In this paper, we discuss the existence and convergence conditions of extreme solutions for a first-order differential boundary value system with an impulsive integral condition. By employing quasi-linear and monotone iterative methods, the existence, uniform convergence, and quadratic convergence for solution sequences are obtained.

This study introduces a novel cortico-thalamo-basal ganglia-pedunculopontine nucleus (PPN) (CTBGP) computational framework to investigate how PPN-related pathways control beta oscillations. We observe that PPN-thalamic pathways exert bidirectional Hopf bifurcation control over thalamic beta oscillations, with coupling strength adjustments shifting stable/oscillatory state boundaries; PPN-cortical projection stabilizes cortical beta oscillations through supercritical/subcritical Hopf transitions dependent on coupling strength; PPN-GPi projection modulates cortico-thalamic beta oscillations via interactions with GABAergic GPi-thalamic pathways, enabling coexistence of supercritical/subcritical bifurcations; PPN-STN projection strongly suppress basal ganglia beta oscillations by elevating STN-GPe network activity to saturated states. Notably, three direct PPN inputs (EPN-PPN, STN-PPN, GPi-PPN) collectively regulate beta oscillations through the PPN-STN-GPe axis. This work provides the computational evidence that PPN pathways dynamically control beta oscillations across CTBG subcircuits via bifurcation mechanisms. The identified PPN-STN-GPe axis and thalamic/cortical projections offer novel targets for DBS and pharmacological interventions.