Figure 1

Examples of distributions of avalanche sizes in dependence on the interaction parameter $\alpha$. Circles: subcritical distribution ($\alpha =0.52$), stars: critical branch ($\alpha =0.56$), triangles: supercritical $\alpha =0.59$) for $N=300$, $\nu =10$, $u_0=0.1$, $I_0=7.5$. Inset: Probability of an avalanche of length $L=40$. Circles are obtained by incrementing $\alpha$ and stars by decrementing $\alpha$.Reuse & Permissions

Figure 2

Deviation of the avalanche size distribution from a power law for different values of the interaction parameter $\alpha$. The deviation is estimated by the minimum of $\sqrt{\sum _{L=1}^{L_{\max }}[\log P(L,N,\alpha )-\log c_{\gamma }{L}^{\gamma }{]}^2}$ with respect to $\gamma$ and $c_{\gamma }$, where $L_{\max }=N/2$ introduces a cutoff. We checked the unbiasedness of our estimation with the maximum likelihood method. Squares represent the model with both facilitating and depressing synaptic dynamics, circles are for depressing synapses only. $N=300$, $\nu =10$, $u_0=0.1$, and $I_0=7.5$. The inset shows the length of the critical interval for dynamical synapses (filled symbols) and static synapses (empty symbols) $N=500$, 700, 1000.Reuse & Permissions

Figure 3

Average synaptic strength $⟨u_{ij}{J}_{ij}⟩$ and interspike interval $⟨{\Delta }^{\mathrm{isi}}⟩$ corresponding to Eq. (8) (virtually straight line) and, respectively, Eq. (6) for $\alpha =0.5$, $\alpha ={\alpha }_c$, $\alpha =0.54$, $\alpha ={\alpha }^c$, and $\alpha =0.6$ (curved lines, from bottom to top). Circles represent numerical results for a network with matching $\alpha$ and parameters $N=300$, $\nu =10$, $u_0=0.1$, and $I_0=7.5$.Reuse & Permissions

Figure 4

Bifurcation diagram representing the solutions of the self-consistency Eq. (9). The effective synaptic strength $⟨u_{ij}{J}_{ij}⟩$ is plotted vs the interaction parameter $\alpha$. The insets assign the graphs of Fig. 3 to the corresponding regions by choosing each time a typical $\alpha$ value. $N=300$, $\nu =10$, $u_0=0.1$, and $I_0=7.5$.Reuse & Permissions

Figure 5

A cusp is formed in the ($\alpha$, $u_0$) plane by the mergence of the two critical branches ${\alpha }_c$ and ${\alpha }^c$ into a critical point. The graphs are obtained from Eq. (9) for $N=300$ (bold), $N=1000$ (semibold), and $N\to \infty$ (dashed). For $N\to \infty$ the cusp is at $(\alpha ,u_0)=(1,\frac{1}{2})$. In the interior of the cusp shape, Eq. (9) has three solutions, one of which corresponds to a critical phase and extends also to larger values of $\alpha$. The other stable solution is subcritical and exists also below the cusp. At values of $u_0$ below the critical point, the critical phase is reached by a first-order phase transition for $N\to \infty$, while for higher $u_0$ values critical behavior is reached by a second-order transition. Further parameters are: $\nu =10$, $I_0=7.5$.Reuse & Permissions