C Gentner

P Berto

J.-F Berret

S Reichman

R Kuszelewicz

G Tessier

Magnetic nanorods for quantitative viscosity imaging using heterodyne holography

Many processes in microfluidics and biology are driven or affected by viscosity. While several methods are able to measure this parameter globally (AFM, surface acoustic waves, DLS, ...), very few can provide high resolution viscosity images, particularly in living cells. Here, we propose to use sub-micrometer magnetic rods to perform high resolution viscosity imaging. An external magnetic field forces the oscillation of superparamagnetic iron oxide nanorods. Under linearly polarized illumination, the rotation of these highly anisotropic optical scatterers induces a blinking which is analyzed by heterodyne holography. The spectral analysis of the rotation dynamics yields a regime transition frequency 𝑓 𝑡 from which the local viscosity is deduced. Holography provides 3D image reconstruction and 3D superlocalization of the nanorods, which allows super-resolved viscosity measurements. Relying on the fast Brownian rotation of nanorods instead of their slower translation therefore allows faster measurements and, crucially, smaller effective voxels for viscosity determination. Viscosity imaging is demonstrated with a 0.5 μm 3 3D resolution.

Intro

Viscosity-driven phenomena are ubiquitous in current biological research concerns [START_REF] Ross | The Dark Matter of Biology[END_REF] . For instance, viscosity has been established as a key factor in cancer cells identification [START_REF] Nematbakhsh | Correlating the viscoelasticity of breast cancer cells with their malignancy[END_REF] , but also in embryonic development through the off-center positioning of the oocyte nucleus during meiosis [START_REF] Nematbakhsh | Correlating the viscoelasticity of breast cancer cells with their malignancy[END_REF][START_REF] Almonacid | Control of nucleus positioning in mouse oocytes[END_REF] . While macroscopic and global viscosity measurement techniques are well established [START_REF] Colin | Active diffusion in oocytes nonspecifically centers large objects during prophase I and meiosis I[END_REF][START_REF] Engmann | Squeeze flow theory and applications to rheometry: A review[END_REF][START_REF] Radmacher | Measuring the viscoelastic properties of human platelets with the atomic force microscope[END_REF]7,[START_REF] Hochmuth | Erythrocyte membrane elasticity and viscosity[END_REF][START_REF] Hochmuth | Micropipette aspiration of living cells[END_REF] , smaller scale mapping methods are needed to unveil cytoplasm and cytoskeleton structures and behaviors.

Viscosity essentially quantifies the response of a medium to mechanical actuation [START_REF] Liu | Quantitatively Mapping Cellular Viscosity with Detailed Organelle Information via a Designed PET Fluorescent Probe[END_REF] . Its measurement therefore requires detectable transducers to i) induce and ii) quantify movements. Although invasive, nanoor microscopic probes are the only current way to produce microscopic viscosity mapping in a closed environment such as a cell [START_REF] Ortega | Interfacial microrheology: Particle tracking and related techniques[END_REF] . External magnetic actuation is probably the most preeminent method to induce probe movements. In several microrheology experiments [START_REF] Jérôme | Orientational behavior of an assembly of superparmagnetic rods[END_REF][START_REF] Frka-Petesic | Dynamics of paramagnetic nanostructured rods under rotating field[END_REF][START_REF] Chevry | Magnetic wire-based sensors for the microrheology of complex fluids[END_REF][START_REF] Chevry | Intracellular micro-rheology probed by micron-sized wires[END_REF][START_REF] Colin | Rotational microrheology of Maxwell fluids using micronsized wires[END_REF][START_REF] Berret | Microrheology of viscoelastic solutions studied by magnetic rotational spectroscopy[END_REF][START_REF] Berret | Local viscoelasticity of living cells measured by rotational magnetic spectroscopy[END_REF] , maghemite rods (a polymorphic iron oxide [START_REF] Berret | Local viscoelasticity of living cells measured by rotational magnetic spectroscopy[END_REF][START_REF] Fresnais | Electrostatic Co-Assembly of Iron Oxide Nanoparticles and Polymers: Towards the Generation of Highly Persistent Superparamagnetic Nanorods[END_REF] ) are rotated using various revolving magnetic field frequencies. The cut-off frequency at which rods cease to follow the actuation provides an accurate measurement of the viscosity. However, since rotation is detected using direct optical imaging, the rods must be resolved, and their sizes typically lie in the 10 µm range. While intracellular measurements have been reported, they can hardly be considered local since the probe size is comparable to that of most cells [START_REF] Berret | Local viscoelasticity of living cells measured by rotational magnetic spectroscopy[END_REF] . Optical tweezers are another tool to induce movement. They can drive particles with high precision, and allow measurement duration as short as tens of µs [START_REF] Madsen | Ultrafast viscosity measurement with ballistic optical tweezers[END_REF][START_REF] Pesce | Viscosity measurements on micron-size scale using optical tweezers[END_REF]22,[START_REF] Robertson-Anderson | Optical Tweezers Microrheology: From the Basics to Advanced Techniques and Applications[END_REF] . Imaging can then be obtained using multiple optical tweezers and/or particle scanning. However, to obtain full images of viscosity, multiple, freely moving probes using Brownian motion as an inherent actuation are potentially faster and simpler. Fluorescence-based probes are therefore widely used for microrheology [START_REF] Holyst | Scaling form of viscosity at all length-scales in poly(ethylene glycol) solutions studied by fluorescence correlation spectroscopy and capillary electrophoresis[END_REF] [25] [START_REF] Chambers | An Optical Technique for Mapping Microviscosity Dynamics in Cellular Organelles[END_REF] . Rather than translation, they rely on fluorescence lifetime imaging (FLIM) and on the rotation of the probe molecules, the latter occurring on shorter time scales than translation processes [START_REF] Ramanujan | Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: Large deformations and the effect of fluid viscosities[END_REF][START_REF] Chevry | Magnetic wire-based sensors for the microrheology of complex fluids[END_REF] . Since probes are permanently in translational motion, viscosity is averaged over the volume which the probe explores during the measurement. With a freely-diffusing probe, the best possible spatial resolution for a viscosity measurement is therefore given by √𝐷 𝑙𝑎𝑡 𝜏, where Dlat is the translational diffusion coefficient and  is the measurement duration. Fluorescence-based methods relying on fast rotational movements offer an interesting way to reduce  and preserve the locality of the measurements. However, in addition to possible photobleaching or phototoxicity issues, fluorescence-based viscosity measurements can hardly be used with single molecule resolution: low photon yields require longer acquisition times , during which Brownian translation counteracts the accuracy of single molecule superlocalization.

Nanoparticle probes therefore appear as promising alternatives for viscosity measurements. In metallic particles, plasmonic resonances can be exploited to obtain relatively strong scattering cross sections. An illumination at the appropriate wavelength (plasmonic resonance) and power (to mitigate thermal effects) allows the detection of particles between a few tens and a few hundreds of nanometers within tens of milliseconds. Although such nanoparticles are 1 or 2 orders of magnitude larger than fluorescent molecules, they can actually lead to better spatial resolutions, owing to lower diffusion coefficients Dlat and shorter detection times. As in the case of fluorescent molecules, relying on Brownian rotation rather than translation is faster and more effective. In the case of nanoparticles, a good option is to use asymmetric objects like nanorods. Optical scattering by a nanorod is characterized by an anisotropic radiation pattern [START_REF] Noguez | Surface plasmons on metal nanoparticles: The influence of shape and physical environment[END_REF] . As the rod rotates, this results in an orientation-dependent modulation of the scattered light [START_REF] Chhetri | Imaging threedimensional rotational diffusion of plasmon resonant gold nanorods using polarization-sensitive optical coherence tomography[END_REF] , which has an optimal contrast when the rod is illuminated near the plasmonic resonance of one of its axes. Nanorods illuminated by a linearly polarized laser beam as they undergo Brownian rotation in a fluid therefore induce a blinking. A spectral analysis of this blinking can lead to local viscosity values, allowing a quantitative viscosity mapping [START_REF] Gentner | Viscosity imaging using heterodyne holographic spectral analysis of Brownian nanorod rotation[END_REF] . However, the random nature of the rotation, and of the associated blinking, calls for a careful averaging and an analysis of the frequency spectrum of the optical fluctuations, which is time-consuming and hardly compatible with fast imaging.

