Chemotactic cell motion in aggregation patterns of the slime mould Dictyostelium discoideum is analyzed by a computerized cross-correlation method. In this excitable medium the movement of amoebae is selforganized by signals of cAMP that propagate as rotating spirals or expanding concentric circles (target patterns). A vortex-like rotation of cells is found close to the core of spiral waves, with a maximum velocity of 15 microns/min. Cell motion and spiral tip orbiting follow an opposite sense of rotation. The calculation of streamlines of the most probable trajectories reveals the existence of a closed curve (radius approximately 130 microns) corresponding to the boundary of the spiral core. This spatial-limit cycle attracts the amoebae on rotational paths and leads to the formation of a cell-free disk in the centre of the pattern. In contrast, pacemakers of target patterns organize cell movement in a radial, star-shaped motion, leading directly to the formation of central mounds.