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==雪崩模型==  

==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: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.]]

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:Figure7.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|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:  

\mathit{p_{ij}}

\mathit{p_{ij}}

</math>

</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.  

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.

==Implications of avalanches==

==Implications of avalanches==