation, like branches in a tree. For a single cylinder or an entire tree, the static input conductance at the base (where the tree meets the cell body, or any such boundary) is defined as

线性拉索理论将神经元的树突杆描述为一个经历有规律分叉的圆柱形结构，就像树上的树枝。对于单个圆柱体或整个树，基部的静态输入电导(树与单元体或任何这样的边界相交处)定义为

[math]\displaystyle{ G_{in} = \frac{G_\infty \tanh(L) + G_L}{1+(G_L / G_\infty )\tanh(L)} }[/math],

G_{in} = \frac{G_\infty \tanh(L) + G_L}{1+(G_L / G_\infty )\tanh(L)},

g { in } = frac { g _ infty tanh (l) + g _ l }{1 + (g _ l/g _ infty) tanh (l)} ,

where L is the electrotonic length of the cylinder which depends on its length, diameter, and resistance. A simple recursive algorithm scales linearly with the number of branches and can be used to calculate the effective conductance of the tree. This is given by

where is the electrotonic length of the cylinder which depends on its length, diameter, and resistance. A simple recursive algorithm scales linearly with the number of branches and can be used to calculate the effective conductance of the tree. This is given by

其中圆柱体的电渗长度取决于其长度、直径和电阻。一个简单的递归算法随着分支数目的线性变化而变化，可以用来计算树的有效电导。这是由

[math]\displaystyle{ \,\! G_D = G_m A_D \tanh(L_D) / L_D }[/math]

\,\! G_D = G_m A_D \tanh(L_D) / L_D

\,\!G _ d = g _ m a _ d tanh (l _ d)/l _ d

where A_D = πld is the total surface area of the tree of total length l, and L_D is its total electrotonic length. For an entire neuron in which the cell body conductance is G_S and the membrane conductance per unit area is G_md = G_m / A, we find the total neuron conductance G_N for n dendrite trees by adding up all tree and soma conductances, given by

where is the total surface area of the tree of total length , and is its total electrotonic length. For an entire neuron in which the cell body conductance is and the membrane conductance per unit area is , we find the total neuron conductance for dendrite trees by adding up all tree and soma conductances, given by

其中是总长度树的总表面积，是它的总电渗长度。对于细胞体电导为，单位面积膜电导为的整个神经元，我们通过将所有树和胞体电导加起来，得到了树枝状树的总电导

[math]\displaystyle{ G_N = G_S + \sum_{j=1}^n A_{D_j} F_{dga_j}, }[/math]

G_N = G_S + \sum_{j=1}^n A_{D_j} F_{dga_j},

1} n a { d j } f { dga j } ,

where we can find the general correction factor F_dga experimentally by noting G_D = G_mdA_DF_dga.

where we can find the general correction factor experimentally by noting .

在那里我们可以通过实验记录找到一般修正因子。

The linear cable model makes a number of simplifications to give closed analytic results, namely that the dendritic arbor must branch in diminishing pairs in a fixed pattern and that dendrites are linear. A compartmental model allows for any desired tree topology with arbitrary branches and lengths, as well as arbitrary nonlinearities. It is essentially a discretized computational implementation of nonlinear dendrites.

The linear cable model makes a number of simplifications to give closed analytic results, namely that the dendritic arbor must branch in diminishing pairs in a fixed pattern and that dendrites are linear. A compartmental model allows for any desired tree topology with arbitrary branches and lengths, as well as arbitrary nonlinearities. It is essentially a discretized computational implementation of nonlinear dendrites.

