Periodic neural activity induced by network complexity

We study a model for neural activity on the small-world topology of Watts and Strogatz and on the scale-free topology of Barabási and Albert. We find that the topology of the network connections may spontaneously induce periodic neural activity, contrasting with nonperiodic neural activities exhibited by regular topologies. Periodic activity exists only for relatively small networks and occurs with higher probability when the rewiring probability is larger. The average length of the periods increases with the square root of the network size.

Figure 1

One starts with a regular chain having nearest- and next-nearest-neighbor connections. With probability $p$ we replace short-range by long-range connections by rewiring. Initially $v$ is connected to $v^{\prime }$. But after rewiring, the connection of $v$ to $v^{\prime }$ is replaced by another connection, say $v$ to $v^{″}$.

Figure 2

Temporal evolution of the neural activity of graphs having 2048 sites (a) for a rewiring probability $p=0$, i.e., a regular network; (b) for a rewiring probability $p=1$, i.e., a random graph.

Figure 3

Renormalized fraction ${\varphi }^{\prime }\left(N\right)$ of graphs with periodic time series as a function of the tanh of the renormalized rewiring probability $p^{\prime }\left(N\right)$. Inset: Fraction $\varphi$ of graphs with periodic time series as a function of the rewiring probability for different network sizes $N$.

Figure 4

Fraction $\varphi$ of graphs with periodic time series as a function of the size $N$ of the graph for different values of $p$. The inset shows the data for the Barabási-Albert network for $m=3$.

Figure 5

Double-logarithmic plot of the average period length for the Watts-Strogatz network as function of the size $N$ of the network averaged over 1000 networks for $p=0.9$. The inset shows the data for the Barabási-Albert network for $m=3$.