Dynamic patterns and their interactions in networks of excitable elements

Formation of localized propagating patterns is a fascinating self-organizing phenomenon that happens in a wide range of spatially extended, excitable systems in which individual elements have resting, activated, and refractory states. Here we study a type of stochastic three-state excitable network model that has been recently developed; this model is able to generate a rich range of pattern dynamics, including localized wandering patterns and localized propagating patterns with crescent shapes and long-range propagation. The collective dynamics of these localized patterns have anomalous subdiffusive dynamics before symmetry breaking and anomalous superdiffusive dynamics after that, showing long-range spatiotemporal coherence in the system. In this study, the stability of the localized wandering patterns is analyzed by treating an individual localized pattern as a subpopulation to develop its average response function. This stability analysis indicates that when the average refractory period is greater than a certain value, there are too many elements in the refractory state after being activated to allow the subpopulation to support a self-sustained pattern; this is consistent with symmetry breaking identified by using an order parameter. Furthermore, in a broad parameter space, the simple network model is able to generate a range of interactions between different localized propagating patterns including repulsive collisions and partial and full annihilations, and interactions between localized propagating patterns and the refractory wake behind others; in this study, these interaction dynamics are systematically quantified based on their relative propagation directions and the resultant angles between them before and after their collisions. These results suggest that the model potentially provides a modeling framework to understand the formation of localized propagating patterns in a broad class of systems with excitable properties.

Figure 1

Snapshots of localized patterns emerging from the simple three-state network model. Red (dark gray) dots indicate the positions of the elements that are activated at that time, black dots for those in the refractory state, and green (light gray) dots for ones in the resting state. (a) When $p_g=0.98$, individual patterns have their activated elements fuzzily distributed in a near-circular area. (b) When $p_g=0.26$, there are multiple, localized propagating patterns, whose propagating directions are indicated by arrows. Note that each pattern has crescent-shaped, coherent front. Given the random nature of the system, e.g., the random transition from the refractory state to the ready states, the emergence of the coherent moving front is a fascinating example of self-organization in our simple system. Each propagating localized pattern leaves a comet-like refractory tail in its wake, which is characterized analytically below.

Figure 2

Magnitude of the order parameter $\left|\phi \right|$ as a function of $p_g$. Two localized excitation patterns (the dots) above and below the symmetry breaking point are shown, for $p_g=0.68$ and $p_g=0.13,$ from which one can see how rotational symmetry of these patterns is reduced by symmetry breaking.

Figure 3

Mean-squared-displacement $D\left(\tau \right)$ versus time interval τ for different $p_g$. From upper to lower curves, the value of $p_g$ is 0.3, 0.4, 0.5, 0.6, respectively. The levelling off in these curves is due to the finite size of the system. A fit to the straight-line parts using the maximum likelihood method gives the best fit $\left(\tau \right)\sim {\tau }^{\alpha }$; the red (dark gray) line is fitted line with an exponent of $\alpha$. For the cases $p_g=0.3$ and $p_g=0.4$, the diffusion exponent α is 1.78 and 1.6, respectively. When $p_g=0.5,D\left(\tau \right)\sim {\tau }^{\alpha }$, with $\alpha =1.56$ when $\tau$12, and $\alpha =0.92$ when $\tau <12$, indicating a crossover from superdiffusion to subdiffusion around the critical point of symmetry breaking.

Figure 4

The average response function of subpopulation in a circular area. The dots show the results of the calculated average response values, and the solid line depicts a best fit [Eq. (11)] to them.

Figure 5

Graphical illustration of the solutions of Eq. (13). When $p_g=0.4994$, the evolution of the map goes to infinity. The map starts from the point $x_0=0.31$, and its next iterate value $x_1$ according to the Eq. (13) is located at the point A.

Figure 6

Collisions between different localized propagating patterns and between a propagating pattern and the refractory wake of another one. Red (dark gray) dots indicate the positions of the elements that are activated at that time, black dots for those in the refractory state, and green (light gray) dots for ones in the resting state. (a) Front-front collision when $p_g=0.25$. Paths before and after the collision are highlighted. Due to the lateral inhibitory coupling, each propagating pattern is inhibited by another one when they approach each other, so there is no contact between the patterns during the collision. (b) A front-wake collision happened within the area marked by the box with dashed lines, when $p_g=0.18$.

Figure 7

A typical repulsive collision between two localized propagating patterns when $p_g=0.1$. The incoming angle formed by their propagating paths before the collision is ${\theta }_{\mathrm{in}}$, and the corresponding outgoing angle formed by those after the collision is ${\theta }_{\mathrm{out}}$. (a) The relative ratio of different types of collisions with different collision angles, which is obtained by tracking many repulsive collisions with the different pairs of ${\theta }_{\mathrm{in}}$ and ${\theta }_{\mathrm{out}}$, and then calculating the relative ratio of each pair. The ratio is represented by different color (different gray levels), as indicated by the color (gray) bar. (b) A schematic of the repulsive collision happening in Region A. (c) A schematic of the repulsive collision happening in Region B. (d) A schematic of the repulsive collision happening in Region C.

Figure 8

The density of the elements that are still in the refractory state versus their distance to the front of a localized propagating pattern. The blue line is for the analytical result for the case when $p_g=0.3$, and the dots are for simulated results.

Figure 9

The rate $r_a$ of annihilations versus $p_g$. The dots show the calculated results and the red line (vertical line) indicates the point $p_g=0.21$; when $p_g\le 0.21$, the annihilation rate increases significantly at low $p_g$.

Figure 10

(a) The relative ratio of different types of collisions when there is a time delay between two localized propagating patterns; e.g., two patterns do not come to an interacting spot at the same time moment. The proximity of collision is the distance between the center of a wave front and that of a refractory wake. The ratio is represented by different color (different gray levels), as indicated by the colored (gray) bar. (b) The repulsive collision when the time delays between two patterns are small. (c) Elastic interaction between a propagating front and a refractory wake. (d) A propagating front passes through the refractory wake of another one.