Antoine G Godin

Juan A Varela

Zhenghong Gao

Noémie Danné

Julien P Dupuis

Brahim Lounis

Laurent Groc

Laurent Cognet

Single-nanotube tracking reveals the nanoscale organization of the extracellular space in the live brain

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

sideband [START_REF] Santos | All-optical trion generation in single-walled carbon nanotubes[END_REF] ) and emission (986 nm for (6,5) SWCNTs) are in the transparency window of the studied brain tissues (Figure 1c) and as such, minimize tissue phototoxicity (Figures S2 andS3).

Phonon sideband excitation also ensures better emission photostability [START_REF] Santos | All-optical trion generation in single-walled carbon nanotubes[END_REF] as compared to high photon-energy excitation commonly used to excite SWCNT at their second order electronic transition (Figure S4).

Single luminescent (6,5) SWCNTs were detected in diverse areas of sagittal brain slices (e.g. neocortex, hippocampus, striatum) (Figure S5). Imaging was performed at different depths in the brain slices, from a few tens of µm to avoid the damaged superficial layers up to ~100 µm (Figure S6). The broad SWCNT dissemination away from cerebroventricular injection site was surely facilitated by their nanometer-size diameters, and also suggests that SWCNTs were primarily located in the ECS since such a large dissemination would not be possible if SWCNTs were internalized in brain cells. The diffusion of molecules in the ECS is thought to be globally governed by several parameters such as the geometric path length, trapping in dead space domains, and ECS viscosity by molecular crowding and transient molecular interactions. At a macroscopic scale, these parameters participate to the so-called tortuosity of the ECS 2,[START_REF] Thorne | In vivo diffusion analysis with quantum dots and dextrans predicts the width of brain extracellular space[END_REF] . Many SWCNTs were found immobilized away from the ventricular injection zones, likely originating from non-specific interactions with molecules of the ECS (although PL-PEG coating minimizes this effect in live cell culture (Figure S7) and live animals [START_REF] Liu | In vivo biodistribution and highly efficient tumour targeting of carbon nanotubes in mice[END_REF] ), and/or by the existence of deadspace microdomains 2 . The direct observation of PL-PEG-coated SWCNT trapping is thus consistent with the existence dead-space domains in the ECS. To further ascertain extracellular location, we directly loaded SWCNTs within hippocampal pyramidal neurons through a patchclamp pipette. The behaviours of detected intracellular SWCNTs were distinct from cerebroventricular injected ones (Figure S8), further supporting that SWNCTs imaged after icv injection are primarily extracellular.

The movements of individual SWCNTs exploring the ECS could be resolved with video rate fluorescence imaging (Figure 1d). This is primarily due to the SWCNT high aspect-ratio and rigidity, linked to their nanometer diameter, which significantly slows down their diffusion in the ECS maze, as compared to nanometer sized probes or fully flexible polymers. We indeed checked that small isotropic fluorescent probes diffuse too fast in the ECS to be imaged at video rate for a long time (observed trajectories < 1s, Figure S9). In addition, we could record long movies of single SWCNT ( , ) diffusion occurring during more than 20,000 frames within the ~1µm depth-of-focus of the microscope (Movie 1). By means of two-dimensional asymmetric Gaussian fitting analysis, we extracted in each movie frame the nanotube center-of-mass localization ( , ) in the imaging plane with sub-wavelength precision (~ 40 nm) and its axis orientation relative to the axis in the laboratory frame (Figure 1c, inset). Analysis of the fitted 2D Gaussian asymmetry provides a measure of the nanotube lengths L which range here from ~490-780 nm. These lengths match the typical SWCNT lengths of the injected solution determined by AFM imaging (Figure S10). Assuming characteristic ECS dimensions larger than 50 nm, these SWCNTs behave as rigid rods in the ECS and their diffusion should be independent of flexibility [START_REF] Fakhri | Brownian motion of stiff filaments in a crowded environment[END_REF] . The nanotube movements along (∥) and perpendicular (⊥) to their axis were then decoupled to calculate two 1D Mean Squared Displacements ∥ and as a function of time (Figure 2a). The corresponding to the 2D movements in the laboratory frame was also calculated. [START_REF] Schutz | Single-molecule microscopy on model membranes reveals anomalous diffusion[END_REF][START_REF] Tardin | Direct imaging of lateral movements of AMPA receptors inside synapses[END_REF] along and perpendicular to the nanotube axis ( ∥, ) defined for different time lagsτwith τ < (supplemental materials). Along each direction (∥, ⊥), two diffusion behaviors are found characterized by the time dependence of their mean denoted 〈r ∥, 〉 ( ), where = 1 corresponds to the slowest SWCNT diffusive states and = 2 to faster ones (Figure S11). The population of small displacements perpendicular to SWCNT axis is bounded as evidenced by the fast saturation of 〈r 〉 ( ) (Figure 2b) while other states display mainly unrestricted diffusion (Figure 2b andS11). This behaviour can be related to successive diffusion episodes occurring in different ECS confinement domains (see below). The plateau of 〈r 〉 ( ), which equals 〈 〉 6 ⁄ for confined diffusion along a 1D coordinate [START_REF] Kusumi | Confined lateral diffusion of membrane receptors as studied by single particle tracking (nanovid microscopy). Effects of calcium-induced differentiation in cultured epithelial cells[END_REF] , provides a measure of 〈 〉 representing the average confining ECS dimension (Figure 2b). As displayed in Figure 2c, 〈 〉 ranges from 80 to 270 nm (150 ± 40 nm, mean ± SD). This dynamic analysis demonstrates that anisotropic diffusion behaviours of single SWCNTs locally probe the brain ECS and can reveal nanoscale dimensions (Figure 2d).

