on Bacterial Membranes 2013

The first Minisymposium on Bacterial Membranes was organized more than ten years ago and has become a successful tradition, with the 2013 edition taking place last March. This edition covered a variety of topics, including advances in the structural analysis of membrane proteins by solid-state NMR, the structure and function of components of several extracellular protein secretion systems, lipid modification and lipid-protein interactions, insertion of proteins into the outer membrane and uptake of DNA and nutrients. Over fifty young scientists from Germany,

Structural analysis of membrane proteins by solid-state NMR

Structural analysis of membrane proteins is a prerequisite for understanding their function and biogenesis. Although membrane proteins form 30% of the proteome, they still represent only 1% of the structures in the Protein Data Bank. X-ray crystallography has been the main method used for structure determination. Typically produced in small quantities, requiring specific detergents for membrane extraction, flexible and dynamic, membrane proteins have been far fewer for yielding crystals of sufficient quality for X-ray diffraction analysis. Among alternative methods for structure determination, such as nuclear magnetic resonance (NMR) and electron microscopy (EM), solid-state NMR (ssNMR) has proven particularly powerful, providing structures of soluble and membrane proteins with equal success. Importantly, ssNMR is the only high-resolution technique that allows structural Antoine Loquet (Max Planck Institute for Biophysical Chemistry, Göttingen, Germany) described some of the most recently developed labeling techniques that greatly improve the resolution and information content of ssNMR data, using combinations of sparsely labeled carbon sources [START_REF] Loquet | 13C spin dilution for simplified and complete solid-state NMR resonance assignment of insoluble biological assemblies[END_REF]. These approaches have proven extremely powerful when applied to structural analysis of the type III secretion needle from Salmonella typhimurium, revealing the molecular bases of stable polymer formation from flexible building blocks. Typically, polymer structures are deduced by rigid docking of isolated subunit structures into the electron density maps of multimers obtained by cryo-EM. These approaches are not only limited by the resolution of EM structures, but also by the extent of conformational changes that monomers undergo upon assembly within the polymer.

Contrary to previous needle models obtained using rigid docking [START_REF] Fujii | Structure of a type III secretion needle at 7-Å resolution provides insights into its assembly and signaling mechanisms[END_REF], which placed the N-terminus of the monomer in the lumen of the needle, ssNMR analysis showed that these domains, disordered and invisible in X-ray monomer structures, adopt a rigid and defined conformation within the needle. An impressively detailed interaction network could be deduced from ssNMR data with over 150 intersubunit distance restraints. In the new model, the N-terminus of the needle subunit PrgI is stabilized by specific contacts with protomer P -11 and exposed on the needle surface [START_REF] Loquet | Atomic model of the type III secretion system needle[END_REF]. This model was confirmed by immunogold labeling of the N-terminally tagged protein PrgI, not only on needles assembled in Salmonella, but also on needles of the Shigella T3SS formed by MxiH [START_REF] Demers | The Common Structural Architecture of Shigella flexneri and Salmonella typhimurium Type Three Secretion Needles[END_REF].

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3 Important components of NMR analysis are powerful computational methods, such as ARIA or ISD, that allow structural calculations based on the distance data [START_REF] Rieping | ISD: a software package for Bayesian NMR structure calculation[END_REF], Bardiaux et al., 2012). Since not only biological polymers, but also membrane proteins typically exist as symmetric assemblies, the use of symmetry to calculate structures can reduce calculation complexity and cost [START_REF] Bardiaux | ARIA for solution and solid state NMR[END_REF]. Benjamin Bardiaux (Institut Pasteur, CNRS UMR3528, Paris, France) described the power of combining ssNMR and computational methods based on ambiguous distance restraints to solve the structure of the Yersinia adhesin A (YadA), secreted via the type V pathway [START_REF] A N U S C R I P T A C C E P T E D Accepted Manuscript 13 Leyton | From self sufficiency to dependence: mechanisms and factors important for autotransporter biogenesis[END_REF]. YadA belongs to a class of type V secretion systems called autotransporters, which cross the inner membrane of Gram-negative bacteria via the Sec system. Their N-terminal part forms the passenger domain that is secreted via a channel in the outer membrane formed by the Cterminal beta-barrel domain. Cellular Bam machinery facilitates insertion of the C-terminal domain into the outer membrane. Autotransporters are widespread and come in many flavors.

Yersinia YadA is a member of the trimeric autotransporter family, with the outer membrane barrel formed by beta sheets derived from three polypeptides. The passenger domain of YadA has been analyzed by X-ray crystallography of its different fragments. Interestingly, poorly diffracting crystals of the outer membrane domain were used as material for ssNMR analysis, successfully providing the de novo structure of the trimer [START_REF] Shahid | Membrane-protein structure determination by solid-state NMR spectroscopy of microcrystals[END_REF]. NMR analysis provided crucial information on the flexible nature of a short conserved tetrapeptide ASSA between the passenger and the beta-barrel domains. The results support the model where this sequence forms an essential hairpin that initiates secretion of the passenger domain, which is subsequently driven by folding of the passenger domain into a beta-helix structure.

Autotransporter family adhesins play important functions in bacterial colonization. In Helicobacter pylori, outer membrane adhesins SabA and BabA are crucial for bacterial

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4 establishment in the hostile niche of human gastric mucosa. Gaetano Castaldo (Vrije Universiteit Brussels, Brussels, Belgium) uses X-ray crystallography to study the structure of BabA, a 721 aa protein that binds to the Lewis b antigen [START_REF] A N U S C R I P T A C C E P T E D Accepted Manuscript 14 Yamaoka | Roles of Helicobacter pylori BabA in gastroduodenal pathogenesis[END_REF]. Using a series of constructs facilitating BabA fragment production and purification, they revealed that these adhesins have an unusual fold. Although structural analysis is required to confirm this, preliminary biochemical studies strongly suggest that the BabA passenger domain is "inserted" into the beta-barrel-encoding polypeptide, thus defining a novel class of autotransporters.

