Pattern formation and firing synchronization in networks of map neurons

Patterns and collective phenomena such as firing synchronization are studied in networks of nonhomogeneous oscillatory neurons and mixtures of oscillatory and excitable neurons, with dynamics of each neuron described by a two-dimensional (2D) Rulkov map neuron. It is shown that as the coupling strength is increased, typical patterns emerge spatially, which propagate through the networks in the form of beautiful target waves or parallel ones depending on the size of networks. Furthermore, we investigate the transitions of firing synchronization characterized by the rate of firing when the coupling strength is increased. It is found that there exists an intermediate coupling strength; firing synchronization is minimal simultaneously irrespective of the size of networks. For further increasing the coupling strength, synchronization is enhanced. Since noise is inevitable in real neurons, we also investigate the effects of white noise on firing synchronization for different networks. For the networks of oscillatory neurons, it is shown that firing synchronization decreases when the noise level increases. For the missed networks, firing synchronization is robust under the noise conditions considered in this paper. Results presented in this paper should prove to be valuable for understanding the properties of collective dynamics in real neuronal networks.