Dynamics of the dopamine ultradian synaptic regulation model with autapse and time delays

Dopaminergic models at the microscopic scale play a fundamental role in regulating rhythmic brain activity. Existing dopamine regulation models generally neglects autapse and non-instantaneous regulatory process. In this paper, we develop an improved four-dimensional dopamine ultradian synaptic regulation (DUSR) model with time delays and memristive electromagnetic autaptic feedback. According to the center manifold reduction and normal form theory, the analytical closed-form expressions for the Hopf bifurcation stability coefficients in the improved DUSR model are derived. The theoretical analysis and numerical simulation are employed to investigate the dynamical properties. It is found that the improved DUSR model can induce a series of complex dynamical transitions among the resting, spiking, multi-period firing, and chaos based on the one-parameter and two-parameter bifurcations. The results uncover the roles of the electromagnetic autapse and delayed feedback in regulating dopaminergic rhythmicity, providing theoretical insight into rhythm instability in dopamine regulatory systems.

The online version contains supplementary material available at https://doi.org/10.1007/s11571-026-10543-y.