Dr Holm Schneider
email: holm.schneider@uk-erlangen.de

Johanna Hammersen

Sabine Preisler-Adams

Kenneth Huttner

Wolfgang Rascher

Axel Bohring

Sweating ability and genotype in individuals with X-linked hypohidrotic ectodermal dysplasia

Keywords: ectodermal dysplasia, genotype, sweat gland

Background: X-linked hypohidrotic ectodermal dysplasia (XLHED), the most common type of ectodermal dysplasia, is caused by EDA gene mutations. Reduced sweating contributes substantially to XLHED-associated morbidity and mortality. To characterize the genotypephenotype relationship, sweat gland function was assessed non-invasively in XLHED patients and healthy controls.

Subjects and methods:

In 36 genotyped XLHED patients and 29 control subjects aged 0-57 years, pilocarpine-induced sweat volume, palmar sweat pore density and palmar skin conductance before and after stimulation were determined.

Results: Among 31 XLHED males, 14 had neither detectable sweat pores nor inducible sweating, 10 showed a few sweat pores but absent sweating, and 7 produced reduced sweat volumes (1-11 μl) as compared with controls (38-93 μl). Two of the low-sweating XLHED subjects had normal sweat pore counts. In all 5 heterozygous females, some sweat was detected, but generally less than in female controls. Basal and stimulated skin conductance readings were reduced in 23 of 24 non-sweating, but only in 3 of 12 low-sweating XLHED subjects. There was no correlation between sweat production and number of missing teeth.

Conclusions:

In contrast to prior reports on non-genotyped HED populations, this study confirmed a consistent, quantifiable defect of sweat gland function in male XLHED subjects as a disease biomarker. Among 26 different EDA genotypes, specific mutations were shown to be consistently associated with anhidrosis, implying that systematic mapping of EDA mutations together with the analysis of objective clinical data may help to distinguish functionally crucial mutations from those allowing residual activity of the gene product.

Introduction

X-linked hypohidrotic ectodermal dysplasia (XLHED [MIM 305100]), the most common type of ectodermal dysplasia, is characterized by severe hypohidrosis, hypoplasia of sweat, sebaceous, submucous, meibomian and mammary glands, sparse hair and eyebrows, and oligodontia. The disease is caused by EDA [MIM 300451] gene mutations. Reduced sweating is assumed to be a major contributor to XLHED-associated morbidity and mortality. [START_REF] Blüschke | Prevalence and prevention of severe complications of hypohidrotic ectodermal dysplasia in infancy[END_REF] The EDA gene spans approximately 425 kb of genomic sequence on the human Xchromosome and encodes ectodysplasin A, a trimeric type II transmembrane protein belonging to the tumor necrosis factor (TNF) superfamily of ligands. Eight transcriptional variants due to alternative splicing are known. The longest transcript consists of exons 1a, 3a, and 4-9 (nomenclature according to Bayés et al. [START_REF] Bayés | The anhidrotic ectodermal dysplasia gene (EDA) undergoes alternative splicing and encodes ectodysplasin-A with deletion mutations in collagenous repeats[END_REF] and Monreal et al. [START_REF] Monreal | Identification of a new splice form of the EDA1 gene permits detection of nearly all X-linked hypohidrotic ectodermal dysplasia mutations[END_REF]), encoding the 391 amino acids of the ectodysplasin A1 isoform (EDA-A1), lack of which has been demonstrated to underly XLHED. EDA-A1 comprises a small N-terminal intracellular, a transmembrane, and a large C-terminal extracellular domain with a furine cleavage site, a collagen-like domain consisting of [START_REF] Kere | X-linked anhidrotic (hypohidrotic) ectodermal dysplasia is caused by mutation in a novel transmembrane protein[END_REF] Gly-X-Y repeats with a single interruption, and a TNF homology domain. To be able to bind to its receptor EDAR, EDA-A1 has to be released by proteolytic processing at the furin recognition site as a soluble homotrimerized protein consisting of the collagen-like and TNF homology domains. Further multimerization through the collagen-like region has been shown [START_REF] Schneider | Mutations leading to X-linked hypohidrotic ectodermal dysplasia affect three major functional domains in the tumor necrosis factor family member ectodysplasin-A[END_REF] and seems to be important for protein function.

The phenotype of individuals affected by XLHED shows interfamilial as well as intrafamilial variability, and no clear genotype-phenotype correlation has been observed so far. [START_REF] Ferguson | Scarcity of mutations detected in families with X-linked hypohidrotic ectodermal dysplasia: diagnostic implications[END_REF][START_REF] Kobielak | Mutations in the EDA gene in three unrelated families reveal no apparent correlation between phenotype and genotype in the patients with an X-linked anhidrotic ectodermal dysplasia[END_REF][START_REF] Vincent | Mutational spectrum of the ED1 gene in X-linked hypohidrotic ectodermal dysplasia[END_REF][START_REF] Lexner | X-linked hypohidrotic ectodermal dysplasia. Genetic and dental findings in 67 Danish patients from 19 families[END_REF] However, at least 6 EDA missense mutations causing a specific phenotype of non-syndromic selective tooth agenesis have been described. [START_REF] Schneider | Mutations leading to X-linked hypohidrotic ectodermal dysplasia affect three major functional domains in the tumor necrosis factor family member ectodysplasin-A[END_REF][START_REF] Li | Nonsyndromic tooth agenesis in two Chinese families associated with novel missense mutation in the TNF domain of EDA (ectodysplasin A)[END_REF][START_REF] Han | Novel EDA mutation resulting in X-linked non-syndromic hypodontia and the pattern of EDA-associated isolated tooth agenesis[END_REF][START_REF] Rasool | A novel missense mutation in the EDA gene associated with X-linked recessive isolated hypodontia[END_REF][START_REF] Fan | Mutations in the EDA gene are responsible for X-linked hypohidrotic ectodermal dysplasia and hypodontia in Chinese kindreds[END_REF][START_REF] Mues | Functional analysis of Ectodysplasin-A mutations causing selective tooth agenesis[END_REF] Thus, one may hypothesize that the occurrence or severity of particular XLHED-related symptoms might correlate with the EDA genotype. Our present study, therefore, aimed at establishing quantitative skin assessment techniques to evaluate a genotype-phenotype relationship with respect to sweating ability.

Subjects and Methods

Thirty-one male XLHED patients aged from 0 to 57 years and 24 healthy, age-matched male controls with no family history of XLHED or negative testing for an EDA mutation as well as 5 heterozygous females and 5 healthy female controls were studied. All adults gave written informed consent to participate; in the case of minors, parental consent and child assent in >7 year olds was obtained. The study (clinicaltrials.gov NCT01109290) was approved by an independent institutional ethics committee and conducted according to national regulations and GCP/ICH guidelines.

XLHED patients were included only if pathogenic EDA mutations had been detected and liquid intake on the day of the study had been normal. Criteria for exclusion were acute febrile illness, implantable electronic devices, pregnancy or breastfeeding, and hypersensitivity to self-adhesive electrodes. Subjects were studied either at a German-Austrian bi-national family conference in Germany in May 2010 or during an appointment at the Competence Centre for Children with Ectodermal Dysplasias of the University Hospital Erlangen between May and August 2010. Subjects ED-Ch1 and ED-AS4, EDCh2 and ED-AS5, and ED-Ch11 and ED-AS7, respectively, have documented familiar relationships.

Medical and dental history was taken using a specific questionnaire, interviews and orthodontic records of permanent dentition. Parameters determined included palmar sweat pore density in an area of 16 mm 2 , sweat volume produced in a 30-minute period subsequent to stimulation with pilocarpine using the Wescor 3700 device (Wescor Biomedical Systems, Logan, U.S.A.), and palmar skin conductance, an indirect measure of skin hydration, [START_REF] Harrison | Skin conductance as a measure of pain and stress in hospitalised infants[END_REF] taken with the Med-Storm 1001 device (Med-Storm, Oslo, Norway). The examiners were blinded to the genotype of the participant.

First, standard sweat collection was conducted at the forearm site, as an even surface area of 57 mm 2 was needed. Maximum volume that could be collected in the Wescor device was 93 μl (estimated by weight). Then, sweat pore density was determined on graphite prints of the palm, which had been enlarged 16-fold, and calculated per cm 2 . Finally, palmar skin conductance measurements were recorded at baseline and following capillary blood sampling.

For genotyping, DNA was extracted from peripheral blood or buccal swabs (OmniSwab TM , Whatman International Ltd., UK) using standard procedures or following manufacturer's instructions, respectively. PCR was performed with specific primer pairs for exons 1a, 3a, and 4-9 selected with Primer3 and checked with MFOLD and SNPCheck. The PCR products were treated with ExoSAP-IT (USB Corporation, Cleveland, Ohio, USA) according to the manufacturer's instructions and sequenced using the BigDye® Terminator v3.1 Cycle Sequencing kit (Applied Biosystems, Carlsbad, California, USA) and a 3730 DNA Analyzer (Applied Biosystems). Screening for larger deletions/duplications was done by multiplex ligation-dependent probe amplification (MLPA; MRC-Holland_P183, Netherlands) according to the manufacturer's instructions. To validate that any previously unreported nucleotide substitutions are real mutations, at least 100 healthy, anonymized, population-matched males were tested for each alteration by EDA sequence analysis.

Group comparisons, both between affected and control subjects, and between adult and pediatric subsets, were done by Mann-Whitney and Exact Fisher's tests using GraphPadPrism software version 4.0 for Windows (GraphPad Software, Inc., La Jolla, CA, U.S.A) or SPSS software version 17.0 for Windows (SPSS Inc., Chicago, IL, USA).

Results

Assessment and quantification of sweat gland function

As perspiration may change with puberty, which had also been observed in one of our previous studies on ectodermal dysplasia, [START_REF] Bohring | WNT10A mutations are a frequent cause of a broad spectrum of 13 ectodermal dysplasias with sex-biased manifestation pattern in heterozygotes[END_REF] the XLHED cohort and the controls were divided into groups of children and postpubertal individuals (adults) with a cut-off of 16 years.

Individual data for each subject tested are shown in Tables 1 and2. As expected, control children had a significantly higher density of sweat pores than control adults (p<0.001) which is explained by the smaller body surface area of the children. However, control adults produced larger sweat volumes than control children (p=0.0014). When stimulated by pilocarpine, sweat glands of children did not equal the function of adult sweat glands, but seem to reach maturity with puberty.

One of the XLHED patients, ED-AS1, reported that despite being unable to sweat as a child, he had realised some sweating for the first time during puberty with further improvement thereafter, and got to know a few regions of his body, including the margins of both palms, where he can sweat. However, XLHED children and adults did not differ in sweat volume, basal or stimulated skin conductance. Therefore, age groups were pooled where appropriate for further comparisons of XLHED and control subjects.

