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[[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.]]

[[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.]]

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。

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.]]

[[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

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。

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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).

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）。

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).

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）。

===Travelling Waves===

===Travelling Waves===