PRC bisection tests

This communication presents a new method for evaluating phase response curves (PRCs). A PRC describes the phase shifts produced in an oscillator by stimuli applied at different initial phase-states of that oscillator. In the PRC bisection tests, we repeatedly cut in half the circular distribution of the initial phase-states of the oscillator when stimuli are given. Empirically, we locate that optimal diameter which best bisects the circular distribution of phase responses into arcs of relative phase advance and phase delay. We compute a D score reflecting the success of the best bisection. The null hypothesis of a random distribution of phase responses by initial phase is tested with a Monte Carlo procedure, which computes D_r scores from random combinations of phase shifts with initial phases, thus determining the probability, given the null hypothesis, that the observed D score was from a random distribution. The bisection procedure can be extended to examine whether stronger phase shifts are produced in one phase response curve than in contrasting curves. Also, the bisection procedure yields an estimate of the inflection point of the phase response curve. A method is given to estimate the power of the PRC bisection test.     

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.     

Circadian phase shifting induced by clonidine injections in Syrian hamsters

Abstract

The mammalian circadian pacemaker can be phase shifted by photic, pharmacological, and behaviorally‐derived stimuli. The phase‐response curves (PRCs) characterizing these diverse stimuli may comprise two distinct families; a photic PRC typified by the response to brief light pulses, and a non‐photic PRC, typified by the response to dark pulses and to behavioral activation. The present study examined the phase shifting effects of acute systemic treatment with the alpha2‐adrenoceptor agonist, clonidine, in Syrian hamsters. Clonidine injections (0.25 mg/kg, ip) delivered during subjective night mimicked the phase shifting effects of light pulses in animals housed in both constant darkness (DD) and constant red light (RR), but similar effects were not seen in saline‐treated controls. Both clonidine and saline injections resulted in phase advances during subjective day, but only in RR‐housed animals. Clonidine‐induced phase shifting was dose‐dependent, but rather high doses were required to induce phase shifts. Pretreatment with the selective noradrenergic neurotoxin, DSP‐4, blocked clonidine‐induced phase shifting. These results suggest that clonidine acts at presynaptic alpha2‐adrenergic autoreceptors to disinhibit spontaneous and/or evoked activity in the photic entrainment pathway.     

A phase response curve for circannual rhythm in the varied carpet beetle <Emphasis Type="Italic">Anthrenus verbasci</Emphasis>

We know that entrainment, a stable phase relationship with an environmental cycle, must be established for a biological clock to function properly. Phase response curves (PRCs), which are plots of phase shifts that result as a function of the phase of a stimulus, have been created to examine the mode of entrainment. In circadian rhythms, single-light pulse PRCs have been obtained by giving a light pulse to various phases of a free-running rhythm under continuous darkness. This successfully explains the entrainment to light-dark cycles. Some organisms show circannual rhythms. In some of these, changes in photoperiod entrain the circannual rhythms. However, no single-pulse PRCs have been created. Here we show the PRC to a long-day pulse superimposed for 4 weeks over constant short days in the circannual pupation rhythm in the varied carpet beetle Anthrenus verbasci. Because the shape of that PRC closely resembles that of the Type 0 PRC with large phase shifts in circadian rhythms, we suggest that an oscillator having a common feature in the phase response with the circadian clock, produces a circannual rhythm.     

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.     

Transient and steady state phase response curves of limit cycle oscillators

The experiment of phase shifts resulting from discrete perturbations of stable biological rhythms has been carried out to study entrainment behavior of oscillators. There are two kinds of phase response curves, which are measured in experiments, according to as one measures the phase shifts immediately or long after the perturbation. The former is the first transient phase response curve and the latter is the steady state phase response curve. We redefine both curves within the framework of dynamical system theory and homotopy theory. Several topological properties of both curves are clarified. Consequently, it is shown that we must compare the shapes of both two phase response curves to investigate the inner structures of biological oscillators. Moreover, we prove that a single limit cycle oscillator involving only two variables cannot simulate transient resetting behavior reported by Pittendrigh and Minis (1964). In other words, the circadian oscillator of Drosophila pseudoobscura does not consist of a single oscillator of two variables. Finally we show that a model which consists of two limit cycle oscillators is able to simulate qualitatively the phase response curves of Drosophila.     