线性电缆模型进行了大量的简化，给出了封闭的解析结果，即树枝状乔木必须以固定的模式分枝，树枝状晶是线性的。一个分隔模型允许任何具有任意分支和长度以及任意非线性的想要的树拓扑。它实质上是非线性树枝晶的离散化计算实现。

Each individual piece, or compartment, of a dendrite is modeled by a straight cylinder of arbitrary length l and diameter d which connects with fixed resistance to any number of branching cylinders. We define the conductance ratio of the ith cylinder as B_i = G_i / G_∞, where [math]\displaystyle{ G_\infty=\tfrac{\pi d^{3/2}}{2\sqrt{R_i R_m}} }[/math] and R_i is the resistance between the current compartment and the next. We obtain a series of equations for conductance ratios in and out of a compartment by making corrections to the normal dynamic B_out,i = B_in,i+1, as

Each individual piece, or compartment, of a dendrite is modeled by a straight cylinder of arbitrary length and diameter which connects with fixed resistance to any number of branching cylinders. We define the conductance ratio of the th cylinder as , where G_\infty=\tfrac{\pi d^{3/2}}{2\sqrt{R_i R_m}} and is the resistance between the current compartment and the next. We obtain a series of equations for conductance ratios in and out of a compartment by making corrections to the normal dynamic , as

树枝晶的每一个单独的部件或隔间由一个任意长度和直径的直柱体模拟，直柱体与任意数量的分支柱体相连接。定义了圆柱体的电导率为，其中 g _ infty = tfrac { pi ^ {3/2}{2 sqrt { r _ i r _ m } ，是电流区与下一区之间的电阻。通过对常规动力学方程进行修正，得到了室内外电导率的一系列方程

• [math]\displaystyle{ B_{\mathrm{out},i} = \frac{B_{\mathrm{in},i+1}(d_{i+1}/d_i)^{3/2} }{ \sqrt{R_{\mathrm{m},i+1}/R_{\mathrm{m},i}} } }[/math]
• [math]\displaystyle{ B_{\mathrm{in},i} = \frac{ B_{\mathrm{out},i} + \tanh X_i }{ 1+B_{\mathrm{out},i}\tanh X_i } }[/math]
• [math]\displaystyle{ B_\mathrm{out,par} = \frac{B_\mathrm{in,dau1} (d_\mathrm{dau1}/d_\mathrm{par})^{3/2}} {\sqrt{R_\mathrm{m,dau1}/R_\mathrm{m,par}}} + \frac{B_\mathrm{in,dau2} (d_\mathrm{dau2}/d_\mathrm{par})^{3/2}} {\sqrt{R_\mathrm{m,dau2}/R_\mathrm{m,par}}} + \ldots }[/math]

• B_{\mathrm{out},i} = \frac{B_{\mathrm{in},i+1}(d_{i+1}/d_i)^{3/2} }{ \sqrt{R_{\mathrm{m},i+1}/R_{\mathrm{m},i}} }
• B_{\mathrm{in},i} = \frac{ B_{\mathrm{out},i} + \tanh X_i }{ 1+B_{\mathrm{out},i}\tanh X_i }
• B_\mathrm{out,par} = \frac{B_\mathrm{in,dau1} (d_\mathrm{dau1}/d_\mathrm{par})^{3/2}} {\sqrt{R_\mathrm{m,dau1}/R_\mathrm{m,par}}} + \frac{B_\mathrm{in,dau2} (d_\mathrm{dau2}/d_\mathrm{par})^{3/2}} {\sqrt{R_\mathrm{m,dau2}/R_\mathrm{m,par}}} + \ldots

• B_{\mathrm{out},i} = \frac{B_{\mathrm{in},i+1}(d_{i+1}/d_i)^{3/2} }{ \sqrt{R_{\mathrm{m},i+1}/R_{\mathrm{m},i}} }

• B_{\mathrm{in},i} = \frac{ B_{\mathrm{out},i} + \tanh X_i }{ 1+B_{\mathrm{out},i}\tanh X_i }

