Evidence of scaling regimes in the hopfield dynamics of whole brain model

It is shown that the activity of a whole-brain model based on a Hopfield recurrent neural network exhibits a scaling regime whose exponents depend on both the number of parcels and the decay length of the coupling strength. This scaling regime recovers the picture introduced by Deco et al., whereby information transfer within the human brain generates spatially correlated patterns qualitatively similar to those displayed by turbulent flows, although with a lower exponent, 1/2 instead of 2/3, indicating less regularity. Both models employ coupling strengths which decay exponentially with the Euclidean distance between nodes, as informed by experimental work on the brain topology. Given that the Hopf oscillators model and the Hopfield neural network are mathematically very different, their convergence under the same data parameters, suggests an intriguing robustness of the scaling picture. The present analysis further shows that the Hopfield model brain remains functional after removing connections above about five decay lengths, which corresponds to roughly one sixth of the size of the whole brain. This suggests that, in terms of connectivity decay length, the Hopfield brain functions in an intermediate "turbulent liquid"-like regime, in which the essential connections lie between the connectivity decay length and the global size of the brain. The sensitivity of the numerical value of the scaling exponent to both the decay length, and the number of brain parcels employed, invites great caution in formulating a quantitative assessment on the specific nature of underlying mechanisms sustaining the scaling regime.