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.

Oxalis regnellii Mig. is a trifoliate plant, and the three leaflets usually show synchronized up and down movements with a circadian period of 26–27 h. The three leaflets can also perform desynchronized ultradian oscillations, and we report on such rhythms under different conditions. A study of the occurrence of ultradian leaf movement rhythms as a function of irradiance is presented. At an irradiance of approximately 1 μW cm−2, the occurrence was maximal and ca 30%. The periods varied from 5 to 15 h. Four other cases of ultradian rhythms in different conditions are also presented. In one case spontaneous ultradian rhythms occurred, and in another, two of the leaflets showed ultradian rhythms when the third leaflet had received a light pulse. In two more cases, the three leaflets on a leaf were separated by physical cuts along the petiole between the pulvini; in both cases the period was approximately 5 h. Possible mechanisms to explain the ultradian rhythms in Oxalis regnelli are discussed.

A transducer was developed to record the circadian movement of the individual leaflets in Oxalis regnellii Mig. The method can easily be adapted to measure other kinds of plant movements as well. It is based on the detection of the shadow each leaflet casts on the small side of a specially formed Perspex plate. The light is guided through the Perspex and collected by a phototransistor, which provides an electrical signal that is proportional to the light intensity falling onto it. The output signal can be made a linear function of the leaf angle. This equipment was used in experiments to study the coupling between the 3 leaflets in Oxalis. Pulses of 4 h of red light were given to one of the leaflets, the two others were shielded from the light. A phase response curve was determined for each leaflet, but there was no significant difference in the phase response between the 3 leaflets. Experiments were also made in which the 3 leaflets were separated physically by cuts along the petiole between the pulvini. In this case ultradian oscillations were observed.

Abstract The effect of physical loading on period and amplitude of the ultradian oscillating leaflets of Codariocalyx motorius (= Desmodium gyrans) and the circadian oscillating leaves of Oxalis regnellii was investigated. In Codariocalyx the leaflets were loaded with pieces of copper thread which were clamped to the leaflet. Weights of the loads varied from 3 to 12 mg, and a typical leaflet is 1.5 mg. In about 80% of the cases we recorded an increase of the period length due to the loading in the order of 10% (the period length is typically 3-4 min). The amplitude was lowered in all the cases, the magnitude depending on the size of the weight used. In most of the cases, when not too heavily loaded, both amplitude and period returned to (or close to) the previous value when the leaflets were unloaded. In Oxalis the leaves were loaded with metal clips fixed to the leaflets at the midrib position. All three leaflets on a stalk were loaded. The clips used weighed 390 ± 0.4. 640 ± 6 and 810 ± 4 mg respectively, and a typical leaflet is 91 ± 4 mg. For the heaviest weights at least, this resulted in a lowering of the amplitude, but the oscillations continued, and we could not observe any effect on the period of the rhythmic movements.