A mathematical model of the sleep/wake cycle

J Math Biol. 2010 May;60(5):615-44. doi: 10.1007/s00285-009-0276-5. Epub 2009 Jun 26.


We present a biologically-based mathematical model that accounts for several features of the human sleep/wake cycle. These features include the timing of sleep and wakefulness under normal and sleep-deprived conditions, ultradian rhythms, more frequent switching between sleep and wakefulness due to the loss of orexin and the circadian dependence of several sleep measures. The model demonstrates how these features depend on interactions between a circadian pacemaker and a sleep homeostat and provides a biological basis for the two-process model for sleep regulation. The model is based on previous "flip-flop" conceptual models for sleep/wake and REM/NREM and we explore whether the neuronal components in these flip-flop models, with the inclusion of a sleep-homeostatic process and the circadian pacemaker, are sufficient to account for the features of the sleep/wake cycle listed above. The model is minimal in the sense that, besides the sleep homeostat and constant cortical drives, the model includes only those nuclei described in the flip-flop models. Each of the cell groups is modeled by at most two differential equations for the evolution of the total population activity, and the synaptic connections are consistent with those described in the flip-flop models. A detailed analysis of the model leads to an understanding of the mathematical mechanisms, as well as insights into the biological mechanisms, underlying sleep/wake dynamics.

Publication types

  • Research Support, U.S. Gov't, Non-P.H.S.

MeSH terms

  • Circadian Rhythm / physiology*
  • Electroencephalography
  • Humans
  • Intracellular Signaling Peptides and Proteins / physiology
  • Models, Neurological*
  • Neurons / physiology
  • Neuropeptides / physiology
  • Orexins
  • Sleep / physiology*
  • Wakefulness / physiology*


  • Intracellular Signaling Peptides and Proteins
  • Neuropeptides
  • Orexins