Eun Young Choi

Lu Tian

Jason H Su

Matthew T Radovan

Thomas Tourdias

Tammy T Tran

Alexandra N Trelle

Elizabeth Mormino

Anthony D Wagner

Brian K Rutt
email: brutt@stanford.edu

Thalamic nuclei atrophy at high and heterogenous rates during cognitively unimpaired human aging

Keywords: Conceptualization, Methodology, Software, Formal analysis, Data curation, Writing -original draft, Visualization, Supervision, Project administration, Funding acquisition Thalamus, Thalamic nuclei, Atrophy, Cognitively unimpaired aging, White matter nulled imaging, THOMAS segmentation

The thalamus is a central integration structure in the brain, receiving and distributing information among the cerebral cortex, subcortical structures, and the peripheral nervous system. Prior studies clearly show that the thalamus atrophies in cognitively unimpaired aging. However, the thalamus is comprised of multiple nuclei involved in a wide range of functions, and the age-related atrophy of individual thalamic nuclei remains unknown. Using a recently developed automated method of identifying thalamic nuclei (3T or 7T MRI with white-matter-nulled MPRAGE contrast and THOMAS segmentation) and a cross-sectional design, we evaluated the age-related atrophy rate for 10 thalamic nuclei (AV, CM, VA, VLA, VLP, VPL, pulvinar, LGN, MGN, MD) and an epithalamic nucleus (habenula). We also used T1-weighted images with the FreeSurfer SAMSEG segmentation method to identify and measure age-related atrophy for 11 extra-thalamic structures (cerebral cortex, cerebral white matter, cerebellar cortex, cerebellar white matter, amygdala, hippocampus, caudate, putamen, nucleus accumbens, pallidum, and lateral ventricle). In 198 cognitively unimpaired participants with ages spanning 20-88 years, we found that the whole thalamus atrophied at a rate of 0.45% per year, and that thalamic nuclei had widely varying age-related atrophy rates, ranging from 0.06% to 1.18% per year. A functional grouping analysis revealed that the thalamic nuclei involved in cognitive (AV, MD; 0.53% atrophy per year), visual (LGN, pulvinar; 0.62% atrophy per year), and auditory/vestibular (MGN; 0.64% atrophy per year) functions atrophied at significantly higher rates than those involved in motor (VA, VLA, VLP, and CM; 0.37% atrophy per year) and somatosensory (VPL; 0.32% atrophy per year) functions. A proximity-to-CSF analysis showed that the group of thalamic nuclei situated immediately adjacent to CSF atrophied at a significantly greater atrophy rate (0.59% atrophy per year) than that of the group of nuclei located farther from CSF (0.36% atrophy per year), supporting a growing hypothesis that CSF-mediated factors contribute to neurodegeneration. We did not find any significant hemispheric differences in these rates of change for thalamic nuclei. Only the CM thalamic nucleus showed a sex-specific difference in atrophy rates, atrophying at a greater rate in male versus female participants. Roughly half of the thalamic nuclei showed greater atrophy than all extra-thalamic structures examined (0% to 0.54% per year). These results show the value of white-matter-nulled MPRAGE imaging and THOMAS segmentation for measuring distinct thalamic nuclei and for characterizing the high and heterogeneous atrophy rates of the thalamus and its nuclei across the adult lifespan. Collectively, these methods and results advance our understanding of the role of thalamic substructures in neurocognitive and disease-related changes that occur with aging.

Introduction

Atrophy of brain tissue is a classic hallmark of aging. Brain atrophy reflects the shrinkage and loss of neurons and the connections between them [START_REF] Fjell | Structural brain changes in aging: courses, causes and cognitive consequences[END_REF]. While brain atrophy is a prominent feature in many brain diseases, atrophy can also be seen both broadly and focally in the cognitively unimpaired aging brain [START_REF] Fjell | Structural brain changes in aging: courses, causes and cognitive consequences[END_REF]. Characterizing brain structural changes during cognitively unimpaired aging will inform efforts to understand the differences between normal and disease-related atrophy, the structural changes that can be tolerated before they significantly impact function, and the resiliency of the cognitively unimpaired aging brain.

Within the brain, the thalamus is a central integration structure, receiving and distributing information among the cerebral cortex, subcortical structures, and the peripheral nervous system via the brainstem [START_REF] Steriade | Thalamus[END_REF]. This high degree of integration is accomplished in part by thalamic projection neurons that send long-range outputs with collateral projections and large ramifications at their targets [START_REF] Steriade | Thalamus[END_REF]. This processing is organized by intra-thalamic nuclei dedicated to distinct functions, including cognitive, visual, auditory, vestibular, and sensorimotor functions [START_REF] Steriade | Thalamus[END_REF]. Lesions of thalamic nuclei lead to profound disruptions in the modulation of behaviors subserved by these functional domains [START_REF] Carrera | The thalamus and behavior: effects of anatomically distinct strokes[END_REF]. As a highly connected and integrative structure in the brain, the thalamus is in a critical but also vulnerable position that may show prominent structural changes resulting from age-related changes across the brain, such as degeneration due to reduced connectivity or accumulated oxidative stress from high levels of neuronal activity.

Current knowledge of age-related thalamic volumetric changes in humans comes primarily from neuroimaging studies which show that the thalamus as a whole significantly atrophies over the cognitively unimpaired adult lifespan [START_REF] Cherubini | Aging of subcortical nuclei: microstructural, mineralization and atrophy modifications measured in vivo using MRI[END_REF][START_REF] Fjell | Critical ages in the life course of the adult brain: nonlinear subcortical aging[END_REF][START_REF] Goodro | Age effect on subcortical structures in healthy adults[END_REF][START_REF] Hughes | Regional changes in thalamic shape and volume with increasing age[END_REF][START_REF] Long | Healthy aging: an automatic analysis of global and regional morphological alterations of human brain[END_REF][START_REF] Murphy | Sex differences in human brain morphometry and metabolism: an in vivo quantitative magnetic resonance imaging and positron emission tomography study on the effect of aging[END_REF][START_REF] Murphy | Volumetric magnetic resonance imaging in men with dementia of the Alzheimer type: correlations with disease severity[END_REF][START_REF] Narvacan | Evolution of deep gray matter volume across the human lifespan[END_REF][START_REF] Nyberg | Longitudinal evidence for diminished frontal cortex function in aging[END_REF][START_REF] Raji | Age, Alzheimer disease, and brain structure[END_REF][START_REF] Schippling | Global and regional annual brain volume loss rates in physiological aging[END_REF][START_REF] Squarzoni | Reduced gray matter volume of the thalamus and hippocampal region in elderly healthy adults with no impact of APOE varepsilon4: a longitudinal voxel-based morphometry study[END_REF][START_REF] Su | Sparse representation of brain aging: extracting covariance patterns from structural MRI[END_REF][START_REF] Takahashi | Gender and age differences in normal adult human brain: voxel-based morphometric study[END_REF][START_REF] Tullo | MR-based age-related effects on the striatum, globus pallidus, and thalamus in healthy individuals across the adult lifespan[END_REF][START_REF] Van Der Werf | Thalamic volume predicts performance on tests of cognitive speed and decreases in healthy aging. A magnetic resonance imaging-based volumetric analysis[END_REF][START_REF] Walhovd | Effects of age on volumes of cortex, white matter and subcortical structures[END_REF][START_REF] Walhovd | Consistent neuroanatomical age-related volume differences across multiple samples[END_REF][START_REF] Yoo | The importance of aging in gray matter changes within tinnitus patients shown in cortical thickness, surface area and volume[END_REF][START_REF] Zheng | Age-related changes in cortical and subcortical structures of healthy adult brains: a surface-based morphometry study[END_REF]. Postmortem histological studies afford a higher resolution characterization of the volumes of thalamic nuclei. However, while there are a number of postmortem human histological studies showing that thalamic nuclear volumes and neuronal counts are reduced in disease [START_REF] Byne | Postmortem assessment of thalamic nuclear volumes in subjects with schizophrenia[END_REF][START_REF] Popken | Subnucleus-specific loss of neurons in medial thalamus of schizophrenics[END_REF][START_REF] Young | Reduced number of mediodorsal and anterior thalamic neurons in schizophrenia[END_REF], to our knowledge, there are no such studies of cognitively unimpaired human aging. Thus, as yet, we do not know the rates of volumetric change of specific thalamic nuclei over the cognitively unimpaired adult lifespan [START_REF] Fama | Thalamic structures and associated cognitive functions: relations with age and aging[END_REF]. Such studies would identify whether age-related neurodegeneration is occurring heterogeneously within the thalamus, and therefore whether the concept of "regional vulnerability" [START_REF] Small | Isolating pathogenic mechanisms embedded within the hippocampal circuit through regional vulnerability[END_REF] applies to the thalamus.

Recently, our group developed an automated image segmentation method that non-invasively and accurately identifies 10 thalamic nuclei and two adjacent structures. This method uses 3T or 7T MRI with white-matter-nulled magnetization prepared rapid acquisition gradient echo (WMnMPRAGE) contrast, which provides higher contrast in the thalamus than conventional MPRAGE imaging in which the cerebrospinal fluid (CSF) is nulled [START_REF] Saranathan | Optimization of white-matternulled magnetization prepared rapid gradient echo (MP-RAGE) imaging[END_REF][START_REF] Tourdias | Visualization of intra-thalamic nuclei with optimized white-matter-nulled MPRAGE at 7T[END_REF]. WMnMPRAGE image contrast is strong enough to manually identify individual thalamic nuclei and adjacent structures [START_REF] Tourdias | Visualization of intra-thalamic nuclei with optimized white-matter-nulled MPRAGE at 7T[END_REF]. Our automated thalamic segmentation method, known as THOMAS (THalamus Optimized Multi-Atlas Segmentation), was trained using manual delineations of these identifiable structures in a set of 20 training volumes to create a multi-atlas label-fusion segmentation tool that accurately identifies 10 thalamic nuclei and two adjacent structures [START_REF] Su | Thalamus Optimized Multi Atlas Segmentation (THOMAS): fast, fully automated segmentation of thalamic nuclei from structural MRI[END_REF]. THOMAS has been used to characterize the volumes of thalamic nuclei in cognitively unimpaired versus patient populations of multiple sclerosis [START_REF] Planche | Whitematter-nulled MPRAGE at 7T reveals thalamic lesions and atrophy of specific thalamic nuclei in multiple sclerosis[END_REF], alcoholism and HIV [START_REF] Zahr | Sensitivity of ventrolateral posterior thalamic nucleus to back pain in alcoholism and CD4 nadir in HIV[END_REF], and essential tremor [START_REF] Su | Improved Vim targeting for focused ultrasound ablation treatment of essential tremor: a probabilistic and patient-specific approach[END_REF].

In the present study, our principal goal was to characterize the structural changes of thalamic nuclei over the adult lifespan in cognitively unimpaired participants using a cross-sectional approach. This was done by applying THOMAS to segment thalamic nuclei within WMnMPRAGE image volumes acquired from a large group of 198 cognitively unimpaired adult participants with ages ranging from 20 to 88 years. Our secondary goal was to compare the atrophy of thalamic nuclei with those of extra-thalamic regions, with the latter segmented using conventional T1-weighted images of the same participants and the Sequence Adaptive Multimodal SEGmentation (SAMSEG) tool within the FreeSurfer suite of neuroimaging analysis tools. We employed an exponential regression model to estimate atrophy rates (annualized percent change in volume) and differences in atrophy rates among thalamic nuclei and extra-thalamic structures.

Material and methods

Participants

198 Cognitively unimpaired adult participants (86 M, 112 F) were recruited, with a mean age of 59.4 years and an age range of 20 to 88 years at the time of imaging (Table 1). Older participants (>=60 years of age, n = 143, 66 M, 77 F) were recruited at Stanford University as part of the Stanford Aging and Memory Study (SAMS) [START_REF] Trelle | Hippocampal and cortical mechanisms at retrieval explain variability in episodic remembering in older adults[END_REF][START_REF] Trelle | Association of CSF biomarkers with hippocampal-dependent memory in preclinical Alzheimer Disease[END_REF]. Younger participants (<60 years of age) were recruited at Stanford University (n = 17, 9 M, 8 F) and at the University of Bordeaux (n = 38,11 M,27 F). Written informed consent was obtained for all participants and study procedures were approved by each institution's Institutional Review Board. Participants were screened against neurological and psychiatric disease. Older participants were additionally screened to have a Clinical Dementia Rating (CDR) [START_REF] Morris | The Clinical Dementia Rating (CDR): current version and scoring rules[END_REF] score of zero and age-appropriate normal cognitive functioning. The latter was based on a neuropsychological assessment battery that examined multiple domains, including episodic memory, attention, executive function, visuospatial processing, and language. Participants were classified as cognitively unimpaired by a team of neurologists and neuropsychologists, based on test scores falling within 1.5 standard deviations of demographically adjusted means; for details, see [START_REF] Trelle | Hippocampal and cortical mechanisms at retrieval explain variability in episodic remembering in older adults[END_REF].

