Andronov–Hopf and Bogdanov–Takens bifurcations in a Filippov Hindmarsh–Rose system with switching policy for the slow variable

This paper investigates the Andronov–Hopf and Bogdanov–Takens bifurcations in a three-dimensional Hindmarsh–Rose system with switching dynamics. A switching policy is applied to the slow recovery variable, dividing the system into two smooth subsystems by setting the switching threshold to zero. The presence of visible or invisible fold and cusp singularities in the sliding vector field is examined. By examining conditions satisfied by the trace and determinant, this study establishes the existence of Andronov–Hopf and Bogdanov–Takens bifurcations. The necessary and sufficient conditions for the occurrence of Andronov–Hopf and Bogdanov–Takens bifurcations are derived. Through numerical simulations, we have obtained the irregular limit cycle phase portrait with sawtooth-shaped and the corresponding time series of spike peaks gradually attenuating within the firing period, these new firing phenomena attributed to the switching policy. We have gathered statistics on the spike counts of membrane potential during firing events within a specific range of parameter d. Thus, we can achieve a wide variety of bursting patterns through this switching system. Furthermore, single-parameter and two-parameter bifurcation diagrams are generated by varying the key parameters b and d, we found that the chaotic and periodic regions intertwine with each other, with chaotic regions existing within the periodic regions, and periodic states found within the chaotic regions. The number of spikes during a firing period in neurons can provide important clues about the neuronal activity state, information processing, and potential diseases. In summary, this study demonstrates that the Filippov Hindmarsh–Rose system, characterized by its slow variable switching threshold, significantly influences the firing behavior of the system.