Bifurcations and the switching of spiking and bursting in the extended Hindmarsh–Rose neuronal system

This paper investigates a four-dimensional Hindmarsh-Rose model with two slow variables, each of which proportionally regulates the excitability and inhibitory properties of neurons. The proportional regulation of the two slow variables is inspired by the predator dormancy phenomenon in prey-predator systems. Following the theory of normal forms, theoretical analysis demonstrates the conditions and directions of Hopf bifurcation. Hopf bifurcation plays a vital role in neuroscience research to reveal mechanisms related to neuronal transitions, internal oscillations, biological rhythm regulation, encoding mechanisms, and neurological disorders. Through the numerical simulation, it is shown that adjusting parameters related to the slow variables can easily modulate the bursting behavior of the system and generate periodic bursting patterns with different spike counts, as revealed by timeseries plots, phase portraits, and bifurcation diagrams. Meanwhile, the spike count in a burst exhibits a period-adding phenomenon, and it increases (or decreases) by one each time the parameter value changes. Additionally, by selectively filtering irrelevant inputs, switching between different bursting patterns in neural networks helps to suppress interference or noise. This study lays the foundation for future investigations of bursting switching phenomena in neural networks.