We propose a novel output layer activation function, which we name ASTra (Asymmetric Sigmoid Transfer function), which makes the classification of minority examples, in scenarios of high imbalance, more tractable. We combine this with a loss function that helps to effectively target minority misclassification. These two methods can be used together or separately, with their combination recommended for the most severely imbalanced cases. The proposed approach is tested on datasets with IRs from 588.24 to 4000 and very few minority examples (in some datasets, as few as five). Results using neural networks with from two to 12 hidden units are demonstrated to be comparable to, or better than, equivalent results obtained in a recent study that deployed a wide range of complex, hybrid data-level ensemble classifiers.