Phase resetting and phase singularity of an insect circannual oscillator

In circadian rhythms, the shape of the phase response curves (PRCs) depends on the strength of the resetting stimulus. Weak stimuli produce Type 1 PRCs with small phase shifts and a continuous transition between phase delays and advances, whereas strong stimuli produce Type 0 PRCs with large phase shifts and a distinct break point at the transition between delays and advances. A stimulus of an intermediate strength applied close to the break point in a Type 0 PRC sometimes produces arrhythmicity. A PRC for the circannual rhythm was obtained in pupation of the varied carpet beetle, Anthrenus verbasci, by superimposing a 4-week long-day pulse (a series of long days for 4 weeks) over constant short days. The shape of this PRC closely resembles that of the Type 0 PRC. The present study shows that the PRC to 2-week long-day pulses was Type 1, and that a 4-week long-day pulse administered close to the PRC’s break point induced arrhythmicity in pupation. It is, therefore, suggested that circadian and circannual oscillators share the same mode in phase resetting to the stimuli.     

Pulse coupled oscillators and the phase resetting curve

Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.     

Evaluating functional roles of phase resetting in generation of adaptive human bipedal walking with a physiologically based model of the spinal pattern generator

The central pattern generators (CPGs) in the spinal cord strongly contribute to locomotor behavior. To achieve adaptive locomotion, locomotor rhythm generated by the CPGs is suggested to be functionally modulated by phase resetting based on sensory afferent or perturbations. Although phase resetting has been investigated during fictive locomotion in cats, its functional roles in actual locomotion have not been clarified. Recently, simulation studies have been conducted to examine the roles of phase resetting during human bipedal walking, assuming that locomotion is generated based on prescribed kinematics and feedback control. However, such kinematically based modeling cannot be used to fully elucidate the mechanisms of adaptation. In this article we proposed a more physiologically based mathematical model of the neural system for locomotion and investigated the functional roles of phase resetting. We constructed a locomotor CPG model based on a two-layered hierarchical network model of the rhythm generator (RG) and pattern formation (PF) networks. The RG model produces rhythm information using phase oscillators and regulates it by phase resetting based on foot-contact information. The PF model creates feedforward command signals based on rhythm information, which consists of the combination of five rectangular pulses based on previous analyses of muscle synergy. Simulation results showed that our model establishes adaptive walking against perturbing forces and variations in the environment, with phase resetting playing important roles in increasing the robustness of responses, suggesting that this mechanism of regulation may contribute to the generation of adaptive human bipedal locomotion.     

The variance of phase-resetting curves

Phase resetting curves (PRCs) provide a measure of the sensitivity of oscillators to perturbations. In a noisy environment, these curves are themselves very noisy. Using perturbation theory, we compute the mean and the variance for PRCs for arbitrary limit cycle oscillators when the noise is small. Phase resetting curves and phase dependent variance are fit to experimental data and the variance is computed using an ad-hoc method. The theoretical curves of this phase dependent method match both simulations and experimental data significantly better than an ad-hoc method. A dual cell network simulation is compared to predictions using the analytical phase dependent variance estimation presented in this paper. We also discuss how entrainment of a neuron to a periodic pulse depends on the noise amplitude.     

Predicting the Responses of Repetitively Firing Neurons to Current Noise

We used phase resetting methods to predict firing patterns of rat subthalamic nucleus (STN) neurons when their rhythmic firing was densely perturbed by noise. We applied sequences of contiguous brief (0.5–2 ms) current pulses with amplitudes drawn from a Gaussian distribution (10–100 pA standard deviation) to autonomously firing STN neurons in slices. Current noise sequences increased the variability of spike times with little or no effect on the average firing rate. We measured the infinitesimal phase resetting curve (PRC) for each neuron using a noise-based method. A phase model consisting of only a firing rate and PRC was very accurate at predicting spike timing, accounting for more than 80% of spike time variance and reliably reproducing the spike-to-spike pattern of irregular firing. An approximation for the evolution of phase was used to predict the effect of firing rate and noise parameters on spike timing variability. It quantitatively predicted changes in variability of interspike intervals with variation in noise amplitude, pulse duration and firing rate over the normal range of STN spontaneous rates. When constant current was used to drive the cells to higher rates, the PRC was altered in size and shape and accurate predictions of the effects of noise relied on incorporating these changes into the prediction. Application of rate-neutral changes in conductance showed that changes in PRC shape arise from conductance changes known to accompany rate increases in STN neurons, rather than the rate increases themselves. Our results show that firing patterns of densely perturbed oscillators cannot readily be distinguished from those of neurons randomly excited to fire from the rest state. The spike timing of repetitively firing neurons may be quantitatively predicted from the input and their PRCs, even when they are so densely perturbed that they no longer fire rhythmically.     