Using a similar approach as in single-molecule localization microscopy [START_REF] Godin | Super-resolution microscopy approaches for live cell imaging[END_REF] , a large number of nanotube localization coordinates obtained with sub-diffraction precisions were pooled to create a super-resolved image of the ECS (Figure 3a). Such images directly unveil the spatial, nanoscale and structural tortuosity of the live brain ECS. The ECS maze is heterogeneous and consists of many connected submicron structural domains of various sizes. A total of 419 ECS domains were identified with typical dimensions ranging from ~ 50 to 700 nm (Figure 3b and S12). Noteworthy, the ECS nanoscale dimensions and morphologies described here in live brain samples resembles the ECS morphology visualized by electron microscopy following cryofixation of brain tissues [8][9] (Figure S13). Maps of the instantaneous diffusion constants calculated along nanotube trajectories on 300 ms sliding time windows were constructed (Figure 3c) to evaluate the spatial dependence of the local SWCNT diffusion in live brain tissue. From the knowledge of nanotube lengths, high-resolution spatial maps of the ECS viscosity were also obtained (Figure 3c In order to support this hypothesis, we then biochemically altered the ECS structure by injecting hyaluronidase in the brain lateral ventricles of young rats several hours before nanotube injection and imaging, a time period supposedly sufficient to significantly digest hyaluronic acid [START_REF] Kochlamazashvili | The extracellular matrix molecule hyaluronic acid regulates hippocampal synaptic plasticity by modulating postsynaptic L-type Ca(2+) channels[END_REF] . We first confirmed alterations of the ECM components in hyaluronidase-injected brains as attested by strong decrease of both hyaluronan binding protein (HABP) (Figures 4a-d indistinguishable from the one used to fit the histogram of naive brains (Figure 4h). This demonstrates that hyaluronidase effect is in fact not spatially homogeneous in the ECS (see also Figure 4g). Characterizing the ECS dimensions explored by SWCNTs using the DSD analysis, a moderate but significant reduction of mean size values 〈 〉 was found in altered brain (280 ± 60 nm) as compared to naive ones (150 ± 40 nm) (Figure 4i). This indicates that upon profound but non-uniform alteration of the ECM, the ECS topology offers larger territories for SWCNT nanoscale molecular exploration.

Here, we provide a new approach to reveal the ECS dimensions and viscosity at the nanoscale in live brain tissue of young rats. The ECS appears as a maze of interconnected polymorphic compartments structured down to the nanoscale and bearing specific rheological properties. This knowledge will surely influence our understanding of the chemical-based cellular communication in brain physiology and pathology. In addition, detailed knowledge of nanoscale ECS network should also foster new drug delivery strategies.

paper. Reprints and permission information is available online at www.nature.com/reprints.

Correspondence and requests for materials should be addressed to L.C. (laurent.cognet@ubordeaux.fr) & L.G. (laurent.groc@u-bordeaux.fr). Analysis codes used in this study are available upon request to L.C. or L.G. Intraventricular injections. Sprague-Dawley rats (Janvier, France) were used for this work, both male and female, treated according to the guidelines of the University of Bordeaux/CNRS Animal Care and Use Committee. Injections of SWCNTs were performed in living Sprague-Dawley rat pups (1-4 days old) anesthetized by hypothermia. Approximately 5 µl of SWCNTs dispersion were injected in each lateral ventricle (12 rats). Under cold light illumination, the injection place was found by firstly drawing a virtual line between the eye and lambda (easily seen through the skin), secondly finding the midpoint of that line, and thirdly moving 2 mm caudal from that mid-point along the virtual line. The injection was performed in that point, at a depth of 2.6 mm for P1 pups, 2.9 mm for P2 pups, 3.1 mm for P3 pups and 3.5 mm for P4 pups.

For hyaluronidase studies the intraventricular injection was performed in the same way (3 rats), injecting hyaluronidase from bovine testes 20 mg/mL in PBS (Sigma Aldrich #H3506) the night before injecting SWCNT.

Acute

  and Extended Data Movie 1). Viscosity values ranged from ~ 1 to 50 mPa•s which is up to two orders of magnitude larger than that of the cerebrospinal fluid. These viscosity maps reveal that ECS local viscosity is spatially inhomogeneous. Interestingly, within this viscosity range, the instantaneous diffusion constants of quantum dots in the ECS, having hydrodynamic diameters of 35 nm would locally vary from ~ 0.1 to 10 µ . As expected, the average diffusion constant of such quantum dots (~ 0.1 µ ) previously observed on macroscopic spatial scales in the ECS[START_REF] Thorne | In vivo diffusion analysis with quantum dots and dextrans predicts the width of brain extracellular space[END_REF] is constrained towards the smallest local values and cannot divulge the diversity of local viscosities. By comparing Figures 3a&c, the spatial dependence of the ECS rheological properties indicates that locally ECS viscosity values do not correlate with ECS dimensions. Indeed, although ξ values span up to two orders of magnitude, SWCNT instantaneous diffusion constants vary over up to three decades independently of the ECS dimensions in which SWCNTs are diffusing. Although bearing diffusion properties not necessarily identical to those of endogenous molecules diffusing in the brain ECS, our findings demonstrate that SWCNT tracking unveils and quantifies the diversity of nanoscale diffusive environments composing the ECS. Interestingly, considering the ECS as a fluidic connected space with specialized functional regions 2,4 , our results indicate that local structural dimensions are not sole responsible for affecting molecular diffusion properties of the ECS. It follows that other mechanisms including fine-tuned molecular interactions involving constitutive components of the ECS must play an important role for controlling local diffusion of e.g. endogenous signalling molecules or nutrients in local areas of interest.