In this paper, we report on a fluorescence-free method which takes advantage of the simplicity and efficiency of an external magnetic actuation to rotate superparamagnetic rods, but scales down the probe size by more than one order of magnitude. Since the size of these rods is of the same order as the diffraction limit, determining their orientation through direct imaging is difficult, and we chose to detect the variations in their effective scattering cross section as they rotate under a uniform polarized illumination. Using digital holography and a heterodyne lock-in scheme, the blinking signal of multiple or individual rotating probes is efficiently extracted from the background, from which we deduce quantitative, local and 3-dimensional values of the surrounding fluid viscosity. Viscosity measurements with a single 3D-tracked superparamagnetic rod is demonstrated, paving the way to super-resolved viscosity imaging.

Principle

When aiming at high spatial resolution, the size of the probes is a key issue. Since suitably small magnetic nanorods are not commercially available, sub-micrometer maghemite rods (length 𝐿 = 1020 ± 240 nm, diameter 𝐷 = 440 ± 80 nm, mean values and standard deviations measured by imaging 50 rods magnetically aligned in the image plane) were synthesized. As shown in Fig. 1a, the orientation of these rods is driven by an external magnetic field modulated at a frequency 𝑓 𝐵 , superimposed to a perpendicular static field, thus creating an oscillating effective magnetic field 𝐵 ⃗ (𝑓 𝐵 ).

The optical detection of the rods is based on the blinking resulting from rod rotation as described previously for nonmagnetic objects [START_REF] Gentner | Viscosity imaging using heterodyne holographic spectral analysis of Brownian nanorod rotation[END_REF] . The scattering diagram of a nanorod is essentially a torus aligned with the long rod axis, as schematically depicted in Fig. 1a. When illuminated by a 785 𝑛𝑚 laser beam with a polarization 𝐸 𝑂 ⃗⃗⃗⃗ , the intensity and angular distribution of the electric field 𝐸 𝑆 ⃗⃗⃗⃗ scattered by a rod depend on its orientation with respect to 𝐸 𝑂 ⃗⃗⃗⃗ . The portion of this light collected by a microscope objective is therefore modulated as the rod rotates. A Digital Holographic Microscope (DHM) operating in homodyne or heterodyne lock-in mode [START_REF] Gross | Digital holography with ultimate sensitivity[END_REF] selectively measures the amplitude of the collected light modulated at a chosen frequency 𝑓 (see section 2). This interferometric imaging method provides two main advantages. First, DHM exploits the product 𝐸 𝑆 ⃗⃗⃗⃗ . 𝐸 𝑅 ⃗⃗⃗⃗ of the electric field 𝐸 𝑆 ⃗⃗⃗⃗ scattered by the object (i.e. the light scattered by a nanorod) by a reference field 𝐸 𝑅 ⃗⃗⃗⃗ . Although 𝐸 𝑆 is weak in the case of nanorods with small scattering cross section, 𝐸 𝑅 can be easily increased to obtain hologram intensities approaching the saturation level of the camera, thus providing an optimal, shot noise-limited detection of small nanoparticles [START_REF] Robertson-Anderson | Optical Tweezers Microrheology: From the Basics to Advanced Techniques and Applications[END_REF][START_REF] Holyst | Scaling form of viscosity at all length-scales in poly(ethylene glycol) solutions studied by fluorescence correlation spectroscopy and capillary electrophoresis[END_REF][START_REF] Gentner | Viscosity imaging using heterodyne holographic spectral analysis of Brownian nanorod rotation[END_REF][START_REF] Gross | Digital holography with ultimate sensitivity[END_REF] . Second, the two-arm interferometric configuration is particularly suited to single-(homodyne) or dual-(heterodyne) frequency phase modulations. As such, it allows the study of modulated phenomena over a very broad frequency range (from mHz to tens of MHz) [32, 33, 34] [35] , and thus gives access to high rotation/blinking frequencies.

As shown in Fig. 1b, when monitoring a single nanorod, the presence of a magnetic field 𝐵 ⃗ modulated at 𝑓 𝐵 = 10 Hz induces an increase in the DHM signal at the frequency 𝑓 = 𝑓 𝐵 which clearly indicates that the rod rotates and blinks at the frequency 𝑓 𝐵 . However, as shown in the inset, the object does not disappear totally in the absence of modulation: Brownian motion randomly contributes to a component at 𝑓.

A simultaneous scanning of the magnetic field and analysis frequencies 𝑓 𝐵 and 𝑓 yields a spectrum which displays two distinct frequency regimes separated by a transition at a critical frequency 𝑓 𝑡 , as shown in Fig. 1c (see Supplementary):

-𝑓 ≪ 𝑓 𝑡 : the magnetic field oscillation period is longer than the characteristic time taken by the magnetic moment to align. The rod is able to follow the rotation of the magnetic field, and the signal amplitude is then maximum.

-𝑓 ≫ 𝑓 𝑡 : the magnetic field oscillation period is too short to allow the rod to align with it. The average orientation of the rod long axis is thus stuck close to the direction of the permanent magnetic field 𝐵 0 ⃗⃗⃗⃗ , and the amplitude of the modulated component of the DHM signal decreases to zero as 𝑓 𝐵 increases.

This transition frequency 𝑓 𝑡 is clearly related to viscosity 𝜂. Its measurement therefore gives access to 𝜂 through Eq. ( 1), which is obtained by considering the evolution of a superparamagnetic rod, i.e. a macromagnetic moment 𝑚 ⃗⃗ = 𝜒 ̿𝑉 𝜇 0 𝐵 ⃗ [START_REF] Berret | Local viscoelasticity of living cells measured by rotational magnetic spectroscopy[END_REF] , in a field 𝐵 ⃗ = 𝐵 ~𝑐𝑜𝑠(2𝜋𝐹 𝐵 𝑡)𝑢 𝑧 ⃗⃗⃗⃗ + 𝐵 0 𝑢 𝑥 ⃗⃗⃗⃗ . (See Supplementary for derivation).

𝑓 𝑡 = 1 2𝜋𝜇 0 ∆𝜒𝐵 2 𝜂 ℎ ( 𝐿 𝐷 ) ( 1
)
where 𝜂 is the medium dynamic viscosity, ℎ is a function of the rod length 𝐿 and diameter 𝐷 defined as

ℎ ( 𝐿 𝐷 ) = 𝐿 2 𝑁 ∥ +𝐷 2 𝑁 ⊥ 2(𝐿 2 +𝐷 2 )
with spheroidal depolarization factors expressed in terms of the eccentricity ) is the magnetic susceptibility anisotropy factor for a rod composed of an assembly of spherical nanoparticles of massive susceptibility 𝜒 [36] . The factor 𝐵 2 in Eq. ( 1) corresponds to the squared modulus of the average magnetic field,

𝑒 = √ 1 -( 𝐷 𝐿 ) 2 as 𝑁 ∥ = 1-𝑒 2 2𝑒 3 (𝑙𝑛 ( 1+𝑒 1-𝑒 ) -2𝑒), 𝑁 ⊥ = 1-𝑁 ∥ 2 . ∆𝜒 = 𝜒 2 (𝑁 ⊥ -𝑁 ∥ ) (1+𝑁 ∥ 𝜒)(1+𝑁 ⊥ 𝜒
𝐵 2 = 𝐵 0 2 + 𝐵 2 2 .
As shown in Supplementary, the measured spectrum can be modeled by a sigmoidal function, convoluted with a 𝑠𝑖𝑛𝑐 transfer function [START_REF] Gentner | Viscosity imaging using heterodyne holographic spectral analysis of Brownian nanorod rotation[END_REF] . The inflexion point of this sigmoid indicates the transition frequency 𝑓 𝑡 . For a rod of dimensions 𝐿 and 𝐷 in an oscillating magnetic field of amplitude 𝐵, a fit of this spectrum using Eq.

(1) with 𝑓 𝑡 as parameter can therefore provide a local value of the viscosity 𝜂 across the volume explored by the rod during the acquisition.

The expression of 𝑓 𝑡 in Eq. ( 1) is similar to that of the more widely studied case of a rotating field [START_REF] Frka-Petesic | Dynamics of paramagnetic nanostructured rods under rotating field[END_REF][START_REF] Colin | Rotational microrheology of Maxwell fluids using micronsized wires[END_REF][START_REF] Berret | Local viscoelasticity of living cells measured by rotational magnetic spectroscopy[END_REF] , although the transition is broader: as illustrated in Supplementary, when the oscillating field frequency increases, the angular exploration range of the rod decreases smoothly from its maximum (lower than 𝜋) to 0 within the same half-plane; whereas a rotating field produces a sharp cutoff between a full 2𝜋 rotation regime and a regime with oscillation around a fixed (arbitrary) axis. Rotating fields are typically produced using four identical coils [START_REF] Jérôme | Orientational behavior of an assembly of superparmagnetic rods[END_REF][START_REF] Frka-Petesic | Dynamics of paramagnetic nanostructured rods under rotating field[END_REF][START_REF] Chevry | Magnetic wire-based sensors for the microrheology of complex fluids[END_REF][START_REF] Berret | Local viscoelasticity of living cells measured by rotational magnetic spectroscopy[END_REF] . Here, the dark field illumination through an oil-coupled prism strongly constraints the available space around the sample, and we chose a more compact oscillating field configuration.

