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Am J Physiol Regul Integr Comp Physiol 280: R1023-R1030, 2001;
0363-6119/01 $5.00
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Vol. 280, Issue 4, R1023-R1030, April 2001

Functioning of the rat circadian system is modified by light applied in critical postnatal days

M. M. Canal-Corretger, J. Vilaplana, T. Cambras, and A. Díez-Noguera

Department de Fisiologia-Divisió IV, Facultat de Farmàcia, Universitat de Barcelona, 08028 Barcelona, Spain


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
MATERIAL AND METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

Lighting conditions influence biological clocks. The present experiment was designed to test the presence of a critical window of days during the lactation stage of the rat in which light has a decisive role on the development of the circadian system. Rats were exposed to 4, 8, or 12 days of constant light (LL) during the first days of life. Their circadian rhythm was later studied under LL and constant darkness. The response to a light pulse was also examined. Results show that the greater the number of LL days during lactation, the stronger the rhythm under LL and the smaller the phase shift due to the light pulse. These responses are enhanced when rats are exposed to LL days around postnatal day 12. A mathematical model was built to explain the responses of the circadian system with respect to the timing of LL during lactation, and we deduced that between postnatal days 10 to 20 there is a critical period of sensitivity to light; consequently, exposure to LL during this time modifies the circadian organization of the motor activity.

circadian rhythm; light sensitivity


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
MATERIAL AND METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

THE CIRCADIAN SYSTEM is formed by a network of structures that control the circadian rhythms of organisms. In mammals, it is located basically in the brain. It comprises structures such as the suprachiasmatic nuclei (SCN) of the hypothalamus (the main circadian pacemaker in mammals), the retinohypothalamic tract (RHT) (necessary and sufficient for photic entrainment), the geniculohypothalamic tract (GHT) (pathway that brings photic and nonphotic information to the SCN), and the retina (the only known photoreceptor organ in mammals).

The development and maturation of these structures start during gestation and progressively continue until an adult pattern is attained (normally 2 or 3 wk after birth). For example (see Ref. 18 for review), between embryonic day 17 (E17) and postnatal day 10 (P10), the SCN gradually enlarges and takes on an adult appearance; between E20 and P10, there is a gradual maturation of neuronal morphology such that the distinctive separation of neuron types into the SCN subdivisions is evident by P6 and completely developed by P10. At E19 there are few synapses, but their number increases until P10, when the synaptic density in the SCN reaches adult levels. Speh and Moore (23) found that the RHT projection to the SCN and adjacent areas first appears as scattered varicosities in the ventral part of the SCN at P1 and gradually increases until the adult pattern is reached at approximately P10. The ganglion cell projections to the intergeniculate leaflet (IGL) are present before the development of the RHT, and, in the rat, the GHT reaches an adult pattern on P10 (19).

Therefore, during the first weeks of life, although the adult pattern is not completely developed and the definitive structures are still not established, some external factors, such as environmental light conditions, may play a decisive role in the future organization of the circadian system. For instance, it has been shown that short cycles (i.e., of 4 h) have a different effect on the circadian rhythm manifestation in young and adult rats (25). Similarly, rats that are subjected to constant light (LL) throughout their lactation period manifest a circadian rhythm of motor activity under LL in adulthood, whereas adult rats reared under constant darkness (DD) or a light-dark cycle during the lactation period show arrhythmicity under LL (3). Likewise, we recently observed (5) that the expression of the circadian rhythm of motor activity under LL in adult rats depends on the number of days of LL to which rats are exposed during lactation. In the same experiment, we also found that the same number of LL days did not appear to have the same effect depending on their timing during lactation: applied in the first part or in the last part of lactation, they produced distinct responses of the circadian system to light. We then proposed that there may be a critical period of sensitivity of the circadian system to light during the development of rats.

The present experiment aims to verify the hypothesis that there are some critical days during the early life of rats in which exposition to LL influences the further functioning of the circadian system, and it also aims to time this critical period. The identification of such a stage may provide further information about the adaptive response of the circadian system to environmental conditions.