  ) and aggrecan immunostaining (Extended Figure14). Noteworthy, we did not observe modification of the morphology of neuronal soma and overall cellular density in hyaluronidase-injected brains at spinning-disk confocal resolution. Analysis of SWCNT diffusion indicates that they explored wider areas as compared to naive brains within identical time scales (Figures 4e&f). Consistent with hyaluronidase-induced disappearance of ECM components, instantaneous diffusion coefficients were significantly increased (calculated as in Figure3b), indicating a global reduction of medium viscosity (Figures 4g&h, Movie 2). More precisely, the histogram of diffusion coefficients from naive brains was well fitted by a single lognormal distribution whereas in the altered brains two lognormal distributions were needed, with one of them being

Figure caption Figure 1 .

 caption1 Figure caption Figure 1. Single molecule tracking of luminescent SWCNTs in live ECS brain tissue. (a) SWCNTs are injected in lateral ventricles of living rats and diffuse into the neocortex. (b) Luminescent SWCNTs are imaged in brain slices. (c) Absorbance spectrum of 1 mm thick brain slice (black) and photoluminescence spectrum of a single (6,5) SWCNT in the ECS (red) upon 845 nm laser excitation (purple). (c, inset) Photoluminescence image of a (6,5) SWCNT recorded in the ECS with its orientation . (d) Color-coded trajectory of a single SWCNT diffusing in live ECS (20,000 data points). Scale bars = 1 µm.

Figure 2 .

 2 Figure 2. SWCNT diffusion properties in the ECS. (a) , ( ) in the laboratory frame (black squares) and ∥, ( ) calculated along (blue circles) and perpendicular (red diamonds) to the SWCNT axis for a 2D trajectory recorded in the ECS. Purple crosses correspond to ∥ + . (b) Evolution of the mean MSDs for the slow diffusion states 〈 ∥, 〉 ( ) retrieved from the analysis of the DSDs. The plateau of 〈 〉 ( ) gives an average ECS domain size 〈 〉 of 160 nm. (c) Histograms of 〈ξ〉 values measured from all single SWCNT trajectories as in (b). (d) Schematic representation of a SWCNT diffusing in the ECS.

Figure 3 .

 3 Figure 3. Super-resolution imaging of ECS morphology and ECS local viscosity maps. (a) Super-resolved image of live brain ECS morphology obtained from 20,000 localizations of a diffusing SWCNT. Scale bar = 500 nm. (b) Histogram of ECS domain dimensions ξ (N = 419) from 14 ECS super-resolved images as in (a). (c) Spatial map of SWCNT instantaneous diffusion coefficients calculated along the same trajectory using a sliding window of 300 ms. This representation also constitutes a high-resolution spatial map of ECS viscosities (see scale bar).

Figure 4 . 6 , 5 )

 465 Figure 4. ECS local modifications in chemically altered brains. Representative images of HAPB immunostaining in brain slices from naïve (a) and hyaluronidase-injected rats (altered, b). Neuronal nuclei are labelled with DAPI (blue). Scale bar: 250 µm. High magnification spinning disk microscope images of HAPB staining in both conditions are also presented (c, d). Scale bar: 20 µm. (e) Examples of super-resolved ECS images of naive vs. altered live brain tissues. Scale bars: 2 µm. (f) Areas explored normalized by SWCNT trajectory lengths. (g) Corresponding spatial viscosity maps. (h) Histograms of all instantaneous diffusion coefficients. In naive brains, the distribution is fitted with a single log-normal distribution (0.08±0.08 µm 2 /s, median±s.d.)

