Marie-Laure Ancelin
email: marie-laure.ancelin@inserm.fr

PhD Isabelle Carriere

PhD Sylvaine Artero

PhD; Jerome J Maller

PhD; Chantal Meslin

Karen Ritchie

PhD Joanne Ryan

PhD Isabelle Chaudieu

LIFETIME MAJOR DEPRESSION AND GREY MATTER VOLUME SHORT-TITLE: Depression and grey matter volume

Keywords: Depression, grey matter volume, deep nuclei, occipital visual cortex, sex, 5-HTTLPR

Background: While there is evidence of structural brain alterations in major depressive disorder (MDD), little is known about how this may be affected by age at onset or genetic vulnerability. This study examines whether lifetime episodes of MDD are associated with specific alterations in grey matter volumes and whether this varies according to sex and 5-HTTLPR genotype.

Methods

: Structural magnetic resonance imaging was used to acquire anatomical scans from 610 community-dwelling participants. Quantitative regional estimates of 16 subregional volumes were derived using FreeSurfer software. MDD was diagnosed according to DSM-IV criteria. Analyses were adjusted for age, sex, total brain volume, education, head injury, and comorbidities. Results: Lifetime MDD was associated with smaller insula, thalamus, ventral diencephalon, pallidum and nucleus accumbens, and larger pericalcarine volumes in men and women. These associations remained after adjusting for the false discovery rate. Lifetime MDD was also associated with smaller caudate nucleus and amygdala in men, and in women, with larger rostral anterior cingulate. Late onset (over age 50) first episodes were associated with larger rostral anterior 4 cingulate, lingual and pericalcarine volumes whereas early onset cases were associated with smaller ventral diencephalon and nucleus accumbens. Some associations differed by 5-HTTLPR genotype; thalamus was smaller in the LL MDD compared to non-MDD participants, and pericalcarine and lingual were greater in the SL heterozygotes.

Limitations:

The study is limited by its cross-sectional design.

Conclusions:

MDD was associated with persistent volume reductions in deep nuclei and insula and enlargement in visual cortex subregions with variability according to both age of onset and genotype.

Introduction

The identification of sociodemographic, environmental and physiopathological factors associated with onset, recovery and relapse in major depressive disorder (MDD) has become a major public health priority given the association of the disorder with high mortality and co-morbidity rates. While there has been considerable research into the clinical characterization of the disorder and its associated risk factors, the neuroanatomical substrates involved in MDD are still unclear, with meta-analyses of structural and functional imaging reporting inconsistent findings. [START_REF] Geerlings | Late-Life Depression, Hippocampal Volumes, and Hypothalamic-Pituitary-Adrenal Axis Regulation: A Systematic Review and Meta-analysis[END_REF] These are likely due to heterogeneity in study design (case-control vs. cohort), setting (general population, in-or out-patients), population (age, sex), and depression characteristics (such as diagnosis or symptoms, comorbidity, age of onset, recurrent episodes, antidepressant treatment), [START_REF] Geerlings | Late-Life Depression, Hippocampal Volumes, and Hypothalamic-Pituitary-Adrenal Axis Regulation: A Systematic Review and Meta-analysis[END_REF] The most consistent findings suggest that MDD is associated with dysregulation in neural networks implicated in affective and cognitive processing, as well as autonomic system activity resulting in a heterogeneous array of emotive, cognitive and behavioral abnormalities. [START_REF] Jaworska | A review of fMRI studies during visual emotive processing in major depressive disorder[END_REF] Grey matter volume (GMV) changes constitute network nodes in MDD, of which the hippocampus, amygdala, and prefrontal cortex have been extensively

examined. [START_REF] Geerlings | Late-Life Depression, Hippocampal Volumes, and Hypothalamic-Pituitary-Adrenal Axis Regulation: A Systematic Review and Meta-analysis[END_REF][START_REF] Arnone | Magnetic resonance imaging studies in unipolar depression: systematic review and meta-regression analyses[END_REF][START_REF] Koolschijn | Brain volume abnormalities in major depressive disorder: a meta-analysis of magnetic resonance imaging studies[END_REF][START_REF] Schmaal | Subcortical brain alterations in major depressive disorder: findings from the ENIGMA Major Depressive Disorder working group[END_REF][START_REF] Sexton | A systematic review and meta-analysis of magnetic resonance imaging studies in late-life depression[END_REF] Conversely, structural alterations in deep nuclei, notably the pallidum, thalamus, and hypothalamus, as well as in insula and occipital regions have rarely been studied. This is despite accumulating evidence implicating their role in emotion and neuropathology of stress-related affective disorder. [START_REF] Sliz | Major depressive disorder and alterations in insular cortical activity: a review of current functional magnetic imaging research[END_REF][START_REF] Zhang | Brain gray matter alterations in first episodes of depression: A meta-analysis of whole-brain studies[END_REF][START_REF] Glahn | High dimensional endophenotype ranking in the search for major depression risk genes[END_REF][START_REF] Du | Brain grey matter volume alterations in late-life depression[END_REF] The nature and course of volumetric changes may also vary across the life span. [START_REF] Fiske | Depression in older adults[END_REF] Although GM abnormalities within frontal-subcortical and limbic networks are hypothesized to play a key role in the pathophysiology of depression, recent meta-analyses in late-life depression showed that the most consistent evidence for brain volume reductions were found for the hippocampus but not for other brain areas. [START_REF] Geerlings | Late-Life Depression, Hippocampal Volumes, and Hypothalamic-Pituitary-Adrenal Axis Regulation: A Systematic Review and Meta-analysis[END_REF][START_REF] Schmaal | Subcortical brain alterations in major depressive disorder: findings from the ENIGMA Major Depressive Disorder working group[END_REF][START_REF] Sexton | Magnetic resonance imaging in late-life depression: vascular and glucocorticoid cascade hypotheses[END_REF] In these meta-analyses, no distinction was made between current and past (remitted) depression and age of onset was rarely considered. Most studies have been limited to clinical cohorts, which may not be representative of case heterogeneity within the general population. Studies have generally been limited in brain regions and sample size, hence lacking the power to examine modifying factors. More particularly, sex has rarely been examined despite evidence for sexually-dimorphic brain structural and functional differences across the lifespan and their potential implication in sex-biased psychiatric conditions. [START_REF] Ruigrok | A metaanalysis of sex differences in human brain structure[END_REF] This may be particularly important for MDD with prevalence, age of onset, symptomatology and etiology differing between the sexes, and given the influence of steroid hormones on brain development and MDD onset throughout the life. [START_REF] Kockler | Gender differences of depressive symptoms in depressed and nondepressed elderly persons[END_REF] Genetic risk factors may also influence brain volumes and/or depression, [START_REF] Hibar | Common genetic variants influence human subcortical brain structures[END_REF][START_REF] Brouwer | Genetic influences on individual differences in longitudinal changes in global and subcortical brain volumes: Results of the ENIGMA plasticity working group[END_REF] but are seldom considered.

Serotoninergic genes, notably the genetic variant in the promoter region of the serotonin transporter (5-HTTLPR), have been reported to influence the structure and function of certain brain regions in depressed patients. [START_REF] Won | Imaging genetics studies on monoaminergic genes in major depressive disorder[END_REF] Whether GMV alterations could be influenced by age-related characteristics (somatic and psychiatric comorbidity) also remains to be addressed.

