Topological effects on dynamics in complex pulse-coupled networks of integrate-and-fire type

For a class of integrate-and-fire, pulse-coupled networks with complex topology, we study the dependence of the pulse rate on the underlying architectural connectivity statistics. We derive the distribution of the pulse rate from this dependence and determine when the underlying scale-free architectural connectivity gives rise to a scale-free pulse-rate distribution. We identify the scaling of the pairwise coupling between the dynamical units in this network class that keeps their pulse rates bounded in the infinite-network limit. In the process, we determine the connectivity statistics for a specific scale-free network grown by preferential attachment.

Figure 1

Gain curves depicting the pulse rate $m_N$ as a function of the driving strength $f\nu$ in the mean-driven limit of the all-to-all coupled network. The parameter values are (7) and (11). The coupling strengths $SN$ along the gain curves are (left to right) $0.096$, $0.08$, $0.064$, $\tau \ln A=0.04823$, $0.032$, $0.016$, and 0. The gain curves approach straight-line asymptotes; their equations are given by Eq. (19). The asymptote of the gain curve corresponding to the coupling strength $SN=\tau \ln A$ is vertical. Numerical simulations in Ref. [51] indicate that backward-sloping segments of the gain curves are unstable.

Figure 2

Gain curves for the network discussed in Sec. 4c, size $N={10}^4$, initial size $\ell =50$. Top: The dots (red online) represent the dependence of the mean network pulse rates, $\overline{m}$, on the external driving strength, $f\nu$, for an ensemble of 100 network realizations. The network size was ${10}^4$. The solid curve (black online) represents the average $\overline{m}$-versus-$f\nu$ gain curve over the ensemble. The dashed curve (green online) represents the linear gain-curve asymptote predicted by Eq. (38). Bottom: Gain curves depicting the average pulse rate $\overline{m}$ as a function of the external driving strength $f\nu$ for three individual realizations of a network with ${10}^4$ nodes are represented by dots (red online). The dashed lines (green online) represent the linear gain-curve asymptote predicted by Eq. (38) with the variance ${\sigma }^2$ in Eq. (31) replaced by the realization variance ${\sigma }_r^2$ in Eq. (41). The parameter values are those in Eqs. (7) and (11), $S=4·{10}^{-5}$, $f=5·{10}^{-5}$.

Figure 3

Pulse rate $m_k$ as a function of the incoming node-degree $k$ for the network discussed in Sec. 4c, size $N={10}^4$, initial size $\ell =50$. Top: Gray solid line (green online) represents $m_k$ averaged over all $k$-nodes in 500 network realizations. Dashed line (red online) represents $m_k$ as computed numerically from Eq. (8a), truncated at $N={10}^4$, with $T(n,k)$ given by Eq. (B31), using the approximation described at the end of Appendix B. Black solid line represents the exact solution (36) of Eq. (9), with $k\ge \ell /2$. Bottom: Gray solid line (green online) represents $m_k$ averaged over all $k$-nodes in a single network realization of size $N={10}^4$. Black solid line represents the solution (36) with $k\ge \ell /2$ and the variance ${\sigma }^2$ in Eq. (31) replaced by the realization variance ${\sigma }_r^2$ in Eq. (41). On both panels, the oscillations exhibited by the simulation results at large values of $k$ are a consequence of the poor statistics due to the scarcity of large-degree nodes in a finite network. The parameter values are those in Eqs. (7) and (11), $S=4·{10}^{-5}$, $f=1.8·{10}^{-5}$, $\nu =2·{10}^4$ ($f\nu =0.36$).

Figure 4

The power $\gamma$ in the asymptotic relationship (45) as computed from Eq. (46) (dashed line, black online), and extracted from the results of the numerical solution of Eq. (8), truncated at $N={10}^4$ (solid line, green online), and direct numerical simulations of one network realization of size $N=7\times {10}^5$ (dots, red online). The ansatz $m_k=m_0+B{k}^{\gamma }$ was used as a fit. Note that both the solutions of Eq. (8) and the results of the numerical simulations extend beyond $\gamma =2$ ($\lambda =1$). The rest of the parameter values are those in Eqs. (7) and (11), $\nu =2\times {10}^4$, $f=1.8\times {10}^{-5}$ ($f\nu =0.36$).

