Scott A Love

Emmanuelle Haslin

Manon Bellardie

Frédéric Andersson

Laurent Barantin

Isabelle Filipiak

Hans Adriaensen

Csilla Lea Fazekas

Laurène Leroy

Dóra Zelena

M Morisse

Chr F Elleboudt

F Moussu

R Lévy

E Nowak

Chaillou

C Bellardie

Fazekas

Maternal deprivation and milk replacement affect the integrity of gray and white matter in the developing lamb brain

Keywords: statistical analysis, results discussion, secondary writer internship, segmentation MRI protocole, MRI acquisition. I. Filipiak: Diffusion MRI sequences. H. Adriaensen: MRI acquisition. F. Elleboudt, Chr. Moussu: animal preparation, anaesthesia, caregivers. F. Lévy: protocol discussion, interpretation, article reviewing artificial rearing, mothering, sheep, diffusion MRI, T1-weighted MRI, White and Gray matter

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In both human and non-human primates, the quality of motherhood is important for an infant's development, especially for social personality development [START_REF] Pryce | Determinants of motherhood in human and nonhuman primates. A biosocial model[END_REF]. In nonhuman primates, the lack of mothering from birth to adolescence leads to anxiousness for novelty exploration, increased aggressiveness at puberty and, peer-reared primiparous females display disturbance of maternal behaviour [START_REF] Suomi | Early determinants of behaviour: evidence from primate studies[END_REF]. More generally, whether in pets or farm and zoo animals, maternal deprivation at birth or early weaning leads to behavioural problems such as reproductive or maternal deficits, aggressiveness, and stereotypic behaviours such as non-nutritive suckling, head-banging or self-harm in general (Rhesus macaque: [START_REF] Liu | The effects of breastfeeding versus formula-feeding on cerebral cortex maturation in infant rhesus macaques[END_REF]; Lambs: [START_REF] Gaudin | All bonds are not alike: A psychoendocrine evaluation of infant attachment[END_REF]; Piglets: [START_REF] Rzezniczek | Comparison of the behaviour of piglets raised in an artificial rearing system or reared by the sow[END_REF]; and for a review, [START_REF] Latham | Maternal deprivation and the development of stereotypic behaviour[END_REF]). These impacts of maternal deprivation can result simultaneously from the lack of a social model, i.e., the mother [START_REF] Fleming | Mothering begets mothering: The transmission of behavior and its neurobiology across generations 15[END_REF] and from a lack of maternal milk as a source of bioactive factors [START_REF] Bernstein | Bioactive factors in milk across lactation: Maternal effects and influence on infant growth in rhesus macaques ( Macaca mulatta ): Macaque Milk Bioactive Factors[END_REF]. However, the challenge is not to isolate the influence of only one factor (mother or milk) but to consider "the dynamic interactions among the multitude of equally important maternal and non-maternal variables" [START_REF] Tang | On the causes of early life experience effects: Evaluating the role of mom[END_REF]. In sheep (ovis aries), a precocial species, it is difficult to dissociate the impact of maternal deprivation from that of formula-milk feeding; each affecting the infants' development. The reciprocal mother-infant bond between a ewe and her lambs occurs in the first 12 hours of life. It is based mainly on odorant cues and is characterized by an individual recognition of each partner and nursing exclusivity between a mother and her own infant [START_REF] Nowak | Development of a Preferential Relationship With the Mother by the Newborn Lamb: Importance of the Sucking Activity[END_REF][START_REF] Nowak | From birth to colostrum: early steps leading to lamb survival[END_REF]. In this context, it is not surprising that the mother is crucial for the development of the lamb. However, motherless rearing is commonly used in conventional dairy farming with very early separation of lambs from their mother, or in other situations if the mother is non-maternal, has too many lambs or if she has mastitis. But several studies reported deleterious impact of motherless rearing conditions with a level of mortality between 10 and 30% [START_REF] Miller | Metabolic maturity at birth and neonate lamb survival: Association among maternal factors, litter size, lamb birth weight, and plasma metabolic and endocrine factors on survival and behavior1[END_REF][START_REF] Napolitano | Welfare implications of artificial rearing and early weaning in sheep[END_REF][START_REF] Napolitano | Effect of artificial rearing on lamb welfare and meat quality[END_REF]. According to the absence of mother, the milk composition and/or the age of weaning, the impact of early rearing conditions is very variable and affects a wide range of functions and behaviours. At birth, colostrum is crucial to avoid mortality by the passive immune transfer from the mother to their lambs [START_REF] Hernández-Castellano | The effect of colostrum period management on BW and immune system in lambs: from birth to weaning[END_REF][START_REF] Khan | Maternal immunoglobulins transfer and neonatal lamb mortality -a review[END_REF][START_REF] Nowak | From birth to colostrum: early steps leading to lamb survival[END_REF]. The days after, the impact of mother deprivation on immune response is not so strong [START_REF] Napolitano | Influence of artificial rearing on behavioral and immune response of lambs[END_REF] especially if the lambs are fed with ewe's milk [START_REF] Napolitano | Lambs prevented from suckling their mothers display behavioral, immune and endocrine disturbances[END_REF]. However, there is an altered immune response in lambs fed with commercial milk in comparison with mothered lambs or lambs fed with a mix of maternal and commercial milks [START_REF] Sevi | Effect of gradual transition from maternal to reconstituted milk on behavioural, endocrine and immune responses of lambs[END_REF]. During the weeks after birth, the mother plays a very strong role as a social demonstrator that influences the establishment of feeding and social preferences. For example, it has been reported that the influence of the mother for the feeding preferences is stronger than thus of a dry ewe [START_REF] Thorhallsdottir | Brain does not utilize low density lipoproteincholesterol during fetal and neonatal development in the sheep 9[END_REF] and could continue after weaning [START_REF] Black-Rubio | Maternal influence on feeding site avoidance behaviour of lambs[END_REF], this influence being reinforced by the access to the olfactory cues of the food [START_REF] Saint-Dizier | Influence of the mother in the development of flavored-food preference in lambs[END_REF]. Interestingly, the presence of the mother also influences the capacity of lambs to recognize partners or to create social bonds. For example, lambs express preference for their twin only if they are reared together by their mother, this capacity disappears if they are artificially reared in a group without their mother [START_REF] Ligout | Effect of maternal presence on the development of social relationships among lambs[END_REF]. Moreover, the presence of the mother influences the development of social attachment with new partners. For example, even if lambs have positive contacts with a stockperson who gives milk, they prefer their mother and are not appeased by the presence of the familiar human during isolation from conspecifics [START_REF] Boivin | The presence of the dam affects the efficiency of gentling and feeding on the early establishment of the stockperson-lamb relationship[END_REF]. However, in absence of the mother, lambs have the capacity to create a very strong social bond with a familiar human caregiver, this bond presenting similar characteristics as those described in the mother-infant bond [START_REF] Guesdon | Behavioral evidence of heterospecific bonding between the lamb and the human caregiver and mapping of associated brain network[END_REF]. The absence of the mother also has a negative impact on the sexual behaviour of male lambs reared in a uni-gender group from 24-36 hours