Complex dynamics of the dopamine ultradian synaptic regulator model with external electromagnetic stimulus

Microscopic neural systems can exhibit complex firing patterns, such as excitability, spiking, bursting, and chaos. A dopamine ultradian synaptic regulator (DUSR) model characterizes the pattern richness, but the detailed dynamic mechanism under the external electromagnetic stimulus is unclear. Here, we investigate the formation and transition mechanism of complex rhythms in the DUSR model when the constant and alternating stimuli are introduced. Two kinds of excitabilities with respect to the zero-frequency and nonzero-frequency are found in the model subjected to the constant stimuli. The transition mechanism between the resting and firing states is demonstrated by employing the codimension-2 bifurcation and two-parameter numerical scanning plane. Furthermore, we deduced the occurrence conditions of bifurcations such as Hopf, cusp, and Bogdanov–Takens bifurcations according to the normal form theory. Additionally, the model subjected to the alternating stimuli can exhibit abundant firing rhythms involving bursting. The formation mechanism of the bursting is revealed through the fast-slow decomposition method. The different morphologies of bursting oscillation are obtained by modulating the saddle-node, Hopf, and saddle-node on invariant cycle bifurcation. The appropriately external electromagnetic stimulus is conducive to the modulation of ultradian rhythms. These findings not only serve as theoretical support for understanding the role of dopamine and electromagnetic regulation in ultradian rhythms but also provide new perspectives for studying the regulatory mechanisms of neural rhythms in microscopic systems and the treatment of rhythm-related neurological disorders.