Procedures for numerical analysis of circadian rhythms

Abstract

This article reviews various procedures used in the analysis of circadian rhythms at the populational, organismal, cellular and molecular levels. The procedures range from visual inspection of time plots and actograms to several mathematical methods of time series analysis. Computational steps are described in some detail, and additional bibliographic resources and computer programs are listed.

Keywords: Biostatistics; chronobiology; chronomics; circadian rhythm; time series analysis.

Figure 1

Treatment of perioral cancer timed at peak tumor temperature leads to more than doubling of the 24-month survival rate, as compared to reference groups treated 4 or 8 hours before or after the tumor temperature peak. The dashed line indicates the outcome for treatment conducted without regard for circadian time. Data from Halberg et al. (2003a), as documented by a peak test (Savage et al. 1962).

Figure 2

Flowchart of some of the methods available for analysis in chronobiology and for chronomics, with emphasis on extended cosinor rhythmometry. Note on the top of the chart that one can start with a time-specified datum as well as with a time series. This is because the methods of rhythmometry also serve for the preparation of reference intervals. Such time-specified reference intervals, or chronodesms, and their variants are currently a raison d’être of clinical chronobiology, as this discipline resolves the normal range of variation. Use of the procedures outlined in this flowchart has to be based on cost-effectiveness that determines the sampling requirements which in research will be much more demanding for exploring new periodicities even in the circadian spectral region than for dealing with already-mapped rhythms. As biology and medicine involve longer and longer series, it will become essential to resolve more than single components in a given spectral region, including the circadian one. As more and more variables can be studied concomitantly for long spans, procedures that analyze relations among time series will become indispensable. Adapted from Halberg et al. (1987).

Figure 3

Diagram illustrating aliasing (that is, identification of a spurious “rhythm” [alias] because of poor [insufficiently dense] sampling of an actual rhythmic process). The actual rhythmic component in this graph is sinusoidal with a period of 27.43 h. Sampling once a day at a fixed clock hour is shown by the dots. Analysis of the sampled data detects a component with a period of 8 days instead of 27.43 h. More generally, aliasing refers to all artifacts resulting from insufficiently dense sampling that fails to detect rhythm characteristics, such as the amplitude, acrophase or waveform (as well as the period illustrated on this graph). Adapted from Halberg et al. (1977).

Figure 4

Time plots of the records of body temperature of a gerbil collected by radio-telemetry every 6 minutes for 8 consecutive days (A) and of a simulated data set containing temperature values sampled in the same temperature range but temporally randomized (B). An approximately 24-h pattern of oscillation is evident in A but not in B. Data from Refinetti (1998).

Figure 5

Actogram of the running-wheel activity rhythm of a Nile grass rat recorded with 6-min resolution for 35 consecutive days. As indicated by the horizontal bars at the top, a light – dark cycle was in effect for the first 22 days (LD), whereas darkness prevailed for the following 13 days (DD). The rhythm free-ran in constant darkness with a period shorter than 24.0 h. Data from Refinetti (2004a).

Figure 6

Temporal distributions of suicides. (A) Accumulated temporal distribution of train suicides in the Netherlands from 1980 to 1994 (data from Houwelingen & Beersma 2001). (B) Distribution of values sampled in the same range as in A but temporally randomized. (C) Longitudinal data on suicides in Minnesota from 1968 to 2002 shown as monthly means (data from Halberg et al. 2005). (D) Subset of the Minnesota data set showing the number of daily suicides during the first two months of 2001. (E) Spectrum of cosinor amplitudes for the full time series (1968 – 2002).

Figure 7

Mean concentration of cholesterol in the serum of a goat sampled every 3 h for 10 consecutive days (A) and simulated data set with randomized time bins (B). Each vertical bar indicates the mean (+SE) of the 10 measurements conducted at each of the eight daily collection times. The horizontal bar at the top indicates the duration of the light and dark phases of the prevailing light-dark cycle. Analysis by ANOVA rejects the hypothesis of uniformity in both cases, but only A exhibits 24-h rhythmicity. Data from Piccione et al. (2003).

