Thierry Rabilloud
email: thierry.rabilloud@cea.fr

Variations on a theme: changes to electrophoretic separations that can make a difference

Electrophoretic separations of proteins are widely used in proteomic analyses, and rely heavily on SDS electrophoresis. This mode of separation is almost exclusively used when a single dimension separation is performed, and generally represents the second dimension of two-dimensional separations. Electrophoretic separations for proteomics use robust, well-established protocols. However, many variations in almost all possible parameters have been described in the literature over the years, and they may bring a decisive advantage when the limits of the classical protocols are reached. The purpose of this article is to review the most important of these variations, so that the readers can be aware of how they can improve or tune protein separations according to their needs. The chemical variations reviewed in this paper encompass gel structure, buffer systems and detergents for SDS electrophoresis, two-dimensional electrophoresis based on isoelectric focusing and two-dimensional electrophoresis based on cationic zone electrophoresis.

Introduction

Strictly speaking, two-dimensional electrophoresis is the combination of any type of electrophoreses. However, in the vast majority of cases, the second dimension is fixed to SDS-PAGE. Consequently, in order to maximize resolution, the first dimension shall be as remote as possible, in its separation principles, from SDS-PAGE. Although this leaves quite a wide space for non-classical 2D gels, as reviewed in [START_REF] Miller | Other than IPG-DALT: 2-DE variants[END_REF], the first separation of 2D gels generally falls into three main categories, namely: i) denaturing zone electrophoresis [START_REF] Booth | Novel System for 2-Dimensional Electrophoresis of Membrane Proteins[END_REF][START_REF] Macfarlane | Two-Dimensional Benzyldimethyl-N-Hexadecylammonium Chloride-]Sodium Dodecyl-Sulfate Preparative Polyacrylamide-Gel Electrophoresisa High-Capacity High-Resolution Technique for the Purification of Proteins from Complex-Mixtures[END_REF] ,ii) native zone electrophoresis, as pioneered by Blue-Native PAGE [START_REF] Schägger | Blue Native Electrophoresis for Isolation of Membrane-Protein Complexes in Enzymatically Active Form[END_REF] and in most cases iii) denaturing isoelectric focusing, as first described in [START_REF] Macgillivray | The heterogeneity of mouse-chromatin nonhistone proteins as evidenced by two-dimensional polyacrylamide-gel electrophoresis and ion-exchange chromatography[END_REF][START_REF] Farrell | High resolution two-dimensional electrophoresis of proteins[END_REF]. For all of these three basic setups, many changes and adaptations have been devised over the years to improve the performances of the two-dimensional electrophoretic separation and /or to adapt them to the experimental needs. The word performance can encompass resolution, scope of analytes, analysis window in terms of protein parameters, loading capacity, and any type of combination of these parameters. Many of these optimized variants have been described in the literature, and are quite often forgotten, although many can still be very useful. The purpose of this review is therefore to redescribe some of these variants and bring them back to light, hoping that it will be useful to researchers using the various types of 2D gels for their research. However, some areas are deliberately left out of the scope of this review. One of these is the native/denaturing setup, which has been recently reviewed elsewhere [START_REF] Wittig | Features and applications of blue-native and clear-native electrophoresis[END_REF]. The other is the pH gradient engineering in the case of IEF-based separations, as this too has been reviewed thoroughly [START_REF] Chiari | The Immobiline Family -from Vacuum to Plenum Chemistry[END_REF]. Last but not least, the field of protein detection after electrophoresis will be left out of the scope of this review, and readers interested in this aspect should refer to a recent review [START_REF] Miller | Protein stains for proteomic applications: Which, when, why?[END_REF]. Thus, this review will be focused on two types of SDS-PAGE-based 2D separations using respectively denaturing zone electrophoresis and denaturing IEF as the first separation.

Variations in SDS-PAGE

Technically speaking, a SDS-PAGE separation is made in a gel containing a buffer and a detergent. Not surprisingly, numerous variations have been attempted for all three parameters, and will therefore be described below 2.1. Variations in the gel structure Compared to the classical acrylamide-bisacrylamide gel, which makes the vast bulk of SDS gels run nowadays, some variants have been devised to improve the characteristics of the gel itself. Polyacrylamide gels are relatively fragile, and this can be felt especially when large-size gels are used [START_REF] Voris | Very-high-resolution two-dimensional gel electrophoresis of proteins using giant gels[END_REF]. This fragility has indeed hampered the wide use of very large gels, despite their wonderful resolution, as resolution in 2D gels goes with the surface of the gel, i.e. the square of the linear dimensions. Thus, doubling the size of each dimension increases the resolution by a factor of 4. To increase the mechanical resistance of the gels, two approaches have been described. In the first one [START_REF] Patton | A Mechanically Strong Matrix for Protein Electrophoresis with Enhanced Silver Staining Properties[END_REF], a linear polymer (linear, uncrosslinked polyacrylamide) is added to the monomer solution before polymerization, and the ready to use mixed solution was sold under the trade name Duracryl. Considerably enhanced mechanical properties were described. However, with the advent of the mass spectrometry-based protein identification techniques, the presence of a polymer in the gel considerably degraded the performances in terms of protein identification. This led to very limited use of this matrix nowadays. The second way on increasing the mechanical strength of the gel is to change the monomers themselves. One successful attempt in this direction was made by changing the crosslinker of the gels, and namely to replace methylene-bisacrylamide by bis-acryloyl piperazine. Bis-acryloyl piperazine was first described in conjunction with another monomer, acryloyl morpholine [START_REF] Artoni | Fractionation Techniques in a Hydro-Organic Environment .2. Acryloyl-Morpholine Polymers as a Matrix for Electrophoresis in Hydro-Organic Solvents[END_REF], but its successful use in 2D gels was to be described some years later, and in conjunction with regular acrylamide as the main monomer [START_REF] Hochstrasser | Development of Polyacrylamide Gels That Improve the Separation of Proteins and Their Detection by Silver Staining[END_REF]. In addition to greater mechanical properties, bis-acryloyl piperazine was also shown to provide slightly larger pores in the gel and improved performance in some silver staining techniques [START_REF] Hochstrasser | Development of Polyacrylamide Gels That Improve the Separation of Proteins and Their Detection by Silver Staining[END_REF]. This story shows how difficult it is to end up with monomers that are really useful for protein separations in SDS PAGE. In fact, the higher the molecular mass of the monomer, the better in terms of flexibility, as at constant monomer mass percentage a gel with wider pores will be obtained, while at constant pore size more monomer will be added, resulting in a stronger gel. However, the story of acryloyl morpholine shows that the game is not that simple, as heavy monomers are generally more hydrophobic and either lead to phase separation during polymerization, with an acrylic plastic separated from the aqueous phase, or to gels that have strong hydrophobic properties and therefore perform very poorly in protein electrophoresis.

The solution to this problem would be to use heavy but hydrophilic monomers, such as acryloylaminosugars [START_REF] Whistler | 1-Acrylamido-1-Deoxy-D-Glucitol, 1-Deoxy-1-Methacrylamido-D-Glucitol and Their Polymerization[END_REF] or trisacryl [START_REF] Kozulic | Poly-N-Acryloyl Tris Gels as Anticonvection Media for Electrophoresis and Isoelectric-Focusing[END_REF]. However, the use of these monomers is not without problems, and points to a second important feature of polyacrylamide gels, which is their chemical resistance. After polymerization, polyacrylamide can be seen as a polyethylene backbone with numerous pending amido groups conferring to the polymer its hydrophilicity and thus its ability to form hydrogels (see figure 1). The polyethylene backbone is chemically very resistant, but the amido groups are sensitive to hydrolysis, especially under basic conditions. This leads to conversion of polyacrylamide into polyacrylate, with deeply altered properties, such as greater swelling in water, strong electroendosmosis and ion retention properties. These properties are unwanted for electrophoretic applications, and base-resistant monomers should be used if possible. Acrylamide is reasonably stable at the pH present during electrophoresis (see next section on buffers), but the highly hydroxylated monomers such as acryloylaminosugars and trisacryl are much more prone to base-catalyzed hydrolysis [START_REF] Gelfi | Investigation of the Properties of Novel Acrylamido Monomers by Capillary Zone Electrophoresis[END_REF], and this precludes their use in SDS PAGE under standard conditions. Some monomers, however, have been developed specifically to combine hydrophilicity with resistance to hydrolysis, such as acrylamidoethoxyethanol (AAEE) [START_REF] Chiari | Towards New Formulations for Polyacrylamide Matrices -N-Acryloylaminoethoxyethanol, a Novel Monomer Combining High Hydrophilicity with Extreme Hydrolytic Stability[END_REF] and acrylamidopropanol (AAP) [START_REF] Simò-Alfonso | Novel acrylamido monomers with higher hydrophilicity and improved hydrolytic stability .1. Synthetic route and product characterization[END_REF]. The use of these monomers has remained confidential up to now, but a recent paper focused on membrane proteins takes advantage of the base resistance of AAP gels to use them in conjunction with in gel proteolysis at high pH with proteinase K [START_REF] Bendz | Quantification of membrane proteins using nonspecific protease digestions[END_REF].

