In this paper, we study the existence of solution for a class of impulsive integro-differential equations. Different from traditional periodic and anti-periodic boundary value problems, a more general boundary condition introduced in this new system. First, we obtain some new comparison principles. Then, we obtain the expression of the solution for a class of linearized systems. Finally, the existence of extremal solutions for the new boundary value system are obtained by using the monotone iterative technique. The theoretical results obtained have wider applications in practical fields.

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.