intensity-dependent phase-adjustments in the locomotor activity rhythm of the nocturnal field mouse...

444 V.K. SHARMA AND R. CHIDAMBARAMJOURNAL OF EXPERIMENTAL ZOOLOGY 292:444–459 (2002)DOI 10.1002/jez.10084

© 2002 WILEY-LISS, INC.

Intensity-Dependent Phase-Adjustments in theLocomotor Activity Rhythm of the Nocturnal FieldMouse Mus booduga

VIJAY KUMAR SHARMA1* AND R. CHIDAMBARAM2

1Chronobiology Laboratory, Evolutionary and Organismal Biology Unit,Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur,Bangalore 560 064, Karnataka, India

2Department of Animal Behaviour and Physiology, School of BiologicalSciences, Madurai Kamaraj University, Madurai 625 021, Tamil Nadu,India

ABSTRACT The locomotor activity rhythm of the nocturnal field mouse Mus booduga wasmonitored under constant darkness (DD) and free-running periods (τ) were estimated. Following afree-run of about 15 days in DD, the animals were exposed to periodic light pulses (LPs) of variousintensities (1 lux, 10 lux, 50 lux, 100 lux, and 1,000 lux) and 15 minutes duration for 65 days atintervals of 24 hours to investigate the influence of intensity of light on the phase-angle-difference(ψ) between the onset of locomotor activity and the time of LP administration. The experimentallyobserved values of ψ and τ for a LP of 1,000 lux intensity used for 15 minutes every 24 hr, showeda sigmoid shaped relationship with τ. This relationship was similar to that predicted based on thenonparametric model of entrainment, which uses the τ and the LP phase response curve (PRC)constructed using LP of similar duration and intensity. The functional nature of the relationshipbetween ψ and τ was not found to change significantly with increasing intensities of LP used toentrain the locomotor activity rhythm. However, ψ was significantly modulated by the intensity ofLP. These results suggest that the periodic sensitivity of the circadian pacemaker underlying thelocomotor activity rhythm in the nocturnal field mouse M. booduga to LPs plays an important rolein maintaining a characteristic ψ with the zeitgeber and the ψ changes in a light intensity-depen-dent manner. J. Exp. Zool. 292:444–459, 2002. © 2002 Wiley-Liss, Inc.

In addition to the free-running nature of circa-dian rhythms, another characteristic feature ofan organism’s circadian organization is its capac-ity to adjust its phase so as to synchronize its“internal time” with “environmental or externaltime.” In such a state of entrainment, the circa-dian clock acquires a stable phase-relationship (ψ)with the environmental cycle, and each of theovert rhythms adopts a characteristic phase-re-lationship with the zeitgeber. Besides variationsin biotic and abiotic factors, these responses mayvary within and among species.

The presence of a zeitgeber, day-to-day insta-bilities in the free-running period (τ), and acutechanges in photoperiods may present a circadianclock with challenges in preserving an optimal ψ.The general waveform of light-induced phase re-sponse curves (PRCs) suggests that ψ should be anonlinear function of τ. Pittendrigh and Daan (’76)made such a prediction for the relationship be-tween ψ and τ in the rodent Peromyscus leucopus.

According to their prediction, when τ deviates sig-nificantly from the zeitgeber period (Τ), ψ shouldshow small changes for large deviations in τ, whilefor τ close to Τ, changes in ψ should be large.Therefore, ψ not only reflects the relationship be-tween τ and Τ but also suitably amplifies the con-sequences of small day-to-day variations in τ whenit is close to Τ. The prediction was that ψ shouldtherefore be a function of τ in DD, measured be-fore entraining zeitgeber was introduced, Τ, andthe PRC (Pittendrigh and Daan, ’76). However,empirical observations on the relationship be-tween ψ and τ did not generate enough supportfor this prediction. What was indeed observed wasthat in a few species of mammals, ψ and τ were