Another feature of interest in gels used for proteomics applications is the control of the pore size. Ideally, the gel should have a small pore size during electrophoresis to perform an efficient separation, and then a larger pore size to facilitate protease penetration and peptide elution. This means to be able to cleave the crosslinker, ideally only part of the bonds to keep a solid gel structure and not to pollute the extracts with polymers. Regular acrylamide-bisacrylamide gels are quite resistant and require concentrated hydrogen peroxide to be cleaved [START_REF] Donato | Enhancement of polyacrylamide gel slice dissolution in hydrogen peroxide by cupric sulfate[END_REF]. These conditions would be of course completely incompatible with any decent peptide analysis. Thus, it is required to change the crosslinker for manipulating the pore size, and quite different cleavable crosslinkers have been developed over the years (reviewed in [START_REF] Gelfi | Polymerization Kinetics of Polyacrylamide Gels .1. Effect of Different Cross-Linkers[END_REF]). However, these crosslinkers are not always easy to use. Some (e.g. the allyl derivatives) polymerize very poorly [START_REF] Gelfi | Polymerization Kinetics of Polyacrylamide Gels .1. Effect of Different Cross-Linkers[END_REF], Others such as ethylene diacrylate and bisacrylyl cystamine, cleave under the conditions used for SDS PAGE, i.e. during the migration, which is of course to be avoided. Thus, the only successful reports in proteomic-type applications up to now use either DHEBA with subsequent basecatalyzed cleavage [START_REF] Airey | Use of a Reversible Polyacrylamide-Gel Cross-Linker in Western Blotting for Rapid Transfer of a Wide Size Range of Polypeptides[END_REF], or a new acid-cleavable crosslinker [START_REF] Kim | Isolation of intact proteins from acid-degradable polyacrylamide gel[END_REF]. In the case of DHEBA, however, it must be stressed that this report used immunoblotting as the protein probing technique. As cleavage of DHEBA under basic conditions will liberate glyoxal (see figure 1), it can be expected that this technique will not be very useful for mass spectrometry-based proteomics setups. This drives to the last point in the gel structure that may interfere with the subsequent proteomic analysis, i.e. the chemical reactivity of the gel itself on the proteins. A perfect polyacryamide gel would be completely unreactive toward proteins, but real-life polyacrylamide gels are not perfect. First of all, the conversion efficiency of the monomer into the polymer is far from perfect under standard conditions [START_REF] Righetti | Polymerization Kinetics of Polyacrylamide Gels .3. Effect of Catalysts[END_REF][START_REF] Righetti | On the kinetics of monomer incorporation into polyacrylamide gels, as investigated by capillary zone electrophoresis[END_REF]. Moreover, a 90% conversion efficiency in a 7.5% gel still means 0.1M of free acrylamide at the end of the polymerization process. It is therefore not surprising that, under the basic conditions prevailing in SDS PAGE, such a concentration of free acrylamide can lead to addition on the most nucleophilic amino acids, e.g. cysteine [START_REF] Bonaventura | Acrylamide in polyacrylamide gels can modify proteins during electrophoresis[END_REF]. In fact, the extent of aminoacid alkylation observed in proteomics is relatively low when compared to this concentration of acrylamide, and this may be due to the use of a glycine-containing buffer in SDS PAGE, glycine acting as a scavenger for unpolymerized acrylamide [START_REF] Geisthardt | Polyacrylamide-Gel Electrophoresis -Reaction of Acrylamide at Alkaline pH with Buffer Components and Proteins[END_REF].

Another problem in the use of classical polyacrylamide gels for proteomics lies in the chemicals used for inducing polymerization. What is commonly called "catalysts" (i.e. TEMED and persulfate) are not catalysts, because they are not recovered unchanged at the end of the process. Their proper naming is initiators, as they initiate the polymerization process. In fact, TEMED-derived species and persulfatederived species can be found at both ends of the polymer chains, as they will graft covalently at the start of the chain to initiate polymerization (see figure 1), but can also play a role in the termination reactions. Thus, many chemicals used to initiate polymerization, although ionic, will not be migrated away during the electrophoretic process, as they are covalently bound to the gel itself. This has been used purposely to graft thiosulfate in the gel in order to decrease the background during silver staining [START_REF] Hochstrasser | Catalysts' for polyacrylamide gel polymerization and detection of proteins by silver staining[END_REF], but the key point is that standard TEMEDpersulfate systems lead to gels that are oxidizing. This has been very clearly demonstrated for IPG gels [START_REF] Righetti | Oxidation of alkaline immobiline buffers for isoelectric focusing in immobilized pH gradients[END_REF], but it is highly likely that the same holds true (although probably to a lesser extent) in SDS gels [START_REF] Brewer | Artifact Produced in Disc Electrophoresis by Ammonium Persulfate[END_REF], and may explain why methionine sulfoxides are so often observed in any type of gel-based proteomics.

Although not widely used, to say the least, there are however polymerization systems that alleviate most of these problems. These systems are based on photopolymerization. The first photopolymerization systems, based on riboflavin [START_REF] Brewer | Artifact Produced in Disc Electrophoresis by Ammonium Persulfate[END_REF], were poorly efficient and rather unreliable. However, it could be demonstrated that photopolymerization did not induce any oxidizing power in the resulting gels [START_REF] Chiari | Polyacrylamide-Gel Polymerization under Nonoxidizing Conditions, as Monitored by Capillary Zone Electrophoresis[END_REF]. However, improved photopolymerization systems based on methylene blue were also introduced [START_REF] Lyubimova | Photopolymerization of Polyacrylamide Gels with Methylene-Blue[END_REF] resulting in excellent monomer conversion in almost any condition of pH [START_REF] Caglio | On the Ph-Dependence of Polymerization Efficiency, as Investigated by Capillary Zone Electrophoresis[END_REF] and solvents [START_REF] Caglio | On the Efficiency of Methylene-Blue Versus Persulfate Catalysis of Polyacrylamide Gels, as Investigated by Capillary Zone Electrophoresis[END_REF]. It was shown, however, that this system is not fully compatible with SDS electrophoresis, due to the cationic nature of methylene blue [START_REF] Rabilloud | Micropreparative One-Dimensional and 2-Dimensional Electrophoresis -Improvement with New Photopolymerization Systems[END_REF]. The solution found was to replace methylene blue by riboflavin phosphate, but also keeping the co-initiators toluene sulfinate and diphenyliodonium chloride. This resulted in efficient and reliable polymerization [START_REF] Rabilloud | Micropreparative One-Dimensional and 2-Dimensional Electrophoresis -Improvement with New Photopolymerization Systems[END_REF], with decreased protein modification (as assessed by Edman sequencing). However, its is clear that such photopolymerization systems, replacing methylene blue with anionic dyes, result in less artefacts in the subsequent proteomic analyses [START_REF] Sun | Prevention of artifactual protein oxidation generated during sodium dodecyl sulfate-gel electrophoresis[END_REF]. Although these photopolymerization systems are cumbersome to use when large number of gels are to be run, as in 2D gels-based proteomics, they should bring some improvements in other proteomics setups using SDS-PAGE, such as the widely used GeLC system [START_REF] Lasonder | Analysis of the Plasmodium falciparum proteome by high-accuracy mass spectrometry[END_REF].

Variations in the buffer systems

Due to their high resolution, only discontinuous systems are used. In most cases, the original Tris-HCl-glycine system is used [START_REF] Davis | Disc Electrophoresis .2. Method and Application to Human Serum Proteins[END_REF], although many systems can be devised by application of the discontinuous electrophoresis theory [START_REF] Jovin | Multiphasic Zone Electrophoresis .1. Steady-State Moving-Boundary Systems Formed by Different Electrolyte Combinations[END_REF][START_REF] Jovin | Multiphasic Zone Electrophoresis .2. Design of Integrated Discontinuous Buffer Systems for Analytical and Preparative Fractionation[END_REF]. A simple point that must be kept as a reminder for anionic systems, as those used for SDS-PAGE, is that for any given system, the mobility of the ions is pH-dependent. The higher the pH, the faster the moving boundary, and thus the more the system is able to resolve low molecular mass proteins, at the expense of the resolution space allocated to high molecular mass proteins. Conversely, the lower the pH, the more space is allocated to resolve high molecular mass proteins, with the low molecular mass ones comigrating with the ion boundary. In the case of the popular Tris-glycine system, it is possible to play with the pH in mainly one direction, i.e. lowering it. With a gel pH of 8.8, the system operates at a pH of 9.5, and further increase in the pH results in polyacrylamide hydrolysis, with severe consequences on separation performances. Furthermore, 8.8 is already quite far from the pK of Tris (8.05) so that the buffering power of the system drops at higher pH. Thus the upper limit for use of the glycine-based system and its application for resolving low molecular mass proteins has been reached with the ammediol-glycine system [START_REF] Bury | Analysis of Protein and Peptide Mixtures -Evaluation of Three Sodium Dodecyl Sulfate-Polyacrylamide Gel-Electrophoresis Buffer Systems[END_REF]. This system has never reached strong popularity because it operates too close to the basic limits tolerated by acrylamide. Furthermore, it should be kept in mind that basic conditions promote deamidation not only on the polyacrylamide gel, but also at glutamine and asparagine, i.e. conditions that are not favourable for a proteomic analysis. Conversely, lowering the pH in the tris-glycine system has several advantages such as a better buffering power and a better resolution of high molecular mass proteins, as first exemplified by Johnson [START_REF] Johnson | Enhanced Resolution in Two-Dimensional Electrophoresis of Low-Molecular-Weight Proteins While Utilizing Enlarged Gels[END_REF], and further extended to giant proteins (1MDa) by Fritz et al. [START_REF] Fritz | Factors Affecting Polyacrylamide-Gel Electrophoresis and Electroblotting of High-Molecular-Weight Myofibrillar Proteins[END_REF].

