Asuman Nur Karhan

Jamila Zammouri

Martine Auclair

Emilie Capel

Feramuz Demir

Fehmi Ateş

Marie-Christine Verpont

Jocelyne Magre

Bruno Fève

Olivier Lascols

Feramuz Demir Apaydin

Fehmi Ates

Jocelyne Magré

Yusuf Usta

Isabelle Jéru

Corinne Vigouroux
email: corinne.vigouroux@inserm.fr

Biallelic CAV1 null variants induce Congenital Generalized Lipodystrophy with achalasia

Keywords: CAV1, lipodystrophy, autosomal recessive, achalasia, caveolae 9, Table: 1, Supplemental Table: 1

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INTRODUCTION

Caveolae are plasma membrane microdomains that act as signalling platforms in several cell types, including adipocytes, smooth muscle cells, endothelial cells and fibroblasts. The integral membrane protein caveolin-1, encoded by the CAV1 gene, is required for caveolae formation and is the main protein component of caveolae. Caveolin-1 interacts, among others, with the insulin receptor and contributes to the compartmentalization of insulin signaling pathways [START_REF] Saltiel | Insulin signaling in microdomains of the plasma membrane[END_REF].

Caveolin-1 is also a fatty acid-binding protein able to translocate from the plasma membrane to the adipocyte lipid droplets, therefore contributing to the regulation of lipid storage [START_REF] Lay | Cholesterol-induced caveolin targeting to lipid droplets in adipocytes: a role for caveolar endocytosis[END_REF][START_REF] Hodges | Proteomic insights into an expanded cellular role for cytoplasmic lipid droplets[END_REF]. In addition, it plays a role in tumor suppression and oxidative stress-induced cellular senescence [START_REF] Zou | Caveolin-1, cellular senescence and agerelated diseases[END_REF]. Loss of caveolin-1 expression in mice induces several defects including progressive lipodystrophy with insulin resistance and hypertriglyceridemia [START_REF] Razani | Caveolin-1-deficient mice are lean, resistant to diet-induced obesity, and show hypertriglyceridemia with adipocyte abnormalities[END_REF][START_REF] Lay | Filling up adipocytes with lipids. Lessons from caveolin-1 deficiency[END_REF], cardiomyopathy and pulmonary hypertension [START_REF] Drab | Loss of caveolae, vascular dysfunction, and pulmonary defects in caveolin-1 gene-disrupted mice[END_REF][START_REF] Zhao | Defects in caveolin-1 cause dilated cardiomyopathy and pulmonary hypertension in knockout mice[END_REF].

The phenotypic spectrum of the rare CAV1 genetic defects in humans remains difficult to delineate. A single patient has been previously reported with a homozygous CAV1 pathogenic variant. This patient was a young woman described with congenital generalized lipodystrophy (CGL), severe insulin-resistant diabetes, and hypertriglyceridemia, referred to as CGL3 [START_REF] Kim | Association of a homozygous nonsense caveolin-1 mutation with Berardinelli-Seip congenital lipodystrophy[END_REF]. Heterozygous CAV1 null variants, including nonsense and frameshift mutations, have been reported so far in ten patients with different phenotypes, i.e. partial lipodystrophy with neurological involvement [START_REF] Cao | Heterozygous CAV1 frameshift mutations (MIM 601047) in patients with atypical partial lipodystrophy and hypertriglyceridemia[END_REF], precocious and severe pulmonary arterial hypertension [START_REF] Austin | Whole exome sequencing to identify a novel gene (caveolin-1) associated with human pulmonary arterial hypertension[END_REF][START_REF] Marsboom | Aberrant caveolin-1-mediated Smad signaling and proliferation identified by analysis of adenine 474 deletion mutation (c.474delA) in patient fibroblasts: a new perspective on the mechanism of pulmonary hypertension[END_REF][START_REF] Copeland | A disease-associated frameshift mutation in caveolin-1 disrupts caveolae formation and function through introduction of a de novo ER retention signal[END_REF] and/or neonatal progeroid syndromes [START_REF] Garg | Whole exome sequencing identifies de novo heterozygous CAV1 mutations associated with a novel neonatal onset lipodystrophy syndrome[END_REF][START_REF] Schrauwen | A Frame-Shift Mutation in CAV1 Is Associated with a Severe Neonatal Progeroid and Lipodystrophy Syndrome[END_REF][START_REF] Han | Characterization of a caveolin-1 mutation associated with both pulmonary arterial hypertension and congenital generalized lipodystrophy[END_REF].

In the present study, we investigated a large consanguineous kindred referred for CGL and identified a novel homozygous CAV1 frameshift variant in four affected young patients.

Besides variable insulin resistance-associated metabolic abnormalities, esophageal achalasia, leading to severe dysphagia, emerges as a specific comorbidity. One affected patient also displayed atypical retinitis pigmentosa. By studying cultured skin fibroblasts from the index case and from the previously reported patient with a homozygous CAV1 nonsense variant [START_REF] Kim | Association of a homozygous nonsense caveolin-1 mutation with Berardinelli-Seip congenital lipodystrophy[END_REF], we show that CGL3 is associated with a complete loss of protein expression of caveolin-1 and its partners caveolin-2 and cavin-1, and with the absence of caveolae at the plasma membrane.

Fibroblasts from patients also display insulin resistance, increased oxidative stress and premature senescence.

PATIENTS AND METHODS

Patients

This study includes nineteen individuals from a large Turkish consanguineous family investigated at Mersin University, Department of Pediatric Gastroenterology, Hepatology and Nutrition, Turkey. Genetic studies were performed in the Department of Molecular Biology and Genetics, and the disease features were reviewed in the French Reference Center for Rare Diseases of Insulin Secretion and Insulin Sensitivity (PRISIS), both at Assistance-Publique Hôpitaux de Paris, Saint-Antoine Hospital, Paris, France. Clinical and molecular studies, and skin biopsy were performed after full written informed consent, according to the Ethics Committee of Mersin University. The study was approved by a French institutional research ethics board (CPP Ile de France 5). Written informed consent for publication of their clinical details and/or clinical images was obtained from the patients' parents, from patients themselves when aged above 12, and from their relatives.

Genetic analyses

Exons and flanking intronic sequences of a panel of 23 genes involved in lipodystrophic and/or insulin resistance syndromes, including CAV1, were sequenced from genomic DNA in Patient 1, as described (17) (Figure 1). Sanger sequencing was performed with the Big Dye Terminator v3.1 sequencing kit (Thermo Fisher Scientific, Waltham, MA, USA) after polymerase chain reaction. Data were analyzed on a 3500xL Dx device with the SeqScape v2.7 software (Thermo Fisher Scientific). CAV1 variants were described based on the longest isoform (NM_001753.4) using Alamut 2.11 (Sophia Genetics, Switzerland) and Human Genome Variation Society guidelines.

Fibroblast cultures

Primary fibroblast cultures from Patient 1 were established after skin biopsy. Cultured fibroblasts from the previously reported patient with CGL3, carrying the homozygous CAV1 p.Glu38* pathogenic variant [START_REF] Kim | Association of a homozygous nonsense caveolin-1 mutation with Berardinelli-Seip congenital lipodystrophy[END_REF], and from two non-obese, non-diabetic, normotensive women who underwent plastic surgery [START_REF] Salle-Teyssieres | Maladaptative Autophagy Impairs Adipose Function in Congenital Generalized Lipodystrophy due to Cavin-1 Deficiency[END_REF] were studied at similar passages. Cells were grown in DMEM low glucose with pyruvate (#31885049; Thermo Fisher Scientific) supplemented with 10% fetal bovine serum (D. Dutscher, Bernolsheim, France), 1% penicillin/streptomycin, and 2 mM glutamine (Invitrogen, Cergy-Pontoise, France).

Transmission electron microscopy

Cultured fibroblasts were fixed in 2.5% glutaraldehyde, 0.1M cacodylate buffer at 4°C, rinsed in 0.2M cacodylate, post-fixed in 1.5% potassium ferrocyanide 1% osmium tetroxide, dehydrated using graded alcohol series, then embedded in epoxy resin. Semi-fine sections (0.5 µm) were stained with toluidine blue. Ultrathin sections (70 nm) were contrasted with UranyLess and lead citrate (Delta Microscopies, France) and examined using an electron microscope (JEOL 2110 HC, Croissy, France) with a 2Kx2K Veleta CCD camera (Olympus, Rungis, France).

Western blot

Protein expression studies were performed on whole cell extracts using antibodies described in Supplemental Table 1.

Cellular response to insulin

To measure their ability to bind insulin, fibroblasts were maintained for 16 hours in serum-free medium supplemented with 0.1% albumin (Sigma-Aldrich, Saint-Louis, MO, USA), then incubated with 125 I-insulin (0.3 ng/mL, PerkinElmer, Coutaboeuf, France) with or without unlabeled insulin (5.10 -8 M) in Hepes buffer at pH 7.65, 15°C, for 5 hours. Radioactivity was measured in a gamma counter (PerkinElmer 2470 Wizard2) and results were normalized to protein content. Insulin effect on glycogen synthesis was evaluated by the incorporation of 14 Cglucose as previously described [START_REF] Thauvin-Robinet | PIK3R1 mutations cause syndromic insulin resistance with lipoatrophy[END_REF]. Results were normalized to the protein content and expressed as a percentage of the basal value.

Oxidative Stress and Cellular Senescence

The production of reactive oxygen species (ROS) was assayed by quantifying the oxidation of 5-6-chloromethyl-2,7-dichlorodihydrofluorescein diacetate (CM-H2DCFDA). The blue staining produced by hydrolysis of X-gal (5-bromo-4-chloro-3-indolyl-ß-Dgalactopyranoside) by senescence-activated-ß-galactosidase at pH 6.0 was used as a biomarker of cellular senescence, as previously described [START_REF] Dour | A homozygous mutation of prelamin-A preventing its farnesylation and maturation leads to a severe lipodystrophic phenotype: new insights into the pathogenicity of nonfarnesylated prelamin-A[END_REF]. The ratio of pH 6-to-pH 4 staining of blue X-gal was quantified at 630 nm.

Statistical analyses

GraphPad Prism software (GraphPad Software Inc., CA, USA) was used to calculate statistical significance with a threshold at p<0.05. Gaussian distribution was tested with Kolmogorov-Smirnov test. Differences between two groups were assessed by unpaired two-sample t-tests or Mann-Whitney tests and multiple comparisons between more than two groups were conducted by ANOVA with Bonferroni-test or Kruskal-Wallis test for post-hoc analysis. All data are means  SEM of at least three independent experiments.