*Correspondence to: Vijay Kumar Sharma, Chronobiology Labora-tory, Evolutionary and Organismal Biology Unit, Jawaharlal NehruCentre for Advanced Scientific Research, Jakkur, PO Box 6436, Ban-galore 560 064, Karnataka, India. E-mail: [email protected]

Received 3 April 2001; Accepted 28 September 2001

PHASE-ADJUSTMENT IN MUS BOODUGA 445

negatively correlated (Pittendrigh and Daan,’76;Aschoff, ’94; Sharma et al., ’98). It was furtherreported that such dependency became weaker forhigher zeitgeber strength (Hoffmann, ’69; Pitten-drigh and Daan, ’76). However, both these predic-tions are based on data obtained from only a fewstudies.

In this paper, we report results from experi-ments performed on the nocturnal field mouseMus booduga, aimed at investigating the relation-ship between ψ (defined as the duration betweenthe onset of locomotor activity rhythm and theend of light pulse used as zeitgeber) and τ. Wehave also investigated the role of zeitgeberstrength by increasing intensity of light in modu-lating such relationships. Although we do nothave precise information about the activity timeof these mice in their natural habitat, they aregenerally found to come out of their burrows af-ter sunset and return before sunrise. The activ-ity pattern of these mice is therefore that of atypical nocturnal animal. As these animals spendmost of the daylight hours in light-excluding bur-rows, it is likely that they are exposed to lightonly during twilight hours. Estimation of the ψvalues under LD cycles, where the light phaselasts for a brief period, for individuals with dif-ferent τ values, can give us a better insight tothe mechanism of entrainment of the circadianlocomotor activity rhythm of the field mouse M.booduga to natural LD cycles.

MATERIALS AND METHODSAdult male field mice (approximate age greater

than 90 days), M. booduga (n = 72), were cap-tured from fields near the Madurai Kamaraj Uni-versity campus (9°58′ N lat., 78°10′ E long.). Thelocomotor activity of these animals in laboratoryconditions was monitored using an activity run-ning wheel (diameter ∼20 cm) attached to a trans-parent plexiglass cage of dimension 15 × 10 × 8cm, with a small opening of 0.02 m diameter. Theanimals were kept in the same activity-monitor-ing cage throughout the experiments, which in-cluded pre-and post-entrainment free-runs as wellas the duration of entrainment to LP. Reed-relaysattached to the wheels activated the writingstylets of an Esterline Angus A620X Event Re-corder when the running mice caused revolutionsof the wheel. The activity patterns of as many as20 mice in separate running wheels, placed onopen shelves in the experimental room, could beassayed concurrently. This setup was consideredto be safe because we have never observed any

kind of interaction between the animals that werekept on open shelves in any LD regime. The tem-perature inside the experimental rooms (3.05 m× 2.44 m × 4.01 m), which did not have windowsbut were gently ventilated, remained constant at25° ± 1°C and the relative humidity was 75 ± 5%.Food (millet and grain) and water were availablead libitum. The room was entered at irregular in-tervals on an average once in two days for pur-poses of cleaning cages, placing food and water,etc. Care was taken that the animals were notdisturbed except for purposes like feeding andcleaning cages, which seldom lasted beyond 5–10min. Red light of λ > 640 nm obtained with a com-bination of red and orange filters fitted to a bat-tery operated torch light was used inside thecubicle. Double plotted actograms were obtainedby pasting 24 hr activity/rest strips chronologi-cally one below the other.