after birth [START_REF] Damián | Growing without a mother results in poorer sexual behaviour in adult rams[END_REF][START_REF] Damián | Reproductive and sexual behaviour development of dam or artificially reared male lambs[END_REF]. Emotional reactivity, evaluated in a social isolation context by cortisol plasma and behavioural responses, is also impacted by maternal deprivation [START_REF] Napolitano | Lambs prevented from suckling their mothers display behavioral, immune and endocrine disturbances[END_REF][START_REF] Napolitano | Effect of artificial rearing on lamb welfare and meat quality[END_REF][START_REF] Sevi | Effect of gradual transition from maternal to reconstituted milk on behavioural, endocrine and immune responses of lambs[END_REF]. In addition to these behavioural alterations, endocrine troubles are also reported, in relationship with sexual behaviour [START_REF] Damián | Growing without a mother results in poorer sexual behaviour in adult rams[END_REF][START_REF] Damián | Reproductive and sexual behaviour development of dam or artificially reared male lambs[END_REF], infant attachment [START_REF] Gaudin | All bonds are not alike: A psychoendocrine evaluation of infant attachment[END_REF] or nutrition [START_REF] Berry | Neonatal milk supplementation in lambs has persistent effects on growth and metabolic function that differ by sex and gestational age[END_REF]. Together, these studies demonstrate how maternal deprivation has wide ranging, deleterious consequences, affecting the development of the immune system, behavioural expression, endocrine development, growth and cognitive capacities of lambs. Whereas the benefits of breastfeeding for brain development have been reported in humans [START_REF] Deoni | Early nutrition influences developmental myelination and cognition in infants and young children[END_REF][START_REF] Deoni | Breastfeeding and early white matter development: A crosssectional study[END_REF][START_REF] O'muircheartaigh | White matter development and early cognition in babies and toddlers: White Matter Development and Early Cognition[END_REF][START_REF] Schack-Nielsen | Advances in Our Understanding of the Biology of Human Milk and Its Effects on the Offspring[END_REF] and non-human primates [START_REF] Liu | The effects of breastfeeding versus formula-feeding on cerebral cortex maturation in infant rhesus macaques[END_REF], sparse information has been reported in lambs. Mainly studied as a model of the developmental troubles posed by preterm birth [START_REF] Castillo-Melendez | Experimental Modelling of the Consequences of Brief Late Gestation Asphyxia on Newborn Lamb Behaviour and Brain Structure[END_REF][START_REF] Chaillou | MRI Techniques and New Animal Models for Imaging the Brain[END_REF][START_REF] De Matteo | Induced Preterm Birth in Sheep: A Suitable Model for Studying the Developmental Effects of Moderately Preterm Birth[END_REF][START_REF] Dunlop | Repeated prenatal corticosteroids delay myelination in the ovine central nervous system 5[END_REF][START_REF] Huang | Repeated prenatal corticosteroid administration delays myelination of the corpus callosum in fetal sheep[END_REF][START_REF] Malhotra | Advanced MRI analysis to detect white matter brain injury in growth restricted newborn lambs[END_REF], lamb brain development in relation to postnatal early experience is little understood. In relation to maternal deprivation, the only reported data concern a very short period of maternal deprivation (72h) at 2 weeks of age after birth [START_REF] Polkowska | Effects of maternal deprivation on the somatotrophic axis and neuropeptide Y in the hypothalamus and pituitary in female lambs. The histomorphometric study[END_REF][START_REF] Wańkowska | Effects of maternal deprivation on the adrenocorticotrophic and gonadotrophic axes in the hypothalamo-pituitary unit of preweanling female sheep: The histomorphometric approach[END_REF]. These studies examined the distribution of neuropeptide immunoreactivity in different hypothalamic areas in 3 mothered and 3 maternal deprived lambs. The authors noticed greater immunoreactivity in maternal deprived lambs for adrenocorticotropic hormone and gonadotrophin-releasing hormone [START_REF] Wańkowska | Effects of maternal deprivation on the adrenocorticotrophic and gonadotrophic axes in the hypothalamo-pituitary unit of preweanling female sheep: The histomorphometric approach[END_REF], for somatostatin in the periventricular zone and neuropeptide Y in the periventricular zone and arcuate nucleus [START_REF] Polkowska | Effects of maternal deprivation on the somatotrophic axis and neuropeptide Y in the hypothalamus and pituitary in female lambs. The histomorphometric study[END_REF], consistent with the endocrine perturbations described in similar maternal deprivation contexts [START_REF] Damián | Growing without a mother results in poorer sexual behaviour in adult rams[END_REF][START_REF] Damián | Reproductive and sexual behaviour development of dam or artificially reared male lambs[END_REF][START_REF] Sevi | Effect of gradual transition from maternal to reconstituted milk on behavioural, endocrine and immune responses of lambs[END_REF]. Using magnetic resonance imaging (MRI), it is possible to follow the growing brain and various indicators of brain maturation, especially with T1-weighted (T1w) signal intensity [START_REF] Flood | Age-Dependent Signal Intensity Changes in the Structurally Normal Pediatric Brain on Unenhanced T1-Weighted MR Imaging[END_REF][START_REF] Luby | The Effects of Poverty on Childhood Brain Development: The Mediating Effect of Caregiving and Stressful Life Events[END_REF], diffusion MRI and diffusion tensor imaging (DTI) [START_REF] Dubois | The early development of brain white matter: A review of imaging studies in fetuses, newborns and infants[END_REF][START_REF] Dubois | Assessment of the early organization and maturation of infants' cerebral white matter fiber bundles: A feasibility study using quantitative diffusion tensor imaging and tractography[END_REF][START_REF] Hüppi | Diffusion tensor imaging of brain development[END_REF]. Based on studies done in humans [START_REF] Deoni | Breastfeeding and early white matter development: A crosssectional study[END_REF][START_REF] Luby | The Effects of Poverty on Childhood Brain Development: The Mediating Effect of Caregiving and Stressful Life Events[END_REF] and non-human primates [START_REF] Liu | The effects of breastfeeding versus formula-feeding on cerebral cortex maturation in infant rhesus macaques[END_REF], we proposed to examine the impact of maternal deprivation on different morphometric and structural characteristics of the lamb brain. As the impact of early experience could affect the brain differently depending on the age of the subject [START_REF] Bock | Perinatal programming of emotional brain circuits: an integrative view from systems to molecules[END_REF], and because the myelination process is in progress for several months after birth [START_REF] Mcintosh | FOETAL BRAIN DEVELOPMENT IN THE SHEEP[END_REF][START_REF] Thorhallsdottir | Brain does not utilize low density lipoproteincholesterol during fetal and neonatal development in the sheep 9[END_REF], we examined brain development from 1 to 19 weeks of age after birth in mothered and maternal deprived female lambs. In this context, we considered the mothered lambs as the control. In this aim, we investigated morphometric and structural parameters for the grey (GM) and white (WM) matters. For morphometric parameters, we measured, on T1w MRI, brain, GM, WM and cerebrospinal fluid (CSF) volumes as well as those of specific brain structures involved in socio-emotional behaviours. Here, our hypothesis was that the absence of mother's milk would negatively affect the developmental increase of these morphometric parameters. For structural characteristics, we focused on T1w signal intensity and diffusion parameters extracted from DTI in GM, WM and specific WM areas. Here, our hypothesis was that maternal deprivation would affect the evolution of these structural parameters, especially in the WM.

