The topology of phase response curves induced by single and paired stimuli in spontaneously oscillating chick heart cell aggregates

Describing dynamics of driven multistable oscillators with phase transfer curves

Phase response curve is an important tool in the studies of stable self-sustained oscillations; it describes a phase shift under action of an external perturbation. We consider multistable oscillators with several stable limit cycles. Under a perturbation, transitions from one oscillating mode to another one may occur. We define phase transfer curves to describe the phase shifts at such transitions. This allows for a construction of one-dimensional maps that characterize periodically kicked multistable oscillators. We show that these maps are good approximations of the full dynamics for large periods of forcing.

Phase response characteristics of sinoatrial node cells

In this work, the dynamic response of the sinoatrial node (SAN), the natural pacemaker of the heart, to short external stimuli is investigated using the Zhang et al. model. The model equations are solved twice for the central cell and for the peripheral cell. A short current pulse is applied to reset the spontaneous rhythmic activity of the single sinoatrial node cell. Depending on the stimulus timing either a delay or an advance in the occurrence of next action potential is produced. This resetting behavior is quantified in terms of phase transition curves (PTCs) for short electrical current pulses of varying amplitude which span the whole period. For low stimulus amplitudes the transition from advance to delay is smooth, while at higher amplitudes abrupt changes and discontinuities are observed in PTCs. Such discontinuities reveal critical stimuli, the application of which can result in annihilation of activity in central SAN cells. The detailed analysis of the ionic mechanisms involved in its resetting behavior of sinoatrial node cell models provides new insight into the dynamics and physiology of excitation of the sinoatrial node of the heart.

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

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.

Phase-amplitude resetting of the human circadian pacemaker via bright light: a further analysis

We present here an analysis of strong, weak, and critical bright-light resetting trials in humans, and report not only phase but also amplitude data for the first time. For this analysis, an appropriate iterative smoothing procedure for phase transition curves is introduced, in which the data are sequenced so as to minimize the perpendicular distance from the data to the smoothed fit. From these smoothed data, we create polar phase-amplitude resetting maps (PARMs) in order to fully illustrate the effects of the resetting stimuli on both circadian amplitude and phase, and thereby to determine whether these resetting results can be described by a phase-only model or whether a phase-amplitude model is required. Our results indicate that a single 5-hr episode of bright light induces weak type 1 resetting of the human circadian pacemaker. Two cycles of exposure to the same stimulus on consecutive days induce critical resetting, in which significant amplitude reduction may be observed. A three-cycle stimulus induces strong type 0 resetting with different effects on circadian amplitude, depending on the initial phase of the stimulus application. When a three-cycle stimulus is centered near the nadir of the temperature cycle, large phase shifts are achieved via amplitude suppression. However, when this stimulus is centered away from the temperature nadir, smaller phase shifts are achieved in which both small increases and small decreases in circadian amplitude are observed. These data indicate that the human circadian pacemaker is not a simple, phase-only oscillator. Instead, a full description of human circadian resetting responses to light requires analysis of both phase and amplitude data--a finding that is consistent with a phase-amplitude model of the circadian resetting mechanism.