The animals were first introduced into DD forabout 15 days wherein their τ was measured. Fol-lowing a free-run in DD of about 15 days, theseanimals were divided into groups consisting of about15 mice each and introduced to LD 0.25:23.75 hrfor 65 days, wherein LPs of 1 lux, 10 lux, 50 lux,100 lux, and 1,000 lux intensity, respectively, wereadministered, for a duration of 15 min. Fluorescentlamps (Philips, TL 40W/54 6500°K B05) were usedas light sources in all the experiments and the lightintensities at cage level was measured using UnitedDetector Technology (UDT; Hawthorne, CA), optom-eter. These lamps were also used to construct theLP PRC (Sharma et al., ’97), which is used to pre-dict the relationship between ψ and τ, which is usedas the predicted relationship in this paper. The de-sired light intensities were obtained by placing theanimals (along with the activity cage) at an appro-priate distance from the light source. The LPs wereinitiated at 10:00 hr every day which lasted for 15min. Finally the animals were brought back to DDand the post-entrainment τ was measured. Stableentrainment was assumed to occur when the slopeof the regression line passing through the onsets ofactivity in the presence of LD cycle was not signifi-cantly different from the slope of the vertical. Theonset of activity for the last 20 days in the LD cyclewas used to estimate ψ, giving enough time for tran-sients to subside. The onset of activity was used forcomputation of τ during free-runs and ψ duringentrainment. The ψ of the animals in LD is ex-pressed as the difference in time between the ac-tivity onset and end of LP. The ψ is taken as positivewhen the activity onset anticipated the LP, andnegative when it followed the LP.

446 V.K. SHARMA AND R. CHIDAMBARAM

For each of the observed values of τ, the ψ re-quired for stable entrainment was determinedbased on the nonparametric model of entrainment(Pittendrigh and Daan, ’76), using the light-in-duced PRC for the same species of mouse, con-structed using LP of 15 min duration and 1,000lux intensity (Sharma et al., ’97). The PRC hasthe shape of a standard PRC, which can be char-acterized by large phase delays during the earlysubjective night and large phase advances dur-ing the late subjective night. Maximum phase de-lay was observed at CT14 (an average phase delayof ∼ 1.96 hr) and maximum phase advance wasobserved at CT20 (an average phase advance of0.69 hr). The major part of the subjective day wasmarked by small but significant phase delays. ThePRC has no region, which can be referred to as atrue “dead zone.” To quantify the relationship be-tween ψ and τ, we fitted various nonlinear func-tions capable of yielding sigmoid curves to thesedata. The equation that gave the best fit (R2 =0.95; Fig. 1), while keeping the number of inde-pendent parameters relatively low, was:

ψ = a + b / {1+ exp (c – τ) / d} (I).

Here the parameter ‘a,’ which is the period-in-dependent term of the equation, indicates theposition of the sigmoid along the ψ axis, whereas‘b,’ ‘c,’ and ‘d’ indicate vertical spread of the sig-moid, the point of inflection, and the steepnesswith which ψ decreases with increasing τ afterthe point of inflection, respectively. The period-independent term of the equation-I (a) representsthe phase-relationship between the onset of ac-tivity and the LP, mediated through the speciesaverage PRC, whereas, the period-dependent term(b / {1 + exp (c – τ) / d}) appears to maintain afunctional relationship between ψ and τ. The pa-rameters ‘c’ and ‘d’ can be thought of as repre-senting the strength of dependence of ψ on τ,wherein a steeper decline of ψ with increasing τis taken to imply a stronger relationship betweenthese two variables. Moreover, the degree to whichthe relationship between ψ and τ is consistentacross individual animals can be assessed by thecoefficient of determination (R2). Our goal here isto test whether the data on ψ and τ gathered dur-ing the current study are consistent with the pre-dicted relationship (equation-I), and to examinethe effects, if any, of light intensity on the rela-tionship between ψ and τ. We approach this is-sue by fitting equation-I to the data obtained fromthe experiments carried out at each of the four

light intensities, and looking for any pattern oflight-intensity dependent changes in any of thefour parameters of this equation. In order to com-pare the ψ-τ curves obtained from experimentsdone at the five intensities of light and the light-induced PRC, confidence limits were constructedaround the mean values of the parameters (a, b,c, and d) by using the bootstrap (Sokal and Rohlf,’97) to generate multiple data sets of ψ and τ val-ues (six sets for each light intensity) by re-sampling without replacement from the sets ofexperimentally obtained ψ and τ values at eachlight intensity. Equation (I) was then fitted toeach resampled data set, and the parameters a,

