In-phase and anti-phase bursting dynamics and synchronisation scenario in neural network by varying coupling phase

We have analysed the synchronisation scenario and the rich spatiotemporal patterns in the network of Hindmarsh-Rose neurons under the influence of self, mixed and cross coupling of state variables which are realised by varying coupling phase. We have introduced a coupling matrix in the model to vary coupling phase. The excitatory and inhibitory couplings in the membrane potential induce in-phase and anti-phase bursting dynamics, respectively, in the two coupled system. When the off-diagonal elements of the matrix are zero, the system shows self coupling of the three variables, which helps to attain synchrony. The off-diagonal elements give cross interactions between the variables, which reduces synchrony. The stability of the synchrony attained is analysed using Lyapunov function approach. In our study, we found that self coupling in three variables is sufficient to induce chimera states in non-local coupling. The strength of incoherence and discontinuity measure validates the existence of chimera and multichimera states. The inhibitor self coupling in local interaction induces interesting patterns like Mixed Oscillatory State and clusters. The results may help in understanding the spatiotemporal communications of the brain, within the limitations of the size of the network analysed in this study.