RESULTS

Identification of a novel CAV1 pathogenic variant responsible for an autosomal recessive congenital generalized lipodystrophy syndrome

Patient 1, a 15-year-old girl from a large consanguineous Turkish family was referred for genetic investigation of CGL (Figure 1). We identified a novel homozygous CAV1 variant (NM_001753.4) : c.237_238del, p.(His79Glnfs*3) through sequencing of a panel of 23 genes involved in lipodystrophy and/or insulin resistance syndromes. This variant, confirmed by Sanger sequencing, was absent from the gnomAD and ExAC databases that list genetic variants from the general population, and is classified as pathogenic according to the American College of Medical Genetics and Genomics (ACMG) criteria [START_REF] Richards | Standards and guidelines for the interpretation of sequence variants: a joint consensus recommendation of the American College of Medical Genetics and Genomics and the Association for Molecular Pathology[END_REF]. The deletion of 2 nucleotides in exon 3 leads to a shift in the reading frame. If the transcript is expressed, it is predicted to result in the synthesis of an abnormal form of caveolin-1, truncated in its oligomerization domain and deprived from its scaffolding and intra-membrane domains (Figure 2). Sequencing of the gene panel did not reveal any other pathogenic variant. Patient 1 was thus diagnosed with CGL3.

The same CAV1 variant was observed in the homozygous state in Patient 2, who was subsequently referred with similar symptoms, and in Patients 3 and 4, diagnosed with CGL after familial investigations. The parents of Patients 1 to 4 were asymptomatic. Genetic investigations, performed in seven of them, revealed that they were heterozygous carriers of the variant. Unaffected siblings of the patients (n=8) were heterozygous for the variant or had a normal genotype. This family thus showed an autosomal recessive transmission of CGL3 due to a novel p.(His79Glnfs*3) CAV1 pathogenic variant (Figures 1 and2).

Phenotype of the disease

Table 1 summarizes the patients' main phenotypic features.

Patient 1

Patient 1 was referred at the age of 15 to Mersin University with difficulty in swallowing liquids and solids. She was born from first-cousin parents of Turkish origin, at term after an uneventful pregnancy, with a birth weight and height of 2800g and 48cm respectively. At examination, her height was 153 cm, BMI was 17.9, Tanner pubertal stage 5. She showed a triangular and acromegaloid face, with a generalized loss of fat sparing palms and soles, a taut and thin mottled skin with visible dermal vessels, extensive acanthosis nigricans in the armpits and groins, a prominent musculature, and enlarged hands and feet (Figure 3A-C). She also complained about hirsutism, with secondary amenorrhea since the age of 14. The family noticed her dysmorphic facial and body appearance during early infancy. Although her linear growth was delayed with poor weight gain during infancy, she was described with a voracious appetite. She did not present with any skeletal deformity, joint contractures or muscular functional defect. No cognitive delay was observed and she had normal developmental milestones. Neurological examination was normal. Blood pressure was normal.

Laboratory parameters showed hypertriglyceridemia (5.4 mmol/L), low HDL-cholesterol (0.73 mmol/L), major fasting hyperinsulinemia (300 mIU/L) with normal glycemia (5 mmol/L) and HbA1c (5.8%). Liver tests and creatine kinase were normal as well as kidney and thyroid function. Androgen levels were in the normal range. Serum luteinizing hormone (LH, 13.9 IU/L) was higher than follicle-stimulating hormone (FSH, 5.8 IU/L). Estradiol levels were in the normal range for the follicular phase. Calcemia and parathormone serum and urinary levels were normal, with decreased vitamin D level and bone mineral density (Z score -2.78 at lumbar spine).

Generalized lipoatrophy was confirmed by magnetic resonance imaging (MRI) (Figure 4), and by very low leptin levels (0.3 ng/mL), also consistent with hyperphagia. MRI also revealed enlarged polycystic ovaries (Figure 5). Electrocardiogram and echocardiogram, including estimated right ventricular systolic pressure, were normal. Liver steatosis was diagnosed with ultrasonography. Medical history, physical examination and results of metabolic investigations allowed the diagnosis of CGL. A treatment with metformin and medium-chain triglycerides was initiated.

Dysphagia was investigated with several procedures. Barium esophagogram series showed a dilated esophagus with a distal bird's beak deformity (Figure 6A), and a severe stenosis of the distal esophagus was confirmed by endoscopy. High resolution esophageal manometry showed the absence of lower esophageal sphincter relaxation with increased integrated relaxation pressure (51 mmHg) and decreased peristalsis, consistent with esophagogastric junction outflow obstruction (22) (Figure 6 B,C). Based on clinical, endoscopic, radiological and manometric findings, an early stage of achalasia was diagnosed. Peroral endoscopic myotomy (POEM) was performed leading to a significant relief in dysphagia-related symptoms.

Patient 2

Patient 2 was an 18-year-old cousin of Patient 1 (Figure 1) referred for difficulty in swallowing.

Generalized lipodystrophy and mild dysmorphic features were present from early infancy. His symptoms and physical examination were very similar to those of Patient 1 (Table 1, Figure 3D). However, his metabolic abnormalities were milder, with low HDL-cholesterol but normal triglyceridemia. Although glucose tolerance was normal, normal-to-high 2h-postprandial insulin (45.8 mIU/L, N: 4-52.5 at T120 min after oral glucose tolerance test) [START_REF] Chevenne | Proinsulin and specific insulin responses to an oral glucose tolerance test in a healthy population[END_REF] and C-peptide levels (8.32 g/L, N: 1.1-4.4) suggested insulin resistance. Blood pressure, cardiac examination including echocardiography were normal, as well as liver, renal and thyroid function, and testosterone and FSH levels. Type 2 achalasia, diagnosed upon radiological, endoscopic and manometric findings (Figure 6 D-F), was successfully treated by POEM. Patient 2 also complained of trouble seeing at night for two years. Although neurological examination and ophthalmological fundus analysis were normal, optical coherence tomography (OCT) revealed bilateral loss of ellipsoid line and atrophy of the retinal pigment epithelium, and defective outer limiting membrane in the left eye, suggesting atypical retinitis pigmentosa.

Patient 3 and Patient 4

Patient 3, a 10-year-old girl, and Patient 4, an 8-month-old girl, were investigated during the systematic familial screening (Figure 1). As Patients 1 and 2, they were born from consanguineous parents, at term after normal pregnancies, with normal birth weight and height.

Their linear growth was delayed and they presented with a dysmorphic appearance and generalized lipoatrophy, which were noticed by the family during early infancy (Figure 3 E,F).

Their physical examination was otherwise normal. They both achieved normal developmental milestones. Laboratory investigations revealed increased levels of triglycerides and insulin, and decreased HDL-cholesterol (Table 1). The p.(His79Glnfs*3) and the previously described p.(Glu38*) CAV1 pathogenic variants predict to interrupt the caveolin-1 aminoacid sequence at the N-terminal part of the protein, proximally to its scaffolding and intra-membrane domains (24) (Figure 2). As expected, we show that, similarly to the fibroblasts carrying the homozygous CAV1 p.(Glu38*) variant ( 9), cells from Patient 1 with the novel CAV1 p.(His79Glnfs*3) variant do not exhibit detectable expression of caveolin-1 (Figure 7A). We evaluated the protein expression of caveolin-2 and cavin-1, which are binding partners of caveolin-1. We show that caveolin-2 and cavin-1 protein levels are strongly decreased in patients' cells (Figure 7A). Notably, pathogenic variants in CAVIN1 (previously referred to as PTRF), encoding cavin-1, lead to a form of CGL previously associated with achalasia [START_REF] Van Der Pol | Berardinelli-Seip syndrome and achalasia: a shared pathomechanism?[END_REF]. As expected from the major role of caveolin-1 and cavin-1 for caveolae formation, electron microscopy shows that caveolae, abundant in control cells, are completely absent in fibroblasts from patients with CAV1 null variants (Figure 7B).

Patients' relatives

Relatives

Homozygous CAV1 null variants impair insulin signaling

We then investigated the insulin-mediated activation of proximal signalling intermediates from the metabolic and mitogenic insulin pathways (i.e, IRß, IRS1, AKT, and ERK1/2) and show that it is strongly impaired in fibroblasts from patients with CAV1 null variants (Figure 8A). In addition, the effect of insulin on its distal signalling, as evaluated by glycogen synthesis, is also severely inhibited in patients' cells (Figure 8B). Although the total amount of insulin receptors is not significantly decreased in whole protein extracts from patients' cells (Figure 8A), the absence of caveolae, which are normally enriched at the plasma membrane with insulin receptors (1), could impair the initiation of the insulin signal. To test this hypothesis, we evaluated the capacity of fibroblasts to bind insulin, and showed that it was decreased by ~ 20% in cells bearing CAV1 null variants as compared to control cells (Figure 8C). These results suggest that the loss of caveolae could contribute, in part, to the insulin resistance observed in cells from patients carrying CAV1 null pathogenic variants.

Homozygous CAV1 null variants increase oxidative stress and senescence in fibroblasts

Caveolin-1 has been shown to modulate oxidative stress-induced cellular senescence (4), a pathway which has been involved in lipodystrophic diseases [START_REF] Vigouroux | Molecular mechanisms of human lipodystrophies: from adipocyte lipid droplet to oxidative stress and lipotoxicity[END_REF]. We thus aimed to evaluate the effects of CAV1 pathogenic variants on the production of reactive oxygen species (ROS) and on cellular senescence. Patients' fibroblasts, as compared to control cells, display a major increase in oxidative stress (Figure 9A), as well as elevated levels of senescence markers (Figure 9B), and increased senescence-activated--galactosidase activity (Figure 9 C,D).

DISCUSSION

We report here a novel homozygous null pathogenic variant in CAV1 in four patients with CGL3, and show the autosomal recessive transmission of the disease in a large consanguineous family. This study adds important phenotypic data, since CGL3 was previously described in only one patient [START_REF] Kim | Association of a homozygous nonsense caveolin-1 mutation with Berardinelli-Seip congenital lipodystrophy[END_REF]. By revealing the consequences of CAV1 loss-of-function in patients' cells, this study also highlights the involvement of caveolin-1 and caveolae in cellular insulin response, oxidative stress and cellular senescence, that could contribute to specific clinical manifestations.

Insulin resistance and related signs, including acanthosis nigricans, prominent veins, muscular hypertrophy, hypertriglyceridemia, low HDL-cholesterol, hirsutism and/or polycystic ovary syndrome, and hepatic steatosis, are consistently associated with lipodystrophy syndromes (27).

These manifestations, together with generalized lipoatrophy, were present in the previously described patient with CGL3 (9) as well as, with variable severity, in the affected patients from this study. As previously described in other CGL subtypes, with the exception of CGL2 (9, 28), lipoatrophy only spared mechanical adipose tissue (from palms, plantar and retro-orbital regions) and bone marrow fat. It can be hypothesized that different adipose tissue depots, which have specific developmental origin and gene expression, could be differently impacted by CGL-associated pathogenic variants [START_REF] Billon | Developmental origins of the adipocyte lineage: new insights from genetics and genomics studies[END_REF].

Lipodystrophy-associated insulin resistance was shown to result, at least in part, from defective adipocyte lipid storage and leptin deficiency leading to cellular lipotoxicity [START_REF] Semple | Genetic syndromes of severe insulin resistance[END_REF]. Our studies in patients' fibroblasts suggest that deficient caveolin-1 and cavin-1 protein expression and/or the complete absence of caveolae could also directly contribute to insulin resistance.

