Scaling and tuning to criticality in resting-state human magnetoencephalography

Abstract

Scaling laws in biological neural networks have long been investigated. From 1/f noise to neuronal avalanches, evidence of scaling in brain activity has been increasingly linked to tuning to or near criticality. The concept of scaling is intimately related to the renormalization group (RG), in essence providing coarse-grained, simplified descriptions that generalize to classes of diverse physical systems. Following the RG idea, a coarse-graining scheme has recently been proposed for populations of real neurons, and scaling behaviors in collective quantities have been reported in the hippocampus and in different areas of the rat cortex. To bridge the gap between neuronal population scales and species, here we consider large-scale, electrophysiological recordings of human brain activity in the awake resting-state. We demonstrate robust scaling behaviors of collective dynamics across coarse-graining scales, with exponents close to those measured in populations of spiking neurons. Further, we show that dynamics of neuronal avalanches, scale-free cascades of neural activity, are invariant under the proposed coarse-graining approach. Simulations of a non-equilibrium adaptive Ising model inferred from data and apt to reproduce a large repertoire of resting-state brain dynamics indicate that the scaling behaviors of the resting human brain activity emerge close to criticality and depend on the excitation/inhibition (E/I) balance of the network. While extending the range of validity of previous observations at small spatial scales and pointing to common scaling laws in mammals, the results open the way to a robust (currently missing) non-invasive approach to estimate the E/I balance, a key quantity in neuroscience research.

Figures (32)