Effect of Light Intensity on the Phase and Period Response Curves in the Nocturnal Field Mouse Mus booduga

The effect of light intensity on the phase response curve (PRC) and the period response curve (τRC) of the nocturnal field mouse Mus booduga was studied. PRCs and τRCs were constructed by exposing animals free-running in constant darkness (DD), to fluorescent light pulses (LPs) of 100 lux and 1000 lux intensities for 15min duration. The waveform of the PRCs and τRCs evoked by high light intensity (1000 lux) stimuli was significantly different compared to those constructed using low light intensity (100 lux). Moreover, a weak but significant correlation was observed between phase shifts and period changes when light stimuli of 1000 lux intensity were used; however, the phase shifts and period changes in the 100 lux PRC and τRC were not correlated. This suggests that the intensity of light stimuli affects both phase and period responses in the locomotor activity rhythm of the nocturnal field mouse M. booduga. These results indicate that complex mechanisms are involved in entrainment of circadian clocks, even in nocturnal rodents, in which PRC, τRC, and dose responses play a significant role.     

Asymmetric re‐entrainment and phase response of the circadian activity rhythm of a nocturnal marsupial (Petaurus breviceps: Phalangeridae)

Abstract

Sugar Gliders (Petaurus breviceps) re‐entrain faster after 8‐h delay shifts of an LD 12:12 and an LD 8:16 (31–56:0.3 lux each) than after 8‐h advance shifts of these Zeitgeber cycles. In order to test whether this asymmetric re‐entrainment behavior is related to, or even caused by the phase response characteristics of the circadian system, the phase response of the activity rhythm to short and long light pulses was studied. Short light pulses (15 min of 31–56 lux against a background intensity of 0.3 lux) caused only relatively small delay shifts when applied around the onset, and more pronounced advance shifts when given at the end of the activity time (α). Onset and end of activity shifted by different amounts. Long light pulses produced by 8‐h advances and delays of one single lighttime of an LD 12:12 elicited pronounced phase delays when applied at the beginning of the activity time, but only minor phase advances when given at the posterior part of α. These results indicate that in Petaurus breviceps the phase response characteristics to long light pulses exerting parametric effects of light are responsible for the pronounced asymmetry effect in re‐entrainment. Differing phase responses of onset and end of activity point to a two‐oscillator structure of the circadian pacemaker system in this marsupial.     

Resetting biological oscillators—A stochastic approach

We present a stochastic approach to phase-resetting of an ensemble of oscillators. In order to describe stimulation-induced dynamical phenomena we develop a stochastic model which consists of an ensemble of phase oscillators interacting via random forces. Every single oscillator is submitted to a phase stimulus. The ensemble's dynamics is determined by a Fokker-Planck equation. The stationary states are calculated explicitly, whereas the transients are analysed numerically. If the stimulus of a given (non-vanishing) intensity is administered at a critical initial cluster phase for a critical duration T _crit the ensemble's synchronized oscillation is annihilated. A transition from type 1 resetting to type 0 resetting occurs when the stimulation duration exceeds T _crit. Stimulation causes a shift of the mean frequency of every single oscillator. This frequency shift is explicitly calculated by deriving the mean first passage time. The model shows that there is a subcritical intensity which is connected with an enhanced vulnerability to stimulation. The desynchronized states, the so-called black holes, are typically associated with a double peak in the ensemble's phase distribution. This is important for analysing experimental data because simple peak-detection algorithms are not able to extract the underlying dynamics.Our results are discussed from the experimentator's point of view so that the insights derived from our model can improve data analysis and design of stimulation experiments.     

Mirror imaging phase response curves obtained for the circadian rhythm of a bat with single steps of light and darkness∗

Abstract

The circadian rhythm in the flight activity of a tropical microchiropteran bat Taphozous melanopogon responds at all phases with delay phase shifts to single light‐on steps (DD/LL transfers). The circadian rhythm responds at all phases with advance phase shifts to single light‐off steps (LL/DD transfers). Phase shifts were measured from the delays or advances of the onsets of flight activity on days following DD/LL and LL/DD transfers relative to the temporal course of the onsets of activity in controls. The magnitude of the phase shifts was a function of the phases in which the transfers were made. The On‐PRC and Off‐PRC plotted from such data are mirror‐images in their time‐course and wave‐form.

