This paper studies the local stability of Lur'e-form nonlinear systems with static nonlinear feedback implemented using feedforward neural networks (FFNNs). We exploit positivity system constraints and use a localized variant of the Aizerman conjecture to provide sufficient conditions for exponential stability of trajectories confined to a compact set. Building on this, we develop two methods for estimating the Region of Attraction (ROA). First, a less conservative Lyapunov-based approach constructs invariant sublevel sets of quadratic functions satisfying linear matrix inequality (LMI). Second, a novel technique for computing strict local sector boundaries for FFNNs via layer-by-layer propagation of linear relaxations is presented. These boundaries are integrated into a localized Aizerman framework to verify local exponential stability. Numerical results demonstrate significant improvements over existing approaches based on integral quadratic constraints in terms of ROA size and scalability.