• B_\mathrm{out,par} = \frac{B_\mathrm{in,dau1} (d_\mathrm{dau1}/d_\mathrm{par})^{3/2}} {\sqrt{R_\mathrm{m,dau1}/R_\mathrm{m,par}}} + \frac{B_\mathrm{in,dau2} (d_\mathrm{dau2}/d_\mathrm{par})^{3/2}} {\sqrt{R_\mathrm{m,dau2}/R_\mathrm{m,par}}} + \ldots

where the last equation deals with parents and daughters at branches, and [math]\displaystyle{ X_i = \tfrac{l_i \sqrt{4R_i}}{\sqrt{d_i R_m}} }[/math]. We can iterate these equations through the tree until we get the point where the dendrites connect to the cell body (soma), where the conductance ratio is B_in,stem. Then our total neuron conductance for static input is given by

where the last equation deals with parents and daughters at branches, and X_i = \tfrac{l_i \sqrt{4R_i}}{\sqrt{d_i R_m}}. We can iterate these equations through the tree until we get the point where the dendrites connect to the cell body (soma), where the conductance ratio is . Then our total neuron conductance for static input is given by

其中最后一个方程处理分支上的父母和子代，x _ i = tfrac { l _ i sqrt {4R _ i }{ sqrt { d _ i r _ m }。我们可以通过树迭代这些方程，直到我们得到树突连接到细胞体的点，其中电导率是。然后，我们的总神经元电导为静态输入是由

[math]\displaystyle{ G_N = \frac{A_\mathrm{soma}}{R_\mathrm{m,soma}} + \sum_j B_{\mathrm{in,stem},j} G_{\infty,j}. }[/math]

G_N = \frac{A_\mathrm{soma}}{R_\mathrm{m,soma}} + \sum_j B_{\mathrm{in,stem},j} G_{\infty,j}.

g _ n = frac { a _ mathrm { soma }{ r _ mathrm { m，soma } + sum _ j b _ { mathrm { in，stem } ，j } g _ { infty，j }.

Importantly, static input is a very special case. In biology inputs are time dependent. Moreover, dendrites are not always linear.

Importantly, static input is a very special case. In biology inputs are time dependent. Moreover, dendrites are not always linear.

重要的是，静态输入是一种非常特殊的情况。在生物学中，输入是时间依赖的。此外，树突并不总是线性的。

Compartmental models enable to include nonlinearities via ion channels positioned at arbitrary locations along the dendrites. For static inputs, it is sometimes possible to reduce the number of compartments (increase the computational speed) and yet retain the salient electrical characteristics.

See also: Multi-compartment model

Compartmental models enable to include nonlinearities via ion channels positioned at arbitrary locations along the dendrites. For static inputs, it is sometimes possible to reduce the number of compartments (increase the computational speed) and yet retain the salient electrical characteristics.

房室模型能够通过沿树突任意位置的离子通道包括非线性。对于静态输入，有时可以减少隔间的数量(增加计算速度) ，同时保留显著的电气特性。

= = 推测神经元在更广泛的大脑运作原理中的作用 =

The neurotransmitter-based energy detection scheme

The neurotransmitter-based energy detection scheme suggests that the neural tissue chemically executes a Radar-like detection procedure.

The neurotransmitter-based energy detection scheme suggests that the neural tissue chemically executes a Radar-like detection procedure.

以神经递质为基础的能量检测方案以神经递质为基础的能量检测方案表明，神经组织以化学方式执行雷达式的检测程序。

文件:Neural detector nossenson 2013.png

Fig. 6 The biological neural detection scheme as suggested by Nossenson et al.

thumb|611x411px|Fig. 6 The biological neural detection scheme as suggested by Nossenson et al.

611x411px | Fig.6 Nossenson 等人提出的生物神经检测方案。

As shown in Fig. 6, the key idea of the conjecture is to account neurotransmitter concentration, neurotransmitter generation and neurotransmitter removal rates as the important quantities in executing the detection task, while referring to the measured electrical potentials as a side effect that only in certain conditions coincide with the functional purpose of each step. The detection scheme is similar to a radar-like "energy detection" because it includes signal squaring, temporal summation and a threshold switch mechanism, just like the energy detector, but it also includes a unit that emphasizes stimulus edges and a variable memory length (variable memory). According to this conjecture, the physiological equivalent of the energy test statistics is neurotransmitter concentration, and the firing rate corresponds to neurotransmitter current. The advantage of this interpretation is that it leads to a unit consistent explanation which allows to bridge between electrophysiological measurements, biochemical measurements and psychophysical results.