Depending on the investigated medium and on the geometry of the rods, the frequency range of the transition frequency 𝑓 𝑡 imposes the use of either homodyne (low frequency) or heterodyne (higher frequencies) holography, as detailed in the following sections. (c) Numerical model of the DHM signal, a sigmoid with a 𝑠𝑖𝑛𝑐 acquisition transfer function [START_REF] Gentner | Viscosity imaging using heterodyne holographic spectral analysis of Brownian nanorod rotation[END_REF] .

Low frequencies -high viscosities: homodyne holography

The first experiment, shown in Fig. 2, was carried out in two microfluidic chambers separated by a horizontal 300 µm PDMS wall. The first chamber (Fig. 2b-top) contains a 27% glycerol aqueous solution (𝜂 = 2.6 mPa s).

The second chamber contains water (𝜂 = 1.0 mPa s) for comparison. The 1020 × 440 nm maghemite nanorods were synthesized using a low weight concentration (𝑐 = 0.02 wt%) of initial nanoparticle-polymer solution using a synthesis protocol described in [START_REF] Chanteau | Electrosteric Enhanced Stability of Functional Sub-10 nm Cerium and Iron Oxide Particles in Cell Culture Medium[END_REF][START_REF] Sehgal | Precipitation-redispersion of cerium oxide nanoparticles with poly (acrylic acid): Toward stable dispersions[END_REF] . Individual nanorod detection requires a density well below the limit of 1 object per optical Point Spread Function (PSF), i.e. a few 10 6 rods.µL -1 with the X4, NA=0.13 objective we used. Here, the rods were diluted to 10 3 µL -1 in both chambers, 3 orders of magnitude below this limit. The rods were submitted to an oscillating magnetic field 𝐵 ⃗ = 𝐵 0 ⃗⃗⃗⃗ +𝐵 ⃗⃗⃗⃗ , with a modulation amplitude 𝐵 ~= 1.3 ± 0.3 mT, and a static component 𝐵 0 in the mT range.

The holographic detection setup is depicted in Fig. 2a. To allow an intense illumination of the low-scatteringcross-section nanoparticles without blinding the camera, the beam illuminating the microfluidic chamber is sent through a glass prism coupled to the microfluidic chamber, and totally reflected at the glass-air interface.

Only the light scattered by the nanorods is collected by a X4, NA=0.13 objective and sent to the camera. There, it interferes with a reference plane-wave coming from the same 785 nm, single longitudinal mode laser in an off-axis configuration, at an angle ~1°. In order to extract the component of the signal modulated at 𝑓 𝐵 , the camera is triggered at 𝑓 𝑎𝑐𝑞 = 4𝑓 𝐵 , in a 4-bucket integration scheme [START_REF] Absil | Photothermal heterodyne holography of gold nanoparticles[END_REF][START_REF] Atlan | Accurate phase-shifting digital interferometry[END_REF] which filters out fluctuations occurring at other frequencies and selectively produces holograms of the 𝑓 𝐵 -modulated component. After a first Fourier transform, the +1 interferometric order is numerically extracted, propagated, and finally back Fourier-transformed to obtain the image of a chosen plane inside the microfluidic chamber [START_REF] Verrier | Off-axis digital hologram reconstruction: some practical considerations[END_REF] .

For each frequency f in the range [0.5,42] Hz, series of 80 holograms were acquired, yielding 20 holograms after 4-bucket demodulation. Fig. 2b shows images reconstructed from the average of these 20 holograms obtained at 7 and 20 Hz. Similar images and signals averaged in each compartment were acquired in 0.5 Hz steps. The values of 𝑓 𝑡 were extracted from fits using the function derived in detail in the Supplementary section. Note that a simplified sigmoidal function was used here, omitting the convolution by the 𝑠𝑖𝑛𝑐 function for sake of computational speed.

In the high viscosity compartment (𝜂 = 2.6 mPa s, green curve in Fig. 2b), a clear transition frequency 𝑓 𝑡 = 12 ± 1 Hz is observed. With 𝐿 𝐷 = 2.8 ± 0.5 and 𝜒 = 6.5 [START_REF] Berret | Local viscoelasticity of living cells measured by rotational magnetic spectroscopy[END_REF] , Eq. ( 1) predicts 𝑓 𝑡 = 12 ± 7 Hz for 𝐵 0 = 0.9 ± 0.3 mT, in good agreement with the measured value. In the lower viscosity medium, 𝜂 = 1.0 mPa s, we measure a transition at 𝑓 𝑡 = 30 ± 5 Hz, which compares well with the expected value 𝑓 𝑡 = 32 ± 22 Hz. Images acquired between these two values of 𝑓 𝑡 (Fig. 2b, center, 𝑓 = 20 Hz) clearly display a viscosity contrast: while rods do not follow the rotation of the magnetic field in the high viscosity medium (top), they are rotating and appear bright in the low-region (bottom). Similar spectra were acquired to measure 𝑓 𝑡 in various water/glycerol mixtures, with glycerol concentrations in the [0%, 55%] range, corresponding to viscosities in the [1,10] mPa s range (deduced using [START_REF] Cheng | Formula for the viscosity of a glycerol-water mixture[END_REF] ). The measured values 𝑓 𝑡 (𝜂 -1 ) plotted in Fig. 2c , this can be difficult to achieve with most cameras (𝑓 𝑎𝑐𝑞 of a few tens of Hz), particularly at low viscosity values. A possible workaround is to decrease the intensity of the modulated component of the magnetic field 𝐵 ~. However, this approach breaks down when the magnetic driving force becomes of the same order as the random Brownian forces exerted on the particle. Before this, lowering 𝐵 ~ clearly decreases the detection signal-to-noise ratio. If we choose an arbitrary limit of 10 3 for the ratio between the magnetic energy 𝑚𝐵 and the energy of Brownian fluctuations 𝑘 𝐵 𝑇 which allows a satisfactory detection, i.e. where 𝑉 is the rod volume. For our rod size and composition, the minimum magnetic field amplitude needed to overcome Brownian fluctuations is then 𝐵 = 3.5 mT, which, from Eq. ( 1), corresponds to a transition frequency 𝑓 𝑡 = 200 Hz in water. In 4-bucket integration mode, its measurement therefore requires camera frame rates well above 4𝑓 𝑡 = 800 Hz. Cameras operating in the kHz range or above are available, but sub-ms integration times are hardly compatible with nano-object detection. For this reason, we have chosen to use heterodyne holography, which allows the measurement of high frequency phenomena using slow detectors [30, 31-35, 39, 40, 42] .

