Laboratory of Mathematical Physics, Rockefeller University, New York, New York, United States of America.
PLoS One. 2012;7(7):e41419. doi: 10.1371/journal.pone.0041419. Epub 2012 Jul 25.
A number of studies have suggested that many properties of brain activity can be understood in terms of critical systems. However it is still not known how the long-range susceptibilities characteristic of criticality arise in the living brain from its local connectivity structures. Here we prove that a dynamically critically-poised model of cortex acquires an infinitely-long ranged susceptibility in the absence of input. When an input is presented, the susceptibility attenuates exponentially as a function of distance, with an increasing spatial attenuation constant (i.e., decreasing range) the larger the input. This is in direct agreement with recent results that show that waves of local field potential activity evoked by single spikes in primary visual cortex of cat and macaque attenuate with a characteristic length that also increases with decreasing contrast of the visual stimulus. A susceptibility that changes spatial range with input strength can be thought to implement an input-dependent spatial integration: when the input is large, no additional evidence is needed in addition to the local input; when the input is weak, evidence needs to be integrated over a larger spatial domain to achieve a decision. Such input-strength-dependent strategies have been demonstrated in visual processing. Our results suggest that input-strength dependent spatial integration may be a natural feature of a critically-balanced cortical network.
一些研究表明，大脑活动的许多特性可以用临界系统来理解。然而，目前尚不清楚，从大脑的局部连接结构中，如何产生具有临界特征的长程磁化率。在这里，我们证明了在没有输入的情况下，皮层的动态临界平衡模型会获得无限长的磁化率。当输入时，磁化率会随距离呈指数衰减，输入越大，空间衰减常数（即范围越小）越大。这与最近的研究结果直接一致，这些结果表明，猫和猕猴初级视觉皮层中单个尖峰诱发的局部场电位活动波会随特征长度衰减，该特征长度也随视觉刺激对比度的降低而增加。与输入强度相关的磁化率可以改变空间范围，可以认为它实现了依赖输入的空间整合：当输入较大时，除了局部输入之外，不需要额外的证据；当输入较弱时，需要在更大的空间域中整合证据才能做出决策。这种依赖输入强度的策略已在视觉处理中得到证明。我们的结果表明，依赖输入强度的空间整合可能是临界平衡皮层网络的自然特征。
