Jean-Jacques Lemaire
email: jjlemaire@chu-clermontferrand.fr

Jerome Coste

Lemlih Ouchchane

François Caire

Christophe Nuti

Philippe Derost

Vittorio Cristini

Jean Gabrillargues

Simone Hemm

Franck Durif

Jérôme Coste

Jean Chazal

Brain mapping in stereotactic surgery: A brief overview from the probabilistic targeting to the patient-based anatomic mapping

Keywords: Stereotaxy, DBS, MRI, Targeting, Anatomy, Computer-aided surgery

Introduction

Brain mapping in the classical meaning of stereotactic surgery is based on intra operative electrophysiological recordings in order to locate, for a given patient, the so-called invisible targets. Since the pioneering days, the stereotactic targeting was indirect because the targets were de facto arranged in areas relative to ventricular baselines as imaging techniques failed to show the internal anatomy of the brain [START_REF] Talairach | Atlas d'anatomie stéréotaxique. Repérage radiologique indirect des noyaux gris centraux des régions mésencéphalo-sous-optiques et hypothalamiques de l'homme[END_REF]. For most of the deep brain structures, basal ganglia, internal subdivision of thalamus and main white bundles, a reference position in relation to ventricular landmarks was proposed based on anatomic specimen studies transcribed in stereotactic atlases [START_REF] Talairach | Atlas d'anatomie stéréotaxique. Repérage radiologique indirect des noyaux gris centraux des régions mésencéphalo-sous-optiques et hypothalamiques de l'homme[END_REF][START_REF] Schaltenbrand | Introduction to Stereotaxis With an Atlas of the Human Brain[END_REF]. Nowadays in spite of considerable progresses in magnetic resonance imaging (MRI), few surgical teams have shifted to pure direct anatomic targeting with neither ventricular baselines and/or atlas matching nor neuronal activity recordings [START_REF] Lemaire | Direct stereotactic MRI location in the globus pallidus for chronic stimulation in Parkinson's disease[END_REF][START_REF] Coubes | Deep brain stimulation for dystonia: surgical technique[END_REF][START_REF] Caire | Stimulation sous-thalamique dans la maladie deParkinson sévère: Étude de la localisation des contacts effectifs[END_REF][START_REF] Plaha | Stimulation of the caudal zona incerta is superior to stimulation of the subthalamic nucleus in improving contralateral parkinsonism[END_REF]. This direct targeting concept depends exclusively on the visualization of the detailed internal anatomy of each patient's brain providing a personal MRI map.

Here, we briefly review the concept of brain mapping in stereotactic surgery by indirect and direct methods of targeting, in light of recent advances in stereotactic MRI.

Background in classical stereotactic surgery with indirect probabilistic targeting

The indirect probabilistic targeting

The gold standard in classical stereotactic surgery relies worldwide on indirect probabilistic targeting, relative to a stereotactic reference. Historically ventricle landmarks represented this reference because X-ray ventriculography (intraventricular injection of an iodized contrast agent and/or the air) was the only technique visualizing the gross internal brain morphology. With the advent of slice (computerized tomography, CT and MR) imaging, the method has shifted without redefinition of landmarks yet determined on radiographic projections. Because slice imaging can often be anatomically poorly informative in clinical routine, the ventricle landmarks continue to be used widely: the anterior (AC) and the posterior (PC) diencephalic white commissures around the third ventricle, sometimes the floor of the body of the lateral ventricle (corresponding to the superior border of the thalamus) and the width of the third ventricle. The stereotactic coordinates of targets, in relation to the stereotactic reference, are provided by classical stereotactic atlases and stereotactic graphs [START_REF] Talairach | Atlas d'anatomie stéréotaxique. Repérage radiologique indirect des noyaux gris centraux des régions mésencéphalo-sous-optiques et hypothalamiques de l'homme[END_REF][START_REF] Schaltenbrand | Introduction to Stereotaxis With an Atlas of the Human Brain[END_REF]. Thus the probabilistic targeting relies on transposition of the theoretical position of a target, atlas-or graph-based, into the stereotactic space of a given patient's brain through the ventricular stereotactic reference. The use of proportional coordinates and probabilistic location minimizes the influence of individual variability [START_REF] Talairach | Atlas d'anatomie stéréotaxique. Repérage radiologique indirect des noyaux gris centraux des régions mésencéphalo-sous-optiques et hypothalamiques de l'homme[END_REF][START_REF] Schaltenbrand | Introduction to Stereotaxis With an Atlas of the Human Brain[END_REF][START_REF] Velasco | Electrical stimulation of the prelemniscal radiation in the treatment of Parkinson's disease: an old target revised with new techniques[END_REF][START_REF] Benabid | Imaging of subthalamic nucleus and ventral intermedius of the thalamus[END_REF] with continuous effort to optimize this approach [START_REF] Nowinski | Statistical analysis of 168 bilateral subthalamic nucleus implantations by means of the probabilistic functional atlas[END_REF][START_REF] Yelnik | A three-dimensional, histological and deformable atlas of the human basal ganglia: I. Atlas construction based on immunohistochemical and MRI data[END_REF]. Although the method seems universally applicable, basic points can lead to errors of location: (1) the stereotactic locations provided by atlases and graphs can differ (e.g. Fig. 1) as an atlas relies on only one brain per plane [START_REF] Schaltenbrand | Introduction to Stereotaxis With an Atlas of the Human Brain[END_REF] and a geometric figure represents only the area with the highest probability to find a given structure [START_REF] Talairach | Atlas d'anatomie stéréotaxique. Repérage radiologique indirect des noyaux gris centraux des régions mésencéphalo-sous-optiques et hypothalamiques de l'homme[END_REF][START_REF] Schaltenbrand | Introduction to Stereotaxis With an Atlas of the Human Brain[END_REF][START_REF] Benabid | Imaging of subthalamic nucleus and ventral intermedius of the thalamus[END_REF];

(2) the commissural points are defined appreciably differently on their center [START_REF] Schaltenbrand | Introduction to Stereotaxis With an Atlas of the Human Brain[END_REF][START_REF] Benabid | Imaging of subthalamic nucleus and ventral intermedius of the thalamus[END_REF] or their periphery [START_REF] Talairach | Atlas d'anatomie stéréotaxique. Repérage radiologique indirect des noyaux gris centraux des régions mésencéphalo-sous-optiques et hypothalamiques de l'homme[END_REF] or according to different planes for the same atlas [START_REF] Schaltenbrand | Introduction to Stereotaxis With an Atlas of the Human Brain[END_REF]. Moreover the expression of stereotactic coordinates is different even if the orientation is standardized (x represents the laterality relative to the vertical midline plane going through ACPC; y represents the anteriorposterior position along the ACPC line; z represents the superior-inferior position relative to the ACPC axial plane, perpendicular to the midline plane). The units are in millimeters in case of absolute coordinates or in percentage of a standard, the ACPC line (e.g., 1/12 of ACPC) or the height of the thalamus (e.g. 1/4 of the thalamus height), in case of proportional coordinates.

Coordinates are only proportional for y or z, since there is no reliable ventricle landmark to determine a standard in the frontal plane; however, a lateral proportionality can be applied [START_REF] Velasco | Electrical stimulation of the prelemniscal radiation in the treatment of Parkinson's disease: an old target revised with new techniques[END_REF]. Due to this probabilistic approach, it is strongly recommended to adjust the targeting by intra operative tests (electrophysiological neuronal recordings and/or clinical assessments) and multiple-tract explorations. The strength of the indirect targeting is above all the simplicity of the coordinate calculation and the important electrophysiological knowledge harvested. [START_REF] Benabid | Imaging of subthalamic nucleus and ventral intermedius of the thalamus[END_REF]; projections of STN boundaries determined on axial (black dots and lines), sagittal (dark grey dots and lines) and frontal (light grey lines, white circles) slices [START_REF] Schaltenbrand | Introduction to Stereotaxis With an Atlas of the Human Brain[END_REF]

Figure 1. Stereotactic location of the subthalamic nucleus (STN). STN projection area (pale grey surface)

. Frontal (left; white circle, projection of ACPC) and lateral (right; doted line, ACPC line; white circles, AC and PC) views (TH, thalamus height; AC, PC see text for abbreviations).

Imaging for indirect targeting

Clinical MRI, at least on the first generations of generalist machines, has non-negligible image distortion making delicate the ACPC definition and/or the location of surgical fiducials, as well as a low tissue contrast making reliable recognition of deep brain structures difficult. Therefore, matching CT with MRI was proposed [START_REF] Duffner | Relevance of image fusion for target point determination in functional neurosurgery[END_REF] in order to take the best of the two techniques, the geometric accuracy of CT plus the better ventricular anatomy on MRI. Nowadays because of the progresses of MRI machines, teams used more and more MRI to determine ACPC directly [START_REF] Patel | MRI directed bilateral stimulation of the subthalamic nucleus in patients with Parkinson's disease[END_REF].

Indirect anatomic analysis of data

Even though it was originally designed to reach an invisible target for a given patient, the method of indirect location is also applied to analyze the positions of lesions (ablative surgery) or electrode contacts (deep brain stimulation) for different patients regardless of the technique of targeting [START_REF] Benabid | Imaging of subthalamic nucleus and ventral intermedius of the thalamus[END_REF][START_REF] Plaha | Stimulation of the caudal zona incerta is superior to stimulation of the subthalamic nucleus in improving contralateral parkinsonism[END_REF]. The aim is to determine the anatomic structures involved in the therapeutic process. The coordinates of lesions or contacts, for each patient and/or hemisphere, are displayed on an atlas or a graph. It is assumed that locations of lesions or contacts are effectively in the same structure identified during surgery; this depends mainly on surgical technical constraints and on the level of reliability of electrophysiological intraoperative guidance. The surgical constraints are linked up, among other things, with the capabilities of stereotactic instrumentation. The reliability of electrophysiological guidance is not absolute [START_REF] Schiff | Failure of single-unit neuronal activity to differentiate globus pallidus internus and externus in Parkinson disease[END_REF][START_REF] Israel | Microelectrode Recording in Movement Disorder Surgery[END_REF][START_REF] Mcclelland | Subthalamic stimulation for Parkinson disease: determination of electrode location necessary for clinical efficacy[END_REF] in particular because of our partial, and almost exclusively atlas-based, knowledge of "electro-anatomic" processes; a better comprehension of these latter should be allowed by a direct, patient-based, anatomic study.