    MATERIAL AND METHODS
TOP
ABSTRACT
INTRODUCTION
MATERIAL AND METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

Eleven pregnant Wistar rats, supplied by Criffa (Barcelona), arrived at our laboratory on day 16 of gestation and were then subjected to LL (~300 lx of intensity). After delivery, 7 days later, all the pups were cross-fostered and then placed into a transparent Makrolom cage of 50 × 25 × 12 cm. From the day of birth and throughout the lactation period, each dam and her pups were kept under DD (~0.1 lx of dim red light), except for a determined number of days in which they were submitted to LL (~300 lx of intensity) (Fig. 1). The groups of rats were named according to the number of LL days during lactation and their timing, that is to say, "number-L-number". The first number indicates the number of LL days during the lactation period (4, 8, or 12 days), and the second number corresponds to the postnatal day on which LL started. For example, group 12L0 received 12 LL days during lactation, starting on the day of birth, and group 4L16 received 4 LL days during lactation, starting on day 16 after birth. In a more general way, all the groups that had 12, 8, and 4 days of LL during lactation are referred to as 12L, 8L, and 4L groups, respectively.


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Fig. 1.   Scheme of the rat groups and the lighting conditions during the experiment. The first 24 days correspond to the lactation stage and the rest to the days when motor activity was registered. Open areas indicate constant light (LL), and filled areas constant darkness (DD). On day 89 of registration, a 1-h light pulse was applied at circadian time (CT) 15.

On day 25 after birth, the pups were weaned and placed in individual transparent Makrolom cages (25 × 25 × 12 cm). A register of their motor activity was started. From now on, the time of the experiment will be expressed as recording days. The activity was individually measured by means of an activity meter with two crossed perpendicular infrared beams situated on a plane 7 cm above the floor of the cage. Movements produced in successive intervals of 15 min were automatically recorded in a personal computer and stored for further analyses. The total number of registered rats per group varied from 6 to 11 (Fig. 1).

From the day of weaning (day 1 of the recording period), all the rats were subjected to LL for 55 days to study the evolution of the motor activity rhythm under such conditions. On day 56, the rats were transferred to DD to observe their free-running rhythm. Thirty-four days later, a light pulse of 1 h of duration and a mean intensity of 350 lx was given to all the rats at circadian time (CT) 15 to study the response of the pacemaker to light. CT15 was calculated by adding 3 circadian hours to CT12 (time of the beginning of the activity phase), which was estimated visually by four investigators, independently, from the double-plotted actograms. Mean values were used.

Throughout the experiment, rats had free access to food (Rodent Toxicology Diet, B&K Universal) and tap water. Approximately every 7 days, the cages were cleaned and, until day 73 of the register, the rats were weighed.

Mathematical and statistical analysis. In both the LL and DD stages, the period of the circadian rhythm was calculated by means of Sokolove and Bushell's periodogram (22), with a high global level of significance (P = 0.01 with Bonferroni correction) to reject spurious peaks. The percentage of variance (PV) explained by the highest peak in the periodogram was used as an indicator of the importance of the motor activity rhythm.

Due to some registration problems (which did not affect the lighting conditions), only data from days 15 to 35 (in the LL stage) and from days 64 to 89 (in the DD stage) were considered for the statistical analysis.

The phase shift after the light pulse at CT15 was calculated separately by six researchers for the onsets and offsets of activity. Lines were drawn through the daily onsets and offsets for the 10 days before and after the treatment. The difference between the two eye-fitted lines before and after the light pulse was the phase shift value. The mean value obtained by the researchers was used for the statistical analysis.

Graphs and calculations were carried out using the integrated package for chronobiology analysis "El Temps" (A. Díez-Noguera, Barcelona 1999, Universitat de Barcelona).

ANOVA of several linear models (Systat) was carried out. The dependent variables for all the models were period and PV of the rhythm in the LL stage, period and PV of the rhythm in the DD stage, and the phase delay in the onset and offset of activity after the light pulse at CT15. Four types of linear models, which differ in their independent variables, were calculated. In the first model, the independent variable was the number of LL days during the lactation stage (4, 8, or 12). The second, third, and fourth models were built to test the influence of the timing of the same number of LL days during the lactation stage. Thus these models correspond to 12L, 8L, and 4L groups, respectively, and in each case the independent variable was the day on which LL started. The four models were analyzed in two ways: first, by considering the independent variable as qualitative to test differences between groups and, second, by considering the independent variable as quantitative to test a linear regression. When comparing two groups of data, a Student's t-test was used.


