Co-existence of blue-sky catastrophe and bistability in a coupled Leech-heart inter-neuron model with both chemical and electrical coupling.

Neuronal activity is characterized by diverse firing patterns that underlie both normal physiological processes and pathological states of the brain. In this study, we explore diverse firing activities, including tonic, bursting and spiking in a coupled Leech Heart inter-neuron model incorporated with both chemical and electrical synapses. These rich dynamics arise due to the emergence of multiple bifurcations, including codimension-one bifurcations (Hopf, saddle-node, and branch point) and codimension-two bifurcations (Bogdanov-Takens and cusp) in the model. Further, due to the changes of chemical coupling, we demonstrate the bistability of spiking along with the blue sky catastrophe phenomena in different parameter ranges. The long-term behavior of the coupled neurons is examined using synchronization analysis, which reveals complete synchronization under certain conditions. Additionally, numerical simulations are carried out to test the theoretical results and demonstrate the dual nature of neurons for the physiologically relevant neural processes.