To address the limitations of previous studies, we investigated the relationship between lifetime MDD and various fronto-subcortical and limbic subregions in a large communitydwelling elderly population. We tested the hypothesis that regional brain structure abnormalities will be more extensive in the participants with lifetime MDD diagnosis and may persist after recovery. We also hypothesized that these abnormalities will differ according to sex and genetic vulnerability to 5-HTTLPR. We also considered age of onset and the effect of physical and psychiatric comorbidity. In the absence of availability of prospective lifetime birth cohort data, the study has been conducted retrospectively on elderly people for whom both lifetime MDD episodes and genotype have been recorded.

Methods

Participants

MRI protocol and image analysis

All the neuroimaging scans were acquired using the same scanner at the examination centre (Gui de Chauliac Neurology Hospital, Montpellier, France [START_REF] Calati | Repatriation is associated with isthmus cingulate cortex reduction in community-dwelling elderly[END_REF] The FreeSurfer outputs (from 2d and 3d perspectives) of each scan were inspected for error/misclassification. Twenty-eight scans with clear errors were excluded from the analyses. Sixteen regions of interest (ROIs) were defined using Desikan's Atlas. [START_REF] Desikan | An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest[END_REF] The regions of primary interest were the hippocampus, amygdala, orbitofrontal (OFC) and anterior cingulate cortex (AAC), and several subcortical structures (thalamus, caudate, putamen, pallidum, and accumbens nuclei) which may show neuroimaging abnormalities in depressed patients. [START_REF] Geerlings | Late-Life Depression, Hippocampal Volumes, and Hypothalamic-Pituitary-Adrenal Axis Regulation: A Systematic Review and Meta-analysis[END_REF][START_REF] Arnone | Magnetic resonance imaging studies in unipolar depression: systematic review and meta-regression analyses[END_REF][START_REF] Schmaal | Subcortical brain alterations in major depressive disorder: findings from the ENIGMA Major Depressive Disorder working group[END_REF][START_REF] Sexton | A systematic review and meta-analysis of magnetic resonance imaging studies in late-life depression[END_REF][START_REF] Kempton | Structural neuroimaging studies in major depressive disorder. Metaanalysis and comparison with bipolar disorder[END_REF][START_REF] Bora | Gray matter abnormalities in Major Depressive Disorder: a metaanalysis of voxel based morphometry studies[END_REF] We also examined ROIs which have rarely been evaluated despite accumulating evidence for abnormalities in MDD and their potential role in emotional processing and symptom characteristics, e.g. the insula, [START_REF] Sliz | Major depressive disorder and alterations in insular cortical activity: a review of current functional magnetic imaging research[END_REF][START_REF] Zhang | Brain gray matter alterations in first episodes of depression: A meta-analysis of whole-brain studies[END_REF][START_REF] Graham | Metaanalytic evidence for neuroimaging models of depression: state or trait?[END_REF] ventral diencephalon, a region primarily comprising the hypothalamus in FreeSurfer, [START_REF] Glahn | High dimensional endophenotype ranking in the search for major depression risk genes[END_REF][START_REF] Sacchet | Subcortical volumes differentiate Major Depressive Disorder, Bipolar Disorder, and remitted Major Depressive Disorder[END_REF][START_REF] Lebedeva | MRI-Based Classification Models in Prediction of Mild Cognitive Impairment and Dementia in Late-Life Depression[END_REF] and occipital visual cortex. [START_REF] Du | Brain grey matter volume alterations in late-life depression[END_REF][START_REF] Graham | Metaanalytic evidence for neuroimaging models of depression: state or trait?[END_REF]26,27 Total brain volume (grey+white matter) was computed for each participant using the segment m-file of the SPM5 software (Wellcome Department of Cognitive Neurology,UK), SPM showing greater accuracy [START_REF] Malone | Accurate automatic estimation of total intracranial volume: a nuisance variable with less nuisance[END_REF] and consistency [START_REF] Sargolzaei | A practical guideline for intracranial volume estimation in patients with Alzheimer's disease[END_REF] and less systematic bias evaluation [START_REF] Sargolzaei | Estimating Intracranial Volume in Brain Research: An Evaluation of Methods[END_REF] than FreeSurfer for this measure.

Diagnosis of lifetime psychiatric disorder

The diagnosis of current and past MDD and anxiety disorder (phobia, generalized anxiety disorder, panic disorder, obsessive compulsive disorder as well as post-traumatic stress disorder) was made by psychologists and psychiatric nurses according to DSM-IV criteria and using the Mini-International Neuropsychiatric Interview (MINI, French version 5.00), a standardized psychiatric examination validated within the general population. [START_REF] Sheehan | The Mini-International Neuropsychiatric Interview (M.I.N.I.): the development and validation of a structured diagnostic psychiatric interview for DSM-IV and ICD-10[END_REF] Positive cases were reviewed by a panel of independent psychiatrists. [START_REF] Ancelin | Angiotensinconverting enzyme gene variants are associated with both cortisol secretion and late-life depression[END_REF]

Sociodemographic and clinical variables

The standardized interview included information on socio- [START_REF] Ancelin | Sex differences in the associations between lipid levels and incident dementia[END_REF]

5-HTTLPR genotyping

Blood samples were collected after the baseline clinical interview, enabling DNA extraction and 5-HTTLPR genotyping as described [START_REF] Ritchie | Association of adverse childhood environment and 5-HTTLPR genotype with late-life depression[END_REF] and replicate independent genotyping was performed using buccal DNA extracts. [START_REF] Ancelin | Angiotensinconverting enzyme gene variants are associated with both cortisol secretion and late-life depression[END_REF]

Statistical analysis

Brain volume measurements were normally distributed.

Associations between brain regions and lifetime MDD were evaluated using ANCOVA adjusted for age, sex, and total brain volume (Model M0). Where significant associations were observed, exploratory analyses were used to assess the specificity of these findings. The sex by diagnosis interaction was tested by adding lifetime MDD x sex term to the model and where an interaction effect was observed (p<0.10), analyses were stratified by sex. Further adjustment was made for other covariates, which has been reported in the literature to modify the association between MDD and brain volumes, i.e. education level, head injury, cardiovascular ischemic pathologies, and antidepressant (Model M1), as well as lifetime anxiety disorder (Model M2). For significant bivariate associations, the effect of 5-HTTLPR was evaluated by stratification into three genotypes (LL, SL, and SS) due to the absence of consensus regarding the choice of a genetic model and frequent report of heterosis for 5-HTTLPR. [START_REF] Ancelin | Heterogeneity in HPA axis dysregulation and serotonergic vulnerability to depression[END_REF][START_REF] Ancelin | 5-HTTLPR x stress hypothesis: is the debate over?[END_REF] To account for the multiple brain regions examined, we adjusted the significance levels using the false discovery rate (FDR) method. [START_REF] Benjamini | Controlling the false discovery rate: a practical and powerful approach to multiple testing[END_REF] All tests were 2-sided and SAS (v9.4, SAS Institute, Inc.,NC) was used for the statistical analyses. S1). These covariates accounted for a relatively small proportion of the variance in each volume, with the standardized regression coefficients ranging between -0.338 to +0.257 (Supplementary Table S2).

Results

Participant characteristics

Baseline

In our sample, 576 participants also had diagnosis for lifetime anxiety disorder and a similar pattern was observed after controlling for anxiety disorder (data not shown). We conducted sensitivity analysis excluding 41 participants currently depressed or taking antidepressants, and similar patterns were observed except for a slightly weaker association for the nucleus accumbens but stronger for the lingual region (Supplementary Table S3).