Fig. 1. Relationship between τ and ψ obtained by using aPRC constructed using light pulses (LP) of 1,000 lux inten-sity, for a duration of 15 min (a). The smooth curve that ex-pressed τ as a function of ψ is sigmoid, and was chosenamongst many nonlinear equations based on goodness-of-fit.Experimentally obtained data on τ and ψ using recurrent lightpulses (LP) of 1 lux, 10 lux, 50 lux, 100 lux, and 1,000 luxintensities used for a duration of 15 min every 24 hr (b–f).The mean values of the parameters (a, b, c, and d) used todraw the solid curve were estimated by using the bootstrapthat generated multiple data sets of τ and ψ values (six setsfor each light intensity) by resampling without replacementfrom the sets of experimentally obtained τ and ψ values ateach light intensity.


b, c, and d and the R2 values were estimated, us-ing Table Curve (AISN Software, ’96).

RESULTSThe τ in DD was found to lie between 23.30 hr

and 24.25 hr and all mice showed stable entrain-ment to periodic LP administered every 24 hr. Theψ showed a sigmoid relationship with τ at all thefive intensities of LPs used (1 lux, 10 lux, 50 lux,100 lux, and 1,000 lux; Fig. 1). Change in ψ with τwas very small for values of τ deviating from 24hr, whereas ψ changed drastically with τ, for val-ues of τ close to 24 hr. Such a nonlinearity in therelationship between ψ and τ was observed at allintensities of LP (Fig. 1). The experimentally ob-tained relationship between ψ and τ as describedby the parameters a, b, and d did not differ signifi-cantly from the predicted relationship for a LP of1,000 lux intensity and 15 minute duration (P >0.05). The only parameter which was estimatedfrom the experimentally obtained data on ψ and τ,and was found to be significantly different fromits predicted value, was c (P < 0.0001). For all fiveintensities of LP, the data gave a close fit to theabove curve (with R2 values of 0.83, 0.89, 0.85, 0.95,and 0.95, respectively; Fig. 1; Table 1). A linear re-gression of the R2 values, and the model param-eters a and b versus the logarithm of intensity ofLP was found to be significant (P < 0.05), indicat-ing that the goodness of fit of the model (equa-tion-I) and two of the model parameters weresystematically affected by light intensity (Fig. 2).The correlation between the model parameters cand d and the logarithm of intensity of LP was notfound to be significant (P > 0.05). The activity dataof two mice out of a total of about 15, at variousintensities of LP, are illustrated in Figs. 3, 4, 5, 6,and 7 and give details of the responses of the cir-cadian system of M. booduga to light exposure.

TABLE 1. The values of the parameters a, b, c, and d of the equation y = a + b/{1 + exp (c-t)/d}1

Source of the data between τ and ψ Values of the degree of determination (R2) and the(derived either using PRC or parameters of the sigmoid (ψ = a + b/(1 + exp (c-τ)/d))light pulses of intensity given below) R2 a b c d

Predicted 0.95 –11.557 12.3867 24.0942 –0.0029 1 lux 0.83 –7.875 7.37333 23.9533 –0.0177

10 lux 0.89 –8.485 8.285 23.88 –0.0382 50 lux 0.84 –9.245 7.75167 23.9783 –0.0063

100 lux 0.95 –11.685 10.1383 24.03 –0.00671,000 lux 0.95 –10.847 9.25333 23.96 –0.00371These parameters were estimated after fitting the experimentally obtained data on ψ and τ into the above equation. The mean values of theparameters (a, b, c, and d) given in the talble were estimated by using the bootstrap that generated multiple data sets of τ and ψ values (6sets for each light intensity) by resampling without replacement from the sets of experimentally obtained τ and ψ values at each lightintensity.