• Figure 1: Phenomenological Renormalization Group analysis of MEG data. Schematic presentation of the coarse-graining procedure for four MEG sensors. Sensor signals (z-normalized, amplitude in unit of standard deviation (SD); see Appendix \ref{['data']}) are initially binarized by marking as $1$'s the extremes of positive and negative excursions beyond a threshold $e = \pm 3$ SD (horizontal grey lines) and as $0$'s all other points (left). Resulting binary time series (colored bars) are shown at the bottom of each signal, where vertical lines represent tips of above-threshold excursions, which we refer to as extreme events. The case $k = 0$ corresponds to the original MEG signals (left). Signals are sorted in couples, from the most to the least correlated. The first couple is made by the sensors F46 and F56 (dark red dots), the second by the sensors P45 and P44 (light red dots). Location of these sensors on the scalp is shown on the right upper corner. At each step $k$ of the coarse-graining, pairwise correlations between sensor time series are calculated. Each sensor is paired with its most correlated sensor, and their activity is summed and results in a new time-series (middle, $k = 1$). On top of each time series of events we show the signal resulting from summing together the original sensor (or variable) signals, which is the continuous, z-normalized coarse-grained variable. The resulting summed activity defines the coarse-grained variables. At each successive iteration of the procedure, we compute the correlations between coarse-grained variables and group the most correlated pairs by summing their activity. This process is repeated iteratively until only one variable is left (right, $k = 2$).
• Figure 1: PRG analysis of resting state brain activity shows little variability across subjects.A. Average probability distribution for normalized nonzero activity for the cluster sizes $K \geq 8$. Darker blue = Increasing cluster size. B. Average log-probability of silence in coarse-grained variables ($ln P_0$) as a function of cluster size, $K$ (n = 100). The solid black line represents a linear least-squares fit of $log(-ln P_0) = log\cdot K^\beta$ + b, where $\beta = 0.82 \pm 0.002$ (fit $\pm$ error on the fit) The grey dashed line indicates prediction for independent variables. C. Average variance (Var) of the non-normalized coarse-grained variables across n = $100$ subjects as a function of the cluster size, $K$. The solid black line corresponds to the linear least-square fit, $log \ Var(K) = \alpha \cdot logK + b$, where $\alpha = 1.33 \pm 0.005$ (fit $\pm$ error on the fit). Grey dashed lines indicate reference growth rates: linear scaling ($\alpha=1$) characteristic of independent variables and quadratic scaling ($\alpha=2$) expected for fully correlated variables. D.. Average correlation functions (n = $100$) for different cluster sizes, $K$, collapse on a single curve after rescaling time, $\tau$, by the correlation time $\tau_c$. Inset: Original autocorrelation for different $K$. E. The autocorrelation time, $\tau_c$, as a function of $K$, grows as $\tau_c \propto K^z$. The solid black line corresponds to the linear least-square fit, $log \ \tau_c(K) = z \cdot logK + b$, where $z = 0.34 \pm 0.02$ (fit $\pm$ error on the fit). F. Eigenvalues of cluster covariance matrices for different cluster sizes. Results are averaged over $n = 100$ subjects. Larger clusters correspond to darker colors. The solid line is a least squares fit to $log \ \lambda = log \ b (rank/K)^{-\mu}$ performed for $K=128$. The fitting range is $2/128 - 50/128$. All results are averaged over $n = 100$ subjects. Error bars and shaded areas represent standard deviation (SD).
• Figure 2: Scaling in resting-state human MEG.A. Probability distribution of normalized non-zero activity for the cluster sizes $K \geq 8$. Darker blue = Increasing cluster size. B. The log-probability of silence in coarse-grained variables ($ln P_0$) as a function of cluster size ($K$). The solid black line represents a linear least-squares fit of $-ln P_0 = bK^\beta$ in log-log scale, where $\beta = 0.82$ ($R^2=0.99$, $p < 10^{-5}$). The grey dashed line indicates the prediction for independent variables. For each $K$, SEM is calculated across coarse-grained variables. Error bars are always smaller than the marker size. C. Variance (Var) of the non-normalized coarse-grained variables as a function of the cluster size, $K$. The solid black line corresponds to the linear least-square fit, $log \ Var(K) = \alpha \cdot logK + b$, where $\alpha = 1.32$ ($R^2=0.99$, $p < 10^{-5}$). Grey dashed lines indicate reference growth rates: bottom, linear scaling ($\alpha=1$) characteristic of independent variables; top, quadratic scaling ($\alpha=2$) expected for fully correlated variables. D. Average correlation functions for different cluster sizes, $K$, collapse on a single curve after rescaling time, $\tau$, by the auto-correlation time $\tau_c$. Inset: Original autocorrelation for different $K$. E. The autocorrelation time, $\tau_c$, as a function of $K$, grows as $\tau_c \propto K^z$. The solid black line corresponds to the linear least-square fit, $log \ \tau_c(K) = z \cdot logK + b$, where $z = 0.31$ ($R^2=0.97$, $p < 10^{-5}$). F. Eigenvalues of covariance matrices within clusters of different sizes. Larger clusters correspond to darker colors. The solid line is a linear least squares fit to $log \ \lambda = log \ b (rank/K)^{-\mu}$ performed for $K=128$. The fitting range is $[2/128, 50/128]$. The analysis is performed using $e = \pm 3$SD and $\delta = 1T_s = 1.67$ ms, where $T_s$ is the sampling time (Appendix \ref{['data']}).
• Figure 2: Distribution of normalized cluster activity is independent of the binarization threshold, $e$, and bin size $\delta$.Top row. Distribution of normalized cluster activity for $K = 32$, $K = 64$, and $K = 128$ for different values of the binarization threshold $e$. Bottom row. Distribution of normalized cluster activity for $K = 32$, $K = 64$, and $K = 128$ for different bin sizes $\delta$.
• Figure 3: Relationships between scaling exponents in the MEG of the resting state.A. The scaling exponent of the cluster variance is inversely proportional to the exponent $\beta$ that characterizes the probability of silence $P_0 (K)$, i.e. $\beta = -0.54\alpha + 1.53$ (linear least square fit: $R^2 = 0.81$, $p < 10^{-5}$; $n = 98$. Two subjects were excluded based on Cook's distance Cook_1977). B. The exponent $\mu$ characterizing the eigenvalue spectrum of the covariance matrix increases linearly with $\alpha$: $\mu = 0.61\alpha - 0.39$ (linear least square fit: $R^2 = 0.31$, $p < 10^{-5}$; $n=98$). C. As expected from A and B, the exponent $\mu$ decreases for increasing values of $\beta$. The relationship between $\mu$ and $\beta$ is well described by a linear equation, $\mu = -1.0\beta + 1.24$ (linear least square fit: $R^2 = 0.3$, $p < 10^{-5}$; $n=98$). D. The scaling exponent of the largest eigenvalue of the covariance matrix, $\epsilon$, increases for increasing values of $\alpha$, as shown by the linear relationship $\epsilon = 1.46\alpha - 1.54$ (linear least square fit: $R^2 = 0.68$, $p < 10^{-5}$; $n = 98$).