Type VI secretion: killing (or not) by secreted toxins

Type VI secretion system (T6SS) analysis is currently one of the most dynamic areas of research in the field of bacterial protein transport. Although T6SSs are implicated in bacterial pathogenesis and interactions with eukaryotic host cells, their role in interspecies competition in the context of microbial communities and biofilms has been the focus of recent attention. Contact-dependent killing, discrimination between sister cells and members of different species and killing and resistance mediated by T6SSs are novel factors that need to be taken into account when considering seemingly peaceful communities of bacteria growing in close proximity on various surfaces. Yannick Brunet (University Aix-Marseille, Marseille, France) illustrated the role of T6SS in bacterial competition between two strains of the same species, the domesticated Escherichia coli K-12 and a pathogenic enteroaggregative E. coli (EAEC). Co-cultivation of these strains results in killing of the laboratory workhorse in a contact-dependent manner. The "secret weapon" of the EAEC strains, the T6S apparatus, looks like an inverted phage tail with its piercing device pointing outwards. Assembled in the bacterial envelope, it is connected to a cytoplasmic shaft harboring a hollow tube filled with

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5 the panoply of "bullets", effector proteins with various enzymatic activities. Using live fluorescence imaging and the GFP-labeled shaft component TssB, one can observe the assembly and contraction of the shaft in a predator cell. This contraction in an EAEC cell that is in direct contact with E. coli K-12 leads to the lysis of prey within minutes [START_REF] Brunet | Imaging Type VI Secretion-Mediated Bacterial Killing[END_REF]. The tube is built as a stack of Hcp subunit hexamers, which can be assembled in vitro both in a head-to-tail and tail-to-tail orientation. However, the preferred orientation in vivo is head-to-tail, suggesting a mechanism that regulates tube assembly.

Serratia marcescens strain Db10 is a significant cause of hospital-acquired infections, including in neonates. Using S. marcescens strain Db10 as model to study the T6SS function, Grant English (University of Dundee, UK) focused on effector proteins secreted by this system. Genome analysis identified genes encoding small highly acidic and highly basic proteins that seem to come in pairs. They are non-essential for T6SS function and the basic proteins Ssp1 and Ssp2 are indeed secreted by the T6SS. Interestingly, deletion of genes encoding their acidic partners Rap1a and, in particular, Rap2a led to severe toxicity. Bacterial filamentation caused by rap2a deletion was alleviated by T6SS inactivation or by deleting the ssp2 gene, but not ssp1, strongly suggesting that these proteins act as toxin and immunity protein pairs. Consistent with this model, purified Ssp1 and Rap1a form a stoichiometric 2:2 complex with nanomolar affinity. Remarkably, despite similar charge properties of these pairs, binding is highly specific, since Ssp1 does not form a complex with Rap2a [START_REF] English | New secreted toxins and immunity proteins encoded within the Type VI secretion system gene cluster of Serratia marcescens[END_REF]. While structural analysis of other members of this class Rap1b and Rap2b provides possible clues for this specificity, co-crystal structures of the cognate pairs should shed more light on the mechanism by which antitoxins block the active sites as well as on toxin activities. Recent in vitro studies show that Ssp1 is a member of cell-wall-degrading amidases. Its production in the bacterial cytoplasm is tolerated, but its periplasmic production leads to cell death, which is prevented by co-expression of its partner gene rap1a. Similar

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6 results were obtained with the Ssp2 and Rap2a pair and several other toxin-antitoxin modules were found not only in the Db10 genome, but also in other species, including Agrobacterium and Salmonella. These results have interesting implications for the evolution of strains in populations, with continuous acquisition of toxin/antitoxin pairs with different specificities and enzymatic activities, contributing to rapid target killing, but also possibly propagating the same weapons and resistance proteins across species.

Lipid modification and lipid-protein interactions

Bacterial membranes are composed of phospholipids, proteins and modified lipids such as lipoteichoic acids (LTA), lipopolysaccharides (LPS) and lipoarabinomannan (LAM).

Peter Bond (University of Cambridge, Cambridge, UK) provided structural and thermodynamic insights into the interaction between the lipid moiety of LPS and the MD-2 co-receptor of Toll-like receptor 4 (TLR4) using a molecular dynamics simulation approach [START_REF] Paramo | The simulation approach to lipid-protein interactions[END_REF]. Molecular simulation is based on descriptions of energy states of protein structures obtained from X-ray and NMR data (i.e. the TLR4 co-receptor MD-2) in a specific environment, here an asymmetric lipid bilayer composed of phospholipids and LPS. These energy descriptions are translated into forces and then into motion, resulting in a dynamic structural movie of a protein in a membrane. MD-2 binds the lipid A moiety of LPS [START_REF] Park | The structural basis of lipopolysaccharide recognition by the TLR4-MD-2 complex[END_REF]. The simulation data presented by Bond showed that MD-2 induces stronger binding of the agonist (LPS) by TLR4 and that the conformational changes predicted in MD-2 depend on the fatty acids of LPS. Sabina Chalabaev (Institut Pasteur, Paris, France) discussed the regulation of lipid A modification in biofilm-grown bacteria. Different growth conditions have been shown to lead to alterations in the composition of the bacterial cell envelope. For example, bacteria in biofilms are more resistant to antimicrobial agents and

