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

TL;DR

The paper investigates whether Deco2020's turbulence-like scaling of brain information transfer persists when the brain is modeled as a discrete-time Hopfield network whose couplings decay exponentially with distance, $J_{ij}=e^{-d_{ij}/\delta}$. Using Schaefer parcellations and structure-factor analysis of the resulting dynamics, it finds a transient scaling regime with an exponent $\alpha$ that is highly sensitive to the decay length $\delta$ and the parcel count $N$, e.g., $\alpha\approx 2/5$ at $\delta\approx 5.55$ mm and $\alpha\approx 2/3$ at $\delta\approx 5.88$ mm. The study also shows the network remains functional after pruning long-range links beyond roughly $5\delta$, indicating an intermediate turbulent-like state bridging local and global connectivity. Overall, the results demonstrate robustness of the scaling picture across modeling frameworks while highlighting the importance of connectivity length and parcellation resolution for quantitative conclusions.

Abstract

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.

Figures (5)

• Figure 1: Structure factor $S_2(d)$ for $N=1000$ nodes and $N_r=1000$ realizations respectively at $\delta = 5.55$ mm (A), and $\delta = 5.88$ mm (B). The values $S_2(d)$ were binned over equal sized intervals. The dashed lines indicate different $\alpha$ values: $\alpha = 2/3$ which corresponds to turbulence, $\alpha = 1/2$, the value obtained in Deco2020, and $\alpha = 2/5$. We estimate the scaling exponent $\alpha$ with a linear regression of $\log(S(d)) \sim \alpha \log(d) + \beta$ in the interval range $[2.7 - 33.1 ]$ mm, as used in Deco2020, and we obtain $\alpha \approx 2/5$ for $\delta = 5.55$ mm, and $\alpha \approx 2/3$ for $\delta = 5.88$ mm.
• Figure 2: A) The blue line shows the average slope of $\alpha = \log S(d)/ \log d$ for different $\delta$s, given the connectivity matrix $J_{ij}=\exp{(- d_{i,j}/\delta) }$. The horizontal dotted lines correspond to $\alpha_{RW}=1$ and $\alpha_T=2/3$, corresponding to random walk and homogeneous incompressible turbulence, respectively, while the dot-dashed horizontal line corresponds to Deco2020 "brain" value $\alpha_D=1/2$. The orange line shows the function $\alpha(\delta)$ with a connectivity matrix obtained by randomly shuffling all the weights $J_{ij}$, which results into a basically uncorrelated signal, indicating the importance of the metric structure of the weights. The plot stops at $\delta=10$ mm, a region where the signal is highly correlated ($\alpha \sim 1.2$), but still below the smooth regime marked by $\alpha=2$. Conversely, a brain below $4$ mm, would exhibit "chaotic" behaviour, with scaling exponents close to zero, i.e. no scaling regime. The bottom panels show the neuron activity mapped on the Schaefer’s cerebral cortical parcellation atlas Schaefer2018 coordinates for (B) $\delta = 4$ ("chaotic"), (C) $\delta=5.99$, and (D) $\delta=6.66$ (random walk), respectively.
• Figure 3: Pair-correlation functions $B(d)$ for different $\delta$s. At small $\delta$ the signal is largely uncorrelated, while at increasing $\delta$, correlations start to emerge, although affected by significant amount of statistical fluctuations.
• Figure 4: The average slope of $\log S(d)$ for different dilution levels $\alpha(\rho)$, normalized to its threshold-free value $\alpha(0)$ at that given $\delta$. The dilution is measured as the fraction of disconnected pairs $i,j$, $J_{ij}$ elements with value 0. The second $y$-axis on top shows the cutoff distance in mm for which at a certain dilution $\rho$ the pairs at a larger or equal distance are disconnected. The differently colored curves indicate simulations with different $\delta$ values. The graph shows that within the measurement error, the system is not affected by the dilution of the edges, up to the removal of more than the $95\%$ of the connections, in descending order of distance. This shows that, in view of the exponential connectivity $J_{ij} = e^{ -d_{ij}/\delta}$, the onset of collective patterns is sustained mostly by the close connections, within about $4 \delta$.
• Figure 5: The average slope of $\alpha$ as a function of $\delta$ for Schaefer 2018 Parcellations with different sizes $N$Schaefer2018. Given each size $N$, we fitted the points with the sigmoid functions in Eq. (\ref{['eq:exp']}).