The phase shifts of the circadian rhythm in either direction were accompanied by changes in period (for the duration of our recordings after die transfer). The period lengthened following a delay shift and it shortened following an advance shift. The phase shifts are abrupt and discernible in the first cycle after perturbation. There are no transients.     

Effect of light intensity on the phase and period response curves in the nocturnal field mouse Mus booduga

The effect of light intensity on the phase response curve (PRC) and the period response curve (τRC) of the nocturnal field mouse Mus booduga was studied. PRCs and τRCs were constructed by exposing animals free-running in constant darkness (DD), to fluorescent light pulses (LPs) of 100 lux and 1000 lux intensities for 15min duration. The waveform of the PRCs and τRCs evoked by high light intensity (1000 lux) stimuli was significantly different compared to those constructed using low light intensity (100 lux). Moreover, a weak but significant correlation was observed between phase shifts and period changes when light stimuli of 1000 lux intensity were used; however, the phase shifts and period changes in the 100 lux PRC and τRC were not correlated. This suggests that the intensity of light stimuli affects both phase and period responses in the locomotor activity rhythm of the nocturnal field mouse M. booduga. These results indicate that complex mechanisms are involved in entrainment of circadian clocks, even in nocturnal rodents, in which PRC, τRC, and dose responses play a significant role.     

Melatonin-induced phase and dose responses in a diurnal mammal,Funambulus pennantii

ABSTRACT

Melatonin, an essential pineal hormone, acts as a marker of the circadian clock that regulates biological rhythms in animals. The effects of exogenous melatonin on the circadian system of nocturnal rodents have been extensively studied; however, there is a paucity of studies on the phase-resetting characteristics of melatonin in diurnal rodents. We studied the phase shifting effects of exogenous melatonin as a single melatonin injection (1 mg/kg) at various phases of the circadian cycle on the circadian locomotor activity rhythm in the palm squirrel, Funambulus pennantii. A phase response curve (PRC) was constructed. Adult male squirrels (N = 10) were entrained to a 12:12 h light-dark cycle (LD) in a climate-controlled chronocubicle with food and water provided ad libitum. After stable entrainment, squirrels were transferred to constant dark condition (DD) for free-running. Following stable free run, animals were administered a single dose of melatonin (1 mg/kg in 2% ethanol-phosphate buffered saline (PBS) solution) or vehicle (2% ethanol-PBS solution) at circadian times (CTs) 3 h apart to evoke phase shifts. The phase shifts elicited at various CTs were plotted to generate the PRC. A dose response curve was generated using four doses (0.5, 1, 2 and 4 mg/kg) administered at the CT of maximum phase advance. Melatonin evoked maximum phase advances at CT0 (1.23 ± 0.28 h) and maximum phase delays at CT15 (0.31 ± 0.09 h). In the dose response experiment, maximal phase shifts were evoked with 1 mg/kg. In contrast, no significant shifts were observed in control groups. Our study demonstrates that the precise timing and appropriate dose of melatonin administration is essential to maximize the amelioration of circadian rhythm–related disorders in a diurnal model.     

Fault tolerance in the cardiac ganglion of the lobster

Canavier et al. (1997) used phase response curves (PRCs) of individual oscillators to characterize the possible modes of phase-locked entrainment of an N-oscillator ring network. We extend this work by developing a mathematical criterion to determine the local stability of such a mode based on the PRCs. Our method does not assume symmetry; neither the oscillators nor their connections need be identical. To use these techniques for predicting modes and determining their stability, one need only determine the PRC of each oscillator in the ring either experimentally or from a computational model. We show that network stability cannot be determined by simply testing the ability of each oscillator to entrain the next. Stability depends on the number of neurons in the ring, the type of mode, and the slope of each PRC at the point of entrainment of the respective neuron. We also describe simple criteria which are either necessary or sufficient for stability and examine the implications of these results. Received: 2 April 1998 / Accepted in revised form: 2 July 1998     

Phase and period responses to short light pulses in a wild diurnal rodent,Funambulus pennanti

