Mathematical Model of the Human Circadian System with Two Interacting Oscillators

Human subjects during extended isolation from environmental time cues show complex variations in timing and duration of sleep with a progressive pattern, which eventually results in rest-activity and body temperature rhythms having different average periods. We present a model where temperature and rest-activity are each governed by an oscillator of the van der Pol type, denoted x and y, respectively. The oscillators affect one another through "velocity" type coupling, the effect of x on y being about four times greater than y on x. Periodic zeitgeber, z, is modeled as forcing only on y. We find that the entire pattern sequence can be realistically reproduced by causing only the intrinsic period of the y oscillator to increase while that of x remains stable. Desynchronization between x and y is the result of the intrinsic periods of the two oscillators becoming so disparate that the coupling is no longer able to enforce synchrony. Prior to desynchronization both human subjects and our model exhibit "phase trapping" wherein the relative phase of x and y is slowly modulated although the average x and y periods match. The model phase relations between temperature and both the timing and duration of sleep are, throughout, in good agreement with entrained and free-running human data. Most importantly, the model shows that the dramatic change in the length of the rest-activity cycle when desynchronization occurs is actually due to a relatively small variation in the governing variable, y.