Spiking Neural Networks (SNNs), regarded as the third generation of neural networks, emulate the brain's information processing with unparalleled biological plausibility compared to traditional neural networks. However, their non-linear, event-driven dynamics pose significant challenges for training, and existing methods often deviate from neuroscientific principles of cortical learning. Drawing inspiration from predictive coding theory-a leading model of brain information processing-we propose PC-SNN, a novel learning framework that integrates predictive coding with SNNs to enable biologically plausible, local Hebbian plasticity without reliance on backpropagation. Unlike conventional SNN training approaches, PC-SNN leverages only local computations, aligning with the brain's distributed processing and overcoming the biological implausibility of global error propagation. Our classification model achieves competitive performance on the benchmark datasets, including Caltech Face/Motorbike, MNIST, and CIFAR10, surpassing state-of-the-art multi-layer SNNs. Furthermore, our predictive coding-based regression model outperforms backpropagation-based methods while adhering to local plasticity constraints, offering a scalable and biologically grounded alternative for SNN training. PC-SNN drives progress in neuromorphic computing through validating the adaptability of bio-inspired algorithms within spiking neural architectures, but also unveils novel understandings of neurocognitive learning processes, presenting a conceptual framework distinguished by its theoretical originality and functional efficacy.