Interpreting the human phase response curve to multiple bright-light exposures

Czeisler and his colleagues have recently reported that bright light can induce strong (Type O) resetting of the human circadian pacemaker. This surprising result shows that the human clock is more responsive to light than has been previously thought. The interpretation of their results is subtle, however, because of an unconventional aspect of their experimental protocol: They measured the phase shift after three cycles of the bright-light stimulus, rather than after the usual single pulse. A natural question is whether the apparent Type O response could reflect the summation of three weaker Type 1 responses to each of the daily light pulses. In this paper I show mathematically that repeated Type 1 resetting cannot account for the observed Type O response. This finding corroborates the strong resetting reported by Czeisler et al., and supports their claim that bright light induces strong resetting by crushing the amplitude of the circadian pacemaker. Furthermore, the results indicate that back-to-back light pulses can have a cooperative effect different from that obtained by simple iteration of a phase response curve (PRC). In this sense the resetting response of humans is similar to that of Drosophila, Kalanchoe, and Culex, and is more complex than that predicted by conventional PRC theory. To describe the way in which light resets the human circadian pacemaker, one needs a theory that includes amplitude resetting, as pioneered by Winfree and developed for humans by Kronauer.     

Circadian phototransduction: phase resetting and frequency of the circadian clock of Gonyaulax cells in red light

Constant red light (RR) influences the Gonyaulax clock in several ways: (1) Phase resetting by white or blue light pulses is stronger under background RR than in constant white light (WW); (2) frequency of the rhythm is less in RR than in WW; and (3) the amplitude of the spontaneous flashing rhythm is greater in RR than in WW. The phase response curve (PRC) to 4-hr white or blue light pulses is of high amplitude (Type 0) for cells in RR, but is of lower amplitude (Type 1) for cells in WW. In all cases, the PRC is highly asymmetrical: The magnitude of advance phase resetting is far higher than that of delay resetting. Consistent with this PRC, Gonyaulax cells in RR (free-running period greater than 24 hr) will entrain to T cycles of between 21 and 26.5 hr. The bioluminescence rhythms exhibit "masking" by blue light pulses while entrained to these T cycles. The fluence response of phase resetting to light-pulse intensity is not linear or logarithmic--rather, it is discontinuous. This feature is consistent with a limit cycle interpretation of Type 0 resetting of circadian clocks. Light pulses that cause large phase shifts also shorten the subsequent free-running period. The phase angle difference between the clock and the previous LD cycle is within 2 hr of the same phase between 16 degrees C and 25 degrees C, as determined from the light PRCs at various temperatures. Several drugs that inhibit mitochondria and/or electron transport will partially inhibit the phase shift by light.     

Modeling circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators: phase shifts and phase response curves

Circadian rhythm generation in the suprachiasmatic nucleus was modeled by locally coupled self-sustained oscillators. The model is composed of 10,000 oscillators, arranged in a square array. Coupling between oscillators and standard deviation of (randomly determined) intrinsic oscillator periods were varied. A stable overall rhythm emerged. The model behavior was investigated for phase shifts of a 24-h zeitgeber cycle. Prolongation of either the dark or the light phase resulted in a lengthening of the period, whereas shortening of the dark or the light phase shortened the period. The model's response to shifts in the light-dark cycle was dependent only on the extent of the shift and was insensitive to changes in parameters. Phase response curves (PRC) and amplitude response curves were determined for single and triple 5-h light pulses (1000 lux). Single pulses lead to type 1 PRCs with larger phase shifts for weak coupling. Triple pulses generally evoked type 1 PRCs with the exception of weak coupling, where a type 0 PRC was observed.     

The circadian system of the Turkish hamster, Mesocricetus brandti: responses to light

The circadian system of the Turkish hamster controlling wheel-running activity responded to single 1-hr light pulses and to repeated 1-hr pulses in a similar way as that of Syrian hamsters studied previously. At constant light of 100 lx, the period length (tau) of the freerunning activity rhythm of Turkish hamsters was longer and the activity time (alpha) was shorter than that of Syrian hamsters. Among individuals, the ability of the system to be entrained by one 1-hr light pulse per cycle was related to the range (advance plus delay amplitude) of the phase response curve (PRC) derived from single light pulses and to the compression of alpha caused by the pulse Zeitgeber. The data support the hypothesis derived from experiments on Syrian hamsters that the range of the PRC is functionally related with alpha, possibly reflecting the phase relations (coupling) between two oscillators.     

Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators

Gamma oscillations can synchronize with near zero phase lag over multiple cortical regions and between hemispheres, and between two distal sites in hippocampal slices. How synchronization can take place over long distances in a stable manner is considered an open question. The phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike, depending upon where in the cycle it is received. We use PRCs under the assumption of pulsatile coupling to derive existence and stability criteria for 1:1 phase-locking that arises via bidirectional pulse coupling of two limit cycle oscillators with a conduction delay of any duration for any 1:1 firing pattern. The coupling can be strong as long as the effect of one input dissipates before the next input is received. We show the form that the generic synchronous and anti-phase solutions take in a system of two identical, identically pulse-coupled oscillators with identical delays. The stability criterion has a simple form that depends only on the slopes of the PRCs at the phases at which inputs are received and on the number of cycles required to complete the delayed feedback loop. The number of cycles required to complete the delayed feedback loop depends upon both the value of the delay and the firing pattern. We successfully tested the predictions of our methods on networks of model neurons. The criteria can easily be extended to include the effect of an input on the cycle after the one in which it is received.     

Cell Division Cycles and Orcadian Oscillators in Euglena

The algal flagellate Euglena grown photoautotrophically in L:D 3:3 displays a circadian rhythm of cell division. Oscillatory models for cell cycle (CDC) control (particularly those of the limit cycle variety) include the property of phase perturbation, or resetting. This prediction has been tested in synchronous cultures in which the free-running rhythm has been scanned by 3-hr light signals. A strong (Type 0) phase response curve (PRC), yielding both advances and delays as great as 15 hr, has been derived. A second prediction of the limit cycle model is that there exists a pulse of a critical intensity, which, if given at one specific phase of the rhythm (the singularity point), should result in a phaseless, motionless state in which the rhythmicity disappears. Such a point has been found in Euglena in the late subjective night for light pulses having an intensity ranging from 40 to 700 Ix. Finally, circadian oscillators typically display temperature-compensated period lengths within the physiological range of steady-state temperatures, although the length of the CDC is commonly thought to be highly temperature dependent. We have found that over a range of at least 10°C, the period of the division rhythm is only slightly affected, exhibiting a Q₁₀ of about 1.05-1.20. These observations, therefore, collectively implicate a circadian oscillator in the control of the CDC.     

Amplitude model for the effects of mutations and temperature on period and phase resetting of the Neurospora circadian oscillator.

This paper analyzes published and unpublished data on phase resetting of the circadian oscillator in the fungus Neurospora crassa and demonstrates a correlation between period and resetting behavior in several mutants with altered periods: As the period increases, the apparent sensitivity to resetting by light and by cycloheximide decreases. Sensitivity to resetting by temperature pulses may also decrease. We suggest that these mutations affect the amplitude of the oscillator and that a change in amplitude is responsible for the observed changes in both period and resetting by several stimuli. As a secondary hypothesis, we propose that temperature compensation of period in Neurospora can be explained by changes in amplitude: As temperature increases, the compensation mechanism may increase the amplitude of the oscillator to maintain a constant period. A number of testable predictions arising from these two hypotheses are discussed. To demonstrate these hypotheses, a mathematical model of a time-delay oscillator is presented in which both period and amplitude can be increased by a change in a single parameter. The model exhibits the predicted resetting behavior: With a standard perturbation, a smaller amplitude produces type 0 resetting and a larger amplitude produces type 1 resetting. Correlations between period, amplitude, and resetting can also be demonstrated in other types of oscillators. Examples of correlated changes in period and resetting behavior in Drosophila and hamsters raise the possibility that amplitude changes are a general phenomenon in circadian oscillators.     

Probing the circadian pacemaker of a mouse using two light pulses

In two separate sets of experiments, the phases of the locomotor activity rhythm of the nocturnal field mouse Mus booduga were probed using two light pulses (LPs). In the first set of experiments, the circadian pacemaker underlying the locomotor activity rhythm was perturbed at circadian time 14 (CT 14) using a resetting light pulse LP1 of 1000 lux intensity and 15 min duration. The phases of the resetting pacemaker were then probed at all even CTs between CT 16 and CT 14 using a PRC probing light pulse LP2 of equal strength. The "LP2 PRC" thus obtained was then compared with the single light pulse PRC in terms of the area under delay (D) and advance (A) zones of the PRCs. The time course and waveform of the two LP PRCs suggest that the LP2 PRC resembled the single LP PRC, displaced by 2 h toward the right. The LP1 PRC had smaller D compared to the single LP PRC (p = 0.007), whereas both the PRCs had A of equal magnitude (p = 0.23). This suggests that the pacemaker phase shifts rapidly after LP perturbations. In the second set of experiments, the LP1 was administered at CT 14. The phase of the pacemaker was then perturbed on day 1 (next cycle after LP1) either 2 h after activity onset (at ca. CT 14 of the transient cycle) or 8 h after activity onset (at ca. CT 20 of the transient cycle) using an LP2 of equal strength. It was observed that the steady-state phase shifts evoked by positioning an LP2, 2 h after activity onset, were positively correlated with the phase shifts observed on day 1. The steady-state phase shifts observed, when the LP2 was positioned, 8 h after activity onset, were negatively correlated with the phase shifts observed on day 1. These results suggest that the transient cycles do not mirror the state of the pacemaker oscillator.     

