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The Sigmoidally-transformed Cosine Curve: A Mathematical Model for Circadian Rhythms with Non-sinusoidal Shapes

  • Author(s): Marler, Matthew R.
  • Gehrman, Philip
  • Martin, Jennifer L.
  • Ancoli-Israel, Sonia
  • et al.
Abstract

We introduce a family of nonlinear transformations of the traditional cosine curve used in the modeling of biological rhythms. The nonlinear transformation is the sigmoidal family, represented here by three family members: the Hill function, the 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 2 additional harmonics to achive the same fit when modeled 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. We illustrate the sigmoidally transformed cosine curves, and compare them to harmonic regression modeling, in a sample of 8 activity recordings made on patients in a nursing home. We introduce a family of nonlinear transformations of the traditional cosine curve used in the modeling of biological rhythms. The nonlinear transformation is the sigmoidal family, represented here by three family members: the Hill function, the 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 2 additional harmonics to achive the same fit when modeled 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. We illustrate the sigmoidally transformed cosine curves, and compare them to harmonic regression modeling, in a sample of 8 activity recordings made on patients in a nursing home.

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