  Differential equation-based physiological models of sleep-wake networks
describe sleep-wake regulation by simulating the activity of wake- and
sleep-promoting neuronal populations and the modulation of these populations by
homeostatic and circadian ($\sim24$ h) drives. Here, we consider a sleep-wake
flip-flop network model consisting of mutually inhibitory interactions between
wake- and sleep-promoting neuronal populations. Motivated by changes in sleep
behavior during early childhood as babies transition from napping to
non-napping behavior, we vary homeostatic and circadian modulation and analyze
effects on resulting sleep-wake patterns. To identify the types and sequences
of bifurcations leading to changes in stable sleep-wake patterns in this
piecewise-smooth model, we employ multiple mathematical methods, including
fast-slow decomposition and numerical computation of circle maps. We find that
the average daily number of sleeps exhibits a period adding sequence as the
homeostatic time constants are reduced, and that the temporal circadian profile
influences the number of observed solutions in the sequence. These solutions
emerge through sequences of saddle-node and border collision bifurcations,
where the particular sequence depends on parameter values. When the temporal
circadian profile is steep, as is observed with long day lengths, some sleep
patterns are lost and bistability of other patterns may occur. We analyze a
limiting case of the temporal circadian waveform, a circadian hard switch
model, to understand this loss of solutions. Generally, our holistic analysis
approach provides an alternative analysis method for model systems that defy
conventional numerical bifurcation analysis techniques.
