神经雪崩

此词条由神经动力学读书会词条梳理志愿者Spidey0o0Zheng翻译审校，未经专家审核，带来阅读不便，请见谅。

图1：数据表示的示意图。超过三个标准差的局部场电位（LFPs）用黑色方块表示。

图2：急性皮层切片中的神经雪崩。

神经雪崩是神经元网络中的一连串爆发性活动，其大小分布可以用幂律来近似，如临界沙堆模型（Bak等人，1987）。神经雪崩见于培养的和急性皮质切片（Beggs和Plenz，2003；2004）。在这些新皮层切片中，活动的特点是持续几十毫秒的短暂爆发，中间有几秒钟的静止期。当用多电极阵列观察时，在爆发期间被驱动超过阈值的电极数量近似于幂律分布。虽然这种现象具有高度的稳定性和可重复性，但它与完整大脑中的生理过程的关系目前还不清楚。

实验观察

图3：雪崩示例。显示了7个帧，其中每个帧代表电极阵列在一个4毫秒的时间布长中的活动。雪崩是一系列连续的活动帧，开始和结束均是空白帧。雪崩的大小是由活动电极的总数决定的。这里显示的雪崩的大小为9。

幂律尺寸分布

这个片段说明，多通道数据可以被分解为没有活动的帧和至少有一个活动电极的帧，这些电极可能接收来自几个神经元的活动。由非活动帧包围的连续活动帧序列可以称为雪崩。所示的雪崩例子的大小为9，因为这是被驱动超过阈值的电极总数。雪崩大小的分布方式几乎符合幂律分布。由于阵列中的电极数量有限，幂律在阵列大小为60之前就开始向下弯曲切断，但是对于更大的电极阵列，可以看到幂律会延伸得更远。

图4：雪崩尺寸分布。 A，用60个电极阵列记录的急性切片局部场电位的大小分布，以对数空间绘制。实际数据显示为黑色，而泊松模型的输出显示为红色。在泊松模型中，每个电极的发射速度与实际数据中看到的相同，但独立于所有其他电极。注意这两条曲线之间的巨大差异。实际数据在1-35的尺寸下几乎是一条直线；在这一点之后，存在由电极阵列大小引起的截止。直线表示幂律，表明网络在临界点附近运行（未发表的数据由W. Chen, C. Haldeman, S. Wang, A. Tang, J.M. Beggs记录）。 B，尖峰的雪崩大小分布可以用概率超过三个数量级的直线近似，没有A组中看到的尖锐的截止点。数据是用512个电极阵列从浸泡在高钾和零镁的急性皮层切片中收集的（A. Litke, S. Sher, M. Grivich, D. Petrusca, S. Kachiguine, J. M. Beggs未发表的工作）。峰值以-3个标准差设定阈值，并且未进行分类。数据在1.2ms时合并，以匹配60μm的短电极间距离。从培养基中记录的皮层切片培养中也获得了类似于A和B的结果。

幂律分布的公式是:

[math]\displaystyle{ P(S)=kS^{-\alpha}\, }[/math]

其中[math]\displaystyle{ P(S) }[/math]是观察到大小为[math]\displaystyle{ S }[/math]的雪崩的概率，[math]\displaystyle{ \alpha }[/math]是给出对数图中幂律斜率的指数，[math]\displaystyle{ 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]\displaystyle{ \alpha\approx 1.5 }[/math]，但在不同阵列的尖峰记录中，指数[math]\displaystyle{ \alpha\approx2.1 }[/math]。指数差异背后的原因仍在探索中。需要注意的是，如果每个电极上的活动是独立驱动的，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），峰值计数统计中的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，神经雪崩是从分布式网络中的集体过程中产生的。

重复的雪崩模式

图5：急性切片的重复雪崩的家族。每个家庭（1-4）显示了一组三个类似的雪崩。与50组随机数据相比，每组内的相似性高于偶然预期。重复雪崩也发生在皮层切片培养中，其中平均有30±14（平均±标准差）个不同的可重复雪崩家族，每个家族包含约23个雪崩（Beggs和Plenz，2004）。重复雪崩在10小时内是稳定的，时间精度为4毫秒，这表明它们可以作为神经网络中存储信息的基底。

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多巴胺激动剂和NMDA（Beggs和Plenz，2003；Stewart和Plenz，2006）或高K⁺和低Mg²⁺的人工脑脊液中浸泡时，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

图6：分支过程的三个阶段。The three regimes of a branching process. Top, when the branching parameter, [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \sigma }[/math] processing units in the next time step. The number [math]\displaystyle{ \sigma }[/math] is called the branching parameter and can be thought of as the expected value of this ratio:

[math]\displaystyle{ \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]\displaystyle{ \sigma\ , }[/math] as shown in the figure.