High frequencies: heterodyne holography

Heterodyne detection is implemented by adding an acousto-optic modulator (AOM) in each holographic path, as illustrated in Fig. 3a, to introduce a relative phase modulation at a chosen frequency. Since AOMs operate in the 80 MHz range, the object beam is modulated at 80 MHz, while the reference beam is modulated at 80 MHz -∆𝑓 to produce an optical phase beating at ∆𝑓. The 4-bucket integration described above is used to sample and demodulate holograms with a camera triggered at 𝑓 𝑎𝑐𝑞 . By choosing the beating frequency so that ∆𝑓 = 𝑓 𝑎𝑐𝑞 4 + 𝑓 𝑖 , one can selectively isolate holograms of any modulation at 𝑓 𝑖 . Much like a multiplexed lock-in scheme, this process isolates a hologram of component modulated at frequency 𝑓 0 and filters out other fluctuations (including most of the Brownian noise in our case). Even with a slow camera, i.e. 𝑓 𝑎𝑐𝑞 of a few Hz, heterodyning provides high-SNR-holograms of phenomena modulated at frequencies 𝑓 𝑖 up to tens of Hz. To illustrate the ability of this system to image quantitatively the viscosity, a test sample displaying a spatial viscosity gradient was prepared using a concentration 𝐶 𝐴𝑔𝑎𝑟 = 0.5 wt% Agar gel (𝜂 ≈ 5 mPa s [START_REF] Zhang | Rheological behavior of agar solution in relation to the making of instant edible bird's nest products[END_REF] ) and water (𝜂 = 1.0 mPa s). These media were put in contact to obtain a gradient of Agar concentration by diffusion at the interface. In order to visualize this gradient, 1% Alexa 488 fluorophores were added to the water phase. The fluorescence image shown in Fig. 3c-left is therefore a direct image of the water gradient (1 -𝐶 𝐴𝑔𝑎𝑟 ), which is directly related to the viscosity. Both media were homogeneously seeded using the same maghemite nanorods (𝐿 = 1020 ± 240 nm, 𝐷 = 440 ± 80 nm) at a concentration of 10 5 μL -1 . With the microscope objective used here (×4 ON 0.13), this corresponds to about 100 rods per PSF, i.e. rods were not individually resolved. In the experiment shown in Fig. 3, the coils were powered using a 100 V -200 W audio amplifier to produce a modulated field amplitude 𝐵 = 3.9 mT. This stronger field modulation (as compared to 𝐵 = 1.3 mT in the previous experiment) increases SNR, but also leads to higher transition frequencies 𝑓 𝑡 . Heterodyning, however, gives access to a broad frequency range. For each of the 27 measured frequencies 𝑓 𝑖 (i.e. 𝑖 = 1 to 27) ranging from 80 to 1000 Hz, 100 holograms were acquired at 𝑓 𝑎𝑐𝑞 = 10 Hz, with a 𝜏 = 50 ms integration time. After 4-bucket demodulation of the 100 measurements, the 25 resulting holograms were averaged and numerically propagated to the sample plane to produce 27 images of the dynamic blinking for each of the 27 frequencies 𝑓 𝑖 . The 1024 × 1024 pixels image were then resampled into 40 × 40 macropixels, a size corresponding to 65 × 65 µm 2 in the object space, which was chosen as a trade-off between spatial resolution and SNR. The resulting hyperspectral (40 × 40 × 27) cube of (𝑥, 𝑦, 𝑓 𝑖 ) data was then used to obtain spectral fits (i.e. along 𝑓 𝑖 ) using the sigmoidal fitting function described above (see Supplementary section for the derivation of this function). Note that the slope parameter of the sigmoid was set equal to the value of the camera integration time as being the slowest response time of the system. These fits yield transition frequencies 𝑓 𝑡 (𝑥, 𝑦) for each pixel, which are converted into a 40 × 40 image of the local viscosity values 𝜂(𝑥, 𝑦) using Eq. ( 1), as shown in Fig. 3c.

As shown in Fig. 3c, the fluorescence image indicates that a vertical viscosity profile is expected, with quasipure water at the top, and a low Agar concentration (< 5%) at the bottom of the image. This is indeed observed qualitatively in the viscosity image (right), although some evolution of the gradient is likely to occur during the 5 min total acquisition time. The [245,275] Hz range of 𝑓 𝑡 values corresponds to a viscosity in the [1.1,1.3] mPa s range, close to values expected in pure water, i.e. very low Agar concentration. This underlines the sensitivity of the method, which allows to identify weak viscosity variations.

On single particle tracking

The experiments shown in Fig. 3 involve relatively large macropixels (65 µm laterally), each addressing a large number (typ. thousands) of nanorods. With sufficient SNR, this volume can be easily reduced down to the size of the PSF of the microscope. Further reduction is possible provided that less than one particle is present in the volume of the PSF: calculating the center of mass of the holographic image of a rod provides its 3D position with nm-range precision. In liquids, Brownian motion ensures a stochastic exploration of the environment by the rods, and we have shown that this can provide sub-diffraction optical imaging in 3D [START_REF] Martinez-Marrades | Stochastic 3D optical mapping by holographic localization of Brownian scatterers[END_REF] . However, in the context of viscosity measurements, the measurement time 𝜏 𝑚𝑒𝑎𝑠 must be long enough to allow rod rotation by one or more turns. During this time, the Brownian rod is free to travel laterally over a mean distance √𝐷 𝑙𝑎𝑡 𝜏 𝑚𝑒𝑎𝑠 , with 𝐷 𝑙𝑎𝑡 the lateral diffusion coefficient, which determines the size of the probed volume, i.e. the spatial resolution. Fig. 4a shows the trajectories of a nanorod measured by superlocalization on holographic reconstructions recorded at 𝑓 𝑎𝑐𝑞 = 20 Hz with a × 50, 0.6 NA objective, with a magnetic field 𝐵 ~= 3.5 ± 0.3 mT and 𝐵 0 = 3.0 ± 0.4 mT. The volume explored by the rod during 𝜏 𝑚𝑒𝑎𝑠 = 𝑁 𝑓 𝑎𝑐𝑞 ⁄ (with 𝑁 = 40 to 1000 the number of holograms) decreases down to 0.5 μm 3 for 𝑁 = 40. Clearly, the SNR of the heterodyne signal also decreases with 𝑁. However, the transition frequency 𝑓 𝑡 is well identified in measurements obtained at 9 frequencies by averaging 𝑁 = 40 holograms in 3 glycerol solutions of respective viscosities 1.0 ± 0.1, 2.1 ± 0.2 and 3.0 ± 0.4 mPa s. Fits on these data yield viscosities in good agreement with these expected values, showing that 𝑁 = 40 averages provide a reliable measurement of the viscosity. Several frequencies 𝑓 𝑖 are arguably necessary to derive these values, but assuming that only one measurement is taken, which would be possible by heterodyne frequency multiplexing [START_REF] Rubin | Six-pack off-axis holography[END_REF] , the value (0.5 μm 3 ) 1 3 ⁄ = 800 nm therefore indicates the best achievable resolution at this point. This corresponds to the diffraction limit (

1.22𝜆 2𝑁𝐴

= 800 nm) laterally, but clearly surpasses it longitudinally. With higher SNR, by e.g. increasing the optical intensity or 𝐵, optical superresolution by 3D rod superlocalization is therefore achievable.

Conclusion

Although well tolerated and easily internalized by living cells, optically resolved maghemite needles with lengths of the order of 10 μm are not adapted to sub-cellular resolution measurements. Passive, Brownianmotion based methods with sub-100 nm gold nanorods [START_REF] Gentner | Viscosity imaging using heterodyne holographic spectral analysis of Brownian nanorod rotation[END_REF] are a possible alternative since they are also biocompatible, and offer relatively large optical scattering cross sections near their plasmon resonance. However, analyzing random motion to extract viscosity values demands a fine sampling and long acquisition times which reduce the effective spatial resolution of the viscosity image. The sensitivity of the holographic detection proposed here definitely allows a drastic reduction of the probe size compared to the works which inspired it [START_REF] Jérôme | Orientational behavior of an assembly of superparmagnetic rods[END_REF][START_REF] Frka-Petesic | Dynamics of paramagnetic nanostructured rods under rotating field[END_REF][START_REF] Chevry | Magnetic wire-based sensors for the microrheology of complex fluids[END_REF][START_REF] Berret | Local viscoelasticity of living cells measured by rotational magnetic spectroscopy[END_REF] . We have shown here that the magnetically-driven rotation of non-plasmonic rods can be detected selectively at frequencies up to the MHz for 𝐿 ≈ 1 µm. The current photonic budget of the holographic detection indicates that a size reduction to a few hundreds of nm will be achievable as soon as our chemical synthesis protocol [START_REF] Fresnais | Electrostatic Co-Assembly of Iron Oxide Nanoparticles and Polymers: Towards the Generation of Highly Persistent Superparamagnetic Nanorods[END_REF] allows their fabrication. Coating maghemite rods with gold might be an interesting option as it would increase their scattering cross section.

To further improve the locality and the spatial resolution, fast-and high-SNR-measurements are crucial.

