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第1行： − '''神经场'''方程是组织层面的模型，描述了粗粒度变量的时空演变，例如神经元群体中的突触或放电率活动。+ + + + + + + + + + + − 目录+ + + − * 1 简介+ − * 2 生理学动机 − * 3 数学框架 − * 4 动态行为 − ** 4.1 模式形成 − ** 4.2 行波 − * 5 参考文献 − * 6 外部链接 − ** 6.1 近期综述 − * 7 推荐阅读 − * 8 另见 + + + + + + + − 简介+ − 即使是一小块皮层中的神经元和突触的数量也是巨大的。因此，一种流行的建模方法是采用连续极限并研究神经网络（神经网络中，空间是连续的，并且宏观状态变量等于平均发放率）。首次尝试开发神经活动的连续近似，或可追溯到 1950 年代的 Beurle (1956) 和 1960 年代的 Griffith (1963, 1965)。通过关注给定体积的模型大脑中每单位时间被激活的神经元比例组织，Beurle 能够分析大规模大脑活动的触发和传导。然而，这项工作只处理没有难治或恢复变量的兴奋性神经元网络。Wilson和 Cowan(1972,1973) 在 1970 年代将 Beurle 的工作扩展到包括抑制性和兴奋性神经元以及不应期。Amari (1975, 1977) 在对连通性和放电率函数的自然假设下，在对经活动的连续模型中进行了进一步工作，特别是关于模式形成。Amari 考虑了局部激发和远端抑制对于具有典型皮质连接（通常称为“墨西哥帽连接”）的相互作用的抑制性和兴奋性神经元的混合群体，这是一个有效的模型。+ + + − 由于这些对动态神经场理论的开创性贡献，类似的模型已被用于研究EEG节律、视觉幻觉、短期记忆和运动感知机制。+ + + + + + + − 生理学动机+ + + − 神经领域考虑嵌入在粗粒度空间区域中的神经元群体（Wilson & Cowan 1973）。这种区域的颗粒反映了在初级感觉区域中观察到的微观或宏观柱状体，例如啮齿动物的桶状皮层 (Petersen 2007) 或哺乳动物的视觉皮层 (Hubel & Wiesel 1962, Saez et al. 1998)。此外，神经领域考虑瞬时种群放电率（种群编码），即在几毫秒的某个短时间间隔内放电神经元的数量。神经场模型中的时间变量是这个时间间隔的倍数。因此，神经场在时间和空间上是粗粒度的，并表示平均场模型。重要的是要提到与神经质量模型的相似性，后者忽略了空间扩展，但涉及到关于人口编码和时间粒度的相同基本假设。+ + + − Beurle (1956) 和 Wilson & Cowan (1972, 1973) 是最早从数学上推导出神经场模型方程的人之一。最近 Faugeras 等人。 (2009) 和 Bressloff (2009) 考虑到单个神经元的统计特性和随机动力学，给出了不同的推导。有趣的是，他们提出了一个扩展模型，同时考虑了平均活动及其方差的动态，而以前的模型只考虑了平均活动。以下段落给出了神经场模型的更具启发性的推导，其中考虑了神经作用的三个主要要素：输入脉冲在化学突触处引起的树突电流、受到树突电流影响的神经元的放电以及沿轴突分支的动作电位传输。诱发的树枝状电流 I(t) 服从+ + + − （公式1）+ − 具有化学受体对单个传入脉冲和传入瞬时脉冲序列 P(t) 的突触响应函数 h(t)。 单个突触的实验研究表明某些函数 h(t) 可以很好地近似突触反应行为。 在神经领域，人们考虑一个有效的平均突触响应函数，它描述了许多突触对神经群体网络中许多传入脉冲序列的平均响应。 当然，一组突触的平均时间响应可能与单个突触受体的响应函数不同，但形状可能相同。 因此，假设方程（1）也适用于群体树突电流 I（t），其中涉及有效参数和与突触群体响应函数 h（t）相关的群体放电率 P（t）。+ + + + − 在数学上，从 P(t) 到 I(t) 的转换是具有核 h(t) 的 Volterra 积分方程。 考虑微分方程通常比考虑积分方程更方便，因此可以尝试计算积分算子的逆。 这对于某些积分核函数 h(t) 是可能的。 一个简单且应用广泛的模型是 h(t)=exp(-t/τ)/τ，衰减时间常数为 τ。 然后+ − （公式2）+ − 应用链式法则和方程式。 ((1)) 可以写成微分方程+ − （公式3）+ + − 更详细的响应函数及其对应的微分算子是+ + − （公式4）+ + + + − 具有较短的上升时间 τ2 和较长的衰减时间 τ1。 对于τ1=τ2=τ，响应函数就是所谓的阿尔法函数 − （公式5） − 数学框架+ − 在神经组织中电活动传播的许多连续模型中，假设突触输入电流是突触前放电率函数的函数（Wilson & Cowan 1973）。 