When resolution of low molecular mass proteins at a more neutral pH is needed, it becomes necessary to change the trailing ion, and to opt for chemicals with a lower pK, so that their mobility will be higher for the same pH. The extreme in this direction is represented by the MES system [START_REF] Kyte | A Discontinuous Electrophoretic System for Separating Peptides on Polyacrylamide Gels[END_REF] (pK = 6 for MES instead of pK= 9.74 for glycine). However, the most popular systems are based on other Good's buffers, namely Bicine [START_REF] Wiltfang | A New Multiphasic Buffer System for Sodium Dodecyl Sulfate-Polyacrylamide Gel-Electrophoresis of Proteins and Peptides with Molecular Masses 100 000-1000, and Their Detection with Picomolar Sensitivity[END_REF] and Tricine [START_REF] Schägger | Tricine Sodium Dodecyl-Sulfate Polyacrylamide-Gel Electrophoresis for the Separation of Proteins in the Range from 1-Kda to 100-Kda[END_REF], as these system show good performances for low molecular mass proteins at a gel pH lesser that 8.5, i.e. with very good buffering power. In 2D gel systems, the Tris-Tricine system has been successfully used in cases where the resolution of low molecular mass proteins is needed, e.g. for mitochondrial proteins [START_REF] Kruft | Proteomic approach to identify novel mitochondrial proteins in Arabidopsis[END_REF], but also for microorganisms [START_REF] Hernychova | Construction of a Francisella tularensis two-dimensional electrophoresis protein database[END_REF][START_REF] Fountoulakis | Reference map of the low molecular mass proteins of Haemophilus influenzae[END_REF]. Further developments of the Tricine-based system have been made to replace the glycine-based system. In fact, a Tricine system operating at gel pH of 7 has the same separation window than a glycine system operating at gel pH of 8.8 [START_REF] Patton | Tris Tricine and Tris Borate Buffer Systems Provide Better Estimates of Human Mesothelial Cell Intermediate Filament Protein Molecular-Weights Than the Standard Tris Glycine System[END_REF], with the advantages of lesser hydrolysis of amides (and thus longer shelf life for ready to use gels) and of lesser reactivity of nucleophilic amino acids toward spurious chemicals such as residual acrylamide. However, operating at pH 7 also poses buffering problems with Tris, and the situation is further complicated by the fact that many low pK bases, which could replace Tris as gel buffers interfere strongly with polyacrylamide polymerization, either by inhibiting it (e.g. imidazole) or by acting as polymerization catalysts (e.g. triethanolamine), leading to excessive heat production during polymerization and thus irregular gels.

Thus, the two most popular electrophoresis systems operate in their own separate niche, Tricine (pK 8.05) for low molecular mass proteins, and glycine (pK 9.74) for medium and high molecular mass proteins. This suggests that a system using a trailing ion of intermediate pK (i.e. ca. 9) should be able to operate on the whole spectrum with Tris as the only buffer component needed. Indeed, a taurine-based system is able to operate over the whole spectrum, at the expense of a precise control of the gel pH [START_REF] Tastet | A versatile electrophoresis system for the analysis of high-and low-molecular-weight proteins[END_REF] (see figure 2). Here again, 2D electrophoresis of mitochondrial proteins has shown the ability of this system to separate low-molecular mass proteins without increasing acrylamide concentration too much [START_REF] Lescuyer | Progress in the definition of a reference human mitochondrial proteome[END_REF]. However, this system is clearly not limited to low molecular mass proteins [START_REF] Linker | Proteome Profiling in Murine Models of Multiple Sclerosis: Identification of Stage Specific Markers and Culprits for Tissue Damage[END_REF][START_REF] Brechlin | Cerebrospinal fluid-optimized two-dimensional difference gel electrophoresis (2-D DIGE) facilitates the differential diagnosis of Creutzfeldt-Jakob disease[END_REF] An additional, but less understood, phenomenon taking place when changing from one buffer system to another one consists in alterations of the relative mobilities of proteins in the SDS gel. Theoretically, proteins migrate according to their molecular mass in SDS gels. However, there are exceptions to this basic rule in every buffer system, but the ones observed in the widely used glycine system are of course more documented. For example, it has been shown that Tricine and borate-based buffers show an altered migration of some cytoskeletal proteins, and indeed more accurate compared to their molecular mass [START_REF] Patton | Tris Tricine and Tris Borate Buffer Systems Provide Better Estimates of Human Mesothelial Cell Intermediate Filament Protein Molecular-Weights Than the Standard Tris Glycine System[END_REF]. This phenomenon can be used to enhance separation of some proteins in some crowded zones in 2D gels, although the results cannot be predicted.

Variations in the detergent

As the name says, SDS PAGE is carried out in the presence of dodecyl sulfate as the primary solubilizing and charge-conferring agent. The performances of this detergent are documented by more than 40 years of use and thousands of satisfied scientists. However, there are cases in proteomics studies where it would be nice to have even better performances, either in the separation process itself or in the post-separation process, when the SDS gel in interfaced with the process leading to identification by mass spectrometry.

As to the separation itself, there are cases in the course of a proteomics study where the study gets focused on a few proteins, and it is the desirable to separate the proteins of interest from other proteins migrating very close to them, or even to separate variants of the proteins of interest. To this purpose, the detergent used can be altered, on order to induce differential detergent binding and thus differential mobility. This has been observed empirically when comparing different sources of SDS [START_REF] Swaney | Sodium dodecylsulfate-dependent anomalies in gel electrophoresis: alterations in the banding patterns of foot-andmouth disease virus polypeptides[END_REF], and the concept has been rationalized by mimicking the detergent composition most able to induce separation of normally comigrating proteins [START_REF] Brown | Mixed Anionic Detergent Aliphatic Alcohol Polyacrylamide-Gel Electrophoresis Alters the Separation of Proteins Relative to Conventional Sodium Dodecyl-Sulfate Polyacrylamide-Gel Electrophoresis[END_REF]. Such alterations of the detergent composition may prove quite efficient in some cases. It must be kept in mind however, that the resolution and performances of SDS electrophoresis rely on a strong binding of numerous detergent molecules to proteins, so that the space for variations is rather limited, as shown in [START_REF] Lopez | Effect of Various Detergents on Protein Migration in the 2nd-Dimension of 2-Dimensional Gels[END_REF].

As to the post-separation steps, SDS, as most detergents, is not the mass spectrometer best friend, and it would be nice use detergents that can be chemically cleaved at will into products that are more mass spectrometry-friendly. Two families of cleavable anionic detergents have been described to date, acid-cleavable detergents [START_REF] Ross | Identification of proteins from two-dimensional polyacrylamide gels using a novel acid-labile surfactant[END_REF] and photocleavable ones [START_REF] Epstein | The Synthesis of a Photolabile Detergent and Its Use in the Isolation and Characterization of Protein[END_REF]. Only acid-cleavable detergents have been used to a certain extent. However, their separation properties are much poorer than the ones of SDS [START_REF] Ross | Identification of proteins from two-dimensional polyacrylamide gels using a novel acid-labile surfactant[END_REF], as could be anticipated from the stringent constraints applying to detergents for electrophoretic separations [START_REF] Lopez | Effect of Various Detergents on Protein Migration in the 2nd-Dimension of 2-Dimensional Gels[END_REF].

Variations in the first dimension: zone electrophoresis

This section of the review will focus on how to improve and simplify zone electrophoretic separations used in conjunction with SDS PAGE to build twodimensional systems. In order to maximize the resolution, this separation should be as different as possible from the one obtained by SDS PAGE. As the magnitude of this difference is clearly IEF > native zone electrophoresis >> denaturing zone electrophoresis, it is clear that systems using denaturing zone electrophoresis as a first dimension will be used mainly when the other systems do not perform well, e.g. for membrane proteins [START_REF] Hartinger | 16-BAC/SDS-PAGE: A twodimensional gel electrophoresis system suitable for the separation of integral membrane proteins[END_REF][START_REF] Braun | Two-dimensional electrophoresis of membrane proteins[END_REF] As can be seen from [START_REF] Lopez | Effect of Various Detergents on Protein Migration in the 2nd-Dimension of 2-Dimensional Gels[END_REF], electrophoresis in the presence of cationic detergents is clearly the best candidate for denaturing separations, a fact that had been discovered empirically before [START_REF] Macfarlane | Two-Dimensional Benzyldimethyl-N-Hexadecylammonium Chloride-]Sodium Dodecyl-Sulfate Preparative Polyacrylamide-Gel Electrophoresisa High-Capacity High-Resolution Technique for the Purification of Proteins from Complex-Mixtures[END_REF]. Compared to this core protocol, changes and improvements will target the same areas as in SDS PAGE, i.e. gel structure, buffer system and detergent.

Electrophoresis in the presence of cationic detergents proceeds with a reverse polarity compared to SDS gels, and at acidic pH compared to the basic pH used for SDS PAGE. Consequently, polyacrylamide hydrolysis is not a problem in this case. However, gel polymerization cannot proceed with the standard TEMED-persulfate system, as protonated TEMED does not induce persulfate decomposition. In most cases, the Fenton's system described in the original publication [START_REF] Macfarlane | Use of Benzyldimethyl-Normal-Hexadecylammonium Chloride (16-Bac), a Cationic Detergent, in an Acidic Polyacrylamide-Gel Electrophoresis System to Detect Base Labile Protein Methylation in Intact-Cells[END_REF] is used. However, this system is not as reliable as the TEMED-persulfate one, so that the robust methylene blue-based system [START_REF] Lyubimova | Photopolymerization of Polyacrylamide Gels with Methylene-Blue[END_REF] can be used very successfully in this case (see figure 3). As to the detergent, only two cationic detergents have been successfully used to date, the original benzalkonium chloride, either with a pure C16 chain (16-BAC stands for hexadecyl benzyl dimethyl ammonium chloride) or with a mixture of hydrocarbon chains [START_REF] Macfarlane | Use of Benzyldimethyl-Normal-Hexadecylammonium Chloride (16-Bac), a Cationic Detergent, in an Acidic Polyacrylamide-Gel Electrophoresis System to Detect Base Labile Protein Methylation in Intact-Cells[END_REF], and CTAB (hexadecyl trimethylammonium bromide) [START_REF] Navarre | Subproteomics: Identification of plasma membrane proteins from the yeast Saccharomyces cerevisiae[END_REF]. What has changed over the years, however, is the process for interfacing the first dimension on the second one. In the core protocol [START_REF] Macfarlane | Two-Dimensional Benzyldimethyl-N-Hexadecylammonium Chloride-]Sodium Dodecyl-Sulfate Preparative Polyacrylamide-Gel Electrophoresisa High-Capacity High-Resolution Technique for the Purification of Proteins from Complex-Mixtures[END_REF], the cationic gel was first fixed and stained to remove the cationic detergent, and then equilibrated in SDS prior to loading onto the SDS gel. In more recent protocols, this fixation and staining is omitted, and the gel containing the cationic detergent is just rinsed in water prior to equilibration in SDS buffer [START_REF] Navarre | Subproteomics: Identification of plasma membrane proteins from the yeast Saccharomyces cerevisiae[END_REF], or even directly equilibrated in SDS buffer [START_REF] Helling | 2-D differential membrane proteome analysis of scarce protein samples[END_REF].