The "direct" patient-based anatomic mapping in stereotactic surgery

In order to improve the surgical stereotactic method, since targets become more visible with new MRI modalities (hardware and software), a stereotactic reference is not mandatory anymore; the patient's brain is its own reference, a direct patient-based anatomic mapping gets rational as from 1.5 T [START_REF] Lemaire | Direct stereotactic MRI location in the globus pallidus for chronic stimulation in Parkinson's disease[END_REF][START_REF] Coubes | Deep brain stimulation for dystonia: surgical technique[END_REF][START_REF] Plaha | Stimulation of the caudal zona incerta is superior to stimulation of the subthalamic nucleus in improving contralateral parkinsonism[END_REF][START_REF] Derost | Is subthalamic nucleus deep brain stimulation (DBS-STN) appropriate to manage severe Parkinson disease in an elderly population?[END_REF]. Some teams use transitional methods, locating targets with reference to easy recognizable structures, like the red nucleus, coupled with a classic indirect atlas-based approach [START_REF] Starr | Magnetic resonance imaging-based stereotactic localization of the globus pallidus and subthalamic nucleus[END_REF][START_REF] Bejjani | Bilateral subthalamic stimulation for Parkinson's disease by using three-dimensional stereotactic magnetic resonance imaging and electrophysiological guidance[END_REF][START_REF] Patel | MRI directed bilateral stimulation of the subthalamic nucleus in patients with Parkinson's disease[END_REF][START_REF] Rampini | Multiple sequential image-fusion and direct MRI localisation of the subthalamic nucleus for deep brain stimulation[END_REF][START_REF] Andrade-Souza | Comparison of three methods of targeting the subthalamic nucleus for chronic stimulation in Parkinson's disease[END_REF]. Because of the wide range of anatomical quality of MRI sequences and the variability of surgical techniques, analysis of the literature concerning direct targeting reliability can be confusing [START_REF] Andrade-Souza | Comparison of three methods of targeting the subthalamic nucleus for chronic stimulation in Parkinson's disease[END_REF][START_REF] Breit | Pretargeting for the implantation of stimulation electrodes into the subthalamic nucleus: a comparative study of magnetic resonance imaging and ventriculography[END_REF]. However, pure direct targeting, allowed by patient-based anatomic mapping, yields benefits by reducing the number of exploration tracts and duration of intra operative tests due to optimal primary positioning [START_REF] Caire | Stimulation sous-thalamique dans la maladie deParkinson sévère: Étude de la localisation des contacts effectifs[END_REF] and by improving the analysis of relationships between a lesion or a contact and the structures implied in the clinical effect because of detailed anatomic analysis [START_REF] Ulla | Manic behaviour induced by deep brain stimulation in Parkinson's disease: evidence of substantia nigra implication?[END_REF]. The increasing risk of hemorrhage with numerous exploration tracts [START_REF] Hariz | Safety and risk of microelectrode recording in surgery for movement disorders[END_REF] argues for optimizing the primary positioning. A more widespread application of the method depends mostly on two factors: technical, the transfer of adequate MRI sequences; anatomic, the spread of the anatomic knowledge of nuclei and bundles related to the thalamus and basal ganglia, arranged in a complex manner and poorly known in detail. In practice, the identification and targeting of an anatomic structure rely on 3 key-points: namely, an MRI anatomic knowledge, high-quality images and dedicated surgical software. Even though anatomical knowledge is mostly based on anatomy books and stereotactic atlases, the identification of structures is facilitated by a very high field MRI anatomic reference [START_REF] Lemaire | Contribution of 4.7-Tesla MRI in the analysis of the MRI anatomy of the human subthalamic area[END_REF] with contrasts similar to those used in clinical conditions and with 3D topographic analysis (Fig. 2). The high quality of MRI images is achieved with a small voxel size, below 1.5 mm 3 , and a high contrast between white and grey matters achieved with dedicated T2-weighted [START_REF] Lemaire | Deep brain stimulation in the subthalamic area for severe idiopathic Parkinson's disease: location of plots in the preoperative phase and at the three month follow-up[END_REF][START_REF] Patel | MRI directed bilateral stimulation of the subthalamic nucleus in patients with Parkinson's disease[END_REF][START_REF] Slavin | Direct visualization of the human subthalamic nucleus with 3 T MR imaging[END_REF] and inversion-recovery sequences [START_REF] Magnotta | Visualization of subthalamic nuclei with cortex attenuated inversion recovery MR imaging[END_REF][START_REF] Siadoux | IRM stéréotaxique de la région sous thalamique: optimisation d'une séquence de repérage pré opératoire pour la mise en place d'electrodes de stimulation profonde chronique[END_REF]. In clinical routine, there is no evidence, because of the intricate relationships between hardware and software, that 3-T magnetic field offers higher anatomic definition than 1.5-T magnetic field, at least with optimized sequences. If the anatomic images are acquired in stereotactic conditions, i.e. with the stereotactic frame locked in the head coil, then head motion is minimized during the acquisition time of about 10 min per plane, increasing the signal/noise ratio (unpublished data). Anatomic sub structures in the area of interest, e.g. the thalamo-subthalamic region or the lenticular nucleus, can be labeled and manually highlighted (Fig. 3) after identification based on the analysis of their knownrelative positions. The outlines of the structures can be helpful to interpret data on non-conventional interpolated planes reconstructed along the trajectories. Dedicated surgical software simplifies and improves the manipulation of image sets, allowing determination of the detailed anatomy, optimization of the trajectories (Fig. 4) and more detailed study of relationships between the anatomy and electrophysiological and clinical data. Anatomic location of a set of lesions or contacts in a group of patients is still topical. The membership concept could incorporate the fact that a contact or a lesion can involve severalstructures. The membership degree can be weighted by a fuzzy logic method integrating the expert's opinion concerning the level of uncertainty of the membership [START_REF] Caire | Stimulation sous-thalamique dans la maladie deParkinson sévère: Étude de la localisation des contacts effectifs[END_REF].

The membership degree can also be weighted by the level of involvement of a contact into a structure and vice versa [START_REF] Lemaire | Stimulation électrique à haute fréquence du noyau sous thalamique dans la maladie de Parkinson sévère idiopathique: analyse du site optimale de stimulation à partir des données électrophysiologiques per opératoires et de l'IRM anatomique stéréotaxique[END_REF].

Future trends

Patient-based anatomic mapping should benefit from higher magnetic field, although beyond 3 T there are still important technical constraints preventing a routine clinical use, in particular in stereotactic conditions. But as from 1.5 T, diffusion tensor imaging (DTI) and tractography already offer new possibilities. These techniques enable the analysis of the anisotropy of brain tissue as well as fibbers constituting white bundles [START_REF] Mori | Three-dimensional tracking of axonal projections in the brain by magnetic resonance imaging[END_REF][START_REF] Mori | Fiber tracking: principles and strategies-a technical review[END_REF][START_REF] Le Bihan | Looking into the functional architecture of the brain with diffusion MRI[END_REF][START_REF] Wakana | Fiber tract-based atlas of human white matter anatomy[END_REF][START_REF] Hermoye | Pediatric diffusion tensor imaging: normal database and observation of the white matter maturation in early childhood[END_REF] and several clinical applications have been published [START_REF] Mukherjee | Diffusion tensor imaging and fiber tractography in acute stroke[END_REF][START_REF] Nimsky | Preoperative and intraoperative diffusion tensor imaging-based fiber tracking in glioma surgery[END_REF][START_REF] Arfanakis | Optimization of white matter tractography for presurgical planning and image-guided surgery[END_REF][START_REF] Wilde | Diffusion tensor imaging in the corpus callosum in children after moderate to severe traumatic brain injury[END_REF]. As bundles seem to participate in the DBS effects [START_REF] Gabriëls | Deep brain stimulation for treatment-refractory obsessive-compulsive disorder: psychopathological and neuropsychological outcome in three cases[END_REF][START_REF] Hamel | Deep brain stimulation of the subthalamic nucleus in Parkinson's disease: evaluation of active electrode contacts[END_REF],studying their implication becomes possible. Parallel to the knowledge of the tissue anisotropy, also extracted from DTI, might be of interest because it could influence the electric current diffusion [START_REF] Mcintyre | Electric field and stimulating influence generated by deep brain stimulation of the subthalamic nucleus[END_REF]. Until now, it has been difficult to perform such imaging in stereotactic conditions, i.e. with a stereotactic frame in place, because of an important image distortion. Recently, modified surgical software yielded this approach at 1.5 T with routine stereotactic conditions. After matching of DTI with T2-weighted anatomic images and 3D tractography, it is possible to determine the main bundles of the subthalamic region potentially implied in the DBS process like the ansa lenticularis (Fig. 5).

Beyond improving stereotactic targeting, the patient-based anatomic mapping would enable new considerations for functional treatments relying on the spatial location inside specific brain areas, like radiosurgery, or the topographic diagnosis of lesions as during degenerative diseases, at least if the anatomy is not substantially modified. Furthermore, advances in predictive computational modeling [START_REF] Frieboes | An integrated computational/experimental model of tumor invasion[END_REF] might help by reproducing in the computer the complexity and multi-dimensionality of a particular patient's brain structure.

Figure 5. Fiber tracking on stereotactic 1.5-T DTI. Anisotropic diffusion images (diffusion tensor imaging [DTI];

top row, left) matched (mutual information algorithm) with anatomic images (inversion-recovery sequence, voxel size=0.52×0.62×2 mm3; top row, right) plus 3D anatomic structures (objects were created after manual outlining). DTI acquisition was performed on a 1.5-T machine (Siemens Sonata,GmbH,Erlangen,Germany): stereotactic conditions,with a Leksell G frame;6 directions,b value=750 s/mm2,voxel size=1.8×1.8×3 mm3;image post processing with Iplan (BrainLab,Feldkirchen,Germany), color-coded fiber direction (blue for superior-inferior, red for left-right, and green for anterior-posterior) plus color map matched with 3D anatomic structures (intermediate row). Tractography (bottom row): (Top) tracking with a fractional anisotropy threshold ≥0.30 and length of fibers ≥40 mm; (Middle) volume of interest (blue box displayed on a coronal slice; insert bottom right) placed along the right trajectory on the interface between the subthalamic nucleus and the pre rubral Forel's field (same patient as Fig. 2); (Bottom) right lateral view of anatomic structures (see Figs. 2 and3 for labels; the substantia nigra is transparent; left structures are hidden) fused with color-coded fibers according to the direction: ansa lenticularis (Al), subthalamic occipito-parietal bundle (OP b), frontal bundle going trough the anterior limb of the internal capsule (Fr b) and the dento-rubro-thalamic fascicle (Fx DRT).

Figure 2 .

 2 Figure 2. 4.7-T MRI reference of the human thalamus and basal ganglia. Imaging on a Biospec 4.7-T MRI system (Bruker, GmbH, Ettlingen, Germany), anatomic specimen, 3D spin-echo sequence T1-weighted; isotropic voxel=250μm3. Isocentric images, reconstructed in the coronal (left) and axial (right) planes: Raw data (top row): labelled and highlighted structures (bottom row).

Figure 3 .