这些无限维动力系统通常是形式上的变体+ − （公式6）+ + + − 在这里，u(x,t) 被解释为一个神经场，表示在位置 x 和时间 t 的一组神经元的局部活动。右边的第二项代表突触输入，f 被解释为单个神经元的放电率函数。间隔 y 的神经元之间的连接强度表示为 w(y)，函数 w 通常称为突触足迹。该公式假设系统在空间上是均匀的和各向同性的。通常 w 反映全局激发 (w>0)、全局抑制 (w<0)、局部激发 - 侧抑制 (墨西哥帽子) 描述，例如视觉皮层的方向调整（Somers et al. 1995, Ben-Yishai et al. 1995），或反映抑制性中间神经元的短程相互作用和兴奋性锥体细胞的长程相互作用的局部抑制 - 横向激发（逆墨西哥帽） .周期性 w 也引起了一些关注（Ben-Yishai et al. 1995, Laing and Troy 2003）。参数 α 是突触的时间衰减率。空间积分下 u 的延迟参数表示由于信号在距离 y 上传播的有限速度引起的轴突传导延迟（Wilson & Cowan 1972；Nunez 1974；Jirsa & Haken 1997）；即 |y|/v 其中 v 是沿轴突纤维的动作电位的速度。最近的扩展涉及轴突传输速度 v 的分布（Atay 和 Hutt 2006）。+ − （图片1）+ − 发射率函数有多种自然选择，最简单的是 Heaviside 阶跃函数。 在这种情况下，神经元最大程度地激发（以绝对不应期设定的速率）或根本不激发，这取决于突触活动是否高于或低于某个阈值。 在制定平均场神经方程的统计力学方法中，这种全有或全无的响应被平滑的 S 形形式所取代（Wilson & Cowan 1972；Amari 1972）。+ + − 上面的简单数学模型可以自然地扩展到描述多个群体、皮质层、尖峰频率适应、神经调节、慢离子电流以及更复杂的突触和树突处理形式，如下面的评论文章中所述。+ + + + + − 动态行为+ − 在神经场模型中通常观察到的动态行为类型包括超出图灵不稳定性的空间和时间周期性模式 (Ermentrout 1979; Tass 1995)、局部活动区域，例如颠簸 (Kishimoto 1979) 和行波 (Ermentrout 1993; Pinto & Ermentrout 2001)。在后一种情况下，可以使用多电极记录和成像方法通过实验观察到相应的现象。特别是可以电刺激取自皮层的经过药物处理的组织切片 (Chervin et al. 1988; Golomb & Amitai 1997, Wu et al. 1999}、海马体 (Miles et al. 1995) 和丘脑 (Kim et al. 1995)。 1995). 在脑切片中，这些波可以采取癫痫发作期间的同步放电形式 (Connors & Amitai 1993) 和与感觉处理相关的传播兴奋 (Ermentrout & Kleinfeld 2001)。前额叶皮层的工作记忆（大脑中的信息的临时存储）（Colby et al. 1995, Goldman-Rakic 1995），头部方向系统的表征（Zhang 1996），以及视觉皮层的特征选择性，其中肿块的形成与特定神经元反应的调节有关（Ben-Yishai et al. 1995）。+ + − 模式形成+ − 神经场模型是非线性空间扩展系统，因此具有支持模式形成的所有必要成分。 这种行为的分析通常结合线性图灵不稳定性理论、弱非线性微扰分析和数值模拟进行。 在一维中，具有墨西哥帽连通性的单一种群模型可以支持全球周期性静止模式。 对于一个以上的人口，非固定（旅行）模式也是可能的。 在二维中可能会出现许多其他有趣的模式，例如螺旋波（Laing 2005）、目标波和双周期模式。 后面的这些图案采用条纹和棋盘状图案的形式，并且已被 Ermentrout 和 Cowan (1979) 和 Bressloff 等人 (2001) 与药物引起的视觉幻觉联系起来。 − 具有短程激发和长程抑制的神经场模型也能够支持空间局部解决方案，通常称为颠簸或多颠簸。对于触发率函数是具有阈值 h 的 Heaviside 阶跃函数的情况，Amari (1977) 能够构造如下形式的显式单凸点解+ + + + − （公式7）+ − 这样，在某个临界阈值以下，宽和窄的解决方案并存。在这两者中，更广泛的解决方案是稳定的。对于平滑的 sigmoidal 发射率，没有已知的封闭形式的空间局部解决方案，尽管使用最初为研究四阶图案形成系统而开发的技术已经获得了对多凹凸解决方案形式的深入了解 (Laing & Troy 2003)。