In recent years, there have been significant advances in our understanding of the mechanisms underlying chemically directed motility by eukaryotic cells such as Dictyostelium. In particular, the local excitation and global inhibition (LEGI) model has proven capable of providing a framework for quantitatively explaining many experiments that present Dictyostelium cells with tailored chemical stimuli and monitor their subsequent polarization. In their natural setting, cells generate their own directional signals via the detection and secretion of cyclic adenosine monophosphate (cAMP). Here, we couple the LEGI approach to an excitable medium model of the cAMP wave-field that is propagated by the cells and investigate the possibility for this class of models to enable accurate chemotaxis to the cAMP waveforms expected in vivo. Our results indicate that the ultra-sensitive version of the model does an excellent job in providing natural wave rectification, thereby providing a compelling solution to the 'back-of-the-wave paradox' during cellular aggregation.