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[[Image:Figure1.jpg|thumb|400px|right|Schematic of data representation. [[Local field potential]]s (LFPs) that exceed three standard deviations are represented by black squares.]]

[[Image:Figure1.jpg|thumb|400px|right|Schematic of data representation. [[Local field potential]]s (LFPs) that exceed three standard deviations are represented by black squares.]]

[[Image:Beggs_avalanche_movie.gif|frame|right|Neuronal avalanches in an acute cortical slice.]]

[[Image:Beggs_avalanche_movie.gif|frame|right|Neuronal avalanches in an acute cortical slice.]]

A '''神经雪崩''' is a cascade of [[bursting|bursts]] of activity in [[Neurons|neuronal]] networks whose size distribution can be approximated by a [[power law]], as in [[critical sandpile models]] (Bak et al. 1987). Neuronal avalanches are seen in cultured and acute [[cortical slices]] (Beggs and Plenz, 2003; 2004). Activity in these slices of [[neocortex]] is characterized by brief bursts lasting tens of milliseconds, separated by periods of quiescence lasting several seconds. When observed with a [[multielectrode array]], the number of electrodes driven over threshold during a burst is distributed approximately like a power law. Although this phenomenon is highly robust and reproducible, its relation to physiological processes in the intact brain is currently not known.   

'''神经雪崩''' is a cascade of [[bursting|bursts]] of activity in [[Neurons|neuronal]] networks whose size distribution can be approximated by a [[power law]], as in [[critical sandpile models]] (Bak et al. 1987). Neuronal avalanches are seen in cultured and acute [[cortical slices]] (Beggs and Plenz, 2003; 2004). Activity in these slices of [[neocortex]] is characterized by brief bursts lasting tens of milliseconds, separated by periods of quiescence lasting several seconds. When observed with a [[multielectrode array]], the number of electrodes driven over threshold during a burst is distributed approximately like a power law. Although this phenomenon is highly robust and reproducible, its relation to physiological processes in the intact brain is currently not known.   

==Experimental Observations==

==实验观察==

[[Image:Figure3.jpg|thumb|400px|right|Example of an avalanche. Seven frames are shown, where each frame represents activity on the electrode array during one 4 ms time step. An avalanche is a series of consecutively active frames that is preceded by and terminated by blank frames. Avalanche size is given by the total number of active electrodes. The avalanche shown here has a size of 9.]]

[[Image:Figure3.jpg|thumb|400px|right|Example of an avalanche. Seven frames are shown, where each frame represents activity on the electrode array during one 4 ms time step. An avalanche is a series of consecutively active frames that is preceded by and terminated by blank frames. Avalanche size is given by the total number of active electrodes. The avalanche shown here has a size of 9.]]

===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.  

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.

===Repeating avalanche patterns===

===Repeating avalanche patterns重复的雪崩模式===

[[Image:Figure5.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:Figure5.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]].]]  

Power law distributions of sequence sizes have also been observed in spikes from the isolated [[leech ganglion]] (V. Torre, conference talk) and in spikes from [[dissociated cortical cultures]] (L. Bettencourt; R. Alessio,  personal communications), suggesting that the phenomenon of avalanches may be quite general to in-vitro preparations. Preliminary reports also indicate that avalanches are present in the superficial cortical layers of awake, resting primates (Petermann et al, 2006). These reports have not been published yet and are included here only to indicate that researchers are now exploring the avalanche concept in a variety of preparations.   

Power law distributions of sequence sizes have also been observed in spikes from the isolated [[leech ganglion]] (V. Torre, conference talk) and in spikes from [[dissociated cortical cultures]] (L. Bettencourt; R. Alessio,  personal communications), suggesting that the phenomenon of avalanches may be quite general to in-vitro preparations. Preliminary reports also indicate that avalanches are present in the superficial cortical layers of awake, resting primates (Petermann et al, 2006). These reports have not been published yet and are included here only to indicate that researchers are now exploring the avalanche concept in a variety of preparations.   

==Models of avalanches==  

==雪崩模型==  

[[Image:Figure6.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:Figure6.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.]]

[[Category:Computational Neuroscience]]

[[Category:Computational Neuroscience]]

[[Category:Network Dynamics]]

[[Category:Network Dynamics]]