Control subjects had 455 sweat pores/cm 2 on average (range 294-900), which varied by age as expected, and a mean sweat volume of 72 µl following stimulation (range 29-93 µl). There was a clear demarcation between the XLHED males and controls across all measurements of sweat gland function (p<0.0001), that was less consistent in the females studied (Table 3).

However, in 7 of 31 male adults and all female adults with XLHED some sweat was detectable by the Wescor assay. Dermatoglyphic investigation of the graphite prints revealed the complete absence or a significant lack of sweat pores and partial ridge dissociation in the 24 male XLHED patients who were unable to sweat (Figure 1). Palmar skin conductance measurements in these 24 males typically showed both low baseline values and the absence of conductance increase (equivalent to absence of sweat gland function) following capillary blood sampling, whereas in control subjects a transient rise in palmar skin conductance indicated pain-inducible sweat gland function (Figure 1). Among XLHED males (Tables 1 and2), 14/31 had neither detectable sweat pores nor inducible sweating, 15/31 showed a few sweat pores (6-113/cm 2 ) correlating with absence of sweat or low sweat volumes (up to 11.0 µl), and 2/31 had normal sweat pore counts but markedly reduced pilocarpine-stimulated sweat volumes. In general, the number of sweat pores at the palm did not correlate with the ability to sweat. Although no adult male XLHED subject had more than 10 permanent teeth, there was also no correlation between the number of missing teeth and sweat production.

Basal skin conductance, an indirect measure of skin hydration, was reduced (<20.0 µS) in 23 of 24 non-sweating as well as in 2 of 7 low-sweating XLHED males and in 1 of 5 XLHED females, but none of the controls. In healthy subjects, baseline values were lowest in the neonate and two adults known to have dry skin. Similarly, stimulated skin conductance following blood sampling was reduced (<7.0 µS) in 23 of 24 non-sweating XLHED males, 2 of 7 with decreased sweating, 1 of 5 XLHED females, but only 1 of 28 controls. Typically a relatively broad peak was observed within two seconds after the needle-stick associated with capillary blood draw, the maximum of which was taken for the calculation of conductance increase. However, the pain threshold differed remarkably among individuals. In control children, the largest increase was observed, whereas athletic, tall men showed the weakest reaction. Despite large interindividual differences, the skin conductance measurements were reproducible and well tolerated by both age groups. This non-invasive technology, validated for use in newborns, [START_REF] Hernes | Skin conductance changes during the first year of life in full-term infants[END_REF] provided evidence of lack of sweat gland function even in a neonate (subject ED-Ch11) with known family history of XLHED.

Genotype-phenotype correlations

Sequence analysis of the EDA gene in our XLHED cohort revealed a total of 26 different EDA mutations, 11 of which are known from the literature [2-4, 7, 8, 17-19] while [START_REF] Bohring | WNT10A mutations are a frequent cause of a broad spectrum of 13 ectodermal dysplasias with sex-biased manifestation pattern in heterozygotes[END_REF] have not yet been described (Tables 123). None of the missense or splice site mutations was present in a control group of 100 healthy males.

The 25 mutations detected in male XLHED subjects (Figure 2) included 15 mutations expected to cause marked impairment of ectodysplasin A synthesis or release. All patients carrying gross deletions (exon 1, exon 3 or exons 4-9), small out-of-frame deletions (Pro17GlyfsX81, Thr278LeufsX2), or a nonsense mutation (W274X) were anhidrotic. Four EDA mutations altering a splice acceptor (c.527G>T, c.707-2A>G, c.707-2A>T) or splice donor site (c.793+1G>C) were identified, three of which also led to a severe phenotype without any sweating ability (subjects ED-A6, ED-A8, ED-A10). In addition, two small inframe deletions detected in exon 5 (P220_P225del) and exon 9 (N372_S374del) were associated with anhidrosis. Among the 12 missense mutations found in this study, three are known to abolish furin cleavage (R153C, R155C, R156H). These mutations were discovered in non-sweating patients only.

The mutation G291R provided an unexpected example for clear genotype-phenotype correlation; all subjects with this mutation were unable to sweat under test conditions. Moreover, anhidrosis was caused by the mutation G2C within the intracellular domain of ectodysplasin A, by mutations G215R and G224V affecting its collagen triple helix, and by mutation Y304C within the TNF homology domain (Figure 2).

On the other hand, all individuals with the missense mutations V262F, R276C or G299R, which are also situated in the TNF homology domain (Figure 2), showed some residual sweating ability. Another missense mutation outside of a known functional domain (R69L) was associated with moderately reduced sweat gland function and a mild phenotype in general (short stature and low body weight, but dark hair, normal teeth and a face indistinguishable from healthy children).

Interestingly, a small in-frame deletion in exon 5 (N185_P196del) was detected in subject ED-AS1 who was able to produce some sweat.

Discussion

This study demonstrates that sweat collection following pilocarpine iontophoresis on the forearm can discriminate clearly between XLHED males and healthy controls. The sweat pore count also delineated affected males with high sensitivity (94% for children, 92% for adults), whereas skin conductance was less definitive as a diagnostic parameter. Such conductance measurements appear useful as a pediatric screening tool, but have reduced sensitivity in detecting XLHED patients with residual sweat gland function.

Furthermore, the study identified deleterious mutations of the EDA gene resulting in most severe sweat gland dysfunction, whereas other mutations were found to be associated with residual sweating ability.

Sweating could not be induced in any male XLHED subject with an exon deletion, small outof-frame deletion or nonsense mutation in EDA, for all of which premature termination of translation was predicted. Moreover, missense mutations affecting the furin recognition site of ectodysplasin A consistently resulted in anhidrosis. The altered protein sequence at Arg 153 , Arg 155 , or Arg 156 disrupts this recognition site and prevents proteolytic processing and release of the soluble ligand. [START_REF] Chen | Mutations within a furin consensus sequence block proteolytic release of ectodysplasin-A and cause X-linked hypohidrotic ectodermal dysplasia[END_REF] Two other functionally crucial missense mutations (Gly291Arg, Tyr304Cys) and an in-frame deletion (N372_S374del) were identified within the TNF homology domain. As the impact of various mutations on protein structure and receptor binding capacity has been shown in vitro to differ significantly, [START_REF] Schneider | Mutations leading to X-linked hypohidrotic ectodermal dysplasia affect three major functional domains in the tumor necrosis factor family member ectodysplasin-A[END_REF] it is noteworthy that all three subjects carrying the mutation Gly291Arg hemizygously were anhidrotic, suggesting that the affected amino acid may be essential for proper protein folding as already hypothesized. [START_REF] Schneider | Mutations leading to X-linked hypohidrotic ectodermal dysplasia affect three major functional domains in the tumor necrosis factor family member ectodysplasin-A[END_REF] A severe phenotype without any sweating ability would also be expected in all 4 males with splice site mutations in exon 5 of EDA predicted to abolish the regular splice site using the BDGP splice site prediction tool (referenced at the end). Such mutations commonly result in an aberrant, non-functional protein due to out-of-frame exon skipping or use of alternative splice sites, respectively. However, only three of these 4 patients showed a severe phenotype. Surprisingly, subject ED-ChS5 who carries the putative splice-site mutation c.527G>T was able to sweat. In this patient, two of the predicted alternative splice sites within exon 5, which codes mainly for the collagen-like domain, revealed a high acceptor score of 0.67 and 0.83 and would result in an in-frame deletion of 3 or 13 of the 19 Gly-X-Y repeats, respectively.

Although this may explain the milder phenotype seen in this patient, prediction of phenotypic consequences of in-frame deletions within the collagen-like domain seems particularly difficult. FLAG-tagged recombinant EDA-A1 without the collagen domain (EDA-A1 E245) as well as a deletion mutant lacking 4 of the 19 Gly-X-Y repeats (EDA-A1 S160 ∆185-196) have been shown to bind to EDAR equally well in vitro, and the latter did also not affect multimerization. [START_REF] Schneider | Mutations leading to X-linked hypohidrotic ectodermal dysplasia affect three major functional domains in the tumor necrosis factor family member ectodysplasin-A[END_REF] In line with this observation, patient ED-AS1 had a normal number of sweat pores and was able to produce some sweat. In contrast, multimerization was not observed in vitro if the first half of the collagen domain was deleted (in frame), suggesting that certain mutations have specific impact on multimerization. [START_REF] Schneider | Mutations leading to X-linked hypohidrotic ectodermal dysplasia affect three major functional domains in the tumor necrosis factor family member ectodysplasin-A[END_REF] This may provide an explanation for the different findings in subject ED-Ch1 with a small in-frame deletion within the collagen domain as well as for two of our non-sweating patients (ED-Ch3, ED-Ch9) who carry missense mutations altering glycine residues, which leads to an additional interruption of the Gly-X-Y repeats and might disturb protein tri-or multimerization due to reduced solidity of the collagen helix. [START_REF] Monreal | Identification of a new splice form of the EDA1 gene permits detection of nearly all X-linked hypohidrotic ectodermal dysplasia mutations[END_REF] In patients with the missense mutations Val262Phe, Arg276Cys or Gly299Arg in the TNF homology domain, sweat gland function was affected only moderately. Although one of these mutations (Gly299Arg) is located within the most homologous motif of TNF ligands (Leu 293 -Val 309 ), [START_REF] Ezer | Ectodysplasin is a collagenous trimeric type II membrane protein with a tumor necrosis factor-like domain and co-localizes with cytoskeletal structures at lateral and apical surfaces of cells[END_REF] a residual, probably tissue-specific receptor binding activity cannot be excluded and has already been demonstrated for other mutations within the TNF homology domain. [START_REF] Mues | Functional analysis of Ectodysplasin-A mutations causing selective tooth agenesis[END_REF] The missense mutation Arg69Leu, situated outside of a known functional domain but close to the extracellular end of the transmembrane domain, led to a mild phenotype including some sweating ability which may, however, not be sufficient to preclude overheating. This mutation may reduce expression or translocation, and therefore dosage, of otherwise normal ligand, as proposed for two similar mutations (Arg65Gly [START_REF] Tao | A novel missense mutation of the EDA gene in a Mongolian family with congenital hypodontia[END_REF], Glu67Val [START_REF] Fan | Mutations in the EDA gene are responsible for X-linked hypohidrotic ectodermal dysplasia and hypodontia in Chinese kindreds[END_REF]) associated with an oligosymptomatic phenotype (hypodontia). [START_REF] Mues | Functional analysis of Ectodysplasin-A mutations causing selective tooth agenesis[END_REF] However, the mutation Arg69Leu was also detected in patients with apparently typical XLHED, [START_REF] Schneider | Mutations leading to X-linked hypohidrotic ectodermal dysplasia affect three major functional domains in the tumor necrosis factor family member ectodysplasin-A[END_REF][START_REF] Vincent | Mutational spectrum of the ED1 gene in X-linked hypohidrotic ectodermal dysplasia[END_REF][START_REF] Kere | X-linked anhidrotic (hypohidrotic) ectodermal dysplasia is caused by mutation in a novel transmembrane protein[END_REF] making the prediction of its phenotypic consequences rather difficult.