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7 their surface exposed polysaccharides differ in structure compared to free-living planktonic bacteria [START_REF] Ciornei | Biofilm-forming Pseudomonas aeruginosa bacteria undergo lipopolysaccharide structural modifications and induce enhanced infammatory cytokine response in human monocytes[END_REF]. In E. coli biofilms, the lipid A moiety of LPS is modified by addition of palmitate, a reaction catalyzed by PagP. This phenomenon is also observed in other biofilm-forming enterobacteria. Activation of pagP expression occurs via SlyA, which releases the inhibitory effect of H-NS on expression of the pagP gene. The PhoPQ and EvgAS pathways are not involved in induction of pagP expression. Muriel Masi (University Paris XI, Orsay, France) presented results of her work on the molecular basis of recognition of sugar entities within the cell wall of Corynebacterium glutamicum by mycoyltransferase A (MytA). The cell envelope of Corynebacterineae is a complex structure containing lipids, peptidoglycan and arabinogalactan that is itself covalently linked to mycolic acids, the socalled mAGPG. Preliminary structural data on the N-terminus of MytA indicates the presence of an esterase fold, while its C-terminus contains several unique repeats involved in mycolic acid transfer to the sugars of the cell envelope. Masi et al. reported a surprising observation that MytA interacts with peptidoglycan and not arabinogalactan, and that arabinogalactan inhibits binding of peptidoglycan by MytA.

Transport of folded proteins across membranes

Proteins are synthesized in the cytoplasm as linear polypeptides and are typically exported across the inner membrane in an unfolded state through narrow membrane channels, minimizing chances of causing proton leakage that would be detrimental to the energetic state of the cell. An example is the inner membrane channel formed by the essential protein SecY with a constriction and a plug, or the more complex trans-envelope channels in type I, type III and type IV protein secretion systems. However, bacteria have also met the challenge of transporting folded proteins across the inner or outer membrane. In the late 1990s, one of the

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ACCEPTED MANUSCRIPT 8 first Symposia on Bacterial Membranes witnessed the discovery of a novel transport system allowing export of folded proteins, typically those that acquire various cofactors upon synthesis in the cytoplasm [START_REF] Santini | A novel sec-independent periplasmic protein translocation pathway in Escherichia coli[END_REF]. Such proteins are synthesized with Nterminal signal sequences containing a double arginine sequence in a conserved signature motif, lending the name to this export pathway--the twin arginine translocation (Tat) pathway. Three essential protein components of this export system, TatA, TatB and TatC, have been extensively characterized [START_REF] Palmer | The twin-arginine translocation (Tat) protein export pathway[END_REF]. TatA and TatB are single-span membrane proteins with cytoplasmic membrane associated alpha-helical domains. TatC is a polytopic membrane protein that is involved in specific interactions with Tat signal peptides. Sarah Rollauer (University of Oxford, Oxford, UK) described the 3-D structure of TatC from a themophillic bacterium Aquifex aeolicus, recently determined by X-ray crystallography [START_REF] Rollauer | Structure of the TatC core of the twin-arginine protein transport system[END_REF].

The TatC structure is a product of extensive groundwork employed to optimize the expression and solubilization of the protein using an elegant TatC-GFP fusion tool to facilitate the screening process. TatC has 6 transmembrane segments and a large surface area with a concave face oriented towards the cytoplasm. To assess its binding to other Tat components, cysteine cross-linking data provided information on putative interaction sites with TatA and TatB. The dynamic nature of these interactions, however, makes it difficult to provide a clear model of this complex in action or to understand how the translocation proceeds without compromising the membrane impermeability. Nevertheless, together with in vitro binding assays and a vast collection of mutations of E. coli TatC, the structure provides molecular insight into the mechanism of signal sequence recognition and binding. Clearly, this structure is an essential and crucial step forward in structure-function analysis of this fascinating transport system.

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9 Type II secretion systems (T2SSs) provide a unique route for prefolded proteins to the bacterial cell surface or to the surrounding medium, through an outer membrane pore formed by the protein secretin. This unique class of membrane proteins forms multimeric channels that are also present in bacteria producing filamentous phage and type III secretion systems.

One of the best studied is the family of T2SS secretins exemplified by the E. coli GspD and Klebsiella oxytoca PulD. Domains N0, N1 and N2 residing in the periplasm have been crystallized, providing insight into the molecular basis of their flexible nature. However, little is known about the highly conserved module composed of domains N3 and C that seems to contain all information required for formation of multimeric channels, their gating and insertion into the outer membrane [START_REF] Korotkov | Secretins: dynamic channels for protein transport across membranes[END_REF]. While cryoEM has provided a global view of secretins, a high-resolution structure of the channel is still lacking. Even if it were available, structural information is unlikely to be sufficient to allow us to understand the biogenesis of this multimer and its membrane insertion. Gerard Huysmans (Institut Pasteur, Paris, France) described new biophysical approaches used to characterize the folding pathway of PulD from the pullulanase type II secretion system. PulD is a 65 kDa protein that forms multimers in an in vitro transcription-translation system and inserts into liposomes apparently without any accessory factors. Adding liposomes to the in vitro reaction allows one to monitor the kinetics of multimer formation. Assays of resistance to SDS, urea and trypsin are used to discriminate between the formation of different states along the folding pathway. This analysis suggests that multimer formation precedes insertion of PulD into membranes.