 3 Figure 3. Axial 1.5-T MRI slices. Imaging (deep brain stimulation surgery, Parkinsonian) on a Sonata machine(Siemens, GmbH, Erlangen, Germany), inversion-recovery sequence, stereotactic conditions; voxel size=0.52×0.62×2 mm3. Diencephalo-mesencephalic subthalamic structures are outlined and labeled (horizontal white bar=10 mm): substantia nigra (deep blue), red nucleus (orange), subthalamic nucleus (yellow), mammillary bodies (blue-green), nucleus of ansa lenticularis (light orange), substantia Q (green), zona incerta (red), peri peduncular nucleus (pink), lateral (light purple) and medial (deep purple) geniculate bodies.

Figure 4 .

 4 Figure 4. 3D optimization of trajectory. Electrode implantation planned (same patient as Fig. 2) in the posterior part of the subthalamic nucleus (yellow) near the zona incerta (red) and the Forel's fields (pale green), see Fig. 2 for the others labeled structures: 3D volume rendering of the subthalamic structures (frontal view; top left); pseudo coronal, pseudo axial and pseudo sagittal planes reconstructed along the right trajectory (clockwise from top right to bottom left).


Ray El Boustany

Christopher Taveau

Catherine Chollet

Gilberto Velho

Lise Bankir

François Alhenc-Gelas

Ronan Roussel

Nadine Bouby
email: nadine.bouby@crc.jussieu.fr

Antagonism of vasopressin V2 receptor improves albuminuria at the early stage of diabetic nephropathy in a mouse model of type 2 diabetes

Keywords: vasopressin, V2 receptor, antagonist, diabetic nephropathy, type 2 diabetes

Aims: Vasopressin is increased in diabetes and was shown to contribute to development of diabetic nephropathy through V2 receptor (V2R) activation in an experimental model of type 1 diabetes. The role of V2R in type 2 diabetes remains undocumented. This study addresses the issue in a mouse model of type 2 diabetes.

Methods: Male obese diabetic db/db mice were treated for 12 weeks with a selective V2R antagonist (SR121463) and compared to non-treated db/db and non-diabetic db/m mice. All animals were previously uninephrectomized.

Results: The V2R antagonist did not alter glycaemia or glycosuria in db/db mice. It induced a two-fold increase in urine output and a 52% decrease in urine osmolality compared to nontreated db/db mice. After four weeks of treatment urinary albumin to creatinine ratio was 50% lower in treated mice compared to non-treated mice, and remained significantly lower until end of experiment. Glomerular filtration rate increased significantly over time in non-treated db/db mice but remained stable in treated mice.

Conclusions: This study shows that vasopressin contributes to albuminuria and glomerular hyperfiltration via V2R in a mouse model of type 2 diabetes. It documents causality behind the association of vasopressin with renal disease observed in diabetic patients.

Introduction

Diabetic nephropathy is a leading cause of end-stage renal disease (ESRD), and a major contributor to the increased mortality observed in subjects with diabetes [START_REF] Reutens | Epidemiology of diabetic kidney disease[END_REF]. An expanding set of data suggests that the vasopressin -hydration axis plays a role in onset and progression of chronic kidney disease (CKD) [START_REF] Bankir | Vasopressin: a novel target for the prevention and retardation of kidney disease?[END_REF]. Increased plasma osmolarity is the main stimulus for vasopressin secretion, which is thus strongly dependent on the hydration status.

Vasopressin is co-secreted in equimolar amount with copeptin, the C-terminal portion of the preprovasopressin peptide. Cross-sectional and prospective studies in the general population have shown associations between plasma copeptin and albuminuria or renal function decline [START_REF] Meijer | Copeptin, a surrogate marker of vasopressin, is associated with microalbuminuria in a large population cohort[END_REF][START_REF] Enhorning | Copeptin, a marker of vasopressin, in abdominal obesity, diabetes and microalbuminuria: the prospective Malmo Diet and Cancer Study cardiovascular cohort[END_REF][START_REF] Roussel | Plasma Copeptin and Decline in Renal Function in a Cohort from the Community: The Prospective D.E.S.I.R. Study[END_REF]. High circulating copeptin concentration was also associated with the development and progression of diabetic nephropathy in patients with type 2 diabetes [START_REF] Boertien | Copeptin, a surrogate marker for arginine vasopressin, is associated with declining glomerular filtration in patients with diabetes mellitus (ZODIAC-33)[END_REF][START_REF] Velho | Plasma Copeptin and Renal Outcomes in Patients With Type 2 Diabetes and Albuminuria[END_REF][START_REF] Hu | Serum copeptin as a new biomarker in the early diagnosis of decline in renal function of type 2 diabetes mellitus patients[END_REF][START_REF] Pikkemaat | Association between copeptin and declining glomerular filtration rate in people with newly diagnosed diabetes. The Skaraborg Diabetes Register[END_REF][START_REF] Zhu | Serum levels of copeptin are associated with type 2 diabetes and diabetic complications in Chinese population[END_REF], and with prevalence of ESRD in subjects with long-standing type 1 diabetes [START_REF] Bjornstad | Elevated copeptin is associated with atherosclerosis and diabetic kidney disease in adults with type 1 diabetes[END_REF][START_REF] Velho | Plasma Copeptin, Kidney Outcomes, Ischemic Heart Disease, and All-Cause Mortality in People With Longstanding Type 1 Diabetes[END_REF]. Experimental data suggests a causal role of vasopressin in renal dysfunction through activation of the V2receptor (V2R). Indeed, administration of dDAVP, a V2R agonist, to normal rats, induces, besides the well-documented antidiuretic effect, glomerular hyperfiltration and a rise in urinary albumin excretion (UAE) [START_REF] Bouby | Vasopressin increases glomerular filtration rate in conscious rats through its antidiuretic action[END_REF]Bardoux et al., 2003a). Similarly, in human subjects, acute administration of dDAVP increases UAE. This effect of dDAVP does not occur in subjects with loss of function mutations in V2R (Bardoux et al., 2003a). In renal diseases, activation of V2R was shown to participate in the progression of renal failure in rats with five-sixth reduction in renal mass [START_REF] Bouby | Effect of water intake on the progression of chronic renal failure in the 5/6 nephrectomized rat[END_REF]. In a rat model of type 1 diabetes, vasopressin was shown to contribute to microalbuminuria through its V2R activation (Bardoux et al., 2003b). The role of V2R in kidney complications of type 2 diabetes remains however so far undocumented.

In the present investigation, we evaluated the contribution of the V2R to the progression of nephropathy in a murine model of type 2 diabetes. The effect of chronic treatment with a selective non-peptide V2R antagonist on renal function parameters was studied in db/db mice.

Material and methods

Animals and treatment

All animal procedures were conducted in accordance with the Directive 2010/63/eu of the European Union and were approved by the Animal Care and Ethics Committee of the French Ministry of Research. Reporting of this work complies with ARRIVE guidelines. Male diabetic db/db and non-diabetic db/m mice bred on a C57BLKS/J background (Janvier Labs -Le Genest Saint Isle, Mayenne, France) were used. Animals were housed in the registered pathogen free facility (A75-06-12) of the Centre de Recherche des Cordeliers with a 12-hour light/dark cycle and had free access to water.

Seven-week old mice of both strains underwent left nephrectomy under pentobarbital anesthesia (Nembutal®, 0.6mg/10g BW), in order to increase the vulnerability of the remnant kidney [START_REF] Levine | Modulation of single-nephron GFR in the db/db mouse model of type 2 diabetes mellitus. II. Effects of renal mass reduction[END_REF]. Two weeks after the surgical procedure, animals were housed in metabolic cages (Techniplast, Lyon, France) for two consecutive 24-hour periods for establishing basal values for all parameters including body weight (BW), food and fluid intake, urine flow rate and urine osmolality. Three groups of mice were then studied. Diabetic db/db mice treated with the selective V2R antagonist SR121463 (Sanofi-Recherche, Toulouse, France) for 12 weeks were compared to non-treated diabetic db/db mice and to non-diabetic db/m mice (n=7 per group). The V2R antagonist was mixed with standard powdered food (A04, Safe, Augy, France) and a small amount of water (0.5 ml/g food). It was administered at 30 mg/kg/day for the first 9 weeks of the protocol and at 45 mg/kg/day for the remaining 3 weeks. The increase in dose was necessary to maintain the high urine flow rate and low urine concentration in treated animals.

The diabetic db/db control and non-diabetic db/m mice received the same powdered food/water mix without the drug. In order to ensure the same food intake in all groups and total drug intake in the V2R antagonist treated db/db group, all mice were offered a daily amount of standard diet slightly less than their spontaneous intake (3 and 5 g/day for db/m and db/db mice, respectively).

Plasma and urinary parameters

Every second week after initiation of treatment, body weight, water and food intake, and urine volume were measured. Data from two 24-hr urine collections were averaged for each animal.

The following parameters were measured: osmolality (freezing-point osmometer, Roebling, Germany), creatinine, glucose (Konelab® 20i, Ortho-clinical Diagnostics, Thermo Electron Corporation), sodium (flame photometer 943, Instrumentation Laboratory, Bedford, MA) and albumin (ELISA, Albuwell M, Exocell, USA). Urinary albumin excretion was expressed as albumin/creatinine ratio (ACR). Blood samples were taken by retro-orbital puncture after a 6hour fasting period for the measurement of plasma creatinine (Konelab® 20i, Orthoclinical Diagnostics, Thermo Electron Corporation), and blood glucose (OneTouch Vita, Lifescan, Switzerland). At the end of the 12 th week, mice were anesthetized and sacrificed. The remaining kidney and the heart (minus the auricles) were removed and weighed.

Blood pressure measurements

Systolic blood pressure was measured in conscious animals at three successive days at weeks 3, 5 and 7 by tail-cuff plesthymography (Blood Pressure System Analysis, Model BP-2000, Visitech System, USA). Data from the last two days were averaged for each mouse.

Statistical analysis

Data are expressed as mean ± SD unless otherwise specified. Effect of diabetes was assessed by analysis of variance (ANOVA) followed by Fisher post hoc test. Effect of the V2R antagonist was assessed by ANOVA for repeated measures followed by Fisher post hoc test.

Organ weights were compared by Student's t-test. P<0.05 was considered as statistically significant.

Results

Body weight, fasting blood glucose and glycosuria were similar in SR121463 treated and nontreated diabetic db/db mice at baseline, and throughout the study. They were higher in the db/db groups than in the non-diabetic db/m mice. No difference in systolic blood pressure was observed between groups, 3 weeks after initiation of treatment (Table 1), and blood pressure remained stable in all groups throughout the study (data not shown).

As expected, non-treated db/db mice had higher diuresis than db/m mice (ANOVA, p<0.05) (Figure 1). Blockade of V2R with SR121463 resulted in a rapid and significant decrease in urine concentration in db/db mice. At week 4, urine osmolality was two-fold lower in treated db/db than in non-treated db/db mice (894 ± 177 vs. 1855 ± 858 mosm/kg H2O, p<0.05).