The CGL3 phenotype was previously associated with short stature, hypocalcemia, osteopenia and megaesophagus ( 9), but whether these signs were due to the homozygous CAV1 null variant remained elusive. We show that achalasia, diagnosed at age 15 and 18 in two affected patients, should be considered as a main complication of CGL3 requiring careful investigations and specific management. Endoscopic myotomy was successful in the two affected patients, and avoided invasive surgical procedures. Three affected patients from this study had short stature, and one patient (Patient 3) had a height percentile under 25 th , contrasting with the accelerated growth frequently described in patients with CGL1 and CGL2, due to AGPAT2 or BSCL2 pathogenic variants, respectively (27). Whether this could be due to the severe impairment of insulin-activated mitogenic signalling pathways linked to the loss of caveolae requires further studies. The four patients did not present with any clinical bone or joint abnormality. However, we did observe osteopenia in two of the investigated patients, with a Z-score below -2.5 at the lumbar spine level, contrasting with the increased bone density usually observed in patients with CGL1 or CGL2 (27,[START_REF] Lima | Bone Density in Patients With Berardinelli-Seip Congenital Lipodystrophy Is Higher in Trabecular Sites and in Type 2 Patients[END_REF]. Interestingly, osteopenia was confirmed by DEXA at the age of 21 in the previously described patient with CGL3, with a lumbar spine Z-score of -3 (Dr Chong Kim, personal communication). The decreased bone mass observed in Patient 1 and Patient 2 was not explained by vitamin D nor sex steroid hormone levels. In addition, in contrast to hypocalcemia with hypercalciuria observed in the previously reported patient [START_REF] Kim | Association of a homozygous nonsense caveolin-1 mutation with Berardinelli-Seip congenital lipodystrophy[END_REF], and in caveolin-1 knockout mice [START_REF] Cao | Disruption of the caveolin-1 gene impairs renal calcium reabsorption and leads to hypercalciuria and urolithiasis[END_REF], serum and urinary calcium were normal in patients from this study. Detailed investigations of calcium homeostasis remains to be performed in CGL3.

The biallelic pathogenic variants responsible for CGL3, located in exon 2 and proximal exon 3 of CAV1, lead to similar phenotypes associated with a loss of protein expression of caveolin-1 and of its partner caveolin-2 (9, this study). Our results show that these variants also strongly inhibit cavin-1 protein expression, and lead to a complete loss of caveolae at the plasma membrane of patients' fibroblasts. Heterozygous subjects were asymptomatic, suggesting that half amount of functional caveolin-1 is sufficient to avoid any specific pathological consequences. In contrast, CAV1 variants affecting the C-terminal domain of the protein, were reported as pathogenic in the heterozygous state and lead to rare but heterogeneous diseases (Figure 2) that may result from different pathophysiological mechanisms. Functional studies of the CAV1 p.(Leu159Serfs*22) variant, responsible for autosomal dominant pulmonary arterial hypertension, showed that the mRNA and protein expression of the mutated allele is preserved, and that the resulting abnormal protein could act via a dominant-negative effect on the trafficking of caveolin-1 to the plasma membrane [START_REF] Austin | Whole exome sequencing to identify a novel gene (caveolin-1) associated with human pulmonary arterial hypertension[END_REF][START_REF] Marsboom | Aberrant caveolin-1-mediated Smad signaling and proliferation identified by analysis of adenine 474 deletion mutation (c.474delA) in patient fibroblasts: a new perspective on the mechanism of pulmonary hypertension[END_REF][START_REF] Copeland | A disease-associated frameshift mutation in caveolin-1 disrupts caveolae formation and function through introduction of a de novo ER retention signal[END_REF]. The CAV1 p.(Phe160*) variant, reported in one patient with a neonatal complex progeroid syndrome associated with pulmonary artery hypertension and lipodystrophy, was also shown to be transcribed, without increased RNA degradation [START_REF] Garg | Whole exome sequencing identifies de novo heterozygous CAV1 mutations associated with a novel neonatal onset lipodystrophy syndrome[END_REF][START_REF] Schrauwen | A Frame-Shift Mutation in CAV1 Is Associated with a Severe Neonatal Progeroid and Lipodystrophy Syndrome[END_REF]. In accordance, caveolae were present at the cell surface of the patient's fibroblasts, but caveolin-1 oligomers displayed decreased stability and weakened interactions with cavin-1 [START_REF] Garg | Whole exome sequencing identifies de novo heterozygous CAV1 mutations associated with a novel neonatal onset lipodystrophy syndrome[END_REF][START_REF] Schrauwen | A Frame-Shift Mutation in CAV1 Is Associated with a Severe Neonatal Progeroid and Lipodystrophy Syndrome[END_REF][START_REF] Han | Characterization of a caveolin-1 mutation associated with both pulmonary arterial hypertension and congenital generalized lipodystrophy[END_REF].

Cavin-1, encoded by the CAVIN1 gene, is a caveolin-1-interacting protein, which, like caveolin-1, stabilizes caveolae structures. Cavin-1 is located at adipocyte lipid droplets, and contributes to the adipocyte differentiation process [START_REF] Lay | Cholesterol-induced caveolin targeting to lipid droplets in adipocytes: a role for caveolar endocytosis[END_REF][START_REF] Hodges | Proteomic insights into an expanded cellular role for cytoplasmic lipid droplets[END_REF][START_REF] Salle-Teyssieres | Maladaptative Autophagy Impairs Adipose Function in Congenital Generalized Lipodystrophy due to Cavin-1 Deficiency[END_REF][START_REF] Palacios-Ortega | Expression of Caveolin 1 is enhanced by DNA demethylation during adipocyte differentiation. status of insulin signaling[END_REF]. Biallelic CAVIN1 pathogenic variants are responsible for CGL4, a form of CGL also characterized by skeletal and/or cardiac muscular dystrophy [START_REF] Semple | Genetic syndromes of severe insulin resistance[END_REF][START_REF] Lima | Bone Density in Patients With Berardinelli-Seip Congenital Lipodystrophy Is Higher in Trabecular Sites and in Type 2 Patients[END_REF], with achalasia in some patients [START_REF] Van Der Pol | Berardinelli-Seip syndrome and achalasia: a shared pathomechanism?[END_REF][START_REF] Hayashi | Human PTRF mutations cause secondary deficiency of caveolins resulting in muscular dystrophy with generalized lipodystrophy[END_REF][START_REF] Rajab | Fatal cardiac arrhythmia and long-QT syndrome in a new form of congenital generalized lipodystrophy with muscle rippling (CGL4) due to PTRF-CAVIN mutations[END_REF]. Achalasia is due to the dysfunction of nitrergic inhibitory motor neurones that innervate the circular smooth muscle of the distal esophagus, resulting in impaired relaxation of lower esophageal sphincter [START_REF] Boeckxstaens | The lower oesophageal sphincter[END_REF]. We and others have previously shown that CGL4-related CAVIN1 pathogenic variants severely impair cavin-1 and caveolin expression, as well as caveolae formation [START_REF] Salle-Teyssieres | Maladaptative Autophagy Impairs Adipose Function in Congenital Generalized Lipodystrophy due to Cavin-1 Deficiency[END_REF][START_REF] Rajab | Fatal cardiac arrhythmia and long-QT syndrome in a new form of congenital generalized lipodystrophy with muscle rippling (CGL4) due to PTRF-CAVIN mutations[END_REF][START_REF] Liu | Deletion of Cavin/PTRF causes global loss of caveolae, dyslipidemia, and glucose intolerance[END_REF]. Since caveolae have been shown to regulate calcium homeostasis and excitation-contraction coupling in smooth muscle [START_REF] Taggart | Smooth muscle excitation-contraction coupling: a role for caveolae and caveolins?[END_REF][START_REF] Parton | Caveolae as plasma membrane sensors, protectors and organizers[END_REF], their deficiency could promote the development of uncontrolled esophageal contractions, that may evolve to achalasia. Oxidative stress could also contribute to achalasia, as reported in the Triple A (alacrimaachalasiaadrenal insufficiency) syndrome (OMIM 231550) [START_REF] Storr | Deficiency of ferritin heavy-chain nuclear import in triple a syndrome implies nuclear oxidative damage as the primary disease mechanism[END_REF]. In addition, oxidative stress and/or premature senescence could participate in the pathophysiology of lipodystrophy [START_REF] Vigouroux | Molecular mechanisms of human lipodystrophies: from adipocyte lipid droplet to oxidative stress and lipotoxicity[END_REF]27,[START_REF] Semple | Genetic syndromes of severe insulin resistance[END_REF][START_REF] Sollier | Lipodystrophic syndromes: From diagnosis to treatment[END_REF]. Patient 2 from this study was diagnosed with atypical retinitis pigmentosa. Interestingly, caveolin-1 has been shown to modulate retinal neuroprotective signalling [START_REF] Reagan | Retinal Caveolin-1 Modulates Neuroprotective Signaling[END_REF][START_REF] Gu | Caveolins and caveolae in ocular physiology and pathophysiology[END_REF], and retinitis pigmentosa was reported in two patients with a syndrome of partial lipodystrophy, congenital cataracts, and neurological abnormalities, due to the heterozygous CAV1 p.(Lys135Argfs*4) variant [START_REF] Cao | Heterozygous CAV1 frameshift mutations (MIM 601047) in patients with atypical partial lipodystrophy and hypertriglyceridemia[END_REF]. However, ophthalmological fundus examination did not show any sign of retinitis pigmentosa in the previously described patient with CGL3, investigated at the age of 26 (9, and Dr Chong Kim, personal communication). Whether this sign could be due to CGL3-associated CAV1 variants remains hypothetical. Dysmorphic features and/or mottled skin were observed in the affected patients from this study and in the two patients reported with neonatal premature ageing syndrome due to de novo CAV1 variants (14,15) (Figure 2), but no other progeroid sign was observed in patients with CGL3. Although right heart catheterization was not performed, clinical examination and echocardiography did not reveal any sign of pulmonary arterial hypertension in patients with CGL3.

Other phenotypic studies are required to better delineate the clinical relationships between the different CAV1-related diseases, due to loss-of-function or dominant negative mechanisms.

However, in addition to careful metabolic monitoring, we suggest that gastrointestinal, cardiac, ophthalmological and neurological evaluation should also be included in the follow-up of patients with CGL3. with a homozygous CAV1 p.(Glu38*) variant responsible for CGL3 [START_REF] Kim | Association of a homozygous nonsense caveolin-1 mutation with Berardinelli-Seip congenital lipodystrophy[END_REF]. Cells were maintained for 24h in a serum-free medium, then incubated or not with 50 nmol/L human insulin (#I9278, Sigma-Aldrich) for 8 min, and the total protein expression of signaling intermediates as well as their phosphorylated activated forms were evaluated. Representative images of Western blots performed in triplicate are shown. Tubulin is used as an index of the cellular protein level.

  of Patients 1 to 4 (n=15) (Figure 1) had no complaint and did not show any sign of lipodystrophy. Their anthropometric measurements and physical examination were normal, except for the older brother of Patient 1 who presented with a mild mental retardation of unknown origin. Their laboratory evaluation including full blood count, fasting glucose and insulin, HbA1c, liver function tests and serum lipids, were normal. Homozygous CAV1 p.(His79Glnfs*3) and p.(Glu38*) pathogenic variants induce a loss of protein expression of caveolin-1, caveolin-2 and cavin-1, and a loss of caveolae formation in patients' fibroblastsWe compared cultured fibroblasts from Patient 1 and from the previously reported patient with CGL due to the homozygous p.(Glu38*) CAV1 pathogenic variant[START_REF] Kim | Association of a homozygous nonsense caveolin-1 mutation with Berardinelli-Seip congenital lipodystrophy[END_REF], to those from controls.