In conclusion, we have shown that within the EDA gene gross deletions, splice site mutations expected to lead to exon skipping or use of alternative splice sites, nonsense mutations, missense mutations altering furin cleavage as well as mutations affecting specific amino acids of ectodysplasin-A (e.g. glycine residues within the Gly-X-Y repeats or Gly 291 ) are associated with severely compromised sweat gland function, which is a major determinant of XLHEDassociated morbidity and mortality. Genotype-phenotype correlation was seen across all measurements of sweat gland function, but, surprisingly, not with respect to the number of sweat glands (which can be determined very soon after birth) and the degree of oligodontia (diagnosable already by prenatal ultrasonography). This is important since such easily recognizable signs obviously do not allow prediction of the risk of hyperthermia and the associated morbidity and mortality in XLHED patients.

Our results imply that systematic mapping of EDA mutations together with the analysis of objective clinical data may help to distinguish functionally crucial mutations from those allowing residual, possibly tissue-specific activity of the EDA-A1 protein. Furthermore, these data might be useful to improve our understanding of ectodysplasin A function and signaling and to identify putative sites of interaction with other proteins.

Web Resources

The URLs for data presented herein are as follows:

BDGP splice site prediction: http://www.

Licence for Publication statement

The Corresponding Author has the right to grant on behalf of all authors and does grant on behalf of all authors, an exclusive licence (or non exclusive for government employees) on a worldwide basis to the BMJ Publishing Group Ltd to permit this article (if accepted) to be published in JMG and any other BMJPGL products and sublicences such use and exploit all subsidiary rights, as set out in our licence (http://group.bmj.com/products/journals/instructions-for-authors/licenceforms).

Table 1. EDA mutations and clinical symptoms in male children with XLHED and age-matched controls a novel mutations in bold type; b upper detection limit = 93 µl; c within a 1-cm 2 area of the palm; SD, standard deviation

Figure 1 :

 1 Figure 1: Examples of graphite prints and skin conductance measurements. Graphite prints of 16-mm 2 areas of the palm (outlined in black) from a boy with XLHED (a) and a healthy control subject (b) were magnified 16-fold. Sweat pores can be recognized as circular white dots. Skin conductance measurements at the same site show significantly reduced baseline values and indicate absence of sweating after stimulation of the sympathetic nervous system in the XLHED patient (c). The time period of 60 seconds following such stimulation is indicated by gray color. In the control subject, pain-induced sweating leads to a temporary rise in palmar skin conductance (d).

Figure 2 :

 2 Figure 2: Distribution of EDA mutations among 31 male XLHED patients. Numbers of triangles (▼▲) and circles (•) indicate numbers of patients with this mutation. Triangles are used for patients unable to sweat, circles for patients able to sweat. For exon deletions, exact breakpoints were not determined. The position and extension of smaller deletions is indicated by black bars. Predicted splice site mutations: *c.527G>T, **c.707-2A>G and A>T, ***c.793+1G>C

  fruitfly.org/seq_tools/splice.html Genbank: http://www.ncbi.nlm.nih.gov/sites/entrez?db=nucleotide MFOLD: http://mobyle.pasteur.fr/cgi-bin/MobylePortal/portal.py?form=mfold

	Online Mendelian Inheritance in Man (OMIM): http://www.ncbi.nlm.nih.gov/Omim/
	Primer3: http://frodo.wi.mit.edu/primer3/input.htm
	SNPCheck: http://ngrl.man.ac.uk/SNPCheck/SNPCheck.html

Acknowledgements

We thank Stephane Karoubi, Ellen Gattner, Elisabeth Koppmann and Ramsey Johnson for excellent technical assistance and would like to express our gratitude to all individuals who participated in this study.

Code

Age (years)

Competing interests statement:

This study was supported partially by the German-Swiss-Austrian ectodermal dysplasia patient support group and by a grant from Edimer Pharmaceuticals (Cambridge, U.S.). One of the authors (K.H.) is an employee of Edimer Pharmaceuticals. He contributed to study design and interpretation of final results. Otherwise the sponsor was neither involved in collection, analysis or interpretation of data, nor in the writing of the report or the decision to submit it for publication. The first draft of this manuscript was written by H.S. and A.B.. None of the authors has been paid to produce this article.


Judith Drenthen
email: j.drenthen@erasmusmc.nl

Nobuhiro Yuki

Jan Meulstee

Ellen M Maathuis

Doorn Van

Gerhard H Visser

Joleen H Blok

Bart C Jacobs

Pieter A Van Doorn

Guillain-Barré syndrome subtypes related to Campylobacter infection

Keywords: Guillain-Barré syndrome, EMG, Cytomegalovirus, Epstein-Barr virus, Campylobacter jejuni

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Guillain-Barré syndrome (GBS) is a post-infectious polyradiculoneuropathy with variable clinical and electrophysiological subtypes. GBS is commonly subdivided into acute inflammatory demyelinating polyneuropathy (AIDP), acute motor axonal neuropathy (AMAN), and acute motor sensory axonal neuropathy (AMSAN). There is a remarkable difference in geographical distribution of these subtypes. [START_REF] Ogawara | Axonal Guillain-Barre syndrome: relation to anti-ganglioside antibodies and Campylobacter jejuni infection in Japan[END_REF] The axonal variants abound in Asia, while AIDP is predominant in western countries. [START_REF] Ogawara | Axonal Guillain-Barre syndrome: relation to anti-ganglioside antibodies and Campylobacter jejuni infection in Japan[END_REF][START_REF] Ho | Guillain-Barre syndrome in northern China. Relationship to Campylobacter jejuni infection and anti-glycolipid antibodies[END_REF][START_REF] Hadden | Electrophysiological classification of Guillain-Barre syndrome: clinical associations and outcome. Plasma Exchange/Sandoglobulin Guillain-Barre Syndrome Trial Group[END_REF] This geographical variation is unexplained, but may be related to differences in genetic or environmental factors. Previous studies indicated that preceding infections may influence the electrophysiological subtype of GBS. [START_REF] Ang | Differential immune response to gangliosides in Guillain-Barre syndrome patients from Japan and The Netherlands[END_REF][START_REF] Hadden | Preceding infections, immune factors, and outcome in Guillain-Barre syndrome[END_REF][START_REF] Visser | Cytomegalovirus infection and Guillain-Barre syndrome: the clinical, electrophysiologic, and prognostic features. Dutch Guillain-Barre Study Group[END_REF] More specifically, a recent study proposed that in Japan infections with Campylobacter jejuni are exclusively related to AMAN. [START_REF] Kuwabara | Does Campylobacter jejuni infection elicit "demyelinating" Guillain-Barre syndrome?[END_REF] The high incidence of C. jejuni infections in GBS patients in Japan would explain why axonal variants predominate there.

Patients from western countries most frequently report symptoms of preceding respiratory tract infections. Indeed, serological studies have yielded evidence for infections with cytomegalovirus (CMV), Epstein-Barr virus (EBV) and Mycoplasma pneumoniae. [START_REF] Hadden | Preceding infections, immune factors, and outcome in Guillain-Barre syndrome[END_REF] Nevertheless, in these regions C. jejuni is the predominant type of infection preceding GBS, that may affect >30% of patients. [START_REF] Jacobs | The spectrum of antecedent infections in Guillain-Barre syndrome: a case-control study[END_REF][START_REF] Hughes | Guillain-Barre syndrome[END_REF] AMAN in western countries is rare (<10%), [START_REF] Hadden | Electrophysiological classification of Guillain-Barre syndrome: clinical associations and outcome. Plasma Exchange/Sandoglobulin Guillain-Barre Syndrome Trial Group[END_REF][START_REF] Meulstee | Electrodiagnostic criteria for polyneuropathy and demyelination: application in 135 patients with Guillain-Barre syndrome. Dutch Guillain-Barre Study Group[END_REF] however, suggesting that C. jejuni in western patients may also induce other GBS subtypes.

Here, we defined what subtypes of GBS may occur after C. jejuni infection in the Netherlands. C. jejuni infections were defined by specific techniques, including those used in Japanese studies, to exclude methodological bias. Patients with C. jejuni infections were assessed in serial electrophysiological studies and compared with patients with preceding viral infections.

PATIENTS AND METHODS

Patients

The patients for our study were derived from a cohort of 147 GBS patients who participated in a randomised trial comparing the therapeutic effect of intravenous immunoglobulins (IVIg) versus plasmapheresis. [START_REF] Van Der Meche | A randomized trial comparing intravenous immune globulin and plasma exchange in Guillain-Barre syndrome. Dutch Guillain-Barre Study Group[END_REF] All patients fulfilled the diagnostic criteria for GBS, were admitted to the hospital within two weeks of onset of weakness, and were unable to walk ten meters independently (GBS disability score > 2). [START_REF] Asbury | Assessment of current diagnostic criteria for Guillain-Barre syndrome[END_REF] During a follow-up of six months, the patients underwent neurological assessments at standardised time points according to a predefined protocol. During these visits, clinical parameters such as sensory abnormalities, cranial nerve involvement, GBS disability scales, and MRC sum scores were determined. [START_REF] Kleyweg | Interobserver agreement in the assessment of muscle strength and functional abilities in Guillain-Barre syndrome[END_REF][START_REF] Merkies | Connecting impairment, disability, and handicap in immune mediated polyneuropathies[END_REF] Informed consent was obtained from all patients and the study was approved by the local medical ethics committee. Data from serial electrophysiological studies and pretreatment serum samples were available from 123 (84%) of the 147 patients. These patients were not significantly different from the excluded 24 cases regarding baseline and clinical characteristics.

To confirm our findings in patients with positive C. jejuni serology in this cohort, we analysed patients with positive C. jejuni stool cultures from a second cohort who participated in a trial studying the effect of combined IVIg and methylprednisolone treatment. [START_REF] Van Koningsveld | Effect of methylprednisolone when added to standard treatment with intravenous immunoglobulin for Guillain-Barre syndrome: randomised trial[END_REF] For this later cohort, stool specimens were available, allowing isolation of C. jejuni bacteria directly from these specimens (the gold standard for identification of a recent C. jejuni infection). In this second cohort, eight (4%) of 225 GBS patients were identified with a positive stool culture for C. jejuni. All eight patients of this second cohort fulfilled the diagnostic criteria for GBS, underwent an extensive follow-up, and had an EMG within three weeks after randomisation. [START_REF] Van Koningsveld | Effect of methylprednisolone when added to standard treatment with intravenous immunoglobulin for Guillain-Barre syndrome: randomised trial[END_REF] Electrophysiology Electrophysiological findings vary considerably in the acute stage of GBS. This dynamic situation may lead to changes in classification of patients depending on the timing of the EMG. Therefore, in this study, serial EMG studies were performed at three standard time points: within two days of hospital admission, after one week, and after one month. [START_REF] Meulstee | Electrodiagnostic criteria for polyneuropathy and demyelination: application in 135 patients with Guillain-Barre syndrome. Dutch Guillain-Barre Study Group[END_REF] Kuwabara et al. categorized their EMGs based on the time elapsed since the onset of weakness rather than the time of admission. [START_REF] Kuwabara | Does Campylobacter jejuni infection elicit "demyelinating" Guillain-Barre syndrome?[END_REF] To be able to compare our study with this previous study, we chose to analyse the EMGs that were made in the second week after onset of weakness (median 11 days, 95% CI 10-12 days) and more than 4 weeks after onset of weakness (median 33 days, 95% CI 32-38 days).