Fig. 2. Illustrates the association between the R2 valueand the model parameters a, b, c, and d with the intensityof light (expressed in log lux). The values of the parameters(a, b, c, and d) were estimated by using the bootstrap thatgenerated multiple data sets of τ and ψ values (six sets foreach light intensity) by resampling without replacement fromthe sets of experimentally obtained τ and ψ values at eachlight intensity.


Fig. 3a–b. Free-running and entrainment data on thelocomotor activity rhythm of two adult male mice. Thesemice were first maintained in constant darkness (DD) forabout 15 days and then administered recurrent light pulses(LP) of 1 lux intensity for 15 min duration, initiated at10:00 hr every 24 hr for 65 days (indicated by a box) andthen restored back to DD. The lines drawn through the

onsets of activity do not represent the regression line usedfor analysis of data. They are drawn to give a visual im-pression of the approximate trend the locomotor activityrhythm takes during the free-running and entrained con-ditions. The pre- and post-entrainment τ of animals ad-ministered LPs were 23.8 ± 0.20 hr and 23.90 ± 0.07 hr.The ψ of the onset of activity was –0.69 ± 0.15 hr.

a


Fig. 3b.

b


Fig. 4a–b. Free-running and entrainment data on the lo-comotor activity rhythm of two adult male mice. These micewere first maintained in constant darkness (DD) for about15 days and then administered recurrent light pulses (LP)of 10 lux intensity for 15 min duration, initiated at 10:00 hrevery 24 hr for 65 days (indicated by a box) and then re-stored back to DD. The lines drawn through the onsets of

activity do not represent the regression line used for analy-sis of data. They are drawn to give a visual impression ofthe approximate trend the locomotor activity rhythm takesduring the free-running and entrained conditions. The pre-and post-entrainment τ of animals administered LP were23.61 ± 0.48 hr and 23.53 ± 0.41 hr. The ψ of the onset ofactivity was –0.31 ± 0.12 hr.

a


Fig. 4b.

b


Fig. 5a–b. Free-running and entrainment data on the lo-comotor activity rhythm of two adult male mice. These micewere first maintained in constant darkness (DD) for about15 days and then administered recurrent light pulses (LP) of50 lux intensity for 15 min duration, initiated at 10:00 hrevery 24 hr for 65 days (indicated by a box) and then re-stored back to DD. The lines drawn through the onsets of

activity do not represent the regression line used for analy-sis of data. They are drawn to give a visual impression of theapproximate trend the locomotor activity rhythm takes dur-ing the free-running and entrained conditions. The pre- andpost-entrainment τ of animals administered LP were 23.91 ±0.54 hr and 23.43 ± 0.57 hr. The ψ of the onset of activitywas –9.03 ± 0.32 hr.

a


Fig. 5b.

b


Fig. 6a–b. Free-running and entrainment data on the lo-comotor activity rhythm of two adult male mice. These micewere first maintained in constant darkness (DD) for about15 days and then administered recurrent light pulses (LP) of100 lux intensity for 15 min duration, initiated at 10:00 hrevery 24 hr for 65 days (indicated by a box) and then re-stored back to DD. The lines drawn through the onsets of

activity do not represent the regression line used for analy-sis of data. They are drawn to give a visual impression of theapproximate trend the locomotor activity rhythm takes dur-ing the free-running and entrained conditions. The pre- andpost-entrainment τ of animals administered LP were 24.18 ±0.16 hr and 24.41 ± 0.48 hr. The ψ of the onset of activitywas –10.95 ± 0.48 hr.

a


Fig. 6b.