Conversely, urine flow rate was higher in treated db/db than in non-treated db/db mice (10.4 ± 6.1 vs. 5.6 ± 3.7 ml/day, p<0.05). The diuretic effect of SR121463 persisted during the 12 weeks of treatment (p=0.02) (Figure 1). The V2R antagonist had no effect on osmolar excretion (Table 1) and natriuresis: sodium excretion was 177 ± 40 in db/m, 298 ± 37 in nontreated db/db and 310 ± 63 µmol/day in treated db/db mice after 4 weeks of treatment and 147 ± 28, 291 ± 47, 317 ± 66 µmol/day respectively after 8 weeks.

ACR at baseline was similar in treated and non-treated db/db groups. In both groups, it was significantly higher than in the db/m group. In non-treated db/db mice ACR steadily increased over time up to 179 % of baseline values at week 12. By contrast, in SR121463 treated db/db mice, ACR initially decreased by roughly 50%, remained stable for the next four weeks and then increased but remained significantly lower (p<0.02) than in non-treated db/db mice (figure 1).

Compared to db/m mice, db/db mice showed elevated creatinine clearance rate which indicated significant glomerular hyperfiltration. Creatinine clearance, increased over time in non-treated db/db mice (409 ± 95 and 688 ± 96 ml/day, p<0.05, at weeks 4 and 8 respectively) but remained stable in treated db/db mice (437 ± 185 and 530 ± 226 ml/day, NS) (Figure 2).

At the end of the study, no difference in the absolute or relative weight of the heart was observed among the experimental groups. Absolute kidney weight was similar in non-treated db/db and db/m mice, while kidney to body weight ratio was lower in db/db mice, treated and non-treated db/db compared to db/m mice due to the increased body weight of db/db animals. Kidney weight was reduced (p<0.05) in treated db/db mice as compared to the non-treated db/db group (Table 2).

Discussion

We assessed the involvement of vasopressin V2R in the renal dysfunction of an experimental model of type 2 diabetes and diabetic nephropathy, the db/db mouse, 7-week-old with unilateral nephrectomy. The main finding of the study is that treatment with the V2R antagonist rapidly induced a decrease in UAE in db/db mice compared to non-treated db/db mice. These observations are consistent with our previous study in a rodent model of type 1 diabetes, the rat with streptozotocin-induced diabetes (Bardoux et al., 2003b). In db/db mice, the V2R antagonist lowered albuminuria and partially prevented hyperfiltration, two main features of incipient diabetic nephropathy. These results support an important role for V2R activation in development of diabetic nephropathy, at least at early stage of the disease.

The mechanism of action of V2R in diabetic nephropathy is not fully elucidated. V2R is active in the collecting duct. Our previous studies in non-diabetic rats suggest that the deleterious effect of vasopressin in the kidney is mainly secondary to the V2R-mediated urine concentrating action of the hormone [START_REF] Bouby | Effect of water intake on the progression of chronic renal failure in the 5/6 nephrectomized rat[END_REF][START_REF] Bankir | Is the process of urinary urea concentration responsible for a high glomerular filtration rate?[END_REF]. It has been suggested that the activation of V2R in the tubule inhibits the tubuloglomerular feedback as a result of reduction in salt concentration at the macula densa, triggering elevation of intraglomerular capillary pressure, hyperfiltration and glomerular injury [START_REF] Bankir | Vasopressin: a novel target for the prevention and retardation of kidney disease?[END_REF].

Administration of the V2R antagonist attenuated the urine concentrating activity of the kidney without inducing formation of hypo-osmotic urine. Urine osmolality was markedly reduced while urine flow and water intake were increased. Urinary sodium excretion was similar between treated and non-treated db/db groups. These data show that the V2R antagonist was a pure and potent aquaretic agent, even in diabetic condition.

Blockade of V2R stimulates pituitary secretion of vasopressin with subsequent V1a receptor (V1aR) activation. Thus, we cannot rule out that the phenotype observed in treated db/db mice was, at least in part, indirectly related to an activation of renal, vascular and hepatic V1aR [START_REF] Cantau | Reduction in hepatic but not in renal and vascular vasopressin receptor number in hyperinsulinemic mice and rats[END_REF][START_REF] Morita | Expression of arginine vasopressin and vasopressin V1a receptor mRNA in diabetic (db/db) mice[END_REF][START_REF] Taveau | Vasopressin and hydration play a major role in the development of glucose intolerance and hepatic steatosis in obese rats[END_REF]. However, lack of increase in blood pressure or glycaemia during V2R antagonist treatment is not in favor of a strong V1aR stimulation.

Renal hypertrophy is also a feature of diabetic nephropathy. In the present study, absolute kidney weight was similar in non-treated db/db and non-diabetic db/m mice. This could be explained by a masking effect of the compensatory hypertrophy process occurring in the remnant kidney after uninephrectomy, on the hypertrophic effects of diabetes. Similar to our findings, Levine et al found no difference between db/db and wild type mice three weeks after uninephrectomy [START_REF] Levine | Modulation of single-nephron GFR in the db/db mouse model of type 2 diabetes mellitus. II. Effects of renal mass reduction[END_REF]. However, we observed that the V2R antagonist treatment slightly attenuated kidney hypertrophy in the uninephrectomized db/db mice, consistent with its other renal effects and in agreement with previous studies in rats [START_REF] Bouby | Effect of water intake on the progression of chronic renal failure in the 5/6 nephrectomized rat[END_REF]Bardoux et al., 2003b).

Our study has some limitations related to the experimental model. Experiments were performed in the db/db mouse, one of the most widely used rodent models of type 2 diabetes.

It allows for the study of the early stage of diabetic nephropathy but like other genetic or dietary models of type 2 diabetes, it does not reproduce features of evolved human diabetic kidney disease [START_REF] Betz | Recent advances in animal models of diabetic nephropathy[END_REF]. Also, we chose to limit the experiment to 12 weeks in order to avoid hydronephrosis or dehydration related to high urine output caused by diabetes and V2R treatment. Accordingly, we cannot establish whether vasopressin and the V2R are still involved in progression of diabetic nephropathy at more advanced stages of the disease.

Conclusion

We showed that in an animal model of type 2 diabetes, treatment with a selective vasopressin V2R antagonist improves albuminuria and partially prevents hyperfiltration or kidney hypertrophy, which are all features of incipient diabetic nephropathy. In addition to supporting a role of V2R activation in the early phase of diabetic kidney disease, these results provide a pathophysiological basis for the association observed in epidemiological studies of type 2 diabetes subjects between high vasopressin/copeptin levels and albuminuria or kidney function decline. The study also suggests putative new therapeutic strategies for renal protection in patients, based on pharmacological blockade of V2R or inhibition of endogenous vasopressin secretion by high water intake.

Figure legends Figure 1 .

 legends1 Figure legends

Figure 2 .

 2 Figure 2. Creatinine clearance at 4, 6 and 8 weeks of treatment with the V2R antagonist in

Table 1 : Physiological parameters at baseline and at the end of the study

 1 Data are presented as mean ± SD (n=7 per group). Statistics are ANOVA followed by Fisher's post hoc test: *significantly different (p<0.05) from db/m. Blood pressure data were obtained at week 3 of treatment.

	db/m	non-treated db/db	Treated db/db

Acknowledgements

The authors would like to thank the staff of the phenotyping platform (Centre de Recherche des Cordeliers, Paris, France), particularly Georges Zadigue, for their technical assistance. We also thank Claudine Serradeil-Legal for the gift of SR121463 (Sanofi-Recherche, Toulouse, France).

Grants/ Funding

This study was supported by grants from the Société Francophone du Diabète, INSERM, Pierre & Marie Curie, Paris Diderot, and Paris Descartes Universities. R.E.B. was supported by a CIFRE (Conventions Industrielles de Formation par la Recherche) grant from INSERM and Danone Research.

Conflicts of interest

The authors declare that there is no conflict of interest associated with this manuscript. Dr.

Roussel has been a consultant or is on the speakers' bureau for AstraZeneca, Janssen, Eli Lilly, Sanofi, Merck Sharp and Dohme, and Novo-Nordisk, and has received research funding from Sanofi and Amgen.

Contribution statement

REB, CT, CC, GV, RR, FAG and NB contributed to the acquisition and analysis of data and helped to review the article. REB, GV, RR, LB and NB contributed to the design and conception of the study, provided data interpretation and wrote the article. All authors approved the final version of the manuscript.


Aditya Challa

B S Sravan Danda

Daya Sagar

Laurent Najman

Imagery

Walter G ; Kropatsch

Nicole M ; Artner

Ines Janusch

B S Daya Sagar

An Introduction to Gamma-Convergence for Spectral Clustering

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

An Introduction to Gamma-Convergence for Spectral Clustering 1 Introduction

The problem of clustering is defined as -given a set of elements {x i }, partition the set into non overlapping groups such that elements belonging to the same group are "similar", and elements belonging to different groups are "dissimilar". The importance of a solution to the problem of clustering is due to its wide range of applications [START_REF] Jain | Data clustering: a review[END_REF][START_REF] Jain | Data clustering: 50 years beyond k-means[END_REF][START_REF] Aggarwal | Data Clustering: Algorithms and Applications[END_REF]. Clustering in image data is also referred to as image segmentation. There exists several methods to solve the problem of clustering [START_REF] Arthur | k-means++: The advantages of careful seeding[END_REF][START_REF] Dempster | Maximum likelihood from incomplete data via the em algorithm[END_REF][START_REF] Ester | A density-based algorithm for discovering clusters in large spatial databases with noise[END_REF][START_REF] Mclachlan | Finite mixture models[END_REF]. A comprehensive textbook on the subject is [START_REF] Aggarwal | Data Clustering: Algorithms and Applications[END_REF] . One of the most commonly used clustering methods is K-means [START_REF] Arthur | k-means++: The advantages of careful seeding[END_REF][START_REF] Macqueen | Some methods for classification and analysis of multivariate observations[END_REF]. However, it suffers from the problem of not being able to detect the non-globular structures. Spectral clustering methods were proposed to overcome this problem. Loosely speaking spectral clustering methods embed the data in a lower dimensional subspace, in which usual methods K-means clustering would be able to detect the non globular clusters as well.

Recently, in [START_REF] Couprie | Power watershed: A unifying graphbased optimization framework[END_REF], seeded clustering/segmentation methods in [START_REF] Cousty | Watershed cuts: Minimum spanning forests and the drop of water principle[END_REF][START_REF] Cousty | Watershed cuts: Thinnings, shortest path forests, and topological watersheds[END_REF][START_REF] Grady | Random walks for image segmentation[END_REF][START_REF] Sinop | A seeded image segmentation framework unifying graph cuts and random walker which yields a new algorithm[END_REF] were extended by taking the limit of minimizers. This is referred to as the Γ -limit [START_REF] Najman | Extending the powerwatershed framework thanks to Γ -convergence[END_REF]. Γ -convergence is a tool to study the asymptotic behavior of families of minimum problems [START_REF] Braides | Gamma-convergence for Beginners[END_REF]. The aim of Γ -convergence is to replace a family of minimum problems with a single problem whose minima exhibits some interesting properties. For instance in [START_REF] Couprie | Power watershed: A unifying graphbased optimization framework[END_REF] it has been shown that the Γ -limit revealed a new segmentation method which performs at least as well as graph cuts, random walker and shortest paths, if not better.