Increasing the amplitude of the modulated field 𝐵 ~ is an efficient way to overcome rotational Brownian fluctuations. As discussed here, the transition frequency 𝑓 𝑡 increases with 𝐵 2 and this points to high frequency measurements. Up to the kHz, high speed cameras are a solution, but short exposures call for intense illuminations which can affect the local viscosity by heating up the probes. The heterodyning scheme used here can isolate blinking nanorods well above the MHz using mW-range illumination. Since two frequencies are in principle sufficient to determine 𝑓 𝑡 in media where the expected viscosity range is known, holographic multiplexing is also a possible way to improve measurement speed and hence spatial resolution. The geometry of the holographic assembly, the vertical direction of the rotation that it imposes and the presence of the prism (for dark-field illumination) complicate the use of a rotating field scheme by four identical coils [START_REF] Jérôme | Orientational behavior of an assembly of superparmagnetic rods[END_REF][START_REF] Frka-Petesic | Dynamics of paramagnetic nanostructured rods under rotating field[END_REF][START_REF] Chevry | Magnetic wire-based sensors for the microrheology of complex fluids[END_REF][START_REF] Berret | Local viscoelasticity of living cells measured by rotational magnetic spectroscopy[END_REF] . The oscillating field configuration shown in figures 2 and 3 is a good compromise between simplicity of fabrication and high enough field amplitude values. The price to pay is that the critical phenomenon allowing viscosity measurements loses accuracy: with a rotating field, a sharp cutoff between complete rotation and oscillation around a fixed direction is observed, whereas an oscillating field requires the determination of a smoother inflection point. The transition between rod oscillation and static alignment (see Supplementary) is defined with a lower precision.

Holography has the advantage of allowing 3D reconstructions. Although the PSF extension is larger in the axial direction, three-dimensional particle localization (Fig. 4) can provide 3D viscosity mapping. Conversely, without localization, a pixel contains the average of signals from multiple planes (Fig. 3c). There is no axial sectioning, except for the weighting of out-of-focus signals according to their pixel spread. This can be problematic in the case of thick and highly scattering samples. In order to achieve axial sectioning at excitation, a holography-OCT hybridization consisting in the use of a weakly incoherent source on the holographic setup may be a relevant strategy.

SUPPLEMENTARY INFORMATION 1: Superparamagnetic rod motion

An object made of a ferromagnetic material whose size is comparable to or smaller than a domain can be considered as a so called "macro-spin". When averaging the magnetic moment over an assembly of these objects or by ergodicity, in a single object over a long enough period of time, the resulting magnetic moment vanishes in the absence of an external magnetic field, and is referred to as "superparamagnetic".

For this class of objects, which includes maghemite nanorods, superparamagnetism appears under the condition 𝑇 ≫ 𝑇 𝑁 where 𝑇 𝑁 = 𝐸 𝑏 𝑘 𝐵 𝑙𝑛 𝜏 0 𝜏 , the Néel temperature, is the minimum temperature allowing the thermalization of the elementary moments and a resulting zero magnetic moment when averaged over 𝜏 ≫ 𝜏 0 . 𝐸 𝑏 is the energy barrier required to flip the magnetic moment and 𝜏 0 is a characteristic time of the material. The barrier energy 𝐸 𝑏 is therefore negligible in this condition (𝑘 𝐵 𝑇 ≫ 𝐸 𝑏 ) [START_REF] Dormann | Le phénomène de superparamagnétisme. Revue de Physique Appliquee[END_REF] and the magnetic energy then reduces to the Zeeman term 𝐸 𝑝 𝑚𝑎𝑔 = -𝑚 ⃗⃗ • 𝐵 ⃗ . Its minimization leads to the alignment of the magnetic moment 𝑚 ⃗⃗ along the field direction. In a superparamagnetic particle of volume 𝑉, far from saturation, magnetization results in a macro-spin of moment:

𝑚 ⃗⃗ = 𝜒̿ 𝑉 𝜇 0 𝐵 ⃗ (2)
where the susceptibility tensor 𝜒̿ , for a superparamagnetic rod with respective long and short axes susceptibilities 𝜒 ∥ and 𝜒 ⊥ , reads:

𝜒̿ = ( 𝜒 ⊥ 0 0 0 𝜒 ⊥ 0 0 0 𝜒 ∥ ) ( 3
)
This tensor is expressed along 𝑢 3 ⃗⃗⃗⃗ , the long axis unit vector, and 𝑢 1 ⃗⃗⃗⃗ and 𝑢 2 ⃗⃗⃗⃗ , two arbitrary orthonormal vectors of its perpendicular plane. Out of equilibrium, equations ( 2) and (3) imply that the magnetic moment 𝑚 ⃗⃗ of the superparamagnetic rod is given by a linear superposition of the magnetic field 𝐵 ⃗ components, weighed by the elements of the susceptibility tensor:

𝑚 ⃗⃗ = 𝑉 𝜇 0 (𝜒 ⊥ 𝐵 1 𝑢 1 ⃗⃗⃗⃗ + 𝜒 ⊥ 𝐵 2 𝑢 2 ⃗⃗⃗⃗ + 𝜒 ∥ 𝐵 3 𝑢 3 ⃗⃗⃗⃗ ) (4)
In the regime of low Reynolds number, where hydrodynamics is ruled by Stokes equations, one can assume a linear relationship between the particle kinematic torsor (velocity and rotation vectors) ( 𝑈 ⃗ ⃗ 𝛺 ⃗ ) and the effort torsor (resulting force and moment) ( 𝐹 𝜏 ) that it undergoes:

( 𝑈 ⃗ ⃗ 𝛺 ⃗ ) = 𝑀 ( 𝐹 𝜏 ) (5)
Velocity, rotation, force and torque refer to a common arbitrary point of the rod, chosen in general as the hydrodynamic center. The latter, under common symmetry conditions coincides with the center of mass. 𝑀 is the 6 × 6 mobility tensor split into four 3 × 3 respective sub-tensors of namely translation 𝑀 𝑡 , rotation 𝑀 𝑟 and translation-rotation coupling 𝑀 𝑡𝑟 and transposed 𝑀 𝑟𝑡 [START_REF] Happel | Low Reynolds number hydrodynamics[END_REF] :

𝑀 = ( 𝑀 𝑡 𝑀 𝑡𝑟 (𝑀 𝑡𝑟 ) 𝑇 𝑀 𝑟 ) (6)

2: General rate equations

From equation ( 5), the time-evolution equations of a particle with center position 𝑟 are:

𝑑𝑟 𝑑𝑡 = 𝑈 ⃗ ⃗ = 𝑀 𝑡 𝐹 + 𝑀 𝑡𝑟 𝜏 (7) 𝑑𝑢 𝑖 ⃗⃗⃗ 𝑑𝑡 = 𝛺 ⃗ × 𝑢 𝑖 ⃗⃗⃗ = ((𝑀 𝑡𝑟 ) 𝑇 𝐹 + 𝑀 𝑟 𝜏 ) × 𝑢 𝑖 ⃗⃗⃗ (8)
In a spatially uniform magnetic field where 𝛻 ⃗ • 𝐵 ⃗ = 0 and 𝛻 ⃗ × 𝐵 ⃗ = 0 ⃗ , the magnetic force 𝐹 = 𝛻 ⃗ (𝑚 ⃗⃗ • 𝐵 ⃗ ) is equal to zero. Only pure rotational motion is then induced. From symmetry considerations [START_REF] Kim | Microhydrodynamics: Principles and Selected Applications[END_REF] , one also has 𝑀 𝑡𝑟 = 𝑀 𝑟𝑡 = 0. Moreover, 𝑀 𝑟 is diagonal. Equations ( 7) and ( 8) now read: ) [START_REF] Kim | Microhydrodynamics: Principles and Selected Applications[END_REF] .

𝑑𝑟 𝑑𝑡 = 0 ⃗ (9)
The three vector differential equations (10) lead to 9 scalar equations when projecting each onto a reference basis (𝑢 𝑥 ⃗⃗⃗⃗ , 𝑢 𝑦 ⃗⃗⃗⃗ , 𝑢 𝑧 ⃗⃗⃗⃗ ).

Component Rate Equations orthogonal oscillating and fixed components.

In the experimental scheme developed in the article, the magnetic field is defined as 𝐵 ⃗ = 𝐵 ~𝑐𝑜𝑠(2𝜋𝐹 𝐵 𝑡)𝑢 𝑧 ⃗⃗⃗⃗ + 𝐵 0 𝑢 𝑥 ⃗⃗⃗⃗ where (𝑢 𝑥 ⃗⃗⃗⃗ , 𝑢 𝑦 ⃗⃗⃗⃗ , 𝑢 𝑧 ⃗⃗⃗⃗ ) is the laboratory referential.