静止的活动颠簸可以表现出各种动态不稳定性，包括 Hopf 分岔到空间局部振荡解或呼吸器（Folias 和 Bressloff 2004，Coombes 和 Owen 2005）。+ − 活动凸起的一个可能的计算角色是根据空间结构化网络中凸起的峰值位置对一组刺激特征进行编码。在同构网络的情况下，允许的特征集将形成反映网络底层拓扑的连续流形（吸引子）。因此，周期性刺激特征可以由同质环网络中的活动凸起编码（Ben-Yishai 等人 1995，Zhang 1995）。然而，同质网络的后果之一是，相对于与连续流形相切的空间平移，凹凸将略微稳定。这意味着在存在任意小的噪声水平的情况下，活动颠簸会随着时间缓慢漂移（Laing 和 Chow 2001）。构建对噪声具有鲁棒性的神经场模型的一种方法是引入某种形式的细胞双稳态（Camperi and Wang 1998，Fall et al 2004）。+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − 行波+ + − 对于具有 sigmoid 发射率函数和兴奋耦合的一维模型，可以找到将激发态连接到静止态的波前 (Ermentrout & McLeod 1993)。 此外，在具有混合（兴奋性和抑制性）耦合的系统或具有自适应电流的兴奋性系统中，单独的行进脉冲也是可能的。 对于具有阈值 h 的 Heaviside 发射率函数，已经获得了许多关于行波的精确结果。 例如，在 w(x)=exp(-|x|)/2 的纯兴奋性网络中，稳定行进前沿的速度采用显式形式+ + + + + − （公式8）+ − 现在，Richardson 等人在真实的神经组织（浸泡在 GABA_A 阻滞剂印防己毒素中的大鼠皮质切片）中间接确定了波速对阈值 h 的强烈依赖性。 2005. 这些实验利用了这样一个事实，即皮质神经元具有长的顶端树突，并且很容易被电场极化，并且癫痫样爆发可以由刺激电极引发。跨切片施加的正（负）电场增加（降低）波传播速度，这与假设正（负）电场降低（增加）阈值 h 的神经场理论的理论预测一致。+ − 行波的分岔结构可以使用所谓的埃文斯函数来分析。这最初是由 Evans (1975) 在关于霍奇金-赫胥黎类型的可兴奋神经轴突方程的稳定性定理的背景下制定的。这个复解析函数的零点决定了算子的法线谱，该算子是通过将系统关于其行波解进行线性化而获得的。神经场模型的扩展是最近才出现的，对于 Heaviside 放电率函数的特殊情况，现在已经研究了几个模型（Coombes & Owen 2004, 2005）。+ + + − 大多数神经场模型中的一个常见假设是网络是同质的和各向同性的，也就是说，权重分布取决于网络内交互种群之间的距离。然而，由于在皮层表层中发现的长程水平连接的斑块性质，真正的皮层更现实地被建模为各向异性和不均匀的二维介质（Bosking et al 1997）。重量分布的各向异性可能导致不同方向的波速变化，而不均匀性可能导致随时间变化的波剖面，并可能导致波传播失败（Bressloff 2001）。+ + + + − 参考文献+ + − 外部链接+ + − 近期综述+ − 推荐阅读+ − 另见+ + + + + + + + +
无编辑摘要
<strong><nowiki>神经场</nowiki></strong>方程是组织层面的模型，描述了粗粒度变量的时空演变，例如神经元群体中的突触或放电率活动。
==简介==
即使是一小块皮层中的神经元和突触的数量也是巨大的。因此，一种流行的建模方法是采用连续极限并研究神经网络（神经网络中，空间是连续的，并且宏观状态变量等于平均发放率）。首次尝试开发神经活动的连续近似，或可追溯到 1950 年代的 Beurle (1956) 和 1960 年代的 Griffith (1963, 1965)。通过关注给定体积的模型大脑中每单位时间被激活的神经元比例组织，Beurle 能够分析大规模大脑活动的触发和传导。然而，这项工作只处理没有难治或恢复变量的兴奋性神经元网络。Wilson和 Cowan(1972,1973) 在 1970 年代将 Beurle 的工作扩展到包括抑制性和兴奋性神经元以及不应期。Amari (1975, 1977) 在对连通性和放电率函数的自然假设下，在对经活动的连续模型中进行了进一步工作，特别是关于模式形成。Amari 考虑了局部激发和远端抑制对于具有典型皮质连接（通常称为“墨西哥帽连接”）的相互作用的抑制性和兴奋性神经元的混合群体，这是一个有效的模型。
由于这些对动态神经场理论的开创性贡献，类似的模型已被用于研究EEG节律、视觉幻觉、短期记忆和运动感知机制。
==生理学动机==