What remains poorly explored with this technique, however, is the choice of buffers. Because of the presence of the cationic detergent, electrophoresis could theoretically be carried out at any pH. Indeed, in one-dimensional cationic PAGE various buffers have been used, and the very acidic phosphate-glycine buffer used for 16-BAC electrophoresis [START_REF] Macfarlane | Use of Benzyldimethyl-Normal-Hexadecylammonium Chloride (16-Bac), a Cationic Detergent, in an Acidic Polyacrylamide-Gel Electrophoresis System to Detect Base Labile Protein Methylation in Intact-Cells[END_REF] is not the most widely used. Other systems operating at higher pH, such as the acetate-beta alanine buffer [START_REF] Mòcz | Use of Cationic Detergents for Polyacrylamide-Gel Electrophoresis in Multiphasic Buffer Systems[END_REF] or even the Tricine-arginine buffer [START_REF] Akins | Cetyltrimethylammonium Bromide Discontinuous Gel-Electrophoresis -Mr-Based Separation of Proteins with Retention of Enzymatic-Activity[END_REF] have also been described. Indeed, the phosphate-glycine buffer system operates at a pH where the classical calculations for discontinuous systems are no longer valid, so that an electrophoretic system specially designed for electrophoresis in the presence of cationic detergents has been recently proposed [START_REF] Kramer | A new multiphasic buffer system for benzyldimethyl-nhexadecylammonium chloride polyacrylamide gel electrophoresis of proteins providing efficient stacking[END_REF]. This buffer system uses methoxyacetic as the buffering compound. The separating gel is cast at pH 3 and operates at pH 2.5, and the stacking gel is cast at pH 4 (acetate buffer) and operates at pH 2.9. For comparison, in the phoshate buffer system [START_REF] Macfarlane | Use of Benzyldimethyl-Normal-Hexadecylammonium Chloride (16-Bac), a Cationic Detergent, in an Acidic Polyacrylamide-Gel Electrophoresis System to Detect Base Labile Protein Methylation in Intact-Cells[END_REF] the separating gel is cast at pH 2 and operates at pH 1.5, and the stacking gel is cast at pH 4 and operates at pH 2.2. However, it has been recently shown that the operative pH seems to play an important role for protein solubility [START_REF] Rabilloud | Fully denaturing two-dimensional electrophoresis of membrane proteins: A critical update[END_REF], a feature that can be of weak importance when working with soluble proteins, as in [START_REF] Mòcz | Use of Cationic Detergents for Polyacrylamide-Gel Electrophoresis in Multiphasic Buffer Systems[END_REF][START_REF] Akins | Cetyltrimethylammonium Bromide Discontinuous Gel-Electrophoresis -Mr-Based Separation of Proteins with Retention of Enzymatic-Activity[END_REF][START_REF] Kramer | A new multiphasic buffer system for benzyldimethyl-nhexadecylammonium chloride polyacrylamide gel electrophoresis of proteins providing efficient stacking[END_REF], but this is not necessarily the case with membrane proteins, which are the most important field of application of this technique in proteomics [START_REF] Braun | Two-dimensional electrophoresis of membrane proteins[END_REF].

Variations in the first dimension: isoelectric focusing

In a sense, the constraints existing on isoelectric focusing gels are the contrapositive of those existing on SDS PAGE, so that the areas for flexibility and improvement are quite different. In SDS PAGE, the gel is simple to make and many buffers can be used, so that there is a lot of flexibility in these two areas, as shown above. However, there are a lot of constraints on the detergent as the protein-solubilizing agent, so that the flexibility is quite limited in this area.

In isoelectric focusing, the difficult task is to form the pH gradient, and thus the buffers are extremely constrained. Now that immobilized pH gradient dominate the field of isoelectric focusing, these constraints on the buffers have also translated into constraints on the gel, as immobilized pH gradients are by definition grafted into the gel. Just as an example, the standard gel for SDS PAGE is made of just two monomers of quite related structure (acrylamide and Bis). A wide range immobilized pH gradient (e.g. 3-10) uses eight different Immobilines plus the two standard monomers, i.e. ten monomers that must be copolymerized with equal efficiency to ensure the accuracy of the gradient, and this has been shown not to be obvious [START_REF] Righetti | Polymerization Kinetics of Polyacrylamide Gels Containing Immobilized pH Gradients for Isoelectric-Focusing[END_REF]. Changing the main gel-forming monomer (acrylamide) would mean to reassess the polymerization efficiency of all Immobilines with this new monomer, and it is therefore not surprising that such changes in the monomer have been limited to cases where acrylamide was bound to fail, i.e. isoelectric focusing in very basic pH gradients [START_REF] Görg | Very alkaline immobilized pH gradients for two-dimensional electrophoresis of ribosomal and nuclear proteins[END_REF].

Oppositely to the situation of the IEF gels, there is a lot of flexibility in the area of solubilizing agents used for IEF. In fact, on the one hand there is a single chemical constraint, i.e. the solubilizing agents must be non ionic and shall not change the charges of the proteins, and on the other hand it is quite difficult to keep the proteins in solution under such conditions of low ionic strength and with no charge modifications. In fact, isoelectric precipitation is as old as isoelectric focusing (e.g. in [START_REF] Rathnam | Isolation and physicochemical characterization of luteinizing hormone from human pituitary glands[END_REF]). Thus, considerable effort has been devoted to improving this situation, and this has resulted in many additives used to increase protein solubility during isoelectric focusing.

In proteomics, isoelectric focusing is always conducted under denaturing conditions, i.e. inducing protein unfolding. Furthermore, disulfide bridges are also reduced to induce complete dissociation of proteins into separate, unfolded polypeptides. From the very beginning of 2D gels, protein solubilization has been carried out with neutral chaotropes such as urea [START_REF] Macgillivray | The heterogeneity of mouse-chromatin nonhistone proteins as evidenced by two-dimensional polyacrylamide-gel electrophoresis and ion-exchange chromatography[END_REF], and most often used in conjunction with detergents [START_REF] Farrell | High resolution two-dimensional electrophoresis of proteins[END_REF]. Thus, the variations that have been brought to this basic protocol have touched the chaotropes and the detergents.

As to chaotropes, there is relatively little choice, chemically speaking, as the chaotrope must remain neutral over the whole pH range. This limits the choice to urea, alkylureas, thiourea, and combinations thereof. Alkylureas proved completely inefficient [START_REF] Rabilloud | Improvement of the solubilization of proteins in two-dimensional electrophoresis with immobilized pH gradients[END_REF], as could be expected from their weak denaturing power [START_REF] Gordon | Relationship of Structure to Effectiveness of Denaturing Agents for Proteins[END_REF]. Oppositely, mixtures of thiourea and urea proved more efficient than urea alone, here again in line with the denaturing power observed in solution [START_REF] Gordon | Relationship of Structure to Effectiveness of Denaturing Agents for Proteins[END_REF]. This was shown to be true in every isoelectric focusing setup, i.e. carrier ampholytes IEF in agarose [START_REF] Oh-Ishi | Micro-two-dimensional gel electrophoresis with agarose gel in the first dimension[END_REF], IEF with immobilized pH gradients [START_REF] Rabilloud | Improvement of the solubilization of proteins in two-dimensional electrophoresis with immobilized pH gradients[END_REF], and carrier ampholytes IEF in polyacrylamide [START_REF] Rabilloud | Use of thiourea to increase the solubility of membrane proteins in two-dimensional electrophoresis[END_REF]. The latter setup, however, is the most difficult to use, as thiourea inhibits acrylamide polymerization, thereby necessitating the use of the highly efficient methylene blue-based photopolymerization initiator [START_REF] Lyubimova | Photopolymerization of Polyacrylamide Gels with Methylene-Blue[END_REF][START_REF] Rabilloud | Use of thiourea to increase the solubility of membrane proteins in two-dimensional electrophoresis[END_REF].