Changing the composition of the bilayer by variation in the acyl chain length and saturation and the reaction temperature allows one to manipulate the rate-limiting steps. Specific PulD mutations that influence some of these steps have also been very useful tools for monitoring the folding pathway of PulD. Interesting future directions of this work include structural or

Concluding remarks

Bacterial membranes, membrane proteins and cell surface structures involved in bacterial interactions with the environment and their hosts have been an important and dynamic area of research. To understand how bacteria acquire their shape, communicate with their environment and impact the ecosystems, whether in an infected host or in a particular environmental niche, an ever-increasing number of tools are available for biochemical, genetic, structural, visual, biophysical and bioinformatic analysis. Increasingly interdisciplinary and structure-oriented, the studies of membrane phenomena have provided mechanistic insights, but also novel biological concepts. Initiated in the late 1990s, the Minisymposia on Bacterial Membranes continue to foster informal exchanges in these areas and build ties between European groups mastering diverse experimental tools but sharing common interests. Over the years, these meetings have witnessed the substantial development of a dynamic scientific community and have favored its renewal.

  polymers or aggregates, membrane proteins in native-like environments or protein filaments / fibrils, without limitations in size and solubility. Thanks to a number of recent technical improvements, the limitations linked to the size of polymer subunits are progressively being reduced.

  PulD variants blocked at different steps of this unique folding pathway.5. Uptake of DNA and nutrientsAlthough bacteria lack the ability to carry out endocytosis, a variety of molecules of different size and composition can enter bacterial cells utilizing specific transport machineries in the cell envelope. The Com system required for DNA uptake by the Gram-positive organism Streptococcus pneumoniae includes putative surface pili, a DNA translocation system in the membrane and an endonuclease EndA peripherally attached to the external side of the membrane. In this bacterium, competence for DNA transformation occurs during a narrow time span at the beginning of the exponential phase of growth. Using a fusion protein between EndA and YFP, Nathalie Campo (CNRS-LMGM, P. Sabatier University, Toulouse, France) showed that EndA-YFP is uniformly distributed in the membrane of non-competent cells and concentrates into one or two foci located at mid-cell during competence. Relocalization is dependent on membrane-bound DNA-receptor ComEA. They developed an elegant method based on binding of fluorescently labelled (Cy3) PCR products to demonstrate that DNA binding also takes place at midcell. Furthermore, a synthetic fluorescent DNA fragment composed of cell division gene ftsZ and gfp was shown to bind at midcell, was taken up by the cell, recombined into the chromosome at the ftsZ locus and expressed with detection of the resulting functional FtsZ-GFP fusion protein. The process was completed 70 minutes after addition of DNA.Laurent Guillon (University of Strasbourg, Illkirch, France) also used fluorescence microscopy to localize the biosynthetic machinery for the Pseudomonas aeruginosa pyoverdine and to identify the site where this siderophore enters the cell following iron fluorescent proteins fused to pyoverdine biosynthesis enzymes PvdA and PvdQ, Guillon showed different localization sites for pyoverdine synthesis: at the old cell pole and throughout the periplasm. Recent findings suggest that uptake of iron-bound pyoverdine most likely occurs over the entire cell surface and not at a specific cellular location.

Acknowledgements

We thank Tony Pugsley for initiating the Minisymposia on Bacterial Membranes and for organizing this event for many years.


domain: info stat info.info-lg sdv

text:
Nathalie T H Gayraud
email: nathalie.gayraud@inria.fr

Nathanael Foy
email: nathanael.foy@inria.fr

Maureen Clerc
email: maureen.clerc@inria.fr

A Separability Marker Based on High-Dimensional Statistics for Classification Confidence Assessment

This work provides a theoretical analysis framework for features that belong to the high dimensional Riemannian manifold of symmetric positive definite matrices. In non-invasive EEG-based Brain Computer Interfaces, such as the P300 speller, these are sample covariance matrices of the epoched EEG signal that are classified into two classes. An analysis of the class shape on the manifold is performed, and the separability level of the two classes is evaluated. The main contribution is the Separability Marker (SM)-confidence method, a method that appends a confidence marker to the prediction of a binary classifier whose decision function is based on the comparison of Riemannian distances.

I. INTRODUCTION

Non-invasive EEG-based Brain-Computer Interfaces (BCI) are devices that provide a direct communication channel between a human brain and a machine. Many potential BCI applications exist [START_REF] Thomas | An analysis of performance evaluation for motor-imagery based bci[END_REF]; one of them is the P300 speller, an application that uses the P300 Event-Related Potential and enables the user to spell words by observing a screen of flashing letters.

The main motivation behind this research is the development of new machine learning tools that in the future will lead to calibration-free BCIs. The major drawback of EEG-based BCI applications is the low signal to noise ratio of the signal; as a result, advanced machine learning techniques need to be employed to boost the classification accuracy. Moreover, as the cross-session and cross-subject variability of the EEG signal is very high, sessions are usually preceded by a calibration.

One of the most promising approaches is the use a classification method under the Riemannian framework, such as the Minimum Distance to Riemannian Mean (MDRM) algorithm, introduced in 2010 by A. Barachant to classify features in Motor Imagery based BCI [START_REF] Barachant | Riemannian geometry applied to bci classification[END_REF]. The authors use the sample covariance matrix Σ i of an epoched trial X i as a feature, where X i is a C × N matrix, C is the number of electrodes used for the recording, and N depends on the length of the epoch and the sampling frequency. Using the Riemannian distance and the extrinsic definition of the mean by Fréchet, the algorithm estimates the centroid of each class in the training set by calculating the Riemannian mean of all the class features. For each new feature, its Riemannian distance to all centroids is calculated, and the smallest among these distances defines the winning class.

