The brain is able to acquire and store memories of everyday experiences in real-time. It can also selectively forget information to facilitate memory updating. However, our understanding of the underlying mechanisms and coordination of these processes within the brain remains limited. However, no existing artificial intelligence models have yet matched human-level capabilities in terms of memory storage and retrieval. This study introduces a brain-inspired spiking neural model that integrates the learning and forgetting processes of sequential memory. The proposed model closely mimics the distributed and sparse temporal coding observed in the biological neural system. It employs one-shot online learning for memory formation and uses biologically plausible mechanisms of neural oscillation and phase precession to retrieve memorized sequences reliably. In addition, an active forgetting mechanism is integrated into the spiking neural model, enabling memory removal, flexibility, and updating. The proposed memory model not only enhances our understanding of human memory processes but also provides a robust framework for addressing temporal modeling tasks.