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第22行： − where <math>P(S)</math> is the probability of observing an avalanche of size <math>S\ ,</math> <math>\alpha</math> is the exponent that gives the slope of the power law in a log-log graph, and <math>k</math> is a proportionality constant. For experiments with [[slice culture]]s, the size distribution of avalanches of [[local field potential]]s has an exponent <math>\alpha\approx 1.5\ ,</math> but in recordings of spikes from a different array the exponent is <math>\alpha\approx2.1\ .</math> The reasons behind this difference in exponents are still being explored. It is important to note that a power law distribution is not what would be expected if activity at each electrode were driven independently. An ensemble of uncoupled, Poisson-like processes would lead to an exponential distribution of event sizes. Further, while power laws have been reported for many years in neuroscience in the temporal correlations of single time-series data (e.g., the power spectrum from [[Electroencephalogram|EEG]] (Linkenkaer-Hansen et al, 2001; Worrell et al, 2002), [[Fano factor|Fano]] or [[Allan factor]]s in [[Spike Statistics|spike count statistics]] (Teich et al, 1997), [[neurotransmitter]] secretion times (Lowen et al, 1997), [[ion channel]] fluctuations (Toib et al, 1998), interburst intervals in neuronal cultures (Segev et al, 2002)), they had not been observed from interactions seen in multielectrode data. Thus neuronal avalanches emerge from collective processes in a distributed network.+ − + − − ===重复的雪崩模式=== − [[Image:急性切片的重复雪崩的家族.jpg|thumb|200px|left|图5：急性切片的重复雪崩的家族。Families of repeating avalanches from an acute slice. Each family (1-4) shows a group of three similar avalanches. Similarity within each group was higher than expected by chance when compared to 50 sets of shuffled data. Repeating avalanches also occur in cortical [[slice culture]]s, where there are on average 30 ± 14 (mean ± s.d.) distinct families of reproducible avalanches, each containing about 23 avalanches (Beggs and Plenz, 2004). Repeating avalanches are stable for 10 hrs and have a temporal precision of 4 ms, suggesting that they could serve as a substrate for storing information in [[neural networks]].]] − − − [[沙堆模型|临界沙堆模型]]中的雪崩在其形成的模式中是随机的，与之相比，[[局部场电位]]的雪崩发生的时空模式比预期的偶然性更频繁（Beggs和Plenz，2004）。图中显示了一个急性皮层切片的几个这样的模式。这些模式在长达10个小时的时间内是可重复的，其时间精度为4ms（Beggs和Plenz，2004）。这些模式的稳定性和精确性表明，神经雪崩可以被[[神经网络]]用作存储信息的基底。在这个意义上，雪崩似乎与在动物执行认知任务时在体内观察到的动作电位序列相似。目前还不清楚体内数据的重复活动模式是否也是雪崩。 − − ===普遍性=== − − − 在上述示例中，雪崩是在浸泡在培养基中的皮层切片培养中产生的，但当急性皮层切片在含有[https://en.wikipedia.org/wiki/Dopamine 多巴胺]激动剂和[https://en.wikipedia.org/wiki/N-Methyl-D-aspartic_acid NMDA]（Beggs和Plenz，2003；Stewart和Plenz，2006）或高K<sup>+</sup>和低Mg<sup>2+</sup>的人工[https://en.wikipedia.org/wiki/Cerebrospinal_fluid 脑脊液]中浸泡时，也可能在急性皮层切片中产生雪崩。诱发雪崩的不同方式表明，它们不仅仅局限于一组实验条件。 − − ====其他系统中的初步报告==== − − 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== − [[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.]] − − 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: − :<math> − \sigma=\frac{\mbox{Descendants}}{\mbox{Ancestors}} − </math> − where ''Ancestors'' is the number of processing units active at time step ''t'' and ''Descendants'' is the number of processing units active at time step ''t + 1''. There are three general regimes for <math>\sigma\ ,</math> as shown in the figure. − − [[Image:分支模型捕获数据的两个主要特征.jpg|thumb|550px|left|图7：分支模型捕获数据的两个主要特征。A branching model captures the two main features of the data. A, Avalanche size distribution from data and model compared, showing fairly close correspondence. Note that both show a straight line portion in log-log space, extending over avalanche sizes 1-35. Model was tuned to the critical point such that the branching parameter, <math>\sigma\ ,</math> equaled unity. There were no other free parameters. B, Three families of significantly similar avalanches produced by the model. Note similarity to avalanche families produced by actual data shown earlier.]] − − At the level of a single processing unit in the network, the branching parameter <math>\sigma</math> is set by the following relationship: − :<math> − \sigma_i=\sum_{j=1}^\mathit{N} − \mathit{p_{ij}} − </math> − where <math>\sigma_i</math> is the expected number of descendant processing units activated by unit <math>i\ ,</math> <math>N</math> is the number of units that unit <math>i</math> connects to, and <math>p_{ij}</math> is the probability that activity in unit <math>i</math> will transmit to unit <math>j\ .</math> Because some transmission probabilities are greater than others, preferred paths of transmission may occur, leading to reproducible avalanche patterns. Both the power law distribution of avalanche sizes and the repeating avalanches are qualitatively captured by this model when <math>\sigma</math> is tuned to the critical point (<math>\sigma=1</math>), as shown in the figure (Haldeman and Beggs, 2005). When the model is tuned moderately above (<math>\sigma>1</math>) or below (<math>\sigma<1</math>) the critical point, it fails to produce a power law distribution of avalanche sizes. This phenomenological model does not explicitly state the cellular or synaptic mechanisms that may underlie the branching process, and many of this model's predictions need to be tested.