Brownian diffusion effects can be introduced in the simulations by adding to (10) a stochastic term [START_REF] Krahne | Physical Properties of Nanorods[END_REF] and replacing deterministic parameters by their expectation values. However, it can be shown that only the direction transverse to the plane defined by the field rotation is affected by the Brownian contribution. Thus, the Brownian motion only reduces the rotation amplitude in the plane of interest, and thereby the signal amplitude, without changing the rotation frequency. Moreover, the mean direction of the major axis inevitably tends to align in the plane of the oscillating field, whatever the magnitude of the Brownian contribution. Brownian motion will therefore be neglected. We define the orthonormal moving referential 𝑟 . These coefficients depend on the specific shape of the rod. In our experimental conditions the mobility matrix coefficients of the rod of length 𝐿 and diameter 𝐷 can be expressed as [START_REF] Krahne | Physical Properties of Nanorods[END_REF] :

(
𝑀 ⊥ 𝑟 = 1 𝜂 𝑔 ( 𝐿 𝐷 ) = 1 𝜂 𝐿 2 𝑁 ∥ + 𝐷 2 𝑁 ⊥ 2(𝐿 2 + 𝐷 2 )
where 𝑁 ⊥ and 𝑁 ∥ are the depolarization factors. In most cases the rods can be assimilated to a prolate spheroid whose depolarization factors can be calculated in terms of the eccentricity 𝑒 = √ 1 -( 𝐷 𝐿 )

2 as [START_REF] Krahne | Physical Properties of Nanorods[END_REF] :

𝑁 ∥ = 1 -𝑒 2 2𝑒 3 (𝑙𝑛 ( 1 + 𝑒 1 -𝑒 ) -2𝑒) 𝑁 ⊥ = 1 -𝑁 ∥ 2 𝛥𝜒 = 𝜒 2 (𝑁 ⊥ -𝑁 ∥ ) (1 + 𝜒𝑁 ∥ )(1 + 𝜒𝑁 ⊥ )
is the magnetic susceptibility anisotropy factor expressed for a rod composed of an anisotropic assembly of isotropic spherical nanoparticles of massive susceptibility 𝜒 [START_REF] Atlan | Accurate phase-shifting digital interferometry[END_REF] . Therefore, one can express the transition frequency as:

𝑓 𝑡 ′ = 1 2𝜋𝜇 0 𝛥𝜒𝐵 2 𝜂 𝑔 ( 𝐿 𝐷 ) (12)
One must note that the frequency term to be considered equivalent to the cutoff expression in the case of a rotating field should be:

𝑓 𝑡 = 1 2𝜋𝜇 0 𝛥𝜒𝐵 2 𝜂 𝑔 ( 𝐿 𝐷 ) [1 + ℎ 2 𝑐𝑜𝑠²(2𝜋𝑓𝑡)]
This expression requires interpretation as it appears as a time-dependent frequency. This translates in the frequency domain as a -dependent overmodulation of the response spectrum envelope. However, if we consider the time average of this expression, we rather define the effective transition frequency as:

𝑓 𝑡 = 1 2𝜋𝜇 0 𝛥𝜒𝐵 2 𝜂 𝑔 ( 𝐿 𝐷 ) (1 + ℎ 2 2 )
(1)

Supplementary figure 5: excursion of a magnetic rod in a (a) oscillating and (b) rotating magnetic field. In both cases, the blue area represents the angular field covered when 𝑓 ≪ 𝑓 𝑡 . When 𝑓 > 𝑓 𝑡 , the rod becomes unable to follow the whole angle covered by the magnetic field, and the rod only covers the angular range corresponding to the red area. With an oscillating field (a), the angle explored by the rod is thus smoothly reduced from the blue area to the red one, which is centered around the static 𝐵 0 ⃗⃗⃗⃗ field direction. In contrast, with a rotating field (b), the rod angular exploration abruptly changes from a full rotation (blue disk) to the red angular sector which is centered around an arbitrary axis. Therefore, the rotating field (b) case defines a much sharper cut-off that illustrates a real phase transition in the rod evolution.

Figure 1

 1 Figure 1 (a) Principle of the measurement: the orientation of a maghemite nanorod is driven by the superposition of a static magnetic field 𝐵 0 ⃗⃗⃗⃗ (horizontal) and an oscillating field 𝐵 ⃗⃗⃗⃗ (vertical) modulated at a frequency 𝑓 𝐵 . Part of the torus-shaped optical scattering diagram is collected by the microscope objective and sent to a heterodyne Digital Holography Microscope (DHM). (b) Reconstructed DHM images (insets) and average signals acquired at an analysis frequency 𝑓 = 𝑓 𝐵 = 10 Hz. When 𝐵 ⃗ = 𝐵 0 ⃗⃗⃗⃗ (𝐵 ⃗⃗⃗⃗ = 0 ⃗ for 𝑡 ∈ [0, 𝑡 1 [∪ [𝑡 2 , +∞[), the rod is mostly static, aligned with 𝐵 0 ⃗⃗⃗⃗ , and the 𝑓 𝐵 component of the DHM signal is low. When 𝐵 ⃗ = 𝐵 0 ⃗⃗⃗⃗ +𝐵 ⃗⃗⃗⃗ (𝑡 ∈ [𝑡 1 , 𝑡 2 [), the strong DHM signal at 𝑓 𝐵 indicates that the rod oscillates and blinks at 𝑓 𝐵 . (c) Numerical model of the DHM signal, a sigmoid with a 𝑠𝑖𝑛𝑐 acquisition transfer function[START_REF] Gentner | Viscosity imaging using heterodyne holographic spectral analysis of Brownian nanorod rotation[END_REF] .

Figure 2

 2 Figure 2 (a) Holographic setup. A magnetic field oscillating at 𝑓 𝐵 modulates the orientation and optical scattering of rods placed in a microfluidic chamber illuminated in total internal reflection. The off-axis holograms are sampled at 𝑓 𝑎𝑐𝑞 = 4𝑓 𝐵 , demodulated and reconstructed. (b) Images of two chambers containing water-glycerol mixtures of viscosity 2.6 mPa s (top) and 1.0 mPa s (bottom) separated by a horizontal wall, at frequencies 𝑓 𝐵 = 7 Hz and 𝑓 𝐵 = 20 Hz. Spectra of signal averaged in the high viscosity (top, green) and low viscosity regions (bottom, blue) exhibit different transition frequencies 𝑓 𝑡 , at 12 Hz for 𝜂 = 2.6 mPa s and 35 Hz for 𝜂 = 1.0 mPa s. (c) Transition frequency 𝑓 𝑡 measured in solutions of 0 to 55% water-glycerol mixtures. Error bars: dispersion obtained on 10 different rods.

∆𝜒𝐵∆𝜒𝐵

  display a linear behavior, with a best fit slope of 0.026 ± 0.003 Pa. This is in good agreement with the value 𝑓 𝑡 𝜂 = 03 ± 0.01 Pa predicted by Eq. (1) with the parameters used in our experiment, ∆𝜒 = 2.0 ± 0.1, 𝐿 𝐷 = 2.8 ± 0.5, 𝐵 0 ≈ 1 mT, 𝐵 ~= 1.3 ± 0.2 mT. However, the accurate determination of the transition frequency requires measurements at frequencies 𝑓 𝑎𝑐𝑞 4 well above the value of 𝑓 𝑡 . With 𝑓 𝑡 =

  𝑚𝐵𝑘 𝐵 𝑇 > 10 3 , we can evaluate the minimal magnetic modulation 𝐵 which is required. Superparamagnetic rods have a macroscopic magnetic moment amplitude 𝑚 = 𝜒𝑉 1+𝑁 ∥ 𝜒 𝐵 𝜇 0

Figure 3 ( 4 +

 34 Figure 3 (a) Heterodyne holography setup: acousto-optic modulators (AOM) are added in both the object and reference arms of the holographic setup to induce a beating at ∆𝑓 = 𝑓 𝑎𝑐𝑞 4 + 𝑓 𝑖 and selectively retrieve 𝑓 𝑖 components after 4-phase demodulation. (b) Experimental spectrum and sigmoidal fit obtained for one of the pixels of (c). Similar fits conducted on each (x,y) pixels yield (c) images of the local transition frequency ft or of the quantitative viscosity (right). The observed gradient agrees with the qualitative fluorescence viscosity image (left).

Figure 4 (

 4 Figure 4 (a) Volumes explored by a maghemite nanorod during 𝑁 = 40 to 1000 heterodyne holography measurements at a rate 𝑓 𝑎𝑐𝑞 = 20 Hz. The trajectories were determined by superlocalization on 3D reconstructions. (b) Spectra of single magnetic rod rotation in water-glycerol mixtures with viscosities 1.0 ± 0.1, 2.1 ± 0.2 and 3.0 ± 0.4 mPa s with 𝐵 ~= 3.5 mT. The fits are performed by convolution between the 𝑠𝑖𝑛𝑐 instrumental transfer function and a sigmoid function (see Supplementary), with two free parameters: the transition frequency 𝑓 𝑡 and an offset 𝑎.