A number of adaptations of the MDRM algorithm have also been proposed for other BCI applications and problems [START_REF] Lindig-León | Comparison of hierarchical and non-hierarchical classification for motor imagery based bci systems[END_REF], [START_REF] Barachant | Multiclass braincomputer interface classification by riemannian geometry[END_REF], [START_REF] Congedo | A new generation of brain-computer interface based on riemannian geometry[END_REF], [START_REF] Barachant | A plug&play p300 bci using information geometry[END_REF]. In [START_REF] Barachant | A plug&play p300 bci using information geometry[END_REF], Barachant et al. propose a calibration free, Riemannian distance based classification method for a P300 game called "Brain invaders". In this work, an adaptive version of MDRM is implemented, where a new training set is learned during the testing of the algorithm. The reported Area Under Curve (AUC) score for both cross-session and cross-subject experiments is 80% in the worst case.

In the P300 Brain Invader game the target "alien" is selected by the program, on which the player has to focus his attention to destroy it, thus the classification result during the testing phase can be evaluated without feedback from the player. In most BCI applications however, such as the P300 speller, the classification results for the test set cannot be assessed without user feedback. Hence, to develop an adaptive algorithm similar to the one described in [START_REF] Barachant | A plug&play p300 bci using information geometry[END_REF] that can potentially perform cross-session and cross-subject classification, a measure of confidence needs to be obtained for the classification results, so that we can decide whether a trial will be added to the learned training set or not.

Moreover, the MDRM classification method relies on the assumption that the point cloud consisting of the covariance matrices of a class forms a multidimensional sphere on the Riemannian manifold embedded with the chosen Riemannian metric. These assumptions cannot be easily verified, due to the dimensionality of the problem, the small amount of available data, the low signal to noise ratio, and the definition of the chosen Riemannian distance itself.

In the following sections, we present the contribution of our research. In section II we examine the Riemannian metric and the derived distance our feature space is subject to, and provide an intuitive explanation to what it represents. We present relevant geometric properties on high dimensional spaces and get a concept of the shape of the distributions our features are drawn from. Section III begins with an analysis of a specific dataset, and presents the SM -confidence method. The SM -confidence method makes use of the observations of our analysis to provide a marker of confidence that can be outputted along with the result of any classification algorithm that uses the Riemannian distance in its decision function. Sections IV and V close this paper with results and conclusions.

Our results show that the SM -confidence method provides a reliable confidence marker that can be used with an adaptive classification method such as the one proposed in [START_REF] Barachant | A plug&play p300 bci using information geometry[END_REF], in the particular case where the features of the test set are learned and used to update the training set without prior knowledge of the label, or user feedback.

II. THEORETICAL FRAMEWORK

A. The Riemannian Metric

Under the framework of classification methods based on Riemannian geometry [START_REF] Congedo | A new generation of brain-computer interface based on riemannian geometry[END_REF], the sample covariance matrix has been emerging as a feature for the classification of mental tasks. The set of all sample covariance matrices is in fact a subset of the set of n × n Symmetric Positive Definite (SPD) matrices P n (R), which constitutes a Riemannian manifold when it is embedded with the following Riemannian metric, first proposed by Forstner et al. [START_REF] Förstner | A metric for covariance matrices[END_REF].

Σ = log(Σ) 2 F = n i=1 log 2 λ i , (1)
where λ i are the eigenvalues of Σ. The structure of the resulting Riemannian manifold together with the above metric, which is often called the Affine Invariant metric, becomes highly regular, bearing much resemblance to a curved vector space. Under the Affine Invariant metric, the manifold transforms from a high dimensional cone into a regular and complete manifold of non positive curvature [START_REF] Pennec | A riemannian framework for tensor computing[END_REF], [START_REF] Pennec | Statistical computing on manifolds: from riemannian geometry to computational anatomy[END_REF].

Note that the same metric has been obtained in the bibliography under another framework as well. In their work, Skovgaard et al. obtained the Affine Invariant metric by using the Fisher Information and considering the set of all Multivariate Gaussian distributions of dimension n as a differentiable manifold [START_REF] Skovgaard | A riemannian geometry of the multivariate normal model[END_REF].

The geodesic Riemannian distance d R (Σ 1 , Σ 2 ) can be derived from the Affine Invariant metric.

d R (Σ 1 , Σ 2 ) = log(Σ -1 1 Σ 2 ) F = n i=1 log 2 λ i , (2)
where Σ 1 , Σ 2 ∈ P n (R) and λ i are the joint eigenvalues of the two matrices.

We distinguish two important differences between d R and the Frobenius distance. These two differences motivate the choice of covariance matrices as an appropriate feature for the classification of epoched EEG signals.

The first one is that the two metrics reflect different measures of separability. The Frobenius distance between two n × n matrices can be perceived as the action of vectorizing the matrices and computing the Euclidean distance between the resulting d-dimensional vectors, where d = n 2 . The Riemannian metric reflects a different measure of separability. As shown in [START_REF] Skovgaard | A riemannian geometry of the multivariate normal model[END_REF], it projects zero mean multivariate Gaussian distributions, by using their covariance matrices, onto a Riemannian manifold. On this manifold, the distance between two matrices corresponds to the informational difference between the Gaussians they represent.

Assuming that each column (time sample) of two epoched trials X 1 i , X 2 j that correspond to two different mental processes is a random variable drawn from two different Gaussian distributions, N (0, Σ 1 ), N (0, Σ 2 ), the Riemannian distance between their sample covariance matrices will be related to the informational difference between the two distributions.

