Coherence-Resonance Chimeras in a Network of Excitable Elements

We demonstrate that chimera behavior can be observed in nonlocally coupled networks of excitable systems in the presence of noise. This phenomenon is distinct from classical chimeras, which occur in deterministic oscillatory systems, and it combines temporal features of coherence resonance, i.e., the constructive role of noise, and spatial properties of chimera states, i.e., the coexistence of spatially coherent and incoherent domains in a network of identical elements. Coherence-resonance chimeras are associated with alternating switching of the location of coherent and incoherent domains, which might be relevant in neuronal networks.

Figure 1

Space-time plots of activator $u_i$ (left column), local order parameter $Z_i$ (middle column, ${\delta }_Z=25$; coherence is identified by $Z_i>1-{10}^{-6}$ for numerical purposes), and line scan of $Z_i$ at fixed time (right column) for different noise intensities (a) $D=0$: steady state; (b) $D=0.0002$: coherence-resonance chimera (initial conditions: randomly distributed on the circle $u^2+v^2=4$). The black rectangle marks the incoherent domain. (c) $D=0.0004$: incoherent in space but periodic in time; (d) $D=0.1$: incoherent in space and time. Horizontal dashed line indicates the value $Z_k=1$. Parameters: $ϵ=0.05$, $a=1.001$, $\sigma =0.4$, $r=0.12$.

Figure 2

Time evolution of coherence-resonance chimera: Snapshots (left column) and corresponding phase space (right column) for (a) $t=976.89$, (b) $t=977.75$, (c) $t=978.13$, (d) $t=979.81$, (e) $t=981.87$, (f) $t=982.48$. Other parameters: $ϵ=0.05$, $a=1.001$, $D=0.0002$, $\sigma =0.4$, $r=0.2$.

Figure 3

Dynamic regimes in the $(r,\sigma )$ parameter plane: (a) steady state (orange dotted), (b) incoherent in space and periodic in time (yellow plain), (c) coherence-resonance (CR) chimera with one incoherent domain (blue crosshatched), (d) CR chimera with two incoherent domains (green crosshatched), (e) CR chimera with three incoherent domains (purple vertically hatched). Multistability is also indicated. Other parameters: $ϵ=0.05$, $a=1.001$, $D=0.0002$, $N=500$.

Figure 4

Characterization of CR chimera: (a) Space-time plot defining active time $\mathrm{\Delta }$ and size $\delta$ of the incoherent domain. (b),(c) Dependence of $\delta /N$ and $\mathrm{\Delta }$ upon excitation threshold $a$ for $D=0.0002$ (a) and upon noise intensity $D$ for $a=1.001$ (c). The gray region corresponds to the existence of CR chimeras. Other parameters: $ϵ=0.05$, $N=1000$, $\sigma =0.4$, $r=0.2$.