如图6所示，该猜想的关键思想是将神经递质浓度、神经递质生成率和神经递质去除率作为执行检测任务的重要量，而将测量到的电势作为一种副作用，只在某些条件下符合每一步的功能目的。该检测方案类似于雷达式的“能量检测”，因为它包括信号平方、时间总和和阈值开关机制，就像能量检测器一样，但它也包括一个强调刺激边缘和可变记忆长度的单元(可变记忆)。根据这个猜想，能量测试统计数据的生理等效物是神经递质浓度，而放电率对应于神经递质电流。这种解释的优点是，它导致了一个单位一致的解释，允许在电生理测量，生化测量和心理物理结果之间的桥梁。

As shown in Fig. 6, the key idea of the conjecture is to account neurotransmitter concentration, neurotransmitter generation and neurotransmitter removal rates as the important quantities in executing the detection task, while referring to the measured electrical potentials as a side effect that only in certain conditions coincide with the functional purpose of each step. The detection scheme is similar to a radar-like "energy detection" because it includes signal squaring, temporal summation and a threshold switch mechanism, just like the energy detector, but it also includes a unit that emphasizes stimulus edges and a variable memory length (variable memory). According to this conjecture, the physiological equivalent of the energy test statistics is neurotransmitter concentration, and the firing rate corresponds to neurotransmitter current. The advantage of this interpretation is that it leads to a unit consistent explanation which allows to bridge between electrophysiological measurements, biochemical measurements and psychophysical results.

如图所示。6、推测的核心思想是将神经递质浓度、神经递质生成和神经递质去除率作为执行探测任务的重要量，而将测量的电位作为一种副作用，只有在特定条件下才与每个步骤的功能目的相一致。检测方案类似于雷达式的“能量检测”，因为它包括信号平方、时间求和和和阈值开关机制，就像能量检测器一样，但它也包括一个强调刺激边缘的单元和一个可变记忆长度(可变记忆)。根据这一推测，能量测试统计量的生理等价物是神经递质浓度，放电频率与神经递质电流相对应。这种解释的优点是，它导致了一个单位一致的解释，允许在电生理测量、生物化学测量和心理物理学结果之间架起桥梁。

The evidence reviewed in suggest the following association between functionality to histological classification:

1. Stimulus squaring is likely to be performed by receptor cells.
2. Stimulus edge emphasizing and signal transduction is performed by neurons.
3. Temporal accumulation of neurotransmitters is performed by glial cells. Short term neurotransmitter accumulation is likely to occur also in some types of neurons.
4. Logical switching is executed by glial cells, and it results from exceeding a threshold level of neurotransmitter concentration. This threshold crossing is also accompanied by a change in neurotransmitter leak rate.
5. Physical all-or-non movement switching is due to muscle cells and results from exceeding a certain neurotransmitter concentration threshold on muscle surroundings.

The evidence reviewed in suggest the following association between functionality to histological classification:

1. Stimulus squaring is likely to be performed by receptor cells.
2. Stimulus edge emphasizing and signal transduction is performed by neurons.
3. Temporal accumulation of neurotransmitters is performed by glial cells. Short term neurotransmitter accumulation is likely to occur also in some types of neurons.
4. Logical switching is executed by glial cells, and it results from exceeding a threshold level of neurotransmitter concentration. This threshold crossing is also accompanied by a change in neurotransmitter leak rate.
5. Physical all-or-non movement switching is due to muscle cells and results from exceeding a certain neurotransmitter concentration threshold on muscle surroundings.

