Humans can learn continuously grasping various objects dexterously. This ability is enabled partly by underlying neural mechanisms. Most current works of anthropomorphic robotic grasping learning lack the capability of continual learning (CL). They utilize large dataset to train grasp models and the trained models are difficult to improve incrementally. By incorporating several discovered neural mechanisms supporting CL, we propose a neuro-inspired continual anthropomorphic grasping (NICAG) approach. It consists of a CL framework of anthropomorphic grasping and a neuro-inspired CL algorithm. Compared with other methods, our NICAG approach achieves better CL capability with lower loss and forgetting, and gets higher grasping success rate. It indicates that our approach performs better on alleviating forgetting and preserving grasp knowledge. The proposed system offers an approach for endowing anthropomorphic robotic hands with the ability to learn grasping objects continually, and has great potential to make a profound impact on robots in households and factories.