Motor and sensory nerve conduction studies were performed on the ulnar and median nerves.

Stimulation and recordings were performed at standardised sites. [START_REF] Van Der Meche | Patterns of conduction failure in the Guillain-Barre syndrome[END_REF][START_REF] Buschbacher | Manual of nerve conduction studies[END_REF] Recordings from motor nerves were quantified using peak-to-peak distal and proximal compound motor action potential amplitudes (dCMAP and pCMAP), distal motor latency (DML), motor nerve conduction velocity (mNCV), and F-wave latencies. For sensory nerves, the peak-to-peak sensory nerve action potential (SNAP) amplitude and sensory nerve conduction velocity (sNCV) were measured. Needle EMG was performed in the anterior tibial muscle, abductor pollicis brevis muscle or the abductor digiti minimi muscle to determine the presence of denervation activity (fibrillation potentials and positive sharp waves). In all investigations, the electromyographers were blinded for the results of infection serology. The EMGs were classified as demyelinating, axonal, inexcitable, equivocal, or normal according to the Hadden criteria. [START_REF] Hadden | Electrophysiological classification of Guillain-Barre syndrome: clinical associations and outcome. Plasma Exchange/Sandoglobulin Guillain-Barre Syndrome Trial Group[END_REF] : Normal: no abnormalities in any single nerve. Primary demyelinating: at least one of the following in each of at least two nerves, or at least two of the following in one nerve if all others are inexcitable and dCMAP >10% of lower limit of normal (LLN): DML > 110% Reference values were defined previously. [START_REF] Buschbacher | Manual of nerve conduction studies[END_REF] The temperature of the recording-and stimulation sites were registered.

Serological studies

Pre-treatment serum samples were examined for evidence of a recent C. jejuni infection using tests from both Japanese and Dutch laboratories (Dokkyo Medical University in Tochigi, Japan, and SSDZ, Delft, The Netherlands). [START_REF] Ang | Validation of an ELISA for the diagnosis of recent Campylobacter infections in Guillain-Barre and reactive arthritis patients[END_REF][START_REF] Koga | Comparative study of preceding Campylobacter jejuni infection in Guillain-Barre syndrome in Japan and The Netherlands[END_REF] Both tests were based on an ELISA, and have been described in detail previously. [START_REF] Koga | Close association of IgA anti-ganglioside antibodies with antecedent Campylobacter jejuni infection in Guillain-Barre and Fisher's syndromes[END_REF][START_REF] Herbrink | Human serum antibody response in Campylobacter jejuni enteritis as measured by enzyme-linked immunosorbent assay[END_REF] In the Japanese laboratory, serological evidence of a recent C. jejuni infection was defined as anti-C. jejuni IgG titres of 2,000 or more; in the Dutch laboratory as the presence of anti-C. jejuni IgM or IgA antibodies at higher levels than an internal standard. [START_REF] Jacobs | The spectrum of antecedent infections in Guillain-Barre syndrome: a case-control study[END_REF] Differences between the Japanese and Dutch tests were described previously. [START_REF] Koga | Comparative study of preceding Campylobacter jejuni infection in Guillain-Barre syndrome in Japan and The Netherlands[END_REF] Both tests were used in the current study to exclude the possibility that the serology method introduced a bias. Patients were regarded C. jejuni-positive if they:

(1) suffered from diarrhea in the three weeks preceding the onset of weakness, and (2) were tested positive for C. jejuni serology in either one of the two independent laboratories. The C. jejuni-related patients were compared with a group of GBS patients from the same trial cohort who had a positive serology for a recent infection with CMV or EBV according to previously defined criteria. [START_REF] Jacobs | The spectrum of antecedent infections in Guillain-Barre syndrome: a case-control study[END_REF] Furthermore, the pretreatment serum samples were tested for IgM and IgG antibody reactivity against GM1 and GD1a (in The Netherlands), and against GM1, GM1b, and GalNAc-GD1a (in Japan) using ELISA as previously described. [START_REF] Jacobs | Campylobacter jejuni infections and anti-GM1 antibodies in Guillain-Barre syndrome[END_REF][START_REF] Jacobs | Serum anti-GQ1b IgG antibodies recognize surface epitopes on Campylobacter jejuni from patients with Miller Fisher syndrome[END_REF]

Statistical analysis

Normally distributed data were analysed using the two-sided t-test for independent samples. Ordinal data and not normally distributed data were analysed using the two-sided Wilcoxon rank sum score test. The χ 2 test or Fisher exact test was used for categorical data depending on sample size. Data are presented as means and standard deviations (SD) or as medians and 95% confidence intervals (95% CI). For categorical variables, frequencies and percentages are given. All calculations were performed using SPSS 15.0 (SPSS Inc, Chicago, IL). A p-value of < 0.05 was considered to be statistically significant.

RESULTS

Comparison of patients with C. jejuni-versus CMV/EBV-related GBS

The electrophysiology of the patients with positive C. jejuni serology was compared with that of patients with a positive CMV or EBV serology. Seventeen (14%) of the cohort of 123 GBS patients met our strict criteria for a recent infection with C. jejuni (diarrhea and positive C. jejuni serology). In the same cohort of patients 24 (20%) had a positive serology for either CMV or EBV. The electrophysiological characteristics of the CMV-and EBV-related patients were very similar; therefore, both groups were combined for further analysis. The mean age of the C. jejuni-related patients (56 ± 17 years) was significantly higher than of the CMV/EBV-related patients (36 ± 15years) (p < 0.001). Sensory deficits were less frequent in the C. jejuni-related patients (41%) compared to the CMV/EBV-related patients (78%) (p = 0.02). The median MRC sum score after six months was lower in the C. jejuni-related patients (44 with 95% CI 31-58) than in the CMV/EBV-related patients (58 with 95% CI 52-60) (p = 0.004). The median GBS disability score at six months was significantly higher for the C. jejuni-related patients (3 with 95% CI 1-4) compared to the CMV/EBV-related patients (1 with 95% CI 1-2) (p = 0.02), indicating a poorer outcome. Anti-ganglioside antibodies were present in 88% of the C. jejuni-related patients and in 4% of the CMV/EBV-related patients (p < 0.001).

The two groups differed considerably with respect to electrophysiological features and subtypes, as shown in Table 1. The C. jejuni-related patients had significantly lower CMAP amplitudes and lower DMLs, but higher SNAP amplitudes. These differences were found in both the ulnar and median nerves, although the differences in the ulnar nerves were more pronounced.

The frequency of denervation activity (fibrillation potentials and/or positive sharp waves) in the acute phase and at 4 weeks did not differ significantly between the two groups. Within 2 weeks, 20% of the C. jejuni patients and 21% of the CMV/EBV patients showed denervation activity (p=0.95). After 4 weeks, 69% of the C. jejuni patients versus 56% of the CMV/EBV patients showed active denervation (p=0.44).

In five patients from the CMV/EBV-related group, an EMG was performed only within two weeks and not after four weeks. At four weeks, the majority of EMGs in the C. jejuni-related group were classified as the axonal type or as inexcitable, but two patients (12%) had a demyelinating type of EMG. By contrast, at that time 85% of the EMGs of the CMV/EBVrelated patients were classified as demyelinating.

The distal temperature during all EMG registrations was not decreased, except for two Nerves with a distal CMAP (dCMAP) < 10% LLN did not change the classification.

Classification of patients with positive C. jejuni serology

Nine (53%) of the 17 patients with positive C. jejuni serology tested positive in both the Dutch and the Japanese assay; the other eight patients tested positive in the Dutch assay only.

These two subgroups were indistinguishable in terms of clinical and electrophysiological characteristics and, therefore, were combined for further analysis. Table 2 provides an overview of the electrophysiological and serological data for all C. jejuni-serology positive patients.

Three of these 17 patients with positive C. jejuni serology fulfilled the electrophysiological criteria of Hadden for demyelination at one or both time points (Table 3). Four weeks after the onset of weakness, when most patients have reached a stable situation, two of these three patients fulfilled the criteria for demyelination. The first patient (study number 6) also had high titers of IgG antibodies to GM1 in the Japanese and Dutch assays, normal SNAPs, and a purely motor form of GBS. Needle EMG did not show any signs of axonal damage in the form of fibrillation potentials and/or positive sharp waves. The second patient (study number 15) had high titers of IgG antibodies to GM1 in the Dutch assay and of GM1, GM1b and GalNAc-GD1a in the Japanese assay. This patient had abnormal SNAPs. Needle EMG also showed signs of axonal involvement (fibrillation and positive sharp waves). The third patient (study number 16), tested negative for anti-ganglioside antibodies, had abnormal SNAPs, and no denervation potentials in needle EMG.

Classification of patients with positive C. jejuni-stool culture

Two (25%) of the eight patients from the second cohort with a stool sample that was positive for C. jejuni fulfilled the demyelination criteria (Table 3). Patient A was a 63-year-old male who was admitted with a progressive burning sensation and paresthesias in hands and feet and, one day later, progressive muscle weakness. His EMG was performed 19 days after weakness onset and showed prolonged DMLs in combination with low NCVs, a conduction block, and absent SNAPs. Needle EMG showed no denervation activity. He had a positive serology for IgG anti-GM1 antibodies. Patient B was a 36-year-old female, with pronounced muscle weakness and mild sensory disturbance. Her EMG was performed 12 days after weakness onset and showed prolonged DMLs in median, ulnar and peroneal nerve, an abnormal mNCV in the median nerve, and abnormal SNAPs of median and ulnar nerve.