回顾的证据表明以下功能与组织学分类之间的联系: # 刺激平方可能是由受体细胞执行的。# 刺激边缘强调和信号转导是由神经元执行的。# 神经递质的时间累积由神经胶质细胞完成。短期的神经递质积累也可能发生在某些类型的神经元中。# 神经胶质细胞进行逻辑切换，超过神经递质浓度的阈值。这个阈值的跨越也伴随着神经递质泄漏率的变化。# 身体的全部或非运动转换是由于肌肉细胞和肌肉环境超过一定的神经递质浓度阈值的结果。

Note that although the electrophysiological signals in Fig.6 are often similar to the functional signal (signal power / neurotransmitter concentration / muscle force), there are some stages in which the electrical observation is different from the functional purpose of the corresponding step. In particular, Nossenson et al. suggested that glia threshold crossing has a completely different functional operation compared to the radiated electrophysiological signal, and that the latter might only be a side effect of glia break.

Note that although the electrophysiological signals in Fig.6 are often similar to the functional signal (signal power / neurotransmitter concentration / muscle force), there are some stages in which the electrical observation is different from the functional purpose of the corresponding step. In particular, Nossenson et al. suggested that glia threshold crossing has a completely different functional operation compared to the radiated electrophysiological signal, and that the latter might only be a side effect of glia break.

注意，虽然图6中的电生理信号通常与功能信号(信号功率/神经递质浓度/肌肉力量)相似，但在某些阶段，电观察与相应步骤的功能目的不同。尤其是诺森森森等人。提示胶质细胞跨越与辐射电生理信号有着完全不同的功能操作，后者可能只是胶质细胞破裂的副作用。

General comments regarding the modern perspective of scientific and engineering models

• The models above are still idealizations. Corrections must be made for the increased membrane surface area given by numerous dendritic spines, temperatures significantly hotter than room-temperature experimental data, and nonuniformity in the cell's internal structure. Certain observed effects do not fit into some of these models. For instance, the temperature cycling (with minimal net temperature increase) of the cell membrane during action potential propagation not compatible with models which rely on modeling the membrane as a resistance which must dissipate energy when current flows through it. The transient thickening of the cell membrane during action potential propagation is also not predicted by these models, nor is the changing capacitance and voltage spike that results from this thickening incorporated into these models. The action of some anesthetics such as inert gases is problematic for these models as well. New models, such as the soliton model attempt to explain these phenomena, but are less developed than older models and have yet to be widely applied. 上述模型仍然是理想化的。必须对大量树突棘所增加的膜表面积、温度明显高于室温实验数据以及细胞内部结构的不均匀性进行校正。某些观察到的效应并不符合这些模型。例如，在动作电位传播过程中，细胞膜的温度循环(净温度增加最小)与依赖于将细胞膜建模为一个电阻的模型不兼容，当电流流过它时，它必须消耗能量。这些模型也没有预测动作电位传播过程中细胞膜的瞬时增厚，也没有预测由增厚引起的电容和电压峰值的变化。一些麻醉剂的作用，如惰性气体是这些模型的问题。新的模型，如孤子模型，试图解释这些现象，但没有旧的模型那么发达，还没有被广泛应用。
• Modern views regarding of the role of the scientific model suggest that "All models are wrong but some are useful" (Box and Draper, 1987, Gribbin, 2009; Paninski et al., 2009). 关于科学模型的作用，现代观点认为“所有的模型都是错误的，但有些是有用的”(Box and Draper, 1987, Gribbin, 2009;Paninski等人，2009)。
• Recent conjecture suggests that each neuron might function as a collection of independent threshold units. It is suggested that a neuron could be anisotropically activated following the origin of its arriving signals to the membrane, via its dendritic trees. The spike waveform was also proposed to be dependent on the origin of the stimulus. 最近的猜想表明，每个神经元可能作为一个独立的阈值单元的集合。这表明，神经元可能是各向异性激活后，其信号的来源到达膜，通过其树突树。脉冲波形也被提出依赖于刺激的起源。