5 CaCl 2 , 2 . 5 5 KCl, 2 . 2

 5225522 brain slices preparation. Between 10 min and up to 3 h after SWCNT injection, pups were decapitated and parasagittal brain slices (0.5 mm thick) were prepared in an ice-cold artificial cerebrospinal fluid (ACSF) solution containing (in mM): 126 NaCl, 3.5 KCl, 2 CaCl 2 , 1.3 MgCl 2 , 1.2 NaH 2 PO 4 , 25 NaHCO 3 , and 12.1 glucose (gassed with 95% O 2 / 5% CO 2 ; pH 7.35). Slices were then imaged in oxygenated ACSF at 33°C perfused in the imaging chamber with a peristaltic pump.Immunostainings. Staining of Iba1, Aggrecan and HABP were performed in coronal brain slices of 80 μm thickness. Before antibody incubation, slices were incubated with PBS-Triton-X100 1% and 4% BSA for 2h under agitation at room temperature. Slices were subsequently washed three times with PBS for 5 min each time. Slices were incubated overnight at 4 °C with primary antibodies in PBS-Triton-X100 0.2% and 2% BSA solution. Anti-Iba1 rabbit antibody (Wako Pure Chemical Industries #019-19741) was used at a 1:200 dilution. Anti-Aggrecan rabbit antibody (Millipore #AB1031) was used at a 1:200 dilution. Biotinilated HABP (Amsbio #AMS.HKD-BC41) was also used at a 1:200 dilution. Slices were subsequently washed three times with PBS for 5 min each time, and incubated with secondary reagents. Anti-Iba1 and anti-Aggrecan antibodies were labelled with goat anti-rabbit Alexa568 (ThermoFisher Scientific #A-11011) in PBS-Triton-X100 0.2% and 2% BSA solution for 2h at room temperature under agitation, at a 1:1000 dilution. Biotinilated HABP was labelled with Alexa568 streptavidin (ThermoFisher Scientific #S11226) in PBS-Triton-X100 0.2% and 2% BSA solution for 2h at room temperature under agitation, at a 1:1000 dilution. Slices were finally washed three times with PBS and mounted in glass slides with Vectashield+DAPI (Vector Labs).Organotypic slice preparation. Organotypic slice cultures were prepared as previously described3 . Briefly, 350 µm thick hippocampal slices were obtained from postnatal day 5 to postnatal day 7 Sprague-Dawley rats using a McIlwain tissue chopper, and were placed in a preheated (37°C) dissection medium containing (in mM): 175 sucrose, 25 D-glucose, 50 NaCl, 0.KCl, 0.66 KH 2 PO 4 , 2 MgCl 2 , 0.28 MgSO 4 -7H 2 O, 0.85 Na 2 HPO 4 -12H 2 O, 2.7 NaHCO 3 , 0.4 HEPES, 2 x 10-5 % phenol red, pH 7.3 (all products from Sigma unless specified). After 25 min of incubation, slices were transferred on white FHLC membranes (0.45 µm) set on Millicell Cell Culture Inserts (Millipore, 0.4 mm; Ø 30 mm), and cultured for up to 14 days on multiwell-plates at 35°C / 5% CO 2 in a culture medium composed of 50 % Basal Medium Eagle, 25 % Hank's balanced salt solution 1X (with MgCl 2 / with CaCl 2 ), 25 % heat-inactivated horse serum, 0.45 % D-glucose, 1 mM L-glutamine (all products from Gibco unless specified). The medium was changed every 2-3 days. Single cell electroporation. Electroporation of single pyramidal neurons was performed as previously described 4 . Briefly, individual CA1 pyramidal neurons from 4-6 days in vitro hippocampal slices were electroporated to transfect cDNA encoding EGFP. Plasmids (5 µl at 1 µg/µl) were dissolved in a filtered cesium-based solution containing (in mM): 135 cesiummethanesulfonate, 8 NaCl, 10 HEPES, 0.2 EGTA, 4 Na 2 ATP, 0.33 Na 3 GTP, 5 tetraethylammonium chloride, pH 7.3 (all products from Sigma unless specified). This solution was supplemented with 10 µL of filtered endotoxin-free buffer TE (Qiagen), then centrifuged twice to pull down potential debris (10000 rpm, 15 min, 4°C) and used to fill 5-6 MΩ borosilicate patch pipettes. Electroporation was performed in 2 mL of pre-warmed (37°C / 5% CO 2 ) HEPES-based artificial cerebrospinal fluid (ACSF) containing (in mM): 130 NaCl, 2.CaCl 2 , 1.5 MgCl 2 , 10 HEPES, 10 D-glucose. Plasmid transfer was allowed by the delivery of 50 µs-width square-pulses at 100 Hz (1 s duration; -14 V current amplitude). COS-7 cell culture. COS-7 cells were grown in Dulbecco's Modified Eagle Medium (DMEM) without phenol-red (PAN Biotech, P04-01515) supplemented with 10% v/v foetal calf serum (FCS-Dominique Dutscher, 500105), 1% v/v penicillin/streptomycin (Dominique Dutscher, P06-07100) in 25 mL flasks (Falcon, 353082) at 37 °C, 5% CO 2 . They were kept below 90% confluence, at which cells were washed in sterile filtered PBS (PAN Biotech, P04-36500), detached in a small volume of trypsin (PAN Biotech, P10-021100), washed in full warm medium and split 1/10 in a new flask. For imaging, 90% confluent cells were similarly detached with trypsin and diluted 1/10 in full warm medium. COS7 cells were plated in uncoated petri dish. 24 hours after the plating, nanotubes were added (10 µl of the 3 mg/L PL-PEG nanotube solution and 60 µl of the 0.4 mg/L PL-PEG-NH 2 nanotube solution) to 1 mL of medium for 1h. The samples were rinsed 3 times for 5 minutes with medium and imaged under the fluorescent microscope.Wide-field Fluorescence Microscopy Setup. Single SWCNT photoluminescence imaging was performed with an inverted microscope equipped with an electron-multiplying CCD camera (Princeton Instruments ProEm) and a 1.45 NA 60x objective in a wide-field geometry. The excitation source consisted of a tunable Ti:Sa laser emitting at a wavelength of 845 nm to preferentially excite (6,5) SWCNTs at the resonance of the dark K-momentum exciton5 . The excitation intensity was kept at 10 kW/cm 2 with circularly polarized light. A dichroic mirror (FF875-Di01, Semrock) and a long-pass emission filter (ET900LP, Chroma) were used in order to illuminate and detect the SWCNT emitted fluorescence. Images of SWCNTs were recorded at 40 or 50 frames per second (typically 20,000 frames). GFP/Alexa488 (Excitation : FF02-472/30, Dichroic : FF495-Di03 and emission : FF01-520/35) and DAPI (Excitation : FF01-387/11, Dichroic : FF409-Di03 and emission : FF02-447/60) were imaged using epifluorescence white light excitation with appropriate filter sets.Spinning disk microscope. Alexa568 imaging in acute slices was performed in a Leica DMI6000 inverted microscope (Leica Microsystems) with a Yokogawa spinning disk unit CSU-X1. The setup was equipped with live cell chamber and temperature was constantly kept at 37 °C. Tissue scans were obtained with either 10x air, 63x oil and 100x oil objectives, with exposure times of 50-100ms per frame. DAPI staining in fixed tissue was excited with a 405nm laser line, and detected through a standard DAPI emission filter. Images were acquired using an Evolve EM-CCD camera (Photometrics), setting the EM gain at 600. NIR Spectrometer Setup. Emission spectra of single SWCNTs were collected using a cryogenically cooled 1D InGaAs detector (OMA V, Roper Scientific) placed at the output of a 150 mm spectrometer. A Chameleon (Coherent) pumped with a Ti:Sapphire laser (spectral range of 530-700 nm) and a Ti:Sa laser (spectral range of 700 to 850 nm) were used to generate a two-dimensional photoluminescence excitation/emission map of the PL-PEG SWCNT solution.Individual spectra of the nanotube solution, in a 1 cm long quartz cuvette, were normalized by the incident power. The spectrum of a solution containing only surfactant, acquired under the same experimental condition, was subtracted for each laser wavelength. The proportion of nanotubes(6,5) in the solution was estimated by comparing the peak value for detected semiconducting chiralities and assuming ~30% of metallic chiralities in HiPco samples.Singlecell loading. Single CA1 pyramidal neurons from 4-8 days in vitro (div) hippocampal organotypic slices were loaded with dilutions of carbon nanotubes to allow monitoring of their intracellular motion patterns. Briefly, nanotubes were dissolved (1/100 dilution) in a filtered cesium-based solution containing (in mM): 135 cesium-methanesulfonate, 8 NaCl, 10 HEPES, 0.2 EGTA, 4 Na 2 ATP, 0.33 Na 3 GTP, 5 tetraethylammonium chloride, pH 7.3 (all products from Sigma unless specified). This solution was supplemented with Alexa-488 (20 µg/mL) to allow further visual detection of the neurons of interest and used to fill 5-6 MΩ borosilicate patch pipettes. Single-cell loading was performed in 2 mL of pre-warmed (37°C / 5% CO 2 ) HEPESbased artificial cerebrospinal fluid (ACSF) containing (in mM): 130 NaCl, 2.5 KCl, 2.2 CaCl 2 , 1.5 MgCl 2 , 10 HEPES, 10 D-glucose. Nanotube transfer from the cesium-based solution to the cytoplasm was allowed by patching pyramidal neurons in whole-cell configuration.Determination of SWCNT localization and orientation. Two-dimensional Gaussian functionswith an arbitrary orientation were used to fit SWCNT images using homemade MatLab routines and determine SWCNT localizations ( , ), orientation , as well as large and small radii of the SWCNT image. Four consecutive images were averaged for each fit to improve localization precision (40 nm). SWCNT coordinates are then interconnected to reconstruct nanotube trajectories. When available, the positions of an immobile SWCNT was used to correct eventual small stage drifts over the 10 min recordings. Orientations (θ ϵ 0, π ) obtained from the Gaussian fits are by construction doubly degenerate. The SWCNT orientation on the first image of the movie is assumed to be between 0, π . This choice sets which end of the SWCNT is considered as the front of the tube. For all other orientations, continuity arguments are used to remove this degeneracy and to allow the orientation to be a real unbounded physical quantity (θ ϵ -∞, ∞ ).

