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 both models employ a coupling strength which decays exponentially with the euclidean distance between the nodes, their mathematical nature is widely different, Hopf oscillators versus Hopfield neural network. Hence, their convergence suggests a remarkable robustness of the aforementioned scaling picture. Furthermore, 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. This "turbulent-like liquid" appears to be more spiky than actual turbulent fluids, with a scaling exponent around $2/5$ instead of $2/3$.

翻译：研究表明，基于实验获知的大脑拓扑结构构建的Hopfield递归神经网络，能够复现Deco等人近期提出的标度图像——人类大脑内部信息传递过程呈现出与湍流定性相似的空间相关模式。尽管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，但它们的数学本质截然不同（Hopf振荡器与Hopfield神经网络），因此二者的收敛性彰显了前述标度图像的显著稳健性。进一步分析表明，当移除超出约五个衰减长度（相当于全脑尺寸的六分之一）的连接时，Hopfield模型大脑仍可维持功能。这暗示在连接衰减长度维度上，Hopfield大脑运行于一种类似"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。这种"类湍流液体"比实际湍流流体更具尖峰特征，其标度指数约为$2/5$而非$2/3$。