Numerical simulations of two-dimensional time-dependent thermal convection in a Boussinesq, isoviscous, infinite Prandtl number fluid with isothermal, stress-free boundaries have been performed in large aspect-ratio configurations, in which the fluid is heated from below as well as internally. The value of the basal heated Rayleigh number ranged from 16000 to 800 000 and the Rayleigh number based on internal heat generation was varied from zero to 4 500 000. Large aspect-ratio cells are found to exist, however, they are time-dependent even at small values of the Rayleigh number. In the absence of internal heating, the onset of time-dependence occurs as a regular oscillation in the flow characteristics (Nusselt number, kinetic energy), and is accompanied by the presence of boundary layer instabilities (BLI) which exist within a large aspect-ratio circulation. At high values of the Rayleigh number the BLI are powerful features which leave the confines of the boundary layer and strongly perturb the large aspect-ratio circulation giving the flow a multi-scale character. Convective mixing of these powerful BLI results in heterogeneity in the cell interior which plays a role in the excitation of new BLI and establishes a negative temperature gradient in the cell interior. We have developed a fluid loop model which gives a qualitative explanation for the variation of the onset of time-dependence with aspect-ratio. The addition of internal heating tends to destabilize the large aspect-ratio cell configuration. Multi-cellular states last longer and occur more frequently with increasing amounts of internal heating. These calculations shed new light on a variety of time-dependent phenomena in geodynamics such as subduction, back-arc spreading, intraplate deformation, and the average geotherm. Recently, Jeanloz and Morris proposed that the seismic inhomogeneity parameter (η) can be used to measure the importance of internal heating in mantle convection. However, our calculations indicate that η cannot be used to gauge the importance of internal heating in mantle convection.