Image acquisition

MRI parameters for image acquisition are summarized in Table 2. Stanford participants were scanned with a 3D T1-weighted inversion-prepared fast spoiled gradient echo protocol at 3T and a WMnMPRAGE protocol at 7T, using 32-channel head coils in both cases. Bordeaux participants were imaged with 3D T1-weighted inversion-prepared fast spoiled gradient echo and WMnMPRAGE sequence protocols, both acquired at 3T using a 32-channel head coil. T1-weighted images were acquired from Stanford older participants using the following scan parameters: 3T GE MR750 scanner, 3D BRAVO base sequence, sagittal acquisition, echo time (TE) 2.8 ms, repetition time (TR) 7.3 ms, inversion time (TI) 450 ms, flip angle (FA) 12°, receive bandwidth (RBW) +/-41.67 kHz, spatial resolution 0.9 mm isotropic, 186 slices per volume, ARC parallel imaging acceleration: 1.75, scan time 4.8 min.

T1-weighted images were acquired from Stanford younger participants using the following scan parameters: 3T GE MR750 scanner, 3D MPRAGE base sequence, coronal acquisition, TE 3.9 ms, TR 8.5 ms, TI 1200 ms, sequence time (time between inversion pulses: TS) 3700 ms, FA 6°, RBW +/-19.2 kHz, spatial resolution 0.8 mm isotropic, 250 slices per volume, ARC parallel imaging acceleration: 3, scan time 4 min. T1-weighted images were acquired from the Bordeaux cohort using one of two scan protocols: 1) for 22 participants of this cohort: 3T GE MR750w scanner, 3D BRAVO base sequence, sagittal acquisition, TE 3.2 ms, TR 8.5 ms, TI 400 ms, FA 11°, RBW +/-31.25 kHz, spatial resolution 1 mm isotropic, 180 slices per volume, scan time 10 min; or 2) for 16 participants of this cohort: 3T Canon Galan scanner, 3D MPRAGE base sequence, sagittal acquisition, TE 3.0 ms, TR 7.0 ms, TI 950 ms, FA 9°, RBW +/-51.3 kHz, spatial resolution 0.6 mm isotropic, 234 slices per volume, SPEEDER parallel imaging acceleration: 1.5; scan time 4.5 min.

WMnMPRAGE images were acquired from all Stanford participants using the following parameters: 7T GE MR950 scanner, 3D MPRAGE base sequence, coronal acquisition, TE 4.6 ms, TR 10.1 ms, TI 680 ms, TS 5500 ms, views per segment 240, FA: 4°, RBW +/-11.9 kHz, spatial resolution 1 mm isotropic, 220 slices per volume, k-space ordering; 2D radial fanbeam, scan time 11.1 min. WMnMPRAGE images were acquired from the Bordeaux cohort using one of two scan protocols: (1) for 22 participants of this cohort: 3T GE MR750w scanner, MPRAGE base sequence, coronal acquisition, TE 4.7 ms, TR 11.1 ms, TI 500 ms, TS 4500 ms, views per segment 240, FA: 7°, RBW +/-11.9 kHz, spatial resolution 1 mm isotropic, 220 slices per volume k-space ordering; 2D radial fanbeam, ARC parallel imaging acceleration: 1.25×1.25, scan time 7.2 min; or (2) for 16 participants of this cohort: 3T Canon Galan scanner, 3D MPRAGE base sequence, axial acquisition, TE 3.6 ms, TR 7.8 ms, TI 470 ms, TS 4500 ms, FA 7°, RBW +/-53.4 kHz, spatial resolution 0.8 mm isotropic, 200 slices per volume, SPEEDER parallel imaging acceleration: 1.3 × 1.6; scan time 11.3 min. See [START_REF] Saranathan | Optimization of white-matternulled magnetization prepared rapid gradient echo (MP-RAGE) imaging[END_REF] for further details.

T1-weighted and WMnMPRAGE image volumes were visually inspected to ensure that scans were of sufficient quality for analysis and were not corrupted by motion artifact.

Segmentations

Whole-brain WMnMPRAGE volumes were processed with the THOMAS thalamic segmentation tool with no preprocessing. The THOMAS algorithm applied to WMnMPRAGE images has been validated against manual segmentation [START_REF] Su | Thalamus Optimized Multi Atlas Segmentation (THOMAS): fast, fully automated segmentation of thalamic nuclei from structural MRI[END_REF]. Using version v0 of this tool (https://github.com/sujason/thomas), we segmented and extracted the volumes of 12 lateralized structures in each hemisphere of the brain: whole thalamus, ten thalamic nuclei (anteroventral ), and one adjacent epithalamic structure, the habenula (Hb) (Fig. 1A). Note that THOMAS segments the whole thalamus separately from the thalamic nuclei; this whole thalamus encompasses all these preceding structures, as well as the mammillothalamic tract and some additional unlabeled thalamic areas (i.e., between segmented thalamic nuclei). See [START_REF] Tourdias | Visualization of intra-thalamic nuclei with optimized white-matter-nulled MPRAGE at 7T[END_REF], [START_REF] Saranathan | Optimization of white-matternulled magnetization prepared rapid gradient echo (MP-RAGE) imaging[END_REF], and Su et al. (2019) for additional details.

Whole-brain T1-weighted gradient echo volumes were processed with the FreeSurfer 7.1.1 pipeline with no preprocessing. Using the FreeSurfer tool SAMSEG (https:// surfer.nmr.mgh.harvard.edu/fswiki/Samseg), we segmented and extracted the volumes of 11 structures on each side of the brain: nucleus accumbens, amygdala, caudate, cerebellar cortex, cerebellar white matter, cerebral cortex, cerebral white matter, hippocampus, lateral ventricle, pallidum, and putamen (Fig. 1B). SAMSEG produces volumes via both probabilistic and discrete voxel segmentation summations; we used the volumes calculated via the probabilistic method. The SAMSEG algorithm has been validated using T1-weighted images from different scanners and acquired with different scan parameters, using expert manual segmentation of these images as ground truth. SAMSEG was shown to have the highest average Dice score of the five segmentation methods studied (SAMSEG, FreeSurfer, PICSL MALF, BrainFuse, and majority voting) [START_REF] Puonti | Fast and sequence-adaptive whole-brain segmentation using parametric Bayesian modeling[END_REF]. Note that SAMSEG produces a whole thalamus segmentation, but we used the THOMAS segmentation of the whole thalamus because it includes the LGN and MGN, two major thalamic nuclei, whereas the SAMSEG whole thalamus does not. In addition, the thalamic contrast is higher in WMnMPRAGE images than in conventional T1-weighted images; thus we expect the THOMAS segmentation of thalamic boundaries to be more accurate.

Intracranial volume (ICV) estimates were obtained from 3T whole-brain T1-weighted gradient echo volumes using SPM12's Segment and Tissue Volume Utility modules (https:// www.fil.ion.ucl.ac.uk/spm/software/spm12) [START_REF] Malone | Accurate automatic estimation of total intracranial volume: a nuisance variable with less nuisance[END_REF] (Fig. 1C). The accuracy of all segmentations was checked visually for each subject.

Statistical analyses

For each structure in the left and right hemispheres, we employed an exponential regression model to quantify the association between brain structural volume and age, adjusting for education, cohort, sex, and ICV, yielding the rate of age-related atrophy. In particular, we estimated the volume of each structure as a function of age, years of education, cohort, sex, and ICV using the following model:

E(Y | age, education, coℎort, sex, ICV ) = exp β 0 + β 1 age + β 2 education +β 3 coℎort + β 4 sex + γ 1 logICV = exp β 0 + β 1 age + β 2 education +β 3 coℎort + β 4 sex × ICV γ 1 , which is equivalent to E Y ICV γ 1 | age, education, coℎort, sex = exp β 0 + β 1 age + β 2 education + β 3 coℎort + β 4 sex ,
where Y is the volume; education is in years; cohort = 0 for Stanford and 1 for Bordeaux, sex = 0 for female and 1 for male; and ICV is the total intracranial volume. Note: our principal analysis did not include education and cohort as covariates, but we did examine the influence of including these variables on our primary conclusions, described in more detail in Results and Discussion sections. An advantage of this exponential regression model is that the structure's relative (percent) volumetric change per year (defined as atrophy rate) is constant over the lifespan and represented by exp(β 1 ) -1. A Wald test was performed to test against the null hypothesis that there is no association between volumetric change and age (i.e., H 0 : β 1 = 0). In the Wald test, the standard error of the estimated coefficient was assessed by nonparametric bootstrap. We compared the atrophy rates of two structures by estimating the difference in the β 1 coefficients from the regression models of the two structures. The standard error of the estimated difference was assessed by nonparametric bootstrap, accounting for correlations between the volumes of structures from the same individual. Significant differences between the atrophy rates of structures were assessed with paired t-tests, adjusted for multiple comparisons. Hemisphere-and sexspecific differences in atrophy rates were similarly assessed by comparing the corresponding regression coefficients of age.

For the analysis assessing the functional specificity of thalamic nuclei atrophy, we grouped thalamic nuclei into six broad functional groups, based on their known anatomical connectivity [START_REF] Steriade | Thalamus[END_REF]; that is, based on the known functions of the brain regions connected to each nucleus. These broad functional groups are cognitive (left+right AV and MD), visual (left+right LGN and pulvinar), auditory and vestibular (left+right MGN), motor (left+right VA, VLA, VLP, and CM), and somatosensory (left+right VPL).

For the proximity-to-CSF analysis, we grouped thalamic nuclei into two groups based on their proximity to the nearest ventricular surface: an "adjacent" group (left+right AV, MD, MGN, LGN, and pulvinar) and a "remote" group (left+right CM, VA, VLA, VLP, and VPL).

For each of these group analyses, we summed the volumes of all nuclei within a group and across both hemispheres, to create a single "composite" volume, and fit this composite volume with our regression model to obtain its atrophy rate. We then performed the Wald test against the null hypothesis that the atrophy rate was zero, and pairwise comparisons between groups to test for differences in their atrophy rates.

To adjust for multiple comparisons, the simultaneous confidence region for parameters of interest (e.g., regression coefficient of age corresponding to a group of structural volumes of interest) was constructed, where the covariance between relevant estimates was estimated by nonparametric bootstrap, and the corresponding p-values controlling the family-wise error rate via the supremum Wald test statistic were reported. The statistical analyses were conducted using R 4.1.1 (The R Foundation for Statistical Computing).

Data/code availability statement

Data and code are available upon request.

Results

Different thalamic nuclei show widely different age-related atrophy rates across the cognitively unimpaired adult lifespan

We found significant age-related atrophy of most of the thalamic nuclei (Fig. 2). Absolute thalamic nuclear volumes varied approximately linearly with age and with different atrophy rates for different nuclei. The following lists the THOMAS structures in descending order of annualized atrophy rate (Table 3). Left hemisphere: AV (1.18%), LGN (0.72%), MGN (0.67%), pulvinar (0.65%), CM (0.62%) VA (0.51%), MD (0.43%), VPL (0.36%), and VLP (0.30%). Right hemisphere: AV (1.08%), LGN (0.77%), CM (0.62%), MGN (0.61%), pulvinar (0.58%), VA (0.50%), MD (0.40%), VLP (0.34%), and VPL (0.28%). Atrophy rates of left and right VLA and habenula were not significantly different from zero. AV was notable in having a significantly greater atrophy rate than all other ipsilateral thalamic nuclei, except for the LGN in the right hemisphere (Supp. Fig. 1). These atrophy rates were not significantly changed after adjusting for years of education or cohort (Supp. Fig. 2). We also qualitatively observed that the older cohort had higher atrophy rates compared to those of the younger cohort (Fig. 2). We therefore assessed whether the atrophy rates for younger (Bordeaux and Stanford) or older cohorts differed from those of the whole group. We found that the younger cohorts together showed low, mostly non-significant atrophy rates, while the older cohort showed mostly significant atrophy rates that were similar to those of the whole group (Supp. Fig. 4 and Supp. Table 1).