b


Fig. 7a–b. Free-running and entrainment data on the lo-comotor activity rhythm of two adult male mice. These micewere first maintained in constant darkness (DD) for about15 days and then administered recurrent light pulses (LP) of1,000 lux intensity for 15 min duration, initiated at 10:00 hrevery 24 hr for 65 days (indicated by a box) and then re-stored back to DD. The lines drawn through the onsets ofactivity do not represent the regression line used for analy-

a

sis of data. They are drawn to give a visual impression of theapproximate trend the locomotor activity rhythm takes dur-ing the free-running and entrained conditions. The pre- andpost-entrainment τ of animals administered LP were (a) 23.82± 0.16 hr and 23.60 ± 0.28 hr; and (b) 24.16 ± 0.31 hr and24.08 ± 0.93 hr. The ψ values for a, and b were –1.2 ± 0.42 hrand –8.95 ± 0.28 hr respectively.


Fig. 7b.

b


DISCUSSIONThe relationship between the phase-angle-dif-

ference (ψ) of the locomotor activity rhythm andthe free-running period (τ) was found to follow asigmoid relationship that can be described byequation-I. The change in ψ was drastic (of theorder of 8 hr) for τ close to 24 hr, suggesting thatwhen τ was close to 24 hr, small variations in τwere sometimes associated with big alterationsin ψ, whereas ψ changed very little due to compa-rable changes in τ, for τ deviating from 24 hr. Theresults of our experiment closely match the na-ture of the relationship predicted between ψ andτ based on the nonparametric model of entrain-ment, using LP PRC and the distribution of τ (Fig.1). The difference between the experimentally ob-tained value of c (which reflects the point at whichψ changes drastically for a small change in τ) andits predicted value, indicates that perhaps thereis some role of the period response curve in addi-tion to the contributions of PRC and τ, in main-taining a stable phase relationship with thezeitgeber during entrainment (Beersma et al., ’99).Such changes in τ can also be seen in our experi-ment. The τ values estimated for the post-entrain-ment free-run were found to be different from thepre-entrainment τ; however, it did not show anyconsistent pattern. Neither did it depend on thepre-entrainment τ nor on the intensity of LP. Af-ter a comparison of the parameters a, b, c, and d,we found that the shape of the predicted curveconstructed using the LP-PRC (Fig. 1) qualita-tively resembles the experimentally observed re-lationship using 1 lux, 10 lux, 50 lux, 100 lux,and 1,000 lux intensities (Figs. 1,2; Table 1). TheR2 value and the parameters a and b showed asignificant increase/decrease with increase in lightintensity between a range of 1 lux to 1,000 lux.This would imply that, although the form of thefunctional relationship between ψ and τ remainsunchanged with increase in light intensity, at leastup to 1,000 lux, its consistency across individualanimals increases with increasing intensity oflight. Such changes suggest that the strength ofrelationship between ψ and τ (Pittendrigh andDaan, ’76) does not get modified because of in-creasing light intensity. The relationship also re-flects the waveform of a general PRC, suggestingthat the LP PRC does have a major role to playin determining ψ (Figs. 1,2). Based on the rela-tionship between ψ and τ as given in the equa-tion-I, the contributions to ψ can also be split intotwo parts: the period-independent part given as

the parameter a and the period-dependent partgiven as b / {1 + exp (τ – c) / d}. Although thechanges in the period-independent part of theequation, a, were significant, the period-dependentpart did not show any significant change withlight intensity. The changes in ψ can stem fromtwo sources: (1) light-intensity dependent increasein the amplitude of LP PRC (Sharma et al., ’99);and (2) modification in τ during entrainment (anafter-effect of entrainment). The fact that thetime-independent part of the equation-I showedsignificant change with the intensity of LP indi-cates that light intensity changes the ψ, and thatsuch changes may be mediated through the LPPRC. The functional nature of the relationshipbetween ψ and τ showed no significant changewith light intensity. Previous reports, which at-tempted to investigate such relationships only,showed a negative correlation between ψ and τ(Pittendrigh and Daan, ’76; Aschoff, ’94; Sharmaet al., ’98). Moreover, the fact that such a nonlin-ear relationship between ψ and τ also holds goodat higher intensities of LP (at least up to 1,000lux) is in some way contrary to previous findingsthat the relationship between ψ and τ becomesweaker for stronger stimuli (Hoffmann, ’69;Pittendrigh and Daan, ’76). However, in thepresent experiment, the intensity of the zeitge-ber, rather than the duration of LPs, was varied.In one of our previous studies on the same spe-cies of mice, the relationship between ψ and τ wasfound to be negatively correlated under semi-natu-ral conditions (Sharma et al., ’98). This suggeststhat entrainment of the locomotor activity rhythmto LD cycle, in this species of mouse, can not beentirely explained by the nonparametric entrain-ment model, and entrainment to brief light pulsesof 15 min duration cannot completely explain theentrainment to complete light phase of a naturalLD cycle.