Subsegmental brain regions according to the age at first major depressive episode

Of the participants with lifetime MDD, 56 [START_REF] Yuan | Regional gray matter changes are associated with cognitive deficits in remitted geriatric depression: an optimized voxel-based morphometry study[END_REF] as well as larger ACC in medication wash-out young adult MDD patients. [START_REF] Zhao | Brain grey matter abnormalities in medication-free patients with major depressive disorder: a meta-analysis[END_REF] In healthy adults, a negative association was shown between GMV and stress-related brain activity in perigenual ACC. [START_REF] Boehringer | Neural Correlates of the Cortisol Awakening Response in Humans[END_REF] Basal nuclei

In our study, lifetime MDD was associated with decreased volumes of several basal nuclei. Smaller caudate nuclei have been reported in adult patients [START_REF] Arnone | Magnetic resonance imaging studies in unipolar depression: systematic review and meta-regression analyses[END_REF][START_REF] Bora | Meta-analysis of volumetric abnormalities in cortico-striatal-pallidalthalamic circuits in major depressive disorder[END_REF] and in late-life depression, the effect sizes increased with age and with smaller percentage of women. [START_REF] Sexton | A systematic review and meta-analysis of magnetic resonance imaging studies in late-life depression[END_REF] Consistently, we found that lifetime MDD was associated with smaller caudate in men specifically and the association was strengthened after adjusting for several confounders including antidepressants and anxiety disorder (-8.6%, p=0.0025 in M2).

Some meta-analyses on the putamen reported significant volume reduction in lifetime MDD, [START_REF] Arnone | Magnetic resonance imaging studies in unipolar depression: systematic review and meta-regression analyses[END_REF][START_REF] Sexton | A systematic review and meta-analysis of magnetic resonance imaging studies in late-life depression[END_REF] others in early but not late onset, [START_REF] Bora | Meta-analysis of volumetric abnormalities in cortico-striatal-pallidalthalamic circuits in major depressive disorder[END_REF] nor current MDD. [START_REF] Schmaal | Subcortical brain alterations in major depressive disorder: findings from the ENIGMA Major Depressive Disorder working group[END_REF] Other studies found that associations were limited to severe or persistent subtypes. [START_REF] Lorenzetti | Structural brain abnormalities in major depressive disorder: a selective review of recent MRI studies[END_REF] We found a marginal association between lifetime MDD and smaller putamen (p=0.055 after FDR correction) and a greater reduction in those with recurrent episodes (global p-value=0.042).

We also found an association between lifetime MDD and smaller volumes of nucleus accumbens and pallidum. These regions were rarely examined and meta-analyses of studies mainly including adult patients with acute or lifetime MDD failed to report significant associations. [START_REF] Schmaal | Subcortical brain alterations in major depressive disorder: findings from the ENIGMA Major Depressive Disorder working group[END_REF][START_REF] Kempton | Structural neuroimaging studies in major depressive disorder. Metaanalysis and comparison with bipolar disorder[END_REF] In an analysis of a high-dimensional set of over 11,000 traits, pallidum volume was reported to be a main endophenotype related to recurrent depression. [START_REF] Glahn | High dimensional endophenotype ranking in the search for major depression risk genes[END_REF] We also observed a greater reduction in the participants having reported multiple episodes compared to those with only one or no previous episode (global p-value=0.013).

Lifetime MDD was also associated with smaller thalamus and ventral diencephalon, a region primarily comprising the hypothalamus. Small meta-analyses of the thalamus reported moderate [START_REF] Sexton | A systematic review and meta-analysis of magnetic resonance imaging studies in late-life depression[END_REF][START_REF] Kempton | Structural neuroimaging studies in major depressive disorder. Metaanalysis and comparison with bipolar disorder[END_REF] or no significant [START_REF] Arnone | Magnetic resonance imaging studies in unipolar depression: systematic review and meta-regression analyses[END_REF][START_REF] Schmaal | Subcortical brain alterations in major depressive disorder: findings from the ENIGMA Major Depressive Disorder working group[END_REF][START_REF] Bora | Meta-analysis of volumetric abnormalities in cortico-striatal-pallidalthalamic circuits in major depressive disorder[END_REF] volume reduction, and none has included hypothalamus despite its crucial role in emotional behavior and stress response as part of the hypothalamic-pituitary-adrenal axis. Smaller ventral diencephalon volume was however reported to be the first topranked neuroimaging endophenotype related to recurrent MDD in adults [START_REF] Glahn | High dimensional endophenotype ranking in the search for major depression risk genes[END_REF] and to correlate with the number of depressive episodes in late-life depression. [START_REF] Lebedeva | MRI-Based Classification Models in Prediction of Mild Cognitive Impairment and Dementia in Late-Life Depression[END_REF]

Insula

The insula plays a role in emotional, sensorimotor and interoceptive processing but has rarely been examined, despite some evidence for abnormalities in MDD and related phenotypes (sadness, irritability, sleep disorders). [START_REF] Sliz | Major depressive disorder and alterations in insular cortical activity: a review of current functional magnetic imaging research[END_REF][START_REF] Zhang | Brain gray matter alterations in first episodes of depression: A meta-analysis of whole-brain studies[END_REF] A small casecontrol study reported reduced anterior insular cortex in current and remitted young adult patients compared with healthy controls. [START_REF] Takahashi | Volumetric MRI study of the insular cortex in individuals with current and past major depression[END_REF] A meta-analysis of VBM studies also showed left insula reductions in young adults with first episode depression. [START_REF] Zhang | Brain gray matter alterations in first episodes of depression: A meta-analysis of whole-brain studies[END_REF] Our study is the first ROI study showing a significant smaller insula volume in elderly general population with lifetime and remitted MDD.

Visual cortex

Another original finding concerns the association of MDD with larger volumes of pericalcarine and lingual ROI, which appeared especially marked for late-onset MDD. Visual cortex has a central role in the fear conditioning paradigm in humans. [START_REF] Lithari | Limbic areas are functionally decoupled and visual cortex takes a more central role during fear conditioning in humans[END_REF] The pericalcarine is the initial region of visual processing and lingual gyrus is associated with high level visual processing and visual memory. The associations were highly significant after excluding currently depressed or treated participants (p=0.006 and 0.004, respectively in M2).

The link between visual cortex and MDD has only been described

in VBM studies and a meta-analysis reported significant GMV increase in the right lingual gyrus in late-life depression. [START_REF] Du | Brain grey matter volume alterations in late-life depression[END_REF] A larger lingual gyrus volume was found to predict early antidepressant response in adults and this was linked to better performance in visual neuropsychological tests. [START_REF] Jung | Impact of lingual gyrus volume on antidepressant response and neurocognitive functions in Major Depressive Disorder: A voxel-based morphometry study[END_REF] Greater sensory reactivity in visual cortex could also predict resilience against depressive relapse. [START_REF] Farb | Mood-linked responses in medial prefrontal cortex predict relapse in patients with recurrent unipolar depression[END_REF] In healthy young adults, facilitation of processing of aversive stimuli was associated with a disconnection of subcortical limbic connections (insula, putamen, amygdala, hippocampus) together with a compensatory emerging centrality of visual cortex. [START_REF] Lithari | Limbic areas are functionally decoupled and visual cortex takes a more central role during fear conditioning in humans[END_REF] Whether pericalcarine and lingual region could participate to neuronal compensatory process to facilitate processing of aversive stimuli and fear or emotional learning in response to abnormal input from other structures, or reflect resilience against relapse, remains to be examined.

