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Two young subjects (1111 and 1105) returned for a classical free-running study in which they self-selected their light-dark cycle (12 13) (Fig. 1 center panel). Two subjects (1111 and 1507) later returned for a 20-hour forced desynchrony protocol (Fig. 1 left panel). One of those subjects (1507) also returned for yet another forced desynchrony protocol with a 42.85-hour imposed day length. The ratio of scheduled bedtime to scheduled wake time was maintained at 1:2 for each forced desynchrony protocol.

Using the NOSA technique core-temperature data collected throughout the month-long experiment were modeled as a harmonic regression model with continuous first-order autoregressive [AR(1)] noise defined as yt=st+xt+vtwhere y t is the core-temperature measurement at time t s t is the circadian signal x t is the forced desynchrony component and v t is the AR(1) noise. We define st=μ+∑r=12Ar cos2πrtτ+Br sin2πrtτ xt=∑k=18Ck cos2πkt28+Dk sin2πkt28 vt=exp(−αΔ)vt−1+ɛt     where μ is mean temperature τ is the intrinsic period of the circadian pacemaker A r and B r are respectively the cosine and sine coefficients of the r th harmonic of the circadian signal C k and D k are respectively the cosine and sine coefficients of the k th harmonic of the forced desynchrony component Δ is the sampling interval α is the approximate time constant of the thermoregulatory system and the ε t 's are independent identically distributed Gaussian noise with zero mean and variance σ ε 2 . The variance of v t is σ v 2 = σ ε 2 [1 – exp(–2αΔ)] –1 . The choice of two harmonics to model the circadian component is based on Brown and Czeisler (32) whereas the choice of eight harmonics to model the forced desynchrony component was determined empirically. The model was fit to the data by an exact maximum likelihood method [R. H. Jones Longitudinal Data with Serial Correlation: A State-Space Approach (Chapman & Hall New York 1993); E. N. Brown and C. H. Schmid in Methods in Enzymology Numerical Computer Methods Part B L. Brand and M. L. Johnson Eds. (Academic Press Orlando FL 1994) pp. 171–181]. The standard deviation of the period estimate used to compute the 95% confidence intervals for τ was computed as στ= 6τ4 σv2 Δ[1−exp(−2αΔ)]π2T3∑r=12Σfr2(Ar2+Br2)1−2 exp(−αΔ)cos2πrΔτ+exp(−2αΔ)12where T is the study length [E. N. Brown V. Solo Y. Choe Z. Zhang Tech. Rep. 95-01 (Statistics Research Laboratory Department of Anesthesia and Critical Care Massachusetts General Hospital April 1996; revised November 1997)]. The hormone measurements have no thermoregulatory component; hence for their analyses we used the model in Eq. 2 with v t = ε t and the formula in Eq. 3 with α = ∞. For the free-running studies only core body temperature was sampled and the fitting did not include a forced period. NOSA can include periodic terms that may arise from nonlinear interactions between the basic periodic signals s t and x t . We have explored such additional terms and identified those that rise above the level of noise in the temperature data. However when these are included the effect on the average endogenous period reported here is minimal (<1 min) and not statistically significant. We therefore report the endogenous periods estimated without interaction periodicities which can be used to investigate related nonlinear processes that are beyond the scope of this report.

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We thank the subject volunteers; research technicians; senior research technicians A. Fergus K. Foote G. Jayne E. Martin and A. Ward; S. Driscoll and the staff of the General Clinical Research Center (GCRC) of Brigham and Women's Hospital; subject recruiters J. Daley J. Kao D. Margolis and R. McCarley; research assistants D. Chen and J. Whittemore Jr.; the Clinical Core Laboratory of the GCRC and Elias USA Inc. for hormonal assays; T. Ding for analytic software; W. Freitag for management of bioengineering systems; L. Rosenthal for illustrations; J. Zeitzer for comments; J. K. Wyatt for provision of comparative data; and G. H. Williams for overall support. Supported by the U.S. Public Health Service National Institute on Aging Award PO1-AG09975 (C.A.C.) National Institute of General Medical Sciences Award R01-GM53559 (E.N.B.) and NASA Cooperative Agreement NCC9-58 with the National Space Biomedical Research Institute. The studies were performed in a GCRC supported by MO1-RR02635. This paper is dedicated to the memory of Professor Jürgen Aschoff (1913–1998) who pioneered the modern science of circadian biology and established the first continuously operational laboratory shielded from external time cues to study human circadian physiology.