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第39行： − + − Models that explicitly predicted avalanches of neural activity include the work of Herz and Hopfield (1995) which connects the reverberations in a neural network to the power law distribution of earthquake sizes. Also notable is the work of Eurich, Hermann and Ernst (2002), which predicted that the avalanche size distribution from a network of globally coupled nonlinear threshold elements should have an exponent of <math>\alpha=1.5\ .</math> Remarkably, this exponent turned out to match that reported experimentally (Beggs and Plenz, 2003). + + + − A branching process model is described here in more detail (Harris, 1989; Beggs and Plenz, 2003; Haldeman and Beggs, 2005; reviewed in Vogels et al, 2005), because it captures both the power law distribution of avalanche sizes and the reproducible activity sequences observed in the data. In this model, a processing unit which is active at one time step will produce, on average, activity in <math>\sigma</math> processing units in the next time step. The number <math>\sigma</math> is called the ''branching parameter'' and can be thought of as the expected value of this ratio:
→雪崩模型 Models of avalanches
在离体水蛭神经节（V.Torre，conference talk）的棘波和分离皮层培养的棘波（L.Bettencourt；R.Alessio，personal communications）中也观察到序列大小的幂律分布，这表明雪崩现象可能在体外制剂中相当普遍。初步报告还表明，在清醒和休息的灵长类动物的表层皮层中存在雪崩（Petermann等人，2006年）。这些报告尚未发表，在此仅表明研究人员目前正在探索各种制剂中的雪崩概念。
在离体水蛭神经节（V.Torre，conference talk）的棘波和分离皮层培养的棘波（L.Bettencourt；R.Alessio，personal communications）中也观察到序列大小的幂律分布，这表明雪崩现象可能在体外制剂中相当普遍。初步报告还表明，在清醒和休息的灵长类动物的表层皮层中存在雪崩（Petermann等人，2006年）。这些报告尚未发表，在此仅表明研究人员目前正在探索各种制剂中的雪崩概念。
==雪崩模型 Models of avalanches==
==雪崩模型==
[[Image:分支过程的三个阶段.jpg|thumb|200px|right|图6：分支过程的三个阶段。The three regimes of a branching process. Top, when the branching parameter, <math>\sigma\ ,</math> is less than unity, the system is subcritical and activity dies out over time. Middle, when the branching parameter is equal to unity, the system is critical and activity is approximately sustained. In actuality, activity will die out very slowly with a power law tail. Bottom, when the branching parameter is greater than unity, the system is supercritical and activity increases over time.]]
[[Image:分支过程的三个阶段.jpg|thumb|200px|right|图6：分支过程的三个阶段。The three regimes of a branching process. Top, when the branching parameter, <math>\sigma\ ,</math> is less than unity, the system is subcritical and activity dies out over time. Middle, when the branching parameter is equal to unity, the system is critical and activity is approximately sustained. In actuality, activity will die out very slowly with a power law tail. Bottom, when the branching parameter is greater than unity, the system is supercritical and activity increases over time.]]
明确预测神经活动雪崩的模型包括Herz和Hopfield（1995）的工作，该模型将神经网络中的混响与地震大小的幂律分布联系起来。同样值得注意的是Eurich、Hermann和Ernst（2002）的工作，他们预测来自全局耦合的非线性阈值元素网络的雪崩大小分布应该有一个<math>\alpha=1.5</math>的指数。值得注意的是，这个指数与实验报告相吻合（Beggs和Plenz，2003）。
这里更详细地描述了一个分支过程模型（Harris, 1989; Beggs and Plenz, 2003; Haldeman and Beggs, 2005; reviewed in Vogels et al, 2005），因为它既能捕捉到雪崩大小的幂律分布，又能观察到数据中可重复的活动序列。在该模型中，在一个时间步长处于活动状态的处理单元将在下一个时间步长中平均产生<math>\sigma</math>处理单元中的活动。<math>\sigma</math>被称为==分支参数==，可以被认为是这个比率的预期值。
:<math>
:<math>
\sigma=\frac{\mbox{Descendants}}{\mbox{Ancestors}}
\sigma=\frac{\mbox{Descendants}}{\mbox{Ancestors}}