The sigmoidally transformed cosine curve: a mathematical model for circadian rhythms with symmetric non-sinusoidal shapes.

We introduce a family of non-linear transformations of the traditional cosine curve used in the modelling of biological rhythms. The non-linear transformation is the sigmoidal family, represented here by three family members: the Hill function, the anti-logistic function, and the arctangent function. These transforms add two additional parameters that must be estimated, in addition to the acrophase, MESOR, and amplitude (and period in some applications), but the estimated curves have shapes requiring many more than two additional harmonics to achieve the same fit when modelled by harmonic regression. Particular values of the additional parameters can yield rectangular waves, narrow pulses, wide pulses, and for rectangular waves (representing alternating 'on' and 'off' states) the times of onset and offset (hence duration, as when modelling the duration of the large night-time melatonin secretory epoch). We illustrate the sigmoidally transformed cosine curves, and compare them to harmonic regression modelling, in a sample of eight activity recordings made on patients in a nursing home.