    RESULTS
TOP
ABSTRACT
INTRODUCTION
MATERIAL AND METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

At first sight and in general, we can observe a distinct manifestation of the motor activity rhythm of the rats under LL or DD (Fig. 2). Under LL, rats showed distinct motor activity patterns: some rats had a clear circadian rhythm, with a mean tau of 25.45 h (SE 0.56 h), whereas some others had an arrhythmic pattern. Under DD, all the rats showed a similar circadian rhythm with a mean period of 24.53 h (SE 0.14 h). No statistically significant differences were found in the period values between groups, either in LL or DD. Therefore, the period of the free-running rhythm after weaning was not significantly affected either by the number of days under LL during lactation or by the initial day of LL during lactation begins. The actograms also show that all the rats responded to the light pulse at CT15 with a phase delay.


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Fig. 2.   Double-plotted actograms showing the motor activity rhythm of 2 representative rats. In the ordinates, days after weaning are represented. The arrowhead indicates the 1-h light pulse at CT15.

Regarding the PV explained by the circadian rhythm in the LL and DD stages, differences were found related to the number of LL days to which the animals were submitted during lactation; however, such differences change depending on the stage we consider. Analysis of the rhythm in the LL stage showed that the greater the number of LL days during lactation, the more consistent the rhythm in the LL stage is (12L groups > 8L groups > 4L groups, P < 0.05, Fig. 3Aa). In contrast, in the DD stage, this tendency was inverted: the more LL days during lactation, the lower the manifestation of the circadian rhythm in the DD stage (12L groups < 8L groups < 4L groups, P < 0.01, Fig. 3Ab).


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Fig. 3.   A: variables studied depending on the number of LL days during the lactation stage (4, 8, or 12) (mean ± SE). B: variables studied for the 12L groups (groups that had 12 LL days during the lactation stage) depending on the day LL started (0, 4, 8, or 12 after birth) (mean ± SE). C: variables studied for the 8L groups (groups that had 8 LL days during the lactation stage) depending on the day LL started (4, 8, or 12 after birth) (mean ± SE). D: variables studied for the 4L groups (groups that had 4 LL days during the lactation stage) depending on the day LL started (4, 8, 12, or 16 after birth). In A, B, C, and D, the variables studied are: a, the percentage of variance (PV) explained by the circadian rhythm of motor activity in the LL stage; b, the PV explained by the circadian rhythm of motor activity in the DD stage; and c, the phase delay (in hours) after the light pulse at CT15 in the DD stage.

The initial day of LL during lactation also influenced the PV explained by the circadian rhythm, but only in the LL stage. In the case of 12L groups, we found several statistically significant differences depending on the LL onset (Fig. 3Ba): group 12L8 had the highest PV values (Student's t-test, P < 0.05), whereas groups 12L0 and 12L12 had similar values. In the 8L group, a statistically significant correlation (P < 0.05) was observed between the PV and the initial day of LL. Because the sign of the regression coefficient is positive, then the earlier LL starts, the lower the PV (Fig. 3Ca). Finally, in 4L groups, no statistically significant positive correlation between the beginning of the LL stage during lactation and the PV was observed (Fig. 3Da). Among these groups, group 4L4 had the lowest PV values (Student's t-test, P < 0.05, Fig. 3Da).

The application of a light pulse at CT15 induced phase delays in all the rats, as expected, with slight differences in their magnitude depending on the groups (Fig. 2). The animals subjected to longer LL during lactation responded with shorter phase shifts, both in the onset and offset of the motor activity profile (P < 0.01, Fig. 3Ac). In the three groups (4, 8, and 12 days under LL), the offsets were longer than the onset delays (Student's t-test, P < 0.001), and these values were positively correlated when all the animals were considered individually (P < 0.01). An analysis of the light-induced phase shifts within each group shows several statistically significant differences in the offsets but not in the onsets. In the 12L group, the offset phase shifts were shorter in those animals that started the LL stage later during lactation (P < 0.01, Fig. 3Bc). In the 8L group, no differences (P > 0.05) were found in the light-induced phase shifts (onset and offset), whereas in the 4L group, the higher offset phase shifts observed corresponded to the animals subjected to continuous light starting on P4 (Student's t-test, P < 0.05, Fig. 3Dc).

No other factors were observed to affect any other dependent variable.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
MATERIAL AND METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

The current experiment was designed based on the hypothesis that the lighting conditions during the early days of the rat's life may affect the further functioning of the circadian pacemaker. We consider the expression of the circadian rhythm of motor activity of the adult rat under LL as an indicator, because it has been widely described that adult rats become arrhythmic when submitted to constant bright light (7, 9, 11, 12, 24). We found that rats reared under LL develop a stable circadian rhythm under LL and that this rhythm is maintained throughout the lifespan of the animal (4). Specifically, we observed that the development of the adult's circadian rhythm under LL depended on the length of the exposure to LL during lactation (5) and that at least 12 days under LL during lactation were needed for it to develop.

With the present experiment, we aimed to identify whether there is a critical interval of time in the developmental stage of the rat in which LL might modify the further expression of the circadian system. As 12 LL days during lactation were found to be the minimum number of days needed for an adult submitted to LL to develop a circadian rhythm, newborn rats were kept under LL for only 12 or fewer days during the first 24 days of life; light treatments started on different days depending on the group. Thus, when interpreting the results, we must take into consideration that as the number of LL days is small, the effects will also be small, but if, even at this level, differences between groups are noticeable, then the presence of a critical window is validated.

Our results confirm that there is an effect of the number of LL days during the lactation stage on the responses of the circadian rhythm of the adult rat to light: more LL days produced a marked manifestation of the rhythm under LL (higher PV values) and smaller phase shifts due to the light pulse. Second, our findings show for the first time, that the same number of LL days has a distinct effect on the circadian system depending on their timing, as can be seen from the differences encountered within 12L, 8L, and 4L groups. For instance, in our previous experiment (5), we observed that the manifestation of the motor activity rhythm under LL and the phase shift after the light pulse of adult rats that had been submitted to 4 LL days starting on P0 or on P20 did not differ from the group that was submitted to DD throughout its whole lactation stage. However, in this study we observed that the rats that received 4 LL days after P8 differed in the expression of rhythm and in phase shifts as adults from those that received LL days on P4 (group 4L4). Group 4L4 had the least-marked rhythm in LL and showed the longest phase shift induced by the light pulse; therefore, we suggest that these rats had the weakest circadian pacemaker. Following the same reasoning, the 12L8 group had the most robust pacemaker, as it had the most marked rhythm in LL, together with one of the shortest phase shift values in response to a light pulse.

The distinct functioning of the circadian pacemaker or distinct sensitivities of adult rats to light may explain the various manifestations of motor activity rhythm under LL and the differing responses to the light pulse under DD between groups. Hamsters neonatally treated with monosodium glutamate, which induces acute degeneration of the retina, optic nerve, visual pathways, and some areas of the brain, will phase shift after a light pulse in the same way as untreated counterparts (6). Moreover, mutant rd/rd mice, which experience a massive degeneration of rods and cones during development, show the same phase shift in response to a light pulse as wild-type mice (10). This indicates that alterations in the retina are not sufficient to induce changes in the response of the circadian pacemaker to light. However, as we detected differences in the phase shift between groups, we hypothesize that the circadian pacemaker differs between groups, although alterations in the retina and visual pathways must not be excluded. Moreover, rdta mice, whose rods degenerate during ontogeny because of a fusion gene integrated in the genome, show greater shifts than wild-type mice at irradiances that produce saturating phase shifts in the latter (14). The explanation proposed was a varying amplitude of the clock response to light and not a distinct clock sensitivity to light. Retina and clock alterations are not excludable, both affect the processing of light information and the manifestation of the motor activity rhythm. It is worth noting that the only difference between rd/rd and rdta mice is that, in the latter, retina degeneration occurs 1 wk earlier. This observation supports the idea that the effect of altered retinal signals on the circadian system depends on the stage of the development of the animal. In the present experiment, we were unable to elucidate whether the distinct responses to light and the rhythm expression of rats are due to alterations in the retina or in the SCN; however, it is clear that LL during the first days of life of rats affects their circadian organization of the motor activity. Taking into account that it is not known what photoreceptors are responsible for regulating the circadian photosensitivity, up to now it is impossible to know whether the state of the retina of our animals is functionally equivalent to any retinal dystrophy, especially considering that the different groups of rats receive a different number of days, at different ages, of LL.

