Instability and Death of Spiral Wave in a Two-Dimensional Array of Hindmarsh-Rose Neurons

Spiral wave could be observed in the excitable media, the neurons are often excitable within appropriate parameters. The appearance and formation of spiral wave in the cardiac tissue is linked to monomorphic ventricular tachycardia that can denervate into polymorphic tachycardia and ventricular fibrillation. The neuronal system often consists of a large number of neurons with complex connections. In this paper, we theoretically study the transitionfrom spiral wave to spiral turbulence and homogeneous state (death ofspiral wave) in two-dimensional array of the Hindmarsh-Rose neuron with completely nearest-neighbor connections. In our numerical studies, a stable rotating spiral wave is developed and selected as the initial state, then the bifurcation parameters are changed to different values to observe the transition from spiral wave to homogeneous state, breakup of spiral wave and weak change of spiral wave, respectively. A statistical factor of synchronization is defined with the mean field theory to analyze the transition from spiral wave to other spatial states, and the snapshots of themembrane potentials of all neurons and time series of mean membranepotentials of all neurons are also plotted to discuss the change ofspiral wave. It is found that the sharp changing points in the curvefor factor of synchronization vs. bifurcation parameter indicatesudden transition from spiral wave to other states. And the resultsare independent of the number of neurons we used.