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]\displaystyle{ \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]\displaystyle{ \sigma }[/math] is set by the following relationship:

[math]\displaystyle{ \sigma_i=\sum_{j=1}^\mathit{N} \mathit{p_{ij}} }[/math]

where [math]\displaystyle{ \sigma_i }[/math] is the expected number of descendant processing units activated by unit [math]\displaystyle{ i\ , }[/math] [math]\displaystyle{ N }[/math] is the number of units that unit [math]\displaystyle{ i }[/math] connects to, and [math]\displaystyle{ p_{ij} }[/math] is the probability that activity in unit [math]\displaystyle{ i }[/math] will transmit to unit [math]\displaystyle{ 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]\displaystyle{ \sigma }[/math] is tuned to the critical point ([math]\displaystyle{ \sigma=1 }[/math]), as shown in the figure (Haldeman and Beggs, 2005). When the model is tuned moderately above ([math]\displaystyle{ \sigma\gt 1 }[/math]) or below ([math]\displaystyle{ \sigma\lt 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.

雪崩的影响

当一个可调谐系统运行在一个产生幂律分布的系统中时，它被说成是运行在临界点。严格地说，只有无限大的系统才能在临界点上运行，但这里的"临界"一词是用来描述有限系统的行为，如果它们被扩展到无限大，就会接近临界点。幂律雪崩大小分布对这四个领域的神经网络的信息处理有潜在的影响。

• 信息传输。当神经网络调整到临界点时，它们具有最佳的信息传输（Beggs和Plenz，2003；Bertschinger和Natschlager，2004；Kinouchi和Copelli，2006），因为在强信号传播和抗饱和之间存在平衡。
• 信息存储。当一个基于分支过程的递归网络被调整到临界点时，显著重复的雪崩模式的数量被最大化（Haldeman和Beggs，2005）。在临界点，存在着强连接和弱连接的混合，允许各种独立稳定的活动模式。
• 计算能力。通过改变尖峰网络模型中突触重量的变化，Bertschinger和Natschlager（Bertschinger和Natschlager 2004）能够产生显示出衰减、持续和扩张活动的网络。这些区域分别对应于亚临界、临界和超临界动力学。他们发现，在广泛的计算任务中，调谐到临界点的网络比调谐到亚临界或超临界动态的网络更有效。
• 稳定性。当一个循环分支网络模型被调整到临界点时，它产生的轨迹基本上是平行的，这意味着该网络处于稳定的边缘（Bertschinger和Natschlager，2004；Haldeman和Beggs，2005）。在这种情况下，轨迹仍然是稳定的，但可以通过较小的校正输入进行控制。

当网络在发生神经雪崩的临界点附近运行时，可以同时优化所有这些信息处理任务。

神经雪崩与其他系统的关系

Power law distributions of event sizes are often seen in complex phenomena including earthquakes, phase transitions, percolation, forest fires, financial market fluctuations, avalanches in the game of life and a host of others (Bak, 1996). In some specific cases, this similarity appears to be more than superficial. For example, earthquake models incorporate local rules in which forces at one site are distributed to nearest neighbors without dissipation. This conservation of forces is similar to the conservation of probabilities in the critical branching model described above. This suggests that conservation of synaptic strengths, as reported in (Royer and Pare, 2003) could be a mechanism responsible for maintaining a network near the critical point. In a related idea, simulations indicate that networks can be kept nearly critical when the total sum of synaptic strengths hovers near a constant value (Hsu and Beggs, 2006). This could be accomplished through a mechanism like synaptic scaling (Turrigiano and Nelson, 2000), which has been observed experimentally. Finally, recently "burned" areas in forest fire models are refractory, while unburned areas are more likely to ignite. This balance of refractoriness and excitability combine to maintain the system near the critical point. Recent models of neuronal avalanches (Levina, Herrmann and Geisel, 2005) have suggested that short-term synaptic depression and facilitation may also serve to drive neuronal networks toward the critical point where avalanches occur. Thus, an understanding of power laws in diverse complex systems can suggest mechanisms that might underlie criticality in neuronal networks.

图8：在电容耦合霓虹灯的二维晶格中对雪崩产生的实验观察。

A simple electronic model of avalanche generation consists of a two-dimensional array of neon lamps, each one connected to a resistor towards a global DC control voltage and capacitively coupled to its von Neumann neighbors. Neon lamps possess rich dynamical properties: as the applied voltage changes, the transition between the "on" and "off" phases is at the same time significantly hysteretic and stochastic (Dance, 1968). The system displays two phases, [math]\displaystyle{ I }[/math] and [math]\displaystyle{ II }[/math], respectively characterized by low and high event rate and spatiotemporal order: the transition between them is strongly hysteretic, hence unequivocally first-order. Nevertheless, close to the spinal point of the [math]\displaystyle{ I\rightarrow II }[/math] transition, critical precursors emerge in the form of avalanches (Fig. 8) having the same scaling exponents characterizing neural activity, namely [math]\displaystyle{ \alpha\approx3/2 }[/math] for size and [math]\displaystyle{ \alpha\approx2 }[/math] for duration (Minati et al., 2016).

外部链接和致谢

作者的个人网页

这项工作的撰写和图中的实验得到了美国国家科学基金会的资助和印第安纳大学的资助，约翰·贝格斯（John Beggs）的资助号为0343636。关于神经雪崩的最初工作是在迪特玛·普伦茨（Dietmar Plenz）的实验室完成的，由美国国立卫生研究院的院内研究项目资助。

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参见

复杂性, 复杂系统, 生命游戏, Neuropercolation, 自组织临界性, Statistical Mechanics of Neocortex