The atrophy of thalamic nuclei segregates by function

We next examined whether the atrophy rates of thalamic nuclei were specific to function. This was hypothesized based on literature demonstrating that age-related atrophy occurs heterogeneously across functionally distinct cortical areas, with the greatest atrophy seen in frontal and temporal cortices and the least atrophy seen in sensorimotor and occipital cortices [START_REF] Fjell | Structural brain changes in aging: courses, causes and cognitive consequences[END_REF]Fjell et al., 2009a;[START_REF] Raz | Selective aging of the human cerebral cortex observed in vivo: differential vulnerability of the prefrontal gray matter[END_REF]. Thalamic nuclei were grouped into broad functional groups: cognitive (AV and MD), visual (LGN and pulvinar), auditory and vestibular (MGN), motor (VA, VLA, VLP, and CM), and somatosensory (VPL). We found a significant segregation of atrophy rates by function.

The cognitive, visual, and auditory/vestibular groups showed the strongest atrophy rates, which ranged from 0.53% to 0.64% (Table 4). These were significantly different (adjusted p<0.006; Table 5) from the atrophy rates of the motor and somatosensory groups, which were 0.37% and 0.32%, respectively.

Thalamic nuclei that are adjacent to the ventricles show greater age-related atrophy

Prior work has found evidence in multiple sclerosis that thalamic nuclei adjacent to the ventricles show greater atrophy than nuclei that are located farther away from the ventricles [START_REF] Lassmann | Pathogenic mechanisms associated with different clinical courses of multiple sclerosis[END_REF][START_REF] Planche | Whitematter-nulled MPRAGE at 7T reveals thalamic lesions and atrophy of specific thalamic nuclei in multiple sclerosis[END_REF]. Here, we examined whether there was evidence that a similar effect occurs in cognitively unimpaired aging. To test this, thalamic nuclei were grouped into two groups -those that were adjacent to the ventricles (AV, MD, LGN, pulvinar, and MGN) and those that were non-adjacent (i.e., remote) to the ventricles (CM, VA, VLA, VLP, and VPL). We found a significant difference of atrophy rates based on ventricular proximity: the adjacent group showed a higher atrophy rate (0.59% per year), which was significantly different (adjusted p<0.001) from that of the remote group (0.36% per year).

The thalamus and its nuclei have a high rate of atrophy in comparison to other brain structures across the cognitively unimpaired adult lifespan

For the extra-thalamic structures, we found that the absolute volumes of certain gray matter structures decreased approximately linearly with age, those of white matter structures did not significantly change with age, and those of the lateral ventricles increased nonlinearly with age (Fig. 3). Fig. 4 shows the percent change in volume per year for each structure; Table 3 shows these atrophy rates and their confidence bounds and p-values, adjusted for multiple comparisons. These atrophy rates were not significantly changed after adjusting for years of education or cohort (Supp. Fig. 2).

Significant differences between atrophy rates were assessed with paired t-tests, adjusted for multiple comparisons (Supp. Fig. 3). Notably, the left and right whole thalamic atrophy rates were higher than those of all extra-thalamic structures with the exception of the nucleus accumbens. Specifically, the atrophy rate of the left thalamus (0.45%) was significantly greater than that of all other significantly atrophying structures in the left hemisphere (Table 3), those being cerebellar cortex (0.24%), putamen (0.19%), and cerebral cortex (0.15%). The exception to this was the left nucleus accumbens (0.44%), which had an atrophy rate similar to (but not significantly different from) that of the left thalamus. The following left hemispheric structures had no statistically significant atrophy with age: cerebral white matter, cerebellar white matter, amygdala, hippocampus, caudate, and pallidum. We note that we do observe significant atrophy of these structures in the older cohort alone (Supp. Table 1). Similarly, the atrophy rate of the right thalamus (0.45%) was significantly greater than all other significantly atrophying right hemispheric structures, those being putamen (0.29%), cerebellar cortex (0.25%), caudate (0.21%), and cerebral cortex (0.15%). The exception to this was the right nucleus accumbens, whose atrophy rate (0.54%) was similar to that of the right thalamus. The following right hemispheric structures had no statistically significant atrophy with age: cerebral white matter, cerebellar white matter, amygdala, hippocampus, and pallidum; again, we note that the atrophy rates of these structures are significantly greater than zero when analyzing the older cohort alone (Supp. Table 1).

Roughly half of the thalamic nuclei had higher rates of atrophy than all of the extra-thalamic structures examined in both the left and right hemispheres (Fig. 4).

We also found that the left and right lateral ventricles significantly increased in volume with age (left: 1.74% volume increase per year; right: 1.61%). The volumetric expansion of the lateral ventricle was nonlinear with age, increasing strongly for ages greater than 60 years.

Virtually no hemispheric or sex differences in the age-dependent changes of thalamic and extra-thalamic structures

Based on pairwise t-tests between the left and right volumes for each nucleus, adjusted for multiple comparisons, we did not observe any significant hemisphere-specific differences in atrophy rates for any thalamic or extra-thalamic structures.

Sex was entered as an independent variable in the exponential regression models described above for both intra-and extra-thalamic structures. Only one structure showed a sex-specific difference in atrophy rate. For the left CM nucleus of the thalamus, male participants showed a significantly greater atrophy rate than female participants (male, 0.86%; female, 0.41%; p = 0.034, adjusted for multiple comparisons).

Discussion

Prior literature examining the volumetric changes of brain structures during aging has consistently reported that the thalamus, along with other brain structures, significantly atrophies with age [START_REF] Cherubini | Aging of subcortical nuclei: microstructural, mineralization and atrophy modifications measured in vivo using MRI[END_REF][START_REF] Coupé | Towards a unified analysis of brain maturation and aging across the entire lifespan: a MRI analysis[END_REF][START_REF] Narvacan | Evolution of deep gray matter volume across the human lifespan[END_REF][START_REF] Raji | Age, Alzheimer disease, and brain structure[END_REF][START_REF] Walhovd | Effects of age on volumes of cortex, white matter and subcortical structures[END_REF][START_REF] Walhovd | Consistent neuroanatomical age-related volume differences across multiple samples[END_REF][START_REF] Zheng | Age-related changes in cortical and subcortical structures of healthy adult brains: a surface-based morphometry study[END_REF]. Our principal finding is that thalamic nuclei have high and heterogeneous atrophy rates during cognitively unimpaired aging. Our results further suggest that a number of thalamic nuclei atrophy at greater rates than most of the extra-thalamic structures examined here. The high atrophy rates of thalamic nuclei may be the result of several underlying causes. One involves the "metabolism hypothesis", proposed in the context of cortical hubs, which are brain regions with high degrees of connectivity and integration of information. Neuroimaging work has shown that these cortical hubs overlap with disease loci in Alzheimer's disease -sites of amyloid beta deposition and hypometabolism [START_REF] Buckner | Cortical hubs revealed by intrinsic functional connectivity: mapping, assessment of stability, and relation to Alzheimer's disease[END_REF][START_REF] Drzezga | Neuronal dysfunction and disconnection of cortical hubs in non-demented subjects with elevated amyloid burden[END_REF]. The metabolism hypothesis proposes that cortical hubs are susceptible to disease due to their high levels of neuronal activity, which demand high levels of metabolism and create an excess of harmful free radical byproducts as well as pathological changes in gene and protein expression [START_REF] Buckner | The brain's default network: anatomy, function, and relevance to disease[END_REF][START_REF] Buckner | Molecular, structural, and functional characterization of Alzheimer's disease: evidence for a relationship between default activity, amyloid, and memory[END_REF]. Similarly, the highly connected and integrative role of thalamic nuclei may make them susceptible to metabolic and molecular changes that disrupt normal neuronal processes and ultimately lead to structural changes.

Another hypothesis is that nuclei adjacent to ventricles may be more susceptible or accessible to CSF-mediated factors or immune cells that cause tissue damage and subsequent structural atrophy. Such CSF-mediated damage has been reported in multiple sclerosis [START_REF] De Meo | Effect of BDNF Val66Met polymorphism on hippocampal subfields in multiple sclerosis patients[END_REF][START_REF] Lassmann | Pathogenic mechanisms associated with different clinical courses of multiple sclerosis[END_REF][START_REF] Planche | Whitematter-nulled MPRAGE at 7T reveals thalamic lesions and atrophy of specific thalamic nuclei in multiple sclerosis[END_REF][START_REF] Blyau | Differential vulnerability of thalamic nuclei in multiple sclerosis[END_REF] and recently in traumatic brain injury [START_REF] Sandry | Global hippocampal and selective thalamic nuclei atrophy differentiate chronic TBI from Non-TBI[END_REF]. We speculate that the same may occur in healthy aging. Indeed, our analysis shows that thalamic nuclei that are adjacent to ventricular surfaces (AV, MD, MGN, LGN, and pulvinar) atrophy at significantly higher rates than those that are more remote from CSF (CM, VA, VLA, VLP, and VPL), supporting the CSF-mediated damage hypothesis and providing the first evidence of this proximity-to-CSF hypothesis in healthy aging.

We note an alternative hypothesis to the proximity-to-CSF hypothesis that could also explain our observations. Our analysis of the functional specificity of thalamic atrophy rates showed that cognitive, visual, and auditory/vestibular nuclei exhibited higher atrophy rates than the sensorimotor nuclei. However, these functional groupings are also consistent with the proximity-to-CSF groupings, with the cognitive, visual, and auditory/vestibular nuclei in the adjacent group and the sensorimotor nuclei in the remote group. With the present data and analyses, we are unable to lend support for one hypothesis over the other; indeed, the biology underlying volumetric atrophy of thalamic nuclei is undoubtedly complex and may allow for both hypotheses.

Despite the observed significant decline in thalamic volumes with age, all participants in our study had normal cognitive function. What might explain this finding? One possibility is that the brain can compensate for structural/volumetric changes via alternative circuitry, at least up to a point, such that cognition is largely preserved [START_REF] Park | The adaptive brain: aging and neurocognitive scaffolding[END_REF]. Postmortem histological and patient neuroimaging studies indicate that thalamic volumes, as well as those of other brain structures, are smaller for patients with a variety of neurological conditions than for cognitively unimpaired age-matched control participants [START_REF] Azevedo | Thalamic atrophy in multiple sclerosis: a magnetic resonance imaging marker of neurodegeneration throughout disease[END_REF][START_REF] Bishop | Analysis of ageing-associated grey matter volume in patients with multiple sclerosis shows excess atrophy in subcortical regions[END_REF][START_REF] Callen | Beyond the hippocampus: MRI volumetry confirms widespread limbic atrophy in AD[END_REF][START_REF] Cash | Patterns of gray matter atrophy in genetic frontotemporal dementia: results from the GENFI study[END_REF][START_REF] Fischl | Whole brain segmentation: automated labeling of neuroanatomical structures in the human brain[END_REF][START_REF] Jernigan | Cerebral structure on MRI, Part II: specific changes in Alzheimer's and Huntington's diseases[END_REF][START_REF] Lee | Quantification of thalamic nuclei in patients diagnosed with temporal lobe epilepsy and hippocampal sclerosis[END_REF][START_REF] Li | Discriminative analysis of early-stage Alzheimer's disease and normal aging with automatic segmentation technique in subcortical gray matter structures: a multicenter in vivo MRI volumetric and DTI study[END_REF][START_REF] Schönecker | Atrophy in the thalamus but not cerebellum is specific for C9orf72 FTD and ALS patientsan atlas-based volumetric MRI study[END_REF]. Thus, while there is significant observable atrophy in the cognitively unimpaired aged brain, an even greater amount of atrophy occurs in the diseased brain and may be necessary before observable clinical signs occur [START_REF] Fjell | Structural brain changes in aging: courses, causes and cognitive consequences[END_REF]. Alternatively, the atrophy observed in the cognitively unimpaired aged versus diseased brains may reflect different underlying biological processes. For example, neurodegenerative disease-related atrophy typically involves neurotoxic proteins and neuronal death. However, studies of cognitively unimpaired aging suggest that age-related atrophy generally does not involve neuronal death, but rather a shrinkage in neuronal size and reductions of the neuropil (see Fjell and Walhovd 2010 for review).

