Extreme events in excitable systems and mechanisms of their generation

We study deterministic systems, composed of excitable units of FitzHugh–Nagumo type, that are capable of self-generating and self-terminating strong deviations from their regular dynamics without the influence of noise or parameter change. These deviations are rare, short-lasting, and recurrent and can therefore be regarded as extreme events. Employing a range of methods we analyze dynamical properties of the systems, identifying features in the systems’ dynamics that may qualify as precursors to extreme events. We investigate these features and elucidate mechanisms that may be responsible for the generation of the extreme events.

Figure 1

(left) Exemplary temporal evolutions of $\overline{x}$ for the investigated systems. (right) Excerpt centered around the first extreme event (at $t_{\mathrm{ext}}$) of the respective time series. For system A (first row) the individual time series of $x_1$ (red, dotted line) and $x_2$ (green, solid line) are shown.

Figure 2

Temporal evolutions of $x_i$ for each unit of system B around an extreme event. The units are indexed by their value of $b_i$. For reference, the temporal evolution of $\overline{x}$ is shown on top.

Figure 3

(left) Snapshots of the spatial distribution of $x_i\left(t\right)$ for one realization of system C around an extreme event (at $t_{\mathrm{ext}}$). Units are represented by pixels, which are arranged according to the lattice underlying the small-world network. The respective value of the dynamical variable $x$ is color coded [53]. Top right: Temporal evolution of $\overline{x}$, with snapshot times shown by red vertical lines. The event is the first one in Fig. 1, third row.

Figure 4

(left) Histogram of the interevent intervals $t_{\mathrm{IEI}}$ for two of the investigated systems. The observation time was $9\times {10}^9$ for system A and ${10}^8$ for system B. The dashed lines are multiples of $\exp \left(-r{t}_{\mathrm{IEI}}\right)$. Here $r=9.8\times {10}^{-5}$ (system A) and $r=1.0\times {10}^{-4}$ (system B), respectively, are the rates of an exponential distribution fitted to the data (for $t_{\mathrm{IEI}}>200$).

Figure 5

Results of the perturbation analysis for an exemplary extreme event (at $t_{\mathrm{ext}}=0$) of system A (left) and system B (right), taking into account events that happen within 1.0 time units of $t_{\mathrm{ext}}$ in the perturbed systems. Top: Temporal evolution of $\overline{x}$ for the unperturbed system. Bottom: Relative frequencies $q(t_{\mathrm{per}},\varepsilon )$ that the shown event is “prevented” by a perturbation of amplitude $\varepsilon$ at $t_{\mathrm{per}}$, estimated using 64 realizations (i.e., different perturbations). For reference, we show $\exp \left({\Lambda }_1{t}_{\mathrm{per}}\right)$ (green, dashed line), with ${\Lambda }_1$ being the largest Lyapunov exponent of the system. For system B, the time series of the number of excited units $e\left(t\right):=\left|\right\{i|x_i\left(t\right)>0.6\}|$ is shown as additional reference.

Figure 6

Top: Temporal evolution of $\overline{x}$ for system A. Bottom: Estimated phase difference $\Delta \varphi \left(t\right)$ between the two units. Inset: Detailed behavior of the system around one example of the phase slips.

Figure 7

Temporal evolutions of $\overline{x}$ (solid, black line) for system A and, on a logarithmic scale, of the distances $D_{\mathrm{s}}$ (green, dashed line) and $D_{\mathrm{u}}$ (red, dotted line) of the system's state from the stable or unstable manifold, respectively.

Figure 8

Projection of the attractor of system A on the $x_2$–$y_2$ plane (red, dotted line). In the bottom zoom, two extreme-event trajectories are shown in black and green thick solid lines, illustrating the channel-like structure leading to the extreme event.

Figure 9

Histogram (with non-overlapping bars) of $e$ for the proto-events from $2·{10}^8$ time units of observation, separated into those that are followed by no extreme event or a single or double extreme event, respectively. There are no more than 200 proto-events for all $e$ between 23 and 60. Also, we observed no extreme event of either kind that was not preceded by a proto-event.