Needle EMG showed no denervation activity. No serum antibodies to gangliosides were found. To be able to clarify which GBS subtypes can be elicited by a C. jejuni infection, our first priority was to prevent the inclusion of false-positive C. jejuni cases. For that reason, an extensively validated serological assay was used, which has a sensitivity of 96% and a specificity of 93% for detecting a recent C. jejuni infection in patients with GBS. [START_REF] Ang | Validation of an ELISA for the diagnosis of recent Campylobacter infections in Guillain-Barre and reactive arthritis patients[END_REF] To further increase the specificity, we added the criterium of presence of preceding diarrhea. This combination of strict criteria explains the low frequency of C. jejuni-related cases compared to previous studies. [START_REF] Hadden | Preceding infections, immune factors, and outcome in Guillain-Barre syndrome[END_REF][START_REF] Jacobs | Campylobacter jejuni infections and anti-GM1 antibodies in Guillain-Barre syndrome[END_REF][START_REF] Rees | Campylobacter jejuni infection and Guillain-Barre syndrome[END_REF] The specificity of the C. jejuni serology may differ among assays. [START_REF] Ho | Guillain-Barre syndrome in northern China. Relationship to Campylobacter jejuni infection and anti-glycolipid antibodies[END_REF][START_REF] Koga | Comparative study of preceding Campylobacter jejuni infection in Guillain-Barre syndrome in Japan and The Netherlands[END_REF] To make our results as comparable as possible to previous studies from Japan, [START_REF] Kuwabara | Does Campylobacter jejuni infection elicit "demyelinating" Guillain-Barre syndrome?[END_REF] serological testing for C. jejuni was performed in the Netherlands as well as in Japan. The findings that anti-ganglioside antibodies were almost exclusively found in the patients with positive C. jejuni serology, lend further support to a high specificity of the assays. Furthermore, in two GBS patients with a positive stool culture for C. jejuni, the gold standard for identifying a recent infection, demyelinating forms were identified. Taken together, it is highly unlikely that the observed coincidence of C. jejuni infection and AIDP in five out of the total 25 C. jejuni-positive patients can be explained by false-positive cases of C. jejuni.

+ + + + CB - - - - - - - - Ulnar nerve mNCV (m/s) 58 (255%) 7.6 (173%) 4.8 (102%) 6.0 (136%) F-wave (ms) - - § § 54.9 (164%) 37.3 (113%) - 39.5 (128%) abnormal SNAP - - + +
The second issue that needs to be addressed in this context is whether the electrophysiological features observed in these five patients can be attributed rightly to the process of demyelination. In the current study we have used the criteria of Hadden et al. [START_REF] Hadden | Electrophysiological classification of Guillain-Barre syndrome: clinical associations and outcome. Plasma Exchange/Sandoglobulin Guillain-Barre Syndrome Trial Group[END_REF] These criteria were developed in Caucasian GBS patients, but are similar to the criteria of Ho et al., [START_REF] Ho | Guillain-Barre syndrome in northern China. Relationship to Campylobacter jejuni infection and anti-glycolipid antibodies[END_REF] that were used in the previous study from Japan. [START_REF] Kuwabara | Does Campylobacter jejuni infection elicit "demyelinating" Guillain-Barre syndrome?[END_REF] All patients in the current study that were classified as AIDP with the Hadden's criteria, including the two C. jejuni-culture positive patients, also fulfilled these criteria for AIDP of Ho et al. Over the years, multiple sets of electrophysiological (research) criteria have been developed to identify demyelination in GBS patients. [START_REF] Ho | Guillain-Barre syndrome in northern China. Relationship to Campylobacter jejuni infection and anti-glycolipid antibodies[END_REF][START_REF] Hadden | Electrophysiological classification of Guillain-Barre syndrome: clinical associations and outcome. Plasma Exchange/Sandoglobulin Guillain-Barre Syndrome Trial Group[END_REF][START_REF] Meulstee | Electrodiagnostic criteria for polyneuropathy and demyelination: application in 135 patients with Guillain-Barre syndrome. Dutch Guillain-Barre Study Group[END_REF][START_REF] Albers | Sequential electrodiagnostic abnormalities in acute inflammatory demyelinating polyradiculoneuropathy[END_REF][START_REF] Cornblath | Electrophysiology in Guillain-Barre syndrome[END_REF][START_REF] Italian | The prognosis and main prognostic indicators of Guillain-Barre syndrome. A multicentre prospective study of 297 patients[END_REF] No set is universally accepted, however, and no comparative studies with nerve biopsies are available for GBS including the current study. Furthermore, a few studies have shown the existence of conduction blocks, conduction slowing, and increased DMLs in AMAN patients. [START_REF] Capasso | Acute motor conduction block neuropathy Another Guillain-Barre syndrome variant[END_REF][START_REF] Uncini | Electrophysiologic and immunopathologic correlates in Guillain-Barre syndrome subtypes[END_REF] These phenomena were presumably caused by alteration of the resting membrane potential and sodium channel inactivation. Along the same lines, it should be noted that three of the five C. jejuni-positive patients with AIDP in our study had high titres of serum IgG antibodies to GM1 or other gangliosides. Only one of these patients also had signs of axonal involvement in the form of fibrillation potentials and positive sharp waves on EMG. Usually, these antibodies are strongly associated with axonal degeneration. [START_REF] Jacobs | Campylobacter jejuni infections and anti-GM1 antibodies in Guillain-Barre syndrome[END_REF][START_REF] Yuki | Acute axonal polyneuropathy associated with anti-GM1 antibodies following Campylobacter enteritis[END_REF] Possibly, these findings point to the existence of an as yet unrecognized subtype, which merges axonal and demyelinating characteristics on EMG. Within the limits of the currently available criteria (whether those of Hadden, Ho or others), however, the EMGs of the five patients unmistakably show primary demyelinating features. Moreover, the majority had marked sensory abnormalities on the EMG, which are very uncommon in AMAN. [START_REF] Uncini | Electrophysiologic and immunopathologic correlates in Guillain-Barre syndrome subtypes[END_REF] The current study is based on the EMGs of median and ulnar nerves, the only two motor nerves that were studied in this patient cohort systematically. In the study of Kuwabara et al, four motor nerves were examined. [START_REF] Kuwabara | Does Campylobacter jejuni infection elicit "demyelinating" Guillain-Barre syndrome?[END_REF]. The current criteria require the presence of a demyelinating feature in at least two nerves, which enabled us to classify patients with AIDP.

Performing EMG in a third (or fourth) nerve will only increase the sensitivity to detect demyelination.

A third potential confounder for the subtyping of GBS patients is the timing of the EMG, since the electrophysiological findings vary considerably in the acute stage of GBS. [START_REF] Cornblath | Electrophysiology in Guillain-Barre syndrome[END_REF] This dynamic situation may lead to changes in the classification of patients during follow-up, as was found in our study as well as that of Japan. [START_REF] Kuwabara | Does Campylobacter jejuni infection elicit "demyelinating" Guillain-Barre syndrome?[END_REF] Four weeks after weakness onset, the situation may be expected to have stabilized. Even at that time, some of our patients fulfilled the electrophysiological criteria for demyelination. In this context, it should also be noted that none of our C. jejuni patients with a demyelinating EMG had the previously reported quickly reversible conduction blocks and increased DMLs that normalize within two weeks after weakness onset. [START_REF] Kuwabara | Does Campylobacter jejuni infection elicit "demyelinating" Guillain-Barre syndrome?[END_REF] More extensive studies are required to establish the relations between the geographical and ethnical origin of the patient, the type of preceding infections, and the subsequent electrophysiological and clinical phenotype. If the patient's origin indeed influences the phenotype of GBS after infection with C. jejuni, this raises the question which factors could explain such an association. Previous studies showed that the type of C. jejuni strain and related ganglioside mimicry or additional virulence factors highly influence the subsequent neurological deficits. [START_REF] Nachamkin | Campylobacter species and Guillain-Barre syndrome[END_REF] Possibly, these characteristics vary between C. jejuni strains derived from different geographical areas. In addition to these pathogen factors, host factors may codetermine the phenotype by influencing the specificity and type of immune response triggered by a preceding C. jejuni infection. Alternatively, the antigenic make-up of the peripheral nervous system or accessibility of these targets to the immune system may depend on the origin of the host. Various genetic factors are associated with disease occurrence and severity. [START_REF] Yuki | HLA-B35 and acute axonal polyneuropathy following Campylobacter infection[END_REF][START_REF] Yuki | Association of Campylobacter jejuni serotype with antiganglioside antibody in Guillain-Barre syndrome and Fisher's syndrome[END_REF][START_REF] Geleijns | Mannose-binding lectin contributes to the severity of Guillain-Barre syndrome[END_REF] Extensive international collaboration and comparisons of large and standardized cohorts of GBS patients from various geographical and ethnical origins are required to resolve these issues. In addition, the current classification into demyelinating and axonal subtypes based on EMG characteristics may be too crude to account for the diversity in clinical, serological, and electrophysiological findings. A more refined classification of patients may lead to more targeted and individualized therapies.

  patients. Patient number 11 in the C. jejuni-related group had a temperature at the wrist of 29 o C and was classified as axonal in the first EMG and as equivocal in the second EMG. One patient from the CMV/EBV-related group had a temperature at the wrist of 29 o C and had normal electrophysiology at both time points.

  Electrophysiological findings in patients with positive C. jejuni serology (patient 6, 15 and 16) or stool culture (patient A and B). Absolute values of nerve conduction velocity (NCV), distal motor latency (DML) and F-wave latency are shown. Between brackets are given the percentages of lower limits of normal (LLN) for mNCV and percentages of the upper limit of normal (ULN) for DML and F-waves. Normal values (and related LLN and ULN) depend on sex, age, and length. Abnormal SNAP amplitude and presence of conduction blocks (CB) are indicated by '-' and '+'. At the time points in bold face (second row), the EMGs were classified as demyelinating. The numbers in bold face represent the abnormalities in the "demyelinating range". [3] Nerves with a distal CMAP (dCMAP) < 10% did not change the classification. § = dCMAP < 10% LLN DISCUSSION The current study confirmed the previous study of Kuwabara et al. that in GBS patients, apreceding infection with C. jejuni is frequently followed by AMAN, even in western countries where this axonal subtype is relatively rare.[START_REF] Hadden | Electrophysiological classification of Guillain-Barre syndrome: clinical associations and outcome. Plasma Exchange/Sandoglobulin Guillain-Barre Syndrome Trial Group[END_REF] The majority of Dutch patients with a preceding C. jejuni infection developed a variant with inexcitable nerves or features of axonal degeneration in serial EMG studies. In contrast, Japanese and Dutch patients with a preceding EBV or CMV infection rarely develop AMAN, indicating that preceding infections influence the electrophysiological subtype of GBS. However, even when very strict clinical and serological criteria for a recent infection with C. jejuni were applied, three (18%) of 17 GBS patients with a C. jejuni fulfilled the electrophysiological criteria for AIDP. Furthermore, in another cohort, C. jejuni was cultured from the stools of two patients with a typical demyelinating form of GBS. These findings demonstrate that at least in Dutch patients, C. jejuni infections do not exclusively elicit the AMAN variant of GBS. This finding is in accordance with the relatively high frequency of both C. jejuni infections and AIDP in the Netherlands.

