All of the functionality of the brain is through the coupling of the neurons. In this manuscript, two neurons with different types of excitability, a spiking neuron of type I excitability and an excitable one of type II excitability, couple through a linear form of gap junction. By increasing the coupling strength both neurons tend to tonic spiking, that is the appearance of two-torus in the coupled system. Further increasing in the coupling strength results in the torus break-down through “fold limit cycle bifurcation” on topological torus, then a stable limit cycle appears. This limit cycle corresponds to the emergent bursting oscillations in the neuron of type II excitability while the other neuron is in tonic spiking. Further increasing in the coupling strength makes the coupled system undergo infinitely many cascades of periodic-doubling bifurcations. Which results in bursting Oscillations with different lengths in the neuron of type II excitability. Through these cascades transient and robust chaos of Smale horseshoe type can be observed in the coupled system. Further increasing in the coupling strength leads to the burst synchronization of the neurons. In this manuscript it is approved that burst synchronizations with different lengths can be observed through a sequence of “blue sky catastrophe” bifurcations. This sequence eventually leads to the synchrony.