The Lomb-Scargle Periodogram in Biological Rhythm Research: Analysis of Incomplete and Unequally Spaced Time-Series

This paper investigates the utility of the Lomb-Scargle periodogram for the analysis of biological rhythms. This method is particularly suited to detect periodic components in unequally sampled time-series and data sets with missing values, but restricts all calculations to actually measured values. The Lomb-Scargle method was tested on both real and simulated time-series with even and uneven sampling, and compared to a standard method in biomedical rhythm research, the Chi-square periodogram. Results indicate that the Lomb-Scargle algorithm shows a clearly better detection efficiency and accuracy in the presence of noise, and avoids possible bias or erroneous results that may arise from replacement of missing data by interpolation techniques. Hence, the Lomb-Scargle periodogram may serve as a useful method for the study of biological rhythms, especially when applied to telemetrical or observational time-series obtained from free-living animals, i.e., data sets that notoriously lack points.     

The Lomb-Scargle Periodogram in Biological Rhythm Research: Analysis of Incomplete and Unequally Spaced Time-Series

This paper investigates the utility of the Lomb–Scargle periodogram for the analysis of biological rhythms. This method is particularly suited to detect periodic components in unequally sampled time-series and data sets with missing values, but restricts all calculations to actually measured values. The Lomb-Scargle method was tested on both real and simulated time-series with even and uneven sampling, and compared to a standard method in biomedical rhythm research, the Chi-square periodogram. Results indicate that the Lomb–Scargle algorithm shows a clearly better detection efficiency and accuracy in the presence of noise, and avoids possible bias or erroneous results that may arise from replacement of missing data by interpolation techniques. Hence, the Lomb–Scargle periodogram may serve as a useful method for the study of biological rhythms, especially when applied to telemetrical or observational time-series obtained from free-living animals, i.e., data sets that notoriously lack points.     

Searching for biological rhythms: peak detection in the periodogram of unequally spaced data

The classical power spectrum, computed in the frequency domain, outranks traditionally used periodograms derived in the time domain (such as the chi2 periodogram) regarding the search for biological rhythms. Unfortunately, classical power spectral analysis is not possible with unequally spaced data (e.g., time series with missing data). The Lomb-Scargle periodogram fixes this shortcoming. However, peak detection in the Lomb-Scargle periodogram of unequally spaced data requires some careful consideration. To guide researchers in the proper evaluation of detected peaks, therefore, a novel procedure and a computer program have recently become available. It is recommended that the Lomb-Scargle periodogram be the default method of periodogram analysis in future biomedical applications of rhythm investigation.     

科学确立奥林帕斯Au640 全自动生化分析仪校准周期

目的:利用校准周期图科学确立奥林帕斯Au640全自动生化分析仪校准周期。方法:生化分析仪定标校准后,每隔2小时分别测定三种浓度水平的质控血清一次,24小时后每天测定一次,总计测定30天。所有检测项目均重复测定四次取平均值。以积变异系数(CV)大于1/6 CLIA’88为评价标准,绘制校准周期图。结果:碱性磷酸酶(ALP)L1校准周期最短每12 h一次。酶类项目较稳定,校准周期可达30天以上,低水平质控血清必不可少,这样有利于测得更为准确的校准周期。结论:科学、合理地确立各检测项目的校准周期,对生化分析仪检测系统进行定期校准,既能够避免不必要的校准、又能够节省检验工作的时间和成本,保证生化检验项目检测结果可靠性。     

Use of chi square periodogram in the analysis of estrous rhythmicity

The usefulness of the chi square periodogram procedure for the analysis of estrous rhythmicity was investigated. Sokolove-Bushell's Q statistic was found to have a chi square distribution at the small degress of freedom involved in estrous rhythmicity in rodents (i.e. rhythmicity with periods of 2–7 days). Consequently, the significance of the peaks in the periodogram can be effectively evaluated. The effects of multiple-period rhythmicity and of random noise added to periodical data were also investigated. Overall, the analysis of simulated as well as empirical data indicated that the chi square periodogram is an excellent tool for the evaluation of estrous rhythmicity.     

