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Mild cognitive impairment (MCI) is a nervous system disease, and its clinical status can be used as an early warning of Alzheimer's disease (AD). Subtle and slow changes in brain structure between patients with MCI and normal controls (NCs) deprive them of effective diagnostic methods. Therefore, the identification of MCI is a challenging task. The current functional brain network (FBN) analysis to predict human brain tissue structure is a new method emerging in recent years, which provides sensitive and effective medical biomarkers for the diagnosis of neurological diseases. Therefore, to address this challenge, we propose a novel Deep Spatiotemporal Attention Network (DSTAN) framework for MCI recognition based on brain functional networks. Specifically, we first extract spatiotemporal features between brain functional signals and FBNs by designing a spatiotemporal convolution strategy (ST-CONV). Then, on this basis, we introduce a learned attention mechanism to further capture brain nodes strongly correlated with MCI. Finally, we fuse spatiotemporal features for MCI recognition. The entire network is trained in an end-to-end fashion. Extensive experiments show that our proposed method significantly outperforms current baselines and state-of-the-art methods, with a classification accuracy of 84.21%.