Evidence of Scaling Regimes in the Hopfield Dynamics of Whole Brain Model

It is shown that a Hopfield recurrent neural network exhibits a scaling regime, whose specific exponents depend on the number of parcels used and the decay length of the coupling strength. This scaling regime recovers the picture 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, informed by experimentally derived brain topology. Nevertheless, 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.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. The evident sensitivity of the scaling exponent to the value of the decay length, as well as to the number of brain parcels employed, leads us to take with great caution any quantitative assessment regarding the specific nature of the scaling regime.