Figure 1 .

 1 Figure 1. Genealogical tree of the studied family

Figure 2 .Figure 3 .

 23 Figure 2. Schematic representation of CAV1 gene and caveolin-1 protein

Figure 4 .

 4 Figure 4. Magnetic resonance imaging of Patient 1 showing generalized lipoatrophy

Figure 5 .

 5 Figure 5. Magnetic resonance imaging of Patient 1 showing enlarged polycystic ovaries

Figure 6 .

 6 Figure 6. Barium esophagogram series and high-resolution manometry showing achalasia

  Numbers on the left correspond to the expected protein molecular weight. (B) The effect of insulin on glycogen synthesis and (C) the ability of cells to bind insulin to its receptor, were evaluated as described in Methods. Results are expressed as the percentage of the first control without insulin. **: p <0.01, ns: not significant.

Figure 9 .

 9 Figure 9. Homozygous CAV1 null variants promote oxidative stress and senescence in

Table 1 . Main phenotypic features of Patients 1 to 4

 1 are indicated as present (+) or absent (-) in each patient. NA, not applicable; ND, not determined Height, weight and BMI centiles are determined according to CDC (Centers for Disease Control and Prevention, USA); Mid-parental target height is calculated according to Tanner et al.[START_REF] Tanner | Normal growth and techniques of growth assessment[END_REF]

	661
	662
		Patient 1	Patient 2	Patient 3	Patient 4
	Age at evaluation	15 years	18 years	10 years	8 months
	Gender	Female	Male	Female	Female
	Height (centile)	153 cm (<	155 cm ( 3 rd )	134 cm (<	62 cm (
	(mid-parental target height	10 th )	(170 cm)	25 th )	3 rd )
	in postpubertal patients)	(160 cm)
	Weight (kg) (centile)	42 kg (10 th )	49 (3 th )	30.5 (40 th )	8.6 (75 th )
	Body Mass Index (kg/m 2 ) (centile)	17.9 (25 th )	20.4 (25 th )	17 (50 th )	22.4 (ND)
	Lipodystrophy onset	Early infancy	Early infancy	Early infancy	Early infancy
	Generalized lipoatrophy sparing palms and soles	+	+	+	+
	Failure to thrive in infancy	+	+	+	+
	Increased appetite	+	-	-	-
	Triangular face	+	+	+	+
	Muscular hypertrophy	+	+	+	-
	Acanthosis nigricans	+	+	-	-
	Hirsutism	+	NA	-	-
	Polycystic ovaries	+	NA	-	-
	Liver steatosis (ultrasonography)	+	-	-	-
	Blood pressure, cardiac
	examination and	Normal	Normal	Normal	Normal
	electrocardiogram
	Echocardiography	Normal	Normal	ND	ND
	Estimated right ventricular
	systolic pressure (N: 12-57	25	27	ND	ND
	mmHg)
	Megaesophagus	+	+	-	-
	Muscular strength and neurological examination	Normal	Normal	Normal	Normal
	Ophthalmological examination	Normal	Atypical pigmentosa retinitis	ND	ND
	Fasting glucose (N: 3.5-5.6 mmol/L)	5.0	4.8	5.3	4.7
	Fasting insulin (N: 2-9 mIU/L)	300	8.1	9.6	9.4
	HbA1c (N: 4.8-6%)	5.8	5.3	5.4	5.1
	Triglycerides (N: 0.3-1.5 mmol/L)	5.4	0.9	3.4	2.3
	Total cholesterol (N: 3-5.1 mmol/L)	4.0	3.8	3.1	4.2
	HDL-chol (N: 0.9-1.8 mmol/L)	0.73	0.68	0.75	0.65

ACKNOWLEDGEMENTS

The authors thank the patients and their families for their participation in this study. They thank Audrey Geeverding and Michael Trichet from Sorbonne University, Department of Electron Microscopy, Paris-Seine Biology Institute, Paris, France for expert advice and help with electron microscopy, Dr Chong Kim, Department of Pediatrics, da Criança Institute, University of Sao Paulo, Brazil for updated clinical data of the previously reported patient with CGL3, and Dr Sonja Janmaat, from the National Reference Center for Rare Diseases of Insulin Secretion and Insulin Sensitivity (PRISIS), Assistance Publique-Hôpitaux de Paris, France, for editorial assistance. Corinne Vigouroux is a member of the European Reference Network on Rare Endocrine Conditions -Project ID No 739527.

DISCLOSURE

The authors declare that there is no conflict of interest that could be perceived as prejudicing the impartiality of the research reported. This study was supported by institutional fundings from Inserm, Sorbonne Université, Centre National de la Recherche Scientifique, and Assistance-Publique Hôpitaux de Paris, and by Fondation pour la Recherche Médicale (grant number EQU201903007868).

This work was supported by the French Ministry of Solidarity and Health, Assistance-Publique Hôpitaux de Paris, Sorbonne University, Centre National de la Recherche Scientifique, the Institut National de la Santé et de la Recherche Médicale (Inserm), and the Fondation pour la Recherche Médicale (EQU201903006878), France.


Thomas Oberlin

Mathieu Verm

REGULARIZATION VIA DEEP GENERATIVE MODELS: AN ANALYSIS POINT OF VIEW

Keywords: inverse problems, regularization, generative models, deep regularization, data-driven priors

This paper proposes a new way of regularizing an inverse problem in imaging (e.g., deblurring or inpainting) by means of a deep generative neural network. Compared to end-to-end models, such approaches seem particularly interesting since the same network can be used for many different problems and experimental conditions, as soon as the generative model is suited to the data. Previous works proposed to use a synthesis framework, where the estimation is performed on the latent vector, the solution being obtained afterwards via the decoder. Instead, we propose an analysis formulation where we directly optimize the image itself and penalize the latent vector. We illustrate the interest of such a formulation by running experiments of inpainting, deblurring and super-resolution. In many cases our technique achieves a clear improvement of the performance and seems to be more robust, in particular with respect to initialization.

INTRODUCTION

Inverse problems are ubiquitous in imaging, because many image acquisition pipelines involve degradation operators that need to be inverted afterwards, such as radial projections in tomography or spatial invariant blur in any image or signal acquisition sensor [START_REF] Bertero | Introduction to inverse problems in imaging[END_REF]. This task can be seen as an estimation problem, which is often solved via optimization algorithms or Monte Carlo sampling [START_REF] Idier | Bayesian approach to inverse problems[END_REF]. To circumvent the ill-posedness of the problem, practicioners need priors or regularizers that will promote some behavior of the solution.

Famous generic priors have been proposed in imaging in the past decades, such as Total Variation [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF] and variants [START_REF] Bredies | Total generalized variation[END_REF], which favour piecewise-smooth images. This is done by minimizing the 1 norm of some linear differentiation operator of the image. If we know a priori a basis in which the data is sparse, the 1 norm can also be placed on the representation coefficients in that basis, which remains an efficient technique in some applications [START_REF] Monier | Fast reconstruction of atomic-scale STEM-EELS images from sparse sampling[END_REF]. More recently, significant improvements have been achieved by considering plug-and-play pri-Part of this work has been funded by the Institute for Artificial and Natural Intelligence Toulouse (ANITI) under grant agreement ANR-19-PI3A-0004.

ors and variants [START_REF] Venkatakrishnan | Plug-and-play priors for model based reconstruction[END_REF][START_REF] Chan | Plug-andplay ADMM for image restoration: Fixed-point convergence and applications[END_REF], often based on highly efficient denoisers such as BM3D [START_REF] Dabov | Image denoising by sparse 3-D transform-domain collaborative filtering[END_REF].

But in many situations such a basis is not known, although it can be learned from some available training data. This idea has been first introduced with the problem of dictionary learning [START_REF] Aharon | K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation[END_REF], which simultaneously learns the dictionary and the noise-free image. It has then been applied to many inverse problems by training the dictionary beforehands on a representative dataset. More recently, numerous works considered to train a deep neural networks (DNN) instead of a dictionary, which appeared to be a more efficient and effective model for images. Some works proposed to learn a denoising networks [START_REF] Zhang | Learning deep CNN denoiser prior for image restoration[END_REF][START_REF] Zhang | FFDNet: Toward a fast and flexible solution for CNN-based image denoising[END_REF] and use it as a plug-and-play prior; or to learn a 1class classification DNN [START_REF] Chang | One network to solve them all-solving linear inverse problems using deep projection models[END_REF] which is used as a projection operator onto the set of natural images.

We will focus on a different approach introduced by [START_REF] Bora | Compressed sensing using generative models[END_REF], which simply learns a generative DNN and use it to constrain the solution to live in its range. In this work, authors showed recovery results similar to the ones obtained in compressed sensing, and demonstrated the effectiveness of the approach with a variational autoencoder (VAE) and a Deep Convolutional Generative Adversarial Network (DCGAN). Impressive results have been obtained by considering another kind of generative model termed Glow [START_REF] Asim | Invertible generative models for inverse problems: mitigating representation error and dataset bias[END_REF], which is invertible and thus enables to better control the distribution of the latent factor by optimizing the likelihood. Our aim is to show that with such a network, the result can be dramatically improved by optimizing the image itself instead of the latent code, the network being used only for regularization purpose. We name our approach deep analysis regularization, as opposed to the synthesis regularization proposed in [START_REF] Bora | Compressed sensing using generative models[END_REF], and following the formulation used in sparse recovery [START_REF] Elad | Analysis versus synthesis in signal priors[END_REF][START_REF] Nam | The cosparse analysis model and algorithms[END_REF].

Note that apart from the "deep plug-and-play approaches" discussed above, which use deep learning only in the regularization, there are several lines of works that tackle inverse problems with deep learning in very different ways. To cite a few, this includes unrolled deep networks [START_REF] Bertocchi | Deep unfolding of a proximal interior point method for image restoration[END_REF], unsupervised approaches [START_REF] Ulyanov | Deep image prior[END_REF] or other end-to-end approaches [START_REF] Yu | Generative image inpainting with contextual attention[END_REF]. Each line of methods has its pros and cons, and comparing those techniques is beyond the scope of the current paper. In a nutshell, plug-and-play approaches seem more generic (one prior for different problems) and somehow more grounded (we can control the optimization algorithm), but they might be outperformed by problem-dependant methods [START_REF] Arridge | Solving inverse problems using data-driven models[END_REF].