Table 1 Electrophysiological findings in the ulnar nerve of GBS patients with positive C. jejuni serology versus GBS patients with positive CMV/EBV serology.

 1 Classification criteria according to Hadden et al.[3]:Normal: no abnormalities in any single nerve.Demyelinating: one of the following in two nerves (or two of the following in one nerve if all

		C. jejuni (n = 17)	CMV/EBV (n = 24)*	p-value
	EMG < 2weeks
	Ulnar CMAP (mV)	0.3 (0.0-3.0)	4.0 (1.9-5.0)	0.01
	Ulnar SNAP (µV)	10.0 (3.6-18.5)	1.9 (0-8.0)	0.02
	Ulnar mNCV (m/s)	52 (46-60)	47 (42-53)	0.08
	Ulnar DML (ms)	3.4 (2.7-4.6)	4.6 (3.9-5.7)	0.04
	Classification (%)			< 0.001
	Demyelinating	2 (12%)	21 (88%)
	Axonal	5 (29%)	1 (4%)
	Inexcitable	4 (24%)	0 (0%)
	Equivocal	6 (35%)	1 (4%)
	Normal	0 (0%)	1 (4%)
	EMG > 4weeks
	Ulnar CMAP (mV)	0.4 (0.0-3.5)	5.4 (2.0-10.0)	0.01
	Ulnar SNAP (µV)	16.5 (4.0-26.0)	2.5 (0.0-6.3)	0.01
	Ulnar mNCV (m/s)	49 (37-59)	44 (39-50)	0.12
	Ulnar DML (ms)	3.6 (2.8-5.0)	5.6 (4.3-7.1)	0.003
	Classification (%)			< 0.001
	Demyelinating	2 (12%)	16 (85%)
	Axonal	5 (29%)	0 (0%)
	Inexcitable	4 (24%)	1 (5%)
	Equivocal	6 (35%)	1 (5%)
	Normal	0 (0%)	1 (5%)
	The values for the ulnar CMAP, SNAP, DML and mNCV are presented as medians (and 95%
	CI between brackets). EMGs were made within two weeks (median 11 days) and after 4
	weeks (median 33 days) after weakness onset. Classifications are presented as numbers (and
	percentages between brackets).

* EMG at > 4 weeks was missing in five (21%) of 24 CMV-EBV-related GBS patients.

Inexcitable: dCMAP absent in all nerves (or present in 1 nerve with dCAMP<10% LLN). , Equivocal: does not fit criteria for any other group.

Table 2 Electrophysiological classification of GBS patients with diarrhea and positive C. jejuni serology.

 2

	Nr	C. jejuni serology	Anti-ganglioside	EMG classification
					antibodies
		Dutch	Japa-	Dutch	Japanese	< 2 weeks	> 4 weeks
			nese
	1	+	+	GM1	GM1	Axonal	Axonal
	2	+	+	GM1	GM1, GM1b,	Axonal	Axonal
					GalNAc-GD1a
	3	+	+	-	GM1b,	Axonal	Equivocal
					GalNAc-GD1a
	4	+	+	GM1,	GM1, GM1b,	Equivocal	Axonal
				GD1a	GalNAc-GD1a
	5	+	+	GD1a	GM1, GM1b,	Equivocal	Equivocal
					GalNAc-GD1a
	6	+	+	GM1	GM1, GM1b	Equivocal	Demyelinating
	7	+	+	GM1	GM1, GM1b	Inexcitable	Inexcitable
	8	+	+	GM1	GM1, GM1b,	Inexcitable	Inexcitable
					GalNAc-GD1a
	9	+	+	GM1	GM1, GM1b	Inexcitable	Inexcitable
	10	+	-	-	GalNAc-GD1a	Axonal	Axonal
	11	+	-	GM1	GM1b,	Axonal	Equivocal
					GalNAc-GD1a
	12	+	-	-	GM1b,	Equivocal	Axonal
					GalNAc-GD1a
	13	+	-	-	-	Equivocal	Equivocal
	14	+	-	-	GM1b	Equivocal	Equivocal
	15	+	-	GM1	GM1, GM1b,	Demyelinating	Demyelinating
					GalNAc-GD1a
	16	+	-	-	-	Demyelinating	Equivocal
	17	+	-	GM1,	Not tested	Inexcitable	Inexcitable
				GD1b

Inexcitable: dCMAP absent in all nerves (or present in 1 nerve with dCAMP<10% LLN).

Equivocal: does not fit criteria for any other group.

Table 3 Patients positive for C. jejuni-serology or -culture fulfilling the criteria for a demyelinating form of GBS. C. jejuni-serology positive C. jejuni-stool culture positive

 3

	Patient		6		15		16	A	B
	No.
	Time	< 2 wks	> 4 wks	< 2 wks	> 4 wks	< 2wks	> 4 wks	19 days	12 days
	point
	Median nerve
	mNCV	50	38	§	§	59	53	38	37
	(m/s)	(98%)	(75%)			(111%)	(100%)	(81%)	(70%)
	DML	3.2	3.7	§	§	11.2
	(ms)	(73%)	(84%)

ACKNOWLEDGEMENTS

We would like to thank the members of the Dutch GBS Study Group for their participation in the study and gratefully acknowledge financial support from the Prinses Beatrix Fonds (research grant WAR06-215). COMPETING INTERESTS Joleen Blok and Bart C. Jacobs received a research grant from the Princess Beatrix Fund (WAR06-215) Judith Drenthen has nothing to disclose. Nobuhiro Yuki has nothing to disclose. Jan Meulstee has nothing to disclose. Ellen M. Maathuis has nothing to disclose. Pieter A. van Doorn has nothing to disclose. Gerhard H. Visser has nothing to disclose. Joleen H. Blok has nothing to disclose. Bart C. Jacobs has nothing to disclose.

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Raja Giryes

Nam Sangnam

Rémi Gribonval

Mike E Davies
email: mike.davies@ed.ac.uk

ITERATIVE COSPARSE PROJECTION ALGORITHMS FOR THE RECOVERY OF COSPARSE VECTORS

Recently, a cosparse analysis model was introduced as an alternative to the standard sparse synthesis model. This model was shown to yield uniqueness guarantees in the context of linear inverse problems, and a new reconstruction algorithm was provided, showing improved performance compared to analysis ℓ 1 optimization. In this work we pursue the parallel between the two models and propose a new family of algorithms mimicking the family of Iterative Hard Thresholding algorithms, but for the cosparse analysis model. We provide performance guarantees for algorithms from this family under a Restricted Isometry Property adapted to the context of analysis models, and we demonstrate the performance of the algorithms on simulations.

INTRODUCTION

Many natural signals and images have been observed to be inherently low dimensional despite their possibly very high ambient signal dimension. It is by now well understood that this phenomenon lies at the heart of the success of numerous methods of signal and image processing.

The sparse synthesis data model, which has enjoyed much popularity in recent years, is the leading model associated to this observation. To show more concretely how useful such a model can be, the following generic linear inverse problem is considered in this paper : For some unknown signal x 0 ∈ R d , an incomplete set of linear observations y ∈ R m is available via y = Mx 0 + e, where e ∈ R m is an additive bounded noise that satisfies ∥e∥ 2 2 ≤ ε 2 . The task is to recover or approximate x 0 . In the noiseless setting where e = 0, this amounts to solving y = Mx.

(1.1)

Of course, a simple fact in linear algebra tells us that (1.1) admits infinitely many solutions. Therefore, when all we have is the observation y and the measurement/observation matrix M, we are in a hopeless situation to recover x 0 . Sparse recovery guarantees. This is where the utility of what we may call in general the 'sparse signal models' comes into play. In the sparse synthesis model, the signal x 0 is assumed to have a very sparse representation in a given fixed dictionary D ∈ R d×n . In other words, there exists z 0 with few nonzero entries as counted by the "ℓ 0 -norm" ∥z 0 ∥ 0 such that x 0 = Dz 0 , and k := ∥z 0 ∥ 0 ≪ d. (1.2) This work was supported in part by the EU FP7-ICT-FET-Open program, project SMALL, grant number: 225913

In more realistic cases x 0 ≈ Dz 0 . Having this knowledge we solve (1.1) using x S = Dz S where z S = argmin z ∥z∥ 0 subject to ∥y -MDz∥ 2 2 ≤ ε 2 . (1.3) The upshot of the model (1.2) in the context of the inverse problem (1.1), in the case that e = 0, is that we know exactly when we can guarantee the uniqueness of sparse solution to the resulting equation y = MDz. Precisely, it is shown [START_REF] Donoho | Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization[END_REF][START_REF] Gribonval | Sparse representations in unions of bases[END_REF] that there exists k 0 such that if y = MDz 0 with k = ∥z 0 ∥ 0 < k 0 , then the linear system y = MDz has a unique k-sparse solution, which is necessarily z 0 . The largest such constant k 0 is exactly known to be spark(MD)/2, where spark(MD) is the minimum number of rows of MD that are linearly dependent. Consequently, if x 0 is sufficiently sparse-in the sense of (1.2)-then we can recover it by solving (1.3) with ε = 0. Convex optimization algorithms. Unfortunately, solving (1.3) was shown to be NP hard. A convex relaxation using an ℓ 1 -norm instead of an "ℓ 0 -norm" can overcome this computational issue [START_REF] Donoho | Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization[END_REF][START_REF] Gribonval | Sparse representations in unions of bases[END_REF]. For example, one can solve

z S = argmin z ∥z∥ 1 subject to ∥y -MDz∥ 2 2 ≤ ε 2 . (1.4)
The ℓ 1 -minimization has gained much popularity both in theory and practice. Interestingly, a slightly different form of ℓ 1 -minimization has been used in practice and studied [START_REF] Elad | Analysis versus synthesis in signal priors[END_REF]. In this 'altered form,' one solves

x A = argmin x ∥Ωx∥ 1 subject to ∥y -Mx∥ 2 2 ≤ ε 2 , (1.5)
where Ω ∈ R p×d is an analysis operator. Examples of such operators include finite difference operators, curvelet transforms, and undecimated wavelet transforms. Despite apparent similarities between (1.4) and (1.5), some fundamental differences have been demonstrated [START_REF] Elad | Analysis versus synthesis in signal priors[END_REF].

Cosparse signal model and recovery guarantees. Recently, it has been pointed out that a new signal model called cosparse analysis model [START_REF] Nam | Cosparse analysis modeling -Uniqueness and algorithms[END_REF] is more relevant to (1.5) than the sparse synthesis model. In particular, the uniqueness property in the context of linear inverse problem (1.1), in the noiseless case, has been obtained [START_REF] Nam | Cosparse analysis modeling -Uniqueness and algorithms[END_REF] for this model, exploiting general results on union of subspace models [START_REF] Lu | A theory for sampling signals from a union of subspaces[END_REF]. Therefore, one has a partial but theoretically firm foundation for using (1.5) for signal recovery.