The situation is quite different for detergents, where any detergent non ionic over the pH range of interest can be used in IEF. Furthermore, the incentive of solubilizing membrane protein in 2D gels has always been a driving force to test non-classical detergents in conjunction with urea as protein solubilizers for 2D gels. As Triton X-100 is known to be one of the best solubilizers of the polyethylene glycol-based detergents [START_REF] Umbreit | Relation of Detergent HLB Number to Solubilization and Stabilization of D-Alanine Carboxypeptidase from Bacillus-Subtilis Membranes[END_REF], the interest has turned to other classes of uncharged detergents, such as glucosides [START_REF] Witzmann | Dodecyl Maltoside Detergent Improves Resolution of Hepatic Membrane-Proteins in 2-Dimensional Gels[END_REF], but also members of the sulfobetaine class. The first one to be used, and by far still the most popular, was CHAPS [START_REF] Perdew | The Use of a Zwitterionic Detergent in Two-Dimensional Gel-Electrophoresis of Trout Liver-Microsomes[END_REF]. Linear sulfobetaines, i.e. the most potent nonionic detergents [START_REF] Navarrete | Solubilization of Yeast Plasma-Membranes and Mitochondria by Different Types of Non-Denaturing Detergents[END_REF], have also been tested. However, as they are not fully compatible with urea, this required decreasing the urea concentration [START_REF] Gyenes | Effect of Stacking on the Resolving Power of Ultrathin-Layer Two-Dimensional Gel-Electrophoresis[END_REF]. This poor compatibility (which is temperature-dependent) has considerably impaired their use in 2D gels [START_REF] Satta | Solubilization of Plasma-Membranes in Anionic, Non-Ionic and Zwitterionic Surfactants for Iso-Dalt Analysis -a Critical-Evaluation[END_REF]. To alleviate this problem, special sulfobetaines with improved compatibility were prepared and used [START_REF] Gianazza | Additives for Immobilized pH Gradient Two-Dimensional Separation of Particulate Material -Comparison between Commercial and New Synthetic Detergents[END_REF][START_REF] Rabilloud | Amidosulfobetaines, a Family of Detergents with Improved Solubilization Properties -Application for Isoelectric-Focusing under Denaturing Conditions[END_REF], at that time mainly with descriptive results. However, when protein identification techniques developed, it became possible to assess more thoroughly the real gain in protein solubilization obtained by various detergents, and especially for membrane proteins. The gains were assessed for sulfobetaines [START_REF] Chevallet | New zwitterionic detergents improve the analysis of membrane proteins by twodimensional electrophoresis[END_REF][START_REF] Rabilloud | Analysis of membrane proteins by two-dimensional electrophoresis: Comparison of the proteins extracted from normal or Plasmodium falciparum -infected erythrocyte ghosts[END_REF], glucosides [START_REF] Taylor | Enhanced resolution of glycosyl-phosphatidylinositolanchored and transmembrane proteins from the lipid-rich myelin membrane by twodimensional gel electrophoresis[END_REF][START_REF] Luche | Evaluation of nonionic and zwitterionic detergents as membrane protein solubilizers in two-dimensional electrophoresis[END_REF], phosphocholines [START_REF] Babu | Solubilization of membrane proteins for two-dimensional gel electrophoresis: identification of sarcoplasmic reticulum membrane proteins[END_REF] and even more classical, polyethylene glycol-based detergents [START_REF] Luche | Evaluation of nonionic and zwitterionic detergents as membrane protein solubilizers in two-dimensional electrophoresis[END_REF]. While the positive trends observed earlier on glucosides and sulfobetaines were confirmed by these studies (see an example in figure 4), it was shown quite surprisingly that some of the polyethylene glycol-based detergents perform better than CHAPS in urea plus thiourea chaotrope, while the reverse is true in urea alone. Despite the documented gains in protein solubility, it is now quite clear that all these variations have proven unable to solubilize a high proportion of the membrane proteins expressed by cells [START_REF] Rabilloud | Membrane proteins and proteomics: love is possible, but so difficult[END_REF].

Besides solubilization of proteins by chaotropes and detergents, the problems caused by cysteines have brought their share of modifications compared to the original protocols. As mentioned earlier, denaturing isoelectric focusing implies reduction of the cysteines in order to separate polypeptidic chains. No problems were encountered as long as carrier ampholytes-based IEF was used, i.e. not extending much higher than pH 7, or when transient focusing systems were used for basic proteins [START_REF] O'farrell | High-Resolution 2-Dimensional Electrophoresis of Basic as Well as Acidic Proteins[END_REF]. However, with the use of immobilized pH gradients, it became obvious that cysteine oxidation was a real problem in the basic pH intervals [START_REF] Altland | Isoelectric-Focusing of Basic-Proteins -the Problem of Oxidation of Cysteines[END_REF]. Indeed, the thiol compounds used for disrupting disufide bridges and keeping cysteines in their reduced form behave as weak acids, and migrate toward the anode when the pH is high enough. This fact has been used in SDS gels to keep giant proteins reduced [START_REF] Fritz | Factors Affecting Polyacrylamide-Gel Electrophoresis and Electroblotting of High-Molecular-Weight Myofibrillar Proteins[END_REF]. Thus, in an IEF setup, the basic portion of the pH gradient becomes devoid of reducing agents, so that cysteines can reoxidize in many ways, including disulfide bridges, which induces a lot of trailing of the proteins in the basic region of the gels. To solve this problem, three strategies have been proposed. The first one consists in the continuous infusion of thiol-containing reducers from the cathode [START_REF] Hoving | Preparative two-dimensional gel electrophoresis at alkaline pH using narrow range immobilized pH gradients[END_REF], so that the proteins are kept in a reducing environment. The obvious problem with this strategy is its absolute need for a careful tuning and timing, in other words its lack of robustness. The second strategy consists in alkylating the cysteines, so that they become chemically inert and cannot either make spurious disulfide bridges nor react on unreacted monomers. Although this strategy seems straightforward and efficient, it has been shown that it does work properly [START_REF] Galvani | Alkylation kinetics of proteins in preparation for two-dimensional maps: A matrix assisted laser desorption/ionizationmass spectrometry investigation[END_REF]. The issue is that various cysteines on different proteins show quite different reactivities, depending on their environment and thus on the neighbouring amino acids. Consequently, it is almost impossible to find conditions where all the cysteines will be alkylated and none of the other nucleophilic amino acids (e.g. lysine, tyrosine). This unfortunately holds true for a wide range of alkylating agents [START_REF] Luche | About thiol derivatization and resolution of basic proteins in two-dimensional electrophoresis[END_REF]. Thus, only the third strategy has proven efficient. It consists in blocking the cysteines of the proteins as disulfide bridges, using a vast excess of a low molecular mass disulfide [START_REF] Olsson | Organic disulfides as a means to generate streak-free two-dimensional maps with narrow range basic immobilized pH gradient strips as first Product dimension[END_REF]. Because of the very high specificity of the reaction, high concentrations of the disulfide can be used, thereby ensuring maximum blocking of the protein thiols. As an added benefit of this strategy, it is possible to perform a simplified equilibration process, in which no alkylation of the cysteines is needed, opposite to the standard process with immobilized pH gradients [START_REF] Görg | Elimination of Point Streaking on Silver Stained Two-Dimensional Gels by Addition of Iodoacetamide to the Equilibration Buffer[END_REF]. Furthermore, as this process is reversible, either the cysteines can be left protected as disulfides until the mass spectrometry step, using the mass of the adduct as a fixed modification, or the cysteines can be alkylated after spot excision, prior to digestion, extraction and mass spectrometry.

Concluding remarks

Prefractionation of samples is generally a pre-requisite in proteomics, and protein fractionation by electrophoresis is quite a versatile and reliable toolbox to perform such prefractionation. As this toolbox has now been used for decades by biochemists and proteomicists, it now relies on a corpus of well-established, robust and efficient techniques and protocols. However, in the course of a proteomic study, there are often cases where it would be very advantageous to be able to tune the separation. This means to increase the resolution and/or the solubilization for proteins of interest, even if it means loosing resolution or solubilization for other proteins that are no longer of interest when the initial proteomics screening process has been achieved. Numerous ways of tuning electrophoretic separations have been described over the years in the literature, and this review cannot claim to be comprehensive for all these modifications. However, many of them are not widely known nowadays. Thus, the main paths for modifications have been reviewed here, in order to provide the reader the rationale and references to be able to design his own separation system tuned to his own needs, and to adapt to the demands risen by different proteomic projects. 3: separation of membrane proteins by double zone electrophoresis A membrane preparation from B. subtilis was used as the sample. 100µg proteins, solubilized in 3.75M urea, 2% 16-BAC, 5mM tris-carboxyethylphosphine, 100mM phosphate buffer pH3, were loaded on the first dimension rod gel, containing 7.5 % acrylamide, 3.75M urea, 0.1% 16-BAC, 200mM phosphate buffer pH 2.1, and photopolymerized with 1mM sodium toluene sulfinate, 20 µM diphenyliodonium chloride and 30µM Methylene Blue. The electrode buffer contained phosphate, glycine and 0.1% 16-BAC Pyronin Y was used as a tracking dye for the end of the first migration (polarity indicated on the figure). The rod gel was then extruded and equilibrated in high SDS buffer (2.5%) for 15 minutes, before sealing on top of the second dimension, SDS PAGE gel (10% acrylamide). Detection by silver staining. The limited streaking and numerous spots show the efficiency of the whole method, including the polymerization process. 16-BAC stands for hexadecyl benzyl dimethyl ammonium chloride.

Figure 1 :

 1 Figure 1: scheme of a polyacrylamide gel

Figure 2 :

 2 Figure 2: Separation modulation in the SDS dimension by a pH change 120 µg of proteins extracted from HeLa cells were used as a sample, and separated on a linear 4-8 immobilized pH gradient (70,000 Vh). The strips were then loaded on top of a 10% polyacrylamide gel and electrophoresed in a Tris-HCl-Taurine system. Detection by silver staining. Molecular mass markers were loaded on the side of the SDS gel. Panel A : gel buffer: Tris-HCl pH 8.05 Panel B: gel buffer: Tris-HCl pH 7.75 A few equivalent proteins on both gels are shown by a box and arrows

Figure

  Figure 3: separation of membrane proteins by double zone electrophoresis A membrane preparation from B. subtilis was used as the sample. 100µg proteins, solubilized in 3.75M urea, 2% 16-BAC, 5mM tris-carboxyethylphosphine, 100mM phosphate buffer pH3, were loaded on the first dimension rod gel, containing 7.5 % acrylamide, 3.75M urea, 0.1% 16-BAC, 200mM phosphate buffer pH 2.1, and photopolymerized with 1mM sodium toluene sulfinate, 20 µM diphenyliodonium chloride and 30µM Methylene Blue. The electrode buffer contained phosphate, glycine and 0.1% 16-BAC

Figure 4 :