The remainder of this paper is structured as follows. Section 2 recalls some background and the works upon which we build our analysis deep regularization technique, presented in Section 3. The experimental setting and the corresponding results are then presented in Sections 4 and 5, respectively, while a summary and some perspectives are discussed in Section 6 which concludes the paper.

BACKGROUND

Problem statement

Many imaging problems consist in sensing the image y from an underlying true scene x * according to

y = Ax * + n, (1)
where A is some observation operator which is often known with good accuracy, and n an error term accounting for random effects and potential model mismatch. This model encompasses denoising (A is the identity matrix), inpainting (A is a mask), deblurring (A is Toeplitz), etc. The noise term is often assumed to be independent and Gaussian, although in many imaging modalities is contains a mixing of Gaussian and Poisson components [START_REF] Makitalo | Optimal inversion of the generalized Anscombe transformation for Poisson-Gaussian noise[END_REF].

The inverse problem consists in estimating the true scene x, which amounts to inverting the linear operator A. This problem is in general ill-posed, because A is often singular or at least badly conditioned. To overcome this, one needs to bring additional information by means of a regularizer ϕ. The formulation of the inverse problem then writes

x = arg min x 1 2
Axy

2 2 + λϕ(x) (2)
where parameter λ tunes the level of regularization. Such a formulation can be related to Bayesian estimation: if the noise n is assumed to be Gaussian, then (2) is the maximum a posteriori (MAP) estimate under the prior p(x) ∝ e -ϕ(x) . Note that this remains valid if the noise follows a different statistics, as long as we replace the least-square by the adequate divergence measure. For convenience, we will restrict our study to the least-square formulation, which is the most convenient and the most widely used.

Priors via deep generative neural networks

Auto-encoders are special kinds of neural networks. They are composed of an encoder which computes the latent code z = E(x), and a decoder or a generator D(z), both being trained so that x i ≈ D(E(x i )) for all data samples x i of a given training dataset. While autoencoders have been introduced a long time ago, there has been a stunning renewal with the introduction of so-called variational auto-encoders (VAEs) in [START_REF] Kingma | Auto-encoding variational Bayes[END_REF][START_REF] Rezende | Stochastic backpropagation and approximate inference in deep generative models[END_REF]. The aim of these works was to constrain the latent space to exhibit a proper structure, which can be seen as a regularization of the learning problem. The first purpose was to be able to generate realistic images by sampling from a simple distribution in the latent space. Recall that the loss function of a VAE is intractable and requires the use of approximate inference techniques. To overcome this, some works [START_REF] Kingma | Improved variational inference with inverse autoregressive flow[END_REF][START_REF] Kingma | Glow: Generative flow with invertible 1x1 convolutions[END_REF] followed a different approach inspired by normalizing flows, where each layer of the network remains invertible, its gradient being efficiently computed to allow for backpropagation. Once learned, such networks can be use as priors to regularize any inverse problem, as described in the following paragraph.

Synthesis-based regularization

We describe here our baselines [START_REF] Bora | Compressed sensing using generative models[END_REF][START_REF] Asim | Invertible generative models for inverse problems: mitigating representation error and dataset bias[END_REF]. They both assume that a generative model has been learned beforehand on a representative database. Then, they propose to solve [START_REF] Idier | Bayesian approach to inverse problems[END_REF], by constraining the sought image x to be in the range of the generator: x = D(z) for some latent vector z. Both works add an extra regularizer, based on the normal distribution assumed for the latent vector, which finally gives

x = D arg min z 1 2 AD(z) -y 2 2 + λ z 2 2 . (3)
3. AN ANALYSIS FORMULATION

Limits of the synthesis regularization

The synthesis formulation (3) operates in the latent space, which brings some clear benefits but also several drawbacks. On the one hand, the solution of such a problem is likely to be visually consistent, since it has been generated by the decoder. But on the other hand, the found solution is often very different from the ground truth, because of the strong nonconvexity of the problem which makes the solution highly dependent to initialization. To reduce this effect one should properly initialize the optimization algorithm, for instance by setting z 0 = E(x 0 ), with x 0 an initial guess. According to our experience, this trick however does not always produce an image which is fully consistent with the observations.

An analysis formulation

To circumvent the drawbacks discussed above, we propose here a straightforward solution which consists in optimizing directly the image. The regularization is performed in the latent space, capitalizing on the Gaussian distribution which is assumed for the latent vector. This writes:

x = arg min x 1 2 Ax -y 2 2 + λ E(x) 2 2 . (4)
We named this formulation analysis. Note that, contrary to the synthesis formulation, the role of parameter λ is fundamental, since it controls the only regularization brought about by the network.

The two formulations are the counterpart of what was already studied concerning sparse representations in redundant dictionaries [START_REF] Elad | Analysis versus synthesis in signal priors[END_REF]. But in the case of deep regularization, both approaches always lead to different results, even if the network is invertible, because of the strong non-convexity of the problem. The remainder of the paper is devoted to illustrate the pros and cons of the two formulations, on various simple inverse problems. To this end, we describe the experimental setting and the results in the next two sections.

EXPERIMENTAL SETTING

We restrict our study to the comparison between the proposed analysis formulation and the more standard synthesis, as implemented by [START_REF] Asim | Invertible generative models for inverse problems: mitigating representation error and dataset bias[END_REF]. Note that this recent work competes favorably with several state of the art techniques, we thus tried to reproduce a close experimental setting.

For the generative DNN, we use the Glow network as introduced in [START_REF] Kingma | Glow: Generative flow with invertible 1x1 convolutions[END_REF]. Glow is inspired by flow-based generative models [START_REF] Dinh | Nice: Non-linear independent components estimation[END_REF], whose particularity is to exhibit a tractable loglikelihood for the generative process, which avoids the use of approximate inference techniques. This is achieved by means of a particular architecture, composed of invertible layers with (fast) tractable Jacobian. Instead of standard convolutions, the layers of Glow are thus composed of split, 1 × 1 convolutions and actnorm operations, we refer to [START_REF] Kingma | Glow: Generative flow with invertible 1x1 convolutions[END_REF] for more details. We trained Glow on the CelebA dataset [START_REF] Liu | Large-scale celebfaces attributes (celeba) dataset[END_REF], composed of colored images of faces with size 64 × 64. We used 4 blocks with 32 steps of flow each and additive coupling layers. The optimizer chosen for the training was Adam with a learning rate of 10 -4 (β 1 = 0.9 & β 2 = 0.999).

Insofar as Glow is invertible, it can be used both as an Encoder E(.) or a Decoder D(.) for (respectively) the analysis and the synthesis formulations. Because of the Gaussian prior on the latent space, the regularizer ϕ will be the squared 2 norm. The optimisation problems (3) and ( 4) are solved by gradient descent (either on the latent space or on the image). The initialization of the gradient descent is done, as explained by [START_REF] Asim | Invertible generative models for inverse problems: mitigating representation error and dataset bias[END_REF], by z 0 = 0 and x 0 = D(0) for the synthesis and analysis formulations, respectively. Parameter λ is chosen empirically around 10 -4 for each method and every inverse problems, so as to obtain the best performance.

In the experiments, we consider the three following inverse problems:

• Super-resolution: increasing the resolution of an image which has been previously downsampled by a factor 2 or 4 with local averaging (uniform filter).

• Deblurring: removing the blur caused by a 7 × 7 uniform filter.

• Inpainting: filling the gaps in an image caused by the application of a mask. We consider two different settings, either 60% of uniformly random missing pixels or a squared mask centered on the image of size 1 9 × 1 9 .

The metrics used to evaluate the effectiveness of the result are the widely used PSNR and SSIM. For the sake of reproducibility, our code and additional results are available at https://github.com/Mathieu-Verm/Analysis_ Synthesis_GLOW.

RESULTS

We run experiments on 50 images from the test set of CelebA. Let us first show visual comparisons between the synthesis and analysis frameworks, to highlight the pros and cons of both formulations. The result of the deblurring and the superresolution experiments are depicted on Fig. 1 ing the deblurring, the results of the synthesis method exhibit artefacts, that are particularly visible in the background and do not disappear if we increase λ. The analysis formulation instead shows a far better result, both for the background and the face. To explain this, we believe that the synthesis approach struggles to represent the background, in particular for the first image because its pattern is a bit unusual and might not have been seen during training. In the analysis approach instead there is no need to generate the background, since the optimization takes place in the image space. The super-resolution experiment investigates a more difficult setting where fewer information is available, and for this the synthesis approach might seem preferable. Indeed, the faces reconstructed by synthesis are much more pleasant than the analysis counterpart, which suffer from strong artefacts (the prior does not seem to work well enough, even for a large value of λ). Note however that the synthesis results, although visually better, are outperformed by analysis in terms of PSNR or SSIM, as we will see later.

Let us now move to the inpainting results, depicted on Figure 2 for the random and the structured masks. In the first case, the analysis technique seems to achieve the best results, while it is outperformed by its synthesis counterpart for the structured mask. We have the same explanations as in the previous experiment: we believe that in general, the analysis formulation stays closer to the true solution because it does not suffer from the intrinsic bias given by the generative network, and its optimization seems safer (there might be more spurious local minima in the latent space than in the image space). However when the problem becomes too difficult, the analysis does not seem to regularize enough, and the synthesis formulation obtains visually better results, although not clearly better in terms of PSNR or SSIM. We also computed the PSNR and SSIM for the 50 images and the 5 different inverse problems, and gathered the results in Table 1. It confirms the previous findings: in most cases, the analysis formulation enables an image recovery which is dramatically closer to the ground truth than with the synthesis formulation. But for the most difficult problem, i.e., inpainting with a structured mask (and to a lower extent, superresolution with factor 4), the synthesis formulation seems better. Another interesting point is the variance of the metrics across the 15 images, which is significantly lower for the analysis formulation. This means that the optimization landscape seems somehow safer in the image space that in the latent space, which is an expected benefit of the analysis formula-tion.

CONCLUSION

We presented here a variant of the deep learning regularization via generative models, were the solution is searched directly in the image space. According to our experiments, our analysis formulation obtains an estimation of the image which is more robust and often closer to the ground truth, even if sometimes visually less pleasant, particularly when the observation does not contain enough information.

Further work is required to better understand what is the best formulation for a given setting, and to extend these findings to improved learning strategies such as in [START_REF] Gonzalez | Solving inverse problems by joint posterior maximization with a VAE prior[END_REF] or [START_REF] Jalal | Robust compressed sensing of generative models[END_REF]. Concerning the applications, the main bottleneck up to now is the limited ability of generative networks to represent heterogeneous datasets, such as images with various size and content. Forthcoming contributions in generative models should soon overcome this, opening a huge number of possible realworld applications.

Fig. 1 .

 1 Fig. 1. Results of deblurring (left) and 2x super-resolution (right) for three images of the test set. From top to bottom: truth, degraded image, results of synthesis and analysis formulations.

Fig. 2 .

 2 Fig. 2. Inpainting with random mask (left) and structured mask (right), for the same three images of the test set.

Table 1 .

 1 Average performance over a test set of 50 images and corresponding standard deviations. The best score between analysis and synthesis are highlighted in bold, for both metrics.