To estimate the changes in rat sensitivity to light as a function of age (days after birth), we pooled the results of a previous experiment (5) that studied the effect of the number of LL days during lactation with the results obtained in this study. The composite analysis includes data from 204 rats, males and females, thus making the conclusions more reliable. We took the PV value of the adult in LL as an indicator of the response of the circadian system to light. There are two aspects of time to be examined: the duration of LL exposure and the time at which light is applied. If it is assumed that the effect of light has a cumulative effect through the time (see mathematical description in APPENDIX), then we suggest that these two aspects do not correspond with two distinct variables but to a single one. Therefore, the expression calculated is the value of PV as a function of time. This curve is crucial because it indicates the evolution of the instantaneous sensitivity of the animal to light during lactation. The curve of the pooled data from the two experiments is shown in Fig. 4. It can be seen that the maximum of the curve is around day 16 after birth, indicating that the period of highest sensitivity to light is between P10 and P20. Integrating the curve around P16, one can calculate that an interval of 6 days is enough to generate, in an adult rat under LL, a PV >8% (equivalent to a significant circadian rhythm), whereas around P6, a 10-day interval is necessary. It is worth noting that the ANOVA of the linear model used to calculate the function shows that the model used is statistically highly significant (P < 0.001). This significance can also be appreciated by the visual inspection of the fiducial limits of the function: it is clearly visible that a straight line cannot be traced inside the 95% confidence band, thus demonstrating the presence of a window of sensitivity to light during lactation.


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Fig. 4.   Sensitivity to light is plotted vs. time. Time is expressed as days after birth, and sensitivity (see MATERIAL AND METHODS for explanation) is represented as the instantaneous contribution to the PV explained by the circadian rhythm in adults under LL. The thick line represents expected values according to the estimated polynomial function (see APPENDIX), and the thin lines are the 95% fiducial limits of the estimation.

The same statistical process was applied to the phase shifts induced by a light pulse at CT15 (separately for the onset and the offset of motor activity). Although the calculated functions are not statistically significant, it is worth noting that the shorter phase shifts (the minimum of the curve) occurred between P10 and P20, just when the specific sensitivity to light reaches its maximum. In both experiments, we observed that the rats with the most consistent rhythm under LL were those with shorter phase shifts after a light pulse under DD.

As we mentioned above, although we cannot disregard the possibility that our results are due to alterations in the circadian photosensitivity, we may also consider that the different light treatment affects the circadian pacemaker itself. Considering that the circadian clock is formed by several groups of coupled oscillators (2, 8, 17, 26), the motor activity arrhythmicity of adult animals under LL can be explained by an uncoupling effect of light on these oscillators. When light is present from the day of birth, the animal adapts accordingly, and the manifestation of the circadian rhythm is permitted even under LL. This may lead to a stronger circadian pacemaker that is not easily affected by environmental light. Externally, this strong clock may manifest a marked circadian rhythm under LL and a low response to a light pulse in darkness. It has been proposed that glial cells are responsible for the coupling between oscillators (26), therefore the mechanisms that regulate the connection between such cells and neurons may be those altered by light. The effect of light on the circadian system may be, in part, mediated by glutamate, which is the main neurotransmitter of the RHT. We suggest that if a rat is submitted to LL when the RHT is developing its connections, the levels of glutamate will be modified, and therefore the intracellular calcium levels of both neurons and astrocytes will be altered. This could modify the functioning and the coupling between neurons and astrocytes, because changes in intracellular calcium levels during early development have been shown to alter the rate of neurite outgrowth (16), synapse formation (20), neural migration (13), and neural phenotype (15).

We conclude that the lighting conditions under which a rat is reared during its first days of life are decisive for the future response of the animal to light. Light has several effects on the circadian organization of the motor activity depending on the stage of maturation of the rat. From the first embryonic stages and until an animal reaches an adult pattern some weeks after birth, a long process of maturation of the nervous system takes place. The SCN are completely developed by P10. Likewise, the pathways that carry environmental information to the SCN develop postnatally: the RHT projection to the SCN and adjacent areas and the GHT also reach an adult pattern also on approximately P10 (19, 23). From the curve calculated, we can observe that the sensitivity of the circadian system to light increases throughout the lactation period, reaching its maximum at about P16, which coincides with the time in which nearly all the significant structures of the circadian system (SCN, RHT, GHT, pineal gland) have already developed and reached their adult pattern. Thus it appears that the circadian system must be developed before it can be influenced and molded by external factors such as light. The internal and external influences that the animal receives during the maturation period are of great importance for the complete development of the nervous system. From studies of the development of the visual cortex (see Ref. 1 for review), it is known that although the developmental processes that lead to the functional pattern of connections that underlie normal vision are genetically determined, they can be interfered with or impaired by abnormal visual experience in the "sensitive period" of the first few months of life. Environment influences can cause large-scale changes in the anatomy and/or functional response properties of neurons that can even persist throughout the lifetime of an animal. For instance, experiments on the presence and type of orientation-selective cells in the primary visual cortex of kittens reared under different visual environments (see Ref. 21 for review) have shown that sensory experience plays a decisive role in the future appearance and orientation of these cells. The effect of environmental conditions is only effective on kittens during the critical period of their development and not on adult cats, because during the critical period the neural circuitry responsible for orientation detector cells in the primary visual cortex is in a "plastic" state, i.e., open to environmental molding and responsive to use and/or disuse. Likewise, we could expect a plastic state of the circadian system between P10 and P20, during which the circadian system would be predisposed to be instructed and shaped by experience, and, therefore, the best input pathways would be selected.

Perspectives

Here we report that the sensitivity of the circadian system to light changes in the first few days of life. This finding may help us to understand the response of the circadian system to environmental factors. This system generates rhythmicity in the various functions of the organism, which allows synchronization to the environment. Thus the circadian rhythm needs to be manifested, whatever the conditions of the environment. This implies that the development of the circadian system may depend on the environment, although the functioning of the system is expected to be similar in all adult animals, which would guarantee adaptation.

Because light is the main zeitgeber, the threshold of the sensitivity of mammals to light may depend on the basal level of light intensity. However, several questions remain; for instance, how does light affect the developing circadian system? Are all the structures of the circadian system (retina, visual pathways, and the circadian pacemaker itself) affected by early exposure to LL? Moreover, might stimuli other than light influence the development of the circadian system? Our findings indicate that the external conditions during ontogeny may be decisive, because brain structures are plastic and external inputs from the environment can act on the connections between nerve cells and modify them structurally and/or functionally. These modifications lead to the acquisition of the mature pattern of the brain.


    APPENDIX A
TOP
ABSTRACT
INTRODUCTION
MATERIAL AND METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

Using the same notation as Sokolove and Bushell (22), the PV can be derived from Qp following the formula
PV<IT>=100 </IT><FR><NU>SS<SUB><A><AC>X</AC><AC>&cjs1171;</AC></A></SUB></NU><DE>SS<SUB>X</SUB></DE></FR><IT>=100 </IT><FR><NU>K <LIM><OP>∑</OP><LL>h<IT>=1</IT></LL><UL>P</UL></LIM> (<A><AC>X</AC><AC>&cjs1171;</AC></A><SUB>h</SUB><IT>−</IT><A><AC>X</AC><AC>&cjs1171;</AC></A>)<SUP><IT>2</IT></SUP></NU><DE><LIM><OP>∑</OP><LL>i<IT>=1</IT></LL><UL>N</UL></LIM> (X<SUB>i</SUB><IT>−</IT><A><AC>X</AC><AC>&cjs1171;</AC></A>)<SUP><IT>2</IT></SUP></DE></FR>

<IT>=</IT><FR><NU><IT>100</IT></NU><DE>N</DE></FR> <FR><NU>K <LIM><OP>∑</OP><LL>h<IT>=1</IT></LL><UL>P</UL></LIM> (<A><AC>X</AC><AC>&cjs1171;</AC></A><SUB>h</SUB><IT>−</IT><A><AC>X</AC><AC>&cjs1171;</AC></A>)<SUP><IT>2</IT></SUP></NU><DE>N<SUP>−<IT>1</IT></SUP> <LIM><OP>∑</OP><LL>i<IT>=1</IT></LL><UL>N</UL></LIM> (X<SUB>i</SUB><IT>−</IT><A><AC>X</AC><AC>&cjs1171;</AC></A>)<SUP><IT>2</IT></SUP></DE></FR><IT>=</IT><FR><NU><IT>100</IT></NU><DE>N</DE></FR> Q<SUB>p</SUB>
Data must be arranged in a matrix of K rows and P columns for each T (N = K · P).


