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Anomaly detection is a very effective method to extract useful information from abundant data. Most existing anomaly detection methods are based on normal region or some specific algorithms, which ignore the fact that many actual datasets are mainly imbalanced, resulting in not function properly or effectively in practical, especially in the medical field. On the other hand, imbalanced dataset is also a frequently encountered problem in the learning of neural network because the lack of data in a minority class may lead to uneven classification accuracy. In this paper, inspired by these observations, a novel anomaly detection approach by using classical echo state network (ESN), a brain-inspired neural computing model, is presented. The entire dataset of the proposed method obeys an extremely imbalanced distribution, that is, anomalies are much rarer than normal data. And the training dataset has only the normal data. When the ESN is well trained, the parameters in ESN are the memory of normal data. If the normal data are added into the well-trained network, the error between the input data and the corresponding output is smaller compared with the error between abnormal input data and its corresponding output. Then anomaly behavior is detected if the error between the input data and the corresponding predictive value exceeds a certain threshold. Different from setting an invariable threshold arbitrarily for all of the data, the threshold value used in the proposed method is determined from the analysis of information theory and can be adjust adaptively according to different datasets. Experiments on abnormal heart rate detection are conducted to demonstrate and verify the effectiveness of the proposed detection algorithm and theory.