With the recent development of advanced thalamic segmentation methods [START_REF] Iglesias | A probabilistic atlas of the human thalamic nuclei combining ex vivo MRI and histology[END_REF][START_REF] Su | Thalamus Optimized Multi Atlas Segmentation (THOMAS): fast, fully automated segmentation of thalamic nuclei from structural MRI[END_REF], studies have begun to show that specific thalamic nuclei are altered in disease [START_REF] Aleman-Gomez | Partial-volume modeling reveals reduced gray matter in specific thalamic nuclei early in the time course of psychosis and chronic schizophrenia[END_REF][START_REF] Bocchetta | Thalamic nuclei in frontotemporal dementia: mediodorsal nucleus involvement is universal but pulvinar atrophy is unique to C9orf72[END_REF][START_REF] Chipika | MRI data confirm the selective involvement of thalamic and amygdalar nuclei in amyotrophic lateral sclerosis and primary lateral sclerosis[END_REF][START_REF] Huang | Thalamic Nuclei volumes in psychotic disorders and in youths with psychosis spectrum symptoms[END_REF][START_REF] Jonak | Decreased volume of lateral and medial geniculate nuclei in patients with LHON disease-7 tesla MRI study[END_REF][START_REF] Lee | Quantification of thalamic nuclei in patients diagnosed with temporal lobe epilepsy and hippocampal sclerosis[END_REF][START_REF] Low | Asymmetrical atrophy of thalamic subnuclei in Alzheimer's disease and amyloid-positive mild cognitive impairment is associated with key clinical features[END_REF][START_REF] Planche | Whitematter-nulled MPRAGE at 7T reveals thalamic lesions and atrophy of specific thalamic nuclei in multiple sclerosis[END_REF][START_REF] Shin | Alterations of individual thalamic nuclei volumes in patients with migraine[END_REF]. Several studies have also found evidence of volumetric differences in thalamic nuclei between asymptomatic genetic carriers and non-carriers in frontotemporal dementia [START_REF] Bocchetta | Differential early subcortical involvement in genetic FTD within the GENFI cohort[END_REF] and their association with PET-based levels of amyloid and tau in Alzheimer's disease [START_REF] Pardilla-Delgado | Associations between subregional thalamic volume and brain pathology in autosomal dominant Alzheimer's disease[END_REF], highlighting the possibility that the volumes of thalamic nuclei may represent preclinical markers for disease. In our study, AV was the brain structure with the highest atrophy rate. We note that while the THOMAS method segments AV, this segmentation likely also includes some or all of the small, adjacent anterodorsal (AD) nucleus. The strong age-related atrophy of the AV nucleus observed in our study is consistent with prior work showing that anterior thalamic structures are implicated in Alzheimer's disease [START_REF] Aggleton | Thalamic pathology and memory loss in early Alzheimer's disease: moving the focus from the medial temporal lobe to Papez circuit[END_REF]Braak and Braak, 1991a).

Postmortem histology shows that the AD nucleus is abnormal relatively early in Alzheimer's disease, with dense neurofibrillary tangles observed in stages II-III of the disease [START_REF] Braak | Pattern of brain destruction in Parkinson's and Alzheimer's diseases[END_REF]. AV is spared early in the disease, but sustains amyloid deposition in later stages of the disease (Braak and Braak, 1991a, b). Thus, our observation of AV's strong atrophy in this cognitively unimpaired sample may be an early indicator of processes contributing to the development of Alzheimer's disease. Future studies are needed to longitudinally correlate the atrophy of AV and other thalamic nuclei with functional and disease markers, such as neuropsychological test scores, fMRI or PET activity, or cerebrospinal fluid levels of disease-related proteins, such as amyloid beta and tau.

We observed no hemisphere-specific differences and almost no sex-specific differences in age-related thalamic atrophy. There has been disagreement in the literature concerning the existence of hemisphere-specific [START_REF] Hughes | Regional changes in thalamic shape and volume with increasing age[END_REF][START_REF] Pieperhoff | Deformation field morphometry reveals age-related structural differences between the brains of adults up to 51 years[END_REF][START_REF] Serbruyns | Subcortical volumetric changes across the adult lifespan: subregional thalamic atrophy accounts for age-related sensorimotor performance declines[END_REF][START_REF] Squarzoni | Reduced gray matter volume of the thalamus and hippocampal region in elderly healthy adults with no impact of APOE varepsilon4: a longitudinal voxel-based morphometry study[END_REF][START_REF] Walhovd | Consistent neuroanatomical age-related volume differences across multiple samples[END_REF][START_REF] Xu | Gender effects on agerelated changes in brain structure[END_REF] or sex-specific [START_REF] Coupé | Towards a unified analysis of brain maturation and aging across the entire lifespan: a MRI analysis[END_REF]Fjell et al., 2009b;[START_REF] Goodro | Age effect on subcortical structures in healthy adults[END_REF][START_REF] Király | Male brain ages faster: the age and gender dependence of subcortical volumes[END_REF][START_REF] Li | Regional specificity of sex effects on subcortical volumes across the lifespan in healthy aging[END_REF][START_REF] Murphy | Sex differences in human brain morphometry and metabolism: an in vivo quantitative magnetic resonance imaging and positron emission tomography study on the effect of aging[END_REF][START_REF] Sullivan | Effects of age and sex on volumes of the thalamus, pons, and cortex[END_REF][START_REF] Wang | Effects of age and sex on subcortical volumes[END_REF][START_REF] Xu | Gender effects on agerelated changes in brain structure[END_REF] differences in whole thalamic atrophy rates. Discrepancies may arise due to methodological differences between the studies, such as the sample size, ethnic background of the participants, and differing accuracies of the segmentation methods. In the present study, using a comparatively large sample size and the most accurate existing method for automatically segmenting thalamic nuclei, our data show virtually no hemisphereor sex-specific differences in age-dependent atrophy rates for either the whole thalamus or individual thalamic nuclei. Future studies with larger sample sizes and state-of-theart thalamic segmentation methods are needed to systematically tease apart thalamic atrophy in the context of different demographic factors, genetic backgrounds, and cognitive capabilities.

Although the focus of our study was on thalamic nuclei, we note several differences in the atrophy of some extra-thalamic structures compared to prior literature. We observed low or no significant atrophy of cerebral and cerebellar white matter and the hippocampus over the adult lifespan. [START_REF] Coupé | Towards a unified analysis of brain maturation and aging across the entire lifespan: a MRI analysis[END_REF] examined volumetric changes in a survey of brain structures (including the whole thalamus, but not thalamic nuclei) during development and over the adult lifespan and found robust cerebral and cerebellar white matter atrophy and moderate hippocampal atrophy over the older age range (>60 years), while for these brain regions the average atrophy rate over the middle age range (20-60) was near zero. Our data showed similar trends, with near-zero atrophy rates when the entire age range was analyzed, but increased atrophy rates (cerebral and cerebellar white matter: 0.14% to 0.31% per year; hippocampus: 0.48% to 0.57% per year; Supp. Fig. 4B; Supp. Table 1) when only the older age range was analyzed. Similar (though not as extreme) observations of increased atrophy rates in the older cohort compared to the full cohort were seen for other brain structures (Supp. Fig. 4B). Notwithstanding these nonlinear effects, the thalamus and its nuclei remain among the highest atrophying structures with age, even in the older cohort. Future studies will need to examine this issue using a larger number of participants analyzed with both the SAMSEG and volBrain pipelines.

Caveats and limitations

In the present study, we combined an older cohort with two younger cohorts to create a large sample spanning seven decades of adult age. As a result, there were several groupspecific differences in scan acquisition parameters, such as scan protocol, field strength, and institution. For example, the WMnMPRAGE imaging for the Stanford younger and older adults was acquired at 7T, while for the Bordeaux younger adults the WMnMPRAGE imaging was obtained at 3T. We first note that our analysis of brain volumes made exclusive use of fractional (or relative) volumes; this means that we analyzed only brain volumes normalized by ICV. We also adjusted for ICV in all regression analyses. Brain volumes and ICVs were obtained using the same segmentation tools for all participants. Therefore, if there is a multiplicative scaling bias as a result of a site-specific field strength, scanner, or software effect, then this scaling bias would be canceled in the formation of the fractional volumes. To further rule out a significant influence of field strength or institution, we tested for thalamic volume differences between Stanford (7T) and Bordeaux (3T) participants within the same age group and found no significant differences. We also found a similar pattern of atrophy after adjusting for cohort (Supp. Fig. 2B) or after analyzing solely the Stanford cohort (single site, same field strength) (Supp. Fig. 2C).

In addition, while the education level of our study population was uniformly high (mean 17.8 years, standard deviation 2.6 years), the older cohort had a somewhat lower group education level (mean 16.7 years) compared to the younger cohorts (mean 20.8 years). It is known that education level correlates with brain health and performance of certain cognitive abilities [START_REF] Capitani | Does education influence the age-related cognitive decline? A further inquiry[END_REF][START_REF] Mortiner | Do psychosocial risk factors contribute to Alzheimer's disease?[END_REF]. Indeed, [START_REF] Trelle | Hippocampal and cortical mechanisms at retrieval explain variability in episodic remembering in older adults[END_REF] reported poorer episodic memory retrieval with lower education level in the present older cohort. Thus, we may have observed lower age-related atrophy rates if our average education levels had been more similar between the younger and older age cohorts. However, we found a similar pattern of atrophy across brain structures when we adjusted for years of education in our regression analysis (Supp. Fig. 2A andB). Thus, the confound of varying education level did not significantly impact the observed atrophy rates across the adult lifespan. Nonetheless, future work should include longitudinal studies following the same participants as they age to obtain more sensitive and controlled data on the atrophy of the thalamus and its nuclei with aging.

As an alternative to THOMAS, [START_REF] Iglesias | A probabilistic atlas of the human thalamic nuclei combining ex vivo MRI and histology[END_REF] have created a thalamic nuclear segmentation method that is available in the FreeSurfer package. While the FreeSurfer thalamic segmentation method has a number of advantages, such as operating on conventional T1 images and offering segmentations of 25 nuclei, we used WMnMPRAGE and THOMAS in the present study, motivated by prior work [START_REF] Majdi | Automated thalamic nuclei segmentation using multi-planar cascaded convolutional neural networks[END_REF][START_REF] Su | Thalamus Optimized Multi Atlas Segmentation (THOMAS): fast, fully automated segmentation of thalamic nuclei from structural MRI[END_REF] showing that the combination of WMnMPRAGE imaging and THOMAS segmentation results in higher accuracy in defining the boundaries of thalamic nuclei (as measured by Dice and VSI overlap metrics) compared to that obtained with the FreeSurfer method.

In the present study, we used two different methods to segment structural volumes -THOMAS for thalamic nuclei using WMnMPRAGE images and SAMSEG for extrathalamic structures using conventional T1-weighted images. This approach was unavoidable, as no single method produces both sets of segmentations. Consequently, we note that there may be systematic software-specific effects that could affect comparisons between the atrophy rates of thalamic versus extra-thalamic structures. We note, however, that both THOMAS-derived and SAMSEG-derived whole thalamus volumes showed significant non-zero atrophy with age (0.45% per year for THOMAS, 0.23% per year for SAMSEG), indicating comparable volumetry for these two segmentation methods.

Another caveat is that we compared brain structures of very differently sized volumes: the smallest structures we analyzed (habenula with an average volume of 25 mm 3 ) were almost 10,000 times smaller in volume than the largest structures (cerebral cortex with an average volume of nearly 250,000 mm 3 ). This huge dynamic range is the reason that we used an exponential model whose fundamental characteristic is a constant percent change in volume per unit time over the lifespan, allowing us to directly compare the relative atrophy rates of small and large structures; however, this did constrain our analysis to one particular nonlinear model. That said, our principal findings about the age-related atrophy rates of thalamic nuclei would not be affected by this issue, as the thalamic nuclei are relatively similar in volume. Future studies that more closely compare thalamic and extra-thalamic atrophy would likely benefit from examining subregions of large structures, such as cortical areas within the cerebral cortex.