    APPENDIX B
TOP
ABSTRACT
INTRODUCTION
MATERIAL AND METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

To quantify the effect induced by light during lactation, we used several variables measured in the adult rat (PV explained by the rhythm or phase shifts). In accordance with our experimental hypothesis, these variables (effects) are dependent on the specific time of light application. We consider the "instantaneous" effect of light as a continuous function of time, but as the nature of this function is unknown, we will use a third-order polynomial expansion of it
f(<IT>t</IT>)<IT>≈</IT>b<SUB><IT>0</IT></SUB><IT>+</IT>b<SUB><IT>1</IT></SUB><IT>t+</IT>b<SUB><IT>2</IT></SUB><IT>t<SUP>2</SUP>+</IT>b<SUB><IT>3</IT></SUB><IT>t<SUP>3</SUP>=</IT><LIM><OP>∑</OP><LL>i<IT>=0</IT></LL><UL><IT>3</IT></UL></LIM> b<SUB>i</SUB>t<SUP>i</SUP> (1)
In real situations, light is not applied just for an instant but during an interval of time (days), consequently, the powers of the instantaneous values of t in Eq. 1 must be substituted by the integrals Ii, corresponding to the distinct intervals during which the animals received light. The integrals are defined as
I<SUB>i<IT>,</IT>a<IT>−</IT>b</SUB><IT>=</IT><LIM><OP>∫</OP><LL>a</LL><UL>b</UL></LIM><IT> t</IT><SUP>i</SUP>d<IT>t=</IT><FENCE><FR><NU><IT>t</IT><SUP>i<IT>+1</IT></SUP></NU><DE>i<IT>+1</IT></DE></FR></FENCE><SUP>b</SUP><SUB>a</SUB><IT>=</IT><FR><NU>b<SUP>i<IT>+1</IT></SUP><IT>−</IT>a<SUP>i<IT>+1</IT></SUP></NU><DE>i<IT>+1</IT></DE></FR> (2)
where a and b are the initial and final time values corresponding to the interval. Therefore, we can rewrite f(t) as
f(<IT>t</IT>)<IT>≈</IT>b<SUB><IT>0</IT></SUB><IT>+</IT>b<SUB><IT>1</IT></SUB>I<SUB><IT>1</IT></SUB><IT>+</IT>b<SUB><IT>2</IT></SUB>I<SUB><IT>2</IT></SUB><IT>+</IT>b<SUB><IT>3</IT></SUB>I<SUB><IT>3</IT></SUB><IT>=</IT><LIM><OP>∑</OP><LL>i<IT>=0</IT></LL><UL><IT>3</IT></UL></LIM> b<SUB>i</SUB>I<SUB>i</SUB> (3)
This function was used to calculate the coefficients bi, for the polynomial linear regression analysis. Once the coefficients were calculated, one can deduce a function fd(t) to express the effect of the application of light for an interval of length d as a function of time (the time of application). If time in the function represents the initial day of the interval, the final day will be t+d, and the function will be expressed as
f<SUB>d</SUB>(<IT>t</IT>)<IT>=</IT><LIM><OP>∑</OP><LL>i<IT>=0</IT></LL><UL><IT>4</IT></UL></LIM> b<SUB>i</SUB>I<SUB>i<IT>,</IT>t<IT>∼</IT>t<IT>+</IT>d</SUB><IT>=</IT><LIM><OP>∑</OP><LL>i<IT>=0</IT></LL><UL><IT>4</IT></UL></LIM> b<SUB>i</SUB> <FR><NU>(<IT>t+</IT>d)<SUP>i<IT>+1</IT></SUP><IT>−t</IT><SUP>i<IT>+1</IT></SUP></NU><DE>i<IT>+1</IT></DE></FR> (4)
The maximum of the function f(t) is estimated by solving the first derivative of Eq. 1
b<SUB><IT>1</IT></SUB><IT>+2</IT>b<SUB><IT>2</IT></SUB><IT>t+3</IT>b<SUB><IT>3</IT></SUB><IT>t<SUP>2</SUP>=0</IT> (5)


    ACKNOWLEDGEMENTS

This work was financially supported by the "Ministerio de Educación y Cultura" (DGESIC, PM98-0186).


    FOOTNOTES

Address for reprint requests and other correspondence: A. Díez-Noguera, Departament de Fisiologia-Divisió IV, Facultat de Farmàcia, Av. Joan XXIII, s/n, 08028 Barcelona, Spain (E-mail: andiez{at}farmacia.far.ub.es).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Received 10 July 2000; accepted in final form 8 November 2000.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
MATERIAL AND METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
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Am J Physiol Regul Integr Comp Physiol 280(4):R1023-R1030
0363-6119/01 $5.00 Copyright © 2001 the American Physiological Society



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