In the analysis model we aim at solving the following minimization problem:

x A = argmin x ∥Ωx∥ 0 subject to ∥y -Mx∥ 2 2 ≤ ε 2 . (1.6)
The model and the uniqueness result can be summarized as follows: For a fixed analysis operator Ω ∈ R p×d , a signal x 0 ∈ R d is said to satisfy the cosparse analysis model if

ℓ := p -∥Ωx 0 ∥ 0 is large. (1.7)
The quantity ℓ is the cosparsity of x 0 and x 0 is said to be ℓ-cosparse, or simply cosparse. As the definition of cosparsity ℓ suggests, the emphasis of the cosparse analysis model is on 'many zeros' of the analysis representation Ωx 0 . This contrasts to the emphasis on 'few non-zeros' of a synthesis representation z 0 in the synthesis model (1.2). The uniqueness result for the analysis model reads as expected: There exists ℓ 0 such that if y = Mx 0 for ℓ-cosparse x 0 with ℓ ≥ ℓ 0 , then the problem (1.6) when ε = 0 has a unique ℓ-cosparse solution. We refer the readers to [START_REF] Nam | Cosparse analysis modeling -Uniqueness and algorithms[END_REF] for more discussion.

New algorithms for cosparse recovery. The two ℓ 1minimization principles (1.4) and (1.5) can now be seen as methods to solve or approximate the solutions of (1.3) and (1.6) for the recovery of sparse and cosparse signals, respectively. Having this parallelism, we ask immediately: what about a plethora of algorithms that also aims to solve (1.3)? Are there analogous counterparts for the analysis model? For some algorithms, the answer is definitely yes. Indeed, a greedy algorithm called Greedy Analysis Pursuit (GAP) has been developed [START_REF] Nam | Cosparse analysis modeling -Uniqueness and algorithms[END_REF]. It somehow mimics Orthonormal Matching Pursuit [START_REF] Pati | Orthonormal matching pursuit : recursive function approximation with applications to wavelet decomposition[END_REF], and its effectiveness was demonstrated for the cosparse signal recovery problem.

Contributions. Another avenue exists for an analysis algorithm in the direction of the iterative hard thresholding algorithms [START_REF] Blumensath | Iterative hard thresholding for compressed sensing[END_REF][START_REF] Foucart | Hard thresholding pursuit: an algorithm for compressive sensing[END_REF][START_REF] Cevher | On accelerated hard thresholding methods for sparse approximation[END_REF]. The contributions in this paper are: • Algorithms in the cosparse analysis framework are defined in Section 2 in the spirit of Iterative Hard Thresholding (IHT, [START_REF] Blumensath | Iterative hard thresholding for compressed sensing[END_REF]) and Hard Thresholding Pursuit (HTP, [START_REF] Foucart | Hard thresholding pursuit: an algorithm for compressive sensing[END_REF]). Note that the main novelty is in the definition of an appropriate cosparse projection replacing the hard thresholding step used in the synthesis framework. • A success guarantee based on an RIP-like property for the analysis model is provided in Section 3. As far as we know, no uniform guarantees exists for a practical algorithm in the analysis model. This guarantee, unlike the one in [START_REF] Candès | Compressed sensing with coherent and redundant dictionaries[END_REF] gives a bound for all cosparse signals under the appropriate conditions. • Empirical performance is demonstrated in Section 4 in the context of the cosparse signal recovery problem.

ALGORITHMS DESCRIPTION

It is quickly observed that the solution z 0 of the ideal problem (1.3) for the case ε = 0 satisfies: a) ∥z 0 ∥ 0 = k (assuming prior knowledge of the optimum sparsity k); and b) ∥y -MDz 0 ∥ 2 2 = 0. This observation somehow motivates the basic ideas of IHT and HTP as follows: We look for a coefficient vector z 0 that minimizes an objective function f (z) := ∥y -MDz∥ 2 2 under the constraint ∥z 0 ∥ 0 = k. The main difficulty here is that the constraint is not only nonconvex but also non-smooth.

Quick review of IHT and HTP.

Fortunately, there are two very simple tasks intimately related to the considered problem; the first one is when we have an estimate, say ẑ1 , of z 0 , it is easy to find a better estimate in terms of reducing the objective f . This can be done by a simple gradient step. The second task is when a new estimate, say ẑg 1 , is not k-sparse, it is straightforward to find the best/closest k-sparse signal ẑ2 . This is achieved by a simple hard thresholding, leading to the IHT algorithm [START_REF] Blumensath | Iterative hard thresholding for compressed sensing[END_REF]. Of course, in the second projection step, we can be mindful of the objective function as well, and instead of simple hard thresholding, we may only take the support of ẑ2 obtained from the hard thresholding but optimize f (z) over all z's with the same support in order to get a new estimate, let us call it again ẑ2 . The HTP algorithm implements this idea [START_REF] Foucart | Hard thresholding pursuit: an algorithm for compressive sensing[END_REF]. Going further, Cevher [START_REF] Cevher | On accelerated hard thresholding methods for sparse approximation[END_REF] proposes to optimize the stepsize in order to minimize f (z) while taking into account a possible change in the support selected by hard thresholding. We refer the readers to [START_REF] Blumensath | Iterative hard thresholding for compressed sensing[END_REF][START_REF] Foucart | Hard thresholding pursuit: an algorithm for compressive sensing[END_REF][START_REF] Cevher | On accelerated hard thresholding methods for sparse approximation[END_REF] for more details and turn our attention to analysis counterparts.

Overview of proposed cosparse recovery algorithms

The discussion above given the cosparsity ℓ of a sufficiently cosparse signal x 0 , in order to solve the ideal problem (1.6), we can look for a solution x A that minimizes an objective f (x) := ∥y -Mx∥ 2 2 under the constraint p -∥Ωx∥ 0 = ℓ. From this, we are led to algorithms for the cosparse signal recovery in the spirit of IHT and HTP. We present a group of algorithms that share similar ideas:

xg i = xi-1 + µ i M T (y -M xi-1 ) (2.1)
for some appropriate µ i > 0. This corresponds to a gradient descent with respect to the objective function f (x). • Third, by an appropriate 'projection,' the intermediate estimate xg i is projected to an ℓ-cosparse element xi .

Some notations

The 'projection' step is the most specific novelty of the proposed algorithms. To explain it, let us fix some notations. For an index set Λ ⊂ 1, p , Ω Λ is the submatrix obtained from Ω by taking the rows indexed by Λ. Hence, if x 0 is an ℓ-cosparse signal, then there are ℓ rows Λ of Ω such that Ω Λ x 0 = 0. This means that x 0 is orthogonal to range(Ω T Λ ). In other words, x 0 belongs to the 'cosparse subspace' W Λ = Null(Ω Λ ), which is the orthogonal complement of range(Ω T Λ ). We denote the orthogonal projection onto range(Ω T Λ ) and W Λ by P Λ and Q Λ , respectively.

Cosparse projections

From xg i which is not necessarily cosparse, we wish to obtain a new estimate xi which is ℓ-cosparse. How we obtain such a cosparse estimate xi is interesting and important.

Going back to the synthesis case, recall that given a coefficient vector z, we obtain the k-sparse vector z k closest to z through the so-called hard-thresholding operation z k = H k (z): by retaining only the k largest coefficients of z -excluding possible ties. This is not so in the cosparse analysis model: When we apply a hard thresholding to an analysis representation Ω xg i , we obtain a vector z := H p-ℓ (Ω xg i ) with ℓ zero entries, but we can no longer assert that z is an admissible analysis representation in general, i.e., there is no x such that z = Ωx.

To overcome this, we can instead estimate the cosupport as the set of ℓ smallest entries of Ω xg i , Λi := argmin

Λ:| Λ|=ℓ Ω Λ xg i 2 2 , ( 2.2)
and then 'project' xg i onto the associated cosparse subspace W Λ to obtain a 'good' estimate. How we project on W Λ is the main difference between the proposed algorithms.

Orthogonal projection. A straightforward solution is to simply perform an orthonormal projection of xg i . This is the approach we take for the algorithm of IHT type, which we name A-IHT. Hence, in A-IHT, we carry out the steps (2.2)

and xi = Q Λi xg i . (2.3)
Note that the quality of this projection heavily depends on the analysis operator Ω. The projection is in general not the optimal projection onto the union of analysis cosparse subspaces. Further implication of this is discussed in section 3.

Best data fidelity projection. From the description above, the cosparse subspace W Λi , plays the role of the support for the k largest coefficients of synthesis representations. In the algorithm of HTP type, which we call A-HTP, we replace the orthogonal projection onto W Λi , (2.3), with a best data fidelity projection: we look for an element of W Λi that minimizes ∥y -Mx∥ 2 2 . That is, Λi is given by (2.2) and xi = argmin

x ∥y -Mx∥ 2 2 subject to Ω Λi x = 0. (2.4)

Choice of the stepsize

Note that the choice of gradient stepsize µ i is crucial: If µ i 's are chosen too small, the algorithm gets stuck at a wrong solution. If too large, the algorithm diverges. For our algorithms, we consider two options for µ i . In the first option, we can choose µ i = µ for some constant µ for all iterations. A theoretical discussion on how to choose µ properly is given in Section 3. Another way is to solve the following problem

µ i := argmin µ y -M( xi-1 + µM T (y -M xi-1 )) 2 2 , (2.5)
that has a simple closed form solution. Algorithm 1 summarizes the proposed family of algorithms.

Targeted cosparsity level

Just as in the synthesis counterpart of the proposed algorithms, where a target sparsity level k must be selected before running the algorithms, we have to choose the targeted cosparsity level ℓ which will dictate the projection steps. In the synthesis case it is known that it may be interesting to over-estimate the sparsity k. Similarly in the analysis framework the question arises: In terms of recovery performance, does it help to under-estimate the cosparsity ℓ ? A tentative yes comes from the following heuristic: Let Λ be a subset of the cosupport Λ of signal x 0 with l := | Λ| < ℓ = |Λ|. Note that if the rank of Ω Λ is greater than or equal to dm, then in general it is sufficient to identify Λ in order to recover x 0 Algorithm 1: Iterative Projection Algorithms: A-IHT, A-HTP Input: y, M, Ω, l, i max , δ term Set i = 0, x0 = 0

repeat i := i + 1 xg i := xi-1 + µ i M T (y -M xi-1 ) Λi := argmin Λ:| Λ|= l Ω Λ xg i 2 2 case A-IHT xi := Q Λi xg i case A-HTP xi := argmin x ∥y -Mx∥ 2 2 s.t. Ω Λi x = 0 until i = i max or ∥ xi -xi-1 ∥ 2 < δ term ∥ xi ∥ 2
from the relations y = Mx 0 and Ω Λx 0 = 0. Therefore, we can use l as the effective cosparsity in the algorithms. The effect of varying the cosparsity level is shown in section 4.