Two coupled oscillators as a model for the coordinated finger tapping by both hands

Recently, it was found that rhythmic movements (e.g. locomotion, swimmeret beating) are controlled by mutually coupled endogeneous neural oscillators (Kennedy and Davis, 1977; Pearson and Iles, 1973; Stein, 1974; Shik and Orlovsky, 1976; Grillner and Zangger, 1979). Meanwhile, it has been found out that the phase resetting experiment is useful to investigate the interaction of neural oscillators (Perkel et al., 1963; Stein, 1974). In the preceding paper (Yamanishi et al., 1979), we studied the functional interaction between the neural oscillator which is assumed to control finger tapping and the neural networks which control some tasks. The tasks were imposed on the subject as the perturbation of the phase resetting experiment. In this paper, we investigate the control mechanism of the coordinated finger tapping by both hands. First, the subjects were instructed to coordinate the finger tapping by both hands so as to keep the phase difference between two hands constant. The performance was evaluated by a systematic error and a standard deviation of phase differences. Second, we propose two coupled neural oscillators as a model for the coordinated finger tapping. Dynamical behavior of the model system is analyzed by using phase transition curves which were measured on one hand finger tapping in the previous experiment (Yamanishi et al., 1979). Prediction by the model is in good agreement with the results of the experiments. Therefore, it is suggested that the neural mechanism which controls the coordinated finger tapping may be composed of a coupled system of two neural oscillators each of which controls the right and the left finger tapping respectively.     

Modeling the dual pacemaker system of the tau mutant hamster

Circadian pacemakers in many animals are compound. In rodents, a two-oscillator model of the pacemaker composed of an evening (E) and a morning (M) oscillator has been proposed based on the phenomenon of "splitting" and bimodal activity peaks. The authors describe computer simulations of the pacemaker in tau mutant hamsters viewed as a system of mutually coupled E and M oscillators. These mutant animals exhibit normal type 1 PRCs when released into DD but make a transition to a type 0 PRC when held for many weeks in DD. The two-oscillator model describes particularly well some recent behavioral experiments on these hamsters. The authors sought to determine the relationships between oscillator amplitude, period, PRC, and activity duration through computer simulations. Two complementary approaches proved useful for analyzing weakly coupled oscillator systems. The authors adopted a "distinct oscillators" view when considering the component E and M oscillators and a "system" view when considering the system as a whole. For strongly coupled systems, only the system view is appropriate. The simulations lead the authors to two primary conjectures: (1) the total amplitude of the pacemaker system in tau mutant hamsters is less than in the wild-type animals, and (2) the coupling between the unit E and M oscillators is weakened during continuous exposure of hamsters to DD. As coupling strength decreases, activity duration (alpha) increases due to a greater phase difference between E and M. At the same time, the total amplitude of the system decreases, causing an increase in observable PRC amplitudes. Reduced coupling also increases the relative autonomy of the unit oscillators. The relatively autonomous phase shifts of E and M oscillators can account for both immediate compression and expansion of activity bands in tau mutant and wild-type hamsters subjected to light pulses.     

Circadian rhythms in Neurospora: a new measurement, the reset zone

The authors define a new feature of a circadian rhythm, the reset zone, and point out its usefulness for predictions concerning oscillator behavior. The reset zone measures the responses of a circadian system to resetting pulses. It can be easily determined from a phase transition curve (PTC), which is simply a phase response curve (PRC) replotted as new phase versus old phase (Winfree's format). The reset zone is the range of new phases seen in such a plot and has two potentially useful characteristics: its size and its midpoint. A series of experiments with Neurospora involving temperature pulses indicated that the size of the reset zone changed in a nonlinear way in response to both the duration of 40 degrees C pulses and to the magnitude of temperature change for 3-h pulses. Other existing data are replotted to show how the reset zone size varies with growth temperature and with the period of different clock mutants. Employing exclusively reset zone data within the framework of a limit cycle displacement model, an equation is formulated that predicts the relative changes in the values of state variables of the oscillator for changes in any given environmental condition, such as temperature. Examples are also drawn from other organisms, such as hamsters, Gonyalaux, Kalanchoe, and Drosophila, illustrating the usefulness of the reset zone measurement. It can be used as a numerical scale for assessing the strength of a pulse, for comparing the relative effects of a given pulse applied to different organisms or mutants, for determining the directionality of the changes in state variables produced by various types of pulses, and possibly for measuring clock amplitude.     