Material and Methods

Ethics

All experimental procedures were approved (00821.03 and 02272.01) by the Val de Loire Ethics Committee (CEEA VdL, Nouzilly, France).

Experimental Animals

Twenty, Ile de France, female lambs (Ovis aries) took part in this study. The animals were born and reared at a French National Research Institute for Agriculture, Food and Environment experimental unit (PAO, INRAE, 2018. Animal Physiology Experimental Facility, DOI: 10.15454/1.5573896321728955E12). After parturition the lambs were allocated to one of two groups: 10 were kept with their mother for 24 hours after birth, which gave them access to colostrum, before being separated from their mothers (maternal deprived lambs, MD); the 10 others were kept with their mothers for the entire study (control mothered lambs, MC). The MD group had ad libitum access to water and formula-milk (Solvor 2015 Instant, composed of whey, whey protein concentrate, wheat gluten, palm oil, copra, rapeseed, flaxseed, soybeans, partially delactosed whey and extracts of yeast from inactive beer; 23% of crude protein and 25% of crude fat level). The formula-milk was distributed via an automatic milk feeder, which the lambs were trained to use by caregivers during the first 72-96h of life. To follow lambs' growth they were weighed at birth, before each of 3 MRI acquisitions and at around 2 months of age (Figure 1). During the experiment 3 lambs died in the MD group at 35, 42 and 45 days of age and 1 MC lamb was excluded because of mother's mastitis. Final group sizes were N=7 for MD and N=9 for MC.

MRI Acquisition

MRI of the brain was performed with a 3T Siemens Magnetom Verio® whole body MR imaging system (Erlangen, Germany) located at the platform CIRE (Chirurgie et Imagerie pour la Recherche et l'Enseignement, INRAE Nouzilly, France). MRI was acquired from each lamb at three different time points. The first point of acquisition (PA) was collected around 1 week after birth (0.86-2.57 weeks; PA0.86-2.57w), the second at around 1.5 months after birth (4.71-6.57 weeks, PA4.71-6.57w) and the third at around 4.5 months after birth (17.57-20.5 weeks, PA17.57-20.5w). Note that within each point of acquisition the individual ages are variable because lambs were born at term without induction of labour between the 1 st and 12 th of September 2014 and were submitted to MRI acquisitions at the rate of two animals per day. The MC lambs (n=9) were on average 1.6 (+/-0.6), 5.5 (+/-0.4) and 19.1 (+/-1) weeks old and the MD lambs (n=7) were 1.6, 5.5 and 18.3 weeks old, respectively for the three points of acquisition.

Before each acquisition lambs were placed under general anaesthesia to prevent stress and avoid motion artefacts. To prevent nausea and the associated risks of food entering the lungs during anaesthesia, for the last 2 acquisitions (PA4.71-6.57w & PA17.57-20.5w) lambs were fasted from the night before. Anaesthesia was induced by an intra-venous co-injection of rompun® and ketamine®. After this injection and the loss of muscular tonus, a tracheal tube was inserted to maintain the anaesthesia with Isoflurane®. The drug doses and tracheal tube sizes were adapted to each animal according to its age, body weight and size (Table 1). The duration of anaesthesia was limited to 2.5 hours and during that time cardiac rhythm and Sp02 levels were continuously monitored. Due to their young age, during the first PA (PA0.86-2.27w), lambs were also perfused with a solution of Ringer-Glucose. At the end of the acquisition, Isoflurane® was stopped and the tracheal tube removed after the first signs of autonomous respiration.

During acquisition lambs lay on their stomachs (prone position) with a 4-channel Siemens FLEX coil wrapped around their head and, for their comfort and safety, were covered with a cloth and wore ear plugs (Figure 1). T1-weighted images were acquired with the 3D-MPRAGE sequence with the following parameters at all points of acquisition: TR=2500ms, TE=3.44ms, TI=756ms, section thickness=0.4mm, FOV=152mm, matrix=384x384mm 2 , 4NEX, TA=45min. DTI-images were acquired with a spin-echo echo-planar sequence with the following parameters at all points of acquisition: TR=13000ms, TE=80ms, FOV=152mm, matrix=128x128mm 2 , TA=6-10min. Images were acquired in 30 directions with a b-value of 1000s/mm 2 (similar to those used in childhood studies, [START_REF] Mukherjee | Diffusion-Tensor MR Imaging of Gray and White Matter Development during Normal Human Brain Maturation 12[END_REF]) and 6 images were acquired without diffusion weighting (b-value=0s/mm 2 ). GRE field mapping sequences were performed in magnitude and in phase with the following parameters: TR=500ms, TE short/long=4.92/7.38ms.

MRI Analysis

Native MR images were converted from DICOM to NIFTI with dcm2nii (mricron, http://www.mccauslandercenter.sc.edu/mricro/mricron).

Morphometric variables

T1w MRI for volume calculation of brain, gray and white matters and cerebrospinal fluid T1-weighted images were first denoised using the DenoiseImage function of the Advanced Normalization Tools (ANTs, Avants et al., 2014;Manjon et al., 2008). They were then cropped in all three dimensions to isolate the cerebral tissue as closely as possible from all non-brain tissue. To reduce the time required for manual brain mask segmentation, the original resolution (0.4mm 3 isotropic) of the T1w images was linearly subsampled to 0.8mm 3 isotropic using the resample-mm function of the Convert3D (ITK-SNAP, www.itksnap.org, Yushkevich et al., 2006). The manual brain mask segmentation was performed with the ITK-SNAP software taking care to exclude the meninges, olfactory bulb and tentorium cerebelli (Figure 2). The segmented brain mask was then upsampled to the original T1w resolution of 0.4mm 3 isotropic before being manually inspected and corrected when necessary. Finally, all brain masks were controlled by experts in sheep neuroanatomy (EC, SAL). The automatic segmentation of gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF) was conducted within each individual's brain mask using FAST (FMRIB Automated Segmentation Tool, https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FAST, Zhang et al. 2001;Smith et al., 2004) (Figure 2).

T1w MRI for volume calculation of specific brain structures

To calculate the volumes of specific brain structures involved in socioemotional behavior (olfactory bulbs [OB], periaqueductal gray [PAG], hippocampus [Hipp], caudate nucleus [CN], Figure 2c) regions-of-interest (ROI) were manually segmented on the denoised T1w images for PA4.71-6.57w and PA17.57-20.5w using ITK-SNAP. The OB, Hipp and CN were segmented bilaterally and the mean volume was calculated. All ROI were controlled by experts in sheep neuroanatomy.

Structural variables T1w signal intensity

The mean T1w signal intensity was calculated for each GM, WM and CSF segmentation. As previously done in a pediatric study [START_REF] Flood | Age-Dependent Signal Intensity Changes in the Structurally Normal Pediatric Brain on Unenhanced T1-Weighted MR Imaging[END_REF], the mean CSF value was used to normalise that of GM and WM.

