Visual attention is a fundamental mechanism in the human brain, and it inspires the design of attention mechanisms in deep neural networks. However, most of the visual attention studies adopted eye-tracking data rather than the direct measurement of brain activity to characterize human visual attention. In addition, the adversarial relationship between the attention-related objects and attention-neglected background in the human visual system was not fully exploited. To bridge these gaps, we propose a novel brain-inspired adversarial visual attention network (BI-AVAN) to characterize human visual attention directly from functional brain activity. Our BI-AVAN model imitates the biased competition process between attention-related/neglected objects to identify and locate the visual objects in a movie frame the human brain focuses on in an unsupervised manner. We use independent eye-tracking data as ground truth for validation and experimental results show that our model achieves robust and promising results when inferring meaningful human visual attention and mapping the relationship between brain activities and visual stimuli. Our BI-AVAN model contributes to the emerging field of leveraging the brain's functional architecture to inspire and guide the model design in artificial intelligence (AI), e.g., deep neural networks.