Figure 8

Records of locomotor activity of a horse monitored by actigraphy at 1-min intervals for 10 consecutive days (top) and the periodogram generated by Fourier analysis of the time series (bottom). The dashed line in the periodogram indicates the 0.05 level of significance. Visual inspection of the raw data suggests the existence of 24-h rhythmicity, and the Fourier periodogram confirms it by showing a large peak at 24.0 h. Other, smaller peaks can also be seen (arrows). Data from Piccione et al. (2005).

Figure 9

Educed rhythm of systolic blood pressure of a healthy woman collected at irregular intervals over nine days (points) and cosine wave fitted by the method of the least squares (curve). The small rectangle denotes the width of the 95% confidence interval of curve fitting. Data from Cornélissen and Halberg (2005).

Figure 10

Expression of the per1 gene in liver and testis of mice. Each data point corresponds to the mean (+SE) of three mice. The dashed lines indicate the overall means for the whole data sets. The rectangles denote the width of the 95% confidence intervals of curve fitting. The confidence interval overlaps the mean in the testis but not in the liver. Data from Yamamoto et al. (2004).

Figure 11

Temporal distribution of activity onsets of Siberian hamster pups a few days after weaning. Each dot denotes one pup. (A) uniform distribution; (B) actual distribution. The estimated time of the activity onset of the dams (time 12 of the LD cycle preceding constant darkness) is indicated by the arrowhead. Data from Duffield and Ebling (1998).

Figure 12

Ten-day segment of the records of running-wheel activity of a mouse maintained under a 24-h light-dark cycle and three periodograms generated by analysis of these data. The dashed lines in the periodograms indicate the 0.01 level of significance. Visual inspection of the time plot suggests the presence of strong rhythmicity with a period of 24.0 h. The three periodograms support this interpretation. Data from Refinetti (2004a).

Figure 13

Probability under the null hypothesis (H_o) as a function of the signal-to-noise ratio of the data sets for tests of 24.0-h rhythmicity conducted by Fourier analysis, by the cosinor procedure, by the Enright periodogram, and by the Lomb – Scargle periodogram. The horizontal line indicates p =0.05. The cosinor procedure detected significant rhythmicity in data sets with signal-to-noise ratios as low as 0.01 (1% signal and 99% noise).

Figure 14

Diagram of an oscillatory process identifying four parameters of the oscillation: mesor, period, amplitude, and phase.

Figure 15

Diagram depicting oscillations with different waveforms (A-C), unstable period (D), unstable waveform (E), and linear trend (F).

Figure 16

Artificial data sets of square waves with periods of 12, 24, and 48 h (left column) and their analyses by the chi square periodogram (Q_P) and the Lomb – Scargle periodogram (PN). The dashed lines in the periodograms indicate the 0.001 level of significance. The chi square periodogram identified a large 24 h component in the data set that contained only a 12-h square wave. Both methods correctly identified the 24-h component of the 24-h wave, although the Lomb – Scargle periodogram erroneously identified other smaller components (arrows). The chi square periodogram correctly failed to identify circadian rhythmicity in the 48-h wave, whereas the Lomb – Scargle periodogram identified a small 16-h component (arrow), which corresponds to the third harmonic of the 48-h component.

Figure 17

Periodograms describing sinusoidal time series with periods of 24 h. Fourier analysis (A) and chi square periodogram (B) were used to analyze segments of a time series generated with temporal resolution of 6 minutes. Notice that greater resolution of analysis is attained for longer segments of the time series. Lomb – Scargle periodogram (C) was used to analyze 10-day segments of times series generated with different temporal resolutions. Notice that statistical power (as indicated by the fraction of the curve above the significance line), but not resolution of analysis, is greater when the resolution of the data is greater. Notice also that the level of the line indicating statistical significance changes as a function of the number of degrees of freedom, which increases with the number of data available as the sampling interval gets shorter.