Diffusion-parameters

The DTI-images were corrected for geometric deformations due to inhomogeneities in the magnetic field using the Fieldmap tool from the Statistical Parametric Mapping toolbox. Eddy-current correction was performed with FSL (FMRIB Software Library v.5.0.4). Using FSL, the FA-and RD-maps were calculated. The mean FA-and RD-values were calculated for GM, WM, and for specific WM-areas sampled by the segmentation of 3 (corpus callosum, optic chiasm and cerebellum; see Figure 3) or 4 (frontal, parietal, temporal and occipital lobes; see Figure 3) successive sections. This segmentation was done manually on the T1w images and controlled by experts in sheep neuroanatomy. The ROI were then re-sampled to the DTI-image resolution (2x2x2mm 3 ) and the mean FA-and RD-values were calculated.

Statistical Analysis

Missing data Body weight data was not collected for 3 MC PA0.86-2.57w lambs. Due to a lack of T1w MR image contrast in a few individuals, the olfactory bulbs were only segmented for n=6 MC and n=6 MD lambs and the hippocampus for n=7 MC lambs. At the resolution of the DTIimages, some specific WM-areas, that had been segmented on T1w images, contained very few or even zero voxels for some individuals. We only include WM-areas of 3 voxels or more in the statistical analysis.

Statistical analyses

To investigate the impact of early experience (EE) on lamb growth, the coefficient of correlation between age (in days) and body weight (in kg), measured 4 times during the experiment (Figure 1), were compared between the groups using a t-test (JASP version 0.13.0, https://jasp-stats.org/, JASP Team, 2020).

ANCOVA analyses with the covariate age at acquisition

To investigate the impact of early experience (EE) and/or the point of acquisition (PA), each measure (the different morphological and structural parameters and body weight) was analysed independently using an ANCOVA with age at acquisition as a covariate of no interest and Type II Sum of Squares (JASP). Post-hoc analyses were conducted with Bonferroni correction. Summary statistics for the different measures are presented as means ± standard deviation.

Results

Body weight

Body weight and age of animals are positively correlated in MC (r2=0.995±0.005, n=9) and in MD (r2=0.993±0.003, n=7). T-test analysis highlighted no significant effect of EE on the coefficient of correlation (t=-0.969, p=0.381). The ANCOVA investigating the impact of EE & PA on body weight after controlling for age at acquisition highlighted no significant effect of EE (F(1,52) = 0.81, p = 0.37, η 2 = 0.013), PA (F(3,52) = 2.20, p = 0.10, η 2 = 0.108) or their interaction (F(3,52) = 0.63, p = 0.6, η 2 = 0.031) (Figure 4).

Morphometric data Brain volume

The total brain volume ANCOVA highlighted effects of EE (F(1,41) = 12.20, p = 0.001, η 2 = 0.161) and PA (F(2,41) = 10.85, p < 0.001, η 2 = 0.287) but not for their interaction (F(2,41) = 0.38, p = 0.69, η 2 = 0.01). Brain volume increased with the PA and MD lamb brain volume was higher than that of MC lambs: PA0.86-2.57w MD=61.79±4.57cm 3 , MC=57.63±4.57cm 3 ; PA4.71-6.57w MD=77.08±4.84cm 3 , MC=69.95±5.32cm 3 ; PA17.57-20.5w MD=89.75±5.01cm 3 , MC=85.05±5.89cm 3 (Figure 5).

Gray, white matters and CSF volumes

The GM volume ANCOVA highlighted effects of EE (F(1,41) = 9.05, p = 0.004, η 2 = 0.143) and PA (F(2,41) = 6.15, p = 0.005, η 2 = 0.194) but not for their interaction (F(2,41) = 0.29, p = 0.747, η 2 = 0.009). GM volume increased with the PA and MD lamb GM volume was higher than that of MC lambs: PA0.86-2.57w MD=30.15±2.05cm 3 , MC=28.60±2.09cm 3 ; PA4.71-6.57w MD=35.36±2.43cm 3 , MC=32.48±2.84cm 3 ; PA17.57-20.5w MD=39.59±2.32cm 3 , MC=36.59±2.88cm 3 (Figure 6a).

The WM volume ANCOVA highlighted effects of EE (F(1,41) = 13.66, p = < 0.001, η 2 = 0.169) and PA (F(2,41) = 11.98, p < 0.001, η 2 = 0.296) but not for their interaction (F(2,41) = 0.52, p = 0.60, η 2 = 0.013). WM volume increased with the PA and MD lamb WM volume was higher than that of MC lambs: PA0.86-2.57w MD=15.94±1.46cm 3 , MC=14.61±1.32cm 3 ; PA4.71-6.57w MD=21.15±1.53cm 3 , MC=19.03±1.26cm 3 ; PA17.57-20.5w MD=26.98±1.61cm 3 , MC=26.05±1.46cm 3 (Figure 6b).

The CSF volume ANCOVA highlighted effects of EE (F(1,41) = 8.1 p = 0.007, η 2 = 0.114) and PA (F(2,41) = 10.67, p < 0.001, η 2 = 0.3) but not for their interaction (F(2,41) = 0.31, p = 0.74, η 2 = 0.009). CSF volume increased with the PA and MD lamb CSF volume was higher than that of MC lambs: PA0.86-2.57w MD=15.70±1.96cm 3 , MC=14.41±1.56cm 3 ; PA4.71-6.57w MD=20.58±1.53cm 3 , MC=18.45±1.77cm 3 ; PA17.57-20.5w MD=23.71±1.78cm 3 , MC=22.41±2.16cm 3 (Figure 6c).

However, the examination of the GM, WM and CSF percentages in the brain showed that they evolved in the same way in MC and MD lambs during their growth (Figure 6).

For the proportion ofWMin the brain, the ANCOVA highlighted an effect of timepoint (F(2,41) = 4.64, p = .015, η2 = 0.153) but not of early experience (F(1,41) = 0.37 p = .55, η2 = 0.006) or their interaction (F(2,41) = 0.73, p = .49, η2 = 0.024). Regardless of early experience the proportion of WM in the brain increased with timepoint.

For the proportion of CSF in the, brain the ANCOVA highlighted no significant effects: early experience (F(1,41) = 0.35 p = .56, η2 = 0.007), timepoint (F(2,41) = 2.89, p = .067, η2 = 0.122), or their interaction (F(2,41) = 0.07, p = .93, η2 = 0.003).

GM-brain structures

For the caudate nucleus volume, the ANCOVA highlighted effects of EE (F(1,27) = 14.86, p < 0.001, η 2 = 0.276) and PA (F(1,27) = 6.97, p = 0.014, η 2 = 0.129) but not for their interaction (F(1,27) = 0.69, p = 0.413, η 2 = 0.013). The volume of the caudate nucleus increased with PA and was larger for MC than MD lambs.

For the hippocampal volume, the ANCOVA highlighted a significant effect of PA (F(1,23) = 39.09, p = 0.001, η 2 = 0.422) but not of EE (F(1,23) = 0.02, p = 0.891, η 2 < 0.001) or their interaction (F(1,23) = 0.35, p = 0.562, η 2 = 0.004). The volume of the hippocampus increased with PA.

