Classifying non-stationary and imbalanced data streams encompasses two important challenges, namely concept drift and class imbalance. Concept drift is changes in the underlying function being learnt, and class imbalance is vast difference between the numbers of instances in different classes of data. Class imbalance is an obstacle for the efficiency of most classifiers. Previous methods for classifying non-stationary and imbalanced data streams mainly focus on batch solutions, in which the classification model is trained using a chunk of data. Here, we propose two online classifiers. The classifiers are one-layer NNs. In the proposed classifiers, class imbalance is handled with two separate cost-sensitive strategies. The first one incorporates a fixed and the second one an adaptive misclassification cost matrix. The proposed classifiers are evaluated on 3 synthetic and 8 real-world datasets. The results show statistically significant improvements in imbalanced data metrics.