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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。

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。

[[Image:图1 数据表示示意图.jpg|thumb|400px|right|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。]]

[[Image:图1 数据表示示意图.jpg|thumb|400px|right|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。]]

[[Image:Beggs_avalanche_movie.gif|frame|right|图2:急性皮层切片中的神经雪崩。]]

[[Image:Beggs_avalanche_movie.gif|frame|right|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[http://www.scholarpedia.org/article/Bursting 爆发性]活动，其大小分布可以用[[幂律分布 power law|幂律]]来近似，如[[沙堆模型|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[http://www.scholarpedia.org/article/Multielectrode_array 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。

'''神经雪崩'''是[[神经元]]网络中的一连串[http://www.scholarpedia.org/article/Bursting 爆发性]活动，其大小分布可以用[[幂律分布 power law|幂律]]来近似，如[[沙堆模型|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[http://www.scholarpedia.org/article/Multielectrode_array 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 Experimental Observations==

==实验观察==

[[Image:雪崩示例.jpg|thumb|400px|right|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。]]

[[Image:雪崩示例.jpg|thumb|400px|right|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 Power law size distribution===

===幂律尺寸分布===

The movie illustrates that multi-channel data can be broken down into frames where there is no activity and where there is at least one active electrode, which may pick up the activity from several neurons. A sequence of consecutively active frames, bracketed by inactive frames, can be called an avalanche.  

The movie illustrates that multi-channel data can be broken down into frames where there is no activity and where there is at least one active electrode, which may pick up the activity from several neurons. A sequence of consecutively active frames, bracketed by inactive frames, can be called an avalanche.  

The example avalanche shown has a size of 9 because this is the total number of electrodes that were driven over threshold. Avalanche sizes are distributed in a manner that is nearly fit by a [[power law]]. Due to the limited number of electrodes in the array, the power law begins to bend downward in a cutoff well before the array size of 60. But for larger electrode arrays, the power law is seen to extend much further.  

The example avalanche shown has a size of 9 because this is the total number of electrodes that were driven over threshold. Avalanche sizes are distributed in a manner that is nearly fit by a [[power law]]. Due to the limited number of electrodes in the array, the power law begins to bend downward in a cutoff well before the array size of 60. But for larger electrode arrays, the power law is seen to extend much further.  

[[Image:雪崩尺寸分布.jpg|thumb|500px|right|图4:雪崩尺寸分布。 A, Distribution of sizes from acute slice [[LFP]]s recorded with a 60 electrode array, plotted in log-log space. Actual data are shown in black, while the output of a [[Poisson model]] is shown in red. In the Poisson model, each electrode fires at the same rate as that seen in the actual data, but independently of all the other electrodes. Note the large difference between the two curves. The actual data follow a nearly straight line for sizes from 1- 35; after this point there is a cutoff induced by the electrode array size. The straight line is indicative of a power law, suggesting that the network is operating near the [[self-organized criticality|critical point]] (unpublished data recorded by W. Chen, C. Haldeman, S. Wang, A. Tang, J.M. Beggs). B, Avalanche size distribution for spikes can be approximated by a straight line over three orders of magnitude in probability, without a sharp cutoff as seen in panel A. Data were collected with a 512 electrode array from an acute cortical slice bathed in high potassium and zero magnesium (unpublished work of A. Litke, S. Sher, M. Grivich, D. Petrusca, S. Kachiguine, J.M. Beggs). Spikes were thresholded at -3 standard deviations and were not sorted. Data were binned at 1.2 ms to match the short interelectrode distance of 60 μm. Results similar to A and B are also obtained from cortical slice cultures recorded in culture medium.]]

[[Image:雪崩尺寸分布.jpg|thumb|500px|right|图4：雪崩尺寸分布。 A, Distribution of sizes from acute slice [[LFP]]s recorded with a 60 electrode array, plotted in log-log space. Actual data are shown in black, while the output of a [[Poisson model]] is shown in red. In the Poisson model, each electrode fires at the same rate as that seen in the actual data, but independently of all the other electrodes. Note the large difference between the two curves. The actual data follow a nearly straight line for sizes from 1- 35; after this point there is a cutoff induced by the electrode array size. The straight line is indicative of a power law, suggesting that the network is operating near the [[self-organized criticality|critical point]] (unpublished data recorded by W. Chen, C. Haldeman, S. Wang, A. Tang, J.M. Beggs). B, Avalanche size distribution for spikes can be approximated by a straight line over three orders of magnitude in probability, without a sharp cutoff as seen in panel A. Data were collected with a 512 electrode array from an acute cortical slice bathed in high potassium and zero magnesium (unpublished work of A. Litke, S. Sher, M. Grivich, D. Petrusca, S. Kachiguine, J.M. Beggs). Spikes were thresholded at -3 standard deviations and were not sorted. Data were binned at 1.2 ms to match the short interelectrode distance of 60 μm. Results similar to A and B are also obtained from cortical slice cultures recorded in culture medium.]]

[[幂律分布]]的公式是:

[[幂律分布]]的公式是:

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.

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 Repeating avalanche patterns===

===重复的雪崩模式===

[[Image:急性切片的重复雪崩的家族.jpg|thumb|200px|left|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]].]]  

[[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]].]]  

==雪崩模型 Models of avalanches==  

==雪崩模型 Models of avalanches==  

[[Image:分支过程的三个阶段.jpg|thumb|200px|right|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.]]

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).  

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).  

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.  

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|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.]]

[[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:  

At the level of a single processing unit in the network, the branching parameter <math>\sigma</math> is set by the following relationship:  

Optimizing all of these information processing tasks may occur simultaneously when a network operates near the critical point, where neuronal avalanches occur.  

Optimizing all of these information processing tasks may occur simultaneously when a network operates near the critical point, where neuronal avalanches occur.  

==神经雪崩与其他系统的关系 Relationship of neuronal avalanches to other systems==

==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.   

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.   

[[Image:Neon lattice avalanche.gif‎|frame|220px|right|F8|Experimental observation of avalanche generation in a 2D lattice of capacitively-coupled neon lamps.]]

[[Image:Neon lattice avalanche.gif‎|frame|220px|right|F8|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>I</math> and <math>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>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>\alpha\approx3/2</math> for size and <math>\alpha\approx2</math> for duration (Minati et al., 2016).

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>I</math> and <math>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>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>\alpha\approx3/2</math> for size and <math>\alpha\approx2</math> for duration (Minati et al., 2016).