Most commonly, atrial fibrillation is triggered by rapid bursts of electrical impulses originating in the myocardial sleeves of pulmonary veins (PVs). However, the nature of such bursts remains poorly understood. Here, we propose a mechanism of bursting consistent with the extensive empirical information about the electrophysiology of the PVs. The mechanism is essentially non-local and involves the spontaneous initiation of non-sustained spiral waves in the distal end of the muscle sleeves of the PVs. It reproduces the experimentally observed dynamics of the bursts, including their frequency, their intermittent character, and the unusual shape of the electrical signals in the pulmonary veins that are reminiscent of so-called early afterdepolarizations (EADs).