Sustained dynamics of a weakly excitable system with nonlocal interactions

We investigate a two-dimensional spatially extended system that has a weak sense of excitability, where an excitation wave has a uniform profile and propagates only within a finite range. Using a cellular automaton model of such a weakly excitable system, we show that three kinds of sustained dynamics emerge when nonlocal spatial interactions are provided, where a chain of local wave propagation and nonlocal activation forms an elementary oscillatory cycle. Transition between different oscillation regimes can be understood as different ways of interactions among these cycles. Analytical expressions are given for the oscillation probability near the onset of oscillations.

Figure 1

(a) Snapshot of sustained dynamics in our model with $u_M=50,\rho =0$, and $L=100$, where nonlocal links are randomly distributed. An initial perturbation is given to a chosen receiver node $i$ as $u_i=u_M$, with all other cells are set to zero. The color bar indicates the activity of each cell relative to the maximum value $u_M$. (b) Example of a two-link oscillatory cycle. Arrows 1 and 2 represent directed nonlocal links from the sender node to the receiver node. Shaded square regions represent the activation range of local excitation waves from the receiver nodes, determined by $(1-\rho )u_M$. Coordinates of nodes: (104,114)$\to$(158,68) (link 1, blue) and (121,78)$\to$(27,140) (link 2, magenta). (c) Periodic oscillations observed in the two-link cycles described in (b), with $u_M=80,\rho =0$, and $L=2$. Solid line: receiver node of link 1; triangles: sender node of link 1; broken line: receiver node of link 2; circles: sender node of link 2. (d) Diagram of a minimal oscillatory cycle. Solid arrows represent directed links; dotted arrows represent possible interactions through propagation of excitable waves; $x_1$ and $x_2$ represent the distance between the sender node of one link and the receiver node of the other, and $y$ represents the distance between the two receiver nodes.

Figure 2

Phase diagram of sustained dynamics with $L=100$ nonlocal links in the parameter space of $u_M$ and $\rho =u_{\mathrm{th}}/u_M$. (a) Oscillation probability defined as the proportion of oscillatory samples of randomly distributed $L$ links to all samples. (b) Probability of periodic oscillations. (c) Number of master nodes. (d) Number of slave nodes. (e) Oscillation periods $T_{\mathrm{per}}$ for active nodes, normalized by the refractory time $u_M$. (f) Degree of synchronization $\sigma$ for active nodes. In (c), (d), (e), and (f), values are averaged over all link distribution samples and all initial conditions with which sustained oscillations are observed.

Figure 3

Three typical examples of sustained dynamics for $L=100$ contained in the phase diagram of Fig. 2: Broken lines represent the activity of a selected active node, and solid lines represent the average over all active nodes. (a) Incoherent periodic oscillations: $u_M=20,\rho =0.2$; (b) complex oscillations: $u_M=50,\rho =0$; (c) coherent periodic oscillations: $u_M=90,\rho =0.2$.

Figure 4

Oscillation boundaries defined as the contour of the oscillation probability equal to 0.2, for different $L$ values indicated adjacent to each curve. The curve for $L=100$ corresponds to data in Fig. 2, and all other curves are calculated by performing the same numerical simulations as in Fig. 2 with different $L$ values.

Figure 5

(a) Oscillation probability as a function of $z=u_M/N$ for $\rho =0$ with different $L$ values. Circles are numerical data and solid curves are theoretical prediction given by $p_L^{\mathrm{tot}}(z,\rho )$. (b) Oscillation probability as a function of $L^{1/2}z$ for three different $\rho$ values, for each of which data for different $L$ values from $L=10$ to $L=1000$ are plotted. Solid curves are given by $p_L^{\mathrm{tot}}(z,\rho )$, where for each $\rho$ value only one curve with $L=1000$ is presented: For each $\rho$ value, curves of $p_L^{\mathrm{tot}}(z,\rho )$ with different $L$ values converge to one curve as $L\to \infty$ when plotted as a function of $L^{1/2}z$, and for $L\ge 10$ they are almost indistinguishable. For each parameter set, the oscillation probability is calculated in the same way as in Fig. 2.

Figure 6

Conditional probability of the formation of an excitable $n$-cycle when $n$ links form either an oscillatory or an excitable cycle, given by $q_0^{\left(n\right)}/[p_0^{\left(n\right)}+q_0^{\left(n\right)}]$, for $n=2,3$, and 4.