更改

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.

==雪崩的含义 Implications of avalanches==

==雪崩的影响==

* ''Information transmission.'' When neural networks are tuned to the critical point, they have optimal information transmission (Beggs and Plenz, 2003; Bertschinger and Natschlager, 2004; Kinouchi and Copelli, 2006), because there is a balance between strong signal propagation and resistance to saturation.

* '' Information storage.'' When a recurrent network based on a branching process is tuned to the critical point, the number of significantly repeating avalanche patterns is maximized (Haldeman and Beggs, 2005). At the critical point, there is a mixture of strong and weak connections, allowing for a variety of independently stable patterns of activity.

* 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。

* ''Computational power.'' By changing the variance in synaptic weights in a [[spiking network]] model, Bertschinger and Natschlager (Bertschinger and Natschlager 2004) were able to produce networks that showed damped, sustained, and expanding activity. These regimes correspond to subcritical, critical, and supercritical dynamics respectively. They found that networks tuned to the critical point performed more effectively on a broad range of computational tasks than networks that were tuned to have either subcritical or supercritical dynamics.

* 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。

* ''Stability.'' When a recurrent, branching network model is tuned to the critical point, it produces largely parallel trajectories, meaning that the network is at the edge of [[stability]] (Bertschinger and Natschlager, 2004; Haldeman and Beggs, 2005). In this case, trajectories are still stable and yet are controllable with minor corrective inputs.

* 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。

==神经雪崩与其他系统的关系==

==神经雪崩与其他系统的关系==