Locomotor Rhythms in the Fiddler Crab,Uca subcylindrica

The locomotor activities of individual specimens of Uca subcylindrica (Stimpson) collected from semi-arid, supratidal habitats in south Texas and northeastern Mexico were studied in the laboratory using periodogram analysis. When crabs were placed under constant darkness (DD) or constant illumination (LL), free-running circadian rhythms were observed in the activity recordings. The locomotor activity of strongly rhythmic crabs in LL has an average period length of 24.4 h. Crabs held in DD express motor rhythms with periods of approximately 24.0 h. In LL the most common wave form for activity is unimodal, while under DD it is bimodal. Recordings under natural illumination (NL) revealed that both period length and the time of maximum activity (phasing) varied through the year. During winter months, the crabs are primarily diurnal with peaks in activity occurring between 0900 and 2100 h and possess a circadian rhythm with a 23.9 h period. During summer, crabs were nocturnal with maximal activity between 1300 and 0600 and a circadian period closer to 24.0 h. In these experiments, the rhythmic locomotor activities of U. subcylindrica are best described as “circadian”. This is unusual for a genus known for its expression of circatidal and circalunidian rhythms.     

Locomotor Rhythms in the Fiddler Crab, Uca subcylindrica

The locomotor activities of individual specimens of Uca subcylindrica (Stimpson) collected from semi-arid, supratidal habitats in south Texas and northeastern Mexico were studied in the laboratory using periodogram analysis. When crabs were placed under constant darkness (DD) or constant illumination (LL), free-running circadian rhythms were observed in the activity recordings. The locomotor activity of strongly rhythmic crabs in LL has an average period length of 24.4 h. Crabs held in DD express motor rhythms with periods of approximately 24.0 h. In LL the most common wave form for activity is unimodal, while under DD it is bimodal. Recordings under natural illumination (NL) revealed that both period length and the time of maximum activity (phasing) varied through the year. During winter months, the crabs are primarily diurnal with peaks in activity occurring between 0900 and 2100 h and possess a circadian rhythm with a 23.9 h period. During summer, crabs were nocturnal with maximal activity between 1300 and 0600 and a circadian period closer to 24.0 h. In these experiments, the rhythmic locomotor activities of U. subcylindrica are best described as “circadian”. This is unusual for a genus known for its expression of circatidal and circalunidian rhythms.     

Photic entrainment of circannual rhythms in golden-mantled ground squirrels: role of the pineal gland

Entrainment of circannual rhythms of body mass and reproduction was monitored for 3 years in female golden-mantled ground squirrels maintained in a simulated natural photoperiod. Both pinealectomized and pineal-intact squirrels generated circannual rhythms of body mass and estrus, but only the intact animals entrained these rhythms to a period of 365 days. In the second and third years after treatment, the period of the body mass rhythm was significantly shorter than 365 days for pinealectomized squirrels, and variance in tau among these animals was significantly greater than for intact squirrels. A similar pattern was evident in the rhythm of reproduction, which was phase-disrupted in pinealectomized squirrels but entrained in intacts. Seasonal changes in duration of nocturnal melatonin secretion by the pineal appear to be necessary to produce phase-delays required to entrain the circannual clock to a period of 12 months.     

Renal excretion of 8-oxo-7,8-dihydro-2'-deoxyguanosine in Wistar rats with increased O(2) consumption due to cold stress

DNA damage by reactive oxygen species is of special interest in the development of cancer and in aging. The renally excreted amount of 8-oxo-7,8-dihydro-2'-deoxyguanosine (oxo(8)dG) is a potential noninvasive marker of oxidative DNA damage. The respiratory chain of mitochondria is one source for the formation of reactive oxygen species. In the present study we investigated in Wistar rats (n = 7; mean body weight at start, 307.4 +/- 11 g) the effect of an increased O(2) consumption, i.e., energy expenditure, due to cold stress on the renally excreted amount of oxo(8)dG. First, the rats were housed for 4 days at 23.5 degrees C (basic period, BP), and then for 6 days at 10 degrees C (cold stress period, CSP), and finally for 3 days at 23.5 degrees C (recovery period, RP). The O(2) consumption (L O(2)/day/kg weight) was significantly (P < 0.0001) on average 50% higher in CSP (69.0 +/- 3.9) than in BP (45.8 +/- 4.8), and similar in BP and RP (44.3 +/- 5.4). The average renal excretion of oxo(8)dG (pmol/day/kg weight) was significantly (P < 0.025) on average 13% higher in CSP (375.5 +/- 27.7) than in BP (333.2 +/- 47. 4) and similar in BP and RP (331.8 +/- 34.3). Maximum increase in oxo(8)dG excretion of on average 17% was on the third to fifth day of the CSP. This study reveals that an increase in O(2) consumption of 50% resulted in a much lower increase in the renal excretion of oxo(8)dG.     