	Task	PSNR (synthesis) PSNR (analysis) SSIM (synthesis) SSIM (analysis)
	Deblurring	23.38 ±2.04	32.16 ±1.56	0.74 ±0.09	0.94 ±0.01
	Super-resolution (x2)	22.26 ±4.21	31.19 ±1.33	0.76 ±0.12	0.93 ±0.01
	Super-resolution (x4)	18.94 ±2.81	24.12 ±1.21	0.61 ±0.11	0.76 ±0.03
	Inpainting (random mask)	21.84 ±3.57	27.89 ±2.24	0.71 ±0.14	0.87 ±0.05
	Inpainting (structured mask)	30.40 ±2.53	27.50 ±3.26	0.94 ±0.02	0.91 ±0.03


MD Greta Mainieri

MD Jean-Baptiste Maranci

MS Pierre Champetier

MD Smaranda Leu- Semenescu

MD Ana Gales

MD Pauline Dodet

MD Isabelle Arnulf
email: isabelle.arnulf@aphp.fr

Are Sleep Paralysis and False Awakenings Different from REM Sleep and from Lucid REM Sleep? A Spectral EEG Analysis

Keywords: sleep paralysis, false awakenings, REM sleep electroencephalography EMG: electromyography EOG: electro-oculography MSLT: multiple sleep latency tests SOREMP: sleep onset in REM period REM: rapid eye movement REMs: rapid eye movements

Study objectives:

To determine the polysomnography characteristics during sleep paralysis, false awakenings and lucid dreaming (which are states intermediate to REM sleep and wake but exceptionally observed in sleep laboratory).

Methods:

In 5 subjects, we captured 5 episodes of sleep paralysis (2 time-marked with the ocular left-right-left-right code normally used to signal lucid dreaming, 1 time-marked by an external noise and 2 retrospectively reported) and 2 episodes of false awakening. The sleep coding (using three seconds mini-epochs) and spectral EEG analysis were compared during these episodes and normal REM sleep as well as wakefulness in the same four among these five subjects, and vs. lucid REM sleep in four other patients with narcolepsy.

Results: During episodes of sleep paralysis, 70.8 % of mini-epochs contained theta EEG rhythm, (vs. 89.7% in REM sleep and 21.2% in wakefulness), 93.8% contained chin muscle atonia (vs. 89.7% in REM sleep and 33.3% in wakefulness) and 6.9% contained rapid eye movements (vs. 11.9% in REM sleep and 8.1% in wakefulness). The EEG spectrum during sleep paralysis was intermediate between wakefulness and REM sleep in the alpha, theta and delta frequencies, whereas the beta frequencies were not different between sleep paralysis and normal REM sleep.

The power spectrum during false awakening followed the same profile as in sleep paralysis.

Conclusions:

The predominant theta EEG rhythm during sleep paralysis and false awakenings (with rare and lower alpha rhythm) suggests that the brain during sleep paralysis is not in an awake but in a dreaming state.

INTRODUCTION

Sleep paralysis is classified within the REM sleep parasomnias, as a transient sleep/wake phenomenon occurring when entering into or awakening from REM sleep, with elements of REM sleep persisting into wakefulness. [START_REF] Fox | The Oxford Handbook of Spontaneous Thought: Mind-Wandering, Creativity, and Dreaming[END_REF][START_REF]The international Classification of Sleep Disorders[END_REF] In particular, the REM sleep-associated muscle atonia seems to persist in an awake brain, leading to the often distressing experience of being incapable to move. [START_REF] Denis | A systematic review of variables associated with sleep paralysis[END_REF] Several historical descriptions and iconic representations delineate a picture of a conscious brain in a still 'sleeping' body.

The EEG, EOG and EMG features (awake, asleep, or admixture of both stages) during sleep paralysis have been challenging to document, as it is difficult to capture this brief and unpredictable experience in a sleep laboratory. However, some rare sleep paralyses have been documented during polysomnography in healthy subjects [START_REF] Takeuchi | Isolated sleep paralysis elicited by sleep interruption[END_REF][START_REF] Takeuchi | Factors related to the occurrence of isolated sleep paralysis elicited during a multi-phasic sleep-wake schedule[END_REF] and in patients with narcolepsy. [START_REF] Terzaghi | Sleep paralysis in narcolepsy: more than just a motor dissociative phenomenon?[END_REF][START_REF] Pizza | Rhythmic movements and sleep paralysis in narcolepsy with cataplexy: a video-polygraphic study[END_REF] In healthy subjects, only episodes elicited by forced awakenings have been studied, using a sleep wake schedule known to facilitate sleep onset REM periods [SOREMPs]. In these instances, elements of REM sleep and wakefulness intermingled in two main patterns: one scored as REM sleep despite abundant trains of alpha rhythms, sometimes interrupted by rapid eye movements (REMs); the other scored as wakefulness with appearance of atonia plus muscle twitches. [START_REF] Takeuchi | Isolated sleep paralysis elicited by sleep interruption[END_REF][START_REF] Takeuchi | Factors related to the occurrence of isolated sleep paralysis elicited during a multi-phasic sleep-wake schedule[END_REF] In patients with narcolepsy, the polysomnography showed features intermediate between REM sleep and wakefulness. [START_REF] Pizza | Rhythmic movements and sleep paralysis in narcolepsy with cataplexy: a video-polygraphic study[END_REF][START_REF] Stefani | Clinical neurophysiology of REM parasomnias[END_REF] In a single case to date, spectral EEG analysis definitely settled the episode as an intermediate state between wake and REM sleep. [START_REF] Terzaghi | Sleep paralysis in narcolepsy: more than just a motor dissociative phenomenon?[END_REF] Sleep paralysis may sometimes be linked to false awakening, [START_REF] Fox | The Oxford Handbook of Spontaneous Thought: Mind-Wandering, Creativity, and Dreaming[END_REF] a state characterized by the perception of waking up in a familiar place and starting a normal daytime routine, only to later discover of having been trapped in one's own dream or to realize of having dreamt. [START_REF] Buzzi | False awakenings in light of the dream protoconsciousness theory: a study in lucid dreamers[END_REF][START_REF] Nielsen | Idiopathic nightmares and dream disturbances associated with sleep-wake transitions[END_REF] This state may also be referred to as a "dreamed" awakening. Only one report describes the polysomnography features of a SOREMP hallucination without sleep paralysis, [START_REF] Takeuchi | Laboratory-documented hallucination during sleep-onset REM period in a normal subject[END_REF] later referred to as a false awakening by other authors. [START_REF] Buzzi | False awakenings in light of the dream protoconsciousness theory: a study in lucid dreamers[END_REF] We incidentally had the opportunity to monitor during polysomnography five episodes of sleep paralysis, one of which continued into a false awakening, as well as an episode of isolated false awakening, and to better characterize the associated EEG pattern.

METHODS

Recordings and polysomnography analysis

EEG monitoring included Fp1-A2, C3-A2, O1-A2 cortical channels, left and right electro-oculogram (EOG), chin and tibialis anterior muscle surface electromyography (EMG) for all participants, electrocardiography, pulse oximetry for almost all patients, nasal pressure, recording of tracheal sounds through a microphone placed at the surface of the trachea, thoracic and abdominal belts, as well as EEG-synchronized infrared video-monitoring and ambiance microphone recordings for Patient #3. In two patients experiencing sleep paralysis (Patient #1 and Patient #3), the exact beginning of the episode could be accurately estimated. For them, we divided the 30-second epochs into 10 mini-epochs of 3 seconds each, analyzing each mini fragment for the presence of alpha/theta background activity on EEG, evaluated on O1-A2 cortical channel (we considered the mini-epoch as having alpha or theta when a rhythm was present for >50% of the mini-epoch), muscle atonia or increased muscle tone on chin and legs EMG, as well as REMs presence or absence on the EOG. In Patient #1 who experienced the sleep paralysis during the multiple sleep latency test (MSLT), the cumulative fragments of the two episodes were of comparable duration with the periods of normal REM sleep and quiet wakefulness in the MSLT. Therefore, we analyzed the totality of normal REM sleep epochs outside the sleep paralysis experiences and all fragments of quiet wakefulness in the MSLT. In Patient #3, the episode of sleep paralysis occurred during the first REM sleep episode of the night, so that we had to match some normal REM sleep and eyes-closed wakefulness period of comparable duration. Analyzing epochs of normal REM sleep and wakefulness all over the night would have resulted in a much greater amount of mini-epochs in comparison with the paralysis episode. For this reason, we assigned a crescent numeration to all mini-epochs of REM sleep and wakefulness throughout the night and randomly draw a corresponding number of miniepochs, exactly matching the duration of the sleep paralysis episode. Arousals during REM sleep, including prominent fast activity and tonic increase in chin EMG, have been excluded from the analysis. We then compared the polysomnography elements associated with sleep paralysis to normal REM sleep and quiet wakefulness of the same subjects. Plus, we compared the elements obtained during sleep paralysis with those obtained during lucid REM sleep in 4 narcolepsy patients with lucid dreaming, chosen as "control" group and previously described in other studies from our centre. [START_REF] Dodet | Lucid dreaming in narcolepsy[END_REF][START_REF] Oudiette | REM sleep respiratory behaviours match mental content in narcoleptic lucid dreamers[END_REF] This particular group of patients carries the advantage of providing a clear time-frame of the episodes (thanks to the ocular coding), giving the opportunity to compare sleep paralysis and false awakenings to another peculiar state of "consciousness while dreaming", unique in its kind, such as lucid dreaming. The same 3-seconds analysis was applied to the fragments of lucid REM sleep. A  2 test was performed to compare the categorical variables (presence of alpha/theta EEG, EMG activity, presence of REMs at the EOG) in the studied sleep states. Statistical significance was set at p value < 0.05.

Statistical analyses were performed with STATA version 16.0.

Spectral EEG analysis

We extracted non-overlapping samplings of EEG recordings (sample frequency of 256 Hz) of sleep paralysis periods for 3 patients and false awakenings for 2 patients, matching them with fragments of quiet closed-eyes wakefulness and of normal REM sleep of corresponding duration. We excluded ocular artifacts by means of visual inspection in all samples. The samples were exported to perform spectral analysis on each electrode channel with Fast Fourier Transform. For each EEG activity, the power spectrum was estimated using the Welch averaged periodogram method using the Matlab Toolbox EEGLAB. For each state (sleep paralysis, false awakening, normal wakefulness and normal REM sleep), we cumulated all the fragments of the patients and performed the mean of the three EEG channels. We also performed a power spectrum for each patient. In addition, we similarly performed a spectral analysis on 4 of the over-mentioned narcoleptic patients recorded during lucid dreaming, for whom we extracted lucid REM sleep samples, following a left-right-left-right ocular signal, as already explained in our previous work, [START_REF] Dodet | Lucid dreaming in narcolepsy[END_REF] and selected equivalent fragments of quiet closed-eyes wakefulness and non-lucid REM sleep.

