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Memory is an intricate process involving various faculties of the brain and is a central component in human cognition. However, the exact mechanism that brings about memory in our brain remains elusive and the performance of the existing memory models is not satisfactory. To overcome these problems, this paper puts forward a brain-inspired spatio-temporal sequential memory model based on spiking neural networks (SNNs). Inspired by the structure of the neocortex, the proposed model is structured by many mini-columns composed of biological spiking neurons. Each mini-column represents one memory item, and the firing of different spiking neurons in the mini-column depends on the context of the previous inputs. The Spike-Timing-Dependant Plasticity (STDP) is used to update the connections between excitatory neurons and formulates association between two memory items. In addition, the inhibitory neurons are employed to prevent incorrect prediction, which contributes to improving the retrieval accuracy. Experimental results demonstrate that the proposed model can effectively store a huge number of data and accurately retrieve them when sufficient context is provided. This work not only provides a new memory model but also suggests how memory could be formulated with excitatory/inhibitory neurons, spike-based encoding, and mini-column structure.