5-HTTLPR genotype

Most previous studies have examined the effect of 5-HTTLPR on GMVs in either depressed or healthy participant groups, but rarely as a modifying factor between MDD risk and GM alterations. [START_REF] Won | Imaging genetics studies on monoaminergic genes in major depressive disorder[END_REF] There are generally size-limited and focused on the hippocampus and amygdala but rarely on striatum or thalamus where 5-HTT is expressed in high density. The vast majority of the structural imaging genetic studies did not consider SL genotype individually despite a lack of consensus regarding genetic model and frequent report of heterosis for 5-HTTLPR. [START_REF] Ancelin | 5-HTTLPR x stress hypothesis: is the debate over?[END_REF] Heterogeneity in age is another potential source of concern; [START_REF] Uher | The moderation by the serotonin transporter gene of environmental adversity in the aetiology of mental illness: review and methodological analysis[END_REF] unlike in younger populations where the S allele is a risk factor, the LL genotype appears to be a risk factor for mental and physical distress in elderly people highly exposed to chronic disorders and severe stressors. [START_REF] Grabe | Update on the 2005 paper: moderation of mental and physical distress by polymorphisms in the 5-HT transporter gene by interacting with social stressors and chronic disease burden[END_REF] We found no significant volumetric differences according to 5-HTTLPR in each group (with or without lifetime MDD) (data not shown) but significant between-group differences.

Particularly, the thalamus was smaller in the LL homozygotes with lifetime MDD whereas the SL heterozygotes only, had larger pericalcarine and lingual volumes compared to their non-MDD counterparts. The thalamus is rich in serotonergic neurons and reduced 5-HTT availability has been described in the thalamus of depressed patients 50 but data on the effect of 5-HTTLPR genotype are lacking and no studies have examined the pericalcarine or lingual regions despite some indication for serotonergic occipital dysfunction in depression. [START_REF] Fernandez | Increased occipital delta dipole density in major depressive disorder determined by magnetoencephalography[END_REF] The same data were found after excluding currently depressed or treated participants, suggesting that these findings were related to serotonergic vulnerability to the disorder.

In a subsample of the Esprit study, we reported that past MDD and stressful events were risk factors for current depression in LL homozygotes specifically, whereas the SL heterozygotes were more resilient to these factors. [START_REF] Ancelin | Heterogeneity in HPA axis dysregulation and serotonergic vulnerability to depression[END_REF] We also found that some adverse events during childhood, e.g. sexual or physical abuse and having had a mother with mental problems were associated with higher risk of late-life depression [START_REF] Ritchie | Association of adverse childhood environment and 5-HTTLPR genotype with late-life depression[END_REF] but decreased risk of cognitive decline notably in visual memory. [START_REF] Ritchie | Adverse childhood environment and late-life cognitive functioning[END_REF] This suggests possible cognitive adaptation or resilience effect. Although speculative, this may suggest that the pericalcarine and lingual region could participate in a persistent neuronal compensatory process in the SL heterozygotes with a history of MDD.

Sex differences

We found some evidence for sexually dimorphic alterations, with smaller caudates and amygdalae in men with lifetime MDD, and larger rostral ACC in women specifically. Specific sex differences in depression symptomatology of older adults have been described, with women showing more mood-related symptoms and appetite disturbance, and men more motivation-related symptoms and psychomotor changes [START_REF] Kockler | Gender differences of depressive symptoms in depressed and nondepressed elderly persons[END_REF] and this may involve different biological correlates. However, only a few neuroimaging studies have investigated sex effects, with many including predominantly females and, likely as a consequence, meta-analyses have seldom reported sex differences. [START_REF] Schmaal | Subcortical brain alterations in major depressive disorder: findings from the ENIGMA Major Depressive Disorder working group[END_REF][START_REF] Du | Brain grey matter volume alterations in late-life depression[END_REF] In a healthy sample mainly consisting of older women, larger ACC was associated with higher levels of anhedonia. [START_REF] Mclaren | Dimensions of depressive symptoms and cingulate volumes in older adults[END_REF] Valence-dependent sex differences in emotional reactivity with divergent activation patterns, notably in the ACC and amygdala, have been reported suggesting difference in recognizing, expressing, or responding to emotions. [START_REF] Filkowski | Sex differences in emotional perception: Meta analysis of divergent activation[END_REF] Rostral ACC and caudate nucleus were shown to be involved in impulse inhibition in young adults in a sex-specific manner suggesting different processing strategies (e.g. inhibiting inappropriate response for males vs. eliciting appropriate response for females). [START_REF] Liu | Sex differences in anterior cingulate cortex activation during impulse inhibition and behavioral correlates[END_REF] In ADHD adults, smaller caudate was associated with impulsive/hyperactive symptoms in males but not females. [START_REF] Onnink | Brain alterations in adult ADHD: effects of gender, treatment and comorbid depression[END_REF] Besides, increased resting rostral ACC activity has been linked to adaptive cognitive aspects of rumination and could predict better antidepressant response and recovery. [START_REF] Pizzagalli | Frontocingulate dysfunction in depression: toward biomarkers of treatment response[END_REF] Whether sex may influence the nature of changes in some structures or be associated with specific symptoms, processing strategies, or characteristics of depression (e.g. rumination, irritability, impulsivity) remains to be examined.

Context of the findings

Retrospectively determined incidence of lifetime MDD was found to be associated in our study with smaller striatum, pallidum, thalamus, hypothalamus, and insula but larger pericalcarine and lingual regions even many years after recovery. These findings are partly consistent with a neurobiological model of current depression that posits dysfunction of the cortico-striatal-pallidal-thalamic network involved in emotion, cognition and motor control, reward and stress systems, as well as sensorimotor and interoceptive processing [START_REF] Graham | Metaanalytic evidence for neuroimaging models of depression: state or trait?[END_REF][START_REF] Price | Neural circuits underlying the pathophysiology of mood disorders[END_REF][START_REF] Tadayonnejad | Brain network dysfunction in late-life depression: a literature review[END_REF] with some evidence for sex difference regarding emotion production and regulation. We did not observe reduced volumes of hippocampus or frontal subregions reported in currently depressed adults. In a meta-analysis of fMRI studies, Graham and colleagues suggested frontal areas as state markers of MDD whereas striatal regions are trait vulnerability markers which may be less affected by treatment. [START_REF] Graham | Metaanalytic evidence for neuroimaging models of depression: state or trait?[END_REF] They also stressed the potential key roles of regions not included within the prevailing models of MDD, such as insula and occipital subregions, the latter showing overactivity in MDD. [START_REF] Graham | Metaanalytic evidence for neuroimaging models of depression: state or trait?[END_REF] Sustained remission from MDD has been associated with a combination of the normalization of reactivity of certain prefrontal and limbic regions and greater sensory reactivity in visual cortices [START_REF] Farb | Mood-linked responses in medial prefrontal cortex predict relapse in patients with recurrent unipolar depression[END_REF] as well as hyperactivity and/or reduced deactivation in the rostral ACC. [START_REF] Pizzagalli | Frontocingulate dysfunction in depression: toward biomarkers of treatment response[END_REF] Our data further suggest a key role of visual corticostriatal loop in remitted elderly with volume enlargement of visual occipital regions and reduction in subcortical structures. [START_REF] Seger | The visual corticostriatal loop through the tail of the caudate: circuitry and function[END_REF] They also suggest a link between certain sensory/visual function (thalamus for sensory relay as well as pericalcarine and lingual region for visual processing and visual memory)

Limitations and strengths

Limitations include the cross-sectional design of the study and thus we cannot determine whether volume alterations

and 5 -

 5 HTTLPR vulnerability (to stress-induced relapse) or resilience to MDD. The question however, remains as to whether these abnormalities represent a biological long-term vulnerability (endophenotype as intermediate expression of genetic vulnerability factors).