Prior studies examining linear and nonlinear models of thalamic volume changes versus age have generally found linear relationships [START_REF] Fjell | Critical ages in the life course of the adult brain: nonlinear subcortical aging[END_REF][START_REF] Goodro | Age effect on subcortical structures in healthy adults[END_REF][START_REF] Hasan | Multimodal quantitative magnetic resonance imaging of thalamic development and aging across the human lifespan: implications to neurodegeneration in multiple sclerosis[END_REF][START_REF] Sullivan | Effects of age and sex on volumes of the thalamus, pons, and cortex[END_REF]. A few recent studies have reported nonlinear relationships [START_REF] Coupé | Towards a unified analysis of brain maturation and aging across the entire lifespan: a MRI analysis[END_REF][START_REF] Narvacan | Evolution of deep gray matter volume across the human lifespan[END_REF][START_REF] Tullo | MR-based age-related effects on the striatum, globus pallidus, and thalamus in healthy individuals across the adult lifespan[END_REF][START_REF] Wang | Effects of age and sex on subcortical volumes[END_REF]. Using an exponential model with the capability of modeling linear and nonlinear atrophy, we also observed a consistently near-linear decrease throughout the adult lifespan. However, we note that our sample was not uniformly distributed over the age range -we had fewer participants in the 50-60 year age range, which is an age at which certain cognitive abilities such as verbal memory and visuospatial processing begin to decline [START_REF] Salthouse | Decomposing age correlations on neuropsychological and cognitive variables[END_REF][START_REF] Schaie | Developmental Influences on Adult Intelligence: The Seattle longitudinal study[END_REF]. Similarly, we note that while certain structures (e.g., whole thalamus, AV, and nucleus accumbens) clearly showed a near-linear reduction of volume throughout the age-span, this linearity is not so clear for other structures (e.g., cerebral and cerebellar cortices) as a result of a higher degree of scatter in our cross-sectional dataset, combined with the non-uniform sample distribution. These limitations could be addressed in future longitudinal studies with more participants, ideally distributed uniformly over the lifespan, to better detect any potential nonlinearities in the age-related atrophy of thalamic nuclear volumes.

Finally, the cross-sectional design adopted here is susceptible to variation in the volumes of these structures between individuals and does not provide a direct measure of withinindividual age-related atrophy. Nonetheless, the present findings provide important initial evidence for heterogenous rates of volume loss among thalamic nuclei across the lifespan, a pattern that should be confirmed in future longitudinal studies.

Conclusions

Here, we showed that the thalamic nuclei have high and heterogeneous age-related atrophy in the cognitively unimpaired human brain. Thalamic nuclei involved in cognition (AV, MD), visual/visual association (LGN, pulvinar), and auditory/vestibular (MGN) function showed the highest atrophy rates. These five nuclei also happen to be adjacent to a ventricular surface, which may make them susceptible to CSF-mediated factors that affect brain structure. The next tier of atrophy rates involved the nuclei related to motor (VA, VLA, VLP, and CM) and somatosensory (VPL) functions, and these five nuclei happen to be located at some distance from the nearest ventricular surface. We found no significant hemispheric, sex-specific, or education-specific differences in these rates of change for thalamic nuclei. These results provide a baseline measure of atrophy rates occurring in thalamic nuclei across the cognitively unimpaired adult lifespan. Given that our participants were all cognitively unimpaired, our results indicate the range of structural atrophy that can be tolerated by the resilient cognitively unimpaired aged brain while preserving cognitive functioning. These findings indicate that the thalamus and its nuclei are a component of age-related brain atrophy in the cognitively unimpaired human brain. A detailed understanding of such atrophy may provide insights into the transition from cognitively unimpaired brain aging to pathological neurodegeneration and other neurological and psychiatric disease. Comparison of annualized atrophy rates of extra-thalamic and thalamic structures. Lines indicate 95% confidence bands, adjusted for multiple comparisons. Note that thalamic nuclei atrophy at rates higher than most other structures in the brain. Red=right hemisphere. Black=left hemisphere.

Fig. 1 .

 1 Fig. 1.Segmentations in representative younger and older subjects. (A) Whole thalamus, thalamic nuclei, and habenula segmentations using THOMAS and white matter-nulled MPRAGE imaging. (B) Extra-thalamic segmentations using FreeSurfer's SAMSEG tool. (C) Intracranial cavity segmentation using SPM12's Segment and Tissue Volume Utility modules.

Fig. 2 .

 2 Fig. 2.Absolute volumes of THOMAS-derived thalamic structures plotted versus age, showing heterogeneous atrophy between nuclei. Curves show best fits to the defined exponential regression model. Red=right hemisphere. Black=left hemisphere.

Fig. 3 .

 3 Fig. 3. Absolute volumes of the whole thalamus and extra-thalamic structures plotted versus age. Curves show best fits to the defined exponential regression model. Red=right hemisphere. Black=left hemisphere.

Fig

  Fig. 4.

  manuscript; available in PMC 2022 December 23.

Table 1

 1 Participant demographics.

	All Participants		198	86 M, 112 F	59.4 / 17.4 / 20-88		17.8 / 2.6 / 10-21
	Stanford Older Participants (>=60	years of age)	143	66 M, 77 F	69.1 / 6.1 / 60-88		16.7 / 2.0 / 10-20
	Stanford Younger Participants (<60	years of age)	17	9 M, 8 F	37 / 11.4 / 24-58		20.2 / 1.8 / 16-21
	Bordeaux Younger Participants (< 60	years of age)	38	11 M, 27 F	33.1 / 10.4 / 20-56		21 / 0 / 21-21
			Number of Participants	Sex (male, female)	Age (mean / standard deviation/range /	range) (yrs)	Education Level (mean / standard	deviation / range) (yrs)

Neuroimage. Author manuscript; available in PMC 2022 December 23.

Table 2

 2 Image acquisition parameters.

	Bordeaux	Younger	Subjects

Table 3

 3 Annualized atrophy rates (percent change in volume per year) of thalamic and extra-thalamic structures with 95% confidence bounds and p-values adjusted for multiple comparisons. Bold numbers indicate change that is significantly different from zero.

	Hemisphere

Structure Mean Volumetric Change Per Year 95% Confidence Bound-Lower Limit 95% Confidence Bound-Upper Limit Adjusted P-Value

	<0.001	0.052	<0.001	1	1	1	0.31	<0.001	<0.001	1	<0.001	<0.001	<0.001	<0.001	<0.001	<0.001	<0.001	<0.001	0.999	<0.001	<0.001	<0.001	0.704	<0.001	0.075	<0.001
	-0.08%	0.00%	-0.14%	0.14%	0.09%	0.10%	0.03%	-0.09%	-0.30%	0.14%	2.42%	-0.35%	-0.93%	-0.30%	-0.50%	-0.38%	-0.50%	-0.30%	0.34%	-0.14%	-0.39%	-0.15%	0.10%	-0.07%	0.00%	-0.14%	0.20%
	-0.23%	-0.19%	-0.35%	-0.25%	-0.14%	-0.10%	-0.24%	-0.29%	-0.58%	-0.12%	1.10%	-0.55%	-1.43%	-0.55%	-0.79%	-1.05%	-0.83%	-0.71%	-0.67%	-0.45%	-0.84%	-0.57%	-0.38%	-0.23%	-0.19%	-0.36%	-0.20%
	Cerebral Cortex -0.15%	Cerebral White Matter -0.10%	Cerebellar Cortex -0.24%	Cerebellar White Matter -0.06%	Amygdala -0.03%	Hippocampus 0.00%	Caudate -0.11%	Putamen -0.19%	Nucleus Accumbens -0.44%	Pallidum 0.01%	Lateral Ventricle 1.74%	Thalamus -0.45%	AV -1.18%	MD -0.43%	Pulvinar -0.65%	LGN -0.72%	MGN -0.67%	VA -0.51%	VLA -0.18%	VLP -0.30%	CM -0.62%	VPL -0.36%	Habenula -0.15%	Cerebral Cortex -0.15%	Cerebral White Matter -0.09%	Cerebellar Cortex -0.25%	Cerebellar White Matter 0.00%
	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Left	Right	Right	Right	Right

1 Neuroimage. Author manuscript; available in PMC 2022 December 23.

Table 5

 5 Functional specificity in the atrophy of thalamic nuclei with age. Significant between-group differences in atrophy rates were tested with paired t-tests. P-values were adjusted for multiple comparisons. Bold numbers indicate statistical significance.

	Between Group Comparison	Adjusted P-Value
	Cognitive > Visual	0.039
	Cognitive > Auditory/Vestibular	0.026
	Cognitive > Motor	<0.001
	Cognitive > Somatosensory	0.006
	Visual > Auditory/Vestibular	0.998
	Visual > Motor	<0.001
	Visual > Somatosensory	<0.001
	Auditory/Vestibular > Motor	<0.001
	Auditory/Vestibular > Somatosensory <0.001
	Motor > Somatosensory	0.929

Neuroimage. Author manuscript; available in PMC 2022 December 23.

Acknowledgements

We thank the following individuals for their contributions to data collection: Jeff Bernstein, Nicole Corso, Wanjia Guo, Madison Hunt, Manasi Jayakumar, Anna Khazenzon, Celia Litovsky, Hannah Lloyd, Ayesha Nadiadwala, Natalie Tanner, and Monica Thieu. We also thank Dr. Garrett Drayna for expert technical assistance and discussion. This work was supported by the National Institutes of Health (P30 AG066515, P41 EB01589125, R01 AG048076, R01 AG065255, R21 AG058111, R21 AG058859, UH3 NS095554, UL1 TR003142), Department of Defense (W81XWH-20-PRARP-CSRA), Alzheimer's Association (AARFD-21-852597), Stanford Precision Health and Integrated Diagnostic Center (PHIND), Stanford Wu Tsai Neurosciences Institute, Stanford Center for Clinical and Translational Education and Research (Spectrum), GE Healthcare, ARSEP Foundation, and Idex Bordeaux University (IMPACT program).

Data/code availability statement

Data and code are available upon request.

Data Availability

Data will be made available on request.

Supplementary Material

Refer to Web version on PubMed Central for supplementary material. Annualized atrophy rates of thalamic nuclei grouped by function. Group atrophy rates were tested with t-tests for significant difference from zero. 95% confidence bands and p-values were adjusted for multiple comparisons. Nuclei are color-coded by functional group: yellow = cognitive; green = visual; pink = auditory/vestibular; blue = motor; and purple = somatosensory.

Abbreviations:

THOMAS


Laurent Bus É
email: laurent.buse@inria.fr

AND Guillaume Ch Èze
email: guillaume.cheze@math.ups-tlse.fr

Salah Najib
email: najib@mpim-bonn.mpg.de

NOETHER'S FORMS FOR THE STUDY OF NON-COMPOSITE RATIONAL FUNCTIONS AND THEIR SPECTRUM

Keywords: Spectrum, GCD, Noether's theorem, Ostrowski's theorem, Bertini's theorem, composite rational function

. With this approach, some new effective results are obtained. In particular, we show that the reduction modulo p of the spectrum of a given integer multivariate rational function r coincides with the spectrum of the reduction of r modulo p for p a prime integer greater or equal to an explicit bound. This bound is given in terms of the degree, the height and the number of variables of r. With the same strategy, we also study the decomposability of r modulo p. Some similar explicit results are also provided for the case of polynomials with coefficients in A = K[Z].

Introduction

Consider a non-constant polynomial f ∈ K[X 1 , . . . , X n ], n ≥ 2, where K is a field. Denoting by K the algebraic closure of K, the spectrum of f is the set

σ(f ) := {λ ∈ K : f -λ is reducible in K[X 1 , . . . , X n ]} ⊂ K.
We recall that a polynomial f (X 1 , . . . , X n ) ∈ K[X 1 , . . . , X n ] is said to be absolutely irreducible if it is irreducible in K[X 1 , . . . , X n ].

It is customary to say that f is non-composite if f cannot be written in the form u(h(X)) with h(X) ∈ K[X], u(t) ∈ K[t] and deg(u) ≥ 2. A famous theorem of Bertini gives that f is non-composite if and only if σ(f ) is finite; see for instance [START_REF] Schinzel | Polynomials with special regard to reducibility[END_REF]Theorem 37]. Furthermore, Stein proved in [START_REF] Stein | The total reducibility order of a polynomial in two variables[END_REF] that the cardinality of σ(f ) is at most equal to deg(f ) -1; see also [START_REF] Najib | Une généralisation de l'inégalité de Stein-Lorenzini[END_REF][START_REF] Lorenzini | Reducibility of polynomials in two variables[END_REF][START_REF] Cygan | Factorization of polynomials[END_REF][START_REF] Kaliman | Two remarks on polynomials in two variables[END_REF].