  𝑢 ~𝑥 ⃗⃗⃗⃗ , 𝑢 ~𝑦 ⃗⃗⃗⃗ , 𝑢 ~𝑧 ⃗⃗⃗⃗ ) aligned with the magnetic field such that 𝑢 ~𝑥 ⃗⃗⃗⃗ is the unitary vector of 𝐵 ⃗ , 𝑢 ~𝑦 ⃗⃗⃗⃗ is upward vertical and , where 𝑖 ∈ {1,2,3} and 𝛼 ∈ {𝑥, 𝑦, 𝑧}, inserting in (9) and projecting leads to the following set of 3 differential equations for 𝑣 3𝛼 , (omitting those for 𝑣 1𝛼 and 𝑣 2𝛼 ) :

	𝑢 ~𝑦 ⃗⃗⃗⃗ = 𝑢 ~𝑧 ⃗⃗⃗⃗ × 𝑢 ~𝑥 ⃗⃗⃗⃗ . Defining 𝑣 𝑖∝ = 𝑢 ~𝑖 ⃗⃗⃗ . 𝑢 ~∝ ⃗⃗⃗⃗ , ℎ =	𝐵 B 0
			𝑑𝑣 3𝑥 𝑑𝑡	= 2𝜋𝑓 𝑡 ′ [1 + ℎ 2 𝑐𝑜𝑠 2 (2𝜋𝑓𝑡)]𝑣 3𝑥 (1 -𝑣 3𝑥 2 ) + 𝑣 3𝑦	2𝜋ℎ𝑓𝑠𝑖𝑛(2𝜋𝑓𝑡) [1 + ℎ 2 𝑐𝑜𝑠 2 (𝜔𝑡)]
			𝑑𝑣 3𝑦 𝑑𝑡	= -2𝜋𝑓 𝑡 ′ [1 + ℎ 2 𝑐𝑜𝑠 2 (2𝜋𝑓𝑡)]𝑣 3𝑥 2 𝑣 3𝑦 -𝑣 3𝑥	2𝜋ℎ𝑓𝑠𝑖𝑛(2𝜋𝑓𝑡) [1 + ℎ 2 𝑐𝑜𝑠 2 (𝜔𝑡)]
						(11)
						𝑑𝑣 3𝑧 𝑑𝑡	= -2𝜋𝑓 𝑡 ′ [1 + ℎ 2 𝑐𝑜𝑠 2 (2𝜋𝑓𝑡)]𝑣 3𝑥 2 𝑣 3𝑧
	where 𝑓 𝑡 ′ =	𝜇 0 2𝜋	𝛥𝜒𝐻 𝑥 2 𝑀 ⊥ 𝑟 =	1 12𝜋𝜇 0	𝛥𝜒𝐵 𝑥 2 𝑀 ⊥ 𝑟 and 𝑀 ⊥ 𝑟 = 𝑀 11 𝑟 = 𝑀 22


Sanaa Hmaida
email: sanaa.hmaida.sh@gmail.com

Hocine Cherifi
email: hocine.cherifi@u-bourgogne.fr

Mohammed El Hassouni
email: mohamed.elhassouni@flsh.um5.ac.ma

Backbone Extraction of Weighted Modular Complex Networks based on their Component Structure

Keywords: Community structure, Component Structure, Backbone extraction, Multi-Core Structure

This work introduces a generic backbone extraction framework exploiting the mesoscopic network structure. Indeed, numerous real-world networks are made of dense groups of nodes called communities, multi-core or local components. To deal with these groups' heterogeneity, we propose to extract the backbones independently from their various components and fuse them. Experimental investigations on real-world networks demonstrate the effectiveness of the proposed approach compared to the classical techniques' agnostic of the mesoscopic structure of real-world networks.

Introduction

As data production grows, network analysis is becoming more challenging. Therefore, preserving relevant information while minimizing the network size is crucial. Backbones simplify the network's underlying structure, making identification easier for patterns, communities, and other essential network features. They have many applications in various fields, such as social, transportation, biological, and telecommunication networks. Researchers are developing new techniques to make networks smaller while maintaining their essential structure. The backbone extraction process identifies critical edges and nodes while removing irrelevant data [1,2,[START_REF] Ghalmane | 9th International Conference on Complex Networks and Their Applications[END_REF][START_REF] Gemmetto | [END_REF][START_REF] Ghalmane | [END_REF][START_REF] Rajeh | Network Science: 7th International Winter Conference, NetSci-X 2022[END_REF][START_REF] Yassin | IEEE Workshop on Complexity in Engineering (COMPENG)[END_REF][START_REF] Yassin | Complex Networks and Their Applications XI: Proceedings of The Eleventh International Conference on Complex Networks and their Applications: COMPLEX NETWORKS[END_REF][START_REF] Dai | [END_REF] . One can distinguish two main approaches in backbones extraction techniques. Structural methods tend to remove nodes or links while preserving critical topological properties of the network [START_REF] Rajeh | Network Science[END_REF]. In contrast, statistical techniques eliminate noisy edges or nodes based on statistical significance [START_REF] Yassin | Complex Networks and Their Applications XI[END_REF]. Here, we introduce an approach based on the mesoscopic properties of networks.

These node subgroups within the network are commonly apprehended through the community or the core-periphery structure. This work builds on the component structure recently introduced due to its adaptability and flexibility [START_REF] Diop | [END_REF]. However, using alternative mesoscopic representations is straightforward. We propose a generic framework for backbone extraction that can use any backbone extraction technique developed for weighted networks. Experimental investigations with the influential threshold and Disparity Filter methods on real-world networks from various origins demonstrate the effectiveness of exploiting the component structure to extract the backbone.

Multiscale Backbone extraction framework

The backbone extraction framework is based on simple ideas. It is well-recognized that real-world networks contain various groups of nodes with different densities of interactions. They can be called communities or multi-core groups. A backbone technique ignoring this mesoscopic structure treats equally groups that can be quite different. Indeed, for example, the weights of the links in the various groups may vary over many orders of magnitude, making extracting the truly relevant connections challenging. To overcome this drawback, we propose tailoring the backbone extraction technique to the group of nodes rather than the entire network. The component structure is a very flexible representation for this purpose. Indeed, it splits the networks into two types of components: local and global components; The local components are the dense parts of the network. The intercommunity links and the nodes to which these links are connected form the global component. So local components are the networks' communities or cores, while the global components include the nodes and links joining the core or communities. The proposed framework contains three steps. First, one needs to build the network component structure. Then, one applies a backbone extraction technique to each component rather than the original network. The third step consists in merging the backbones of the global and local components to uncover the overall network backbone. This strategy allows the backbone extraction technique to adapt to the component's topology heterogeneity.

Experimental results

We perform an extensive experimental evaluation. It includes real-world networks selected from different domains with numbers of nodes and edges, ranging from hundreds to thousands, to cover a variety of situations. To evaluate the backbone extractor's effectiveness, we use classical measures such as average weighted degree, average link weight, average betweenness, weighted modularity, and qualitative comparisons. The component structure uncovering process is similar to the one proposed in the literature. 1) The Louvain algorithm allows the detection of the network community structure. 2) The communities form the local components; the global components contain all the nodes and links joining the communities. 3) The backbones of the components are extracted. We use the influential Disparity Filter and the classical global thresholding method. Indeed, our goal is to evaluate the framework rather than the backbone extraction technique. 4) Finally, the union of the various backbones forms the overall network backbone.

In the experiments, we set the backbone size to 30% of the edges of the original network. Figure 1 illustrates the process with the Wind Surfers network. Firstly, we visualize at the top the original network and its community structure uncovered by the Louvain algorithm. The figures in the middle represent its component structure. It contains two local components corresponding to the two communities colored in pink and green, respectively, and one global component with nodes belonging to each community and the links joining these nodes. The figures at the bottom represent the classical and the multiscale backbones extracted using the Global Threshold algorithm. We observe that the multiscale backbone preserves all the components. In contrast, the classical backbone eliminates the global component. Indeed, only one link between node 1 and 7 is preserved, which connect the two local components. All the other links of the global component are filtered out because of their low weights. This figure illustrates that the Multiscale framework acts as a multi-threshold technique tuned to each component weight distribution rather than the overall network weight distribution.