Fig. 1 :

 1 Fig. 1: Association between lifetime major depressive disorder

Table 1 ;

 1 the median age was 70.7 years and 47.5%

		characteristics	of	the	610	participants	are
	summarized in were male, 26.6% had lifetime MDD of whom 38.3% had recurrent
	episodes. Only 2.1% had current MDD and 5.1% were currently
	taking an antidepressant. Men and women differed in most
	characteristics; women being younger, more frequently alone,
	and	reporting	lifetime	psychiatric	disorder	but	less

Subsegmental brain regions according to lifetime MDD

Adjusted for age, sex, and total brain volume, lifetime MDD was associated with smaller volumes of insula, diencephalic structures and deep nuclei, as well as larger volumes of rostral ACC and two visual cortex subregions (Table

2

). correction. Similar data were found in the multivariate model further adjusted for education, head injury, cardiovascular ischemic pathologies, and antidepressant (Supplementary Table

Table 1 : Characteristics of the 610* participants

 1 Unless specified otherwise. †Kruskal-Wallis tests for continuous variables and Chi-square tests for categorical variables. ‡Diagnosis of past and current major depression or anxiety disorder (phobia, generalized anxiety disorder, post-traumatic stress disorder, panic disorder or obsessive compulsive disorder) according to DSM-IV criteria and using the MINI.[START_REF] Sheehan | The Mini-International Neuropsychiatric Interview (M.I.N.I.): the development and validation of a structured diagnostic psychiatric interview for DSM-IV and ICD-10[END_REF] §History of cardiovascular ischemic pathologies (angina pectoris, myocardial infarction, stroke,

	precede or are subsequent to MDD. Data related to lifetime MDD
	were retrospective. This may introduce recall bias and lead to
	an underestimation of the associations, even if we have
	excluded	participants	diagnosed	with	probable/possible
	dementia to minimize inaccuracies. The volume variations
	associated with lifetime MDD ranged between 2-5% for deep
	nuclei and insula and 5-8% for visual cortex ROIs suggesting a
	relatively small effect size. State-like characteristics were
	not examined due to the low prevalence of current MDD in this

*cardiovascular surgery, arteritis). ¶Fasting glucose ≥ 7.0 mmol/L or treatment.

Table 2 : Association of subsegmental brain region volumes with lifetime major depressive disorder among 610 participants

 2 Mean (SD) values expressed as mm 3 and adjusted for age, sex, and total brain volume. †Raw p-values not adjusted for multiple comparisons. ‡p-values after FDR correction.

		No lifetime MDD (n=448) Lifetime MDD (n=162)
		Mean*	SD	Mean*	SD	p †	p ‡
	Medial orbitofrontal	8962.94	43.20	8939.55	73.61	0.786 0.786
	Lateral orbitofrontal	11652.40	46.37	11718.13	79.01	0.477 0.545
	Rostral anterior cingulate	3347.51	28.37	3458.73	48.34	0.050 0.089
	Caudal anterior cingulate	2993.71	25.67	3065.27	43.73	0.163 0.217
	Hippocampus	7061.54	33.88	7033.17	57.73	0.675 0.720
	Amygdala	2634.05	14.88	2580.65	25.34	0.072 0.115
	Insula	11954.83	48.80	11722.33	83.14	0.017 0.045
	Thalamus	11806.32	43.44	11575.64	74.01	0.008 0.038
	Ventral diencephalon	6794.71	28.62	6624.44	48.76	0.003 0.038
	Caudate nucleus	6852.97	52.58	6669.85	89.58	0.081 0.118
	Putamen	9453.97	52.75	9215.18	89.87	0.024 0.055
	Nucleus accumbens	997.39	7.14	960.92	12.17	0.011 0.038
	Pallidum	2983.73	16.17	2893.75	27.55	0.005 0.038
	Cuneus	4765.02	30.27	4839.00	51.58	0.221 0.272
	Pericalcarine	3404.66	28.46	3547.02	48.48	0.012 0.038
	Lingual	10714.10	67.51	10984.78 115.01	0.045 0.089
	FDR = false discovery rate; MDD = major depressive disorder; SD = standard deviation.

*

Table 3 : Association of subsegmental brain region volumes with lifetime major depressive disorder according to 5-HTTLPR genotype

 3

	5-HTTLPR			LL (n=160)					SL (n=250)					SS (n=124)
		No lifetime	Lifetime		No lifetime	Lifetime		No lifetime	Lifetime
		MDD (n=112)	MDD (n=48)		MDD (n=184)	MDD (n=66)		MDD (n=93)	MDD (n=31)
		Mean*	SD	Mean*	SD	p †	Mean*	SD	Mean*	SD	p †	Mean*	SD	Mean*	SD	p †
	Rostral
	anterior	3394.88 60.80	3537.88 94.35	0.220	3345.48 43.42	3400.87 74.31	0.521 3346.62 63.48	3498.30 112.93 0.243
	cingulate
	Insula	11847.14 99.97	11963.30 155.14 0.544	11996.04 78.88	11628.15 134.99 0.020 11833.83 103.71 11520.94 184.48 0.141
	Thalamus	11966.39 98.90	11376.42 153.47 0.002	11843.10 65.47	11700.58 112.05 0.274 11685.44 89.60	11717.36 159.39 0.862
	Ventral
		6726.12 57.73	6524.76 89.59	0.070	6869.44 46.83	6659.23 80.15	0.025 6765.45 59.92	6731.95 106.59 0.784
	diencephalon
	Putamen	9530.94 131.13 9093.75 203.49 0.083	9591.21 75.65	9323.70 129.47 0.076 9257.61 108.03 8955.26 192.18 0.172
	Nucleus
		996.60	14.19	966.80	22.02	0.273	1003.50 11.03	984.31	18.88	0.382 980.00	16.80	921.09	29.89	0.088
	accumbens

Table S1 : Association of subsegmental brain region volumes with lifetime major depressive disorder among 610 participants in multivariable model adjusted for age, sex, total brain volume, education level, head injury, cardiovascular ischemic pathologies, and antidepressant use

 S1

		No lifetime MDD	Lifetime MDD
		(n=448)	(n=162)
		Mean*	SD	Mean*	SD	p †
	Rostral anterior cingulate	3364.37	72.64	3471.48	78.44	0.059
	Insula	12053.26	125.78	11813.70 135.83	0.015
	Thalamus	11597.33	111.61	11381.19 120.52	0.013
	Ventral diencephalon	6704.75	73.67	6540.86	79.55	0.004
	Putamen	9409.72	135.46	9173.56	146.28	0.025
	Nucleus accumbens	970.78	18.34	935.73	19.80	0.014
	Pallidum	3003.19	41.68	2911.07	45.01	0.005
	Pericalcarine	3462.15	73.24	3601.43	79.09	0.015
	Lingual	10628.71	173.68	10901.94 187.55	0.044
	MDD = major depressive disorder; SD = standard deviation.
	*Mean (SD) values expressed as mm 3 .
	†Raw p-values

Acknowledgements:

The ESPRIT project is financed by the regional government of Languedoc-Roussillon, the Agence Nationale de la Recherche Project 07 LVIE 004, and an unconditional grant from Novartis. This work was also supported by France Alzheimer. The funders had no role in the design and conduct of the study; in data collection, management, analysis or interpretation of the data and were not involved with the writing, preparation, review or approval of the manuscript.


Arthur Scolari Fagundes
email: arthur.fagundes@ufrgs.br

João Manoel Gomes Da Silva

Marc Jungers
email: marc.jungers@univ-lorraine.fr

Stability of control systems under aperiodic sampling and input saturation withclock-dependent Lyapunov functions

Keywords: Aperiodic sampling, Clock-dependent Lyapunov, Control saturation, Sampled-data system

INTRODUCTION

Much attention has been devoted to the stability analysis of sampled-data control systems in the past decades [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF]. A sampled-data system consists in a plant evolving in continuous-time and a controller setting inputs to that plant in discrete-time [START_REF] Seuret | Taking into account period variations and actuator saturation in sampled-data systems[END_REF]. On one side, in the last years of the previous century an intensive research on sampled-data systems under periodic sampling has been developed (see [START_REF] Aström | Computer Controlled Systems: Theory and design[END_REF] and [START_REF] Chen | Optimal sampled-data control systems[END_REF]).

On the other side, the rising importance of the field of networked control systems has brought new challenges to the application of control theory (P. [START_REF] Antsakils | A special issue on technology of networked control systems[END_REF], one very important being the feature of aperiodic sampling (i.e., sampling with variable sampling intervals) imposed by the network. For a sensor to transmit a continuoustime signal over a network, the signal must be sampled, converted to a digital format, transmitted, and finally the data must be converted again to the analog format to be received at the controller side. The total delay between sampling and eventual reception is subject to significant variations due to network conditions such as congestion and eventual packet drop-outs [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF], and motivates the study of aperiodic sampled-data systems.

The study of aperiodic sampled-data systems is quite underdeveloped compared to the periodic counterpart [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF], but has been receiving attention recently in the context of control theory. One way to represent it is This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -(CAPES/Brazil) Finance Code 001, by the CNPq/Brazil (grants PQ:305979/2015-9 and Universal: 422992/2016-0), and ANR/France (project HANDY, number ANR-18-CE40-0010). by an impulsive system model. This system framework emerges as a generalization of the discrete-time model which considers the inter-sampling system behaviour using signal lifting (see [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF] and references therein). For instance, in [START_REF] Geromel | On an LMI approach to optimal sampled-data state feedback control design[END_REF] an impulsive system framework is used to provide conditions for the stabilization of sampled-data linear systems, aiming at optimizing H 2 and H ∞ criteria. More recently, in [START_REF] Geromel | On an LMI approach to optimal sampled-data state feedback control design[END_REF], the synthesis of dynamic output feedback using differential matrix inequalities is addressed. It should be highlighted that the above references do not consider the possibility of actuator saturation.

In the present paper, the almost ubiquitous feature of input saturation is taken into account. Numerous works consider the problems of stability analysis of systems with input saturation (see [START_REF] Hu | Control systems with actuator saturation: analysis and design[END_REF] and [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]). Many of these approaches have been proposed considering either continuous or discrete-time systems. Considering the periodic sampling case, the problem of assessing stability of a samped-data system with dynamic output feedback control law has been addressed in [START_REF] Dai | Output feedback design for saturated linear plants using deadzone loops[END_REF]. Aperiodic sampling in the presence of control saturation is addressed in Seuret and[START_REF] Seuret | Taking into account period variations and actuator saturation in sampled-data systems[END_REF][START_REF] Gomes Da | Regional stability analysis of discrete-time dynamic output feedback under aperiodic sampling and input saturation[END_REF] considering a looped-functional approach. In [START_REF] Fiacchini | Stability of sampled-data control systems under aperiodic sampling and input saturation[END_REF] this problem is addressed considering an impulsive system framework and the discrete-time dynamics of the closed-loop system cast as a difference inclusion obtained from a partition of the intersampling interval.

In the present paper we are interested in the problem of stability analysis of an aperiodic sampled-data system subject to input saturation. From an impulsive system rep-resentation of the closed-loop system it is shown that, by satisfying stability conditions based on a clock-dependent Lyapunov function, a monotonically decreasing quadratic form of this class of function ensures the local stability of the system equilibrium point. Moreover, the conditions can be cast in LMI-based optimization problems to generate piecewise quadratic estimates of the region of attraction of the origin.

The content of this paper is organized as follows: In section 2, the impulsive representation of the sampled-data controlled system is presented. Section 3 regards Lyapunov stability theorem and its statement for the clockdependent class of functions. In section 4, the saturation effect is included in the modeling, and stability conditions dealing with this nonlinearity are derived. In section 5 an optimization problem to generate estimates of the region of attraction of the origin is stated, and a numerical example is exposed in section 6.

Notation: N is the set of natural numbers, R is the set of real numbers, and R + is the set of positive real numbers. For a function x : R → R n , we define

x(t -) = lim t→t, t<t x( t)
The i th element of a vector v is denoted by v (i) , the element in the i th row and j th column of a matrix M is denoted by M (i,j) , while M (i) denotes the entire row i of M . Note that the indexation of lines and elements only occurs between parentheses -in M m , m is simply a subscript. S n is the domain of symmetric matrices of order n, and for a symmetric matrix S ∈ S n , S > 0 means that S is positive definite. M denotes the transpose of M . I denotes a identity matrix of appropriate dimensions.

CONTROL UNDER APERIODIC SAMPLING

Consider the continuous-time plant described by the following linear time-invariant model:

ẋp (t) = Ax p (t) + Bu(t)
where x p ∈ R np represents the state of the plant, u ∈ R m represents its input, and t ∈ R + is time. Matrices A and B have appropriate dimensions and are constant.

It is considered that the input signal is calculated from samples, as if coming from a digital controller, and obeys the control law given by

u(t) = Kx p (t k ) ∀t ∈ [t k , t k+1 ) , ∀k ∈ N with t k , k ∈ N,
being the sampling instants. u(t) is supposed to be kept constant between two successive sampling instants through a zero order holder.

Since we are considering an aperiodically sampled control, the time between two successive samples δ k = t k+1 -t k is not necessarily constant. In particular, we assume that 0 < τ ≤ δ k ≤ τ

The bounds τ and τ are supposed to be imposed by constraints on the networked control implementation. As stated in the introduction, they represent network conditions that affect sampling rate e.g. a lag induced by the communication protocol.

Note that assuming τ > 0 is sufficient to avoid Zeno effect. The sequence of admissible sampling intervals Θ can be formalized by the set below

Θ = {t = t k ∈ R + : t k+1 = t k + δ k , δ k ∈ [τ , τ ] , ∀k ∈ N} (1)
The closed loop system can thus be described by the following system.

ẋp (t) = Ax p (t) + BKx p (t k ) ∀t ∈ [t k , t k+1 )
If we want to describe it for all t ≥ 0 and a particular sampling sequence θ ∈ Θ, the following impulsive model can be considered:

       ẋ(t) = A f x(t) ∀t / ∈ θ; t = 0 (a) x(t) = J j x(t -) ∀t ∈ θ; t = 0 (b) x p (0) = x 0 t = 0 (c) u(0) = Kx 0 t = 0 (d) (2)
where

x(t) = [x p (t) u(t) ] ∈ R n , with n = n p + m,
is the overall system state, and A f and J j ∈ R n×n are given as follows

A f = A B 0 0 , J j = I 0 K 0 3. CLOCKED-LYAPUNOV STABILITY THEOREM
The following Theorem provides conditions to assess the stability of the hybrid system (2). [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF] Theorem 1. Consider system (2) and denote

τ = τ k (t) = t -t k , ∀t ∈ [t k , t k+1 ) , ∀k ∈ N the clock variable. Let c 1 , c 2
and c 3 be positive constants, and V be a function such that

c 1 ||x|| 2 ≤ V (x, τ ) ≤ c 2 ||x|| 2 for all x ∈ R n , τ ∈ [0, τ ] .
Suppose that for any impulse sequence θ ∈ Θ the corresponding solution x(t) to (2) satisfies:

dV (x(t), τ k (t)) dt < -c 3 ||x(t)|| 2 ∀t ∈ [t k , t k+1 ) , ∀k ∈ N (3) and V (x(t k ), τ k (t k )) < V (x(t - k ), τ k (t - k )) ∀k ∈ N (4)
then, the equilibrium point x = 0 of system (2) is globally exponentially stable.

Note that for t ∈ [t k , t k+1 ), i.e., t / ∈ θ ∪ {0}, it follows from (1) that τ k (t) ∈ [0, τ ), and also that τ k (t k ) = 0. Moreover, one has

τk (t) = dτ k (t) dt = 1 ∀t / ∈ θ ∪ {0} (5)
Based on these facts the following corollary can be stated.

Corollary 1. If there exists a differentiable function V (x, τ ) = x(t) P (τ )x(t), P : [0, τ ] → S n , P (τ ) > 0, ∀τ ∈ [0, τ ), such that

A f P (τ ) + P (τ )A f + Ṗ (τ ) < 0, ∀τ ∈ [0, τ ) (6) and J j P (0)J j -P (τ ) < 0, ∀τ ∈ [τ , τ ] (7)
are satisfied, then the equilibrium point x = 0 of system (2) is globally exponentially stable.

Proof. From ( 5), τ = 1 for t / ∈ θ ∪ {0}. Hence, inequality (3) in Theorem 1 is given by

d V (x(t), τ k (t)) dt = d V dx ẋ + d V dτ τ = ẋ P (τ k (t))x + x P (τ k (t)) ẋ + x Ṗ (τ k (t))x < 0. ( 8
)
As τ k (t) ∈ [0, τ ], and taking into account (2)-a, in order to satisfy (8) it suffices that

A f P (τ ) + P (τ )A f + Ṗ (τ ) < 0 ∀τ ∈ [0, τ ) which implies that exists c 4 = λ max (A j P (τ ) + P (τ )A j + Ṗ (τ )), for τ ∈ [0, τ ],
where λ max (•) is the maximum eigenvalue of a matrix.

On the other hand, for t ∈ θ, we have τ k (t) = 0 and thus, from (2)-b, inequality (4) in Theorem 1 is satisfied if

J j P (0)J j -P (τ ) < 0 ∀τ ∈ [τ , τ ]
which concludes the proof.

STABILITY ANALYSIS WITH SATURATION

We consider now the case in which the control signal is constrained in amplitude by means of a saturating actuator. The actual plant input in this case is given as follows

u(t) = sat(K(x p (t k ))) (9)
where sat(•) : R m → R m is a vector valued saturation function, i.e.

sat(v)

(i) = sign(v (i) )min(|v (i) |, 1) ∀i = 1, • • • , m
With saturation in the controller output, the closed-loop system (2) must be rewritten as follows:

           ẋ(t) = A f x(t) ∀t / ∈ θ; t = 0 (a) x(t) = A j x(t -) + B j sat(K j x(t -)) t ∈ θ; t = 0 (b) x p (0) = x 0 t = 0 (c) u(0) = sat(Kx 0 ) t = 0 (d) (10)
where A f , A j ∈ R n×n , B j ∈ R n×m and K j ∈ R m×n are given as follows

A f = A B 0 0 , A j = I 0 0 0 , B j = 0 I , K j = [K 0]
Although the origin of a linear closed-loop system is supposed to be either globally asymptotically stable or not stable in the absence of input constraints, this is not the case when saturation is present. In the case of control systems with input saturation, the stability of an equilibrium point is not necessarily global. In this case it is important to characterize the region of attraction of the origin and its estimates. The approach used here to estimate the region of attraction of the origin can be found in [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF].

Aiming to express stability conditions specifically for system (10), the quadratic function V given in Corollary 1 will be used to rewrite conditions (3) and ( 4) of Theorem 1 as matrix inequalities and therefore to formulate a convex optimization problem to maximize the estimated region of attraction of the system. With that in mind, the first step concerns the treatment of the saturation term of system (10). The idea in this case is to use a deadzone nonlinearity and a corresponding generalized sector condition as a mean of relaxing the problem. A deadzone function can be defined from the saturation function given in (9) as follows dz

(v) = sat(v) -v that is dz(v) (i) = sign(v (i) ) 1 -max(|v (i) |, 1) (11
) Regarding the deadzone nonlinearity, the following lemma can be stated: Lemma 1. [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] Consider K j , G j ∈ R m×n and define the set

S = {x ∈ R n ; |(K j(i) -G j(i) )x| ≤ 1, i = 1, • • • , m} (12)
If x ∈ S, then the deadzone nonlinearity dz(K j x) satisfies the following inequality dz(K j x) T (dz(K j x) + G j x) ≤ 0 (13) for any diagonal positive definite matrix T ∈ R m×m .

In order to apply the property stated in Lemma 1 in the stability analysis, saturation function in (10)-b must be substituted by the deadzone function. Note that:

sat(K j (x(t - k )) = K j (x(t - k )) + dz(K j (x(t - k ))) (14)
The substitution of ( 14) in ( 10)-b results in

x(t) =A j x(t -) + B j dz(K j (x(t - k )) ∀t ∈ θ A j = A j + B j K j (15)
Next we aim to provide a condition to ensure the local exponential stability of the origin of system (10) along with an estimate of its region of attraction.

For this, we will consider the application of Theorem 1 to system (10) with a quadratic Lyapunov candidate function that depends affinely on τ , defined as follows:

V (x(t), τ k (t)) = x (t)P (τ k (t))x(t) = x(t) (P 1 + τ k (t)P 2 )x(t) (16)
In order to state the main result, the following instrumental result is recalled Lemma 2. [START_REF] Finsler | Über das vorkommen definiter und semidefiniter formen in scharen quadratischer formen[END_REF] For y ∈ R 2n+m , Λ = Λ ∈ R 2n+m×2n+m and Γ ∈ R n×2n+m with rank(Γ) < n. The following statements are equivalent:

(1) y Λy < 0 ∀y = 0 : Γy = 0 (2) ∃N ∈ R 2n+m×n : Λ + N Γ + Γ N < 0

Based on Lemmas 1 and 2, and the conditions in Theorem 1 with the quadratic function ( 16), we can now state a testable condition for the stability analysis of system (10).

