Emergence of self-sustained oscillations in excitable Erdös-Rényi random networks

We investigate the emergence of self-sustained oscillations in excitable Erdös-Rényi random networks (EERRNs). Interestingly, periodical self-sustained oscillations have been found at a moderate connection probability $P$. For smaller or larger $P$, the system evolves into a homogeneous rest state with distinct mechanisms. One-dimensional Winfree loops are discovered as the sources to maintain the oscillations. Moreover, by analyzing these oscillation sources, we propose two criteria to explain the spatiotemporal dynamics obtained in EERRNs. Finally, the two critical connection probabilities for which self-sustained oscillations can emerge are approximately predicted based on these two criteria.

Figure 1

Dependence of oscillation proportion $p_{\text{os}}$ on connection probability $P$ in excitable Erdös-Rényi random networks (EERRNs). One thousand independent samples are performed for each $P$. Two thresholds, the lower threshold (LT) of $P \left(P_{\text{LT}}=0.006\right)$ and the upper threshold (UT) of $P \left(P_{\text{UT}}=0.038\right)$, are indicated, between which self-sustained oscillations emerge in EERRNs.

Figure 2

Space-time plots of $u$ for different connection probabilities $P$ in EERRNs: (a) $P=0.004$, (b) $P=0.010$, and (c) $P=0.040$. The figures are plotted in grayscale from black (lowest value at 0.0) to white (highest value at 1.0). This grayscale will be used throughout this paper. (d) Trajectories of $\langle u\left(t\right)\rangle =\frac{1}{N}{\sum }_{i=1}^{N=100}{u}_i\left(t\right)$ corresponding to (a) direct damping at small $P$, shown by the black dashed curve, (b) oscillation at moderate $P$, shown by the red curve, and (c) damping after a collective spiking at large $P$, shown by the blue dotted curve. The inset is the local amplification of $\langle u\left(t\right)\rangle$ for (b). Periodical self-sustained oscillation emerges at moderate $P$.

Figure 3

(a) The DPAD pattern corresponding to the self-sustained oscillation of Fig. 2. (b) Evolution of variable $u$ of nodes in the source loop. These nodes are ordered according to the sequence in the source loop shown in (a). The one-dimensional (1D) Winfree loop which plays the role of the oscillation source is shown explicitly. (c) The network structure corresponding to the oscillation of Fig. 2.

Figure 4

(a) The artificial 1D regular excitable ring containing six nodes. Proper initial condition can initiate unidirectional wave propagation along the pathway $1\to 2\to 3\to 4\to 5\to 6\to 1$ to form a 1D Winfree loop (indicated by the black arrowed lines). The red dashed link between nodes 3 and 5 is added at $t=30$. A shorter 1D Winfree loop composed of $1\to 2\to 3\to 5\to 6\to 1$ should self-organize to dominate the oscillation (indicated by the red dashed arrowed lines). (b) Trajectory of $\langle u\left(t\right)\rangle =\frac{1}{N}{\sum }_{i=1}^{N=6}{u}_i\left(t\right)$ as the red dashed link in (a) is added (denoted by the black dotted line). A distinct oscillation emerges. (c) The DPAD pattern corresponding to the emerging oscillation. A shorter 1D Winfree loop containing five nodes self-organizes to support the oscillation. (d) The same as (a) with five nodes. (e) Trajectory of $\langle u\left(t\right)\rangle =\frac{1}{N}{\sum }_{i=1}^{N=5}{u}_i\left(t\right)$ and (f) the space-time plot as the red dashed link between nodes 3 and 5 in (d) is added at $t=30$. Oscillation damps gradually after several transient periods and the system evolves into a homogeneous rest state. The minimal length of the topological loop to form a 1D Winfree loop sustaining the oscillation is found to be 5 (i.e., $L_{\text{min}}=5$).

Figure 5

The network structures obtained at (a) $P=0.004$ and (b) $P=0.040$, which induce the spatiotemporal dynamics of Figs. 2 and 2, respectively.

Figure 6

Proportion of network structures containing topological loops with five or more nodes $p_{L\ge 5}$ (a) and the average path length $d_{\text{APL}}$ (b) vs connection probability $P$ in ERRNs. One thousand independent samples are performed for each $P$ in (a) and (b). Topological loops with $L\ge 5$ emerge at $P=0.004$. It reveals the lower critical value (LCV) of $P \left(P_{\text{LCV}}=0.004\right)$, beyond which self-sustained oscillations can emerge in EERRNs. The average path length $d_{\text{APL}}$ can be larger than 4 as $P\le 0.040$. It reveals the upper critical value (UCV) of $P \left(P_{\text{UCV}}=0.040\right)$, below which self-sustained oscillations can emerge in EERRNs. The two critical values, the lower critical value of $P \left(P_{\text{LCV}}=0.004\right)$ and the upper critical value of $P \left(P_{\text{UCV}}=0.040\right)$, are predicted explicitly based on the two criteria. The two insets show the dependence of the theoretical predictions ($P_{\text{LCV}}$ and $P_{\text{UCV}}$, shown by open circles) and numerical results ($P_{\text{LT}}$ and $P_{\text{UT}}$, shown by red squares) on system size $N$. The circles show fitting with $P_{\text{LCV}}\sim N^{-1.03}$ and $P_{\text{UCV}}\sim N^{-0.84}$.