神经领域考虑嵌入在粗粒度空间区域中的神经元群体（Wilson & Cowan 1973）。这种区域的颗粒反映了在初级感觉区域中观察到的微观或宏观柱状体，例如啮齿动物的桶状皮层 (Petersen 2007) 或哺乳动物的视觉皮层 (Hubel & Wiesel 1962, Saez et al. 1998)。此外，神经领域考虑瞬时种群放电率（种群编码），即在几毫秒的某个短时间间隔内放电神经元的数量。神经场模型中的时间变量是这个时间间隔的倍数。因此，神经场在时间和空间上是粗粒度的，并表示平均场模型。重要的是要提到与神经质量模型的相似性，后者忽略了空间扩展，但涉及到关于人口编码和时间粒度的相同基本假设。
Beurle (1956) 和 Wilson & Cowan (1972, 1973) 是最早从数学上推导出神经场模型方程的人之一。最近 Faugeras 等人。 (2009) 和 Bressloff (2009) 考虑到单个神经元的统计特性和随机动力学，给出了不同的推导。有趣的是，他们提出了一个扩展模型，同时考虑了平均活动及其方差的动态，而以前的模型只考虑了平均活动。以下段落给出了神经场模型的更具启发性的推导，其中考虑了神经作用的三个主要要素：输入脉冲在化学突触处引起的树突电流、受到树突电流影响的神经元的放电以及沿轴突分支的动作电位传输。诱发的树枝状电流 I(t) 服从
<math>
I(t) = \int_{-\infty}^t {\rm d}\tau^\prime h(t-t^\prime) P(t^\prime)\label{eqn_1}
</math>
具有化学受体对单个传入脉冲和传入瞬时脉冲序列 P(t) 的突触响应函数 h(t)。 单个突触的实验研究表明某些函数 h(t) 可以很好地近似突触反应行为。 在神经领域，人们考虑一个有效的平均突触响应函数，它描述了许多突触对神经群体网络中许多传入脉冲序列的平均响应。 当然，一组突触的平均时间响应可能与单个突触受体的响应函数不同，但形状可能相同。因此，Eq.\ref{eqn_1} 也适用于群体树突电流 I（t），其中涉及有效参数和与突触群体响应函数 h（t）相关的群体放电率 P（t）。
Mathematically, the conversion from P(t) to I(t) is a Volterra integral equation with kernel h(t). It is often more convenient to consider differential equations than integral equations and thus one may try to calculate the inverse of the integral operator. This is possible for certain integral kernel functions h(t). A simple and widely-applied model is <math>h(t)=\exp(-t/\tau)/\tau</math> with the decay time constant <math>\tau</math>. Then
<math>
\begin{array}{lcl}
\frac{dI(t)}{dt}&=&-\frac{1}{\tau}\int_{-\infty}^t {\rm d}t^\prime \exp\left(-(t-t^\prime)/\tau\right)/\tau P(t^\prime) + \int_{-\infty}^t \left({\rm d}t^\prime/dt\right)|_{\infty}^t \exp\left(-(t-t^\prime)/\tau\right)/\tau P(t^\prime)\\
&=&-\frac{1}{\tau}I(t) + \frac{1}{\tau} P(t)
\end{array}
</math>
applying the chain rule and Eq. (\ref{eqn_1}) can be written as a differential equation
<math>
\hat{L} I(t)=P(t)\quad,\quad \hat{L}=\tau\frac{d}{dt}+1 .
</math>
A more detailed response function and its corresponding differental operator is
<math>
h(t)=\frac{1}{\tau_1-\tau_2}\left(e^{-t/\tau_1}-e^{t/\tau_2}\right)
