  We introduce new quantitative approaches to study sleep-stage transitions
with the goal of addressing the two following questions: (i) Can the new
approaches provide more information on the structure of sleep-stage
transitions? (ii) How does sleep fragmentation in patients with sleep apnea
affect the structure of sleep-stage transitions? Our new results show that the
distribution of sleep and wake duration have different functional forms,
indicating fundamental differences in the dynamics between sleep and wake
control. The difference remains even in the fragmented sleep of sleep apnea.
The fragmentation of sleep in sleep apnea results in a shorter wake duration
and interrupts the structure of sleep-stage transitions of sleep apnea
subjects, causing the loss of certain particular transition paths.