→幂律尺寸分布
P(S)=kS^{-\alpha}\,
P(S)=kS^{-\alpha}\,
</math>
</math>
The reasons behind this difference in exponents are still being explored. It is important to note that a power law distribution is not what would be expected if activity at each electrode were driven independently. An ensemble of uncoupled, Poisson-like processes would lead to an exponential distribution of event sizes. Further, while power laws have been reported for many years in neuroscience in the temporal correlations of single time-series data (e.g., the power spectrum from [[Electroencephalogram|EEG]] (Linkenkaer-Hansen et al, 2001; Worrell et al, 2002), [[Fano factor|Fano]] or [[Allan factor]]s in [[Spike Statistics|spike count statistics]] (Teich et al, 1997), [[neurotransmitter]] secretion times (Lowen et al, 1997), [[ion channel]] fluctuations (Toib et al, 1998), interburst intervals in neuronal cultures (Segev et al, 2002)), they had not been observed from interactions seen in multielectrode data. Thus neuronal avalanches emerge from collective processes in a distributed network.
其中<math>P(S)</math>是观察到大小为<math>S</math>的雪崩的概率，</math> <math>\alpha</math>是给出对数图中幂律斜率的指数，<math>k</math>是比例常数。对于切片培养实验，[[局部场电位]]雪崩的大小分布的指数<math>\alpha\approx 1.5 </math>，但在不同阵列的尖峰记录中，指数<math>\alpha\approx2.1\。指数差异背后的原因仍在探索中。需要注意的是，如果独立驱动每个电极上的活性，则幂律分布不是预期的分布。非耦合类泊松过程的集合将导致事件大小的指数分布。此外，虽然神经科学多年来在单个时间序列数据的时间相关性中报告了幂律（例如，脑电图的功率谱（Linkenkaer-Hansen等人，2001；Worrell等人，2002），峰计数统计中的Fano或Allan因子（Teich等人，1997），神经递质分泌时间（Lowen等人，1997），离子通道波动（Toib等人，1998），神经元培养中的爆发间期（Segev等人，2002）），从多电极数据中观察到的相互作用中未观察到。因此，神经元雪崩是从分布式网络中的集体过程中产生的。
其中<math>P(S)</math>是观察到大小为<math>S</math>的雪崩的概率，<math>\alpha</math>是给出对数图中幂律斜率的指数，<math>k</math>是比例常数。对于切片培养实验，[[局部场电位]]雪崩的大小分布的指数<math>\alpha\approx 1.5 </math>，但在不同阵列的尖峰记录中，指数<math>\alpha\approx2.1\。指数差异背后的原因仍在探索中。需要注意的是，如果独立驱动每个电极上的活性，则幂律分布不是预期的分布。非耦合类泊松过程的集合将导致事件大小的指数分布。此外，虽然神经科学多年来在单个时间序列数据的时间相关性中报告了幂律（例如，脑电图的功率谱（Linkenkaer-Hansen等人，2001；Worrell等人，2002），峰计数统计中的Fano或Allan因子（Teich等人，1997），神经递质分泌时间（Lowen等人，1997），离子通道波动（Toib等人，1998），神经元培养中的爆发间期（Segev等人，2002）），从多电极数据中观察到的相互作用中未观察到。因此，神经元雪崩是从分布式网络中的集体过程中产生的。
==雪崩的含义 Implications of avalanches==
==雪崩的含义 Implications of avalanches==