• The models above are still idealizations. Corrections must be made for the increased membrane surface area given by numerous dendritic spines, temperatures significantly hotter than room-temperature experimental data, and nonuniformity in the cell's internal structure. Certain observed effects do not fit into some of these models. For instance, the temperature cycling (with minimal net temperature increase) of the cell membrane during action potential propagation not compatible with models which rely on modeling the membrane as a resistance which must dissipate energy when current flows through it. The transient thickening of the cell membrane during action potential propagation is also not predicted by these models, nor is the changing capacitance and voltage spike that results from this thickening incorporated into these models. The action of some anesthetics such as inert gases is problematic for these models as well. New models, such as the soliton model attempt to explain these phenomena, but are less developed than older models and have yet to be widely applied.
• Modern views regarding of the role of the scientific model suggest that "All models are wrong but some are useful" (Box and Draper, 1987, Gribbin, 2009; Paninski et al., 2009).
• Recent conjecture suggests that each neuron might function as a collection of independent threshold units. It is suggested that a neuron could be anisotropically activated following the origin of its arriving signals to the membrane, via its dendritic trees. The spike waveform was also proposed to be dependent on the origin of the stimulus.

= = 关于科学和工程模型的现代观点的一般性评论 =

• 上述模型仍然是理想化的。必须对树突棘增加的膜表面积、比室温实验数据高得多的温度以及细胞内部结构的不均匀性进行校正。某些观察到的效应并不符合这些模型。例如，动作电位传播过程中细胞膜的温度循环(净温度升高最小)与依赖于将细胞膜建模为电阻的模型不相容，因为电流通过细胞膜时，细胞膜必须耗散能量。这些模型也没有预测到动作电位传播过程中细胞膜的瞬时增厚，也没有预测到这种增厚所导致的电容和电压尖峰的变化。对于这些模型来说，某些麻醉剂如惰性气体的作用也是有问题的。新的模型，例如孤立子模型试图解释这些现象，但是比旧的模型发展得较慢，尚未得到广泛应用。
• 关于科学模型作用的现代观点表明,”所有模型都是错误的，但有些是有用的”(Box 和 Draper，1987年，Gribbin，2009年; Paninski 等人，2009年)。
• 最近的猜想表明，每个神经元可能作为一个独立的阈值单位集合起作用。这表明，神经元可以通过其树突被各向异性激活，随着其到达膜的信号的起源而激活。尖峰波形也被认为是依赖于原产地的刺激。

= = 外部链接 =

• Neuronal Dynamics: from single neurons to networks and models of cognition (W. Gerstner, W. Kistler, R. Naud, L. Paninski, Cambridge University Press, 2014). In particular, Chapters 6 - 10, html online version.
• Spiking Neuron Models (W. Gerstner and W. Kistler, Cambridge University Press, 2002)

• Neuronal Dynamics: from single neurons to networks and models of cognition (W. Gerstner, W. Kistler, R. Naud, L. Paninski, Cambridge University Press, 2014). In particular, Chapters 6 - 10, html online version.
• Spiking Neuron Models (W. Gerstner and W. Kistler, Cambridge University Press, 2002)

• 神经元动力学: 从单个神经元到网络和认知模型(w. Gerstner，w. Kistler，r. Naud，l. Paninski，剑桥大学出版社，2014)。特别是第6-10章，html 的网上版本。
• 尖峰神经元模型(w. Gerstner and w. Kistler，Cambridge University Press，2002)

See also

• Binding neuron
• Bayesian approaches to brain function
• Brain-computer interfaces
• Free energy principle
• Models of neural computation
• Neural coding
• Neural oscillation
• Quantitative models of the action potential
• Spiking Neural Network

• Binding neuron
• Bayesian approaches to brain function
• Brain-computer interfaces
• Free energy principle
• Models of neural computation
• Neural coding
• Neural oscillation
• Quantitative models of the action potential
• Spiking Neural Network

脑-计算机接口自由能原理神经计算模型神经编码神经震荡动作电位定量模型神经网络

= 参考文献 =

Category:Biophysics Category:Computational neuroscience Category:Neuroscience

类别: 生物物理学类别: 计算神经科学科学类别: 神经科学

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