In this article our aim is to calculate the Γ -limit of the ratio cut spectral clustering. The Γ -limit of the ratio cut is referred to as PRcut and an algorithm to calculate the PRcut has been proposed. Due to various numerical precision errors and other constraints, a variant of the algorithm is implemented instead. Thus, during the exposition, we stick to the philosophy that theory is developed to be as general as possible, while the experiments are conducted with slightly modified theory based on practical considerations such as "small" clusters described later.

Background

Let {v i } be the given set of points in R n which we would like to cluster. Taking each of these points as vertices, one can construct a similarity graph G = (V, E, W ) with vertex set V , edge set E, and W : E → R + denotes weights assigned to each edge. Here R + denotes the set of positive real numbers. With slight abuse of notation, we can write the weights as a |V | × |V | matrix, with w ij denoting the edge weight between v i and v j . The degree of a vertex, d i is given by

d i = j w ij (1)
Let D be the diagonal matrix diag(d

1 , d 2 , • • • , d n ).
The Laplacian of a graph is then defined by

L = D -W (2)
We know that the Laplacian is a symmetric positive-semi definite matrix, and hence has non negative real eigenvalues, represented by 0 [START_REF] Lütkepohl | Handbook of Matrices[END_REF]. The corresponding eigenvectors are denoted by (e 0 , e 1 , • • • , e n-1 ). Let A ⊆ V . Then the vector 1 A (x) is given by

= λ 0 ≤ λ 1 ≤ • • • ≤ λ n-1
1 A (x) = 1 if x ∈ A 0 otherwise (3)
Let p be a real parameter. Let W (p) be the matrix such that

W (p) ij = w p ij . Let D (p) denote the matrix as constructed in (1) with weights w p ij . Let L (p) = D (p) -W (p) .

Spectral Clustering

This section briefly reviews spectral clustering methods. For more details please refer to [START_REF] Shi | Normalized cuts and image segmentation[END_REF][START_REF] Luxburg | A tutorial on spectral clustering[END_REF][START_REF] Ng | On spectral clustering: Analysis and an algorithm[END_REF]. As noted before, spectral clustering methods work by embedding the data into a lower dimensional subspace. The three main steps are -1)

Given a set of points {v i } (dataset), construct a similarity graph with each point as a vertex. 2) Construct the Laplacian for the obtained graph and calculate the first k eigenvectors. The value of k is fixed based on the number of clusters one would like to obtain. Let K be the matrix such that the i th column of K is the i th eigenvector e i-1 . 3) Using rows of the matrix K as new representation of the points v i , use traditional clustering methods such as K-means to obtain the final clusters. Note that as a part of K-means step, the algorithm is run several times with random initialization of seeds. Why does spectral clustering work? Although the definitive answer to this question still remains open, there exists several analyses which provide insights into this question [START_REF] Luxburg | A tutorial on spectral clustering[END_REF]. One approach is to interpret the spectral clustering in an optimization framework. One of the measures to validate the appropriateness of the clusters is

cut(A 1 , A 2 , • • • , A k ) = 1 2 k i=1 W (A i , A i ) (4)
where W (A, B) = i∈A;j∈B w ij , A denotes the complement of A in the vertex set V . cut(., .) measures how dissimilar the clusters are by taking the sum of the weights of the edges connecting distinct clusters. In practice minimizing the cut(., .) does not give good results, since it generally separates one vertex, and gives degenerate solutions. To solve this, it was proposed to use a slight modification of the above cost function. Ratio-cut, [START_REF] Luxburg | A tutorial on spectral clustering[END_REF], is given by

Ratio -cut(A 1 , A 2 , • • • , A k ) = 1 2 k i=1 W (A i , A i ) |A i | (5)
where |A i | is the cardinality of set A i . It can be shown that minimizing the Ratio-cut(., .) for k clusters is approximately equivalent to solving the following optimization problem. minimize

H∈R n×k T r(H t LH) subject to H t H = I (6)
Here I is the identity matrix and L is the laplacian as defined in [START_REF] Arthur | k-means++: The advantages of careful seeding[END_REF]. From the Rayleigh-Ritz theorem [START_REF] Lütkepohl | Handbook of Matrices[END_REF] we know that the solution to this optimization problem is obtained by considering the first k eigenvectors of L as columns of H.

Gamma Convergence

Let min{F p (x) : x ∈ X ⊂ R n } be a family of minimum problems. Let x * p be a minimum of F p (x). We are interested in calculating the limit

x * = lim p→∞ arg min F p (7)
In other words, we are interested in the limit of a sequence of minimizers of the family {F p (x)}. Note that there could be many such sequences. Now, consider a special case where

F p (x) = n i=1 α p i Q i (x) (8)
where

1 ≥ α n > α n-1 > • • • > α 1 > 0.
, and Q i (x) are smooth functions. We also assume that there exists a compact set C such that x * p belongs to C for all p. It turns out that, in this case, one can find a simple algorithm to calculate the limit of minimizers as described in the following Thm. 1 [START_REF] Najman | Extending the powerwatershed framework thanks to Γ -convergence[END_REF]. Define

M n = arg min Q n (x) x ∈ C (9)
Recursively define for

k = n -1, n -2, • • • , 0 M k = arg min Q k (x) x ∈ M k+1 (10)
Theorem 1. Let F p (x) be as defined in (8) and x * be the Γ -limit. Then x * ∈ M 1 .

Refer to [START_REF] Najman | Extending the powerwatershed framework thanks to Γ -convergence[END_REF] for proof of Thm. 1. The main consequence of Thm. 1 is that it provides a method to calculate an "approximate" Γ -limit. One starts at the highest scale, optimizes the cost function at this scale, then moves on to the lower scale and repeats the process. Theorem 1 essentially states that one can obtain an approximate solution to the Γ -limit by this process. Approximate, in the sense that the Γ -limit belongs to the set M 1 . The question of how good is the approximation, needs to be analyzed in the specific case.

Gamma Limit of Ratio -Cut

In this section, we calculate the approximate Gamma limit of the Ratio-cut. A few more notations are required. For a given graph G, let G k denote the graph (V, E k , W k ). E k ⊆ E denotes the set of edges whose weight is w k . According to the existence of the edge with weight w k , W k takes the values in {0, 1}. We refer to this graph as scale graph/level graph at level k. Just as for the original graph, one can construct a laplacian, L k , for G k . A point to observe is that the entries in weight matrices of a level graph are either 1 or 0. We assume that the graph has distinct weights w 1 < w 2 < • • • < w j , where j < |E|. Given the notation as above, we have

T r(H t LH) = j k=1 w k T r(H t L k H) (11)
Drawing a parallel with the Γ -convergence framework, we have that Q k (H) = T r(H t L k H) and alpha k = w k . We are thus interested in calculating the limit of minimizers of

j k=1 w p k T r(H t L k H) as p → ∞, subject to H t H = I Let P k denote the following optimization problem. minimize H∈R n×m T r(H t L k H) subject to H t H = I (12)
Thanks to Thm. 1, we have the following method to calculate the Γ -limit for spectral clustering.

1. Let G be a graph with distinct weights w 1 < w 2 < • • • < w j . Let M j+1 be the set of the all n × m matrices. 2. For each k going from j to 1, let M k be the set of solutions for arg min P k which belong to M k+1 .

The set M 1 is the output of the method. Note that the above steps are not implementable.

The main problem in finding an implementable version is to characterize all the solutions for the problem in [START_REF] Grady | Random walks for image segmentation[END_REF]. Let λ (m) denote the m th smallest eigenvalue. Let A k be the matrix [e 1 , e 2 , • • • e l ], where e i is the i th eigenvector, ordered in increasing order of the corresponding eigenvalue. The number of eigenvectors considered, l, are the number of eigenvalues less than or equal to λ (m) . Let K be the matrix

K = I l1×l1 0 0 k l2×m-l1 (13)
where K t K = I. l 2 is the number of eigenvalues equal to λ (m) , and l = l 1 + l 2 . Let X be an orthogonal matrix such that X t X = I. Then, Proposition 1. The set of all solutions to the optimization problem in (12) is of the form A k KX.

Proposition 1 results in an algorithm which is implementable. However, one requires to calculate all the eigenvectors at every stage, which is computationally expensive. Proposition 2 results in an efficient implementation of the algorithm as given in Alg. 1. We refer the result obtained by this algorithm as Power Rcut. Also, G ≥k denotes the graph with vertex set V and edge set E ≥k = ∪ i≥k E i .

Proposition 2. Given a graph G, Let G ≥k denote the graph, whose vertex set is V and edge set E k containing all the edges whose weight is greater than or equal to w k . At stage k, if A is a maximal connected component of G ≥k , then 1 A is an eigenvector with eigenvalue 0 of the optimization problem [START_REF] Grady | Random walks for image segmentation[END_REF].

The proofs for the propositions and details of simplification to obtain the efficient Alg. 1 will be discussed in detail in a complete version of the paper [START_REF] Challa | Power spectral clustering[END_REF]. Observe that Alg. 1 only gives an element of the set to which the Γ -limit actually belongs. The appropriateness of the solution thus obtained must be proved, which Prop. 3 gives. Proposition 3. Let x be the solution obtained by algorithm 1 and let x * be a Γ -limit. Then

F p (x * ) = F p (x) for all p ( 16
)
Where F p is as given in [START_REF] Cousty | Watershed cuts: Thinnings, shortest path forests, and topological watersheds[END_REF], where

Q i (x) = T r(x t L i x).
The above proposition implies that any Γ -limit, and the approximate one calculated have the same cost. In the context of spectral clustering, this implies that the solutions are equivalent and thus the solution obtained from Alg. 1 is a good approximation. The proof of the above proposition will be detailed in later work.

Algorithm 1 Efficient algorithm to compute Γ -limit for ratio-cut.

Analysis with Experiments

In the rest of the article, we discuss how the algorithm works, its practical implementation and its similarities and dissimilarities with the MST based clustering and spectral clustering methods.

How the algorithm works?

Recall that the output of spectral clustering is a projection onto a subspace, and thus the algorithm produces a representation of the points in a subspace (denoted by N in algorithm 1) which is a gamma limit. We assume that we need m clusters. In steps 2-4, we progressively add all the edges while there are at least m connected components in the threshold graph G ≥k . Thanks to Prop. 2 we know the first eigenvectors of the laplacian of the threshold graph are the indicator vectors as in [START_REF] Braides | Gamma-convergence for Beginners[END_REF] where A denotes each of the connected components. We construct the initial representation of the points N by taking the eigenvectors in step 5. Steps 7-14 update the representation N with respect to the lower weight edges. Note that, once the number of columns of the matrix N is equal to m, we need not update the representation anymore, since any other representation would just be an orthogonal transformation of the points and thus the clustering results would not change. This condition is checked in steps 15-19.

