Fig 1.
Order parameter and its variance.
(A) Analytical dependence of the firing rate per neuron at the fixed point on the value of w0, for different values of h. (B) Normalized variance σRR = N〈(R − R0)2〉 as a function of w0. (C) Fano factor σRR/R0. (D) Square coefficient of variation , that is equal to N times the variance of the ratio R/R0. Other parameters: α = 0.1 ms−1, β = 1 ms−1, wE + wI = 13.8.
Fig 2.
Divergence of the correlation time at criticality.
(A) Analytical result for the decay time τ1 in the linear approximation, for the same parameters of Fig 1. (B) Contour plot of τ1 as a function of both h and w0. The graphs show the divergence of the decay time at the critical value w0c = 0.1.
Fig 3.
Firing rate for neuron at the critical point and far from it.
Firing rate measured in numerical simulations as a function of time for wE + wI = 13.8, h = 10−5. Upper row: w0 = 1 (E dominates) (A), middle row: w0 = 0.2 (E dominates) (C), lower row: w0 = 0.1 (E/I balance) (E), left column: N = 103, right column: N = 105 (B,D,F). Blue lines represent the firing rate of the network, while gray dots represent single neuron spikes. Red lines show the value of the firing rate R0 at the fixed point of the dynamics. Note that this value can be quite different from the mean firing rate (green lines), when large non linear effects are present.
Fig 4.
Size and duration avalanche distributions.
(A) Distribution function of the avalanche sizes on the whole observed range. (B) Distribution function of the avalanche sizes on the region where robust power law behaviour is observed. (C) Distribution function of the avalanche duration as a function of the number of bins. (D) Distribution function of the avalanche duration as a function of time. Parameters w0 = 0.1, h = 10−6, N = 106. Different curves correspond to different values of the bin width δ, introduced to define the avalanche (see Methods). (E) Exponents of size and duration distributions, with error bars, computed using the estimator introduced in [38, 39], for different values of the lower bound of the fitting window. Error bars are not shown if they are smaller than the symbol size. (F) Size distribution function of the avalanches for w0 = 0.1, h = 10−6, and different values of the system size N = 103 − 107. As expected for a critical behaviour in finite systems, the exponential cut-off scales with N.
Fig 5.
Shape of the avalanche distributions.
Scaling of the avalanche size S as a function of its duration T for networks with 50% (A) and 20% (C) inhibitory neurons. Collapse of the avalanche shape for avalanche size in the scaling regime for notworks with 50% (B) and 20% (D) inhibitory neurons. Parameters w0 = 0.1, h = 10−6, N = 107.
Fig 6.
Temporal decay of the firing rate autocorrelation, role of the system size N.
Time correlation function of the firing rate for several values of N, for α = 0.1 ms−1, β = 1 ms−1, wE + wI = 13.8, h = 10−6, and w0 = 0.2 (A), w0 = 0.1 (B). Dots correspond to the correlation function of the model simulated with the Gillespie algorithm, while the continuous line corresponds to the linear approximation, that is valid for large values of N.
Fig 7.
Temporal decay of the firing rate autocorrelation, role of the external input h.
Time correlation function of the firing rate for several values of h, for N = 107 and w0 = 0.2 (A), w0 = 0.1 (B), other parameters as in Fig 6.
Fig 8.
Long time scale behaviour as a function of the system size N and the external input h.
(A) Maximum correlation time extracted from an exponential fit of the long time tail of the correlation function, as a function of h and for different values of N, for w0 = 0.1. The continuous red line corresponds to the linear approximation. (B) Maximum correlation time as a function of N for h = 10−6 and w0 = 0.1, 0.2.
Fig 9.
Avalanche size and duration distributions measured from the analysis of the continuous-time series of the firing rate signal.
The avalanche is defined as a continuous interval of time in which the firing rate is greater than a zero threshold and its duration is the width of the time interval. (A) Distribution function of the avalanche sizes. (B) Distribution function of the avalanche duration. Parameters: w0 = 0.1, h = 10−6, N = 106. Exponents of size and duration distributions, computed using the estimator introduced in [38, 39] with lower bounds of the fitting windows Smin = 10, tmin = 10 ms, are τS = 1.54 ± 0.03, τT = 2.04 ± 0.04.
Fig 10.
Avalanche size and duration distributions measured from a finite threshold.
(A) Size distribution and (B) duration distribution according to definition 3 for w0 = 0.1 and h = 10−6.
Fig 11.
Avalanche size distributions measured from a finite threshold.
Distribution of the sizes according to definition 2 for w0 = 0.1 and h = 10−6.
Fig 12.
Equivalence of different numerical simulation methods.
Comparison between the correlation function calculated by the Gillespie algorithm (red dots) and fully non-linear Langevin equation (blue line), for N = 107, w0 = 0.1, h = 10−6.