Figure 5

Pulse rate $m_k$ as a function of the incoming node degree $k$ for the network discussed in Sec. 4d. Dark gray solid line (red online) represents the results of direct numerical simulations averaged over 500 realizations of the network of size $N={10}^4$; light gray solid line (green online) represents averages over 50 realizations of the network of size $N={10}^5$. Dashed line (black online) represents $m_k$ as computed numerically from a finite truncation of Eq. (8a), with $T(n,k)$ given by Eq. (43). (Inset) The pulse rate $m_k$ on the incoming node-degree $k$, as computed from a finite truncation of Eq. (8a), does not depend on the truncation size. The truncation sizes used in the computations were $N={10}^4$ (dashes, red online), $N={10}^5$ (solid gray line, green online), $N={10}^6$ (dashes, black online). The parameter values are those in Eqs. (7) and (11), $S={10}^{-3}$, $f=1.8\times {10}^{-5}$, $\nu =2\times {10}^4$ ($f\nu =0.36$).

Figure 6

The dependence of the coefficient $B$ in Eq. (45) on the external drive through the feedforward pulse rate $\psi$ in Eq. (12a) and the network coupling constant $\lambda$ in Eq. (12b), as computed from the mean-field model (8a). The ratios $B/\lambda$ and $B/\psi$ are used to test for linearity. (Left) The values of $\lambda$ along the curves from top to bottom are 0.21, 0.41, 0.62, and 0.83, corresponding to the values of the network coupling $S=1\times {10}^{-3}$, $2\times {10}^{-3}$, $3\times {10}^{-3}$, and $4\times {10}^{-3}$. The bottom two curves lie on top of each other. (Right) The values of $\psi$ along the curves from top to bottom are 387, 283, 180, 76, and 47, corresponding to the values of the external driving strength $f\nu =2$, 1.5, 1, 0.5, and 0.36. The top four curves lie on top of each other. The truncation size is $N={10}^4$. The rest of the parameter values are those in Eqs. (7) and (11).

Figure 7

For the same value of the coupling parameter $S$ and two different values of the external drive $f\nu$, the logarithmic dependence of the pulse rate $m_k$ on the node degree $k$ approaches a pair of parallel straight lines for large $k$. This verifies that the exponent $\gamma$ in the asymptotic relationship (45) is independent of $f\nu$. Displayed is the verification for two different values of $S$. Gray solid line (green online) represents the results of direct numerical simulations of one realization of the network of size $N=7\times {10}^5$. Dashed line (black online) represents numerical solutions of the mean-field equations (8), truncated at $N={10}^4$. From top to bottom, the values of $f\nu$ and $S$ along the curves are $f\nu =2.0$, $S=2\times {10}^{-3}$; $f\nu =2.0$, $S={10}^{-3}$; $f\nu =0.36$, $S=2\times {10}^{-3}$; and $f\nu =0.36$ $S={10}^{-3}$. The rest of the parameter values are those in Eqs. (7) and (11), and $\nu =2\times {10}^4$.

Figure 8

The dependence of the conditional probability $P_u\left(n\right|k)$ in the undirected network with $\ell =50$, as derived from the edge-distribution function $T_u(n,k)$ and the incoming-degree distribution $P_u\left(k\right)$ using Eq. (3). The dots (red online) represent averages over ${10}^3$ network realizations using Monte Carlo simulations of the network of size $N={10}^4$. The dashed line (green online) represents the analytical result of Eqs. (B21) and (B24) with $n,k\ge \ell$. (Recall that $P_u\left(n\right|k)=0$ for $n,k<\ell$.) Top to bottom on the right-hand side of the figure: $k=50$, 400, and 4000.

Figure 9

The dependence of the conditional probability $P_d\left(n\right|k)$ in the directed network with $\ell =50$, as derived from the edge-distribution function $T_d(n,k)$ and the incoming-degree distribution $P_{\text{in}}\left(k\right)$ using Eq. (3). The dots (red online) represent averages over ${10}^3$ network realizations using Monte Carlo simulations of the network of size $N={10}^4$. The solid line (black online) represents a numerical evaluation using Eqs. (B31) and (B26) as described in the text. The dashed line (green online) represents the analytical result of Eqs. (30) and (32) with $\ell /2<n,k<N/2$. Top to bottom on the right-hand side of the figure: $k=25$, 200, and 2000. We do not display the transition layer near $n=N/2$.