For the olfactory bulb volume, the ANCOVA highlighted no significant effects: EE (F(1,19) = 0.68, p = 0.42, η 2 = 0.034), PA (F(1,19) = 0.02, p = 0.885, η 2 = 0.001) and their interaction (F(1,19) = 0.12, p = 0.73, η 2 = 0.006).

For the PAG volume, the ANCOVA highlighted no significant effects: EE (F(1,27) = 2.45, p = 0.129, η 2 = 0.077), PA (F(1,27) = 0.07, p = 0.79, η 2 = 0.002) and their interaction (F(1,27) = 1.36, p = 0.249, η 2 = 0.044).

Microstructural data

Examination of the microstructural data highlighted an impact of early experience in both gray and white matter, with more parameters being impacted in white matter (Figure 7). Indeed, T1w signal intensity and RD were higher in the gray and white matter of motherdeprived lambs than in mother-reared lambs. FA was also impacted in white matter with higher values observed in mother-reared than in mother-deprived lambs. Moreover, the microstructural data changed between the first two timepoints in white matter (increased T1w signal intensity and FA) and in gray matter (increased FA). FA and RD in different white matter areas indicate an impact of early experience and/or timepoint in a restricted number of areas. In the frontal white matter, there was higher FA in mother-reared than in motherdeprived lambs, and in parietal white matter, there was increased FA throughout the experiment. Detailed statistical results are described below.

Gray matter

For the T1w signal intensity of GM relative to CSF, the ANCOVA highlighted a significant effect of EE (F(1,41) = 20.885, p < 0.001, η 2 = 0.034) but not of PA (F(2,41) = 2.09, p = 0.136, η 2 = 0.058) or their interaction (F(2,41) = 2.89, p = 0.067, η 2 = 0.08). GM intensity was higher for MD compared to MC lambs.

For FA in GM, the ANCOVA highlighted a significant effect of PA (F(2,41) = 10.72, p < 0.001, η 2 = 0.319) but not of EE (F(1,41) = 0.588, p = 0.448, η 2 = 0.009) or of their interaction (F(2,41) = 1.914, p = 0.16, η 2 = 0.057). The FA values in GM increased significantly only between PA0.86-2.57w and PA4.71-6.57w (T = -2.53, p = 0.046).

For RD in GM, the ANCOVA highlighted a significant interaction between EE and PA (F(2,41) = 4.41, p = 0.018, η 2 = 0.103) and a significant effect of EE (F(1,41) = 32.68, p < 0.001, η 2 = 0.38) but not of PA (F(2,41) = 1.68, p = 0.198, η 2 = 0.039). The main effect of EE was driven by higher RD values for MD lambs than MC lambs. Investigating the interaction effect by observing the data however, showed that the higher RD values for MD lambs were present at PA0.86-2.57w and PA4.71-6.57w but not PA17.57-20.5w; however, none of the Bonferroni corrected post-hoc tests to investigate this interaction were significant.

White matter:

For the T1w signal intensity of WM relative to CSF, the ANCOVA highlighted significant effects of EE (F(1,41) = 4.67, p = 0.037, η 2 = 0.086), PA (F(2,41) = 3.92, p = 0.028, η 2 = 0.144) but not their interaction (F(2,41) = 0.535, p = 0.59, η 2 = 0.02). The T1w signal intensity in WM was higher for MD than MC lambs. Observation of the data showed that the T1w signal intensity in WM increased between PA0. ; however, none of the Bonferroni corrected post-hoc tests to investigate the effect of PA were significant.

For FA in WM, the ANCOVA highlighted significant effects of EE (F(1,41) = 5.78, p = 0.021, η 2 = 0.065), PA (F(2,41) = 12.02, p < 0.001, η 2 = 0.271) and their interaction (F(2,41) = 4.47, p = 0.018, η 2 = 0.101). Observation of the data showed that MC lambs had higher FA values in WM than MD lambs for PA0.86-2.57w and PA4.71-6.57w but not PA17.57-20.5w; however, none of the Bonferroni corrected post-hoc tests to investigate this interaction were significant.

For RD in WM, the ANCOVA highlighted a significant interaction between EE and PA (F(2,41) = 5.68, p = 0.007, η 2 = 0.157) and a significant effect of EE (F(1,41) = 14.9, p < 0.001, η 2 = 0.206) but not of PA (F(2,41) = 0.27, p = 0.764, η 2 = 0.007). Observation of the data showed that MD lambs had higher RD values in WM than MC lambs for PA0.86-2.57w and PA4.71-6.57w but not PA17.57-20.5w; however, none of the Bonferroni corrected post-hoc tests to investigate this interaction were significant.

WM-brain areas

For FA in the WM-brain area of the frontal lobe, the ANCOVA highlighted a significant effect of EE (F(1,36) = 5.05, p = 0.031, η 2 = 0.117) but not of PA (F(2,36) = 0.80, p = 0.456, η 2 = 0.037) or their interaction (F(2,36) = 0.26, p = 0.771, η 2 = 0.012). MC lambs had higher FA values than MD lambs in the WM-brain area of the frontal lobe. For RD in the WM-brain area of the frontal lobe, the ANCOVA….

For FA in the WM-brain area of the parietal lobe, the ANCOVA highlighted a significant effect of PA (F(2,41) = 4.91, p = 0.031, η 2 = 0.012) but not of EE (F(1,41) = 0.02, p = 0.889, η 2 = 0.002) or their interaction (F(2,41) = 0.85, p = 0.436, η 2 = 0.025). Observation of the data showed that the FA values in this area were higher at PA4.71-6.57w than at PA0.86-2.57w and PA17.57-20.5w; however, none of the Bonferroni corrected post-hoc tests to investigate the effect were significant. For RD in the WM-brain area of the parietal lobe, the ANCOVA highlighted no significant effects: EE (F(1,41) = 0.83, p = 0.368, η 2 = 0.016), PA (F(2,41) = 2.74, p = 0.077, η 2 = 0.104) and their interaction (F(2,41) = 0.79, p = 0.458, η 2 = 0.03).

For FA in the WM-brain area of the temporal lobe, the ANCOVA highlighted no significant effects: EE (F(1,41) = 3.78, p = 0.059, η 2 = 0.08), PA (F(2,41) = 0.37, p = 0.70, η 2 = 0.016) and their interaction (F(2,41) = 0.58, p = 0.566, η 2 = 0.025). Similarly, for RD in the WM-brain area of the temporal lobe, the ANCOVA highlighted no significant effects: EE (F(1,41) = 3.51, p = 0.068, η 2 = 0.07), PA (F(2,41) = 0.57, p = 0.58, η 2 = 0.022) and their interaction (F(2,41) = 1.57, p = 0.220, η 2 = 0.063).

For FA in the WM-brain area of the occipital lobe, the ANCOVA highlighted no significant effects: EE (F(1,41) = 0.69, p = 0.410, η 2 = 0.015), PA (F(2,41) = 1.64, p = 0.207, η 2 = 0.07) and their interaction (F(2,41) = 0.12, p = 0.89, η 2 = 0.005). Similarly, for RD in the WM-brain area of the occipital lobe, the ANCOVA highlighted no significant effects: EE (F(1,41) = 1.15, p = 0.29, η 2 = 0.023), PA (F(2,41) = 2.94, p = 0.06, η 2 = 0.118) and their interaction (F(2,41) = 0.07, p = 0.928, η 2 = 0.003).