\quad , \quad \hat{L}=\tau_1\tau_2\frac{d^2}{dt^2}+\left(\tau_1+\tau_2\right)\frac{d}{dt}+1
</math>
with the short rise time <math>\tau_2</math> and the long decay time <math>\tau_1</math>.
For <math>\tau_1=\tau_2=\tau</math>, the response function is the so-called alpha-function
<math>
h(t)=te^{-t/\tau}/\tau\quad , \quad \hat{L}=\tau^2\frac{d^2}{dt^2}+2\tau\frac{d}{dt}+1~.
</math>
==Mathematical Framework==
In many continuum models for the propagation of electrical activity in neural tissue it is assumed that the [[Synapse|synaptic input]] current is a function of the pre-synaptic firing rate function (Wilson & Cowan 1973). These
infinite dimensional dynamical systems are typically variations on the form
<math>
\frac{1}{\alpha} \frac{\partial u(x,t)}{\partial t} = -u + \int_{-\infty}^\infty {\rm d} y w(y) f(u(x-y,t - |y|/v)) .
</math>
[[Image:neuralfield.jpg|thumb|left|380px|A caricature of a neural field. Such models can include not only the effects of axonal delays, but the filtering of firing rate signals by both synapses and dendrites.]]
Here, u(x,t) is interpreted as a neural field representing the local activity of a population of neurons at position x and time t.
The second term on the right represents the synaptic input, with f interpreted as the firing rate function of a single neuron. The strength of connections between neurons separated by a distance y is denoted w(y), and the function w is often referred to as the synaptic footprint. This formulation assumes that the system is spatially homogeneous and isotropic. Typically w reflects global excitation (w>0), global inhibition (w<0), local excitation - lateral inhibition (Mexican hat) describing, e.g. [[orientation tuning]] in the visual cortex (Somers et al. 1995, Ben-Yishai et al. 1995), or local inhibition - lateral excitation (inverse Mexican hat) reflecting short-range interactions of inhibitory [[interneurons]] and long-range interactions of excitatory [[pyramidal cells]]. Periodic w have also attracted some attention (Ben-Yishai et al. 1995, Laing and Troy 2003).
