Discriminating targets moving against a cluttered background is a huge challenge, let alone detecting a target as small as one or a few pixels and tracking it in flight. In the insect's visual system, a class of specific neurons, called small target motion detectors (STMDs), have been identified as showing exquisite selectivity for small target motion. Some of the STMDs have also demonstrated direction selectivity which means these STMDs respond strongly only to their preferred motion direction. Direction selectivity is an important property of these STMD neurons which could contribute to tracking small targets such as mates in flight. However, little has been done on systematically modeling these directionally selective STMD neurons. In this paper, we propose a directionally selective STMD-based neural network for small target detection in a cluttered background. In the proposed neural network, a new correlation mechanism is introduced for direction selectivity via correlating signals relayed from two pixels. Then, a lateral inhibition mechanism is implemented on the spatial field for size selectivity of the STMD neurons. Finally, a population vector algorithm is used to encode motion direction of small targets. Extensive experiments showed that the proposed neural network not only is in accord with current biological findings, i.e., showing directional preferences but also worked reliably in detecting the small targets against cluttered backgrounds.