  Photoluminescence

Ackowledgements. We warmly thank E. Bézard, M. Blanchard-Desce, B. Bontempi and P. Bon for helpful discussions. We thank J.P. Salvetat for experimental help for AFM imaging, and J. Ferreira for support with histological experiments. This work was supported by CNRS, Agence Nationale de la Recherche (ANR-14-OHRI-0001-01), IdEx Bordeaux (ANR-10-IDEX-03-02), Labex Brain (ANR-10-LABX-43), Conseil Régional d'Aquitaine (2011-1603009) and the France-BioImaging national infrastructure (ANR-10-INBS-04-01). A.G.G. acknowledges financial support from the Fondation pour la Recherche Médicale and the Fonds Recherche du Québec -Nature et Technologies. J.A.V. acknowledges Marie Curie Individual Fellowship 326442 funding.

Competing financial interest statement. We declare no competing financial interests.

Super-resolved images of the ECS from SWCNT localizations. For each trajectory, a superresolved image of the ECS was computed by cumulating all SWCNT localizations (typically 20,000 points). In super-resolved images, each localization is displayed a two-dimensional Gaussian of 50 nm width and unit amplitude as commonly used in localization microscopy 6 .

Estimation of SWCNT lengths. For each SWCNT trajectory imaged in live brain ECS, the distribution of eccentricity, obtained from the 2D asymmetric Gaussian fits of the nanotube images, was calculated when nanotube movements were undetectable (displacements between images < 40 nm). For each trajectory, the mean of the distribution was then used to determine the nanotube length (Figure S10) by comparison to the following model: the photoluminescence profile of a rigid nanotube of varying length L emitting at 986 nm was modelled as previously 7 using exciton diffusion length of ≈100 nm to provide a look-up table. S10b&c). PL-PEG SWCNTs were spin-coated on plasmacleaned glass coverslips using a G3P-8 Spin coater (Specialty Coating Systems) at 930 rpm for 3 min. To remove excess PL-PEG, samples were rinsed using Milli-Q water. Atomic forced microscopy (AFM) images were recorded using an 8 nm diameter Si3N4 tip. For each well isolated SWCNT, the length was quantified using the NeuronJ plugin in ImageJ 8 . 2a). For each trajectory, three Mean Squared Displacement (MSD) curves were calculated as exemplified in Figure 2a: (i) MSD in the laboratory frame corresponding to 2D movements in the imaging plane, and (ii) MSD ∥ and MSD by decoupling 1D movements along (∥) and perpendicular (⊥) to the nanotube axis.

Atomic force microscopy (Figure

Mean squared displacements (MSD) analysis (Figure

In order to decouple the diffusion of SWCNTs along and perpendicular to their principal axis, the displacement vector (x -x , y -y ) was determined at time t from the projection on the nanotube orientation vector obtained at time t (see inset of Figure 2a). This procedure to decouple the movements along and perpendicular to anisotropic object axis was presented previously [9][10] . One dimensional MSD ∥ and MSD could then be calculated from decoupled displacements (Figure 2a). and equals c(x, t) = √ e ⁄ , where N is the number of particles in the system.