The parameter <math>\alpha</math> is the temporal decay rate of the synapse.
The delayed argument to u under the spatial integral represents the [[axonal conduction delay]] arising from the finite speed of signals travelling over a distance y (Wilson & Cowan 1972; Nunez 1974; Jirsa & Haken 1997); namely |y|/v where v is the velocity of an [[action potential]] along axonal fibres. Recent extensions involve distributions of axonal transmission speeds v (Atay and Hutt 2006).
<br style="clear:both;"/>
There are several natural choices for the firing rate function, the simplest being a Heaviside step function. In this case a neuron fires maximally (at a rate set by its absolute refractory period) or not at all, depending on whether or not synaptic activity is above or below some threshold. In a [[Statistical_Mechanics_of_Neocortex|statistical mechanics]] approach to formulating mean-field neural equations this all or nothing response is replaced by a smooth sigmoidal form (Wilson & Cowan 1972; Amari 1972).
The simple mathematical model above can be naturally extended to describe multiple populations, cortical sheets, [[spike frequency adaptation]], [[neuromodulation]], slow ionic currents and more sophisticated forms of [[Synaptic Transmission|synaptic]] and [[dendritic processing]] as described in the review articles [[#Recent review articles|below]].
==Dynamic behaviour==
The sorts of dynamic behaviour that are typically observed in neural field models include spatially and temporally periodic patterns beyond a Turing instability (Ermentrout 1979; Tass 1995), localised regions of activity such as bumps (Kishimoto 1979) and [[Traveling_Wave|travelling wave]]s (Ermentrout 1993; Pinto & Ermentrout 2001). In the latter case corresponding phenomena may be observed experimentally using [[Multielectrode Array|multi-electrode]] recordings and [[Neuroimaging|imaging methods]]. In particular it is possible to electrically stimulate slices of pharmacologically treated tissue taken from the [[Neocortex|cortex]] (Chervin <i>et al</i>. 1988; Golomb & Amitai 1997,Wu <i>et al</i> 1999}, [[Hippocampus|hippocampus]] (Miles <i>et al</i>. 1995) and [[Thalamus|thalamus]] (Kim <i>et al</i>. 1995). In brain slices these waves can take the form of synchronous discharge seen during an [[Epilepsy|epileptic]] seizure (Connors & Amitai 1993) and spreading excitation associated with sensory processing (Ermentrout & Kleinfeld 2001). Interestingly, spatially localised bumps of activity have been linked to working memory (the temporary storage of information within the brain) in prefrontal cortex (Colby <i>et al</i>. 1995, Goldman-Rakic 1995), representations in the head-direction system (Zhang 1996), and feature selectivity in the visual cortex, where bump formation is related to the <i>tuning</i> of a particular neuron's response (Ben-Yishai <i>et al</i>. 1995).
===Pattern formation===
[[Image:hexagon.jpg|thumb|right|200px|An activity profile of a hexagonal pattern emerging beyond a Turing instability in a two-dimensional neural field model with short-range excitation and long-range inhibition.]]
Neural field models are nonlinear spatially extended systems and thus have all the necessary ingredients to support [[Pattern_formation|pattern formation]].
The analysis of such behaviour is typically performed with a mixture of linear [[Morphogenesis|Turing]] instability theory, weakly nonlinear perturbative analysis and numerical simulations. In one dimension single population models with Mexican-hat connectivity can support global periodic stationary patterns. With more than one population non-stationary (travelling) patterns are also possible.
