  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.