 4 Figure 4: differential protein extraction by detergents in IEF-based 2D electrophoresis A membrane preparation from B. subtilis was used as the sample. 100µg proteins, solubilized in 7M urea, 2M thiourea, 2% detergent, 0.4% carrier ampholytes and 50mM DTT, were separated by IEF in immobilized pH gradients (same buffer as for extraction, linear 4-8 pH gradient). Migration for 70,000 Vh Second dimension: 10% acrylamide gel. Panel A: extraction and migration using CHAPS as the detergent Panel B: extraction and migration using Brij56 as the detergent Spots solubilized and focused by Brij56 and not by CHAPS are shown by arrows


Lefort Mathieu

Boniface Yann

Girau Bernard

Self-organization of neural maps using a modulated BCM rule within a multimodal architecture

Human beings interact with the environment through different modalities, i.e. perceptions and actions. Different perceptions as view, audition or proprioception for example, are picked up by different spatially separated sensors. They are processed in the cortex by dedicated brain areas, which are self-organized, so that spatially close neurons are sensitive to close stimuli. However, the processings of these perceptive flows are not isolated. On the contrary, they are constantly interacting, as illustrated by the McGurk effect. When the phonetic stimulus /ba/ is presented simultaneously with a lip movement corresponding to a /ga/, people perceive a /da/, which does not correspond to any of the stimuli. Merging several stimuli into one multimodal perception reduces the ambiguities and the noise of each perception. This is an essential mechanism of the cortex to interact with the environment. The aim of this article is to propose a model for the assembling of modalities, inspired by the biological properties of the cortex. We have modified the Bienenstock Cooper Munro (BCM) rule to include it in a model that consists of interacting maps of multilayer cortical columns. Each map is able to self-organize thanks to a continuous decentralized and local learning modulated by a high level signal. By assembling different maps corresponding to different modalities, our model creates a multimodal context which is used as a modulating signal and thus it influences the self-organization of each map.

Introduction

The cortex is divided into several areas, each one having a specific function. Some of them are dedicated to compute a perceptive flow as visual areas V1 to V5. Each area is made up of multilayer cortical columns. Thanks to this generic structure, areas are Lefort Mathieu, Boniface Yann, Girau Bernard LORIA, Campus Scientifique -BP 239 -54506 Vandoeuvre-lès-Nancy Cedex e-mail: {mathieu.lefort,yann.boniface,bernard.girau}@loria.fr able to process other functions, depending on the type of inputs they receive, as it can be seen in perceptive cortical areas of disabled people.

An object is defined by a set of perceptions and affordances. This term, defined in Gibson's theory [START_REF] Gibson | The theory of affordances[END_REF], means the possible interactions between an object and a living being. Examples of affordances of a tree are "lie down" or "climb on" for a human or "perch on" for a bird. In order to perceive an object, the cortex has to compute this set of modalities as a single multimodal stimulus.

Although perceptions from an object reach the cortex through different sensors, the cortex merges them in areas named associative maps to form a unified representation. Associating perceptions and actions to make a unique multimodal stimulus is an essential mechanism to interact with the environment. This makes it possible to recall one perception from another one. For example, an adult is able to identify an object he only touches among a set of previously seen objects.

Perceptual flows are not only associated in the cortex but they interact with each other. These influences are easily detectable. For example, when someone is watching a ventriloquist show, his ears perceive a sound coming from the ventriloquist and his eyes see the lip movements of the puppet. Visual perception influences the auditory one that appears to come from the puppet. Bonath and al. [START_REF] Bonath | Neural basis of the ventriloquist illusion[END_REF] have shown that, during the ventriloquist illusion, activity in the auditory area first matches the real sound perception and secondly, after the associative feedback, activity raises in the zone that corresponds to the perceived sound location.

Moreover, perceptive merging is useful to increase our perception threshold. Goldring and al. [START_REF] Goldring | Combined eye-head gaze shifts to visual and auditory targets in humans[END_REF] show that the reaction time to a monomodal stimulus (visual or auditory) is longer than to the multimodal one. The gain may be the consequence of the reduction of noise in multimodal stimuli due to an increase of information quantity.

The context of our study is the design of a multimap, multimodal and modular architecture to process perception/action tasks. The aim is not to model the cortex but to be inspired by its properties such as the genericity of its architecture, the self-organization and plasticity of perceptive areas, the merging of perceptions and their reciprocal influences to form a unified view of the environment. This paradigm raises the question of how to associate different modalities, especially how to obtain the self-organization of a perceptive map in a multimodal context thanks to a continuous learning? To answer this question, we propose a model of perceptive neural map which self-organizes thanks to a continuous decentralized and local learning modulated by a high level signal. By assembling several perceptive maps around an associative one, we create a multimodal context which is used as a modulating signal to influence the self-organization of each map.

We will first explain more precisely the objectives of our study and then describe the association architecture named Bijama [START_REF] Ménard | Model of multi-modal cortical processing: coherent learning in self-organizing modules[END_REF], on which our study is based. In the third section, we will describe the Bienenstock Cooper Munro (BCM) rule that we introduce in the Bijama model to solve some of its problems. In section 4, we modify the BCM rule by modulating it with a multimodal high level signal. Section 5 shows results that confirm the validity of our model.

Objectives

Our goal is to develop a multimap and multimodal architecture inspired by the properties of the cortex to perform perception/action tasks or missing modality recall (see figure 1). The system consists of neural maps which may be modality maps processing all kinds of perceptions or actions, or associative maps linking several modality maps. Each map has a 2D architecture and intra and extra map connections, which all are generic. Each modality map self-organizes in the multimodal context thanks to a continuous decentralized and local learning. The design of this architecture raises a lot of problems. The first one is how to obtain the self-organization of a map with a continuous decentralized and local learning, especially in a multimodal context. For example, the self-organizing map model proposed by Kohonen [START_REF] Kohonen | Self-organization and associative memory[END_REF] works with a centralized learning and a decreasing learning distance. The second main problem is how to associate modality maps to influence perceptions and to provide a recall mechanism.

This article focuses on the first point and it answers by modulating the BCM rule by a multimodal high level signal. This modification is introduced within the Bijama model (see section 3) which provides a solution to the second question.

3 Bijama architecture

General description

In [START_REF] Ménard | Model of multi-modal cortical processing: coherent learning in self-organizing modules[END_REF], Ménard and Frezza-Buet propose Bijama: a multimap and multilayer model for multimodal processing. Two types of maps are available: perceptive and associative. Both are round maps made up of generic multilayer cortical columns. To compute its output, each layer of a cortical column has access to the previous layer output and to the upper layer outputs of the columns it is connected to. Perceptive maps self-organize thanks to a continuous decentralized local learning of input weights. Perceptive maps are reciprocally connected with strips to an unique associative map which represents the unified multimodal context. These strip connections influence the sensory perceptions and continuously learn modalities correspondences.

We first describe the role of each layer of a cortical column in a perceptive and associative map in the two next sections respectively. Then, we present how maps are interconnected and the learning of the system. We conclude by an analysis of the inherent problems of the multimodal association and of the limitations of this model.

Perceptive map

A cortical column of a perceptive map is made up of four layers (see figure 2 (A)). The first one (at the bottom) is a perceptive layer based on Kohonen's model, its value depending on the difference between its input, corresponding to a perception, and its weights. The next layer is a cortical layer integrating multimodal constraints. Its output is the value of the strip connection to the associative map, i.e. the weighted sum of the outputs of the cortical columns that lie in the connection (see subsection 3.4 for precisions on inter map connectivity). The third layer merges the two previous ones to influence the perception by the multimodal context. This is done by the product of the cortical layer activity, plus a leaking term, with the perceptive activity. The leaking term is useful when multimodal context is not yet consistent. Finally, the upper layer takes the previous layer as input and it computes a competition between the different cortical columns of the same map. In this upper layer, all cortical columns are connected with a difference of gaussian shape. It is based on Amari's equation of continuous neural field theory (CNFT) [START_REF] Amari | Dynamics of pattern formation in lateral-inhibition type neural fields[END_REF], providing an activity bump in the map as output.

Associative map

An associative cortical column has a various number of layers depending on how many perceptive maps are connected to it (see figure 2 (B)). Each one of the first layers, named cortical layers, reflect the activity in a strip of the corresponding perceptive map by a weighted sum of the outputs of the cortical columns (see subsection 3.4 for precisions on inter map connectivity). The next layer integrates all the perceptive information by multiplying the values of the previous layers plus a leaking term. The leaking term, as in the perceptive cortical column, is useful when the different perceptions are inconsistent. The upper layer is the same as in a perceptive cortical column, taking the previous layer as input and providing a competitive activity bump, that represents the multimodal context, as map output.

Connectivity

Perceptive maps are reciprocally connected to an associative map as described in figure 3. These connections have strip shapes, inspired by the cortical column model of Burnod [START_REF] Burnod | An adaptive neural network: the cerebral cortex[END_REF].

(A) (B)
Fig. 3 The different perceptive maps are reciprocally connected to an unique associative map with a strip connectivity topographically organized. (A) The associative map is connected to all perceptive maps. The first layers (in grays) of an associative cortical column are each connected to a different perceptive map with a strip connectivity depicted in corresponding gray. (B) Each perceptive map is connected to the associative map using the same strip connectivity. See subsection 3.2 (resp. 3.3) for precisions on the perceptive (resp. associative) cortical column architecture.

Strip connectivity introduces constraints in the multimodal integration. Indeed, if the perceptive activity bumps are not wholly consistent, meaning that they are not located in strips that cross in a single point (see figure 4 (B)), activity in the associative map remains low and is badly located. In this case, the relaxation of the strips constraints, through the dedicated layer in each cortical column, leads the perceptive bumps to move until they reach an equilibrium point (see figure 4 (A)).

Learning

Learning in the Bijama model is continuous and decentralized. This allows the system to adapt to any change of its environment. Learning terms are modulated by the associative map associative map (B) inconsistent activity bumps (A) consistent activity bumps Fig. 4 (A) The activity in the associative map is high if the different perceptions in the perceptive maps are consistent, i.e. the activities are located in strips with an unique intersection point. (B) In other cases, integrating the perceptions provides a badly located and low level activity bump.

upper activity of the column, so that only the active columns can learn. There are two parallel learning processes: one for the self-organization of the perceptive maps and one for the correspondence between perceptions.