RESULTS

Case-reports

Patient 1: Two episodes of sleep paralysis framed by ocular signals

This 18-year-old man was referred for a 5-year history of hypnagogic sleep paralysis, sometimes accompanied by intruder hallucinations as well as visual, auditory or tactile hallucinations, a mild sleepiness (Epworth sleepiness score: 11/24) and no cataplexy or sleep attacks. He was eventually diagnosed with a borderline form of narcolepsy type 2 (mean sleep onset latency of 10.4 min at the MSLT-when normal values are greater than 8 min-3 of them containing REM sleep). However, as his main problem was recurrent sleep paralyses, and as they may occur during daytime naps, we asked him to signal any sleep paralysis via the ocular code commonly used in lucid dreaming (i.e., left-rightleft-right large horizontal eye movements), as we previously did in patients with narcolepsy. [START_REF] Dodet | Lucid dreaming in narcolepsy[END_REF] After the three naps containing REM sleep, he reported having experienced a sleep paralysis, which had been signaled in two of the three naps by the ocular code decided before the nap. In particular, 30 sec after entering REM sleep, he made a left-right-left-right ocular signal (Figure 2), marking the beginning of the sleep paralysis episode, which lasted 30 seconds before awakening. He had an analogous episode during the fourth nap. Seventy seconds after entering into REM sleep, he signaled the beginning of the paralysis, which lasted 113 seconds before awakening.

Afterwards, the patient explained that he was not dreaming but had sleep paralysis.

Patient 2: A sleep paralysis reported upon awakening

A 48-year-old woman with a history of obstructive sleep apnea syndrome adequately treated with continuous positive airway pressure was referred for residual excessive daytime sleepiness (Epworth sleepiness score: 17/24). She also reported rare hypnagogic hallucinations and episodes of sleep paralysis with an intruder hallucination (feeling the presence of unknown people in her room). She was eventually diagnosed with a borderline form of narcolepsy type 2 (no cataplexy, mean sleep latency at the MSLT: 8.9 minutes; 4 SOREMPs /5 naps). At the end of the second nap, she reported to the nurse having experienced a sleep paralysis just before awakening.

Patient 3: Isolated sleep paralysis followed by a false awakening, time-marked by the National Day's fireworks

A 27-year-old woman presented a 5 year-long history of hypnagogic (and rarely hypnopompic) hallucination, accompanied by sleep paralysis, in absence of other sleep complaints. During the sleep paralysis episodes, she reported seeing a black demon moving and speaking with an incomprehensible language, feeling a thoracic oppression and sometimes feeling a presence in her room (like a thief). She met the diagnosis of recurrent isolated sleep paralysis. During the first night of sleep monitoring, at the end of the first cycle of REM sleep, we recorded a sleep paralysis lasting 5 minutes (300 seconds) before awakening (Figure 3). The exact time of sleep paralysis was accurate, because the episode started (as later reported) at time of the first detonation of the fireworks of the National Day on the Eiffel tower of Paris, France (the exact time of fireworks is scheduled and can also be heard on the audio of the polysomnography). At the video inspection during sleep paralysis, she was lying in dorsal position, had initial lips movements without sound emission, followed by moaning at different times during the episode, as if she was trying to call someone with the mouth closed. In the middle of the episode, she briefly opened her eyes. Finally, she managed to perform some small jerky movements of the feet, followed by the arms and then woke up. After the episode, she recalled having experienced a burning sensation widespread over the body, having tried to emit vocal sounds because of the impossibility to move. She also mentioned to have experienced a false (dreamed) awakening at the end of her sleep paralysis episode, (falsely) recalling that she succeeded to push the button for calling the nurse, who finally came and touched her hand, achieving to wake her up.

Patient 4. Feeling crushed by the physician

An 18-year-old woman was referred for excessive daytime sleepiness (Epworth sleepiness score: 16/24) and sleep attacks. She sometimes experienced hypnopompic hallucinations with feelings of presence of people or shadows. She had a normal (12.5 minutes) sleep latency at the MSLT but 3 SOREMPs, and a SOREMP during the "ad libitum" day of recording (patients are monitored during a first night, followed by MSLT and followed again by ad libitum night and day sleep monitoring, (see the bed rest procedure in [START_REF] Vernet | Idiopathic hypersomnia with and without long sleep time: a controlled series of 75 patients[END_REF] ). When awakening from the daytime nap's SOREMP, she reported having experienced a sleep paralysis associated with the hallucination of a physician crushing her and trying to suffocate her.

Patient 5. False awakening as an isolated sleep state misperception

A 22-year-old woman complained of excessive daytime sleepiness (Epworth sleepiness score: 18/24) and irresistible sleep attacks at school or when driving a car. She suffered from cataplexies, disturbed night sleep and frequent nightmares, lucid dreaming, false awakenings, frequent sleep paralyses as well as visual, auditory and presence hypnagogic hallucinations (including out of body experiences). She accurately distinguished these various dissociate sleep-wake states. As an example of false awakening, she reported: "I wake up and leave my bed. I eat my breakfast, wash my hair, I prepare my bag. I leave my flat, go down in the street, take my bus. And then in the bus, I realize that something is wrong, something about the time is wrong. Still in my bus, I wake up and realize this was only a dream. In other instances, I do all this morning routine until taking my bus and arriving at school, and then at school I realize that I am alone, the school is empty. This looks so weird that I wake up and realize this was a dream". Sleep paralysis are described as hypnagogic episodes, often associated with a weird tinnitus: "I try to fall asleep when hearing this tinnitus, turn to be supine in the bed, and then I think that I open my eyes, and despite it is dark around in the room, I am still able to see a person arriving in front of me. I start to be very afraid, try to move, shout and call for help. I feel that I cannot move of shout. Then I try to calm myself, by I see a passing shadow that comes from the ceiling or the door (this is generally a shadow, but sometimes it can be Lord Voldemort from the Harry Potter book himself, or just a flying face with a wide-open mouth coming on me). The shadow stays close to me or in the wood of my bed, or is on me and tries to catch me". She met the criteria of narcolepsy type 1 (MSLT latency: 7.9 minutes; 4 SOREMPs). At the beginning of the first nap, she reported a false awakening. She dreamed of lying in the bed at the sleep laboratory, during the sleep monitoring, trying to fall asleep and saying to herself: "You have to sleep, you have to sleep". She realized when she woke up that it was a dream (a "false waking") and that she was already sleeping. During this nap (sleep onset latency of 6.5 min), she had a late and brief SOREMP, entering from wakefulness directly into REM sleep for a duration of 1 minute, before awakening and realizing she had experienced a false awakening. background for all patients showed a theta predominance, with some bursts of alpha rhythm; in particular, these were in correspondence with REMs in Patient #1 and with the attempts of vocalizations in Patient #3. The EOG showed the presence in all patients of isolated REMs, sometimes occurring in brief bursts. The only exception was Patient #1, which performed the voluntary left-right-left-right signal, marking the beginning of the sleep paralysis, followed by bursts of usual REMs before the definite awakening (Figure 2). The EMG recorded on the chin muscle displayed a complete muscle atonia in all subjects, with rare and subtle activations in some cases (in coincidence with the ocular code in Patient #1, and during the last seconds of the episode in Patient #3, when she began to move before definitely awakening.

Visual polysomnographic aspects of the episodes

The 3 seconds mini-epochs analysis during the sleep paralysis

The results of the 3-second mini-epochs (over a total of 680 mini-epochs) analysis are shown in Table 1. As regards to EEG visual inspection of the 3-second mini-epochs, the relative percentage of the alpha band during sleep paralysis was lower than during wakefulness and higher than during normal REM sleep (P <0.001), setting the sleep paralysis condition in an intermediate position between the two states. This intermediate position was also found when comparing the spectral analysis between the three conditions (Figure 1). Compared to lucid REM sleep, there were increased amounts of EEG alpha rhythm during sleep paralysis (P = 0.001). Concerning the chin muscle tone, beyond the expected differences regarding enhanced muscle tone between sleep paralysis vs. wakefulness and REM sleep vs. wakefulness (P < 0.001), no further difference was found between sleep paralysis and normal REM sleep. However, there were more frequent mini-epochs with enhanced muscle tone during lucid REM sleep than during sleep paralysis (P = 0.019). Regarding the number and type of ocular movements, no difference was identified between the different states.

Spectral EEG analysis of sleep paralyses, false awakenings and lucid REM sleep

Regarding spectral analysis (Figure 1), the patients experienced sleep paralyses during fragments of different durations (113 seconds in Patient #1, 300 seconds in Patient #3 and 60 seconds in Patient #4). We could not analyse the measures in Patient #2 because of an artefact on the EEG channels. As illustrated in Figure 1A, the global power spectrum of the experience of sleep paralysis (using a group analysis in these 3 patients) showed an intermediate power spectrum between wakefulness and REM sleep in the low frequencies and in the alpha band, whereas the more rapid beta EEG frequencies were not different between sleep paralysis and normal REM sleep.

The spectral analysis of false awakenings (Figure 1B) was performed during the last 30 seconds before awakening in Patient #3 (even if we are aware of the arbitrariness of this choice), and during the 60 seconds of the isolated false waking in Patient #5. The spectral power appears in all respects similar to that of sleep paralysis, with the same intermediate position (between wakefulness and normal REM sleep) of low frequencies and alpha band and a profile of the beta band similar to that of normal REM sleep.

Eventually, the spectral analysis of lucid REM sleep was not different from that of nonlucid REM sleep (Figure 1C).

DISCUSSION

The current classification of sleep disorders places sleep paralysis as a dissociated state between normal REM sleep and wakefulness, mostly considering muscle atonia (a characteristic of REM sleep), in contrast with an "awake" subject, which is conscious of the surrounding environment. However, it is quite common during the episode to experience concomitant multi-sensorial hallucinations, which are considered as fragments of dream contents intruding into wakefulness. [START_REF] Dodet | Lucid dreaming in narcolepsy[END_REF] This widely diffused concept of a preserved alertness in an atonic body [START_REF] Hishikawa | Physiology of REM sleep, cataplexy, and sleep paralysis[END_REF] during sleep paralysis can however now be challenged in many aspects.

The brain is not awake during sleep paralysis

First of all, at visual inspection, in contrast with some previous reports describing an almost awake EEG tracing with bursts of REMs and muscle atonia, [START_REF] Takeuchi | Isolated sleep paralysis elicited by sleep interruption[END_REF] the five recorded sleep paralyses here were associated with EEG features typical of REM sleep (not of wakefulness), including predominant theta rhythms, and occasional bursts of alpha rhythms (concomitant with rapid eye movements) as well as muscle atonia.

Consequently, the concept of the sleep paralysis as an awake brain in a paralyzed body might be reviewed in favour of a rather partially aware but dreaming state, as already postulated in some studies. [START_REF] Fox | The Oxford Handbook of Spontaneous Thought: Mind-Wandering, Creativity, and Dreaming[END_REF] Both the spectra and the 3-seconds analysis of the recorded paralysis episodes here display comparable features with the previous case-report by Terzaghi et al., [START_REF] Terzaghi | Sleep paralysis in narcolepsy: more than just a motor dissociative phenomenon?[END_REF] setting EEG during sleep paralysis in an intermediate state between REM sleep and wakefulness, with an intermediate low frequencies EEG spectrum, an intermediate power in the alpha band and a profile more similar to that of normal REM sleep regarding higher (beta) frequencies. Our results confirm their initial findings in a higher number of sleep paralysis episodes, some of them being more accurately signalled than in previous works, thanks to an external (the firework) or an internal (the eye code) markers.

