Lava dome eruptions are commonly characterized by large fluctuations in discharge rate with cyclic behaviour on time-scales ranging from hours to decades. Examples include Bezymianny volcano (Russia), Merapi (Java), Santiaguito (Guatemala), Mt St Helens (USA), Mt Unzen (Japan), and Soufri¿re Hills volcano (Montserrat). Previous models have assumed simple cylindrical conduits for magma transport, but extrusions are mainly fed by dykes, with cylindrical geometries developing only at shallow levels. The widths of dykes embedded in an elastic medium are influenced by local magma pressure, affecting flow rates and system dynamics strongly. We develop a model for magma flow in dykes, which predicts intense pulsations of magma extrusion for the case of a constant source pressure. The period time scale is determined by the elastic deformation of the dyke walls and the length-to-width ratio of the dyke. The dyke acts like a volumetric capacitor, storing magma as pressure increases and then releasing magma in a pulse of extrusion. For the Soufri¿re Hills volcano, cyclic extrusions with time-scales of a few weeks are predicted for dykes 300--500 m long and 3--6 m wide, matching observations. The model explains the sharp onset of tilt pulsations and seismic swarms.