The second difference concerns one of the properties that are specific only to d R , In particular, it is invariant with respect to affine transformations on covariance matrices. This means that for any invertible matrix

W , d R (Σ 1 , Σ 2 ) = d R (W Σ 1 W T , W Σ 2 W T ).
In [START_REF] Congedo | Approximate joint diagonalization and geometric mean of symmetric positive definite matrices[END_REF], Congedo et Al. demonstrate the significance of that property in particular for BCI classification problems. Note that in in [START_REF] Barachant | A plug&play p300 bci using information geometry[END_REF], the authors hypothesize that the affine invariant property is what allows for the obtained classification results, under the assumption that cross-session and cross-subject variability can be described in terms of linear transformations.

B. High Dimensional Spaces

While the Affine Invariant metric provides the Riemannian manifold of n × n symmetric positive definite matrices with some very interesting properties, this manifold is a high dimensional space; its embedded dimension is equal to d = n(n + 1)/2. In non-invasive EEG-based BCI where the classification feature is the sample covariance matrix, n corresponding to the number of electrodes used for the recording. For instance, the dimension of the manifold will be d = 2080 for n = 64 electrodes.

The geometry of high dimensional spaces presents us with a very counter intuitive set of phenomena; many examples can be found in [START_REF] Hopcroft | Computer science theory for the information age[END_REF]. Fortunately, this does not forbid the existence of some valuable properties that will allow us to gain significant insight on the shape of feature space. Under the assumption that the feature space of sample covariance matrices, embedded with the Riemannian metric, can be approximated as a set of random variables drawn from multidimensional Gaussian distributions, we use established properties of multidimensional Gaussians to develop our separability marker.

Regarding the separability of features that come from different classes, [START_REF] Hopcroft | Computer science theory for the information age[END_REF] provide a helpful insight on the distances between points that belong to different unit Gaussian distributions. They demonstrate that a mixture of two unit Gaussian distributions with centers µ 1 and µ 2 that are separated by a distance δ, is separable when the distance between the two centers is δ ∈ Ω(d 1/4 ).

All of the above properties are proven in [START_REF] Hopcroft | Computer science theory for the information age[END_REF] for Euclidean high dimensional spaces. Nonetheless, our data lives in a Riemannian manifold, and we have to know whether the curvature of this particular space allows us to make the same assumptions. Since our previously presented theorems regarding distances between Gaussian distributions use trigonometric properties, we will use the works of [START_REF] Berger | A panoramic view of Riemannian geometry[END_REF] to get an insight on the effect of the curvature of the manifold.

Let T be a a geodesic triangle, that is, a triangle on the manifold whose edges are minimizing geodesics. Because geodesics are uniquely defined on the Riemannian manifold of symmetric positive definite matrices under the Riemannian distance d R , T can be uniquely mapped onto the tangent space T pM of the manifold, by fixing p to one of its three vertices. The Topogonov theorem states that the edges and angles of T ∈ T pM have upper and lower bounds with respect to the bounds of the sectional curvature of the manifold [START_REF] Berger | A panoramic view of Riemannian geometry[END_REF].

It has been shown that the lower and upper bounds [START_REF] Pennec | Statistical computing on manifolds: from riemannian geometry to computational anatomy[END_REF]. This means that, fortunately, the Riemannian distance between two features on the Riemannian manifold can be approximated with little error by the Euclidean distance between their projection on the tangent space, provided that we choose an appropriate reference point. Typically, that reference point is the Riemannian mean of all the features.

K - C , K + C of the sectional curvature K C of the SPD manifold are K - C = -1/2 ≤ K C ≤ K + C = 0 [14],
Therefore, we will use the above described properties of multidimensional Gaussian distributions to establish a separability marker between two sets of features, in order to assess the confidence of a classification result under a Riemannian classifier.

III. MATERIALS AND METHODS

A. Dataset Description

The dataset consists of EEG signals recorded during P300 speller sessions that were conducted by adult patients suffering from Amyotrophic Lateral Sclerosis. For the recording, an Refa-8 amplifier (ANT) with 12 electrodes (Fz,C3,Cz,C4,P7,P3,Pz,P4,P8,O1,Oz,O2) was used, and the sampling rate equal to 256 Hz. The experiment took place in the premises of the Nice University hospital, and had been approved by the local ethics committee CPP Sud Méditerrannée [START_REF] Clerc | Verbal communication through brain computer interfaces[END_REF].

Each subject participated in three free-spelling sessions, each one preceded by a calibration session. Only the calibration sessions are used in this paper. Initially, 20 patients participated in the experiment. The patients where at least one session contained a great number of artifacts were discarded from the dataset and the remaining sessions correspond to 14 patients.

Concerning the pre-processing of the signals, we first apply a 5th order Butterworth filter between 5 and 20Hz to the EEG signals. Each signal is then separated into epochs X i of size C × N , where C = 12 is the number of electrodes and N = 128 to account for a 0.5sec epoch starting at the time of the flash.

The goal of a classification method for a P300 speller is to differentiate between Target and Nontarget trials. Target trials are those who contain the elicited P300 component. As mentioned previously, the feature that is used under the Riemannian framework is the sample covariance matrix Σ i = 1 N X T i X i . This is the feature that we use in the rest of the document, for both the analysis of the shape of the classes, and the evaluation of the SM -confidence method.

B. Geometrical Analysis of the Feature Space

Regarding the theoretical framework we have presented on section II, we want to assert whether the two classes in this application, c ∈ {T, N } (Target/Nontarget) can be approximated as high dimensional Gaussian distributions on the Riemannian manifold.