For FA in the corpus callosum, the ANCOVA highlighted a significant effect of PA (F(2,32) = 3.29, p = 0.05, η 2 = 0.157) but not of EE (F(1,32) = 0.47, p = 0.498, η 2 = 0.011) or their interaction (F(2,32) = 1.28, p = 0.293, η 2 = 0.061). None of the Bonferroni corrected post-hoc tests investigating the effect of PA were significant. For RD in the corpus callosum, the ANCOVA highlighted no significant effects: EE (F(1,32) = 0.44, p = 0.513, η 2 = 0.012), PA (F(2,32) = 1.12, p = 0.339, η 2 = 0.059) and their interaction (F(2,32) = 0.37, p = 0.695, η 2 = 0.02).

For FA in the optic chiasm, the ANCOVA highlighted no significant effects: EE (F(1,29) = 2.96, p = 0.096, η 2 = 0.077), PA (F(2,29) = 0.46, p = 0.639, η 2 = 0.024) and their interaction (F(2,29) = 2.86, p = 0.074, η 2 = 0.148). Similarly, for FA in the optic chiasm, the ANCOVA highlighted no significant effects: EE (F(1,29) = 2.40, p = 0.132, η 2 = 0.067), PA (F(2,29) = 0.56, p = 0.579, η 2 = 0.031) and their interaction (F(2,29) = 1.44, p = 0.252, η 2 = 0.081).

For FA in the cerebellum, the ANCOVA highlighted no significant effects: EE (F(1,41) = 0.10, p = 0.749, η 2 = 0.002), PA (F(2,41) = 3.07, p = 0.057, η 2 = 0.119) and their interaction (F(2,41) = 0.12, p = 0.892, η 2 = 0.004). Similarly for RD in the cerebellum, the ANCOVA highlighted no significant effects: EE (F(1,41) = 0.10, p = 0.756, η 2 = 0.002), PA (F(2,41) = 2.17, p = 0.127, η 2 = 0.089) and their interaction (F(2,41) = 0.25, p = 0.783, η 2 = 0.01).

Discussion

Brain development and maturation in mothered control lambs

Considering the mothered lambs as the healthy condition of rearing, we expected an increase of brain volume as well as of grey and white matter volumes throughout the experiment. In humans, typical paediatric development is characterized by an increased WM volume and an inverted U-shaped trajectory for the total cerebral and GM volumes in developing children and adolescents [START_REF] Giedd | Structural MRI of Pediatric Brain Development: What Have We Learned and Where Are We Going[END_REF]. Increased total brain volume and percentages of WM and GM have also been observed in rhesus macaques imaged between 2 and 6 months of age [START_REF] Liu | The effects of breastfeeding versus formula-feeding on cerebral cortex maturation in infant rhesus macaques[END_REF]. In our study, we observed the increase of the brain, GM, WM and CSF volumes with similar trajectory-shapes, and that the increase appears less important after 1.5 month especially for the CSF. According to the age of puberty for Ile-de-France ewes, that occurs between 8-10 months depending on the rearing conditions (Perret, 1986), the development of the brain, GM and WM are probably unfinished at the end of our experiment, around 4.5 months. However, focusing on specific brain structures, we noticed that the volumes of olfactory bulbs and PAG are not statistically different between the two last points of acquisition (around 1.5 and 4.5 months) suggesting that the growth of these structures may be almost complete. The examination of GM, WM and CSF percentages relative to the total brain volume revealed that the trajectory for the CSF is stable whereas the trajectories of GM and WM percentages are in mirror. This finding could suggest that GM percentage decreases in favour of an increase in WM percentage. In our study, these values have been evaluated after automatic segmentation based on the MRI T1w signal intensity and contrast, which varied during brain development. These changes of signal intensity are explained by the decrease of water content in GM and WM, and the increase of myelin [START_REF] Barkovich | Normal maturation of the neonatal and infant brain: MR imaging at 1.5[END_REF] which can result in mirror evolutions of GM and WM. Thus, structural parameters may be more informative than the total or percentage volume of brain regions. As suggested by several studies performed in primates, it is relevant to study the brain development/maturation by examining MRI parameters extracted from different types of MRI sequences. First, with the examination of the T1w signal intensity, we hypothesized higher values in WM than GM, and decreasing values in both brain tissues throughout infant development. We found the highest intensity values in the WM that increased between the first two points of acquisition before decreasing. The shape of T1w signal intensity in the GM followed the same shape as that of the WM but with an overall lower intensity. Second, we examined the FA-and RD-values in GM and WM, hypothesizing an increased FA-and decreasing RD-values in both GM and WM. Moreover, we predicted higher FA-values and smaller RD-values in the WM than in the GM. As expected, the evolution of diffusion parameters in WM is coherent with reported data in children [START_REF] Dubois | The early development of brain white matter: A review of imaging studies in fetuses, newborns and infants[END_REF][START_REF] Dubois | Assessment of the early organization and maturation of infants' cerebral white matter fiber bundles: A feasibility study using quantitative diffusion tensor imaging and tractography[END_REF][START_REF] Hüppi | Diffusion tensor imaging of brain development[END_REF] and non-human primate infants [START_REF] Liu | The effects of breastfeeding versus formula-feeding on cerebral cortex maturation in infant rhesus macaques[END_REF]. The FA increased from the first to the last point of acquisition, whereas the RD decreased. These concomitant evolutions in the WM could be the consequences of the myelination of the brain, whereas in the GM the increased FA could be due to a decreasing of free water [START_REF] Mukherjee | Diffusion-Tensor MR Imaging of Gray and White Matter Development during Normal Human Brain Maturation 12[END_REF]. In summary, our study described for the first time the brain development and maturation of mothered lambs, showing that the most evident modifications were found between about 1-2 weeks and 1.5 months. Moreover, our observations asked the question of the best method to evaluate brain development and maturation. In our opinion, it is better to follow the percentage of different brain matter since the brain volume is tightly linked to the body weight in children [START_REF] Giedd | Structural MRI of Pediatric Brain Development: What Have We Learned and Where Are We Going[END_REF], even if the question of the best method to evaluate the brain size could be always questioned [START_REF] Hasboun | Évaluation du volume cérébral : Reproductibilité et précision d'une technique 3D IRM[END_REF]. Interestingly, our results demonstrated that brain maturation is still in progress after 4 months of age, whatever the MRI parameters. Indeed, the T1w-signal intensity value decreased in both grey and white matter, suggesting a modification of brain structure. This idea is also supported by the increased FA in both grey and white matter.