In two dimensions many other interesting
patterns can occur such as spiral waves (Laing 2005), target waves and doubly periodic patterns. These latter patterns take the form of stripes and checkerboard like patterns, and have been linked by Ermentrout & Cowan (1979) and Bressloff et al (2001) to drug-induced visual [[hallucination]]s.
[[Image:threebump.jpg|thumb|left|200px|A spatially localised 3-bump solution in a two-dimensional neural field model.]]
Neural field models with short-range excitation and long-range inhibition are also able to support spatially localised solutions, commonly referred to as <i>bumps</i> or <i>multi-bumps</i>. For the case that the firing rate function is a Heaviside step function with threshold h Amari (1977) was able to construct an explicit one-bump solution of the form
<math>
u(x) = \int_0^\Delta w(x-y) {\rm d y}, \qquad u(0)=h=u(\Delta) ,
</math>
such that below some critical threshold there co-exists both a wide and a narrow solution. Of the two, it is the wider solution that is stable. For smooth sigmoidal firing rates no closed-form spatially localised solutions are known, though much insight into the form of multi-bump solutions has been obtained using techniques first developed for the study of fourth order pattern forming systems (Laing & Troy 2003). A stationary activity bump can exhibit a variety of dynamical instabilities including a Hopf bifurcation to a spatially localized oscillatory solution or <i>breather</i> (Folias and Bressloff 2004, Coombes and Owen 2005).
One possible computational role for an activity bump is to encode a set of stimulus features in terms of the peak location of the bump within a spatially-structured network. In the case of a homogeneous network, the set of allowed features will form a continuous manifold (attractor) that reflects the underlying topology of the network. Thus, a periodic stimulus feature can be encoded by an activity bump in a homogeneous ring network (Ben-Yishai et al 1995, Zhang 1995). However, one of the consequences of a homogeneous network is that the bump will be marginally stable with respect to spatial translations tangential to the continuous manifold. This means that the activity bump will slowly drift over time in the presence of arbitrarily small levels of noise (Laing and Chow 2001). One way to construct neural field models that are robust to noise is to introduce some form of cellular bistability (Camperi and Wang 1998, Fall et al 2004).
===Travelling Waves===
[[Image:front_pulse_contract.jpg|thumb|right|200px|A travelling pulse in a one dimensional neural field model with [[spike frequency adaptation]].]]
For one-dimensional models with sigmoidal firing rate functions and excitatory coupling it is possible to find wave fronts joining an excited state to a resting state (Ermentrout & McLeod 1993). Moreover, in systems with mixed (excitatory and inhibitory) coupling or excitatory systems with adaptive currents, solitary travelling pulses are also possible.
For a Heaviside firing rate function with threshold h many exact results about travelling waves have been obtained. For example the speed of a stable travelling front in a purely excitatory network with w(x)=exp(-|x|)/2 takes the explicit form
<math>
c = \frac{v(2h-1)}{2h-1 - 2 h v/\alpha}
</math>
<br style="clear:both;"/>
[[Image:Pinto.jpg|thumb|left|300px|A propagating wave of electrical activity in a slice experiment from the [http://www.bme.rochester.edu/bmeweb/faculty/pinto.html Pinto Lab] (with permission). Left: local field potential recordings from a linear array of 16 electrodes. Right: a smooth colour coded version of the field.
]]
The strong dependence of the wave speed on the threshold h has now been indirectly established in real neural tissue (rat cortical slices bathed in the [[GABA_A]] blocker [[picrotoxin]]) by Richardson <i>et al</i>. 2005. These experiments exploit the fact that cortical neurons have long apical [[dendrites]] and are easily polarisable by an electric field and that [[epileptiform]] bursts can be initiated by a stimulation electrode. An applied positive (negative) electric field across the slice increased (decreased) the speed of wave propagation, consistent with the theoretical predictions of neural field theory assuming that a positive (negative) electric field reduces (increases) the threshold h.
