Finite-size effects and dynamics of giant transition of a continuum quorum percolation model on random networks

We start from a continuous extension of a mean field approach of the quorum percolation model, accounting for the response of in vitro neuronal cultures, to carry out a normal form analysis of the critical behavior. We highlight the effects of nonlinearities associated with this mean field approach even in the close vicinity of the critical point. Statistical properties of random networks with Gaussian in-degree are related to the outcoming links distribution. Finite size analysis of explicit Monte Carlo simulations enables us to confirm the relevance of the mean field approach on such networks and to show that the order parameter is weakly self-averaging; dynamical relaxation is investigated. Furthermore we derive a mean field equation taking into account the effect of inhibitory neurons and discuss the equivalence with a purely excitatory network.

Figure 1

Evolution of $log\left(\rho \right)$ as a function of $log\left(M\right)$. A point corresponds to a numerical simulation, $N$ goes from ${510}^2$ to ${10}^5$. Each network has been simulated ten times. The in-degree distribution is a Gaussian with $\overline{k}=25$ and $\sigma =7$.

Figure 2

Superposition of the histogram of the out-degree probability distribution density and the Poisson law, $N=4{10}^5,\overline{k}=25$, and $\sigma =7$.

Figure 3

Comparison between explicit Monte Carlo simulations (crosses), solutions of the mean field equation (4) (solid lines), and the mean field equivalent excitatory network $\overline{k}-2k_i$ (dotted lines). Upper figures: $\overline{k}=25,\sigma =5,m=\left\{3,5,15,25\right\}$ (from left to right), with $\eta =0.05$ (left) and $\eta =0.25$ (right). Bottom figures: $\overline{k}=75,\sigma =15,m=\left\{9,15,45,75\right\}$ (from left to right), with $\eta =0.05$ (left) and $\eta =0.10$ (right).

Figure 4

(a) Dots correspond to the superposition of explicit Monte Carlo simulations ($525000$ in total) of the asymptotic equilibrium value of $\varphi$ as a function of $f$, starting with ${\varphi }_m(t=0)=f,N=1\times {10}^5,\overline{k}=100$, and $\sigma =10$. Three values of $m$ are considered: $m=75,m=80$, and $m=85$ (from left to right). Lines are the value of $\varphi$ obtained by solving the self-consistency equation (mean field approximation). (b) Same as in (a) for $N=2.5\times {10}^5$ and $125000$ runs.

Figure 5

Distribution of gap size $g_m$ as a function of $f$ for $m=75$ (left), $m=80$ (middle), and $m=85$ (right), $N=1\times {10}^5$, for a total of $125000$ runs, $\overline{k}=100$ and $\sigma =10$.

Figure 6

Histogram $H\left(g_m\right)$ of the gap size $g_m$ for $m=85,N={10}^5$, for a total of $500000$ runs.

Figure 7

Quantile plots of the gap size distribution for $m=75,N={10}^5$ (red-left below) and $N=2.5\times {10}^5$ (blue-right below).

Figure 8

Top curves: power law behavior of ${\sigma }_g=〈\mathrm{\Delta }{g}_m〉/〈g_m〉=a{N}^{-\gamma }$ as a function of $N$ in the cases $\overline{k}=100$ and $\sigma =10$. We obtain those scaling laws for $m=75,m=80$, and $m=85$. Bottom curves: power law for ${\sigma }_{f^{☆}}=〈\mathrm{\Delta }{f}^{☆}〉/〈f^{☆}〉=b{N}^{-\zeta }$, same parameters.

Figure 9

Cusp lines in the $(\delta m,\delta f)$ plane, the region interior to the curves exhibit bistability between ${\varphi }_{-}$ and ${\varphi }_{+}$, the curves merge together at the cusp point. The full and the dashed lines correspond respectively to the normal form and mean field approximations. Curves are plotted in the case $\sigma =3$ and $\overline{k}=30$. $m_c=26.497,f_c=0.7335$.

Figure 10

Asymptotic values for $\varphi \left(f\right)$ as a function of $f$ for $m=26$, i.e., $\delta m=0.497$ with $m_c=26.497$. The upper and the lower lines correspond respectively to the normal form and mean field approximations. Curves are plotted in the case $\sigma =3$ and $\overline{k}=30$.

Figure 11

Power law behavior in log-log plot for $g_m\left(f\right)$ as a function of $\epsilon ={\delta }_m/m_c$ close to the critical point $g_m\left(f\right)=g_0{\epsilon }^{\alpha }$ where $\alpha =0.4937$ for the normal form theory and $\alpha =0.4865$ for the mean field approach. Curve plot in the case $\sigma =8$ and $\overline{k}=50$.

Figure 12

Relaxation time corresponding to spinodal slowing down around $f^{*}$; the power law exponent is $-0.5037$.