Photic phase response curves (PRCs) have been extensively studied in many laboratory-bred diurnal and nocturnal rodents. However, comparatively fewer studies have addressed the effects of photic cues on wild diurnal mammals. Hence, we studied the effects of short durations of light pulses on the circadian systems of the diurnal Indian Palm squirrel, Funambulus pennanti. Adult males entrained to a light–dark cycle (12?h–12?h) were transferred to constant darkness (DD). Free-running animals were exposed to brief light pulses (250 lux) of 15?min, 3 circadian hours (CT) apart (CT 0, 3, 6, 9, 12, 15, 18 and 21). Phase shifts evoked at different phases were plotted against CT and a PRC was constructed. F. pennanti exhibited phase-dependent phase shifts at all the CTs studied, and the PRC obtained was of type 1 at the intensity of light used. Phase advances were evoked during the early subjective day and late subjective night, while phase delays occurred during the late subjective day and early subjective night, with maximum phase delay at CT 15 (?2.04?±?0.23?h), and maximum phase advance at CT 21 (1.88?±?0.31?h). No dead zone was seen at this resolution. The free-running period of the rhythm was concurrently lengthened (deceleration) during the late subjective day and early subjective night, while period shortening (acceleration) occurred during the late subjective night. The maximum deceleration was noticed at CT 15 (?0.40?±?0.09?h) and the maximum acceleration at CT 21 (0.39?±?0.07?h). A significant positive correlation exists between the phase shifts and the period changes (r?=?0.684, p?=?0.001). The shapes of both the PRC and period response curve (τRC) qualitatively resemble each other. This suggests that the palm squirrel’s circadian system is entrained both by phase and period responses to light. Thus, F. pennanti exhibits robust clock-resetting in response to light pulses.     

Message from the President of the Society

The circadian rhythm of locomotor activity of the field mouse Mus booduga was studied and single animal phase response curves (PRCs) (n = 8) were constructed for 15-min daylight pulses of 1000 lux intensity. The light pulses, presented at different phases of the circadian cycle, evoked advancing and delaying phase shifts (ΔPHs) depending on the circadian time (CT) of light pulse application. ΔPHs by light pulses applied at the same phase are strongly correlated with the animals' circadian period (τ). The results indicate a significant correlation between (i) τ and the area under the advance zone of the PRC (A) (r = +0.72, p > 0.05), (ii) τ and the area under the delay zone of the PRC (D) (r = -0.98, p > 0.00001), (iii) τ and the difference between the area under delay and advance zone of PRC (D-A) (r = -0.97, p > 0.00001), and (iv) between τ and ΔpHs (at various phases of the circadian cycle) and further suggest that the waveform and time course of PRC depend on the animals' endogenous period (τ). (Chronobiology International, 13(6), 401–409, 1996)     

Phase response curves for social entrainment

Summary Phase shifts in free-running activity rhythms of male golden hamsters,Mesocricetus auratus, often occur when they establish a new territory and home after a cage change. Similar shifts also often occur after pairs of animals interact with each other for half an hour. When these events take place during the middle of the hamsters' subjective day, they produce phase advances: when late in the subjective night, they produce phase delays. Repeated social interactions at the same time of day can entrain activity rhythms in a way consistent with the shape of the phase response curves. Not all individuals become entrained, as is predictable from the modest amplitude of the phase response curve. The effects of social interactions and of other disturbances may be mediated through an oscillator phased by general arousal. The present findings have implications for the interpretation of drug-induced changes in biological rhythms.     

Davydov Soliton Dynamics in Proteins: I. Initial States and Exactly Solvable Special Cases

For the Davydov Hamiltonian several special cases are known which can be solved analytically. Starting from these cases we show that the initial state for a simulation using Davydovs |D₁> approximation has to be constructed from a given set of initial lattice displacements and momenta in form of a coherent state with its amplitudes independent of the lattices site, corresponding to Davydovs |D₂> approximation. In the |D₁> ansatz the coherent state amplitudes are site dependent. The site dependences evolve from this initial state exclusively via the equations of motion. Starting the |D₁> simulation from an ansatz with site dependent coherent state amplitudes leads to an evolution which is different from the analytical solutions for the special cases. Further we show that simple construction of such initial states from the expressions for displacements and momenta as functions of the amplitudes leads to results which are inconsistent with the expressions for the lattice energy. The site-dependence of coherent state amplitudes can only evolve through the exciton-phonon interactions and cannot be introduced already in the initial state. Thus also in applications of the |D₁> ansatz to polyacetylene always |D₂> type initial states have to be used in contrast to our previous suggestion [W. Förner, J. Phys.: Condens. Matter 1994, 6, 9089-9151, on p. 9105]. Further we expand the known exact solutions in Taylor serieses in time and compare expectation values in different orders with the exact results. We find that for an approximation up to third order in time (for the wave function) norm and total energy, as well as displacements and momenta are reasonably correct for a time up to 0.12-0.14 ps, depending somewhat on the coupling strengh for the transportless case. For the oscillator system in the decoupled case the norm is correct up to 0.6-0.8 ps, while the expectation values of the number operators for different sites are reasonably correct up to roughly 0.6 ps, when calculated from the third order wave function. The most important result for the purpose to use such expansions for controlling the validity of ansatz states is, however, that the accuracy of S(t) and H(t) (constant in time, exact values known in all cases) is obviously a general indicator for the time region in which a given expansion yields reliable values also for the other, physically more interesting expectation values.     

Behavioral responses to jamming and ‘phantom’ jamming stimuli in the weakly electric fish <Emphasis Type="Italic">Eigenmannia</Emphasis>

The jamming avoidance response (JAR) of the weakly electric fish Eigenmannia is characterized by upward or downward shifts in electric organ discharge (EOD) frequency that are elicited by particular combinations of sinusoidal amplitude modulation (AM) and differential phase modulation (DPM). However, non-jamming stimuli that consist of AM and/or DPM can elicit similar shifts in EOD frequency. We tested the hypothesis that these behavioral responses result from non-jamming stimuli being misperceived as jamming stimuli. Responses to non-jamming stimuli were similar to JARs as measured by modulation rate tuning, sensitivity, and temporal dynamics. There was a smooth transition between the magnitude of JARs and responses to stimuli with variable depths of AM or DPM, suggesting that frequency shifts in response to jamming and non-jamming stimuli represent different points along a continuum rather than categorically distinct behaviors. We also tested the hypothesis that non-jamming stimuli can elicit frequency shifts in natural contexts. Frequency decreases could be elicited by semi-natural AM stimuli, such as random AM, AM presented to a localized portion of the body surface, transient changes in amplitude, and movement of resistive objects through the electric field. We conclude that ‘phantom’ jamming stimuli can induce EOD frequency shifts in natural situations.     

Critical pulses of anisomycin drive the circadian oscillator inGonyaulax towards its singularity

Summary Dose and phase response curves for phase shifting the circadian oscillator in the dinoflagellateGonyaulax polyedra were measured with pulses of the antibiotic anisomycin (an inhibitor of protein synthesis on 80 S ribosomes), using the bioluminescent glow rhythm as the assay. The three dimensional surface of final phase, initial phase, and concentration was found to be a right handed helix, with the axis at a critical initial phase near circadian time 12 h, and critical concentration near 0.2 micromolar anisomycin (for 1 h pulses). The normally rhythmic glow of populations ofGonyaulax was significantly disrupted by pulses with these critical parameters, and in many instances appeared nearly arrhythmic.With increasing drug concentration, phase response curves appear to move bodily to earlier phases, and no saturation is evident in the phase shifting effect. These results are interpreted as indicating that anisomycin at sufficiently high doses causes an immediate strong (type 0) phase shift, then holds the clock stationary for a time interval that increases with concentration.the possibility that the 80 S ribosomal complex may be centrally involved in the fundamental circadian oscillation is put forward.Abbreviations DRC dose response curve - PRC phase response curve     

The effects of the general anaesthetic isoflurane on the honey bee (Apis mellifera) circadian clock

General anaesthesia administered during the day has previously been shown to phase shift the honey bee clock. We describe a phase response curve for honey bees (n=105) to six hour isoflurane anaesthesia. The honey bee isoflurane PRC is “weak” with a delay portion (maximum shift of –1.88 hours, circadian time 0 – 3) but no advance zone. The isoflurane-induced shifts observed here are in direct opposition to those of light. Furthermore, concurrent administration of light and isoflurane abolishes the shifts that occur with isoflurane alone. Light may thus provide a means of reducing isoflurane–induced phase shifts.