Phase resetting of embryonic chick atrial heart cell aggregates. Experiment and theory.

The influence of brief duration current pulses on the spontaneous electrical activity of embryonic chick atrial heart cell aggregates was investigated experimentally and theoretically. A pulse could either delay or advance the time of the action potential subsequent to the pulse depending upon the time in the control cycle at which it was applied. The perturbed cycle length throughout the transition from delay to advance was a continuous function of the time of the pulse for small pulse amplitudes, but was discontinuous for larger pulse amplitudes. Similar results were obtained using a model of the ionic currents which underlie spontaneous activity in these preparations. The primary ion current components which contribute to phase resetting are the fast inward sodium ion current, INa, and the primary, potassium ion repolarization current, IX1. The origin of the discontinuity in phase resetting of the model can be elucidated by a detailed examination of the current-voltage trajectories in the region of the phase response curve where the discontinuity occurs.     

Phase shifting the hamster circadian clock by 15-minute dark pulses

The mammalian circadian pacemaker can be phase shifted by exposure to a period of darkness interrupting otherwise continuous light. Circadian phase shifting by dark pulses was interpreted originally as reflecting a photic mirror-image mechanism, but more recent observations suggest that dark pulse-induced phase shifting may be mediated by a nonphotic, behavioral state-dependent mechanism. The authors recently presented evidence indicating that the dark-pulse phase response curve (PRC) is in fact a complex function, reflecting both photic mirror image and nonphotic mechanisms at different phases of the circadian cycle. Previous studies of dark pulse-induced phase shifting have universally employed relatively long (2 to 6 h) pulses, which complicates PRC analysis due to the extended segment of the underlying PRC spanned by such a long pulse. The present study was therefore designed to examine the phase-shifting effects of brief 15-min dark pulses presented at both mid-subjective day and subjective dusk, and to explore the possible activity dependence of these effects by using physical restraint to prevent evoked locomotor activity. The results indicate that 15-min dark pulses are effective phase-shifting stimuli at both midday and dusk. Furthermore, as with longer dark pulses, phase shifting by 15-min dark pulses is completely blocked by physical restraint during subjective day but combines in a simple additive manner with the independent phase-shifting effect of restraint at subjective dusk.     

Light-Induced resetting of the circadian pacemaker: quantitative analysis of transient versus steady-state phase shifts.

The suprachiasmatic nuclei of the hypothalamus contain the major circadian pacemaker in mammals, driving circadian rhythms in behavioral and physiological functions. This circadian pacemaker's responsiveness to light allows synchronization to the light-dark cycle. Phase shifting by light often involves several transient cycles in which the behavioral activity rhythm gradually shifts to its steady-state position. In this article, the authors investigate in Syrian hamsters whether a phase-advancing light pulse results in immediate shifts of the PRC at the next circadian cycle. In a first series of experiments, the authors aimed a light pulse at CT 19 to induce a phase advance. It appeared that the steady-state phase advances were highly correlated with activity onset in the first and second transient cycle. This enabled them to make a reliable estimate of the steady-state phase shift induced by a phase-advancing light pulse on the basis of activity onset in the first transient cycle. In the next series of experiments, they presented a light pulse at CT 19, which was followed by a second light pulse aimed at the delay zone of the PRC on the next circadian cycle. The immediate and steady-state phase delays induced by the second light pulse were compared with data from a third experiment in which animals received a phase-delaying light pulse only. The authors observed that the waveform of the phase-delay part of the PRC (CT 12-16) obtained in Experiment 2 was virtually identical to the phase-delay part of the PRC for a single light pulse (obtained in Experiment 3). This finding allowed for a quantitative assessment of the data. The analysis indicates that the delay part of the PRC-between CT 12 and CT 16-is rapidly reset following a light pulse at CT 19. These findings complement earlier findings in the hamster showing that after a light pulse at CT 19, the phase-advancing part of the PRC is immediately shifted. Together, the data indicate that the basis for phase advancing involves rapid resetting of both advance and delay components of the PRC.