It is shown that a Hopfield recurrent neural network, informed by experimentally derived brain topology, recovers the scaling picture recently introduced by Deco et al., according to which the process of information transfer within the human brain shows spatially correlated patterns qualitatively similar to those displayed by turbulent flows, although with a more singular exponent, 1/2 instead of 2/3. Both models employ a coupling strength which decays exponentially with the euclidean distance between the nodes, but their mathematical nature is very different, Hopf oscillators versus a Hopfield neural network, respectively. Hence, their convergence for the same data parameters, suggests an intriguing robustness of the scaling picture. The present analysis shows that the Hopfield model brain remains functional by removing links above about five decay lengths, corresponding to about one sixth of the size of the global brain. This suggests that, in terms of connectivity decay length, the Hopfield brain functions in a sort of intermediate ``turbulent liquid''-like state, whose essential connections are the intermediate ones between the connectivity decay length and the global brain size. Finally, the scaling exponents are shown to be highly sensitive to the value of the decay length, as well as to number of brain parcels employed. As a result, any quantitative assessment regarding the specific nature of the scaling regime must be taken with great caution.

翻译：研究表明，基于实验获取的脑拓扑结构的Hopfield递归神经网络，重现了Deco等人近期提出的标度图像。该图像指出，人脑内信息传递过程呈现空间相关模式，其性质与湍流显示的模式定性相似，但奇异性指数更小（1/2而非2/3）。两种模型均采用随节点间欧氏距离呈指数衰减的耦合强度，但其数学本质截然不同：前者基于Hopf振荡器，后者则为Hopfield神经网络。因此，两者在相同数据参数下的收敛性，暗示了标度图像具有引人注目的鲁棒性。本分析表明，当移除约五个衰减长度（相当于全脑尺寸约六分之一）以上的连接时，Hopfield模型脑仍能维持功能。这意味着，就连接衰减长度而言，Hopfield脑处于一种介于“湍流液体”的中间态，其关键连接位于连接衰减长度与全脑尺寸之间的中间尺度。最后，标度指数对衰减长度取值及所采用的脑分区数量均高度敏感。因此，任何关于标度区具体性质的定量评估都必须极为审慎。