Self-Induced Switchings between Multiple Space-Time Patterns on Complex Networks of Excitable Units

We report on self-induced switchings between multiple distinct space-time patterns in the dynamics of a spatially extended excitable system. These switchings between low-amplitude oscillations, nonlinear waves, and extreme events strongly resemble a random process, although the system is deterministic. We show that a chaotic saddle—which contains all the patterns as well as channel-like structures that mediate the transitions between them—is the backbone of such a pattern-switching dynamics. Our analyses indicate that essential ingredients for the observed phenomena are that the system behaves like an inhomogeneous oscillatory medium that is capable of self-generating spatially localized excitations and that is dominated by short-range connections but also features long-range connections. With our findings, we present an alternative to the well-known ways to obtain self-induced pattern switching, namely, noise-induced attractor hopping, heteroclinic orbits, and adaptation to an external signal. This alternative way can be expected to improve our understanding of pattern switchings in spatially extended natural dynamical systems like the brain and the heart.

Popular Summary

Many systems in nature exhibit distinct types of activity (or patterns) and apparently spontaneous switching between such states. For example, one can distinguish several normal activities of the brain (e.g., wakefulness or different sleep stages) and several harmful ones (e.g., migraine attacks or epileptic seizures). Here, we present a novel theoretical mechanism for self-induced switching between multiple patterns that may help to shed light on how, when, and why switching between patterns occurs.

We analyze the dynamics of a spatially extended excitable system consisting of 10,000 oscillators that are predominantly coupled to their neighbors but also have shortcuts to distant oscillators (i.e., a “small-world network”). This system is capable of generating three distinct spacetime patterns: low-amplitude oscillations, nonlinear waves, and extreme events. More importantly, we discover that the system can spontaneously switch between these patterns. Although the dynamics of this model is deterministic, we demonstrate that the sequence of patterns is indistinguishable from a sequence generated by a stochastic process without long-term memory. Our analysis shows that this pattern switching is fairly robust against variations in several parameters. We uncover the backbone of the switching dynamics, namely, a specific kind of chaotic saddle. Such a saddle, which has not been described previously, contains the three spacetime patterns and channels between them that facilitate the switching.

We expect that our findings will pave the way for studies of pattern switching in the brain and heart to better understand, and eventually prevent, harmful states such as epileptic seizures and atrial or ventricular fibrillations.

Figure 1

First to third row: Exemplary temporal evolutions of $\overline{x}$, of the number $e$ of units with $x_i>0.4$ (“excited units”), and of an estimate $\lambda$ of the largest local Lyapunov exponent (temporal evolution smoothed with a Gaussian kernel with a width of 30 to improve readability). The line colors indicate the patterns as automatically classified (blue: A, low-amplitude oscillations; green: B, waves; red: C, extreme events). These patterns are also indicated at the very top with pattern C being indicated with a vertical line. Bottom: Snapshots of the spatial distribution of $x_i(t)$ at times corresponding to selected local minima and maxima of $\overline{x}$ (from left to right): adjacent minimum and maximum around $t=5000$, the maximum during the event around $t=10000$, the minimum before the event around $t=30000$, and the maximum during that event. Units are represented by pixels, which are arranged according to the lattice underlying the small-world network and whose color encodes the value of the respective $x_i$ [65].

Figure 2

Exemplary temporal evolutions of $\overline{x}$ for the attractors $\mathfrak{A}$, $\mathfrak{B}$, and $\mathfrak{C}$. The inset in the middle panel shows a zoom with enlarged amplitudes.

Figure 3

Stacked histograms showing the relative number of trajectories exhibiting either pattern switching (yellow, solid) or convergence to one of the attractors $\mathfrak{A}$, $\mathfrak{B}$, or $\mathfrak{C}$ ($\mathfrak{A}$, low-amplitude oscillations: blue, north-east diagonals; $\mathfrak{B}$, traveling waves: green, south-east diagonals; $\mathfrak{C}$, global oscillations: red, crosshatched). Left: Influence of rewiring probability ($p$, bin size 0.01) with a fixed parameter inhomogeneity ($\mathrm{\Delta }b=0.008$). Right: Influence of parameter inhomogeneity ($\mathrm{\Delta }b$, bin size 0.0001) with a fixed rewiring probability ($p=0.2$). For each value of $p$ and $\mathrm{\Delta }b$, the total number of trajectories is 80 (except for $p<0.1$ and $p>0.24$, where it is 40). For each of those trajectories, we employ a different realization of the coupling topology (corresponding to that $p$) and of the inhomogeneity (corresponding to that $\mathrm{\Delta }b$). In the right-hand panel, narrow crosshatching indicates $1^0$ global oscillations, and wide crosshatching indicates $1^1$ global oscillations [75]. The black thick line is the event rate (number of times at which $\overline{x}$ rises above $x_{\mathrm{thr}}$ divided by the observation time) averaged over all realizations. Almost all trajectories exhibiting pattern switching feature all three patterns, A, B, and C.

Figure 4

Black: Histograms of pattern durations and waiting times (end to beginning) between subsequent occurrences of one pattern in $100\times (2\times {10}^6)$ time units. For the former, only patterns that begin and end during the observation time are taken into account. Red: The same averaged over 200 simulations of the pattern sequences (of the same length as the originally observed pattern sequences) as second-order Markov chains with durations of patterns randomly assigned from the respective distributions shown in the left half. Occurrences of pattern C that are longer than ${\theta }_E=90$ are due to a few events occurring in short succession. They contain at most six events and make up for less than 5 % of all occurrences of pattern C.

Figure 5

Top left: Scatter plot of $\overline{x}$ and $\overline{y}$ for 100 trajectories (the same as used for Fig. 4 and Table 1) with the point color indicating the respective pattern (blue, A; green, B; red, C), excluding trajectory segments from the attractors. Top middle: Same, only with data from the attractors $\mathfrak{B}$ (traveling waves) and $\mathfrak{C}$ ($1^1$ global oscillations). There are two distinct attractors for traveling waves (${\mathfrak{B}}_1$ and ${\mathfrak{B}}_2$), one manifested as a horizontally traveling wave front and one manifested as a diagonally traveling wave front. Note that their phase-space projections are almost a point, because $\overline{x}$ and $\overline{y}$ change only little over one period. Attractor $\mathfrak{A}$ is not observed for this realization of the coupling topology and control-parameter inhomogeneity. Middle and bottom row: Same with data only from 50 time units before switchings between patterns. Top right: Sketch of stable and unstable manifolds (black) associated with the fix point at the origin and typical trajectories (colors encode patterns as above; trajectories corresponding to switchings are shown in blended colors). All panels: Insets show zooms of the respective plots. Trajectories are generally directed counterclockwise.