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==Mathematical Framework==

==数学框架==

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

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

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[[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.]]

[[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. 

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 <math>\alpha</math> 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）。

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). 

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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).

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）。

==Dynamic behaviour==

==Dynamic behaviour==