A study of the singularities in a mathematical model for circadian rhythms

One of the models that has been suggested for describing circadian rhythms mathematically is an extension of the van der Pol equation given by ÿ + 0.5(y2 + y-2 - 3)y + (1 + 0.6 y) y = z + z + z, where y is the oscillating variable, and z is the light intensity assumed to excite the oscillator. In order for the equation to exhibit self-sustained oscillations, z has to be within the oscillatory range (0.847 < z < 3.189). This equation has been shown to simulate several of the features possessed by circadian systems (Wever, R., 1984, Toward a mathematical model of circadian rhythmicity, in: Mathematical Models of the Circadian Sleep-Wake Cycle, M.C. Moore-Ede and C.A. Czeisler (eds.) (Raven Press, New York) pp. 17-79). Physiological experiments have been performed which show that circadian rhythms can have stable singularities. Therefore, it was of interest to investigate whether or not the equation given above also has this property. We have studied the stability of the two singularities of the model system above. One of the singularities is unstable and corresponds to non-physiological conditions. The other one is an unstable spiral point if the light conditions are such that oscillations can occur in the system. We conclude that the model mentioned above is unsuitable to describe circadian systems which have stable singularities. The model has been simulated, and pulses have been applied to the system by temporarily changing the value of z to find appropriate conditions forcing the system into its singularity. The strategy to find such pulses is discussed.