Imperfect traveling chimera states induced by local synaptic gradient coupling

In this paper, we report the occurrence of chimera patterns in a network of neuronal oscillators, which are coupled through local, synaptic gradient coupling. We discover a new chimera pattern, namely the imperfect traveling chimera state, where the incoherent traveling domain spreads into the coherent domain of the network. Remarkably, we also find that chimera states arise even for one-way local coupling, which is in contrast to the earlier belief that only nonlocal, global, or nearest-neighbor local coupling can give rise to chimera state; this find further relaxes the essential connectivity requirement of getting a chimera state. We choose a network of identical bursting Hindmarsh-Rose neuronal oscillators, and we show that depending upon the relative strength of the synaptic and gradient coupling, several chimera patterns emerge. We map all the spatiotemporal behaviors in parameter space and identify the transitions among several chimera patterns, an in-phase synchronized state, and a global amplitude death state.

Figure 1

Square wave bursting: time series of $x$ of an individual HR neuron given by Eq. (1) ($a=2.8, \alpha =1.6, c=5, b=9$, and $\mu =0.001$).

Figure 2

$\epsilon >r$: Imperfect traveling chimera state of an ensemble of HR-neurons with asymmetric excitatory local coupling. (a) Spatiotemporal plot as a function of the neuron index $i=1,2,\cdots ,N=200$ for $r=0.2$ and $\epsilon =0.7$ ($\epsilon >r$). (b) Snapshot of the variable $x_i$ at $t=2000$, (c) local order parameter with node index $i$. Red (gray) indicates coherent and blue (dark gray) represents incoherent domains. (d) Time series of an individual neuron $x_0$.

Figure 3

$r=0.2$ and $\epsilon =0.56$ ($\epsilon >r$): Imperfect traveling chimera state with increasing imperfection. (a) Spatiotemporal plot as a function of the neuron index $i=1,2,\cdots ,N=200$. Snapshot of the variable $x_i$ at (b) $t=1750$ showing one-headed chimera state, (c) $t=2250$ showing two-headed chimera state. (d) Time series of an individual neuron $x_0$.

Figure 4

$\epsilon >r, r=0.2$: (a) $\epsilon =1.3$, globally synchronized populations of neurons. (b) The corresponding time series of a single neuron. (c) $\epsilon =1.6$, global amplitude death. (d) The corresponding time series of a single neuron.

Figure 5

One-way excitatory coupling $\epsilon =r$: (a)–(c) $\epsilon =r=0.56$: imperfect chimera state; (a) spatiotemporal plot, (b) local order parameter, and (c) snapshot at $t=3500$ shows multiheaded chimera state. (d)–(f) $\epsilon =r=0.6$: traveling chimera state; (d) spatiotemporal plot, (e) local order parameter, and (f) snapshot at $t=3500$ shows one-headed chimera state. (g)–(i) $\epsilon =r=0.7$: synchronized state; (g) spatiotemporal plot, (h) local order parameter, and (i) snapshot at $t=3500$ shows synchronized $x_i$.

Figure 6

$\epsilon <r:$ Traveling chimera state in an ensemble of HR-neuron with excitation-inhibition local coupling $r=8$ and $\epsilon =0.6$. (a) Spatiotemporal plot as a function of node index $i=1,2,\cdots ,N=200$. (b) Snapshot of the variable $x_i$ at $t=3300$. (c) Local order parameter with node index. Red (gray) indicates coherent domains and blue represents incoherent domains. (d) Time series of an individual neuron.

Figure 7

Simultaneous presence of excitatory and inhibitory coupling for $\epsilon <r$: variation of maximum values of $x_i\left(t\right)$ with time $t$, for (a) $r=8.0$ and (c) $r=6.0$. The corresponding Fourier transform of $M\left(t\right)$ for (b) $r=8.0$ and (d) $r=6.0$. (e) Shift of maxima in the Fourier transform of $M\left(t\right)$ for $r=8.0$ (red dashed line) and $r=6.0$ (blue line).

Figure 8

Variation of traveling speed for different values of gradient coupling strength $r$ for $\epsilon =0.6.$

Figure 9

Phase diagram in the $\epsilon -r$ plane. NS: turbulence or unsynchronized state, IC: imperfect chimera state, ITC: imperfect traveling chimera state, TC: traveling chimera state, SYN: global synchronized state, AD: global amplitude death.

Figure 10

$\epsilon =0.58,r=0.2$: (a) Imperfect traveling chimera state and (b) local order parameter $L_i$ for $N=50$. (c) Imperfect traveling chimera state for $N=100$. (d) Phase diagram for $N=100$ oscillators. Other parameters as in Fig. 2.