Similarly, for two dimensional free diffusion, the MSD slope is equal to 4D since the MSD equals MSD(τ) = 〈r 〉( ) = ρ c(ρ, )dρ = 4D where c(ρ, t) is the solution of the two dimensional diffusion equation.

When the MSDs are not linear with time, a diffusion coefficient can be defined from the initial slopes of 〈r 〉( ) determined on short times.

Distribution of squared displacements (DSD) and calculation of 〈 〉 (Figures 2b-c, 4i

and S11). Considering a diffusing particle, the cumulative distribution of squared displacements ( ( ) ) represents the probability of finding the particle at a distance less than from its initial position after a time lag .

In the case of a freely diffusing particle along one coordinate, it is given by:

= erf 4D ⁄ = erf 2〈 〉( ) ⁄ .

In the case of a mixture of particles following two distinct 1D diffusion behaviours (e.g. free diffusion with D 1 and D 2 ), the DSD is characterized summing two independent diffusive processes:

where α are the fraction of particles in the different diffusive states (in the case of free diffusion : 〈r 〉 ( ) = 2D ).

A similar analysis was previously used to retrieve different molecular populations bearing distinct 2D diffusion behaviours from the observation of many short trajectories [11][12] .

Here a single and long trajectory of diffusing SWCNT displaying different temporal diffusive behaviours along its course is analysed. In order to retrieve the different diffusive behaviours, the DSD analysis was performed on the displacements along and perpendicular to SWCNT axes.

From single trajectories, the cumulative distribution of squared displacement along each direction (parallel and perpendicular to the nanotube axis) was computed for every time lag (see examples in Figure S11 for the trajectory analysed in Figures 2 and3 The evolution of 〈r 〉 is the slowest and displays a saturation revealing a confinement of the transverse movements of SWCNTs in the ECS. The typical ECS dimension responsible for this confinement can be estimated from the saturation value of 〈r 〉 : 〈 〉 = 6〈r 〉 13 .

Two-dimensional spatial maps of instantaneous diffusion coefficients and ECS local viscosity (Figures 3c, 4g&h and Extended Data movies 1 and 2).

In order to create diffusion coefficient spatial maps, MSDs were calculated in the laboratory frame using a sliding window of 300 ms along each trajectory instead of the full trajectory as above. Linear fits of the initial slopes of the sliding MSDs were performed on the first 100 ms to obtain the instantaneous diffusion coefficients as a function of time. Values of instantaneous diffusion coefficients were obtained for each subset of detections (t -150 ms ≤ t ≤ t + 150 ms) and associated to the centroid position (Figure 3c, 4e,g&h). Maps were created using pixels of 25 nm.

Instantaneous diffusion coefficients falling within the same pixel were averaged. For visualization purposes, convolution with a two dimensional Gaussian of 50 nm FWHM was applied to smooth the data.

In order to link instantaneous diffusion coefficient values of the SWCNTs to local viscosities of the ECS, we used the relationship between the diffusion coefficient of highly anisotropic rods diffusing in a 2D plane and the viscosity 14 : = 3k T ln(ϕ) / 8πD L . The nanotube length was estimated for each trajectory using the eccentricity of the imaged nanotube (see above).

Estimation of the ECS dimensions from the analysis of super-resolved images (Figure 3b and Figure S12). Super-resolved images revealed the diversity and complexity of the ECS morphology. In order to measure the typical dimensions of the ECS, we applied a global analysis by using a set of 2D Gaussian functions to directly fit the SWCNT localizations used to create the super-resolved images (Figure S12b). Initial guesses for applying Gaussian fitting was based on local density maxima of nanotube localizations (Figure S12a, inset & c). Two maxima that were closer than 75 nm could not induce two different Gaussian functions. The transverse dimensions of the set of 2D Gaussian functions used to fit the nanotube localizations provides the dimensions of the ECS displayed in Figures 3b andS12a.


Xiaojie Guo

Sophie Quinton

Pascal Fradet

Jean-François Monin

Work In Progress: Toward a Coq-certified Tool for the Schedulability Analysis of Tasks with Offsets

This paper presents the first steps toward a formally proven tool for schedulability analysis of tasks with offsets. We formalize and verify the seminal response time analysis of Tindell by extending the Prosa proof library, which is based on the Coq proof assistant. Thanks to Coq's extraction capabilities, this will allow us to easily obtain a certified analyzer. Additionally, we want to build a Coq certifier that can verify the correctness of results obtained using related (but uncertified), already existing analyzers. Our objective is to investigate the advantages and drawbacks of both approaches, namely the certified analysis and the certifier. The work described in this paper as well as its continuation is intended to enrich the Prosa library.

INTRODUCTION

For hard real-time systems used in application domains such as avionics, missing a deadline may have catastrophic consequences. Schedulability analysis, which aims to guarantee the absence of any deadline miss, is therefore of the utmost importance in the verification of such critical systems. Schedulability analysis has been the subject of intensive research over the past decades and many different techniques have been implemented into tools, including several commercial analyzers such as NETCAR-Analyzer [START_REF]NETCAR-Analyzer: the RTaW-Sim plugin for worst-case timing analysis on Controller Area Network[END_REF], SymTA/S [START_REF]SymTA/S: Model-based timing analysis and optimization[END_REF], etc. There is, however, no guarantee that the results provided by these tools are correct, for two reasons:

1. The theory that they build upon may be flawed, as in the original schedulability analysis for CAN messages [START_REF] Davis | Controller Area Network (CAN) schedulability analysis: Refuted, revisited and revised[END_REF];

2. They may contain undetected bugs and thus return erroneous results.

Our general objective is to provide schedulability analysis tools with formal guarantees using the Coq [START_REF]The Coq proof assistant[END_REF] proof assistant. We aim to investigate two options:

1. Write the entire analyzer and its correctness proof in Coq and extract from it a certified1 Caml implementation; 2. Write in Coq a program that can guarantee that a given result is correct, in contrast to computing that result -for example verifying that a given value is indeed a fixed point rather than computing such a fixed point. The extracted Caml tool would then have to be used together with an existing schedulability analysis tool. Which option will prove the best compromise in terms of computational complexity, implementation effort, acceptance from industry is still an open question.