The first learning process occurs on the input weights of the perceptive layer in the perceptive cortical columns. This layer is Kohonen-like [START_REF] Kohonen | Self-organization and associative memory[END_REF] but the difference is that the learning is decentralized. The distance function to the maximum activity is replaced by the activity bump of the upper layer. Technically speaking, input weights are modified towards the input value by adding the difference between the weight and the input, modulated by the upper activity and a learning rate (see equation 1).

∆ w = αA * (w -i) (1)
with w the input weights, α the constant learning rate, A * the activity of the output layer of the column and i the perceptive input.

The second learning process is performed on the lateral weights of the strips that connect the maps. The use of Widrow and Hoff's rule [START_REF] Widrow | Adaptive switching circuits[END_REF] makes the strips learn presences and not values. The activity in the strip, i.e. the weighted sum of the output activities of the cortical columns that lie in the strip, will be high only when an activity bump is present in the strip at some learned positions. Moreover, the number of such learned positions is not limited, so that a single cortical layer may be active (high activity) for multiple distant perceptions. This allows the system to learn multiple correspondences for a single perception, like for example a red box and a blue one. Technically speaking, the cortical activity moves towards the upper activity of the column with equation (2). This learning is modulated by the upper activity plus a leaking term preventing weights from saturation.

∆ w i j = α(A * i + β )(A * i -A c i )A * j ( 2
)
with w i j the weight from a column i to a distant column j in the strip, α the learning rate, A * i the output activity of the column i, β a leaking term and A c i the activity of the cortical layer of column i.

Analysis

We tested Ménard and Frezza Buet's model in several multimodal contexts and we observed a non convergence of the perceptive layers in perceptive maps. The model as to be parametrized as a whole because an isolated map is not able to self-organize. This lack of convergence may be explained by inappropriate parameters but some other reasons have been pointed out.

Firstly, the learning process that is used in the perceptive layer replaces the centralized winner-take-all with decreasing neighborhood that is used in Kohonen's maps by the bump activity of the upper layer of cortical columns with a constant width. Moreover, with a constant learning rate due to a continuous learning, a constant width neighborhood may cause instability.

Secondly, when a new example is presented, the system has to move from an equilibrium point to another by relaxing strips constraints. This takes time especially because of the spatial and temporal inertia of the competitive activity bump. This results in an inconsistent internal state of the system. As the activity bump drives the perceptive learning, inconsistent bumps cause unlearning and lead to instability.

We consider that these two points have a common cause which is the important perceptive learning dependency on the activity bump location. As it drives completely the learning process, the incoherence of the bumps is transfered to the selforganization. We propose to reduce this dependency by using a perceptive learning that can self-organize more independently, the activity bump only influencing this learning. This idea is consistent with Wallace's observations [START_REF] Wallace | The development of multisensory processes[END_REF] that show that the neuron ability to integrate multisensory information and so the dependency to multisensory merging, grows with age.

We focus on the problem of a too important perceptive self-organization dependency on the activity bump. We suggest to change the learning rule to have a selforganization not driven but only influenced by the activity bump. We choose the BCM learning rule which has the property to self-organize a neural map when used with lateral connections. We adapt this rule to our multimodal model by introducing a modulation feedback from the activity bump.

BCM rule

Biological inspiration

Hebbian learning is based on the correlation between the pre-synaptic and the postsynaptic neural activities. Moreover when the pre-synaptic activities do not succeed in activating the post-synaptic neuron above a certain threshold, weights decrease (LTD for long term depression), whereas long term potentiation (LTP) occurs when the post-synaptic neuron activity is above the threshold. Bear [START_REF] Bear | Mechanism for a sliding synaptic modification threshold[END_REF] shows that this LTD/LTP threshold is sliding in the opposite direction of the previous activity his-tory of the neuron, meaning that if LTP recently occurred, the threshold increases to favour LTD (see figure 5). The Bienenstock Cooper Munro (BCM) rule (see [START_REF] El Bienenstock | Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex[END_REF]) is based on this biological fact that has the property of self regulating weights contrary to the hebbian rule.

Equations

The output activity u of a neuron is equal to the weighted sum of its input x. The θ value of sliding LTP/LTD threshold is equal to the expectation of u 2 on a τ temporal window:

u = w.x and θ = E τ [u 2 ]
The modification of weights is defined by:

∆ w = η * x * φ (u, θ ) and φ (u, θ ) = u * (u -θ ) (3)
with η the learning rate and φ (u, θ ) a function that defines the proportion of LTP and LTD as an approximation of the biological observations (figure 5).

Properties

The first interesting property is that the modification rule of the weights (equation 3) leads the neuron to develop a selectivity to a specific input. Neuron weights have converged when ∆ w = 0, which corresponds to u = 0 or u = θ . In the case of independent noisy input vectors, the only stable weight vectors contain only one non zero value (see Cooper and al. [START_REF] Cooper | Theory of Cortical Plasticity[END_REF] for more precisions).

The second property is the plasticity of the BCM learning rule. Equation 3 uses a constant learning rate and its stabilization is reached when the neuron is selective to one input. If the input distribution changes, corresponding to an environmental change, neural weights are no more stable and they develop another selectivity.

Self-organization

Autonomous selectivity and plasticity are two sought properties for our system, but the BCM rule only considers an independent neuron. As we want to use the BCM rule for self-organization we may introduce lateral connectivity between neurons. In their book, Cooper and al. present an adaptation of their rule to obtain a selforganization of a neural map. It consists in the addition of a lateral connectivity influence to the output. The lateral connectivity used is a difference of gaussians. It adds excitation to close neurons to favour the learning of close inputs, by raising their output u above θ . Similarly, inhibited neurons learn dissimilar inputs.

5 Modification of the BCM rule

Principle

We propose to use the BCM rule as the learning rule of the perceptive layer of a cortical column (see section 3.2). The activity bump of the upper layer still has to be introduced as an influence upon the self-organization of the perceptive layer. This activity bump results from the relaxation and the competition between local perception and multimodal context. Girod and Alexandre [START_REF] Girod | Effects of a modulatory feedback upon the bcm learning rule[END_REF] showed that it is possible to influence the selectivity of a neuron using the BCM rule by modulating its output. We propose to add a modulation to the BCM rule, depending on the activity of the upper layer. The idea is to favour the LTP (respectively the LTD), by increasing u above θ , when the competitive activity is positive (respectively negative).

We choose to use a multiplicative modulation mechanism. When testing the selforganization with additive lateral connectivity (see section 4.4), we have noticed some problems due to a possible negative output and the modification of the equilibrium points which prevents from high modulation. Moreover, multiplicative modulation occurs in the cortex, especially for coordinate transformation (see for example Salinas and Thier [START_REF] Salinas | Gain modulation: a major computational principle of the central nervous system[END_REF]). By the strip constraints, we create a multimodal common coordinates in which each modality, initially perceived in its own coordinates, is able to self-organize in a spatially consistent way with respect to other modalities.

Equation

We introduce a multiplicative modulation feedback within the BCM learning rule by means of a bump activity-dependent term. The equation is the following:

u = w.x * mod α,β (A * ) and mod α,β (A * ) = α 1 + (α -1) * e -β * A * (4)
with A * the activity of the upper layer and α, β the parameters of the sigmoid. The used modulating term is a sigmoid (figure 6) which is limited by 0, to avoid negative outputs, and mod α,β (A * ) > 1 for A * > 0. As a consequence, when there is an activity bump, the value of the output u increases because of the modulating term. It may then overcome θ and thus learn the current perception (LTP).

Protocol

The model of perceptive map we propose is adapted from the one developed by Ménard and Frezza-Buet (see section 3) using our modified BCM rule as the learning rule for the perceptive layer (equation 4). We first test our modified BCM rule in an isolated perceptive map. This validates our approach and allows to separate the parameter space. Secondly, we put our map in a multimodal context with the Bijama assembling architecture (see section 3.4) to validate the self-organization in a multimodal context. The inputs used for the tests are float values within a finite range, that are coded by a population of neurons. This may represent the orientation coding of a bar in the visual cortex. Technically speaking, the output function of each neuron of the population is a gaussian centered at a fixed preferred input. The preferred inputs are homogeneously distributed on the input interval, so that the output shape of the population is also a gaussian (see figure 7). This output gaussian shape has to be large enough to ensure the spatial continuity of the self-organization of the map.

Results

We first test our map model in an isolated context. To do this, we use the population coded output (see section 6.1) as the input of the perceptive layer. As learning rule of the perceptive layer, we compare our modified rule with the usual BCM rule to see the influence of the modulating feedback on the map organization. For the version with feedback, as the map is isolated, we directly connect the perceptive layer as input of the competitive layer.

Comparing the results of the discriminated input by the perceptive layer of each column of the map, we clearly see, as expected, a self-organization of the map using our modulated BCM rule (see figure 8). This validates our modification of the BCM rule and provides a model of self-organizing map using a local, continuous and decentralized learning. Moreover, the upcoming use of this map within a multisensory context does not require any modification of its existing parametrization. To test our map model in a multimodal context, we use the inter map strip connectivity around an associative map as proposed in the Bijama architecture (see section 3). We use three perceptive maps which receive a population coded input as previously. By simplicity, the three inputs are identical. We compare the selforganization of a map within and out of the multimodal context using the same common parameters and receiving the same input flow (see figure 9). We observe that the multimodal context biased the self-organization of the perceptive map by constraining the localization of the activity bumps of the system so that bumps will be spatially consistent as in figure 4 (A).