Temporal variations of coagulation factor VII activity in mice are influenced by lighting regime

It was recently reported that the circadian clock machinery controls plasma levels of factor (F) VII, the serine protease triggering blood coagulation. Here, by exploiting the mouse model, this study showed that variations of photoperiod (i.e., winter or summer conditions or simulated chronic jetlag conditions) have a strong impact on plasma FVII activity levels. Under conditions mimicking summer or winter photoperiods, FVII activity showed a clear 24 h rhythmicity. Interestingly, mean daily FVII activity levels were significantly reduced in mice exposed to summer photoperiods. Behavioral activity rhythms under both photoperiods were synchronized to LD cycles, and the amount of activity per 24 h was comparable. The authors also investigated the influence of chronic jetlag (CJL) on the FVII activity rhythms, which can be easily mimicked in mice through continuous abrupt shifts in the lighting schedule. The exposure of mice to simulated CJL of either consecutive westward or consecutive westward and eastward flights for 15 days did not abolish the behavioral activity rhythms but was associated with a period significantly different from 24 h. Intriguingly, both types of CJL exerted a strong influence on FVII activity rhythms, which were virtually suppressed. Moreover, the mean daily FVII activity was significantly lower in the CJL than in the winter photoperiod condition. Taken together, these findings in mice provide novel insights into the modulation of FVII activity levels, which might have implications for human pathophysiology.     

Temporal Variations of Coagulation Factor VII Activity in Mice Are Influenced by Lighting Regime

It was recently reported that the circadian clock machinery controls plasma levels of factor (F) VII, the serine protease triggering blood coagulation. Here, by exploiting the mouse model, this study showed that variations of photoperiod (i.e., winter or summer conditions or simulated chronic jetlag conditions) have a strong impact on plasma FVII activity levels. Under conditions mimicking summer or winter photoperiods, FVII activity showed a clear 24 h rhythmicity. Interestingly, mean daily FVII activity levels were significantly reduced in mice exposed to summer photoperiods. Behavioral activity rhythms under both photoperiods were synchronized to LD cycles, and the amount of activity per 24 h was comparable. The authors also investigated the influence of chronic jetlag (CJL) on the FVII activity rhythms, which can be easily mimicked in mice through continuous abrupt shifts in the lighting schedule. The exposure of mice to simulated CJL of either consecutive westward or consecutive westward and eastward flights for 15 days did not abolish the behavioral activity rhythms but was associated with a period significantly different from 24 h. Intriguingly, both types of CJL exerted a strong influence on FVII activity rhythms, which were virtually suppressed. Moreover, the mean daily FVII activity was significantly lower in the CJL than in the winter photoperiod condition. Taken together, these findings in mice provide novel insights into the modulation of FVII activity levels, which might have implications for human pathophysiology.     

Non-stationary time series and the robustness of circadian rhythms

Circadian rhythms are regular oscillations in the value of behavioral and physiological variables of organisms that recur on a daily basis. The purpose of this study was to evaluate the extent of non-stationarity of circadian rhythms over several days, to determine how damaging is the violation of the assumption of stationarity in the analysis of circadian rhythms, and to formalize the concept of "rhythm robustness" as an index of oscillatory ("weak") stationarity. Simulated (computer-generated) and experimental data sets (rhythms of body temperature and running-wheel activity in several rodent species) were analysed. Tests of stationarity based on the variance of the daily means and the variance of the daily variances revealed that most experimental data sets are not stationary. Analysis of linear trends indicated that significant trends are rare in experimental data sets. Although the non-stationarity of the experimental data sets reduced the spectral energy of the Enright periodogram used to assess rhythmicity, detection of circadian rhythmicity was not prevented in any of the rhythmic data sets. The results of the various analyses allow the inference that, after high-frequency noise is filtered out, the value of the periodogram's Q(P) statistic reflects the extent of stationarity of the time series. Thus, the "robustness" of a circadian rhythm (i.e. the magnitude of the empirical Q(P) value as compared to the Q(P) value associated with a perfectly rhythmic time series) can serve as an index of the stationarity of the rhythm.     