One issue with the gamma limit is that the property of non-trivial clusters for the Ratio-cut is not preserved in the limit. In the above algorithm, in practice, we get a lot of outliers and this results in small clusters. To avoid this, in steps 2-4, instead of calculating the number of connected components, we calculate the number of connected components whose size is greater than a given parameter threshComp. At the moment the algorithm jumps out of the while loop, all the components which have less than threshComp number of vertices are ignored. Recall that after the representation of the points is obtained, one has to perform K-means clustering to get the final partition. At this stage the ignored "small" clusters can either be returned as a different cluster, or combined with one of the larger clusters at random.

A simple application of the algorithm is illustrated in Fig. 1, where one can see that the algorithm correctly detects the object (flower in this case). Since small clusters are ignored, for the applications of image segmentation we do not get closed contours for the segments. Thus, for image segmentation the clusters are post-processed with an operator. Observe in Fig. 1(b) a few parts of the flower are missing. In this case, simple operators such as an opening works well, which gives the result as in Fig. 1(c). In general connected operators [START_REF] Serra | A lattice approach to image segmentation[END_REF] preserve contours and are better for the post-processing of the image. Another important property of the Power Rcut clustering method is that, it results in smoother contours compared to the spectral clustering method. This is illustrated in Fig. 1 (d) -(f).

Relation to MST-clustering

MST (Maximum Spanning Tree) based clustering is one of the earliest graph based clustering approach [START_REF] Zahn | Graph-theoretical methods for detecting and describing gestalt clusters[END_REF][START_REF] Grygorash | Minimum spanning tree based clustering algorithms[END_REF][START_REF] Peter | Minimum spanning tree-based structural similarity clustering for image mining with local region outliers[END_REF][START_REF] Chowdhury | Minimal spanning tree based clustering technique: Relationship with bayes classifier[END_REF]. There exists several variations of the method. We consider here the simplest method -(a) Construct an MST (b) Iteratively remove the least weight edges until we get the required number of clusters. One of the most useful properties of spectral clustering is its ability to detect non-convex clusters in the data. This property is shared by the MST based clustering methods as well.

However, the problem with MST based clustering is breaking ties between edges of equal weight, which it does arbitrarily. Spectral clustering on the other hand ensures a single clustering (up to the arbitrariness of k-means step). In this sense, Power Rcut can be considered to be a method between these two clustering methods. Power Rcut follows a similar procedure as MST based clustering, and it breaks ties with spectral clustering on a subgraph. For example consider the graph in Fig. 2(a). Power Rcut segments the graph into two equal clusters, Fig. 2(b). The same behavior is also exhibited by spectral clustering. MST based This is because, Power Rcut takes into consideration the sizes of the cluster while breaking ties. Consider another synthetic example in Fig. 3(a). An MST is highlighted in bold edges in Fig. 3 Power Rcut clustering and MST based clustering are in fact closely related as suggested by the Prop. 4. The proof follows from the following observationfor any two vertices belonging to the same connected component has the same value in the embedded space, and hence belong to the same cluster. Since, we require m clusters, and there are m connected components, the k-means step results in each of these m components as a cluster.

(a) (b) (c) (d) (e) (f)

Relation to Spectral Clustering

The Power Rcut solution can also be interpreted as being obtained by spectral clustering on every level graph G k . Since, the first few eigenvectors are the indicator of the connected components, this gives a heuristic explanation for steps 2-4 in algorithm 1. This points out the similarity between Power Rcut solution and spectral clustering.

In low noise conditions, spectral clustering and Power Rcut clustering results are similar. However, as noise level increases, spectral clustering will not be able to identify the regions anymore. In Fig. 4(a),(b) and (c) data points are sampled from two concentric circles with noise. Fig. 4(c) shows the results obtained by spectral clustering are shown. Notice that the structure of the two circles is not preserved. Fig. 4(a) show the results obtained with Power Rcut. To generate the results of Power Rcut, as a post processing step, we assign each of the points the "small" clusters to the closest cluster, which results in Fig. 4(b) . Although not perfect, Power Rcut results preserve some structure. For a quantitative view we calculate the Fowlkes-Mallows(FW) scores, given by the formula Score = T P

(T P + F P ) (T P + F N ) (17)
Where T P is true positives, F P is false positives and F N is false negatives. The FW scores for various noise levels is plotted in Fig. 4(d). Note that Power Rcut performs better in high noise scenarios compared to spectral clustering.

Conclusion And Future Work

In this article we outlined the basics of spectral clustering and discussed the concept of Γ -convergence. Important results to calculate the Γ -limit of Ratiocut were outlined and the algorithm to calculate the gamma limit was obtained. The correctness of the algorithm was shown via a proposition. The similarities and dissimilarities between Power Rcut, MST clustering and Spectral clustering are analyzed stating the result that Power Rcut clustering is a specific kind of MST clustering. Power Rcut clustering was shown to be superior to MST clustering in dealing with ramp effects of the image, and superior to spectral clustering in noisy scenarios.

Note that none of the steps in spectral clustering methods indicate directly as to why spectral clustering methods could obtain non-convex clusters. This has been an open question for a long time. We believe that, the fact that gamma limit of spectral clustering is a method close to MST clustering provides an insight into this question. In particular, this provides a bridge between spectral clustering and MST clustering and allows us to dive into the question of why spectral methods work.

Note that the algorithm starts with combining the edges of highest weight until the number of clusters are obtained. This can be interepreted as a greedy method of "combining all the points which definitely belong to the same cluster", thus reducing the size of the dataset and allowing for faster computation. This in theory can also be done in parallel. This can allow spectral clustering to be applicable in the case of large datasets and also have efficient implementations. The error bounds and the exact algorithm to do this are a subject of future research.

In [START_REF] Cousty | Hierarchical segmentations with graphs: quasi-flat zones, minimum spanning trees, and saliency maps. Research report[END_REF], the authors study in depth hierarchies and their equivalence with MST. Observe that the Alg. 1 is inherently hierarchical. The question of how Alg. 1 is related to the concepts in [START_REF] Cousty | Hierarchical segmentations with graphs: quasi-flat zones, minimum spanning trees, and saliency maps. Research report[END_REF] is also a subject of future research.

1 :I l 1 ×l 1 0 0 A ( 15 ) 12 :

 11512 Input: A weighted graph, G, with distinct weights w1 < w2 < • • • < wj. Number of clusters, m. Output: N -A representation of the subspace spanned by the Γ -limit of the minimizers. Set k := j. 2: while Number of connected components of G ≥k is greater than or equal to m do 3: Set k := k -1 {We refer to this as an MST-Phase} 4: end while 5: Construct N by stacking the vectors 1A i / |Ai| in columns, where Ai is a connected component of G ≥k . 6: Set l1 := 0 and l2 := number of connected components in G ≥k 7: Consider the graph G k and let L k be the corresponding laplacian. 8: Set C = [N t L k N ] l 2 ,l 2 9: Calculate the first eigenvectors of eigenvalue problem whose eigenvalue is less than or equal to λ (m) . Cx = λx (14) 10: Let A be the matrix obtained by stacking the eigenvectors as columns. 11: Construct  as  = Update l1 and l2. 13: N := N × Â 14: Set k := k -1 15: if k = 0 or number of columns of N is equal to m then 16:

Fig. 1 .

 1 Fig. 1. (a) Original Image (Flower) (b) Power Rcut segment result (c) Power Rcut segment result post processing with opening (d) Original Image (Cameraman) (e) Contours obtained by Power R cut and (f) Contours obtained by Normalized spectral clustering.

Fig. 2 .

 2 Fig. 2. (a) Basic Graph (b) Power Rcut Clustering (c) MST based clustering. Observe that Power Rcut splits the graph into two equal halves, where as MST based clustering does not.

Fig. 3 .

 3 Fig. 3. (a) Basic Graph and MST edges highlighted in bold (b) Clustering with MST (c) Power Rcut Clustering. Observe that Power Rcut clustering tries to split the graph into equal parts.

Proposition 4 .

 4 If at a threshold t, we have that G ≥t has exactly m clusters, then MST clustering and Power Rcut clustering results in the same clusters.

Fig. 4 .

 4 Fig. 4. Data is sampled from two concentric circles with noise. The clusters obtained are represented by red dots and blue circles. (a) Power Rcut clustering. The green stars indicate the "noise" clusters obtained. (b) Power Rcut clustering obtained by adjusting the noise clusters to the nearest cluster. (c) Spectral clustering (d) Fowlkes-Mallows scores for varying noise. The dashed line indicates spectral clustering score and continuous line indicate Power Rcut score. The higher the score, the better the procedure.

Acknowledgements

AC and SD would like to thank Indian Statistical Institute. This work has been partly funded by ANR-15-CE40-0006 CoMeDiC and ANR-14-CE27-0001 GRAPHSIP research grants. BSDS would like to acknowledge the support received from the Science and Engineering Research Board (SERB) of the Department of Science and Technology (DST) with the grant number EMR/2015/000853, and the Indian Space Research Organization (ISRO) with the grant number ISRO/SSPO/Ch-1/2016-17.


J L Till
email: jtill@hi.is

Y Guyodo

F Lagroix

G Morin

N Menguy

G Ona-Nguema

Presumed magnetic biosignatures observed in magnetite

The oriented chains of nanoscale Fe-oxide particles produced by magnetotactic bacteria are a striking example of biomineralization. Several distinguishing features of the magnetite particles that comprise bacterial magnetosomes have been proposed to collectively constitute a biosignature of magnetotactic bacteria (Thomas-Keprta et al.,

2001

). These features include high crystallinity, chemical purity, a single-domain magnetic structure, well-defined crystal morphology, and arrangement of particles in chain structures. Here we show that magnetite derived from inorganic breakdown of nanocrystalline goethite exhibits magnetic properties and morphologies remarkably similar to biogenic magnetite from magnetosomes. During heating in reducing conditions, oriented nanogoethite aggregates undergo dehydroxylation and transform to stoichiometric magnetite. We demonstrate that highly crystalline single-domain magnetite with euhedral grain morphologies produced abiogenically from goethite meets several of the biogenicity criteria commonly used for identification of magnetofossils. Furthermore, the suboxic conditions necessary for magnetofossil preservation in sediments are also conducive to promote reductive alteration of nanogoethite, and to preserve detrital magnetite originally formed from goethite. The findings of this study have potential implications for the identification of biogenic magnetite, particularly in older sediments where diagenetic alteration commonly disrupts the chain structure of magnetosomes. Our results indicate in derived from abiotic reductive alteration of nanogoethite that isolated magnetofossils cannot be positively distinguished from inorganic magnetite on the basis of magnetic properties and morphology, and that intact chain structures remain the only reliable distinguishing feature of fossil magnetosomes.