Impact of maternal deprivation

Our study showed for the first time, in lambs, the impact of maternal deprivation on different parameters of brain growth and maturation. Total brain, WM, GM and CSF volumes were all lower in MC than in MD lambs. This is in contrast to study of Rhesus macaques in which no impact of maternal deprivation on brain volume was found at 2, 4 or 6 months of age [START_REF] Liu | The effects of breastfeeding versus formula-feeding on cerebral cortex maturation in infant rhesus macaques[END_REF]. However, we noticed that early experience did not affect the percentages of GM, WM and CSF, which were modified throughout the study. This observation confirmed our question concerning the best indicator of whole brain growth, or even its relevance. Because of their place in the socio-emotional neuronal network [START_REF] Grahn | The cognitive functions of the caudate nucleus[END_REF][START_REF] Menant | Régulation des émotions chez l'animal d'élevage : focus sur les acteurs neurobiologiques[END_REF][START_REF] White | Some highlights of research on the effects of caudate nucleus lesions over the past 200 years[END_REF]) (Zelena et al., 2018;[START_REF] Guesdon | Behavioral evidence of heterospecific bonding between the lamb and the human caregiver and mapping of associated brain network[END_REF], we examined the impact of early experience on OB, CN, Hipp and PAG volumes evaluated at 1 and 4 months of age. We highlighted two relevant information according to our observations. First, the growth of brain structures was not the same for all of them since increased volumes were found only for the CN and Hipp, suggesting that brain development is heterogeneous and not finished at 4 months. Second, we discovered an impact of maternal deprivation on the CN, which was smaller in MD than in MC lambs. These observations were coherent with our hypothesis of a negative impact of maternal deprivation on these brain structures development. For the CN, regarding its role in motor processes and cognitive functions [START_REF] Grahn | The cognitive functions of the caudate nucleus[END_REF][START_REF] White | Some highlights of research on the effects of caudate nucleus lesions over the past 200 years[END_REF], we expected differences between MC and MD lambs because of the role of mother in lambs' cognitive development and activities. This effect of maternal deprivation on CN development may result from the nutrition in early life, since maternal milk is more adapted than formula-milk. This kind of effect was reported in adolescents of preterm birth who were exposed to high-or standard-nutrient regimes [START_REF] Isaacs | The Effect of Early Human Diet on Caudate Volumes and IQ[END_REF] and in early-life stressed humans [START_REF] Cohen | Early Life Stress and Morphometry of the Adult Anterior Cingulate Cortex and Caudate Nuclei[END_REF]. In our study, the lack of impact of maternal deprivation on the Hipp volumes is similar to previous data reported in rhesus monkeys reared with their mother or in a maternal deprivation condition [START_REF] Sánchez | Differential rearing affects corpus callosum size and cognitive function of rhesus monkeys[END_REF][START_REF] Spinelli | Early-life stress induces long-term morphologic changes in primate brain[END_REF] or in humans submitted to early-life stress [START_REF] Cohen | Early Life Stress and Morphometry of the Adult Anterior Cingulate Cortex and Caudate Nuclei[END_REF], whereas lower hippocampal volumes have previously been reported in children exposed to poverty [START_REF] Luby | The Effects of Poverty on Childhood Brain Development: The Mediating Effect of Caregiving and Stressful Life Events[END_REF]. Moreover, we hypothesized that maternal deprivation leads to delay or affect brain maturation. This effect may result from psychological stress with the absence of maternal care and/or from formula-milk being less nutritious than breast milk. Our results are consistent with our hypotheses for the T1w-signal intensity and the diffusion parameters, and with reported data obtained in primates. The global view of GM and WM allowed to identify a negative impact of maternal deprivation on brain maturation that may be more important between about 1.5 week and 1.5 month, several differences no longer existed at about 4.5 months of age. In GM, we noticed higher T1w-signal intensity and higher RD-values in maternal deprived lambs than in mothered lambs especially before 4.5 months. These higher values suggest less maturation, since this process is associated with decreased T1-w signal intensity and decreased RDvalues induced by the incorporation of macromolecules into membranes of neural cells and the decrease of free water [START_REF] Geeraert | A multiparametric analysis of white matter maturation during late childhood and adolescence[END_REF][START_REF] Mukherjee | Diffusion-Tensor MR Imaging of Gray and White Matter Development during Normal Human Brain Maturation 12[END_REF]. The WM seems to be the most impacted part of the brain since the evolution of all structural parameters showed a delay of maturation in maternal deprived lambs. Indeed, the T1w-signal intensity and RD-values were higher and FA-values were smaller in MD than in MC lambs. Similar results have been reported many times whether in formula-fed children [START_REF] Deoni | Early nutrition influences developmental myelination and cognition in infants and young children[END_REF][START_REF] Deoni | Breastfeeding and early white matter development: A crosssectional study[END_REF]; in bonnet macaques born from stressed mother [START_REF] Coplan | The role of early life stress in development of the anterior limb of the internal capsule in nonhuman primates[END_REF] or in formulafed rhesus macaques [START_REF] Liu | The effects of breastfeeding versus formula-feeding on cerebral cortex maturation in infant rhesus macaques[END_REF]. Based on description of T1w-signal intensity, FA-and RD-values evolution in normal brain [START_REF] Barkovich | Normal maturation of the neonatal and infant brain: MR imaging at 1.5[END_REF][START_REF] Flood | Age-Dependent Signal Intensity Changes in the Structurally Normal Pediatric Brain on Unenhanced T1-Weighted MR Imaging[END_REF][START_REF] Mukherjee | Diffusion-Tensor MR Imaging of Gray and White Matter Development during Normal Human Brain Maturation 12[END_REF][START_REF] Westlye | Differentiating maturational and aging-related changes of the cerebral cortex by use of thickness and signal intensity[END_REF], we conclude that maternal deprivation leads to delay in brain maturation in lambs. Even if the impact we observed on white matter may result from less nutritious milk, it may also result from stress. Indeed, whereas no data has described the impact of early life stress on brain development in lambs, a negative impact of gestational stress on neural cells at birth [START_REF] Petit | Stress during pregnancy alters dendritic spine density and gene expression in the brain of new-born lambs[END_REF] and a negative impact of gestational corticosteroid exposure by increasing the number of unmyelinated neurons in corpus callosum [START_REF] Huang | Repeated prenatal corticosteroid administration delays myelination of the corpus callosum in fetal sheep[END_REF] have been reported. Moreover, our results in specific WM-area showed that they were not impacted by the point of acquisition and/or the early experience in the same manners. These differences suggest that it may exist a gradient of WM maturation in sheep as it is described in humans [START_REF] Dubois | The early development of brain white matter: A review of imaging studies in fetuses, newborns and infants[END_REF]. This idea needs to be further explored to better understand the impact of maternal deprivation in lambs since our results allow to conclude a lesser maturation in the WM of the frontal lobe and tendency in the temporal lobe and optic chiasm. A second point needs to be further explored. It concerns the duration of the effect of maternal deprivation, since some difference between MD and MC lambs disappeared at 4.5 months of age. However, the lesser brain maturation seen in maternal deprived lambs before 4.5 months of age could have long-term effects on animal's life.

Conclusions

We described for the first time the brain development and maturation in mothered lambs. These present the same evolution as thus of primates: increased volumes of brain, gray and white matter, increased values of FA and decreased values of RD. The modifications observed for brain development and maturation are more important before 4.5 months of age. Interestingly, we demonstrated that maternal deprivation, currently used in dairy production, leads to strong modification of brain maturation delaying it from as early as 1.5 weeks of age. These alterations observed may help to explain the behavioural, endocrine and health alterations reported in artificial reared lambs.

