Within biological nervous systems, release and reception of bioelectric signals via synapses involves the complicated process of neurotransmitter release. Synapses exhibit nonlinear interaction characteristics when subjected to multiple input signals. However, traditional artificial neural networks often oversimplify synapses by adopting a linear structure. To address this limitation and account for the nonlinear interactions of multiple input signals in synapses, this paper proposes a dendritic spiking neural network with nonlinear synaptic integration, and an online supervised learning algorithm for learning sophisticate spatio-temporal patterns is also presented. Firstly, the algorithm successfully applied to spike trains learning tasks, the algorithm demonstrates its efficacy in capturing diverse spike patterns. Furthermore, the proposed algorithm has also demonstrated significantly better performance in the classification of medical datasets. This research signifies a notable advancement in accommodating the nonlinear dynamics of synaptic interactions within artificial neural networks, enhancing their capacity for intricate pattern recognition.