We have modelled, for the cases of Milky Way and M31, the effects on the galactic discs, of the arrival at high velocity (≥150 km s-1) of giant HI clouds, with masses of up to 108M&sun;. Predictions are compared with the detailed structure of the observed rotation curves for these two galaxies. The model explains the rises and falls observed at large distances from the centre of each galaxy, distributed with a degree of regularity in radius, in terms of a specific type of perturbations driven by the infall of the high velocity clouds (HVC's) arriving from the intracluster medium of the Local Group. The underlying rotation curve is explained conventionally via the distribution of the baryonic and dark matter components of the galaxy in question. This scenario, though tested here on the two major Local Group objects, is in principle applicable to galaxies undergoing minor mergers with subgalactic mass gas clouds.