Circadian organization of a subarctic rodent, the northern red-backed vole (Clethrionomys rutilus)

Arctic and subarctic environments are exposed to extreme light: dark (LD) regimes, including periods of constant light (LL) and constant dark (DD) and large daily changes in day length, but very little is known about circadian rhythms of mammals at high latitudes. The authors investigated the circadian rhythms of a subarctic population of northern red-backed voles (Clethrionomys rutilus). Both wild-caught and third-generation laboratory-bred animals showed predominantly nocturnal patterns of wheel running when exposed to a 16:8 LD cycle. In LL and DD conditions, animals displayed large phenotypic variation in circadian rhythms. Compared to wheel-running rhythms under a 16:8 LD cycle, the robustness of circadian activity rhythms decreased among all animals tested in LL and DD (i.e., decreased chi-squared periodogram waveform amplitude). A large segment of the population became noncircadian (60% in DD, 72% in LL) within 8 weeks of exposure to constant lighting conditions, of which the majority became ultradian, with a few individuals becoming arrhythmic, indicating highly labile circadian organization. Wild-caught and laboratory-bred animals that remained circadian in wheel running displayed free-running periods between 23.3 and 24.8 h. A phase-response curve to light pulses in DD showed significant phase delays at circadian times 12 and 15, indicating the capacity to entrain to rapidly changing day lengths at high latitudes. Whether this phenotypic variation in circadian organization, with circadian, ultradian, and arrhythmic wheel-running activity patterns in constant lighting conditions, is a novel adaptation to life in the arctic remains to be elucidated.     

A Procedure of Multiple Period Searching in Unequally Spaced Time-Series with the Lomb-Scargle Method

Periodogram analysis of unequally spaced time-series, as part of many biological rhythm investigations, is complicated. The mathematical frameworkis scattered over the literature, and the interpretation of results is often debatable. In this paper, we show that the Lomb-Scargle method is the appropriate tool for periodogram analysis of unequally spaced data. A unique procedure of multiple period searching is derived, facilitating the assessment of the various rhythms that may be present in a time-series. All relevant mathematical and statistical aspects are considered in detail, and much attention is given to the correct interpretation of results. The use of the procedure is illustrated by examples, and problems that may be encountered are discussed. It is argued that, when following the procedure of multiple period searching, we can even benefit from the unequal spacing of a time-series in biological rhythm research.     

Locomotor activity rhythms in cave fishes from Chapada Diamantina, northeastern Brazil (Teleostei: Siluriformes)

Previous studies on the locomotor activity of troglobitic (exclusively subterranean) species have shown that circadian rhythmicity may be reduced in populations evolving in the absence of zeitgebers such as daily cycles of light and temperature; therefore, circadian activity rhythms, although not infradian nor ultradian rhythms, seem to have been selected by external, ecological factors. We studied the locomotor activity of a highly specialized Heptapteridae catfish (undescribed genus and species) from Chapada Diamantina, NE Brazil, compared to another specialized Brazilian troglobitic heptapterid, Taunayia sp. Locomotor activity was continuously measured in the laboratory with an infra-red photocell system. Seven specimens of the new genus were tested, each one during 14 consecutive days according to the following schedule: three days in DD → seven days in LD (12:12 h) → four days in DD. Data were submitted both to fast Fourier transform periodogram followed by Siegel's test of significance and Lombs - Scargle periodogram techniques in order to identify spectral composition of the time series. In general, results were similar to those obtained for Taunayia sp.: (a) for most specimens, absence of significant circadian components in locomotor activity under DD; (b) for all specimens, significant circadian components under LD, with higher levels of activity during the dark phase, as expected for species belonging to nocturnal epigean taxa; (c) for most specimens, no residual oscillations recorded when free-running conditions were reinstalled. Circadian locomotor activity detected under LD may thus be interpreted as a direct, masking effect of the LD cycle. This suggests a pattern for highly specialized troglobitic species, isolated for a long time in the subterranean habitat, with a progressive reduction of circadian time-keeping mechanisms. Our studies also demonstrate the potential of subterranean organisms for investigation of the origin, evolution, functioning and genetics of circadian rhthmicity.     