Conclusion

We modify the BCM learning rule with the introduction of a high level feedback modulation signal. This multiplicative modulation favours or unfavours the increase of the neuron output over the LTD/LTP sliding threshold. Using this modified rule as perceptive learning rule in the multilayer cortical columns of the multimap architecture proposed in Bijama, we obtain a self-organization of the perceptive maps with a continuous decentralized and local learning. Using the Bijama strip connectivity as assembling paradigm around an associative map, we obtain a self-organization influenced by the multimodal context. This architecture may enable to learn modalities associations and recall missing perceptions.

Fig. 1

 1 Fig. 1 Example of use of the multimap architecture.

Fig. 2

 2 Fig.2Generic structure of a cortical column in (A) a perceptive map, (B) an associative map.

Fig. 5

 5 Fig.5LTP/LTD threshold θ is sliding depending on the recent neuron activity. If LTP recently occurred (∆ w > 0), LTD is favored by the increase of θ (adapted from Bear[START_REF] Bear | Mechanism for a sliding synaptic modification threshold[END_REF]).

Fig. 6

 6 Fig. 6 Modulating term mod α,β (A * ) of the BCM's rule.

Fig. 7

 7 Fig. 7 Inputs used to test our map model are successive random floats limited in an interval and coded by a population of neurons. Each neuron of the population has a gaussian centered an a fixed preferred input as output function. The preferred inputs are homogeneously distributed on the input interval.

Fig. 8

 8 Fig.8Representation in gray scale of the discriminated input by the perceptive layer of each column of the map using (A) the usual BCM rule, (B) the modified BCM rule.(A) (B)

Fig. 9

 9 Fig. 9 Representation in gray scale of the discriminated input by the perceptive layer of each column of a map in (A) an isolated context, (B) a multimodal context. (A) (B)


B Caron

G Balik

L Brunetti

Preliminary results of the analysis and of the optimization dedicated to the final focus stabilization

. The first part is dedicated to the analysis of the first method that has been developed and the second part attempts to compare this method with

Analysis of the previous development :

The context : For instance, we consider a very basic process:

• The displacement of the beam (∆y) which needs to be controlled can be obtained using the beam position monitor located after the interaction point. • The disturbance (X) is the mechanical excitation of the QD0 magnet • The transfer function between the mechanical displacement of this QD0 magnet and the beam can be modelled by a constant matrix. • The noise of the sensor (W) is added to the displacement beam.. • The action (k b ) meant to reduce the motion of the beam (or the offset between the two beams at the interaction point) is done by a kicker which is located just next the QD0 quadrupole. The obtained displacement of the beam is proportional to the injected current of the kicker. • The dynamic of the system is due to the frequency of the beam train, so the process can be treated as a first approach as a delay with a gain at a sampling period T e equal to 0,02 s.

Next, the process is represented in the figure 01 with these different components. The closed loop transfer functions taken into account are:

1 -The transfer function between the beam displacement and disturbance:

Y X = G 1GH =F  z -1 

-

The transfer function between the beam displacement and sensor noise:

Y W = 1 1GH
It is important to note that, as G is a pure delay, the effect of the closed loop is the same, in term of amplitude, for the disturbance input or for the sensor noise input.

The analysis:

The first corrector developed by D. Schulte is:

k b  n=g i k b n-1 g p  y  n a  g d  y  n- y n-1 a g d2 k b  n-1-k b y  n-2
Note that we have removed in the original equation the delay introduced in the recursive equation (we use  y n instead of  y n-1 )

Using the back shift operator z -1 :

z -1  y n= y n-1
And the notations below:

 y n= y , k b n=k b
the following transfer function of the corrector can then be considered:

k b  y =H  z= g p  g d -g d z -1 1-g i z -1 -g d2  z -1 -z -2  = 1 1- 2 z -1 1- 1 z -1 1- 2 z -1 
Which can be seen as a lead (or lag) compensator (H 1 (z)) plus a first order filter(H 2 (z)):

k b  y = 1 H 1  z  H 2  z  With H 1  z = 1- 2 z -1 1- 1 z -1  , H 2  z = 1 1- 2 z -1
Using the given set of parameters: g i =1.0, g p =1.0, g d =0.5, g d2 =1.0, we impose the denominator = 1-z -1  2 which leads to a double integrator in the controller and a great attenuation at low frequency.

Optimization with classical controller neglecting sensor noise:

In the previous analysis, one or two integrators were imposed into the corrector. In the following, we do no more imposed an integrator but we only impose to reduce RMS at 0 frequency. The parameters of the corrector are only deduced from this only consideration, neglecting influence of sensor noise. In order to obtain a more realistic simulation, a real measurement of the ground motion was used, thanks to Guralp geophone [START_REF] Bolzon | Etude des vibrations et de la stabilisation à l'échelle sous-nanométrique des doublets finaux d'un collisionneur linéaire[END_REF]. The only things we have to choose is the structure of the controller. Due to previous analysis the controller structure is the following:

H  z = b 0 b 1 z -1 b 2 z -2 1a 1 z -1 a 2 z -2 (1)
Tuning the controller parameters:

The goal is to reduce the RMS(0) (at 0 frequency) of  y  z -1  In order to find the best controller that minimize RMS(0), we have used the following steps :

estimation of the PSD of the measured ground motion signal X  z -1 =Z  x t  scanning the parameter space of the controller computation of the PSD of the obtained output using:

-PSD Y  j =|F  j | 2 PSD X  j 
we keep the parameter set of the controller that gives the minimum RMS(0)

The optimized parameters are :

a 1 =-0,125 , a 2 =-0,875 , b 0 =0,375 , b 1 =1,1625 , b 2 =0.1875
These parameters injected in (1) let appear a single integration at the denominator, as 1 is root of it, but it wasn't imposed, it is only a result of the optimization.

It is important to note that these parameters depend on the PSD of the input disturbance. (The previous set of parameters has been computed for the PSD of ground motion coming from the L.A.P.P. site). If this signal is changed in term of PSD (other site, or thanks to a passive/active isolation), then the optimization will produce another set of parameters.

The transfer function F , also called the sensitivity transfer function, has an important property :

∫  e /2 0 log|F  j |d =0 with  e T e =2
It follows that lowering effects of disturbances at low frequencies will increase effects of disturbances at high frequencies. And, if we use the above optimization procedure for a pure white noise disturbance, the result is a controller with a transfer function equal to 0.

The figures 03 and 04 represent the PSD and the integrated RMS displacements which were obtained with the initial algorithm and the developed one. One can notice that the optimization technique has allowed the decrease of the RMS (0). Note that the observed displacement at very low frequencies is lowered regarding to the real displacement due to the fact that the used sensor has the following transfer function : It is important to note that the values obtained in the above illustration depend on the PSD of the ground motion which depend on the measurement site as well.

Optimization with classical controller and sensor noise considerations:

As explained above, the magnitude of the transfer function between sensor noise and the output is the same. The following plot is a zoom for the above optimized controller (optimized for a disturbance without TMC table ). It is clear that the sensor noise is amplified by a factor 3.5 (or 9.5 dB) around 11 Hz. In order to lower this amplification, we use the same procedure as above for the optimization but we keep the controller that produces a maximum amplification of 1.2 (or 1.5 dB).

In that case, we obtain the following parameters set :

a 1 =-1.8 , a 2 =0.8 , b 0 =0,3 , b 1 =-0.2 , b 2 =0
that gives a RMS at 0 equal to 2.25.10 -8 , greater than previous optimized controller. Note that the optimization technique scans the parameters with a step equal to 0.1, this could be refined. The following figure compares the amplification of the noise in the two cases. As expected, the sensor noise is less amplified with the last controller around 11 Hz. But without other considerations, we are not able to say that this last controller is better than the previous one. On the other side, it is shown that we can have some action on the amplification of the noise.

Conclusions

The proposed method to tune the controller is able to obtain the minimum of the RMS at 0 with a given input disturbance. It is important to note that it is not possible to lower more RMS(0) by the mean of a feedback with a Linear Time Invariant controller after the proposed optimization. Thus, in order to lower RSM at 0 of the output, there are two possibilities :

minimizing the input disturbance by adding more mechanical filters : statically or dynamically adding a feed-forward controller with an estimation of the input disturbance It is also important to note that the attenuation of the input disturbance has to be done in the frequency range were the feedback is not efficient (i.e. were there is no disturbance attenuation and more obvious were there is an amplification) clearly above 10 Hz.

We also have investigated the effect of the loop on the amplification of the sensor noise. It is shown that there is always an amplification at high frequency, this amplification can be minimized but the counterpart is a raise of the RMS at 0.

Fig 01 :

 01 Fig 01 : Feedback scheme of the considered system

Fig 03 :

 03 Fig 03 : PSD displacement of the beam with a real disturbance

Fig 06 :

 06 Fig 06 : PSD displacement of the beam excited by the measured displacement on an active table

Fig 08 :

 08 Fig 08: effect of the closed loop on the sensor noise

Fig 09 :

 09 Fig 09: noise amplification : comparison of two controllers

standard controllers that use a parametric optimization.

The figure 05a represents the bode diagram of this sensor and one can notice the attenuation at low frequencies.

The next stage was to compare these 2 methods by using as disturbance a displacement measurement on an industrial active table [START_REF] Tmc Company | TMC STATIS 2000 stable active control isolation system[END_REF] instead of the displacement of the ground motion. The figure 05b represents the Guralp sensor placed on the TMC table. The aim is to simulate the fact that the QD0 magnet will be placed on an active table.

Fig 05a: Bode diagram of the Guralp's transfer function Fig 05b: Measurement set-up with a Guralp sensor placed on the active table

As the PSD of the disturbance signal has changed, we have to optimized the controller. If we keep the previous parameter set, the RMS is not minimized.

The new parameter set is then :

We can see that the optimization technique leads to a double integrator in the controller. The figures 06 and 07 represent the PSD and the integrated RMS displacements which were obtained with a perturbation which is a real displacement measurement on an active table.