Figure 18

Periodograms describing time series containing two sinusoidal components: 21.0 and 25.0 h (A) or 23.5 and 24.5 h (B). In A, but not in B, the Enright (chi square) and Lomb – Scargle periodograms provide better visualization of the two components than does the Fourier periodogram.

Figure 19

An artificial data set lasting 14 days with hourly observations. The data set, which contains a large 24.0-h component and a smaller 24.8-h component, was constructed according to Y_i =100 + 10 cos (2πti/24−&p_i;) + 2 cos (2πt_i/ 24.8−π) + 5 R, where i =1, . . . , 336 (Δt =1 h) and R is uniformly distributed with zero mean and range =1 (±0.5).

Figure 20

Advantages of the MESOR over the arithmetic mean in the estimation of central tendency of a time series. (A) the MESOR has smaller bias in the presence of unequidistant data. (B) the standard error (SE) of the MESOR reflects the variability of the data after the variability accounted for by the rhythmic pattern has been removed. It differs from the global SE reflecting the total variability of the data (most of which can be ascribed to the rhythmic time structure). From Cornélissen et al. (1999). © Halberg. Reproduced with permission.

Figure 21

Analysis by a moving cosinor of phase-shifting of circadian rhythms in oral temperature of five healthy adult white males following transmeridian flights in two directions. Adapted from Halberg (1969).

Figure 22

Polar representation of the amplitude and acrophase of a rhythm, as computed by the cosinor method. The acrophase (φ) is indicated by the angle of a vector whose length corresponds to the amplitude (A). The ellipse denotes the 95% confidence region for the joint estimation of A and φ. Conservative estimates of 95% confidence intervals for A and φ alone can be obtained by drawing concentric circles tangent to the error ellipse for A, determining their intersection with the vector’s length (A), or by drawing tangents to the ellipse for φ.

Figure 23

Two-week segments of the records of bioluminescence of cells in the suprachiasmatic nucleus of a mouse culture in vitro. Because of suboptimal culturing conditions, the rhythm of bioluminescence showed a reduction in amplitude and mesor during the recording interval. (A) original data; (B) filtered data. Data from Yoo et al. (2004).

Figure 24

Graphic display of temporal pattern of gene expression of a sub-sample of 1,104 genes (out of a total of 14,010 genes) from a DNA microarray study conducted on fruit flies. Samples of RNA were collected at 4 h intervals from flies maintained under a light – dark cycle. Different levels of gene expression (normalized for each gene) are denoted by different shades of grey. Many genes exhibit variation in expression along the day. Data from Lin et al. (2002).

Figure 25

Diagram to guide the selection of procedures for the detection of circadian rhythmicity.

Figure 26

Blunders from time-unspecified comparisons of groups characterized by a difference in amplitude, acrophase or period but not in MESOR are best avoided by including the assessment of rhythms in the design of studies and by assessing all rhythm characteristics. Measurements conducted at arbitrary time points in ignorance of rhythmicity can lead to opposite conclusions about the characteristics of processes A and B.

Figure 27

Confusing results, that could wrongly be interpreted as “stress” or “allergy”, are accounted for by the action of food (offered mornings) and light as competing synchronizers of circadian rhythms in counts of circulating eosinophils in C₃H mice with high breast cancer incidence that can be drastically reduced by calorie restriction. Apart from constituting the beginning of Minnesotan chronobiology, and beyond the demonstration that time-unspecified spotchecks must not underlie frequently made inferences of “up-” and “down-regulation”, these studies convey the need to rely on circadian rhythms (and even broader time structures, chronomes) as the indispensable control information for any rhythmic variable, and hence the need to complement the mapping of genomes with the mapping of chronomes. From Halberg et al. (2003b). © Halberg. Reproduced with permission.