Internal synchronization of the main 0.1-Hz rhythms in the autonomic control of the cardiovascular system

Synchronization parameters of 0.1-Hz rhythms isolated from the heart rate and the oscillations of the blood volume in microcirculatory vessels were studied in 12 healthy subjects and 32 patients with acute myocardial infarction. Recordings of the electrocardiogram and the pulsogram from the distal phalanx of the index finger, as well as mechanical recording of respiration with the body in a horizontal position, were performed. In patients with myocardial infarction, the recordings were performed during the first three to five days and the third week after the infarction. Synchronization was tested by plotting phase differences and calculating the total percentage of phase synchronization. Synchronization parameters of 0.1-Hz rhythms were high in healthy subjects. In patients with acute myocardial infarction, synchronization of 0.1-Hz rhythms was considerably poorer. The data obtained suggest that the studied 0.1-Hz rhythms are two independent oscillatory processes that are synchronized in healthy subjects. However, this interaction may be disturbed in cardiovascular pathologies, e.g., myocardial infarction.     

An Example of Circular Statistics in Chronobiological Studies: Analysis of Polymorphism in the Emergence Rhythms of Schistosoma Mansoni Cercariae

In the not too distant past, it was common belief that rhythms in the physical environment were the driving force, to which organisms responded passively, for the observed daily rhythms in measurable physiological and behavioral variables. The demonstration that this was not the case, but that both plants and animals possess accurate endogenous time-measuring machinery (i.e., circadian clocks) contributed to heightening interest in the study of circadian biological rhythms. In the last few decades, flourishing studies have demonstrated that most organisms have at least one internal circadian timekeeping device that oscillates with a period close to that of the astronomical day (i.e., 24h). To date, many of the physiological mechanisms underlying the control of circadian rhythmicity have been described, while the improvement of molecular biology techniques has permitted extraordinary advancements in our knowledge of the molecular components involved in the machinery underlying the functioning of circadian clocks in many different organisms, man included. In this review, we attempt to summarize our current understanding of the genetic and molecular biology of circadian clocks in cyanobacteria, fungi, insects, and mammals. (Chronobiology International, 17(4), 433–451, 2000)     

The Ultradian Clocks of Eukaryotic Microbes: Timekeeping Devices Displaying a Homeostasis of the Period

Temperature compensation of their period is one of the canonical characteristics of circadian rhythms, yet it is not restricted to circadian rhythms. This short review summarizes the evidence for ultradian rhythms, with periods from 1 minute to several hours, that likewise display a strict temperature compensation. They have been observed mostly in unicellular organisms in which their constancy of period at different temperatures, as well as under different growth conditions (e.g., medium type, carbon source), indicates a general homeostasis of the period. Up to eight different parameters, including cell division, cell motility, and energy metabolism, were observed to oscillate with the same periodicity and therefore appear to be under the control of the same central pacemaker. This suggests that these ultradian clocks should be considered as cellular timekeeping devices that in fast-growing cells take over temporal control of cellular functions controlled by the circadian clock in slow-growing or nongrowing cells. Being potential relatives of circadian clocks, these ultradian rhythms may serve as model systems in chronobiolog-ical research. Indeed, mutations have been found that affect both circadian and ultradian periods, indicating that the respective oscillators share some mechanistic features. In the haploid yeast Schizosaccharomyces pombe, a number of genes have been identified where mutation, deletion, or overex-pression affect the ultradian clock. Since most of these genes play roles in cellular metabolism and signaling, and mutations have pleiotropic effects, it has to be assumed that the clock is deeply embedded in cellular physiology. It is therefore suggested that mechanisms ensuring temperature compensation and general homeostasis of period are to be sought in a wider context. (Chronobiology International, 14(5), 469–479, 1997)     

A Procedure of Multiple Period Searching in Unequally Spaced Time-Series with the Lomb–Scargle Method

Periodogram analysis of unequally spaced time-series, as part of many biological rhythm investigations, is complicated. The mathematical frameworkis scattered over the literature, and the interpretation of results is often debatable. In this paper, we show that the Lomb–Scargle method is the appropriate tool for periodogram analysis of unequally spaced data. A unique procedure of multiple period searching is derived, facilitating the assessment of the various rhythms that may be present in a time-series. All relevant mathematical and statistical aspects are considered in detail, and much attention is given to the correct interpretation of results. The use of the procedure is illustrated by examples, and problems that may be encountered are discussed. It is argued that, when following the procedure of multiple period searching, we can even benefit from the unequal spacing of a time-series in biological rhythm research.