Lower-dimensional Gaussian distributions have their mass concentrated near their expected values. In high dimensions, Hopcroft et al. [START_REF] Hopcroft | Computer science theory for the information age[END_REF] prove that there is very little mass located near the expected value of a multivariate Gaussian distribution. The expected squared Euclidean distance of a random variable from the expected value of the Gaussian distribution it is drawn from, is in fact on the order of dσ 2 , where d is the dimension and σ is the standard deviation of the distribution. They also demonstrate that most of the mass of a unit Gaussian multivariate distribution will be concentrated within an annulus of constant width.

Moreover, Hopcroft et al. [START_REF] Hopcroft | Computer science theory for the information age[END_REF] demonstrate that any two randomly drawn points will almost surely be orthogonal with respect to the expected value of the distribution. According to that, we expect that all features belonging to a class will be almost equidistant, ant that the average distance δ I between two features will be related to the average distance to the distribution center through the equation √ 2δ C = δ I . We consider each class separately and calculate the average distance δ C = avg(d R (Σ c , Σ c i )) between the features Σ c i that belong to class c and the centroid Σ c of that class, which we compute using the method described in [START_REF] Pennec | A riemannian framework for tensor computing[END_REF]. We also compute the average distance between same-class features, Table I displays the average over all subjects for δ C and δ I as well as the standard deviation of that average, calculated on the first session of each subject, for both classes. In this case study, our sample covariance matrices are 12 × 12 matrices, so the dimension is d = 78, and √ d ≈ 8.83. The result of that analysis show that, for both classes, the distance δ C between class centroids and class features appears to be on the order of √ d. Additionally, the features are almost equidistant to each other, which we can deduct by observing the standard deviations on Table I.

δ I = avg(d R (Σ c i , Σ c j )).

TABLE I AVERAGE DISTANCE TO CENTROID AND AVERAGE DISTANCE BETWEEN FEATURES FOR TARGET (T) AND NONTARGET (N) CLASS

Finally, if we compare the averages on Table I for the distances between features to the distances between each feature to its centroid, we can verify that √ 2δ C = δ I appears to be holding; we observe only a small deviation that is on average equal to δ I -√ 2δ C ≈ 0.61; it can be attributed to the effect of the curvature on the manifold.

Overall, we see that the Euclidean high dimensional properties of Gaussian distributions can be applied to our space. We proceed by making a cross-class comparison and try to calculate a marker of the amount of overlap that occurs between the two classes. This will give us a significant tool to evaluate the separability of Target and Nontarget classes. We perform an analysis that is based on the description of the shape of a high dimensional Gaussian distribution, given in sections II-B and III-B. A 2-dimensional schematic of this analysis is presented in Fig. 1b. Our goal is to obtain a measure of the amount of overlap between classes, in the annular regions where their distributions are concentrated.

We begin by scaling the distances between the features and their respective centers by √ d, to obtain distributions that are no longer affected by the dimensionality of the space. We plot a histogram of these distances, and approximate their distribution with a Gaussian kernel, as shown in Fig. 1a and1c. Note that, for the Nontarget class, the distances to the centroid are reversed, so that the histograms are coherent with the representation of Fig. 1b.

We display the results of this analysis for a single subject, which we have randomly chosen; subject 7. We perform the analysis twice, one to see the separability of the two classes within a single session (the first session), and once more for the union of all sessions. Observe that the distance distributions suggest a Gaussian probability density function; this is in accordance with our theoretical assumptions. We can also see that the features of the Nontarget class are closer to their centroid; the radius of the annulus is smaller. This can also be seen on Table I by comparing δ N C to δ T C . This analysis allows us to visualize the width of the annulus in which the features are contained, and use that information to define a region where the possibility of class overlap is increased, by taking into account the distance between the two centroids. This overlap region is represented in 2D for a general case in Fig. 1b; histograms 1a and 1c can be seen as 1D projections of the general case. On Fig. 1a we can observe that, for a single session, the histograms that represent the two classes do not significantly overlap. On the other hand, the overlap is more important in Fig. 1c when the class features come from three different sessions. This is due to the crosssession variability, which is causing an increase in the width of the annulus.

Taking into account the previous observations on the overlap region, we calculate the separability marker SM for any pair of classes in the following way. We estimate the distance distributions of the two classes from the distance distribution histograms using an appropriate Gaussian kernel. The two obtained curves that model the estimated distance distributions are superimposed, so that the distance between their expected values is equal to the distance between the two centroids, δ = d R (C T , C N ); if there is an overlap region, the curves will intersect. Let U be the area under the union of these two curves and I the area under the intersection of the two curves. Then, SM is equal to SM = (U -I)/U .

Intuitively, this marker gives us a comparative measure of the overlap. A small value suggests a big overlap, so that the classes are harder to separate, whereas a large value suggests a small overlap. On Fig. 2, we calculate the SM marker and compare it to the result obtained from training and testing a MDRM classifier; the reported statistic to assess the classification accuracy is Cohen's kappa value. We have also calculated the optimal distance needed for the classes to be perfectly separable. Recall that it is equal to d R opt (Σ T , Σ N ) = δ opt = d 1/4 . Figure 2 shows the ratio of the actual distance over the optimal distance, δ/δ opt . Figure 2a displays the analysis and results for each subject within a single session, where a 5-fold crossvalidation is performed to select a training and testing set. Figure 2b is a cross-session evaluation, where the training set of the classification contains the first two sessions, and the test set contains the third.

