Characterizing the amplitude dynamics of the human core-temperature circadian rhythm using a stochastic–dynamic model

https://doi.org/10.1016/j.jtbi.2005.08.015Get rights and content

Abstract

Two measures, amplitude and phase, have been used to describe the characteristics of the endogenous human circadian pacemaker, a biological clock located in the hypothalamus. Although many studies of change in circadian phase with respect to different stimuli have been conducted, the physiologic implications of the amplitude changes (dynamics) of the pacemaker are unknown. It is known that phase changes of the human circadian pacemaker have a significant impact on sleep timing and content, hormone secretion, subjective alertness and neurobehavioral performance. However, the changes in circadian amplitude with respect to different stimuli are less well documented. Although amplitude dynamics of the human circadian pacemaker are observed in physiological rhythms such as plasma cortisol, plasma melatonin and core temperature data, currently methods are not available to accurately characterize the amplitude dynamics from these rhythms. Of the three rhythms core temperature is the only reliable variable that can be monitored continuously in real time with a high sampling rate. To characterize the amplitude dynamics of the circadian pacemaker we propose a stochastic–dynamic model of core temperature data that contains both stochastic and dynamic characteristics. In this model the circadian component that has a dynamic characteristic is represented as a perturbation solution of the van der Pol equation and the thermoregulatory response in the data that has a stochastic characteristic is represented as a first-order autoregressive process. The model parameters are estimated using data with a maximum likelihood procedure and the goodness-of-fit measures along with the associated standard error of the estimated parameters provided inference about the amplitude dynamics of the pacemaker. Using this model we analysed core temperature data from an experiment designed to exhibit amplitude dynamics. We found that the circadian pacemaker recovers slowly to an equilibrium level following amplitude suppression. In humans this reaction to perturbation from equilibrium value has potential physiological implications.

Introduction

The human endogenous circadian (∼24 -h) pacemaker located in the suprachiasmatic nuclei (SCN) in the hypothalamus of the human brain provides a major influence on sleep content, subjective mood and cognitive performance, hormone levels, rest-activity state, urinary output, and other physiologic functions (Waterhouse and DeCoursey, 2004). The influences of the timing (phase) of the circadian pacemaker on many physiological and neurobehavioral functions are established (Tilley et al., 1982; Strogatz et al., 1986; Czeisler and Khalsa, 2000; Wright et al., 2002; Klerman, 2005). However, very little is known about the effect of the strength (amplitude) of the pacemaker on these functions. It has been hypothesized that low amplitude of the endogenous circadian pacemaker may be responsible for maintaining delayed sleep phase syndrome or non-24-h sleep–wake rhythm syndrome in some patients and for the decrement in performance and alertness due to prolonged wakefulness in elderly subjects (Jewett and Czeisler, 1992). Since amplitude is functionally important in humans it is essential to understand the amplitude changes (dynamics) of the pacemaker.

The studies investigating the significance of amplitude dynamics of the circadian pacemaker on physiological and neurobehavioral functions have not been attempted. This is mainly due to the lack of methods available to detect amplitude dynamics of the pacemaker from experimental data. Understanding amplitude dynamics could have physiologic implications for individuals with reduced amplitude, a group that includes older individuals, individuals with dementia, individuals working night or rotating shift and individuals with jet lag (Czeisler et al., 1992; Dagan, 2002).

The method for observing the behavior of the pacemaker relies on subjecting the circadian system to light stimulus. The sensitivity of the circadian pacemaker to light can be assessed by systematic evaluation of light-induced resetting of phase and change in amplitude throughout the cycle, yielding a phase response curve (PRC) and an amplitude response curve (ARC), respectively (Jewett et al., 1994). These response curves provide a good representation of qualitative characteristics of the pacemaker; however they do not provide information about the dynamics of the pacemaker. The daily alternation of light and darkness is the most important environmental stimulus for entraining circadian rhythms across an array of species (Rusak, 1979; Rusak and Zucker, 1979) and light is a primary synchronizer of the human circadian pacemaker (Czeisler et al., 1981).

The significance of the amplitude in the circadian system was first demonstrated by Winfree, 1971, Winfree, 1973, Winfree, 2000 with Drosophila phase resetting experiments. Winfree applied a two-pulse light of appropriate magnitude and duration on Drosophila to demonstrate a “strong” phase resetting (Type 0). During the Type 0 resetting process, the amplitude of the oscillation temporarily diminishes, eventually undergoing a slow recovery to the equilibrium value (limit cycle). Further findings, established by Peterson (1981), based on the two pulse experiments on adult mosquitoes revealed that the amplitude remained at a low value for one to two weeks. In these experimental protocols amplitude suppression of the pacemaker was achieved by applying two light pulses of appropriate magnitude and duration at a critical phase for an interval approximately equal to the period of the pacemaker.

The Type 0 resetting observed in insects has also been demonstrated in humans (Czeisler et al., 1989; Jewett et al., 1991). However in humans the rate of amplitude recovery to the limit cycle following amplitude suppression of the circadian pacemaker is still not known. Since the circadian pacemaker is located deep in the human brain, its phase and amplitude cannot be measured directly and markers of circadian phase and amplitude are employed to study the pacemaker. Three marker rhythms frequently used to study the pacemaker are core temperature, plasma melatonin, and plasma cortisol (Jewett et al., 1991, Jewett et al., 1994; Shanahan and Czeisler, 1991; Boivin and Czeisler, 1998; Arendt, 2005). Currently methods are not available to characterize the amplitude dynamics from these marker rhythms. Of the three marker rhythms of the human circadian pacemaker, core temperature is the only reliable variable that can be monitored continuously in real time and with a high sampling rate.

