Methods for serial analysis of long time series in the study of biological rhythms

Abstract

When one is faced with the analysis of long time series, one often finds that the characteristics of circadian rhythms vary with time throughout the series. To cope with this situation, the whole series can be fragmented into successive sections which are analyzed one after the other, which constitutes a serial analysis. This article discusses serial analysis techniques, beginning with the characteristics that the sections must have and how they can affect the results. After consideration of the effects of some simple filters, different types of serial analysis are discussed systematically according to the variable analyzed or the estimated parameters: scalar magnitudes, angular magnitudes (time or phase), magnitudes related to frequencies (or periods), periodograms, and derived and / or special magnitudes and variables. The use of wavelet analysis and convolutions in long time series is also discussed. In all cases the fundamentals of each method are exposed, jointly with practical considerations and graphic examples. The final section provides information about software available to perform this type of analysis.

Figure 1

Schematic representation of sections. X is the original data set. Filled circles correspond to the elements of the section Y₂. In this example the step s = 4, and the length l = 10, so they are overlapping sections (s < l). Δt is the sampling interval (see text for details).

Figure 2

Effect of the length of the section on the result series values Z. In the top row, the original series (X), and in the lower rows the resulting Z variables, using sections with different lengths. In A, X is a sinusoidal function with period T = 12. The graphs below, show the mean value calculated with a step s = 1, and l values of 3 T/4, T and 5 T/4. Although the real mean remains constant, for different values of l, an oscillation is present. In B, X is a sinusoidal function with period T = 12 with a phase delay in the middle of the series. In the lower graphs the phases calculated with sections of length T, 2 T and 4 T, are shown. A smoothing of the real phase shift is visible when increasing the length of the section.

Figure 3

Serial analysis of a 360 days record of motor activity of a rat submitted to different light patterns. Sections are s = l = T = 1440 min. The calculated variables are shown at right: mesor, percentile 95% and standard deviation.

Figure 4

Transfer functions corresponding to the application of a moving average in a series with a sampling interval of 15 min. Periods are represented in the abscissas (in minutes), and in the ordinates is the ratio of pass for different periods. In A, continuous line is for an interval of 5 points (1 h), dotted line for 9 points (2 h) and dashed line for 17 points (4 h). In B, real transfer function for pink noise using a moving average (dotted line) and a moving median with the same interval (continuous line).

Figure 5

Calculation of center of gravity. A: effect of the displacement of the wave with respect to the section in the calculation of the c.o.g. (black triangles). B: serial calculation of the c.o.g. (red line) in a simulated sinusoidal wave, with s = l = T. Red rectangles show two sections to illustrate the displacement of the wave with respect to the section. If this displacement is not compensated, the erroneous estimation is clearly visible on the resulting graph.

Figure 6

Estimation of acrophase. A: evolution of acrophase (red), positive flank using Heaviside function (green), and center of gravity (blue), in a real series of motor activity from a mice submitted to 8 h advances in the light–dark cycle. B: evolution of the acrophase in a simulated series with one bout of activity at the beginning and the end of a square wave, to show the effect of the wave shape on the estimation of the acrophase. In both cases de sections are s = l = T.

Figure 7

Onsets and offsets. The figure is a triple plotted actogram from a rat, with the positive flanks (blue) and the negative flanks (red) calculated by fitting dichotomized data to a squared wave (left) and to a Heaviside function (right). Small differences are visible in the two methods.

Figure 8

Graphic representation of the evolution of power spectra for a synthetic and a real series. In both examples double plotted actogram is at the left of the graphic matrix in which each column corresponds to a harmonic. In A, a simulated series with a rectangular pattern that changes in length becoming symmetric and then changing to a sinusoidal. The characteristic spectrum of rectangular waves is clearly visible. When symmetric, even harmonics are null, and all the harmonics disappear but the 1st, when the sinusoidal pattern appears. In B, a real record of a young rat after weaning under constant light, clearly showing the transition of an ultradian pattern characteristic of an immature animal (harmonics around 8), to the characteristic circadian pattern of an adult (power in the 1st harmonic).

Figure 9

Serial analysis of the motor activity of a rat maintained under LD cycles with a period of 21 hours. The graph shows the simultaneous presence of the animal's own endogenous component (T2 = 1540) and the component entrained by the light (T1 = 1260). Sections are s = 1440 min and l = 7200 min. The analysis consisted in a linear model including simultaneously the two sinusoidal components (T1 and T2). In the graphs are represented: the total variance explained by the model (left), the variance explained by each component (3rd graph, black = T1, red = T2) and the percentage of variance explained by T2 with respect to the sum of two components (right).

Figure 10

Motor activity of a rat submitted to different constant light intensities. The serial periodogram is shown on the right; the different light intensities are shown on the left. The periodogram used was the Lomb-Scargle with a section of 10 days length and a step of 1 day, and it is represented in a gray scale. The red overpainted line corresponds to the daily estimated period (maximum of the periodogram).

Figure 11

Motor activity record of a rat submitted successively to LD: T = 23 and DD. The serial periodogram is shown on the right. The periodogram used was the Sokolove-Bushell with a section of 20 days length and a step of 1 day, and the percentage of explained variance is represented in a color scale shown below. The presence of two components is clearly visualized during T23, changing to a unique rhythm in DD with an intermediate period.

Figure 12

Wavelet analysis: procedure. The original series A, from a real motor activity record D is convolved with a sinusoidal-Gaussian-modulated function B obtaining the curve C after numerical smoothing. For each resulting cycle in C, three characteristic points are defined: when crossing zero upwards (the onset), the maximum (the middle) and when crossing zero downwards (the offset). In E, the evolution of these points cycle by cycle.

Figure 13

Wavelet analysis: result. Shown is the representation of the evolution of the phases (continuous lines) calculated with the convolution described in Figure 12, on 10 registers of motor activity of rats subjected to continuous phase changes of lighting cycle (dashed line). At right, the result of the Rayleigh z test, to verify the homogeneity of the daily phases, dashed line is the threshold for significance (p = 0.05).