Nocturnal animals are rarely exposed to completedaylight intensities. Many night-active animalsspend most of the daylight hours in light-excludingshelters, emerging at dusk and returning to theirshelters at dawn. Depending on the species, timeof the year, and geographical location (Terman etal., ’91), these animals sample light specificallyduring dawn, dusk, or both (Hastings et al., ’91;Roenneberg and Foster, ’97). In the laboratory,under simulated “dawn” and “dusk” conditions,the choice of reference stimulus in the nocturnalfield mouse M. booduga also depends on the du-ration of photoperiod and on the τ of the animals(Sharma et al., ’97; Sharma et al., ’99). When these


mice were maintained under a semi-natural en-vironment by exposing them to natural daylightinside the laboratory, we found that ψ and τ werenegatively correlated (Sharma et al., ’98). Theseresults suggest that individuals selected variousreference intensities in the LD cycle for timingtheir circadian clock (Sharma et al., ’99). The re-sults of the present experiments suggest thatwhen LPs of different intensities (1 lux, 10 lux,50 lux, 100 lux, and 1,000 lux) were used for 15min duration as zeitgeber, the relationship be-tween ψ and τ was nonlinear (Fig. 2). The natureof the relationship between ψ and τ remains thesame for all the intensities. The ψ reflects inter-action of the τ and the periodicity of the externalLD cycle through the light-induced PRC (with re-gions of maximum and minimum sensitivity). Wehave also reported in the same species of mice thatthe stability of the locomotor activity rhythm ofanimals with τ close to 24 hr was more than foranimals whose τ deviated from 24 hr (Sharma andChandrashekaran, ’99). The instability in τ for val-ues deviating from 24 hr appears to be compen-sated by large phase shifts, whereas stability in τfor values close to 24 hr takes care of the fluctua-tions in τ arising due to minimal phase shifts dur-ing the subjective day. As a consequence of this,small fluctuations in τ, when τ is close to 24 hr,would require larger adjustment in ψ, whereaslarge fluctuations in τ when τ is farther from 24hr would require only a slight adjustment to main-tain a constant ψ. Thus, it can be concluded thatthe very circadian nature of τ itself has functionalsignificance in the conservation of ψ (Roennebergand Foster, ’97).

ACKNOWLEDGMENTSWe acknowledge financial support from the In-

dian Academy of Sciences, Department of Science

and Technology, and Jawaharlal Nehru Centre forAdvanced Scientific Research, India.

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Hastings JW, Rusak B, Boulos Z. 1991. Circadian rhythms:the physiology of biological timing. In: Prosser L, editor.Neural and integrative animal physiology. Amsterdam:Wiley-Liss, Inc. p 535–546.

Hoffmann K. 1969. Die relative Wirksamkeit von Zeitgebern.Oecologia (Berl) 3:184–206.

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Sharma VK, Chandrashekaran MK, Singaravel M. 1998.Relationship between period and phase angle differencesin the tropical field mouse Mus booduga under gradualand abrupt light-dark transitions. Naturwissenschaften85:183–185.

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