As a first step, we focus on the schedulability analysis of uniprocessor systems of tasks with offsets dispatched according to the Fixed Priority Preemptive (FPP) scheduling policy. Schedulability analysis in presence of offsets is a nontrivial problem with a high computational complexity. In contrast to the traditional (offset oblivious) analysis, many scenarios must be tested and compared to identify which one represents the worst-case scenario.

In this paper, we formalize and prove in Coq an analysis technique presented in [START_REF] Tindell | Using offset information to analyse static priority pre-emptively scheduled task sets[END_REF] and [START_REF] Tindell | Adding time-offsets to schedulability analysis[END_REF] to bound the response time of tasks with offsets, and thus decide on the schedulability of a system. In addition to paving the way toward a Coq-certified tool for schedulability analysis of task sets with offsets, our formalization effort underlines implicit assumptions made by the author in [START_REF] Tindell | Using offset information to analyse static priority pre-emptively scheduled task sets[END_REF] and eases the generalization of the verified analysis. Our formalization is based on Prosa [START_REF]A Library for formally proven schedulability analysis[END_REF], a library of definitions and proofs for real-time schedulability analysis.

The rest of this paper is laid out as follows: Section 2 presents the system model; in Section 3, we present our formalization of the existing analysis. Section 4 discusses the Coq proof; finally, we present future work in Section 5.

SYSTEM MODEL

The system model considered in this paper is the one presented by Tindell in [START_REF] Tindell | Using offset information to analyse static priority pre-emptively scheduled task sets[END_REF]. It consists of a set of periodic tasks running on a uniprocessor using FPP scheduling. A system S is a set of transactions S := {Tr 1, Tr 2, . . . , Tr m} where each transaction is a set of tasks Tr i := {. . . , τ i,k , . . .} sharing the same period Ti. A task is described as a tuple

τ i,k := {C i,k , O i,k , D i,k , Ti}
where k denotes its priority (a greater k means a higher priority), C i,k its worst-case execution time and D i,k its deadline. There is no global synchronization between transactions so there is an indefinite time shift between any two transactions. Within a transaction Tr i, however, there is a fixed and known relation between task activations: an instance of a task τ i,k is activated O i,k time after each release of Tr i, where O i,k is called an offset.

In hard real-time systems, the duration between the activation and the completion of any instance of a task must be less than its deadline. The j-th instance (or job) of a task τ i,k is characterized by its activation time act i,k (j), its finishing time end i,k (j) and its computation time c i,k (j)with c i,k (j) ≤ C i,k . Its response time RT i,k (j) is defined as end i,k (j) -act i,k (j). The worst-case response time of task τ i,k , denoted wcrt i,k , is the largest possible response time among all instances of task τ i,k . Figure 1 shows an example with two transactions and four tasks. Black upward arrows indicate task activations, black downward arrows show task finishing times while red upward arrows represent the periodic release of transactions. Grey boxes indicate time intervals during which a task is executing and white boxes intervals during which a task is delayed by a higher priority task. Note that task τ1,8, for example, is always activated one time unit after task τ1,10, but the alignment between activations of τ1,8 and τi,9 is unknown and may vary. The analysis considered in this paper relies on the following hypotheses (we discuss this further in the conclusion):

1. Tasks in the same transaction have the same period. Therefore, a task activation can be associated with a unique transaction release, and conversely.

2. (Restricted offsets) The offset of a task is strictly less than its period: ∀τ i,k , O i,k < Ti.

3. (Implicit deadlines) The deadline of a task is equal to its period:

∀τ i,k , D i,k = Ti.
The analysis relies heavily on the concept of busy window which we explain now. An instant t is said to be a level-k quiet time if all tasks of priority higher than or equal to k released strictly before t have completed at t. A level-k busy window is a time interval [t1, t2[ in which:

1. a task with a priority higher than or equal to k is activated at t1; 2. t1 and t2 are level-k quiet times; 3. there is no other level-k quiet time between t1 and t2. In Figure 1, two level-k busy windows are shown using orange boxes marked with BW k .

RESPONSE TIME ANALYSIS (RTA)

In this section, we reformulate the approximate RTA technique presented in [START_REF] Tindell | Using offset information to analyse static priority pre-emptively scheduled task sets[END_REF]. We discuss in Section 4 how it relates to the original proof. From now on, we focus on a task τ i,k and analyze its worst-case response time.

τ i,k t1 t2 φi θ i,k O i,k BW k (a) (O i,k ≥ φ i ). τ i,k t1 t2 φi θ i,k O i,k BWj (b) (O i,k < φ i ).
Assume that there exists a level-k busy window [t1, t2[ in which the j-th instance of τ i,k is released2 . This job finishes at the latest at t2, so the response time of this instance can be bounded by:

RT i,k (j) ≤ t2 -act i,k (j) (1)
which, with the notation BW k := t2 -t1, is the same as

RT i,k (j) ≤ BW k + t1 -act i,k (j) (2)
The key to the analysis is to show that the response time is maximized when: (1) all jobs in the busy window take their worst-case execution time to complete; and (2) t1 is aligned with an activation of each transaction.