THEORETICAL GUARANTEES

We now turn to theoretical performance guarantees for the proposed algorithms. For the general class of Iterative Projection Algorithms (IPA), which combine a gradient step and an optimal projection P A onto a general union of subspaces, uniform guarantees were proved [START_REF] Blumensath | Sampling and reconstructing signals from a union of linear subspaces[END_REF] assuming that M is bi-Lipschitz on the considered union of subspaces, denoted A . In our case, A = ∪ |Λ|≥ℓ W Λ , and the bi-Lipschitz constants of M are the largest α and smallest β where 0 < α ≤ β such that for all ℓ-cosparse vectors x 1 , x 2 :

α ∥x 1 + x 2 ∥ 2 2 ≤ ∥M(x 1 + x 2 )∥ 2 2 ≤ β ∥x 1 + x 2 ∥ 2 2 . (3.1)
Under this assumption1 , one can apply Theorem 2 from [START_REF] Blumensath | Sampling and reconstructing signals from a union of linear subspaces[END_REF] to the idealized algorithm that performs an optimal projection xi = P A ( xg i ) as a third step (instead of the suboptimal projections described in section 2.4).

Theorem 3.1 If β ≤ 1 µ < (1 + 1
2 )α then, given y = Mx + e where x is ℓ-cosparse, after a controlled number of iterations, i * , we have

∥x -xi * ∥ 2 2 ≤ c 1 ∥e∥ 2 2 , (3.2)
where c 1 is a function of α, µ and an accuracy factor.

Near optimal projections. Unfortunately, calculating the optimal cosparse projection seems to be a hard problem due to the a priori combinatorial search for the best cosparse subspace W Λ . Can we do it more efficiently ? This remains an open question, and in practice, we can merely hope to achieve a sub-optimal projection. Denoting x * = P A x the vector resulting from the optimal projection and Λ * the cosupport of z * = Ωx * , we say that P is near optimal with constant C if for any vector x it obeys:

∥Px -x∥ 2 2 ≤ C inf x∈A ∥ x -x∥ 2 2 = C ∥x * -x∥ 2 2 . (3.3)
Near optimality of proposed projections. The practical projection used in the first proposed algorithm is near optimal when Ω obeys yet another RIP-like property:

L ℓ ∥v∥ 2 2 ≤ ∥Ω Λ v∥ 2 2 ≤ U ℓ ∥v∥ 2 2 (3.4)
for all v ∈ range(Ω T Λ ) and every Λ of size ℓ. Lemma 3.2 For Ω that satisfies Condition (3.4), the selection of the cosupport Λ according to (2.2), followed by a projection on the subspace W Λ according to (2.3) provides a projection which is near optimal with constant C = U ℓ /L ℓ .

Proof: 22 , by the RIP (3.4) and the definition of Λ,

Since ∥Ω Λx∥ 2 = ∥Ω ΛΩ † ΛΩ Λx∥ 2 2 = ∥Ω ΛP Λx∥ 2 2 = ∥Ω Λ(Id -Q Λ)x∥
L ℓ ∥Q Λx -x∥ 2 2 ≤ ∥Ω Λx∥ 2 2 ≤ ∥Ω Λ * x∥ 2 2 . In a simi- lar way, ∥Ω Λ * x∥ 2 = ∥Ω Λ * (P A x -x)∥ 2 2 ≤ U ℓ ∥P A x -x∥ 2 2
. By combining the lower and upper bounds we get that

∥Q Λx -x∥ 2 2 ≤ U ℓ L ℓ ∥P A x -x∥ 2 2
, Recovery guarantees with near optimal projections. While the existing result (Theorem 2 from [START_REF] Blumensath | Sampling and reconstructing signals from a union of linear subspaces[END_REF]) is valid for optimal projections, it can be extended to near optimal projections. The detailed proof of the following result is too long to fit in this conference paper and will appear in another place, hence we only state it as a conjecture. Conjecture 3.3 Consider y = Mx + e, where x is ℓcosparse and the projection is near optimal with constant C. Let σ M be the maximal singular value of M, α and β the constant of the Ω-RIP for M. For the noiseless case e = 0,

if β < 1 µ < (1 + √ 1 -b)α, where b = (C-1)σ 2 M αC
, then the ith iteration of A-IHT and A-HTP with a constant step size µ satisfies

∥x -xi+1 ∥ 2 ≤ c 3 ∥x -xi ∥ 2 2 , ( 3.5)
where c 3 < 1, hence the iterates converge geometrically.

The existence of a stepsize leading to convergence is guaran-

teed if β < (1 + √ 1 -b)α
. This is only possible if β < 2α. Vice-versa, if β < 2α, this holds true provided a near optimal projection is achievable with a constant C ≥ 1 sufficiently close to one. Note that for an orthogonal Ω, the analysis model coincides with the synthesis one. In terms of standard RIP for MΩ T , the condition β < 2α reads δ 2k (MΩ T ) < 1/3, and coincides with the results of IHT with a constant step size [START_REF] Foucart | Hard thresholding pursuit: an algorithm for compressive sensing[END_REF].

EXPERIMENTAL RESULTS

To assess their performance, the proposed algorithms were tested on synthetic cosparse signal recovery problems. For a fixed Ω ∈ R p×d , M ∈ R m×d , a noise level ε, and a cosparsity ℓ, a cosparse signal recovery problem is constructed as follows: A random index set Λ of size ℓ of 1, p is generated. An orthonormal basis B for Null(Ω Λ ) is computed. An i.i.d. Gaussian random vector c, of the same length as the number of columns of B, is generated. Finally, a target cosparse signal x 0 is formed as Bc. The observation vector is given by y = Mx 0 + e where e is an i.i.d. random Gaussian vector with norm ε ∥Mx 0 ∥ 2 . The quintet (Ω, M, y, ε, ℓ) constitutes an instance of the cosparse signal recovery problem and is given as an input to the algorithms.

For both A-IHT and A-HTP, we employed a constant step size 2 µ chosen based on the ideas from section 3. The stopping criteria (cf Algorithm 1) was i max = 500 and δ term = 10 -6 . The target cosparsity3 l is specific to each experiment.

Robustness against noise

To assess the behavior of the algorithms against different levels of noise, we chose the setting d = 200, p = 240, m = 160, and ℓ = 180 and varied the noise level in the range from (almost) 0 to 0.5. We generated 20 instances of cosparsity recovery problems as described at the beginning of this section, and fed the algorithms. The average relative error in estimating x was computed. For both A-IHT and A-HTP, the target cosparsity level was set to l = 0.9ℓ. For comparison with existing cosparse recovery algorithms, we run the same experiment for the GAP [START_REF] Nam | Cosparse analysis modeling -Uniqueness and algorithms[END_REF] and the analysis ℓ 1 -minimization (1.5) with ε = 0. For solving the analysis ℓ 1 optimization problem, we used the Matlab cvx package [START_REF] Grant | CVX: Matlab Software for Disciplined Convex Programming[END_REF] with highest precision and the final solution was debiased.

Figure 1 shows the results. Note that neither GAP nor ℓ 1 -minimization (in this experiment) are noise aware, which may explain their poorer performance compared to the proposed algorithms when the input SNR is below 25dB. Still, the comparatively better robustness of the proposed algorithms against noise remains remarkable, given that they do not use any explicit knowledge of the noise level.

Performance for cosparse signal recovery

Next we assessed the algorithm performance in the task of cosparse signal recovery, when the number of measurements and the cosparsity vary. We fixed d = 200 and p = 240, and varied the number m of measurements and the cosparsity ℓ according to the formulae: δ = m/d, ρ = (d -ℓ)/m in order to obtain phase transition diagrams for successful recovery. For each pair of parameters 0 < δ , ρ < 1 on a grid, the following was repeated 50 times: Ω was drawn as the transpose of a random tight frame of size d × p; M was drawn from the Gaussian ensemble; a cosparse signal recovery problem with ε = 0 was constructed and fed to the algorithms. A relative error less than 10 -4 was counted as a perfect recovery. For both A-IHT and A-HTP, two variants were implemented, one with a target cosparsity l = ℓ, the second one with l = 0.75ℓ. The GAP and the analysis ℓ 1 -minimization were again used for comparison. gorithm and analysis ℓ 1 -minimization yield the largest white areas, showing the best recovery performance of all tested algorithms. However, A-HTP was informally observed to perform the recovery faster than the other algorithms when it succeeded.

The effect of the target cosparsity level l can also be seen in Figure 3. Namely, we clearly observe the benefit of targeting cosparsity 0.75ℓ rather than ℓ. Let us point out, however, that the usage of too low l leads to complete failure in terms of recovery (result not shown). Therefore, l is best seen to be chosen slightly smaller than ℓ.

CONCLUSION

This paper introduced a family of new iterative algorithms for stably recovering/approximating cosparse signals from low-dimensional projections. These algorithms are the analogous, in a cosparse modeling framework, of the IHT and HTP algorithms for sparse signal recovery. This work extends the result of [START_REF] Blumensath | Sampling and reconstructing signals from a union of linear subspaces[END_REF] in the cosparse analysis model framework by allowing the use of sub-optimal projections for the general class of Iterative Projection Algorithms (IPA) described in [START_REF] Blumensath | Sampling and reconstructing signals from a union of linear subspaces[END_REF]. We have provided a tentative theoretical foundation for such sub-optimal projections, hence enabling the implementation of two classes of algorithms that can be proven to converge and recover sufficiently cosparse signals in the noiseless setting. Moreover, the robustness of these algorithms against noise and their effectiveness in recovering cosparse signals was demonstrated by experimental evidence. Further investigation of performance guarantees of the algorithms in the presence of noise, their computational complexity, and the effect of cosparsity levels and gradient step size are ongoing work.

Figure 1 :

 1 Figure 1: Output SNR 20 log 10 (∥x 0 ∥ 2 / ∥ x -x 0 ∥ 2 ) as a function of the input SNR -20 log 10 (ε) for the tested algorithms.

Figures 2 -Figure 2 :

 22 Figure 2: Recovery rate of GAP (left) and analysis ℓ 1minimization (right).

Figure 3 :

 3 Figure 3: Recovery rate of proposed algorithms. Top: A-IHT; Bottom: A-HTP; Left: l = ℓ; Right: l = 0.75ℓ.

This is the analysis counterpart to the D-RIP property for the matrix M introduced in[START_REF] Candès | Compressed sensing with coherent and redundant dictionaries[END_REF], where recovery guarantees have been developed for the solution of (1.5) under the sparse synthesis model with the transposed dictionary D = Ω T , in the case that Ω T is a tight frame. Hence, (3.1) will be called the Ω-RIP for M.

For µ i given by (2.5), the results (not shown in this paper) are similar.

we actually use max( l, dm) as the target cosparsity to ensure uniqueness.