Fig 1.
(A) The model consists of excitatory (Ex) and inhibitory (In) neurons arranged in a L × L square lattice with open boundaries. Each neuron may connect locally to a random fraction of its neighbors within a ℓ × ℓ square (see Methods). In gray we exemplify the neurons connected to a central neuron (black dot) with connectivity of 25% (top), 45% (middle), and 70% (bottom). (B) The connection weights (Wij) are fixed and depend only on the nature of the presynaptic (i) and postsynaptic (j) neurons.
Fig 2.
Neuronal avalanches and order parameter definitions.
(A) A neuronal avalanche starts and ends when the fluctuations of the integrated network activity cross a threshold θ. The size of an avalanche can be defined as sg, the total number of spikes, or sθ, the number of spikes minus the threshold value. The avalanche duration T is the time that the fluctuations stay above θ. (B) Detail of the power spectrum around the region where a peak appears. The order parameter φ is given by the ratio between the power peak area ϕp and the total area ϕp + ϕu. Red symbols represent the frequencies that bound the φ area: fmin(diamond), fpeak(circle) and fmax (triangle).
Fig 3.
CROS model dynamics with L = 300 and ℓ = 7.
(A) Network activity for three different regimes (balance between excitation and inhibition): low (blue), intermediate (yellow) and high (green). Inset: zoom in the lowest level of activity. (C) Power spectrum of A(t) for the three regimes described in (A). Inset: zoom in low frequencies. The parameter space suggests a phase transition as shown in heat maps of (B) the power of collective oscillations and (D) the order parameter. Also (E) a larger standard deviation of the order parameter is observed near the transition region. Arrows in (B) and (D) represent parameters described in (A).
Fig 4.
The CROS model presents long-range temporal correlations at the transition line.
(A) and (B) Strong LRTCs (indicated by DFA exponent α > 0.5) emerge at the onset of oscillations for the raw time series, regardless of the system size (see Methods). For the band-filtered time series, (C) networks with L = 50 show strong LRTCs while (D) for L = 300 the effect is less pronounced (though still clearly α > 0.5). For L = 50, (E) αraw and α are close to one near the critical regime (green diamonds and circles, α ≃ αraw ≃ 1.09 respectively). For L = 300, (F) αraw is close to one while α is smaller near the critical regime (green diamonds and circles, αraw = 1.16 and α = 0.66 respectively). Examples from the subcritical region: (E) L = 50, αraw = 0.57 (purple diamond) and α = 0.80 (purple circle), (F) L = 300, αraw = 0.53 (purple diamond) and α = 0.60 (purple circle). For the supercritical region: (E) L = 50, αraw = 0.60 (orange diamond) and α = 0.62 (orange circle), (F) L = 300, αraw = 0.54 (orange diamond) and α = 0.62 (orange circle). Diamonds (circles) represent DFA applied over raw (band-filtered) time series (see Methods). Solid symbols indicate the fitting range where α was estimated. Colored arrows in (A), (B), (C), (D) and colors in (E) and (F) represent parameters: rE = 0.04 (purple), rE = 0.12 (green) and rE = 0.24 (orange) with rI = 0.60.
Fig 5.
(A) and (B) show heat maps of the κθ index in parameter space employing the τ exponent for the 2D-DP and MF-DP universality classes, respectively. In both cases, avalanches were defined with Γ = 0.5. Arrows indicate the connectivity parameters [rE:rI] of networks: [0.080:0.60] (yellow), [0.1225:0.60] (red), [0.1050:0.60] (purple) and [0.1325:0.60] (blue). Representative (single run) distributions for the parameter values indicated by arrows in (A) and (B) are respectively shown in (C) and (D), exemplifying subcritical, critical and supercritical cases.
Fig 6.
Avalanche duration distributions.
(A) and (B) show heat maps of the κT index in parameter space employing the τt exponent for the 2D-DP and MF-DP universality classes, respectively. In both cases, avalanches were defined with Γ = 0.5. Arrows indicate the connectivity parameters of networks, the same described in Fig 5. Representative (single run) distributions for the parameter values indicated by arrows in (A) and (B) are respectively shown in (C) and (D), exemplifying subcritical, critical and supercritical cases.
Fig 7.
Continuously varying exponents.
(A) DFA exponent (α) and order parameter (φ) versus rE, with rI = 0.60 fixed. The shaded area represents parameter space where the data was better fitted by a exponentially truncated power-law than exponential or lognormal distributions, according to the log-likelihood ratio test (see Methods). Inset: zoom around the shaded area. (B) and (C): average exponents for avalanche size and duration, respectively. Avalanche size versus avalanche duration for (D) 5 different runs and (E) averages over the runs. The black triangle and diamond represent the theoretical exponents for 2D-DP and MF-DP universality classes, respectively, whereas the black square is the result obtained by Coleman et al [44] for a variant of the globally coupled Kuramoto model. The same color code applies to (B)-(E). In all figures, error bars represent the standard deviation over 5 runs. Avalanches were defined with Γ = 0.5.
Fig 8.
Threshold dependence of the continuously varying exponents.
(A) and (B): exponents of avalanche size and duration, respectively, as a function of threshold parameter Γ (see Eq 4). The same color code applies to (A) and (B). Exponents for avalanche size and avalanche duration (C) for 5 different runs and (D) averaged over the runs (the black points are the same as described in Fig 7). The color code in (C) and (D) represents different values of Γ, as described in (C). Orange points are experimental M/EEG results extracted from Palva et al. [22]. In all figures, error bars represent the standard deviation over 5 runs of the CROS model.
Fig 9.
Left column: averaged raw avalanche shapes. Middle column: Estimation of 1/(σνz) over the mean scaled avalanche profiles. Right column: scaling relation given by Eq 6. (A), (B) and (C) L = 50 with Γ = 0.50, (D), (E) and (F) L = 300 with Γ = 0.50. (G), (H) and (I) L = 50 with Γ = 0.80, (J), (K) and (L) L = 300 with Γ = 0.80. Gray shadow represents the standard deviation and orange trace the fitted scaling parameter. Model parameters: rE = 0.14 and rE = 0.12 for L = 50 and L = 300, respectively, with rI = 0.60.
Fig 10.
Relation between critical exponents across the transition region.
Left-hand (triangles) and right-hand (squares) sides of scaling relation given by Eq 8 as a function of rE in the transition region for different values of Γ. Dashed horizontal lines represent the values for 2D-DP and MF-DP universality classes, as well as the result obtained by Coleman et al [44] for a variant of the globally coupled Kuramoto model.
Fig 11.
Similarity between spread of DFA and avalanche size exponents in the CROS model and in MEG data.
DFA (α) and avalanche size (τ) exponents show positive correlation in MEG recordings of humans in resting-state (data extracted from Zhigalov et al [23], see Methods) as well as in the CROS model. Avalanches for the CROS model were defined with Γ = 0.85.