Emergence of self-sustained patterns in small-world excitable media

Motivated by recent observations that long-range connections (LRCs) play a role in various brain phenomena, we have observed two distinct dynamical transitions in the activity of excitable media where waves propagate both between neighboring regions and through LRCs. When the LRC density $p$ is low, single or multiple spiral waves are seen to emerge and cover the entire system. This state is self-sustaining and robust against perturbations. At $p=p_c^l$, the spirals are suppressed, and there is a transition to a spatially homogeneous, temporally periodic state. Finally, above $p=p_c^u$, activity ceases after a brief transient.

Figure 1

(Color online) Emergence of spiral waves, recorded in a simulation of a $N^2=128\times 128$ system, with shortcut probability $p=0.25$. After initialization, the dynamics of the system can be characterized by circular waves. At around $t\sim 1500$ time steps, spiral wave are spontaneously created and subsequently takes over the dynamics. (a) and (b) show the state of the system at different times in terms of the fast variable $x$. Colors indicate excitation level of the cells. (c) Time series of the average activity, i.e., the fraction of cells with $x>0.9$.

Figure 2

(Color online) Fraction $F_S$ of systems with spiral waves (averaged over 400 different random realizations of shortcuts) as function of time $t$: (a) Fixed shortcut probability $p=0.05$ with varying system size; (b) fixed system size $N^2=300\times 300$ with varying shortcut probability $p$.

Figure 3

(Color online) Fraction of spiral configurations $F_S$ as a function of $(p-p_c^l)$ normalized by system size $N$, for shortcut probabilities $p$ around $p_c^l\approx 0.553$. $F_S$ is calculated at $t=20000$ time steps, averaged over $400\mathrm{runs}$. Inset: Dependence of the critical value $p_c^l$ on diffusion constant $D$. The circles are simulation results for $N=200$ while the curve shows fitting with $p_c^l\sim D^{-1.14}$.

Figure 4

(Color online) Periodic regime, recorded in a simulation of a $N^2=128\times 128$ system, with shortcut probability $p=0.6$. (a)–(d) Snapshots of the state of the system taken at intervals of $\mathrm{\Delta }t=10$ steps. (e) Time series of the activity (see Fig. 1) and (f) the corresponding power spectrum. The main peak is at $f_0\approx 0.022$, corresponding to a wavelength of $\approx 43$ time steps; other peaks are harmonics at integer multiples of $f_0$.

Figure 5

(Color online) Inset: Fraction of configurations $F_C$ where activity has ceased at $t=20000$ time steps in 100 simulation runs as function of $(p-p_c^u)$. Mainpanel: The upper critical shortcut probability $p_c^u$ as a function of inverse system dimension $1∕N$ $(\circ )$. The solid line displays a fitted quadratic function, where $p_c^u(N\to \infty )\approx 0.86$.