Temporal order and synchronization characterized by the rate of firing and its characteristic correlation time are studied in a spatially extended network system, which is locally modelled by a two-dimensional Rulkov map neuron with noise. For intermediate noise levels, noise-induced ordered patterns emerge spatially, which propagate through the neurons in the form of beautiful circular waves. Moreover, noise-induced temporal order and synchronization in this system can be greatly enhanced in the presence of a subthreshold stimulus with the frequency very close to the natural frequency of the map neuron. This shows that random perturbations and subthreshold stimuli play an important role in temporal order and synchronization. There exists an optimal noise level, where the temporal order and synchronization are maximum. However, it is observed that at the same time the spatial circular patterns are destroyed in the course of the enhancement of temporal order and synchronization.
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