Recently in [START_REF] Bodin | On indecomposable polynomials and their spectrum[END_REF], A. Bodin, P. Dèbes, and S. Najib have studied the behavior of the spectrum of a polynomial via a ring morphism. Here we generalize this study to the spectrum of a rational function and we give explicit bounds.

Suppose given two non-constant relatively prime polynomials f and g in the polynomial ring K[X 1 , . . . , X n ], n ≥ 2. The spectrum of the rational function r = f /g ∈ K(X 1 , . . . , X n ) is the set

σ(f, g) := {(λ : µ) ∈ P 1 K : µf ♯ -λg ♯ is reducible in K[X 0 , X 1 , . . . , X n ]} ⊂ P 1 K with f ♯ := X deg(r) 0 f X 1 X 0 , . . . , X n X 0 , g ♯ := X deg(r) 0 g X 1 X 0 , . . . , X n X 0 ,
where deg(r) := max(deg(f ), deg(g)). That is to say, σ(f, g) := {(λ : µ) ∈ P 1 K : µfλg is reducible in K[X 1 , . . . , X n ] or deg µfλg < deg r}.

In a more geometric terminology, σ(f, g) counts the number of reducible hypersurfaces in the pencil of hypersurfaces defined by the equations µfλg = 0 with (λ : µ) ∈ P 1 K . Again, r is said to be non-composite if r cannot be written in the form u(h(X)) with h(X) ∈ K(X) and u(t) ∈ K(t), deg(u) ≥ 2. Actually, σ(f, g) is finite if and only if r is non-composite and if and only if the pencil of projective algebraic hypersurfaces µf ♯λg ♯ = 0, (µ : λ) ∈ P 1 K , has its general element irreducible (see for instance [10, Chapitre 2, Théorème 3.4.6] or [START_REF] Bodin | Reducibility of rational functions in several variables[END_REF]Theorem 2.2] for detailed proofs). Notice that the study of σ(f, g) is trivial if d = 1, and n = 1. Therefore, throughout this paper we will always assume that d ≥ 2, and n ≥ 2.

The study of the spectrum is related to the computation of the number of reducible curves in a pencil of algebraic plane curves. This problem has been widely studied. As far as we know, the first related result has been given by Poincaré [START_REF] Poincaré | Sur l'intégration algébrique des équations différentielles du premier ordre[END_REF]. Poincaré's bound was improved by a lot of works, see e.g. [START_REF] Ruppert | Reduzibilität Ebener Kurven[END_REF][START_REF] Lorenzini | Reducibility of polynomials in two variables[END_REF][START_REF] Vistoli | The number of reducible hypersurfaces in a pencil[END_REF][START_REF] Shreeram | Translates of polynomials[END_REF][START_REF] Bodin | Reducibility of rational functions in several variables[END_REF]. In [START_REF] Busé | On the total order of reducibility of a pencil of algebraic plane curves[END_REF] the authors study the number (counted with multiplicity) of reducible curves in a pencil of algebraic plane curves. The method used relies on an effective Noether's irreducibility theorem given by W. Ruppert in [START_REF] Ruppert | Reduzibilität Ebener Kurven[END_REF].

In this article, we follow the same strategy as in [START_REF] Busé | On the total order of reducibility of a pencil of algebraic plane curves[END_REF] using in addition basic results of elimination theory. More precisely, in the first section we give some preliminaries which are used throughout this work. In the second section, we show that the spectrum consists of the roots of a particular homogeneous polynomial denoted Spect f,g . If ϕ is a morphism then we get, under some suitable hypothesis, that ϕ(Spect f,g ) = Spect ϕ(f ),ϕ(g) . For two special situations, namely f, g ∈ Z[X], and ϕ is the reduction modulo a prime number p, or f, g ∈ K[Z 1 , . . . , Z s ][X], and ϕ sends Z i to z i ∈ K, we give explicit results in terms of the degree, the height and the number of variables of f /g.

In the last section of this article we study the behavior of a composite rational function. More precisely we show, under some suitable hypothesis, that "f /g is composite over its coefficients field" if and only if "f /g is composite over any extension of its coefficients field". Thanks to the effective Noether's irreducibility theorem, we then show that if r is a non-composite rational function with integer coefficients and p is a "big enough" prime, then r modulo p is also non-composite. An explicit lower bound is given for such an integer p. Finally, with the same approach we also study the reduction of a non-composite rational function with coefficients in K[Z 1 , . . . , Z n ] after the specialization Z i = z i ∈ K, i = 1, . . . , n. We end the paper by showing that after a generic linear change of variables, a non-composite rational function remains non-composite.

Notations

. If f (X 1 , . . . , X n ) = i1,...,in c i1,...,in X i1 1 . . . X in n ∈ Z[X 1 , . . . , X n ]
then we set H(f ) = max i1,...,in |c i1,...,in | and f 1 = i1,...,in |c i1,...,in |. The field with p elements Z/pZ is denoted by F p . Given a polynomial f ∈ Z[X] and a prime integer p ∈ Z, we will use the notation f p for the reduction of f modulo p, that is to say, the class of f in F p [X]. Finally, for any field K we denote by K (one of) its algebraic closure.

Preliminaries

This section is devoted to the statement of some algebraic properties that are deeply rooted to elimination theory.

1.1. Noether's reducibility forms. We recall some effective results about the Noether's forms that give a necessary and sufficient condition on the coefficients of a polynomial to be absolutely irreducible. We refer the reader to [START_REF] Schmidt | Equations over finite fields. An elementary approach[END_REF][START_REF] Schinzel | Polynomials with special regard to reducibility[END_REF][START_REF] Kaltofen | Effective Noether irreducibility forms and applications[END_REF][START_REF] Ruppert | Reduzibilität Ebener Kurven[END_REF] for different types of such forms.

Theorem 1. Let K be a field of characteristic zero, d ≥ 2, n ≥ 2 and f (X 1 , . . . , X n ) = 0≤e1+•••+en≤d c e1,...,en X e1 1 . . . X en n ∈ K[X 1 , . . . , X n ].
There exists a finite set of polynomials:

Φ t (. . . ,
. ) = 0 ⇐⇒ f is reducible in K[X 1 , . . . , X n ] or deg(f ) < d, ⇐⇒ F (X 0 , . . . , X n ) is reducible in K[X 0 , . . . , X n ],
where F is the homogeneous polynomial

X deg(f ) 0 f X1 X0 , . . . , Xn X0 . Furthermore, deg Φ t ≤ d 2 -1 and Φ t 1 ≤ d 3d 2 -3 n + d n 2 d d 2 -1 .
If K has positive characteristic p > d(d -1), the coefficients of the Φ t 's above are to be taken modulo p.

Proof. In characteristic zero this theorem has been proved by Ruppert [START_REF] Ruppert | Reduzibilität Ebener Kurven[END_REF]. Actually Gao showed in [START_REF] Gao | Factoring multivariate polynomials via partial differential equations[END_REF] that this theorem is also true in positive characteristic p providing p > d(d -1). This improvement of original Ruppert's result is contained in [START_REF] Gao | Factoring multivariate polynomials via partial differential equations[END_REF]Lemma 2.4] where it is explained that the non-vanishing of a certain resultant guaranties the statement of this theorem.

It should be noticed that the above theorem is true without any hypothesis on the characteristic of the ground field K, see e.g. [START_REF] Kaltofen | Effective Noether irreducibility forms and applications[END_REF]Theorem 7], but then the degrees and the heights of these forms are with no comparison with the ones given here.

1.2. GCDs of several polynomials under specialization. The following theorem is a classical and old result taken from elimination theory. Modern statements and proofs of this result can be found in [11, §2.10]

A[U, V ] d1+d2-di-1 ϕ -→ A[U, V ] d1+d2-1 : (g 1 , . . . , g n ) → n i=1 g i f i does not have (maximal) rank d 1 + d 2 .
In particular, given a field L and a ring morphism ρ : A → L, ρ(f 1 ), . . . , ρ(f n ) have a common root in the projective line over L if and only if ρ(ϕ) does not have (maximal) rank d 1 + d 2 (notice that we also denoted by ρ its canonical extensions to some suitable matrix and polynomial rings).

This theorem allows to control the behavior of GCDs of several polynomials with coefficients in a UFD ring under specialization. Hereafter, we will always assume that polynomial GCDs over a field are taken to be monic with respect to a certain monomial order (e.g. the lexicographical order).

Corollary 3. Let A be a UFD, f 1 , . . . , f n be n ≥ 2 nonzero homogeneous poly- nomials in A[U, V ] and suppose given a ring morphism ρ : A → L where L is a field. Let α ∈ A be the leading coefficient of gcd(f 1 , . . . , f n ) ∈ A[U, V ]. There ex- ists a finite collection of algorithmically computable elements (c i ) i∈I in A with the following property: if ρ(c i ) = 0 for some i ∈ I then ρ(gcd(f 1 , . . . , f n )) = ρ(α)gcd(ρ(f 1 ), . . . , ρ(f n )) ∈ L[U, V ]. Proof. Set g ρ = gcd(ρ(f 1 ), . . . , ρ(f n )) ∈ L[U, V ],
which is a monic polynomial, and

g = gcd(f 1 , . . . , f n ) ∈ A[U, V ]. For all i = 1, . . . , n there exists a polynomial h i ∈ A[U, V ] such that f i = gh i . It follows that ρ(f i ) = ρ(g)ρ(h i )
and hence that ρ(g) divides g ρ . Furthermore, h 1 , . . . , h n have no homogeneous common factor of positive degree in A[U, V ], so by Theorem 2 there exists a multiplication map, say ϕ, with the property that ρ(h 1 ), . . . , ρ(h n ) have no homogeneous common factor in L[U, V ] if the rank of ρ(ϕ) is maximal. Denoting (c i ) i∈I the collection of maximal minors of a matrix of ϕ, the fact that the rank of ρ(ϕ) is not maximal is equivalent to the fact that ρ(c i ) = 0 for all i ∈ I. Therefore, we deduce that ρ(g) and g ρ are equal in L[U, V ] up to an invertible element if ρ(c i ) = 0 for some i ∈ I. Since g ρ is monic by convention, the claimed equality is obtained by comparison of the leading coefficients.

In this paper we will be mainly interested in two particular cases, namely the case where A = Z and ρ is the reduction modulo p, and the case where A = K[Z 1 , . . . , Z s ] and ρ : A → K is an evaluation morphism. Our next task is to detail Corollary 3 in these two particular situations. (ii) Denoting by α ∈ Z the leading coefficient of gcd

Proposition 4. Suppose given n ≥ 2 homogeneous polynomials f 1 , . . . , f n ∈ Z[U, V ] of positive degree and set d = max i deg(f i ), H = max i H(f i ). (i) If f 1 , . . . , f n have no (homogeneous) common factor of positive degree in Z[U, V ], then f 1 p , . . ., f n p have no (homogeneous) common factor of positive degree in F p [U, V ] for all prime integer p > d d H 2d . 1 the notation A[U, V ] d , d ∈ N,
(f 1 , . . . , f n ) ∈ Z[U, V ],
we have

α p .gcd(f 1 p , . . . , f n p ) = gcd(f 1 , . . . , f n ) p ∈ F p [U, V ]
for all prime integer

p > d d (d + 1) d 2 2d 2 H 2d .
Proof. Denote by

d 1 ≥ d 2 ≥ . . . ≥ d n ≥ 1 the degree of f 1 , . . . , f n respectively. Observe that d = d 1 .
By Theorem 2, the hypothesis implies that the multiplication map

⊕ n i=1 A[U, V ] d1+d2-di-1 → A[U, V ] d1+d2-1 : (g 1 , . . . , g n ) → n i=1 g i f i has maximal rank d 1 + d 2 .
H d1+d2 ≤ d d H 2d .
Therefore, one of these minors remains non-zero modulo p and Theorem 2 implies that f 1 p , . . . , f n p do not have a common root in the projective line over an algebraically closed field extension of F p . We deduce that f 1 p , . . . , f n p do not have a homogeneous common factor of positive degree in F p [U, V ] and (i) is proved.

