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.

This manuscript presents a new method and computer program for evaluating phase response curves (PRCs). A phase response curve describes those phase shifts produced in an oscillator by stimuli applied at different initial phase-states of that oscillator. Analysis of variance (ANOVA) has often been used to evaluate the null hypothesis that resultant phase shifts are randomly related to the initial phase-state of the oscillator at which stimuli are given, but the PRC bisection tests presented here have several advantages. In the PRC bisection tests, we repeatedly cut in half the circular distribution of the initial phase-states of the oscillator when stimuli are given. We locate an optimal diameter which best bisects the circular distribution of phase responses into arcs of phase advance and phase delay. A D score reflecting the success of the best bisection is computed. The null hypothesis of a random distribution of phase responses by initial phase is tested with a Monte Carlo procedure, which bisects random combinations of phase shifts with initial phases, thus determining the probability of the null hypothesis that the observed D score was from a random distribution. The bisection procedure can also be used to examine whether stronger phase shifts are produced in one phase response curve than in a contrasting curve. Further, the bisection procedure yields an estimate of the inflection point of the phase response curve. Finally, a method is given to estimate the power of the PRC bisection procedure.