The disadvantage of using core temperature data is that the other physiological factors such as posture, level of activity and the dynamics of body's thermoregulatory system significantly contribute to the observed temperature data. Reducing these effects requires an experiment protocol called Constant Routine (CR) in which a subject remains awake in dim light in a semi-recumbent posture with frequent small meals. CR minimizes the evoked effects of sleep, light and posture changes and activity on core temperature data (Duffy, 1993) leaving only circadian and thermoregulatory response.

An additional difficulty for assessing the amplitude dynamics of the pacemaker from core temperature data results from the magnitude of the response. The amplitude of the human circadian pacemaker can be suppressed by properly timing a light stimulus near the minimum of core temperature (Jewett et al., 1991). In response to such critical stimulus, the magnitude of the circadian component is greatly reduced both relative to the original value and in comparison with the statistical fluctuations in the data due to thermoregulatory process and other physiological factors. Consequently, it is difficult to assess the amplitude dynamics of the pacemaker from core temperature data.

To accurately estimate the amplitude and phase of the human circadian pacemaker in the presence of thermoregulatory response a statistical model of the core temperature data has been proposed based on a signal-plus-noise modeling approach (Brown and Czeisler, 1992). In this approach the signal is represented as a multiple harmonic model with amplitude of the harmonics as constant (static) and the noise due to the body's thermoregulatory response is represented as an autoregressive process. This proposed model is effective at estimating the amplitude and phase of the circadian pacemaker at the limit cycle. However due to the static representation of the amplitude, this model fails to capture the dynamic nature of amplitude following amplitude suppression.

Another approach that is based on dynamical system theory can be used for understanding the dynamic nature of circadian pacemaker provided there is sufficient length of data (Ortega et al., 1994). The length of data points obtained in CR procedure is very short typically ∼40 h. Hence we propose a new approach, the stochastic–dynamic modeling approach, to characterize the amplitude dynamics of the circadian pacemaker from core temperature data by incorporating a dynamic representation of the amplitude on a statistical modeling framework. This modeling approach was originally developed by Brown (1987) with a potential to study the amplitude dynamics from core temperature data, however this method had never been employed. We employed this method to analyse the core temperature data from an experiment (Jewett et al., 1991) designed to exhibit amplitude dynamics following amplitude suppression.

Section snippets

Model formulation

We define a stochastic–dynamic model of core temperature data using a signal-plus-noise modeling framework. Our primary objective is to provide a reasonable description of the dynamics of the circadian pacemaker from the core temperature data in the presence of thermoregulatory response.

Core temperature data, y(t) recorded during CR at evenly spaced intervals n=1,2,…,N is expressed as y(t)=s(t)+v(t),where the signal s(t) is the circadian component and the noise v(t) is the fluctuation in core

Results

We estimated the parameters θ=(a0,ε,Ψ0) and ρ using nonlinear optimization (Press et al., 2002) of the log likelihood function in Eq. (7) using the data during CR2 along with the estimated γ at CR1. The standard errors (SE) were calculated from the inverse of the observed Fisher information matrices (Brown, 1987). The average normalized amplitude is calculated with the estimated θ with the appropriate normalization for each data set. The primary objective of this normalization procedure is to

Discussion

We developed a stochastic-dynamic model to characterize the amplitude dynamics of core temperature data by incorporating the perturbation solution of van der Pol equation to a statistical modeling frame work. An innovative feature of this signal plus noise model is that the signal here possesses a dynamic characteristic instead of a static characteristic. Furthermore all the formal statistical methods can be employed to determine the characteristics of amplitude dynamics observed in core

Acknowledgments

The authors are grateful to Megan E. Jewett for sharing the insights on experimental protocol and providing data analysed in this work. Thanks to Richard E. Kronauer, Daniel B. Forger and Elizabeth B. Klerman for helpful discussions on mathematical models of the human circadian pacemaker and to David Paydarfar and Bill Schwartz for their generous support provided to one of us (P.I) to work on this project. Support was provided to E. N. B by NIDA grant DA015644 and NIMH grants MH59733 and

References (45)

  • E.N. Brown et al.

    A stochastic differential equation model of diurnal cortisol patterns

    Am. J. Physiol.

    (2001)
  • E.N. Brown et al.

    Measuring the period of the human biological clock

  • C.A. Czeisler et al.

    The human circadian timing system and sleep–wake regulation

  • C.A. Czeisler et al.

    Bright light induction of strong (type 0) resetting of the human circadian pacemaker

    Science

    (1989)
  • C.A. Czeisler et al.

    Entrainment of human circadian rhythms by light-dark cycles: a reassessment

    Photochem. Photobiol.

    (1981)
  • C.A. Czeisler et al.

    Stability, precision, and near-24-hour period of the human circadian pacemaker

    Science

    (1999)
  • Y. Dagan

    Circadian Rhythms Sleep Disorders (CRSD) in psychiatry—a review. Isr

    J. Psychiatry Relat. Sci.

    (2002)
  • J.F. Duffy

    Constant Routine

  • D.B. Forger et al.

    A simpler model of the human circadian clock

    J. Biol. Rhythm.

    (1999)
  • S.F. Glotzbach et al.

    Sleep and thermoregulation

  • A. Gundel et al.

    A circadian pacemaker model based on empirical data

    J. Biol. Rhythms

    (1999)
  • P. Indic et al.

    Comparison of amplitude recovery dynamics of two limit cycle oscillator models of the human circadian pacemaker

    Chronobiol. Int.

    (2005)
  • View full text