The definition of a busy window implies that BW k is the least fixed point of wl t 1 :

BW k = wl t 1 (BW k ) (3)
where wl t 1 (∆) is the workload of priority higher than or equal to k that arrives between t1 and t1 +∆. This workload is computed by adding up the execution times of all jobs of tasks with a priority higher than or equal to k which are activated between t1 and t1 + ∆. In order to bound RT i,k (j), we first focus on upper bounding wl t 1 and then reason about fixed points to bound BW k . Let φi denote the duration between t1 and the latest release of Tr i that precedes it:

φi = t1 mod Ti (4)
Then, the duration θ i,k between t1 and the first activation of τ i,k in BW k is:

θ i,k = (Ti + O i,k -φi) mod Ti (5)
Note that this definition holds in the two cases shown in Figure 2.

Lemma 1.

wl t 1 ≤ Tr j ∈S wl + j,t 1 (6)
where wl + j,t 1 (∆) :=

τ j,l ∈T r j k≤l ∆ -θ j,l Tj × C j,l (7)
with θ j,l = (Tj + O j,l -φj) mod Tj with φj = t1 mod Tj.

Proof sketch. wl + j,t 1 (∆) upper bounds the actual workload wl j,t 1 (∆) induced by tasks of priority higher or equal to k in Tr j between t1 and t1 + ∆ by assuming all jobs take their worst-case execution time to finish.

∆-θ j,l T j
is the number of activations of τ j,l in [t1, t1 + ∆[, knowing that the first activation is at t1 + θ j,l .

The principle of the approximate analysis is to maximize the workload induced by each transaction in isolation. This may introduce an overapproximation in the obtained bound but it greatly reduces the computational complexity of the analysis compared to the exact solution.

Lemma 2. For Tr j ∈ S,

wl + j,t 1 ≤ wl * j ( 8
)
where wl * j := max

O∈O j {wl + j,O } Oj := {O j,l | τ j,l ∈ Tr j ∧ k ≤ l} wl + j,O (∆) := τ j,l ∈T r j k≤l ∆ -θ O j,l Tj × C j,l with θ O j,l = (Tj + O j,l -O) mod Tj. Proof sketch. Let O t 1 = (O t 1 1 , . . . , O t 1 m
) be such that O t 1 j is the offset of the first task of priority higher than or equal to k in Tr j that is activated after t1. It is fairly easy to prove that:

wl + j,t 1 (∆) ≤ wl + j,O t 1 j (∆)
We now differentiate Tr i from the others and obtain: and thus:

wl t 1 (∆) ≤ wl + i,O t 1 i (∆) + Tr j ∈S j =i wl * j (∆) (9)
RT i,k (j) ≤ BW * O t 1 i + t1 -act i,k (j) (10)
We then prove that act i,k (j)-t1 ≥ (Ti +O i,k -O t 1 i ) mod Ti, so:

RT i,k (j) ≤ BW * O t 1 i -(Ti + O i,k -O t 1 i ) mod Ti (11)
The main theorem follows:

Theorem 1.

wcrt i,k ≤ max O∈O i {BW * O -(Ti + O i,k -O) mod Ti} (12)

FORMAL PROOFS IN COQ

We have formalized in Coq the system model described in Section 2 and proved the correctness of the analysis presented in Section 3. Coq is acknowledged as a very reliable (and rich) interactive environment to develop and verify proofs. To be convinced, an external reader only has to understand the problem specification and the main correctness theorem. Coq also allows the extraction of certified tools. Our development made use of the Prosa library where FPP, busy windows, workload and associated properties are mostly already defined (but Equation (3), for example, was not available). However, periodic tasks with offsets and related analyses are not considered. Our work will be used to enrich Prosa.

The interested reader can access the proofs sources at https://team.inria.fr/spades/coq-proofs-offsets. The Coq code follows the content of the previous sections.

Compared to Tindell's informal proof, we had to prove in Lemma 1 that the worst-case response time is achieved when all execution times are equal to their worst-case execution time -this property is not even mentioned in [START_REF] Tindell | Using offset information to analyse static priority pre-emptively scheduled task sets[END_REF]. Also, Tindell does not prove that the worst case happens within a busy window starting with an activation of each transaction. This led us to prove several nontrivial auxiliary properties about fixed point iterations.

The proof, as it is presented in Section 3 and formalized in Coq, looks quite different from the original paper. We used φ, BW and θ instead of W , w and Ô respectively (these notations are used in other offset-related papers). The main result, however, is of course very similar.

CONCLUSION AND FUTURE WORK

In this short paper, we presented the formalization of a response time analysis for tasks with offsets and its correctness proof using Coq. We defined and proved generic lemmas that can be reused in other proofs. Some hypotheses can be removed to generalize our results. For example, the implicit deadline assumption is not used, while the restricted offsets assumption does not seem to be critical in our proofs. We will investigate some extensions and expect their correctness proof to be very lightweight, as we can build on top of the current proof.

Clearly, this work is in progress. The next steps are: 1. Use the Coq extraction feature to generate a certified schedulability analysis tool (in OCaml). This step should not involve much work.

2. Design a Coq certifier for a standard tool (e.g., py-CPA). The certifier should ensure the correctness of results of an uncertified analyzer. This task is more involved and could require further research (e.g., the analyzer might need to be instrumented to produce additional information for the certifier).

3. Compare the runtimes and effort required by the certified analyzer and the certifier. Checking a fixed point is faster than computing it: the respective efficiency of both tools should depend on how much time is spent in fixed point computations.

Figure 1 :

 1 Figure 1: A system with 2 transactions and 4 tasks (k < 8).

Figure 2 :

 2 Figure 2: Relation between O i,k and φi (with = act i,k ).

Certified in this context means formally verified in Coq.

Note that if the utilization is below 100%, it is always possible to compute that busy window.

* This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01).