Theorem 2. If there exist matrices P 1 ∈ S n , P 2 ∈ S n , G j ∈ R m×n , T ∈ R m×m and N ∈ R 2n+m×n that satisfy the following matrix inequalities:

Λ 1 + N Γ + Γ N < 0 (17) Λ 2 + N Γ + Γ N < 0 (18) A f P 1 + P 1 A f + P 2 < 0 (19) A f (P 1 + τ P 2 ) + (P 1 + τ P 2 )A f + P 2 < 0 ( 20
)
P 1 > 0 (21) P 1 + τ P 2 > 0 ( 22
) (P 1 + τ P 2 ) (K j(i) -G j(i) ) 1 ≥ 0 ( 23
) (P 1 + τ P 2 ) (K j(i) -G j(i) ) 1 ≥ 0, i = 1, • • • , m (24)
with

Λ 1 =   P 1 0 0 -P 1 -τ P 2 -G j T -2T   Λ 2 =   P 1 0 0 -P 1 -τ P 2 -G j T -2T   Γ = [-I A j B j ] (25)
then, for all initial conditions of system (10) in the ellipsoid E(P (0)) = {x ∈ R n : x P 1 x ≤ 1}, the corresponding trajectories converge exponentially to the origin ∀θ ∈ Θ.

Proof. Consider the function V (x(t), τ k (t)) defined in (16). In order to satisfy condition (3) of Theorem 1 we must ensure that:

V (x(t), τ k (t)) = x(t) [(P 1 + τ k (t)P 2 )A f +A f (P 1 + τ k (t)P 2 ) + P 2 ]x(t) < 0, ∀t ∈ (t k , t k+1 ) (26)
which is ensured if

(P 1 + τ k (t)P 2 )A f + A f (P 1 + τ k (t)P 2 ) + P 2 ) < 0 ∀t ∈ (t k , t k+1 ) (27) As τ k (t) ∈ [0, τ ) , ∀t ∈ [t k , t k+1
) and ( 16) has affine dependence on τ k (t), a necessary and sufficient condition to satisfy (3) is given by relations ( 19) and (20).

Let us now consider condition (4) in Theorem 1. This condition evaluated on the system trajectories can be stated as follows:

x(t k ) x(t - k ) P (0) 0 -P (τ k-1 (t k )) x(t k ) x(t - k ) < 0 (28) for [x(t k ) x(t - k ) dz(K j (x(t - k ))) ] such that [-I A j B j ]   x(t k ) x(t - k ) dz(K j (x(t - k )))   = 0 (29) Supposing now that x(t - k ) ∈ S, it follows from Lemma 1 that   x(t k ) x(t - k ) dz(K j (x(t - k )))     0 0 0 0 -G j T -2T     x(t k ) x(t - k ) dz(K j (x(t - k )))   ≤ 0 (30)
Hence, applying the S-procedure and Lemma 2, it follows that if

Λ(τ k-1 (t k )) + N Γ + Γ N < 0 (31) with Λ(τ k-1 (t k )) =   P (0) 0 0 -P (τ k-1 (t k )) -G j T -2T   (32)
then ( 28) is verified.

Recalling now that τ k-1 (t k ) ∈ [τ , τ ), as condition ( 32) is affine in τ k-1 (t k ), a necessary and sufficient condition to verify ( 32) is given by matrix inequalities ( 17) and ( 18).

Suppose now that x(0) = x(t 0 ) ∈ E(P (0)). From ( 19) and ( 20) it follows that x(t - 1 ) ∈ E(P (t - 1 )), since x(t - 1 ) P (τ 0 (t - 1 ))x(t - 1 ) < x(t 0 ) P (0)x(t 0 ) ≤ 1 Now, from conditions ( 23) and ( 24), as x(t - 1 )P (τ 0 (t - 1 ))x(t - 1 ) < 1, it follows that x(t - 1 ) ∈ S. On the other hand, conditions ( 23) and (24) ensure that E(P

(τ k-1 (t - k ))) ⊂ S, ∀τ k-1 (t - k ) ∈ [τ , τ ].
Hence, condition (30) is indeed verified and we conclude that x(t 1 ) P (0)x(t 1 ) < x(t - 1 )P (τ 0 (t - 1 ))x(t - 1 ) ≤ 1 Thus, repeating the reasoning, it follows that x(t - k ) ∈ S ∀k ∈ N, and the proof is concluded.

OPTIMIZATION PROBLEM

For a given upper and lower bound on the sampling period, a region of initial conditions for the closed loop system with guaranteed asymptotic stability can be computed, i.e., provided that x(0) ∈ E(P (0)), Theorem 2 guarantees that the trajectory of x converges asymptotically to the origin.

The idea is to maximize region E(P (0)) considering some size criterion. The maximization of the minor axis of the ellipsoid is adopted here as a criterion, leading to the following optimization problem: min P1,P2,Gj ,N,ε ε subject to: (17), ( 18), ( 19), ( 20), ( 21), ( 22), ( 23), ( 24) and P 1 -εI < 0 (33)

The last constraint in problem (33) ensures that the maximal eigenvalue of P 1 is smaller than ε. Hence, the minimization of ε corresponds to the maximization of the minor axis of the ellipsoid E(P (0)), since P (0) = P 1 . Problem (33) is not convex since the product between G j and T , i.e. the constraints are not LMIs. Thus, a way of overcoming this problem and obtaining LMI constraints consists in fixing T . This limitation will be worked around in a future study, but if we choose to deal with a SISO problem, where m = 1, this is not a serious limitation because T is a scalar. Thus, the optimal value of (33) can be easily obtained by solving LMI problems over a grid in T .

Remark: As in Fiacchini and Gomes da Silva Jr. ( 2018), it is important to remark that the region E(P (0)) is defined for x(t) = [x p (t) u(t) ] . Observe that the variables x p (0) and u(0) are related by the following:

u(0) = sat(Kx p (0)) = sat(Kx 0 )
Therefore, a guaranteed set of plant initial states, i.e. the set in the plant subspace such that x(0) = [x p (0) u(0) ] ∈ E(P (0)) is actually described by the set

X = {x p ∈ R n ; x p sat(Kx p ) P 1 x p sat(Kx p ) ≤ 1, τ ∈ [0, τ )} (
The region of stability X is shown in Figure 1. As a comparison, in Figure 1 is also plotted the domain obtained with the conditions proposed by Fiacchini and Gomes da Silva Jr. (2018). It can be observed that for both estimates, since X is defined from a piecewise quadratic function, the region estimated is not necessarily convex. One advantage of the method of this study, however, is that the optimization problem is far simpler in terms of its conditions. Finally, Figure 2 contains several trajectories obtained from initial conditions at the boundary of region X , while considering δ k randomly chosen in the interval δ k ∈ [0.05, 0.1], showing the expected convergence of all trajectories. The new approach for stability analysis proposed by this paper has been tested and also compared with another current method, showing positive results. This study is actually an early stage of an ongoing work considering more generic clock dependent Lyapunov functions, namely depending polynomially on the clock variable. This will require for instance the use of SOS programming [START_REF] Prajna | SOSTOOLS: sum of squares optimization toolbox for matlab[END_REF] to solve the stability conditions. A study in this sense has been already conducted by [START_REF] Briat | Theoretical and numerical comparisons of looped functionals and clock-dependent lyapunov functions-the case of periodic and pseudo-periodic systems https://proceedings.science/p/111553 with impulses[END_REF] without the saturation effect, which is not straightforward to consider. With this generalization, it is expected to obtain less conservative estimates of the region of attraction.

  34)which defines a piecewise quadratic region in R n .6. NUMERICAL EXAMPLEConsider the system (10) with the following matrices: the intersampling times δ k ∈ [0.05, 0.1]. Solving (33) on a grid on T , the minimum is obtained for T = 0.1, leading to matrices:

Figure 1 .

 1 Figure 1. Region X obtained with the proposed method (red) and region obtained with the algorithm of Fiacchini and Gomes da Silva Jr. (blue).

Figure 2 .

 2 Figure 2. Trajectories starting at the boundary X , with random sampling intervals 7. CONCLUSION