Introduction

Magnetotactic bacteria (MTB) are a diverse group of microbes that produce chains of magnetic nanoparticles called magnetosomes for the purpose of navigation. MTB have been identified in an extensive variety of freshwater and marine environments [START_REF] Faivre | Magnetotactic Bacteria and Magnetosomes[END_REF], and the preserved magnetosome components of such bacteria, also known as magnetofossils, have been identified in sediments dating at least as far back as Cretaceous in age [START_REF] Montgomery | The magnetostratigraphy of Coniacian-Late Campanian chalk sequences in southern England[END_REF]. The stoichiometric magnetite that comprises most bacterial magnetosomes consistently exhibits certain features, including a high degree of crystallinity with few crystallographic defects, high chemical purity, a single-domain magnetic structure, well-defined crystal morphology, a tendency for particle elongation along [111] directions, and arrangement of particles in chain structures [START_REF] Kopp | The identification and biogeochemical interpretation of fossil magnetotactic bacteria[END_REF]. These collective attributes have been proposed as a biosignature of magnetotactic bacteria and have been applied as criteria for the identification of magnetofossils in sediments, sedimentary rocks, and even meteorites [START_REF] Thomas-Keprta | Truncated hexa-octahedral magnetite crystals in ALH84001: Presumptive biosignatures[END_REF]. While all of the above criteria are typically observed in cultured strains of MTB and live bacteria sampled from modern aqueous environments, studies of older sediments often fail to observe intact chain structures in fossil magnetosomes due to collapse and disaggregation of the chains either through diagenesis or by laboratory protocols of magnetic mineral extraction for microscopic investigation. In some cases, methods such as ferromagnetic resonance or low-temperature magnetic measurements can be used to infer the presence of magnetic chain structures (Weiss et al., 2004a). However many studies on ancient sediments rely on microscopic observation of magnetic extracts, combined with analysis of sediment magnetic properties to detect single domain (SD) magnetite (e.g. [START_REF] Abrajevitch | Asteroid impact vs. Deccan eruptions: The origin of low magnetic susceptibility beds below the Cretaceous-Paleogene boundary revisited[END_REF][START_REF] Larrasoana | Paleomagnetic and paleoenvironmental implications of magnetofossil occurrences in late Miocene marine sediments from the Guadalquivir Basin, SW Spain[END_REF][START_REF] Savian | Environmental magnetic implications of magnetofossil occurrence during the Middle Eocene Climatic Optimum (MECO) in pelagic sediments from the equatorial Indian Ocean[END_REF]), which has few known inorganic sources in sediments.

Previous studies have demonstrated how inorganic processes may produce certain characteristic morphologies of biogenic magnetite that can explain the occurrence of SD magnetite in the ALH84001 Martian meteorite [START_REF] Barber | Origin of supposedly biogenic magnetite in the Martian meteorite Allan Hills 84001[END_REF][START_REF] Bradley | Epitaxial growth of nanophase magnetite in Martian meteorite Allan Hills 84001: implications for biogenic mineralization[END_REF][START_REF] Golden | Evidence for exclusively inorganic formation of magnetite in Martian meteorite ALH84001[END_REF]. However, inorganic processes are rarely invoked to explain biogenic characteristics of SD magnetite in terrestrial environments. Rather it is assumed that because MTB are widespread in modern aqueous environments, they are likely to have been widespread throughout much of Earth's history and hence much ancient sediment may be expected to carry magnetic signatures of magnetofossils. Here we describe the various magnetosome-like properties of nanoscale magnetite particles produced by inorganic alteration of nanocrystalline goethite. We propose that magnetite produced by this reaction pathway could potentially contribute to the SD magnetite signals in sediment magnetic properties that are commonly attributed to biogenic magnetite.

The Fe-oxyhydroxide goethite occurs in nanocrystalline form in a wide range of soils, aeolian material, and lake and marine sediments (van der [START_REF] Van Der Zee | Nanogoethite is the dominant reactive oxyhydroxide phase in lake and marine sediments[END_REF]. In many sedimentary systems, it is the dominant substrate available for Fe-redox reactions [START_REF] Hansel | Structural constraints of ferric (hydr)oxides on dissimilatory iron reduction and the fate of Fe(II)[END_REF][START_REF] Van Der Zee | Nanogoethite is the dominant reactive oxyhydroxide phase in lake and marine sediments[END_REF]. Nanogoethite is predicted to be thermodynamically unstable with respect to dehydroxylation to Fe-oxide at ambient temperatures on geologic time scales [START_REF] Diakonov | Thermodynamic properties of iron-oxides and hydroxides .1. Surface and bulk thermodynamic properties of goethite (alpha-FeOOH) up to 500[END_REF][START_REF] Langmuir | Particle size effect of the reaction goethite = hematite + water[END_REF], although the kinetics are sufficiently slow that no reaction occurs below 100°C on laboratory time scales [START_REF] Diakonov | Thermodynamic properties of iron-oxides and hydroxides .1. Surface and bulk thermodynamic properties of goethite (alpha-FeOOH) up to 500[END_REF]. Recently, a study by [START_REF] Till | Goethite as a potential source of magnetic nanoparticles in sediments[END_REF] reported that nanogoethite readily alters to sub-micron magnetite under reducing conditions upon moderate heating (T = 210-270°C).

They identified a two-step process involving dehydroxylation of goethite to nanohematite, and subsequent rapid reduction and recrystallization of nanohematite to fine-grained magnetite. We analyzed the magnetite produced in these experiments in detail using transmission electron microscopy (TEM) and magnetic measurements and describe the results below.

Procedures

Synthesis

Synthetic nanogoethite was produced using the protocol outlined in [START_REF] Schwertmann | Iron Oxides in the Laboratory: Preparation and Characterization[END_REF]. A 0.05 M solution of FeCl 2 •4H 2 O was prepared in a glove box using deoxygenated water and was mixed with a 1 M NaHCO 3 solution. A constant flow of air was bubbled through the resulting suspension, which was continuously agitated and became oxidized over 48 h. The goethite precipitate was separated by centrifuging and rinsing with ultrapure (MilliQ) water several times and dried in a vacuum desiccator. The resulting goethite particles are around 10 nm by 50 nm in size, and consist of well-oriented aggregates of crystallites with crystallite sizes around 6 nm [START_REF] Till | Goethite as a potential source of magnetic nanoparticles in sediments[END_REF].

Characterization

The starting material and reaction products were characterized by Rietveld refinement of the X-ray diffraction (XRD) powder patterns and imaged by high-resolution transmission electron microscopy on JEOL 2100F microscope with field-emission gun at 200 kV accelerating voltage. Electron diffraction patterns were calculated by fast Fourier transforms of high-resolution images. Samples for magnetic measurements were prepared using small amounts of undiluted sample powders packed in gelatin capsules. Lowtemperature magnetic measurements of saturation isothermal remanent magnetization (SIRM) curves measured on warming from 10 K after field-cooling (FC) in a 2.5 T field or cooling in zero-field (ZFC), were made on a Quantum Designs Magnetic Properties Measurement System (MPMS XL-5 with EverCool). δ FC /δ ZFC ratios were calculated as δ = (M irm (80)-M irm (150))/M irm (80), where M is the value of magnetic remanence at 80 K and 150 K upon warming after either FC or ZFC pre-treatment. First-order reversal curve (FORC) diagrams and hysteresis loops were measured on a Princeton Measurements Corporation vibrating sample magnetometer (VSM) at room temperature. FORC measurements used a maximum field of 0.3 T, which is greater than the samples magnetic saturating field, and a field increment of 1 mT. FORC diagrams were processed and plotted with the FORCinel software package [START_REF] Harrison | FORCinel: An improved algorithm for calculating first-order reversal curve distributions using locally weighted regression smoothing[END_REF] using the VARIFORC smoothing protocol [START_REF] Egli | VARIFORC: An optimized protocol for calculating non-regular first-order reversal curve (FORC) diagrams[END_REF], with smoothing parameters of S c0 =3, S b0 =2, and

S c1 =S b1 =5.

Alteration experiments

Alteration experiments were performed by heating synthetic nanogoethite powder at temperatures between 210 and 270•C for up to 2.5 hours in a constant flow of a 20%-80% CO-CO 2 gas mixture. The furnace used for heating experiments was enclosed inside an Ar-filled glove box, and samples were prepared and maintained under anoxic conditions to minimize samples oxidation. Magnetic characterization was performed immediately after each alteration experiment. Conditions for each experimental run as well as detailed results of XRD and other magnetic measurements were reported by [START_REF] Till | Goethite as a potential source of magnetic nanoparticles in sediments[END_REF] and are summarized in Table 1.

Results

Magnetite morphology

TEM images of the pure magnetite end product indicate that the majority of grains are rounded and elongated, ranging from about 20 to 60 nm in width (Fig. 1A). Although many of grains have irregular or non-distinct shapes, a small but significant portion exhibit striking similarities to magnetite particles found in magnetosomes. Among the latter are elongated, tapered particles (Fig. 1C) that resemble bullet-shaped magnetosome particles found in certain MTB strains [START_REF] Kopp | The identification and biogeochemical interpretation of fossil magnetotactic bacteria[END_REF]. A number of equant and slightly elongated euhedral particles were also found that strongly resemble cubooctahedral magnetosome morphologies (Fig. 1B andE). Short "chains" of particles are even observed occasionally in TEM (Fig. 1A andD), although the spontaneously formed chain configurations in our samples can be distinguished from chains formed by MTB by the lack of repeated regular grain shapes and close spacing of the particles. In highresolution TEM images, the magnetite crystals are highly crystalline and free of defects, with the exception of occasional twinning.

Magnetic properties

To examine the distribution of magnetic domain states and degree of magnetostatic interactions in goethite-derived magnetite, first-order reversal curve (FORC) diagrams were obtained for altered goethite samples containing magnetite in various stages of the reaction. Partially reacted G04 and G03 samples containing approximately 4 and 5 wt% magnetite, based on saturation magnetization (M s ) values in Table 1 and the theoretical value of 92 Am 2 /kg for stoichiometric magnetite [START_REF] Dunlop | Rock Magnetism: fundamentals and frontiers[END_REF], exhibit relatively small interaction fields and a high-coercivity (H c ) "tail" extending along the center horizontal axis (Fig. 2). The FORC distributions of these samples reflect a grain size distribution consisting of a mixture of superparamagnetic and weakly interacting small SD magnetite grains, as determined by [START_REF] Till | Goethite as a potential source of magnetic nanoparticles in sediments[END_REF]. Magnetite-rich samples, G02 and G05 with approximately 53 and 71 wt% magnetite respectively, display a localized peak with a broader vertical spread indicating higher magnetostatic interaction fields (H u ) and overall higher coercivities, again including a high-coercivity tail with low interaction fields (Fig. 2). The teardrop-shaped FORC pattern for sample G02 is characteristic of interacting SD magnetite and is typical for experimentally disaggregated magnetosome particles [START_REF] Kopp | The identification and biogeochemical interpretation of fossil magnetotactic bacteria[END_REF][START_REF] Moskowitz | Rock magnetic criteria for the detection of biogenic magnetite[END_REF] and some magnetofossil-bearing sediments [START_REF] Roberts | Searching for single domain magnetite in the "pseudo-single-domain" sedimentary haystack: Implications of biogenic magnetite preservation for sediment magnetism and relative paleointensity determinations[END_REF].