Aknowledgment:

We thank Olivier Lasserre, Didier Dubreuil and all the staff of the experimental facilities "Unité expérimentale Physiologie Animale de l'Orfrasière'' at INRAE Val de Loire, France for caregiving to lambs and ewes. We thank Emilie Bled and Dominique Girardeau for help with MRI acquisitions. We thank Cédric Lapouge (financed by the SFR Neuroimagerie Fonctionnelle -FED4426) for help with manual segmentation of the brain masks. This work was supported by a grant from Région Centre Val de Loire "Ovin2A" (N˚ 2013 00083140) obtained by R. Nowak.


Stéphane Balac

Monique Dauge

Zoïs Moitier
email: zois.moitier@univ-rennes1.fr

Asymptotic expansions of Whispering Gallery Modes in graded index optical micro-cavities

Keywords: Helmholtz transmission problem, Resonances, WKB method

published or not. The documents may come

Asymptotic expansions of Whispering Gallery Modes in graded index optical micro-cavities

Problem setting

Resonant modes in an optical micro-cavity are particular time-harmonic solutions to the source free Maxwell equations inside and outside the dielectric cavity. We consider a 2D setting, a situation that arises as a simplification of the 3D resonance problem, e.g. by using the effective index approach. It is well know that the 2D Maxwell setting is the combination of two subsystems of equations referred to as transverse electric (TE) and transverse magnetic (TM). For brevity we report here only for the TM case.

We denote by Ω the bounded domain in R 2 describing the dielectric cavity. The optical index n is 1 outside Ω and coincides with a smooth function n > 1 in Ω. The resonance problem can be formulated as:

Find (k, u) ∈ C × H 2 loc (R 2 ) with u = 0 such that -∆u -k 2 n 2 u = 0 in R 2 , (1a) u(r, θ) = m∈Z c m H (1) m (kr) e imθ r > R 0 . (1b)
Here H

m refers to Hankel function of the first kind of order m. Equation (1b) expresses the ra-diation condition at infinity in polar coordinates for R 0 large enough. For real k, it corresponds to the outgoing Sommerfeld's condition. It is known that the solutions (k, u) to problem [START_REF] Johnson | Theory of morphology dependent resonances: shape resonances and width formulas[END_REF] are such that k has a negative imaginary part.

For a circular cavity with constant index, asymptotic expansions of the resonances can be obtained using expansions of Bessel functions [START_REF] Johnson | Theory of morphology dependent resonances: shape resonances and width formulas[END_REF]. When n is not constant in Ω, this approach is not applicable.

Disk cavity with radially varying index

Let R be the radius of the disk and assume n is a smooth radial function r → n(r). Then problem (1) can be reduced to a family of 1D radial problems depending on an integer m ∈ Z referred as the polar mode index :

- 1 m 2 Lw + V w = λw (4)
where

V (r) = n(R) R n(r) r 2 -1 is an effective poten- tial, k = m Rn(R) √ 1 + λ and Lw = n(R) 2 R 2 n(r) 2 r (rw ) is an elliptic operator. Since V (R -) = 0 and V (R + ) = n(R) 2 -1 > 0, we have a potential barrier at r = R. We note that V (R -) = -2κ where κ = 1 R + n (R) n(R) . (5)
We have identified three typical behaviors [START_REF] Moitier | Étude mathématique des résonances dans une micro-cavité optique[END_REF] for the solutions of (4), depending on the sign of κ, relying on the spectral theory of Schrödinger operators.

a) Half-triangular potential well. If κ > 0 then V is decreasing in a left neighborhood of R and has a local minimum at R. We have obtained an asymptotic expansion of the resonances in the form k = m K a (m -1 3 ) for a function K a in C ∞ ([0, 1]) with determined Taylor expansion at 0. Moreover, the (quasi)mode u localizes at the boundary.

b) Half-quadratic potential well.

If κ = 0, under the additional condition 2 R 2 -n (R) n(R) > 0 the effective potential V has a local minimum at R. Our expansion is now k = m K b (m -1 2 ) for another function K b ∈ C ∞ ([0, 1]
). Again, the (quasi)mode u localizes at the boundary. c) Internal quadratic potential well. If κ < 0, since lim r 0 V (r) = +∞, the effective potential V has at least a global minimum r 0 in (0, R). Under the condition 2 r 2 0 -n (r 0 ) n(r 0 ) > 0 and that r 0 is the unique global minimum of V , our expansion has again the form k = m K c (m -1

2 ) with a different function K c ∈ C ∞ ([0, 1]), and now u localizes inside the cavity around r = r 0 .

General cavity with variable index

In the general case, the phase function is not known and we have used a phase amplitude ansatz, the famous WKB method, to find asymptotic expressions of the resonances in tubular coordinates along the boundary of Ω [START_REF] Moitier | Étude mathématique des résonances dans une micro-cavité optique[END_REF]. This leads to an eikonal equation coupled with a Schrödinger equation. A small parameter h appears naturally, and has to be quantized so that the phase is well-defined, giving rise to a generalized notion of polar mode index. We have constructed quasi-resonances (k, u) in the sense of [START_REF] Tang | From quasimodes to resonances[END_REF] in a similar, but more general, form than in case a) above, when the condition κ + ∂ν n n > 0 is fulfilled all along ∂Ω where κ is the curvature of ∂Ω. The asymptotic expansions are computed using a computer algebra system.

The advantage of the asymptotic formulas we have obtained is twofold. They provide accurate approximations of resonances at high frequencies when the use of standard numerical approximation is difficult. For lower frequencies, combined with finite element (FE) computations, they provide information on the localization of the resonances in the FE matrix spectrum.

Numerical Experiments

For numerical illustration we consider an elliptic cavity with perimeter L and constant index n. Our 4-term asymptotic expansion of k reads

k [4] j (m) = 2πm Ln 1 + a j 2 κ 2 3 h 2 3 m - n κ 2 √ n 2 -1 h m + a 2 j 12 κ 2 3 2 - κ 4 3 10 - 4 45 κ 2 κ -8 3 h 4 3 m - a j n 0 12 √ n 2 -1 κ 2 3 κ + κ 5 3 n 2 -1 h 5 3 m
as m → +∞, where h m = L πm and • is the mean value along ∂Ω. Here j is a natural integer (radial mode index) and a j is the j-th zero of the reverse Airy function x → Ai(-x).

On Figure 1, we have compared our N -terms asymptotic expansion to FE computation for an ellipse of semi-major axis 1 and eccentricity 0.5 with constant index n = 5. We have used a Perfectly Matched Layer with a structured quadrangular mesh of geometric degree 3 and a FE space of degree 7 with 64156 dofs. For N = 0, k

  ) plus one corrective term, etc. Note that, problem (1) has two solutions for each m.

Figure 1 :

 1 Figure 1: Relative difference between FE resonances and N -terms asymptotic expansions, for j = 0. The slopes of the solid lines are -N +2 3 .

This work has been performed within the framework of the CNRS INFINITI project SONATINE in collaboration with FOTON Laboratory. The authors also acknowledge support of the Centre Henri Lebesgue ANR-11-LABX-0020-01.