To prove (ii) we use Corollary 3 and take again the notation of its proof. For all i = 1, . . . , n there exists a homogeneous polynomial h i ∈ Z[U, V ] such that f i = gh i and we know that the claimed equality holds if ϕ(h 1 ), . . . , ϕ(h n ) does not have a homogeneous common factor of positive degree in It should be noticed that a result similar to (i) has been proved in [26, last paragraph of page 136] but with a larger bound for the prime integer p, namely e 2nd 2 H 2d . Also, to convince the reader that the bound given in (ii) is not too rough, we mention that in the case n = 2 it is not difficult to see that (see for instance [START_REF] Von | Modern computer algebra[END_REF]Theorem 6.26] or [25, §4.4])

F p [U, V ]. Since for all i = 1, . . . , n we have deg(h i ) ≤ d and H(h i ) ≤ (d + 1)
(1.1) gcd(f 1 , f 2 ) p = α.gcd(f 1 , f 2 ) ∈ F p [U, V ] if and only if p ∤ Res(h 1 , h 2 ) ∈ Z, where f i = gcd(f 1 , f 2 )h i , i = 1, 2 and Res(h 1 , h 2 )
is the resultant of h 1 and h 2 . Therefore, it appears necessary to bound

H(h i ), i = 1, 2, in terms of H(f i ), i = 1, 2.
Now, we turn to the second case of application of Corollary 3. For that purpose, we introduce a new set of indeterminates Z := Z 1 , . . . , Z s .

(p i ) i∈I of nonzero elements in K[Z] of degree ≤ 2dk such that ρ(f 1 ), . . ., ρ(f n ) have no (homogeneous) common factor of positive degree in K[U, V ] if ρ(p i ), i ∈ I, are not all zero. (ii) Denoting by α ∈ K[Z] the leading coefficient of gcd(f 1 , . . . , f n ) ∈ K[Z][U, V ]
as a homogeneous polynomial in the variables U, V , there exists a finite collection

(q i ) i∈I of nonzero elements in K[Z] of degree ≤ 2dk such that ρ(α).gcd(ρ(f 1 ), . . . , ρ(f n )) = ρ(gcd(f 1 , . . . , f n )) ∈ K[X]
if ρ(q i ), i ∈ I, are not all zero.

Proof. Completely similar to the proof of Proposition 4.

Study of the spectrum of a rational function

Let A be a UFD, K be its fraction field and suppose given a rational function

r = f /g ∈ K(X 1 , . . . , X n ) with f, g ∈ A[X 1 , . . . , X n ] and gcd(f, g) = 1. Set d := deg(r) = max(deg(f ), deg(g)).
We recall that, by definition, the spectrum of r is the set

σ(f, g) := {(λ : µ) ∈ P 1 K : µf ♯ -λg ♯ is reducible in K[X 0 , X 1 , . . . , X n ]} ⊂ P 1 K where f ♯ := X deg(r) 0 f X 1 X 0 , . . . , X n X 0 , g ♯ := X deg(r) 0 g X 1 X 0 , . . . , X n X 0 .
Assume that σ(f, g) is finite and denote by Φ t (U, V ) the Noether's reducibility forms associated to the polynomial

V f ♯ (X 0 , . . . , X n ) -U g ♯ (X 0 , . . . , X n ) ∈ A[U, V ][X 0 , X 1 , . . . , X n ].
These forms are all homogeneous polynomials in A[U, V ] by construction. We will denote their GCD by Spect f,g (U, V ) ∈ A[U, V ]. By Theorem 1, for all (λ : µ) ∈ P 1 K we have Spect f,g (λ : µ) = 0 ⇐⇒ (λ : µ) ∈ σ(f, g). As an immediate consequence of Corollary 3, we have the following property. Given a morphism ρ : A → L, where L is a field, there exists a non-zero element c of A such that if ρ(c) = 0 then ρ(Spect f,g ) = Spect ρ(f ),ρ(g) up to multiplication by a non-zero element in L.

In what follows, we will investigate this property in two particular cases of interest: the case where A = Z and ρ is the reduction modulo p, and the case where A = K[Z 1 , . . . , Z s ] and ρ : A → K is an evaluation morphism.

2.1. Spectrum and reduction modulo p.

Theorem 6. Suppose given f, g two multivariate polynomials in Z[X 1 , . . . , X n ] such that gcd(f, g) = 1. Set d = deg(f /g) = max(deg(f ), deg(g)). For all prime integer p > B with B = 2 2(d 2 -1) 2 d 2d 2 -2 (d 2 -1) d 2 -1 H 2d where H = d 3d 2 -3 n + d n 2 d d 2 -1 d 2 -1 ⌊(d 2 -1)/2⌋ max H(f ), H(g) d 2 -1 we have Spect f p ,g p = κ.Spect f,g p in the polynomial ring F p [U, V ] with 0 = κ ∈ F p .
Proof. From the definition of Spect f,g (U, V ), this is a consequence of Theorem 1. Indeed, straightforward computations show that

H(Φ t (V f ♯ -U g ♯ )) ≤ Φ t 1 H (V f ♯ -U g ♯ ) d 2 -1
and

H (V f ♯ -U g ♯ ) d 2 -1 ≤ d 2 -1 ⌊(d 2 -1)/2⌋ max H(f ), H(g) d 2 -1 .
It follows

H(Φ t (V f ♯ -U g ♯ )) ≤ d 3d 2 -3 n + d n 2 d d 2 -1 d 2 -1 ⌊(d 2 -1)/2⌋ max H(f ), H(g) d 2 -1 .
Now, applying Proposition 4 with degree d 2 -1 and height H 1 we obtain that

Spect f,g (U, V ) p = gcd Φ t (V f ♯ -U g ♯ ) p = κ.gcd Φ t (V f ♯ -T g ♯ ) p if p > B,
where 0 = κ ∈ F p , and therefore that

Spect f,g (U, V ) p = κ.Spect f p ,g p (U, V ) by Theorem 1.
As a consequence of this theorem we obtain an analog of Ostrowski's result. The classical Ostrowski's theorem asserts that if a polynomial f is absolutely irreducible then f p is absolutely irreducible providing p is big enough. In our context we get the Corollary 7. Under the notations of Theorem 6, if σ(f, g) = ∅ then σ(f p , g p ) = ∅ for all prime integer p such that p > B.

Before moving on, we mention that our strategy can be used similarly to deal with the case of polynomials in A[X 1 , . . . , X n ] and reduction modulo a prime ideal in A.

Spectrum of a rational function with coefficients in

K[Z]. Theorem 8. Let f, g be two polynomials in K[Z 1 , . . . , Z s ][X 1 , . . . , X n ] such that deg Z (f ) ≤ k, deg Z (g) ≤ k, deg X (f ) ≤ d and deg X (g) ≤ d. Given z := (z 1 , . . . , z s ) ∈ K s , denote by ev z the ring morphism K[Z 1 , . . . , Z s ] → K that sends Z i to z i for all i = 1, . .

. , s. There exists a finite collection of nonzero polynomials in K[Z]

, say (q i ) i∈I , of degree smaller than 2(d 2 -1) 2 k with the property that: if ev z (q i ) ∈ K are not all zero then ev z (Spect f,g ) = κ.Spect ev z(f ) ,ev z(g) where 0 = κ ∈ K.

Proof. We consider again Noether's forms (Φ t (V f ♯ -U g ♯ )) t∈T . By construction we have deg

Z (Φ t (V f ♯ -U g ♯ )) ≤ (d 2 -1)k and deg U,V (Φ t (V f ♯ -U g ♯ )) ≤ d 2 -1.
Therefore, Proposition 5 shows the existence of a finite collection of polynomials (q j ) j∈J in K[Z] of degree smaller than 2(d 2 -1) 2 k with the property that ev z (Spect f,g ) = κ.Spect ev z(f ) ,ev z(g) where 0 = κ ∈ K, if ev z (q j ) = 0 for some j ∈ J.

This result has the following probabilistic corollary that follows from the well known Zippel-Schwartz's Lemma that we recall.

Lemma 9 (Zippel-Schwartz). Let P ∈ A[X 1 , . . . , X n ] be a polynomial of total degree d, where A is an integral domain. Let S be a finite subset of A. For a uniform random choice of x i in S we have

P {P (x) = 0 | x i ∈ S} ≤ d/|S|,
where |S| denotes the cardinal of S and P the probability.

Corollary 10. With the notations of Theorem 8, let S be a finite subset of K with |S| elements. If σ(f, g) = ∅ then for a uniform random choice of the z i 's in S we have σ(ev

z(f ) , ev z(g) ) = ∅ with probability at least 1 - 2(d 2 -1) 3 k 2 |S| . Proof. As σ(f, g) = ∅, Spect f,g =: c(Z) is a non-zero polynomial in K[Z]
of degree less than k(d 2 -1). Therefore, if q i c(z) = 0 for some i ∈ I, where (q i ) i∈I is the collection of polynomials in Theorem 8, then Spect ev z(f ) ,ev z(g) ∈ K.

Indecomposability of rational functions

In the previous section we have studied the spectrum of a rational function. It turns out that the spectrum of r(X 1 , . . . , X n ) ∈ K(X 1 , . . . , X n ) is deeply related to the indecomposability of r over K. After recalling this link, we will study indecomposability of rational functions. Theorem 11. Let K be a field of characteristic p ≥ 0. Let r = f /g ∈ K(X 1 , . . . , X n ) be a non-constant reduced rational function. The following are equivalent:

(i) r is composite over K, (ii) f -λg is reducible in K[X 1 , . . . , X n ] for all λ ∈ K such that deg(f -λg) = deg(r), (iii) the polynomial f (X 1 , . . . , X n ) -T g(X 1 , . . . , X n ) is reducible in the polyno- mial ring K(T )[X 1 , . . . , X n ].
Remark 12. We recall that this kind of result is already known for the polynomial case (see for instance [START_REF] Chèze | indecomposability of polynomials via jacobian matrix[END_REF]Lemma 7]).

Proof of Theorem 11. (ii) ⇒ (iii): statement (ii) means that for all λ ∈ K such that deg(fλg) = deg r, we have Φ t (fλg) = 0 for all t. Then Φ t (f -T g) has an infinite number of roots in K and thus Φ t (f -T g) = 0 for all t. This gives f -T g is reducible in

K(T )[X 1 , . . . , X n ]. (iii) ⇒ (ii): statement (iii) means that Φ t (f -T g) = 0 ∈ K[T ] for all t. Hence, if λ ∈ K is such that deg(f -λg) = deg r, we can conclude thanks to Theorem 1 that f -λg is reducible in K[X 1 , . . . , X n ].
The following theorem shows that under some hypothesis, r is composite over K if and only if r is composite over K. Therefore, we will sometimes say hereafter that r is composite instead of r is composite over its coefficients field. Theorem 13. Let K be a perfect field of characteristic p = 0 or p ≥ d 2 and let r = f /g ∈ K(X 1 , . . . , X n ), n ≥ 2, be a non-constant reduced rational function of degree d. Then, r is composite over K if and only if r is composite over K.

Proof. Obviously, if r is composite over K then r is composite over any extension of K and thus over K. So, suppose that r = u(h) with deg(u

) ≥ 2, u ∈ K(T ) and h ∈ K(X 1 , . . . , X n ). We set u = u 1 /u 2 where u 1 , u 2 ∈ K[T ] and h = h 1 /h 2 is reduced and non-composite with h 1 , h 2 ∈ K[X 1 , . . . , X n ].
We are going to show that there exist U ∈ K(T ) and H ∈ K(X 1 , . . . , X n ) such that r = U (H).

The notation mdeg(f ) denotes the multi-degree of f associated to a given monomial order ≺ and lc(f ) denotes the leading coefficient of f associated to ≺.

First step: we claim that one can suppose that lc(f ) = 1, lc(g) = 1 and mdeg(f ) ≻ mdeg(g). Indeed, to satisfy the first condition, we just have to take f /lc(f ) and g/lc(g). Then, for the second conditon, if mdeg(f ) ≺ mdeg(g) we take g/f , and if mdeg(f ) = mdeg(g) we set F = f , G = fg and take F/G. Indeed, f /g is composite over K if and only if F/G is composite over K.

Second step: we claim that one can suppose that lc(h 1 ) = 1, lc(h 2 ) = 1 and mdeg(h 1 ) ≻ mdeg(h 2 ). This actually follows from the same trick that we use in the first step. We just have to remark that if r = u(h 1 /h 2 ) then r = v(h 2 /h 1 ) with v(T ) = u(1/T ).