We can immediately see that the three markers are correlated. High SM scores accompany high classification scores, and distances that are near the optimal distance are also a strong marker of separability, as we see for example for subject 6, in both the single-session and the cross-session cases. Now that we have analyzed the information that can be conveyed from calculating Riemannian distances between features and centroids, we proceed to suggest a probabilistic classifier whose methods will be based on principles of Riemannian geometry and high dimensional geometry.

• 1 - δ c new δ c max • SM where δ c(c) new = d R (Σ new , Σ c(c) ) -d R (Σ new , Σ c(c) )
The first two components give us an measure of how probable it is that the new feature belongs to class c, by taking into account the Riemannian distances to both centroids. The fraction δ c(c) new /δ c(c) max , where c is the other label, will return a higher value depending on how close Σ new is to the centroid of class c(c).

However, if the classes are barely separable, that is, the centroids are very close to each other, or the standard deviation of the sample is affecting the maximum distances, the product of the first two components may be higher for the wrong class, and lower for the correct one, due to the overlap of the two classes. To account for that factor, the product of the first two components is scaled by an amount that is equal to the the separability marker SM , which is calculated during the calibration of the classifier.

Since the p SM marker is calculated for both classes, it can be used to classify new trials with a given confidence. It can also be used to select trials that cannot be classified. We see an example of the latter for a cross-classification experiment in the next session, and discuss the results.

IV. RESULTS

We perform a series of cross-session classification experiments, using the dataset presented in section III-A. For each subject, we train and test the MDRM classifier with trials that belong to this subject alone. The training set consists of trials that are drawn from two out of three sessions, and the test set corresponds to the third session. The experiments are carried out with all three possible combinations of sessions, and for each subject the reported result is the average accuracy. We use the SM -confidence method and compute the previously described values during the training step of the classifier. The output of the testing step is the label of each trial, l i = T (N ) along with the corresponding confidence marker p c SM [i].

Fig. 3. Evaluation of the confidence marker p SM for a cross-session experiment. When the threshold over p SM is equal to zero, the result is equivalent to the classification accuracy of the MDRM algorithm. We compare it to the classification accuracy obtained from the trials where p SM > 0.6.

The value of SM is displayed as well.

To assess the level of certainty provided by our method, we display, on Fig. 3, the classification accuracy for each subject and compare it to a subset of the result where p SM > 0.6. We can see that in most cases, the classification accuracy of the selected subset is higher. For instance, while subject 2 has low classification accuracy, approximately equal to acc = 0.5, the selected subset has an accuracy equal to acc = 1.0; these features can be used to efficiently train an adaptive classifier.

In Table II, one can see the improvement of the classification accuracy of the MDRM algorithm over a threshold value for the p SM that varies from thr = 0.0 to thr = 0.9. Because some subjects do not have a high average p SM value, and for such subjects very few features are selected for high thresholds, we display only the subjects where at least 10% of the features were selected.

We can see that the best accuracy values are obtained when the threshold is equal to thr = 0.6 and thr = 0.7. This is mostly due to the percentage of selected features, which becomes very low for thr ≥ 0.8. In almost every case, there is a net improvement in comparison to the classification accuracy reported by the algorithm.

V. CONCLUSIONS

In this research, we have studied the shape of distributions of sample covariance matrices on the Riemannian manifold of symmetric positive definite matrices. When the sample covariance matrices are classification features of a binary MDRM classifier, obtained from BCI applications such as the P300 speller, we can approximate their shape by using theorems that apply to high-dimensional Gaussian distributions. We can also get an insight on how separable the two classes are, and use that knowledge to get a confidence marker on the classification results. The SM -confidence method provides such a marker, and the experimental results we have obtained suggest that in future works it can be used to improve the prediction accuracy of an adaptive MDRM classifier, where each new feature will be added to the learned training set only when the p SM marker is above a certain threshold.

Fig. 1 .

 1 Fig. 1. A visualization of the distribution of distances between the class features and the estimated centroid. The distances have been scaled down by √ d. The histograms represent the scaled distances distribution, approximated by a Gaussian probability distribution function. On 1a, the features are drawn from a single session, whereas 1c shows the distribution of features from three different sessions. 1b provides a 2D visualization of two multidimensional Gaussian distributions; the annulus is where the mass of the Gaussian is concentrated. The area in green represents a region where the two distributions have the highest chance of overlapping.

VI. ACKNOWLEDGEMENTS

The Nice University Hospital is acknowledged for providing the dataset used in this paper.

C. The SM -confidence Method

We propose SM -confidence, a two part method that appends a confidence value to the prediction result of the MDRM classification method. The first part of SM -confidence is based on the analysis presented in section II, and can be embedded to the training step of MDRM. The second part concerns the estimation of a confidence value that can be appended to the output of the prediction process. The SM -confidence method can also be adapted to any classification method that uses symmetric positive definite matrices as features, and the minimum Riemannian distance to Riemannian class mean, to decide on a binary classification problem.

The training step of MDRM consists of calculating the centroids Σ T and Σ N for each class, where T and N are the labels of the two classes, and I c the set of features corresponding to class c. Our method adds the computation of three more values; δ T max , δ N max and SM . δ c max is the maximum value of the differences of distances to each centroid:

where i ∈ {1, 2..., |I c |}, Σ c i ∈ I c , c is the other class. The separability marker SM is calculated according to the method that has been presented in section III-B.

The second part of our method follows the classification part of the MDRM classification method, and appends the confidence marker p c SM to the classification label of a new feature Σ new that has been attributed label c.

The calculation of p c SM is carried out in the following manner: