Cocaine modulates pathways for photic and nonphotic entrainment of the mammalian SCN circadian clock

Cocaine abuse is highly disruptive to circadian physiological and behavioral rhythms. The present study was undertaken to determine whether such effects are manifest through actions on critical photic and nonphotic regulatory pathways in the master circadian clock of the mouse suprachiasmatic nucleus (SCN). Impairment of SCN photic signaling by systemic (intraperitoneal) cocaine injection was evidenced by strong (60%) attenuation of light-induced phase-delay shifts of circadian locomotor activity during the early night. A nonphotic action of cocaine was apparent from its induction of 1-h circadian phase-advance shifts at midday. The serotonin receptor antagonist, metergoline, blocked shifting by 80%, implicating a serotonergic mechanism. Reverse microdialysis perfusion of the SCN with cocaine at midday induced 3.7 h phase-advance shifts. Control perfusions with lidocaine and artificial cerebrospinal fluid had little shifting effect. In complementary in vitro experiments, photic-like phase-delay shifts of the SCN circadian neuronal activity rhythm induced by glutamate application to the SCN were completely blocked by cocaine. Cocaine treatment of SCN slices alone at subjective midday, but not the subjective night, induced 3-h phase-advance shifts. Lidocaine had no shifting effect. Cocaine-induced phase shifts were completely blocked by metergoline, but not by the dopamine receptor antagonist, fluphenazine. Finally, pretreatment of SCN slices for 2 h with a low concentration of serotonin agonist (to block subsequent serotonergic phase resetting) abolished cocaine-induced phase shifts at subjective midday. These results reveal multiple effects of cocaine on adult circadian clock regulation that are registered within the SCN and involve enhanced serotonergic transmission.     

Double-pulse experiments with nonphotic and photic phase-shifting stimuli.

Three-hour pulses of novelty-induced wheel running in the early to middle subjective day of golden hamsters produced phase advances of 2-3 hr. This phase shifting could be almost totally abolished by a light pulse following within 3 hr of the exercise pulse. When light pulses occurred about 8 hr after the exercise pulses, the phase-advancing effects of the latter were enhanced. Consideration of the amplitude of the phase response curve (PRC) for light pulses alone, in the test paradigms used here, showed that nonphotic and photic phase shifts did not combine additively. Antagonistic and synergistic interactions between photic and nonphotic shifts may have to be taken into account if it transpires that exercise in people can be used to assist adjustment to new schedules after crossing time zones, or in shiftwork.     

Photoperiod modulates the effects of norepinephrine on lymphocyte proliferation in Siberian hamsters

Siberian hamsters (Phodopus sungorus) rely on photoperiod to coordinate seasonally appropriate changes in physiology, including immune function. Immunity is regulated, in part, by the sympathetic nervous system (SNS), although the precise role of the SNS in regulating photoperiodic changes in immunity remains unspecified. The goal of the present study was to examine the contributions of norepinephrine (NE), the predominant neurotransmitter of the SNS, to photoperiodic changes in lymphocyte proliferation. In experiment 1, animals were maintained in long [16:8-h light-dark cycle (16:8 LD)] or short days (8:16 LD) for 10 wk, and splenic NE content was determined. In experiment 2, in vitro splenocyte proliferation in response to mitogenic stimulation (concanavalin A) was assessed in spleen cell suspensions taken from long- or short-day hamsters in which varying concentrations of NE were added to the cultures. In experiment 3, splenocyte proliferation was examined in the presence of NE and selective alpha- and beta-noradrenergic receptor antagonists (phenoxybenzamine and propranolol, respectively) in vitro. Short-day animals had increased splenic NE content compared with long-day animals. Long-day animals had higher proliferation compared with short-day animals independent of NE. NE (1 microM) further suppressed splenocyte proliferation in short but not long days. Last, NE-induced suppression of proliferation in short-day hamsters was blocked by propranolol but not phenoxybenzamine. The present results suggest that NE plays a role in photoperiodic changes in lymphocyte proliferation. Additionally, the data suggest that the effects of NE on proliferation are specific to activation of beta-adrenergic receptors located on splenic tissue. Collectively, these results provide further support that photoperiodic changes in immunity are influenced by changes in SNS activity.     

Modeling interactions between photic and nonphotic entrainment mechanisms in transmeridian flights

In transmeridian flights, photic and nonphotic entrainment mechanisms are expected to interact dynamically in the human circadian system. In order to simulate the reentrainment process of the circadian rhythms, the photic entrainment mechanism was introduced to our previous model, which consisted of three coupled oscillators. Regardless of flight direction, a large time difference beyond 10 h tended to induce the antidromic reentrainment. The partition between the oscillators resulted for the eastward flight over a 10-h or longer time difference and the westward over 6 h or longer. The simulated reentrainment processes almost coincided with empirical knowledge. Simulated effects of physical exercise showed that some antidromic reentrainments were switched to the orthodromic ones for the eastward flight and most of the partitions between the oscillators were prevented in the westward flight. These results are due to an augmentation of the entrainment pressure of the rest–activity cycle on the oscillators. The mechanisms underlying these various reentrainment patterns were explained based on the photic response, the interactions between the oscillators, and their adaptive modification. The simulation results suggest that an appropriate selection of departure time and physical exercise could ease the jet lag caused by transmeridian flight.This research was supported by a Grant-In-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology of Japan (Nos. 14015205, 15014204, and 15560372).     

Photoperiod and food restriction differentially affect reproductive and immune responses in Siberian hamsters Phodopus sungorus

1.  Organisms must contend with seasonal fluctuations in energy availability. To maintain a positive energy balance year-round, a number of adaptations have evolved including seasonal changes in reproduction, energetics and immunity. Photoperiod is the primary environmental signal most animals use to predict seasonal events. Despite the established link between energetics and immune function, little is known regarding how changes in energy availability affect immunity.
2.  The goal of the present study was to determine the effects of food restriction on photoperiodic changes in reproduction and immune function in the Siberian hamster ( Phodopus sungorus ). Adult hamsters were housed in long or short days and were food restricted or fed ad libitum . Immune responses were quantified by measuring specific antibody production and bacterial killing capacity.
3.  Food restriction decreased body and relative reproductive masses in long-day animals. Antibody responses, but not bacterial killing ability, were enhanced in food restricted short-day animals as compared with ad libitum fed controls. We also found differential effects of body fat on immune responses depending on the immune measure.
4.  The effects of food restriction on immune function appear to vary based on the restriction regimen, the response measured, and the physiological state of the organism including energy balance, metabolic rate and reproductive status.
5.  In conclusion, these results suggest that a wide range of factors can differentially affect immune function. In addition, these effects may vary based on the specific response examined. Future studies should include a variety of measurements to provide a more integrative and accurate picture of reproductive, energetic, and photoperiodic effects on immune function.     

Vasopressin differentially modulates aggression and anxiety in adolescent hamsters administered anabolic steroids

Adolescent Syrian hamsters (Mesocricetus auratus) treated with anabolic/androgenic steroids display increased offensive aggression and decreased anxiety correlated with an increase in vasopressin afferent development, synthesis, and neural signaling within the anterior hypothalamus. Upon withdrawal from anabolic/androgenic steroids, this neurobehavioral relationship shifts as hamsters display decreased offensive aggression and increased anxiety correlated with a decrease in anterior hypothalamic vasopressin. This study investigated the hypothesis that alterations in anterior hypothalamic vasopressin neural signaling modulate behavioral shifting between adolescent anabolic/androgenic steroid-induced offensive aggression and anxiety. To test this, adolescent male hamsters were administered anabolic/androgenic steroids and tested for offensive aggression or anxiety following direct pharmacological manipulation of vasopressin V1A receptor signaling within the anterior hypothalamus. Blockade of anterior hypothalamic vasopressin V1A receptor signaling suppressed offensive aggression and enhanced general and social anxiety in hamsters administered anabolic/androgenic steroids during adolescence, effectively reversing the pattern of behavioral response pattern normally observed during the adolescent exposure period. Conversely, activation of anterior hypothalamic vasopressin V1A receptor signaling enhanced offensive aggression in hamsters exposed to anabolic/androgenic steroids during adolescence. Together, these findings suggest that the state of vasopressin neural development and signaling in the anterior hypothalamus plays an important role in behavioral shifting between aggression and anxiety following adolescent exposure to anabolic/androgenic steroids.     

Dexras1 potentiates photic and suppresses nonphotic responses of the circadian clock

Circadian rhythms of physiology and behavior are generated by biological clocks that are synchronized to the cyclic environment by photic or nonphotic cues. The interactions and integration of various entrainment pathways to the clock are poorly understood. Here, we show that the Ras-like G protein Dexras1 is a critical modulator of the responsiveness of the master clock to photic and nonphotic inputs. Genetic deletion of Dexras1 reduces photic entrainment by eliminating a pertussis-sensitive circadian response to NMDA. Mechanistically, Dexras1 couples NMDA and light input to Gi/o and ERK activation. In addition, the mutation greatly potentiates nonphotic responses to neuropeptide Y and unmasks a nonphotic response to arousal. Thus, Dexras1 modulates the responses of the master clock to photic and nonphotic stimuli in opposite directions. These results identify a signaling molecule that serves as a differential modulator of the gated photic and nonphotic input pathways to the circadian timekeeping system.     

Serotonergic mediation of constant light-potentiated nonphotic phase shifting of the circadian locomotor activity rhythm in Syrian hamsters

Short-term (1-3 days) constant light exposure (brief LL) potentiates nonphotic phase shifting induced by sleep deprivation and serotonin (5-HT) agonist stimulation. The present assessments reveal that exposure to brief LL markedly alters the magnitude and shape of the 5-HT1A,7 receptor agonist, 8-(+)2-dipropyl-amino-8-hydroxyl-1,2,3,4-tetrahyronapthalene (8-OH-DPAT) phase-response curve, facilitating (approximately 12 h) phase-advance shifts during the early morning when serotonergics have no phase-shifting effect. Brief LL also reduces the threshold for 8-OH-DPAT shifting at midday, evidenced by 5- to 6-h phase-advance shifts elicited by dosages that have no effect without the LL treatment. The brief LL-potentiated phase advances to intraperitoneal 8-OH-DPAT at zeitgeber time 0 (ZT 0) were blocked by the 5-HT1A antagonists, pindolol and WAY 100635, indicating that this shifting is mediated by 5-HT1A receptors. Antagonists with action at 5-HT7 receptors, including ritanserin and metergoline, were without effect. Although autoradiographic analyses of [3H]8-OH-DPAT binding indicate that brief LL does not upregulate suprachiasmatic nucleus (SCN) 5-HT1A receptor binding, intra-SCN microinjection of 8-OH-DPAT at ZT 0 in brief LL-exposed hamsters induced shifts similar to those produced by intraperitoneal injection, suggesting that SCN 5-HT1A receptors mediate potentiated 8-OH-DPAT-induced shifts during the early morning. Lack of shifting by intra-SCN 8-OH-DPAT at ZT 6 or 18 (when intraperitoneal 8-OH-DPAT induces large shifts), further indicates that brief LL-potentiated shifts at these time points are mediated by 5-HT target(s) outside the SCN. Significantly, sleep deprivation-induced phase-advance shifts potentiated by brief LL (approximately 9 h) at ZT 0 were blocked by pindolol, suggesting that these behavioral shifts could be mediated by the same SCN 5-HT1A receptor phase-resetting pathway as that activated by 8-OH-DPAT treatment.     

Photoperiod modulates the inhibitory effect of in vitro melatonin on lymphocyte proliferation in female Siberian hamsters.

In Siberian hamsters (Phodopus sungorus), short days suppress reproductive function and lymphocyte proliferation. To determine whether melatonin influences cell-mediated immunity through a direct action on lymphocyte proliferation, in vitro responsiveness to mitogens and melatonin was assessed in systemic and splenic lymphocytes from adult female Siberian hamsters housed in either long or short days for 13 weeks. Short days provoked reproductive regression and reduced lymphocyte proliferation. Physiological concentrations of melatonin (50 pg/ml) inhibited in vitro proliferation of circulating lymphocytes, whereas higher concentrations (> or = 500 pg/ml) were required to inhibit proliferation of splenic lymphocytes. Immunomodulatory effects of melatonin were restricted to lymphocytes from long-day hamsters-in vitro melatonin had no effect on circulating or splenic lymphocytes from females in short days. Responsiveness to melatonin in short-day lymphocytes may be restrained by the already expanded nightly pattern of melatonin secretion in short days. These data support the hypothesis that melatonin acts directly on lymphocytes from long-day hamsters to suppress blastogenesis.     

Inhibiting cortisol response accelerates recovery from a photic phase shift

Jetlag results when a temporary loss of circadian entrainment alters phase relationships among internal rhythms and between an organism and the outside world. After a large shift in the light-dark (LD) cycle, rapid recovery of entrainment minimizes the negative effects of internal circadian disorganization. There is evidence in the existing literature for an activation of the hypothalamic-pituitary-adrenal (HPA) axis after a photic phase shift, and it is possible that the degree of HPA-axis response is a determining factor of reentrainment time. This study utilized a diurnal rodent, Octodon degus, to test the prediction that the alteration of cortisol levels would affect the reentrainment rate of circadian locomotor rhythms. In experiment 1, we examined the effects of decreased cortisol (using metyrapone, an 11beta-hydroxylase inhibitor) on the rate of running-wheel rhythm recovery after a 6-h photic phase advance. Metyrapone treatment significantly shortened the length of time it took animals to entrain to the new LD cycle (11.5% acceleration). In experiment 2, we examined the effects of increased cortisol on the rate of reentrainment after a 6-h photic phase advance. Increasing plasma cortisol levels increased the number of days (8%) animals took to reentrain running-wheel activity rhythms, but this effect did not reach significance. A third experiment replicated the results of experiment 1 and also demonstrated that suppression of HPA activity via dexamethasone injection is capable of accelerating reentrainment rates by approximately 33%. These studies provide support for an interaction between the stress axis and circadian rhythms in determining the rate of recovery from a phase shift of the LD cycle.     

Temperature dependence of phase response curves for drug-induced phase shifts

The effectiveness of drugs active in phase-shifting the circadian rhythm of bioluminescent glow in the unicellular dinoflagellate Gonyaulax polyedra differs, depending upon the time of drug exposure (as pulses). For two drugs tested--cycloheximide and anisomycin, both inhibitors of cytosolic protein synthesis--this function, referred to as the drug phase response curve (dPRC), differs, depending upon the ambient temperature. Since dPRCs may differ at different drug concentrations, the effects observed may be attributable to differences in the effectiveness of or recovery from the drugs at different temperatures.     

Increased photic sensitivity for phase resetting but not melatonin suppression in Siberian hamsters under short photoperiods

Light regulates a variety of behavioral and physiological processes, including activity rhythms and hormone secretory patterns. Seasonal changes in the proportion of light in a day (photoperiod) further modulate those functions. Recently, short (SP) versus long days (LP) were found to markedly increase light sensitivity for phase shifting in Syrian hamsters. To our knowledge, photoperiod effects on light sensitivity have not been studied in other rodents, nor is it known if they generalize to other circadian responses. We tested whether photic phase shifting and melatonin suppression vary in Siberian hamsters maintained under LP or SP. Select irradiances of light were administered, and shifts in activity were determined. Photic sensitivity for melatonin suppression was examined in a separate group of animals via pulses of light across a 4 log-unit photon density range, with post-pulse plasma melatonin levels determined via RIA. Phase shifting and melatonin suppression were greater at higher irradiances for both LP and SP. The lower irradiance condition was below threshold for phase shifts in LP but not SP. Melatonin suppression did not vary by photoperiod, and the half saturation constant for fitted sigmoid curves was similar under LP and SP. Thus, the photoperiodic modulation of light sensitivity for phase shifting is conserved across two hamster genera. The dissociation of photoperiod effects on photic phase shifting and melatonin suppression suggests that the modulation of sensitivity occurs downstream of the common retinal input pathway. Understanding the mechanistic basis for this plasticity may yield therapeutic targets for optimizing light therapy practices.     

Twilights widen the range of photic entrainment in hamsters

The range of entrainment of the circadian rhythm of locomotor activity was compared in four groups of Syrian hamsters (eight animals per group) initially exposed to daily light-dark (LD) cycles with either abrupt transitions between light and darkness (LD-rectangular) or simulated twilights (LD-twilight). Lighting was provided by arrays of white light-emitting diodes; daytime illuminance (10 lux) and the total amount of light emitted per day were the same in the two conditions. The period (T) of the LD cycles was then gradually increased to 26.5 h or gradually decreased to 21.5 h, at the rate of 5 min/day. Under LD-rectangular, the upper and lower limits of entrainment were 25.0 to 25.5 h and 22.0 to 22.5 h, respectively, whereas under LD-twilight, 50% of the animals exposed to the lengthening cycles were still entrained at T = 26.5 h and 50% of those exposed to the shortening cycles were still entrained at T = 21.5 h. In a second experiment, two groups of hamsters were exposed to fixed T = 25 h LD-rectangular (n = 15) or LD-twilight cycles (n = 7). Only 33% of the animals entrained in LD-rectangular, whereas 86% of the animals entrained in LD-twilight. Free-running periods in constant darkness were longer following successful entrainment to T = 25 h but did not differ between the animals that entrained to LD-rectangular and those that entrained to LD-twilight. The widening of the range of entrainment observed in LD-twilight indicates that twilight transitions increase the strength of the LD zeitgeber. In LD-twilight, successful entrainment to T = 26.5 h was accompanied by an expansion of activity time to 16.52+/-1.22 h, with activity onsets preceding mid-dusk by 12.56+/-2.15 h. Together with earlier data showing similar phase response curves for hour-long dawn, dusk, and rectangular light pulses, these results suggest that the effect of twilights on the range of entrainment may involve parametric rather than nonparametric mechanisms.     

Non-parametric photic entrainment of Djungarian hamsters with different rhythmic phenotypes

To investigate the role of non-parametric light effects in entrainment, Djungarian hamsters of two different circadian phenotypes were exposed to skeleton photoperiods, or to light pulses at different circadian times, to compile phase response curves (PRCs). Wild-type (WT) hamsters show daily rhythms of locomotor activity in accord with the ambient light/dark conditions, with activity onset and offset strongly coupled to light-off and light-on, respectively. Hamsters of the delayed activity onset (DAO) phenotype, in contrast, progressively delay their activity onset, whereas activity offset remains coupled to light-on. The present study was performed to better understand the underlying mechanisms of this phenomenon. Hamsters of DAO and WT phenotypes were kept first under standard housing conditions with a 14:10 h light–dark cycle, and then exposed to skeleton photoperiods (one or two 15-min light pulses of 100 lx at the times of the former light–dark and/or dark–light transitions). In a second experiment, hamsters of both phenotypes were transferred to constant darkness and allowed to free-run until the lengths of the active (α) and resting (ρ) periods were equal (α:ρ = 1). At this point, animals were then exposed to light pulses (100 lx, 15 min) at different circadian times (CTs). Phase and period changes were estimated separately for activity onset and offset. When exposed to skeleton-photoperiods with one or two light pulses, the daily activity patterns of DAO and WT hamsters were similar to those obtained under conditions of a complete 14:10 h light–dark cycle. However, in the case of giving only one light pulse at the time of the former light–dark transition, animals temporarily free-ran until activity offset coincided with the light pulse. These results show that photic entrainment of the circadian activity rhythm is attained primarily via non-parametric mechanisms, with the “morning” light pulse being the essential cue. In the second experiment, typical photic PRCs were obtained with phase delays in the first half of the subjective night, phase advances in the second half, and a dead zone during the subjective day. ANOVA indicated no significant differences between WT and DAO animals despite a significantly longer free-running period (tau) in DAO hamsters. Considering the phase shifts induced around CT0 and the different period lengths, it was possible to model the entrainment patterns of both phenotypes. It was shown that light-induced phase shifts of activity offset were sufficient to compensate for the long tau in WT and DAO hamsters, thus enabling a stable entrainment of their activity offsets to be achieved. With respect to activity onsets, phase shifts were sufficient only in WT animals; in DAO hamsters, activity onset showed increasing delays. The results of the present paper clearly demonstrate that, under laboratory conditions, the non-parametric component of light and dark leads to circadian entrainment in Djungarian hamsters. However, a stable entrainment of activity onset can be achieved only if the free-running period does not exceed a certain value. With longer tau values, hamsters reveal a DAO phenotype. Under field conditions, therefore, non-photic cues/zeitgebers must obviously be involved to enable a proper circadian entrainment.     

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.     

Robust entrainment of circadian oscillators requires specific phase response curves

The circadian clocks keeping time in many living organisms rely on self-sustained biochemical oscillations entrained by external cues, such as light, to the 24-h cycle induced by Earth's rotation. However, environmental cues are unreliable due to the variability of habitats, weather conditions, or cue-sensing mechanisms among individuals. A tempting hypothesis is that circadian clocks have evolved so as to be robust to fluctuations in the signal that entrains them. To support this hypothesis, we analyze the synchronization behavior of weakly and periodically forced oscillators in terms of their phase response curve (PRC), which measures phase changes induced by a perturbation applied at different times of the cycle. We establish a general relationship between the robustness of key entrainment properties, such as stability and oscillator phase, on the one hand, and the shape of the PRC as characterized by a specific curvature or the existence of a dead zone, on the other hand. The criteria obtained are applied to computational models of circadian clocks and account for the disparate robustness properties of various forcing schemes. Finally, the analysis of PRCs measured experimentally in several organisms strongly suggests a case of convergent evolution toward an optimal strategy for maintaining a clock that is accurate and robust to environmental fluctuations.     

Serotonin modulates a photic response in circadian locomotor rhythmicity of adults of the blow fly, Calliphora vicina

Abstract. The present experiments were undertaken to explore a role for serotonin (5-hydroxytryptamine, 5-HT) in modulating photic signal transduction in photoreceptors of the blow fly, Calliphora vicina. Injection of p-chlorophenylalanine (pCPA) into the haemolymph appeared to reduce sensitivity to the photic effects of constant ‘bright’ light (LL hyperactivity and circadian arrhythmicity). After drug injection in bright LL, flies continued with a free-running rhythm as in constant darkness (DD) or with a lengthened period τ as in ‘dim’ LL. When 5-HT was injected into flies kept in dim LL, they became hyperactive and arrhythmic as in bright LL. This finding suggests a potential role for serotonin as mediator of circadian changes in the insect visual system including extraretinal photoreceptors.     

Circadian phase response curves for dark pulses in the hamster

Summary Hamsters maintained under constant illumination were exposed to 2- or 6-h pulses of darkness at various phases of their circadian activity rhythms. When presented around the time of activity onset, the pulses resulted in phase advances, and when presented toward the end of daily activity, they resulted in phase delays. Since others have shown that light pulses presented at the same phases in constant darkness cause phase shifts in the opposite directions, these results indicate that phase response curves for light and dark pulses are mirror images.Dark pulses also caused phase-dependent changes, both transient and long-lasting, in the period of the free-running rhythms, and a few pulses were immediately followed by splitting of the activity rhythms into two components. Such effects may reflect a differential responsiveness of two coupled oscillators to dark pulses.Abbreviations CT circadian time - DD constant dark - LD lightdark - LL constant light - PRC phase response curve - SD subjective day - SN subjective night - period of a circadian rhythm Supported by grants from the NSERC of Canada to B. Rusak and to G.V. Goddard. We are grateful to Dr. Goddard for his support and encouragement     

The Paramecium circadian behavioral rhythm: light phase response curves and entrainment

The population of a ciliate protozoan, Paramecium multimicronucleatum, exhibits a circadian rhythm as measured by the number of the cells traversing an observation point ("traverse frequency," or TF). The present study examined phase shifting of the TF rhythm by administering 2-hr light pulses at different phases of the circadian cycle to cultures free-running in constant darkness (DD). The results were summarized in a phase response curve (PRC), categorized as Type 1. This PRC indicated a relatively narrow phase zone insensitive to the light pulse ("dead zone"). Entrainment of the rhythm to light pulses repeated at 24-hr intervals was also examined, and it was found that the rhythm gradually reached a steady state, following several transient cycles, with the pulses falling at a phase corresponding to the narrow dead zone. Such a steady-state rhythm, with a minimum at approximately 3 hr after the pulse and a maximum at approximately 12 hr after the pulse, was mathematically simulated by superimposing a response function to the pulse on a sinusoidal function representative of the free-running rhythm in DD.