Third step: one can suppose that h 1 (0, . . . , 0) = 0 and h 2 (0, . . . , 0) = 0. Indeed, if h 2 (0, . . . , 0) = 0 then we can consider

H 1 = h 1 - h 1 (0, . . . , 0) h 2 (0, . . . , 0) h 2
and H 2 = h 2 . Then we can write r = v(H 1 /H 2 ) with H 1 and H 2 satisfying the above conditions and the ones of the second step. Now, if h 2 (0, . . . , 0) = 0 then a change of coordinates r(X 1a 1 , . . . , X na n ), with (a 1 , . . . , a n ) ∈ K n such that h 2 (a 1 , . . . , a n ) = 0, gives the desired result. So we just have to show that there exists such an element (a 1 , . . . , a n ) ∈ K n if p ≥ d 2 (it is clear if p = 0). To do this, we observe that deg h 2 ≤ d < p and proceed by contradiction.

Assume that

h 2 (X 1 , . . . , X n ) = c 0 (X 1 , . . . , X n-1 ) + c 1 (X 1 , . . . , X n-1 )X n + • • • + c d (X 1 , . . . , X n-1 )X d n with c i (X 1 , . . . , X n-1 ) ∈ K[X 1 , . . . , X n-1 ], is such that ∀(x 1 , . . . , x n ) ∈ K n , h 2 (x 1 , . . . , x n ) = 0.
Then, given (x 1 , . . . , x n-1 ) ∈ K n-1 we have that h 2 (x 1 , . . . , x n-1 , X n ) ∈ K[X n ] has degree ≤ d and at least p distinct roots in K. It follows that h 2 (x 1 , . . . , x n-1 , X n ) is the null polynomial and hence that ∀i = 0, . . . , d, ∀(x 1 , . . . , x n-1 ) ∈ K n-1 , c i (x 1 , . . . , x n-1 ) = 0. Now, since c i (X 1 , . . . , X n-1 ) has also degree ≤ d, we can continue this process in the same way to end with the conclusion that h 2 = 0 in K[X 1 , . . . , X n ].

Fourth step: we claim that h 2 (X 1 , . . . , X n ) ∈ K[X 1 , . . . , X n ]. To show this, we are going to prove that if r, h 1 and h 2 satisfies the hypothesis of the previous steps then h

2 ∈ K[X 1 , . . . , X n ]. Let λ ∈ K such that deg(f -λg) = deg r = d.
Since r is composite over K then fλg is reducible over K by Theorem 11, and we have:

f + λg = α m i=1 (h 1 + λ i h 2 ),
where λ i are the roots of u 1 + λu 2 = α m i=1 (Tλ i ), see the proof of [START_REF] Bodin | Reducibility of rational functions in several variables[END_REF]Corollary 2.4]. Thanks to step 1 and 2, lc(f + λg) = 1 and lc(h 1 + λ i h 2 ) = 1 so that α = 1. As h 1 /h 2 is non-composite we can find λ ∈ K such that for all i, h 1 + λ i h 2 is irreducible over K because |σ(h 1 , h 2 )| ≤ d 2 -1 and p = 0 or p > d 2 (the bound |σ(h 1 , h 2 )| ≤ d 2 -1 is proved for any field in the bivariate case in [START_REF] Lorenzini | Reducibility of polynomials in two variables[END_REF] and its extension to the multivariate case is easily obtained using Bertini's Theorem; see for instance [5, Proof of Theorem 13] or [START_REF] Bodin | Reducibility of rational functions in several variables[END_REF].)

Now let τ ∈ Galois(L/K), where L is the field generated by all the coefficients of u 1 , u 2 , h 1 , h 2 , we have:

f -λg = τ (f -λg) = m i=1 τ (h 1 ) + τ (λ i )τ (h 2 ) .
As lc τ (h 1 ) + τ (λ i )τ (h 2 ) = 1 and τ (h 1 ) + τ (λ i )τ (h 2 ) is also irreducible over K, we can write:

h 1 + λ 1 h 2 = τ (h 1 ) + τ (λ i1 )τ (h 2 ), (⋆) h 1 + λ 2 h 2 = τ (h 1 ) + τ (λ i2 )τ (h 2 ).
Thus, (λ 1λ 2 )h 2 = (τ (λ i1 )τ (λ i2 ))τ (h 2 ). As lc(h 2 ) = 1, we deduce that (λ 1λ 2 ) = τ (λ i1 )τ (λ i2 ) and then h 2 = τ (h 2 ). This implies that h 2 ∈ K[X 1 , . . . , X n ] because K is a perfect field.

Last step: we claim that h 1 ∈ K[X 1 , . . . , X n ]. Indeed, (⋆) and the hypothesis h 1 (0, . . . , 0) = 0 (see step 3) implies that λ 1 h 2 (0, . . . , 0) = τ (λ i1 )h 2 (0, . . . , 0). As h 2 (0, . . . , 0) = 0 (by step 3 again), we get λ 1 = τ (λ i1 ). Then (⋆) means that h 1 = τ (h 1 ) and this concludes the proof because K is a perfect field. Remark 14. First, notice that the above result remains obviously true when we take any extension of K instead of K. Also, observe that this theorem is false for univariate 2 (n = 1) rational function; see [START_REF] Gutierrez | Building counterexamples to generalizations for rational functions of Ritt's decomposition theorem[END_REF]Example 5]. Finally, mention that if p ≤ d 2 and the field K is not perfect then the theorem is also false. Indeed, in [2, page 27] one can find the following counterexample: f (X, Y ) = X p + bY p = 2 Recall that a non-constant univariate rational function r(X) ∈ K(X) is called composite over a field K if r(X) = u(h(X)), where u, h ∈ K(X) such that deg(u) ≥ 2 and deg(h) ≥ 2.

(X + βY ) p , with b ∈ K \ K p and β p = b, is composite in K(β) (which is clear) but non-composite in K (which is proved in loc. cit.). Theorem 11 and Theorem 13 yield the

Corollary 15. r = f /g is non-composite ⇐⇒ Spect f,g (T ) = 0 in K[T ] ⇐⇒ σ(f, g) is finite.
Corollary 15 clearly implies several results about the indecomposability of r. For instance, if r = f /g is a non-composite rational function where f, g ∈ Z[X 1 , . . . , X n ], and p is a prime integer bigger than H(Spect f,g ) and the bound B of Theorem 6, then r p is non-composite. Indeed, Spect f,g p = Spect f p ,g p = 0 in F p [T ], for all

p > B.
With this strategy we could deduce several similar results but the bounds obtained in this way can be improved. Indeed, when we use the polynomial Spect we have to study the gcd of the Φ t (f -T g)'s. But if r is supposed to be non-composite then there exits an index t 0 such that Φ t0 (f -T g) = 0. In this case it is enough to study the behaviour of one polynomial instead of the gcd of several polynomials. Thus, in what follows we are going to study the indecomposability of a rational function using Noether's forms.

p > H = d 3d 2 -3 n + d n 2 d d 2 -1 d 2 -1 ⌊(d 2 -1)/2⌋ max H(f ), H(g) d 2 -1 ,
then r p is non-composite and f p , g p are coprime.

Proof. Thanks to Theorem 11, we have that f -T g is irreducible in Q(T )[X 1 , . . . , X n ]. Therefore, there exists t 0 such that Φ t0 (f -T g) = 0 in Z[T ]. Now, if p > H then Φ t0 p (f p -T g p ) = 0 (see the proof of Theorem 6). This means that f p -T g p is irreducible in F p (T )[X 1 , . . . , X n ] and hence r p is non-composite by Theorem 11. Of course, f p and g p are coprime because otherwise f p -T g p cannot be irreducible. Proof. Assume that r is non-composite over K(Z). Then, by Theorem 11, we have that f -T g is irreducible in K(Z)[X]. Thus there exists t 0 such that Φ t0 (f -T g) = 0 in K[Z][T ]. We can write Φ t0 (f -T g) = D i=0 a i T i with a i ∈ K[Z] and a D = 0 ∈ K[Z]. Therefore, for all z ∈ K s such that a D (z) = 0 we have r(z, X) is noncomposite. Furthermore Theorem 1 gives deg Z a i ≤ k(d 2 -1). Then, Lemma 9 applied to a D (Z) gives the desired result.

Remark 18. Theorem 17 is false with the hypothesis "r is non-composite over K" instead of "r is non-composite over K(Z)". Indeed, take n = 2 and s = 1 and consider the polynomial f (X, Y, Z) = (XY ) 2 +Z. This polynomial is non-composite over K (because deg Z (f ) = 1) but f (X, Y, z) = (XY ) 2 + z is composite over K for all value z ∈ K.

3.3.

A reduction from n to two variables. We give the following Bertini like result.

Theorem 19. Let K be a perfect field of characteristic 0 or p ≥ d 2 , S be a finite subset of K and let r = f /g ∈ K(X 1 , . . . , X n ) be a reduced non-composite rational function.

For a uniform random choices of the u i 's, v i 's and w i 's in S, the rational function

r(X, Y ) = r(u 1 X + v 1 Y + w 1 , . . . , u n X + v n Y + w n ) ∈ K[X, Y ].
is non-composite with probability at least 1 -3d(d -1) + 1 /|S| where d is the degree of r.

Proof. As we did before, we study f -T g. This polynomial is irreducible over K(T ) by Theorem 11. Then we apply the effective Bertini's Theorem given in [START_REF] Lecerf | Improved dense multivariate polynomial factorization algorithms[END_REF]Corollary 8] to this polynomial. We obtain that f (X, Y ) -T g(X, Y ) is irreducible in K(T )[X, Y ] with a probability at least 1 -3d(d -1) + 1 /|S|. Then by Theorem 11 yields the desired result about r.

Acknowledgment

During the preparation of this paper, the third author was supported by Abdus Salam center (ICTP, Trieste) and after by Max-Planck Institut (Bonn). He wish to thank Pierre Dèbes for encouragements and comments, as well as Enrico Bombieri and Umberto Zannier for their discussions in the beginning of this work.

  stands for the free A-module of homogeneous polynomials of degree d.

1 2 2 1 2 2

 1212 d H by Mignotte's bounds [24, Corollary 6.33] and (i) imply that the above condition is satisfied if p > e d ln(d) (d + 1) d H 2d = e d ln(d) (d + 1) d 2 2d 2 H 2d .

Proposition 5 .

 5 Let f 1 , . . . , f n be n ≥ 2 polynomials in K[Z][U, V ] that are homogeneous with respect to the variables U, V of degree d with coefficients in K[Z]. Also, suppose given a ring morphism ρ : K[Z] → K and assume that all coefficients of f 1 , . . . , f n are polynomials in K[Z] of degree ≤ k.(i) If f 1 , . . . , f n have no (homogeneous) common factor of positive degree in K[Z][U, V ], then there exists a finite collection

  (i) ⇐⇒ (ii): see [3, Theorem 2.2].

3. 1 .

 1 Reduction modulo p. Theorem 16. Let r = f /g ∈ Z(X) be a non-constant reduced and non-composite rational function. If

3. 2 .

 2 Indecomposability of rational function with coefficients in K[Z].Theorem 17. Let d and k be positive integers, K be a perfect field of characteristic0 or p ≥ d 2 , r = f /g ∈ K[Z](X) be a non-constant reduced rational function with 0 < deg X (r) ≤ d, 0 < deg Z (r) ≤ k and let S be a finite subset of K.If r is non-composite over K(Z) then for a uniform random choice of z i in S we haveP {r(z 1 , . . . , z s , X) is non-composite over K | z i ∈ S} ≥ 1k(d 2 -1)/|S|,where |S| denotes the cardinal of S and P the probability.

  C e1,...,en , . . . ) ∈ Z[. . . , C e1,...,en , . . . ]

	such that
	∀t, Φ

t (. . . , c e1,...,en , . .

  Let A be a domain andf 1 , . . . , f n be n ≥ 2 homogeneous polynomials in A[U, V ] of degree d 1 ≥ • • • ≥ d n ≥ 1 respectively. The polynomials f 1 , . .. , f n have a common root in the projective line over the algebraic closure of the fraction field of A if and only if the multiplication map1

	1]. Theorem 2. n i=1	and [14, Corollaire of Théorème

  Using Hadamard's inequality[START_REF] Von | Modern computer algebra[END_REF] Theorem 16.6] we obtain that the absolute value of each (d 1 + d 2 )-minor of the matrix of the above multiplication map is bounded above by

	( √	dH 2 ) d1+d2 = d	d 1 +d 2 2