Characterization of the AS and of the PS one in terms of macroscopic fields and single neuron dynamics. Panel (a): macroscopic attractors in the plane. Panel (b): the average field as a function of time. Panels (c) and (d): raster plots. The data refer to Erdös-Renyi networks with , quenched disorder, and N = 1600, the black (respectively, red) symbols correspond to the asynchronous state observable for p = 0.2 (respectively, PS for p = 0.7).
(a) Minima and maxima of the average field as a function of p, the circles joined by solid lines refer to the annealed disorder, while the stars connected by the dashed line to the quenched case. The dotted-dashed (red) lines indicate the fully coupled results (corresponding to ). (b) Synchronization indicator averaged over time as a function of p for the quenched case. The data refer to Erdös-Renyi networks with CP and size N = 1600 and have been estimated, after discarding a transient of 4 × 107 spikes, by averaging over a train of 1–2 × 107 spikes.
Phase diagram for the macroscopic activity of the network in the plane: (a) annealed disorder and (b) quenched disorder. The (black) asterisks connected by the solid (black) line correspond to the transition values from AS to PS regime estimated for Erdös-Renyi networks with constant probability. The other symbols refer to Erdös-Renyi with γ > 1: solid (respectively, empty) symbols individuate asynchronous (respectively, partially synchronized) states. In particular, (blue) squares refer to γ = 1.1, (red) circles to γ = 1.3, (green) diamonds to γ = 1.5, and (magenta) triangles to γ = 1.7. The reported data are relative to the state of the network after discarding transients ranging from 2 × 107 spikes at the smaller sizes to 3 × 108 spikes for the larger networks.
(a) Macroscopic attractors in the plane for an Erdös-Renyi networks with γ = 1.3 and annealed disorder, the curves from the interior to the exterior correspond to increasing system sizes, from N = 1600 to 25 600. The most external (black) curve refers to a fully coupled network with N = 3200. (b) Minima and maxima values of the average field as a function of N for various γ-values: namely, (black) circles γ = 1.1, (red) triangles γ = 1.3, and (blue) squares γ = 1.5. The empty (respectively, filled) symbols refer to annealed (respectively, quenched) disorder. The dotted-dashed (magenta) lines indicate the fully coupled values.
(a) Average standard deviation versus the system size N for annealed disorder and various γ-values: γ = 1.1 (blue) squares, 1.3 (red) circles, 1.5 (green) diamonds, and 1.7 (magenta) triangles. The dashed line represents best fits with a power-law to the reported data. The data in the inset (black asterisks) refer to a γ = 2.0, the dashed line is a guide for the eyes. (b) Power-law exponents β for annealed (black circles) and quenched (red triangles) disorder in the network as a function of the parameter γ. The dashed (blue) line refers to the linear law β = 1 − γ/2. The reported data have been estimated by averaging over trains made of 2 × 106–108 spikes, after discarding transients of 4 × 105–4 × 106 spikes.
Maximal Lyapunov exponents as a function of the system size N for various γ-values. The data have been obtained by discarding a transient of the order of 108–109 spikes and then by following the dynamics in the real and tangent space for an equivalent duration, moreover the data have been averaged over 3 to 5 different network realization with quenched disorder.
Power law exponents giving the decay of the maximal Lyapunov for the data reported in Fig. 6(a) for quenched disorder.
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