The bifurcation structure of travelling waves can be analysed using a so-called [[Evans function]]. This was originally formulated
by Evans (1975) in the context of a stability theorem about
excitable nerve axon equations of [[Hodgkin-Huxley_Model|Hodgkin–Huxley]] type. The zeros of this complex analytic function determine the normal spectrum of the operator obtained by linearising a system about its travelling wave solution.
The extension to neural field models is more recent and, for the special case of a Heaviside firing rate function, several models have now been studied (Coombes & Owen 2004, 2005).
One of the common assumptions in most neural field models is that the network is homogeneous and isotropic, that is, the weight distribution depends on the distance between interacting populations within the network. The real cortex, however, is more realistically modeled as an anisotropic and inhomogeneous two-dimensional medium due to the patchy nature of long-range horizontal connections found in superficial layers of cortex (Bosking et al 1997). Anisotropies in the weight distribution could lead to variations in wave speed in different directions, whereas inhomogeneities could lead to time-varying wave profiles and possibly wave propagation failure (Bressloff 2001).
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<b>Internal references</b>
* Yuri A. Kuznetsov (2006) [[Andronov-Hopf bifurcation]]. Scholarpedia, 1(10):1858.
* John W. Milnor (2006) [[Attractor]]. Scholarpedia, 1(11):1815.
* Eugene M. Izhikevich (2006) [[Bursting]]. Scholarpedia, 1(3):1300.
* Philip Holmes and Eric T. Shea-Brown (2006) [[Stability]]. Scholarpedia, 1(10):1838.
* S. Murray Sherman (2006) [[Thalamus]]. Scholarpedia, 1(9):1583.
== External links ==
* [http://www.maths.nott.ac.uk/personal/sc/ Stephen Coombes' website]
===Recent review articles===
* [http://www.worldscinet.com/ijmpb/11/1120/S0217979297001209.html P C Bressloff and S Coombes. Physics of the extended neuron. Int. J. Mod. Phys. B, 11:2343-2392, 1997.]
[http://www.maths.nott.ac.uk/personal/sc/Papers/Papers97/extended.pdf Preprint available].
* [http://www.ingentaconnect.com/content/iop/ropp/1998/00000061/00000004/art00002 B Ermentrout. Neural networks as spatiotemporal pattern-forming systems. Reports on Progress in Physics, 61:353-430, 1998.]
[http://www.math.pitt.edu/~bard/pubs/nnetrev.pdf Preprint available].
* [http://humanapress.com/index.php?option=com_journalshome&task=articledetails&category=journals&article_code=NI:2:2:183 V K Jirsa. Connectivity and Dynamics of Neural Information Processing. Neuroinformatics, 2:183-204, 2004.]
[http://www.ccs.fau.edu/~jirsa/JirsaNeuroinformatics2004draft.pdf Preprint available].
* [http://dx.doi.org/10.1146/annurev.neuro.28.061604.135637 T P Vogels TP, K Rajan, and L F Abbott LF. Neural network dynamics. Annual Reviews Neuroscience, 28:357-76, 2005.]
[http://neurotheory.columbia.edu/~tim/publs/VogelsAnnuRev05.pdf Preprint available].
* [http://dx.doi.org/10.1007/s00422-005-0574-y S Coombes. Waves and bumps in neural field theories. Biological Cybernetics, 93:91-108, 2005.]
[http://eprints.nottingham.ac.uk/archive/00000153/01/Preprint.pdf Preprint available].
==Recommended reading==
* Electric Fields of the Brain: The Neurophysics of EEG, by Paul L. Nunez and Ramesh Srinivasan, Oxford University Press, 2006.
== See also ==
[[Propagating Waves]]
[[Pattern formation]]
[[Category:Synchronization]]
[[Category:Oscillators]]
[[Category:Pattern Formation]]
[[Category: Computational Neuroscience]]
[[Category: Dynamical Systems]]
[[Category: Neural Networks]]
参考 http://www.scholarpedia.org/article/Neural_fields
参考 http://www.scholarpedia.org/article/Neural_fields