Bursting frequency versus phase synchronization in time-delayed neuron networks

We investigate the dependence of the average bursting frequency on time delay for neuron networks with randomly distributed time-delayed chemical synapses. The result is compared with the corresponding curve for the phase synchronization and it turns out that, in some intervals, these have a very similar shape and appear as almost mirror images of each other. We have analyzed both the map-based chaotic Rulkov model and the continuous Hindmarsh-Rose model, yielding the same conclusions. In order to gain further insight, we also analyzed time-delayed Kuramoto models displaying an overall behavior similar to that observed on the neuron network models. For the Kuramoto models, we were able to derive analytical formulas providing an implicit functional relationship between the mean frequency and the phase synchronization. These formulas suggest a strong dependence between those two measures, which could explain the similarities in shape between the curves.