Pure intact magnetosome chains and sediments dominated by intact magnetofossils display a narrow horizontal central ridge signifying non-interacting SD magnetite in FORC diagrams. In natural sediments, the addition of strongly interacting detrital magnetite can mask the central ridge, requiring certain measurement procedures to isolate the biogenic component of magnetization [START_REF] Egli | Detection of noninteracting single domain particles using first-order reversal curve diagrams[END_REF]. The high-coercivity tails seen for our samples resemble the central ridge displayed by biogenic magnetite, but do not represent a separate mineral component; rather they likely represent magnetite particles embedded in a matrix of incompletely reacted nanohematite that are sufficiently dispersed to be weakly interacting. Interaction effects may be further reduced for highly dispersed magnetite particles in sedimentary material, in contrast to the high magnetite concentrations in our measured samples. Previous work suggests that well-dispersed, fine inorganic magnetite may also display features of non-interacting SD particles in FORC diagrams [START_REF] Egli | Detection of noninteracting single domain particles using first-order reversal curve diagrams[END_REF], including pedogenic Fe-oxides in soils [START_REF] Geiss | Direct estimates of pedogenic magnetite as a tool to reconstruct past climates from buried soils[END_REF]. The close similarities between the FORC diagrams for goethite-derived magnetite in Fig. 2 and those reported for magnetofossil-bearing sediments (e.g., [START_REF] Roberts | Searching for single domain magnetite in the "pseudo-single-domain" sedimentary haystack: Implications of biogenic magnetite preservation for sediment magnetism and relative paleointensity determinations[END_REF]) may confound the identification of magnetofossils in natural sediments. The close comparison with natural samples also suggests that material similar to that produced from goethite in our experiments could be sufficient to account for the magnetic signature of certain sediments.

The Moskowitz test [START_REF] Moskowitz | Rock magnetic criteria for the detection of biogenic magnetite[END_REF]) is a commonly used magnetic measurement for detecting intact magnetosome chains. The δ FC /δ ZFC ratio is based on the loss in remanence on warming through the Verwey crystallographic transition around 120 K [START_REF] Verwey | Electronic conduction of magnetite (Fe 3 O 4 ) and its transition at low temperature[END_REF] and is greater than 2 for intact magnetosome chains, while values between 1 and 2 indicate that SD magnetite is present in other forms, including disaggregated or oxidized magnetosomes. δ FC /δ ZFC values for our magnetite-bearing samples are between 1.1 and 1.3. These values fall in the same range as the magnetitebearing carbonate globules in Martian meteorite ALH84001 (Weiss et al., 2004b), experimentally disaggregated magnetosomes [START_REF] Li | Magnetic anisotropy, magnetostatic interactions and identification of magnetofossils[END_REF], and marine sediment cores containing partially oxidized magnetofossils [START_REF] Housen | Depth distribution of magnetofossils in nearsurface sediments from the Blake/Bahama Outer Ridge, western North Atlantic Ocean, determined by low-temperature magnetism[END_REF].

The sharpness of the Verwey transition around 120 K for the pure magnetite endproduct (sample G05) indicates a high degree of oxygen stoichiometry and the absence of substitutional impurities (Weiss et al., 2004b), as expected from the high purity of the starting goethite material. Although natural goethite commonly occurs in aluminous form with up to 30 mole% Al substitution [START_REF] Tardy | Geochemistry of laterites, stability of Al-goethite, Alhematite, and Fe3+-kaolinite in bauxites and ferricretes -An approach to the mechanism of concretion formation[END_REF], the stabilizing effect of aluminum [START_REF] Ruan | Dehydroxylation of aluminous goethite -unit-cell dimensions, crystal size and surface-area[END_REF] suggests that Al-free goethite will alter to magnetite more readily and that Al-substituted magnetite produced by this pathway should be less common. Despite the success of some studies in producing magnetosomes doped with small amounts of metals (e.g. [START_REF] Prozorov | Manganese incorporation into the magnetosome magnetite: magnetic signature of doping[END_REF]), cation substitution in magnetofossils remains an important counter-indicator of biogenicity [START_REF] Amor | Chemical signature of magnetotactic bacteria[END_REF], and the occurrence of aluminum substitution in particular would strongly support an origin from detrital or authigenic goethite.

Discussion and conclusions

Till et al. 2015 recently identified various pathways by which altered goethite may contribute nanoscale magnetic particles to sediments and soils, including thermal alteration by low-grade metamorphism, diagenesis in marine sediments and by wildfire in soils.

Elevated temperatures generated in meta-sediments during deep burial will promote breakdown of goethite and may lead to authigenic magnetite formation under reducing conditions. The stability of goethite in anoxic sediments at ambient temperatures is unknown but it is unlikely to be stable under Fe-reducing conditions, so that the possibility of magnetite formation from low-temperature goethite alteration cannot be excluded. This is especially true given that current knowledge of the stability of nanoparticles of goethite and other iron oxides and hydroxides is even sparser [START_REF] Lagroix | Geological Occurrences and Relevance of iron Oxides[END_REF].

Production of fine magnetic particles and soil magnetic enhancement alteration has been documented to result from goethite alteration during wildfire in various soil types [START_REF] Anand | The association of maghemite and corundum in Darling Range laterites, Western Australia[END_REF][START_REF] Clement | The effects of wildfires on the magnetic properties of soils in the Everglades[END_REF][START_REF] Ketterings | Changes in soil mineralogy and texture caused by slash-and-burn fires in Sumatra, Indonesia[END_REF][START_REF] Nornberg | Comparison of the mineralogical effects of an experimental forest fire on a goethite/ferrihydrite soil with a topsoil that contains hematite, maghemite and goethite[END_REF]. Nanoparticles produced by fire have high mobility due to both increased surface runoff and sediment delivery to lake catchments [START_REF] Smith | Discriminating fine sediment sources and the application of sediment tracers in burned catchments: a review[END_REF] and from increased wind erosion and aeolian transport following wildfire events [START_REF] Whicker | Temporal and spatial variation of episodic wind erosion in unburned and burned semiarid shrubland[END_REF]. These processes represent pathways by which detrital goethite-derived magnetite may enter lake and marine sediments, particularly marine settings with substantial continental inputs from aeolian deposition or submarine fans. Aeolian sediments and detrital material from weathered igneous formations have previously been recognized as potential sources of SD magnetite [START_REF] Roberts | Searching for single domain magnetite in the "pseudo-single-domain" sedimentary haystack: Implications of biogenic magnetite preservation for sediment magnetism and relative paleointensity determinations[END_REF]. However, it is often assumed in sediment magnetism studies that the magnetic signature of biogenic magnetite can be distinguished from detrital sources of sedimentary magnetite on the basis of weak magnetic interactions, narrow coercivity and grain size distributions, and magnetosome-like crystal morphologies. Our findings demonstrate that inorganic magnetite particles can exhibit magnetic signatures and crystal morphologies highly similar to disaggregated magnetosomes and isolated magnetofossils in sediments.

Further complicating the problem of magnetofossil identification is that the conditions suitable for preservation of inorganic SD magnetite in sediments should be identical to those required for preservation of magnetofossils. Namely, anoxic or suboxic conditions are required to inhibit oxidation, but must not be so reducing that fine magnetite particles begin to dissolve. Reliable identification of suspected magnetofossils should address the robustness measures outlined by [START_REF] Kopp | The identification and biogeochemical interpretation of fossil magnetotactic bacteria[END_REF], including the assessment of high quality paleomagnetic data, to rule out the possibility of secondary magnetizations that would result from authigenic growth of SD magnetite. Detrital SD magnetite that shares physical characteristics of biogenic magnetite will also produce high quality paleomagnetic records but will not meet the key criterion of long, intact chain structures detected either by direct microscopic observation or by various tests that infer chain configurations.

Our findings underscore the need for careful characterization of potential magnetofossils and reinforce the assertions of previous studies [START_REF] Barber | Origin of supposedly biogenic magnetite in the Martian meteorite Allan Hills 84001[END_REF][START_REF] Golden | Evidence for exclusively inorganic formation of magnetite in Martian meteorite ALH84001[END_REF][START_REF] Wang | Quantified abundance of magnetofossils at the Paleocene-Eocene boundary from synchrotron-based transmission X-ray microscopy[END_REF] that cubo-octahedral morphologies in single domain magnetite are not strictly unique to MTB magnetosomes. We contend that observation of isolated magnetite particles and magnetic properties associated with SD particles in sediments are not sufficient evidence for a biogenic origin. Given the widespread occurrence of nanocrystalline goethite in nature, its role as a potential precursor to sedimentary magnetite should be considered in future studies.

Figure Captions : Figure 1 :

 :1 Figure Captions: Figure 1: Examples of magnetite produced by reductive dehydroxylation of nanogoethite aggregates imaged by TEM. A) Clusters of magnetite particles demonstrating dominantly rounded, elongated shapes. B) Equant grains of highly crystalline magnetite with approximately cubo-octahedral morphologies. C) Elongated tapered magnetite grains resembling bullet-shaped magnetosome particles. D). Magnetite particles arranged in short irregular chains. E). Highly euhedral magnetite grains with cubo-octahedral morphologies.Insets are simulated electron diffraction patterns for particles in each image.

Figure 2 :

 2 Figure 2: FORC measurement diagrams (top) for altered goethite G04, G03, G02 samples and gradient curves of magnetic remanence acquisition obtained from FORC measurements (bottom) for increasing stages of the magnetite-producing reaction from left to right. A smoothing factor of 3 has been applied to each FORC diagram.

Figure 3 :

 3 Figure 3: Measurements of low-temperature saturation isothermal magnetic remanence (SIRM) measured on warming from 10 K after zero-field cooling (ZFC) or field cooling (FC) in a 2.5 T field for various stages of reductive alteration products of nanogoethite. Solid grey lines represent derivatives of FC remanence curves with local minima indicating the temperature of the Verwey transition.

Table 1 .

 1 Room-temperature hysteresis parameters and experimental conditions for altered samples.

	Sample	Temp	Heating time	M s	M r	H c
		(°C)	(min)	(Am 2 /kg)	(Am 2 /kg)	(mT)
	G02	250	150	49	15	17
	G03	230	150	4.3	1.1	10
	G04	210	155	3.8	0.59	4.6
	G05	270	75	65	18	19

Acknowledgments

This work was supported by the Agence National de Recherche of France under project 2010-BLAN-604-01. This is IPGP contribution XXXX.