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Coherent periodic activity in excitatory Erdös-Renyi neural networks: The role of network connectivity
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Image of FIG. 1.
FIG. 1.

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).

Image of FIG. 2.
FIG. 2.

(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.

Image of FIG. 3.
FIG. 3.

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.

Image of FIG. 4.
FIG. 4.

(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.

Image of FIG. 5.
FIG. 5.

(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.

Image of FIG. 6.
FIG. 6.

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.


Generic image for table
Table I.

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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Coherent periodic activity in excitatory Erdös-Renyi neural networks: The role of network connectivity