We present a model of bursting neurons that combines a Fitz-Hugh Nagumo (FHN) model with an additional dynamic variable, which is slower than those in the FHN model and plays an inhibitory role. The effects of noise from the network connections is incorporated in a single parameter. This inhibitory variable enables the neuron firing to be inhibited and generates inter-spike intervals (ISI) with long time scales resulting in bursting. This phenomenon is also observed in cortical neuronal cultures. where the bursting frequency is found to be much slower than the characteristic time scale of a neuron. It is observed that bursting occurs when the mean coordination number of a neuron with the inhibitory element exceeds a threshold value. Furthermore, in the presence of noise, the ISI distribution displays complex and nontrivial patterns which reveal the interesting property of missing spike events in bursting trains. In particular the missing spikes show a power-law decay suggesting that correlations are induced in the system. The coefficient of variation can well characterize the nature of the bursting transitions, and the associated phase diagram is also computed.