'Lucid' sleep paralysis

To our knowledge, this is the first time that the ocular coding (currently used for signalling lucid dreaming in REM sleep), is obtained for signalling an episode of sleep paralysis, thus configuring for the first time a "lucid" sleep paralysis. Of interest, if the muscle atonia spares the voluntary oculomotor muscles during REM sleep, it seems also to spare them during sleep paralysis, confirming objectively some reports of patients being able to move their eyes (and sometimes to open them) during sleep paralysis. This patient, even being "conscious" of his state of paralysis showed a typical REM sleep EEG, similarly to lucid dreaming (when asleep people are aware of dreaming).

Notwithstanding, his EEG power spectrum showed a peak in the alpha band between wakefulness and REM sleep, similarly to the other sleep paralysis episodes, in sharp contrast with lucid REM sleep in lucid dreamers (in whom the spectra appear utterly similar to proper REM sleep here). The experience of lucid dreaming, in fact, seems to display a reduction in delta frequencies in comparison with non-lucid REM sleep, [START_REF] Dodet | Lucid dreaming in narcolepsy[END_REF] but no difference in the alpha band, characteristics which appears peculiar to both experiences of sleep paralysis and, with proper limitations, false awakening. Of interest, lucid dreamers do not feel paralyzed during lucid REM sleep (despite their body is really paralyzed), whereas patients with sleep paralysis feel the real body paralysis (despite their brain is partly in a REM sleep state). The inputs from the muscle sensors to the brain sensory cortex seem to be ungated during sleep paralysis but blocked during lucid dreaming, suggesting local variations in cortex connectivity among dissociated states.

False awakenings

Clinical and neurophysiological descriptions of false awakenings are scarce. The single report by Takeuchi et al., [START_REF] Takeuchi | Laboratory-documented hallucination during sleep-onset REM period in a normal subject[END_REF] later considered as a false awakening by other authors, [START_REF] Buzzi | False awakenings in light of the dream protoconsciousness theory: a study in lucid dreamers[END_REF] depicts an hypnagogic hallucination of unpleasant and fearful feeling of presence in the sleep laboratory with the perception of having rose from the bed. The polysomnography showed abundant trains of alpha rhythm on the EEG, sometimes blocked by REMs mixed with slow eye movements and low muscle tone. Conversely, the two experiences of false awakening monitored here (similarly to sleep paralysis) were close to regular REM sleep. Indeed, the visual EEG inspection and quantitative analysis clearly shows predominant theta rhythms, suggesting that these two experiences are the products of a dreaming rather than a fully conscious brain. This confirms a strong link between the two experiences, which may sometimes continue into one another, as in Patient #3, and as underlined in a previous population survey. [START_REF] Buzzi | False awakenings in light of the dream protoconsciousness theory: a study in lucid dreamers[END_REF] However, given the impossibility to accurately time-mark the beginning of the false awakening in Patient #3, these results need to be reproduced.

What do these results tell us about sleep paralysis?

The intermediate features between REM sleep and wakefulness of both sleep paralysis and false awakening might be explained in light of the currently consolidated theories of local sleep, stating that different sleep phases or wakefulness, especially in the transitional states, can show intermingled features with one another. [START_REF] Mahowald | Insights from studying human sleep disorders[END_REF] In this direction, the night-time sleep of subjects with recurrent idiopathic sleep paralysis contains higher delta power than healthy subjects, suggesting an alteration in wakefulness-NREM-REM sleep regulation in sleep paralysis sufferers. [START_REF] Fox | The Oxford Handbook of Spontaneous Thought: Mind-Wandering, Creativity, and Dreaming[END_REF] Notably, all but Patient #3 experienced the episodes during SOREMPs, a timing where sleep mechanisms of phase transition may be altered, possibly favouring local REM sleep phenomena In fact, previous studies have ascertained that SOREMPs facilitate the onset of sleep paralysis, in patients with narcolepsy as well as in healthy subjects. [START_REF] Takeuchi | Isolated sleep paralysis elicited by sleep interruption[END_REF][START_REF] Takeuchi | Factors related to the occurrence of isolated sleep paralysis elicited during a multi-phasic sleep-wake schedule[END_REF] This aspect may also explain why patients with narcolepsy experience sleep paralysis more frequently than healthy subjects do.

Regarding false awakening, the single previous description was recorded during a SOREMP, [START_REF] Takeuchi | Laboratory-documented hallucination during sleep-onset REM period in a normal subject[END_REF] as here in Patient #5, whereas in Patient #3 the false awakening emerged from a sleep paralysis. Sleep paralysis and false awakenings share the feeling of being in a familiar environment (usually the bed/sofa in one's house) and not in another place as when dreaming, the perception of being awake, and frequent feelings of anxiety, 1 but inability to move during sleep paralysis episodes vs. freedom to move during false awakening episodes (Table 3). In this respect, sleep paralysis, which is often associated with supine body position, has been linked to disturbances in the parietal cortex, where monitoring of motor programs through proprioceptive feedback normally takes place.

During sleep paralysis, the persistent muscle atonia can create a mismatch contributing to abnormal physical perceptions and modifying the self-body image. [START_REF] Fox | The Oxford Handbook of Spontaneous Thought: Mind-Wandering, Creativity, and Dreaming[END_REF][START_REF] Jalal | How to make the ghosts in my bedroom disappear? Focused-attention meditation combined with muscle relaxation (MR Therapy)-A direct treatment intervention for sleep paralysis[END_REF]

Limitations

Because only three EEG channels were used for routine sleep recordings, we cannot topographically localize the power spectra source over the EEG scalp. It will be highly desirable in future studies to capture these experiences with a more extensive EEG montage, in order to better elucidate the spatial scalp EEG correlates of these fascinating phenomena. Another limitation here is the reduced sample, which is expected knowing the difficulty of capturing these rare experiences. Eventually, two of the subjects with false awakenings and sleep paralyses (and the four lucid dreamers) had a narcolepsy, which may restrain the generalization of the findings to the healthy population. Moreover, we are aware that the arbitrary choice of inspecting the 60 seconds before the awakening in two patients, may represent a further limitation.

Eventually, we could not capture the three phenomena (sleep paralysis, false awakening and lucid REM sleep) in the same subjects, preventing more precise within-subjects' comparisons of the three experiences.

Although our results may be speculative in some aspects, their principal strength is that episodes of sleep paralysis in different subjects show a similar visual aspect and spectral profile on EEG, in accordance with a previous single report using spectral analysis. [START_REF] Terzaghi | Sleep paralysis in narcolepsy: more than just a motor dissociative phenomenon?[END_REF] Besides, we provided the first report of a 'lucid' sleep paralysis (a robust, time marked episode) and a spectral analysis of two false awakenings, even if these last results are less reliable.

Legend of the Figures

  An accurate timing of the episodes is stateable for Patient #1 (ocular code), Patient #3 (video-recording with auditory trigger) and Patient #5 (direct entering in REM sleep for 60 seconds before awakening). For Patients #2 and #4, because of the impossibility to document the exact beginning of the episodes, we inspected the 60 seconds before awakening, being aware of the arbitrariness of this choice. All patients had the episodes during REM sleep and ended with awakening, when they reported the experience to the nurses. The episodes occurred during daytime naps in Patients #1, #2, #4 and #5, and during night-time in Patient #3. All episodes occurred during a SOREMP, with the exception of Patient #3. The visual inspection of the EEG

Figure 1 -

 1 Figure 1 -Global EEG power spectrum: a) during five episodes of sleep paralysis (SP,

Figure 2 -

 2 Figure 2 -An episode of sleep paralysis signaled by the left-right-left right (LRLR

Figure 3 -

 3 Figure 3 -Polysomnography monitoring during a 5 min long sleep paralysis. The upper

Table 1 -

 1 Quantification of EEG, EMG and EOG characteristics via analyzing 3 sec-mini-epochs during REM sleep, wakefulness, sleep paralysis and lucid dreaming episodes These 3 sec mini-epochs are extracted from the measures in Patient 1 and in Patient 3; ** These 3-seconds mini-epochs of lucid REM sleep are extracted from 4 patients with narcolepsy. LD: lucid dreaming; REM: REM sleep; REMs: rapid eye movements, SP: sleep paralysis; W: wakefulness

							P value
		SP*	REM*	W*	LD**	SP vs.	SP	REM vs.
									SP vs. LD
						REM	vs. W	W
	Mini-epochs,
		178	185	198	119	NA	NA	NA	NA
	No.
	Electroencephalography, % of the mini-epochs
	Alpha	29.2	10.3	78.8	12.7
						<0.0001	<0.0001	<0.0001	0.001
	Theta	70.8	89.7	21.2	87.3
	Chin surface electromyography, % of the mini-epochs
	Enhanced
		6.2	10.3	66.7	14.3
	muscle tone					0.157	<0.0001	<0.0001	0.019
	Muscle atonia	93.8	89.7	33.3	85.7
	Electrooculography, % of the mini-epochs
	REMs	6.9	11.9	8.1	8.4
						0.291	0.877	0.213	0.669
	No REMs	93.1	88.1	91.9	91.6

*

Table 2 -

 2 Clinical and neurophysiological characteristics of sleep paralysis, false awakenings and lucid dreaming, compared to normal wakefulness and REM sleep

		Wakefulness,	Sleep	False	Lucid	REM sleep
		closed eyes	paralysis	awakening	dreaming
	Clinical features
	Feeling paralyzed	No	Yes	No	No	No
	Feeling the	Yes	Yes	Yes	No	No
	surroundings as
	familiar
	Accurate awareness	Yes	Yes	No	Yes	No
	of the state
	Associated anxiety	None	Frequent	Frequent	None	None
	Mentation	Awake	Feels awake	Feels awake	Dreaming	Dreaming
		thoughts	but may feel
			hallucinations
	EEG
	Delta band	Low	Similar to	Similar to	Lower than	High
			REM sleep	REM sleep	REM sleep
	Theta band	Low	Similar to	Similar to	Similar to	High
			REM sleep	REM sleep	REM sleep
	Alpha band	High	Intermediate	Intermediate	Similar to	Low
			between Rem	between	REM sleep
			sleep and	Rem sleep
			wakefulness	and
				wakefulness
	Beta-Gamma bands	High	Similar to	Similar to	Similar to	Low
			REM sleep	REM sleep	wakefulness
					(frontal lobe)
					in one study
					and similar to
					REM sleep in
					another study

BRIEF SUMMARY

Current Knowledge/Study Rationale: Sleep paralysis is commonly viewed as an awake brain with a complete muscle atonia, but EEG analysis during sleep paralysis is exceptional.

Study Impact: Our polysomnography and spectral analysis of time-marked sleep paralyses suggest that the brain during sleep paralysis is not in an awake but in a dreaming state. This may explain why hallucinations are often associated with sleep paralysis.