One of the most puzzling features of the prehistoric record of hominid stone tools is its apparent punctuation: it consists of abrupt bursts of dramatic change that separate long periods of largely unchanging technology. Within each such period, small punctuated cultural modifications take place. Punctuation on multiple timescales and magnitudes is also found in cultural trajectories from historical times. To explain these sharp cultural bursts, researchers invoke such external factors as sudden environmental change, rapid cognitive or morphological change in the hominids that created the tools, or replacement of one species or population by another. Here we propose a dynamic model of cultural evolution that accommodates empirical observations: without invoking external factors, it gives rise to a pattern of rare, dramatic cultural bursts, interspersed by more frequent, smaller, punctuated cultural modifications. Our model includes interdependent innovation processes that occur at different rates. It also incorporates a realistic aspect of cultural evolution: cultural innovations, such as those that increase food availability or that affect cultural transmission, can change the parameters that affect cultural evolution, thereby altering the population's cultural dynamics and steady state. This steady state can be regarded as a cultural carrying capacity. These parameter-changing cultural innovations occur very rarely, but whenever one occurs, it triggers a dramatic shift towards a new cultural steady state. The smaller and more frequent punctuated cultural changes, on the other hand, are brought about by innovations that spur the invention of further, related, technology, and which occur regardless of whether the population is near its cultural steady state. Our model suggests that common interpretations of cultural shifts as evidence of biological change, for example the appearance of behaviorally modern humans, may be unwarranted.