A spatially stable pattern of two coexisting coherent and incoherent subpopulations in nonlocally coupled dynamical systems is called as chimera states and seen in many paradigmatic limit cycle as well as chaotic models where the coupling interaction is basically diffusive type. In neuronal networks, besides diffusive electrotonic communication via gap junctions, chemical transmission occurs between the pre-synapse and post-synapse of neurons. We consider, in a numerical study, a network of neurons in a ring using the Hindmarsh-Rose (HR) bursting model for each node of the network and, apply attractive gap junctions for local coupling between the nearest neighbors and inhibitory nonlocal coupling via chemical synaptic transmission between the distant neighbors. For a range of gap junctional and chemical synaptic coupling strengths, a subpopulation of the neuronal network, in the ring, bursts asynchronously and another subpopulation remains silent in a synchronous state. The bursting subpopulation of neurons fires sequentially along the ring when the number of firing nodes remains same but change their positions periodically in time. It appears as a traveling chimera pattern in the ring when the dynamics of the individual bursting nodes is chaotic. The chimera pattern travels in a reverse direction for a larger chemical synaptic coupling strength. A purely inhibitory chemical synaptic coupling can produce a similar traveling chimera pattern, however, the dynamics of the firing nodes is then periodic.

**Introduction**

**Network of Neurons**

**Figure 1.**Schematic diagram of a neuronal network. Open circles represent HR neurons connected to their nearest neighbors by local gap junction links in black lines. Red dashed lines represent non-local chemical synaptic links shown for one node, which is true for all other nodes._{0}=x

_{N}and x

_{N+1}=x

_{1}, vsis the reversal potential, \(Γ(x_j )=\frac{1}{1+e^{-λ(x_j-Θ_s ) }}\) defines a chemical synaptic coupling function. k

_{1}and k

_{2}are strengths of gap junctional interaction and chemical synaptic coupling, respectively. P is the coupling radius that denotes the number of nodes on both sides of each node to which it communicates via chemical synaptic interactions (excluding two nearest neighbors). The coupling parameters for each node are chosen as Θ

_{s}=-0.25,v

_{s}=2,λ=10,P=40 and all the nodes (N = 100) are identical. The system parameters are a = 1, b = 3, c = 1, d = 5, r = 0.01, s = 5 and xR=-1.6 when an uncoupled node exhibits periodic bursting as shown in Fig. 2.

**Figure 2.**Periodic bursting of an isolated HR model neuron. Parameters :a = 1, b = 3, c = 1, d = 5, r = 0.01, s = 5 and x_{R}=-1.6.**Results**

_{1}=3 and k

_{2}=2, we find a traveling chimera pattern. Note that this is not limited to this particular set of coupling parameters rather true for a reasonably large range.

**Figure 3.**Traveling chimera pattern for k_{1}=3 and k_{2}=2. (a) Spatiotemporal pattern of the x-variable of all oscillators. The pattern travels from bottom right corner to upper left corner in time. (b) Snapshots of x-variable of all nodes at two different instants of time. Red line is for a later time than the black snapshot implying a traveling pattern from left to right._{1}=3 and k

_{2}=4, we find a similar kind of traveling chimera pattern, however, the traveling direction is reversed compared to the previous case. Figures 4(a) and 4(b) show a spatiotemporal evolution and snapshot of xi-variable, respectively, that clearly confirms the traveling nature of the chimera pattern. The pattern now travels from bottom left to right upper corner in the spatiotemporal plot as clearly seen from the spatiotemeporal plot in Fig. 4(a). Snapshots in Fig. 4(b) (red and black lines) confirm this reversal in the direction of movement. Figure 4(c) shows a time series of x that confirms chaotic bursting behavior of the nodes. A change of dynamics from periodic to chaotic bursting is reflected in the collective behavior of the chimera states. We explain this collective dynamics of the ring as an asynchronous firing of a subpopulation in a bursting state that moves sequentially along the ring periodically in time and this behavior coexists with another subpopulation in synchronous resting states that also changes periodically. Most importantly, the number of two coexisting subpopulations remains fixed for a fixed set of coupling parameters.

**Figure 4.**Traveling chimera for k_{1}=3 and k_{2}=4.(a) Spatiotemporal evolution of the x-variable shows the pattern is traveling from bottom left to upper right corner. (b) Snapshots of x-variable of all nodes at two different instants. (c) Time series shows a chaotic bursting behavior._{2}=9. A spatiotemporal plot in Fig. 5(a) of xi-variable of all the nodes confirms this traveling character of the chimera states. Figure 5(b) represents the time series of xi-variable exhibiting periodic bursting behavior instead of a chaotic bursting.

**Figure 5.**Traveling chimera pattern for purely inhibitory synaptic coupling, k_{1}=0,k_{2}=9. (a) Spatiotemporal plot shows the traveling chimera pattern, (b) time series of xi of a bursting node exhibits periodic behavior.**Conclusion**

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A.M is supported by the University Grants Commission, India. A.M. and S.S also acknowledge local support and hospitality for two week’s visit by the Lobachevsky state University of Nizhny Novgorod, Russia. D.G. and S.K.D. are supported by the SERB-DST (Department of Science and Technology), Government of India (Project No. INT/RUS/RFBR/P-181), GO acknowledges support of Russian Science Foundation (Project N 17-12-01534).

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