Table 1 reports topological properties of the original network and the various backbones for the Wind Surfers network. It concerns the backbones extracted with the classical global threshold, the disparity filter algorithms, and their multiscale counterparts. The global threshold backbone is good at keeping high weights links. Results show that the Multiscale global threshold technique compares favorably with it for these related properties. Furthermore, its weighted Modularity is almost identical to the original network, demonstrating its ability to preserve the community structure of the original network. The classical Disparity filter preserves links with low weights better than the global threshold. Results show that this property is enhanced in its multiscale version. Once again, the community structure is better preserved. Indeed, whatever filtering technique is used, the global component is more impacted in the classical methods.

To summarize, the proposed framework effectively exploits real-world network heterogeneity, enhancing the advantages of the classical techniques. Future work will explore its association with more sophisticated backbone extraction techniques. Additionally, we plan to investigate the impact of alternative component structure detection algorithms.

Fig. 1

 1 Fig.1The backbone extraction of 'Wind Surfers' network using Global Threshold algorithm.

Table 1

 1 The basic properties values calculated of Wind Surfers network backbones.

		Average	Average	Average	Weighted
		Weighted	Link	Between-	Modularity
		Degree	Weight	ness
	Original Network	56.09	3.59	0.03	0.37
	Classical Global Threshold	44.05	8.50	0.07	0.43
	Multiscale Global Threshold	47.14	8.17	0.06	0.38
	Classical Disparity Filter	6.94	1.24	0.04	0.15
	Multiscale Disparity Filter	6.60	1.31	0.04	0.31


Stephany Rajeh

Hocine Cherifi

Identifying Influential Nodes: The Overlapping Modularity Vitality Framework

Keywords: Influential nodes, Modularity Vitality, Overlapping Community structure

This paper proposes an Overlapping Modularity Vitality framework for identifying influential nodes in networks with overlapping community structures. The framework uses a generalized modularity equation and the concept of vitality to calculate the centrality of a node. We investigate three definitions of overlapping modularity and three ranking strategies prioritizing hubs, bridges, or both types of nodes. Experimental investigations involving real-world networks show that the proposed framework demonstrates the benefit of incorporating overlapping community structure information to identify critical nodes in a network.

Introduction

Real-world networks often have a community structure [1,2]. Recent research has focused on designing "community-aware" centrality measures that take advantage of this structure [3]. However, these measures typically assume that a node belongs to a single community, while they may often belong to multiple communities. This overlapping community structure is a significant feature in many real-world networks, including social networks and protein networks [START_REF] Reid | Mining Social Networks and Security Informatics[END_REF].

While there are many existing works on detecting overlapping network communities, few researchers have leveraged it to identify critical nodes [START_REF] Hébert-Dufresne | [END_REF][START_REF] Chakraborty | Computational Social Networks: 5th International Conference, CSoNet 2016[END_REF][START_REF] Kumar | Companion Proceedings of the The Web Conference 2018[END_REF][START_REF] Taghavian | [END_REF][9][10][START_REF] Gupta | 2015 7th international conference on communication systems and networks (COMSNETS)[END_REF][START_REF] Rajeh | [END_REF]. Moreover, the existing methods have some drawbacks. "Membership" is a straightforward approach quantifying the number of communities shared by S. Rajeh Laboratoire d'Informatique de Bourgogne -University of Burgundy -France E-mail: stephany.rajeh@u-bourgogne.fr a node [START_REF] Hébert-Dufresne | [END_REF]. It cannot differentiate between nodes with the same community affiliations. "OverlapNeighborhood" focuses on identifying hubs located in the vicinity of overlapping nodes without considering the nature of their connections [START_REF] Kumar | Companion Proceedings of the The Web Conference 2018[END_REF]. In contrast, "Random Walk Overlap Selection" prioritizes only highly connected overlapping nodes [START_REF] Taghavian | [END_REF]. The Overlapping Modular Centrality measure combines the communities of overlapping nodes to calculate their local importance, which leads to the loss of important information about the overlap [9]. These methods fail to consider the node's role within its shared communities. For example, a node with connections with several communities based on "Membership" may be more influential than another with the same number of links because it belongs to large communities. Moreover, a node can pertain to one community more than another community. In addition, all these methods suffer from incomplete and fluctuating information. For instance, overlapping information may be available for a set of nodes, not all of them, and it may be available in a different format for these sets of nodes.

Overlapping Modularity Vitality

To solve these issues, we propose a flexible framework for identifying influential nodes in networks with an overlapping community structure called Overlapping Modularity Vitality [13]. It is based on the vitality concept [START_REF] Koschützki | Network analysis[END_REF]. Vitality assesses the influence of a node by looking at the variation of a given quality function when removing it from the network. Building on previous work [START_REF] Magelinski | [END_REF], we extend Modularity Vitality to deal with various overlapping community structures. The proposed framework uses a generalized modularity quality function to model multiple overlapping community typologies, including crisp and fuzzy overlapping information. In crisp communities, a node has full membership in communities, while it can have partial membership in a fuzzy community structure.

Experimental results

We investigate three definitions of overlapping modularity with different membership strengths, including reciprocity membership, degree membership, and node similarity. The measures differ in the information they use about the overlaps of communities and their ability to discriminate between nodes. Reciprocity membership is less discriminating since nodes may share the same number of communities, while node similarity and degree membership encode more nuanced differences. Additionally, since the framework is based on vitality, the Overlapping Modularity Vitality framework can have three ranking strategies for prioritizing hubs, bridges, or both. The effectiveness of the proposed framework is evaluated on 21 real-world networks using the SIR epidemic spreading scenario. Results show that the framework's performance depends on the available resources (f o ), with different versions being more effective depending on a budget of nodes. Overall, results suggest that the distance between initially infected nodes is a good indicator of when enough resources are available. Furthermore, the framework that uses degree membership and node similarity with a bridges-first ranking scheme is most effective in this scenario, as seen in the Facebook friends network in Figure 1.

Conclusion

These findings demonstrate the benefit of incorporating overlapping community structure information to identify critical nodes in a network. The strength of the proposed framework lies in its capacity to integrate various types of structural information by using customized overlapping modularity measures. Indeed, as no one-size-fits-all definition applies to all real-world networks, a flexible approach is necessary to address different scenarios.

Fig. 1

 1 Fig. 1 The Facebook Friends network (A and C) with the nodes chosen to be initially infected by the vitality measures: Modularity Vitality (β M V ), and the three different versions of Overlapping Modularity Vitality, namely: reciprocity membership (β R OM V ), degree membership (β D OM V ), and node similarity (β S OM V ). The measures use a hubs-first ranking scheme denoted by a "+" sign and a bridges-first ranking scheme marked by a "-" sign. The top two rows and the bottom two rows represent 2% and 20% of the fraction of initially infected nodes, respectively. In the middle (B), the figure represents the relative difference of the outbreak size (∆R) as a function of the fraction of initially infected nodes (fo) Facebook Friends. The reference centrality (Rr) is degree centrality (β D ). The centrality measures under test are: Membership (β M ), OverlapNeighborhood (β ON ), Random Walk Overlap Selection (β RW OS ), Overlapping Modular Centrality (β OM C ), Modularity Vitality (β - M V ), and the three different versions of Overlapping Modularity Vitality, namely: reciprocity membership (β R- OM V ), degree membership (β D- OM V ), and node similarity (β S- OM V ).


Majda Lafhel
email: majdalafhel1@gmail.com

Mohammed El Hassouni

Benjamin Renoust
email: renoust@gmail.com

Mohammed El Hassouni
email: mohamed.elhassouni@gmail.com

Hocine Cherifi
email: hocine.cherifi@gmail.com

Measuring Movie Script Similarity using Characters, Keywords

Keywords: , Locations, and Interactions Locations, and Interactions , Locations, and Interactions Movie Script Multilayer Network, Inter layer relationships, Multilayer Graph Distance measure, Comparing movies

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

In recent years, multilayer network analysis has achieved widespread use in various fields, including social networks, transportation networks, biological networks, and communication networks. Measuring the similarity between multilayer networks is a complex task. That is because the multilayer network consists of different entities of layers and relationships, making it challenging to define a distance measure that captures the overall multilayer network structures. Brodka et al. (2018) [1] have proposed a property matrix that represents a multiplex network. The property matrix maps layers and nodes into structures. Brodka et al. have used three methods to compare multiplex networks: aggregations(min, max, entropy), layer distributions(Jensen-Shannon Divergence), and similarity functions(Jaccard, cosine, correlation). Giordano et al. (2019) [2] used factorial methods for quantifying multiplex networks visually. Ghawi et al. (2022) [3] have used community detection to quantify the similarity between multilayer networks. In previous works [START_REF] Lafhel | International Conference on Complex Networks and Their Applications[END_REF][5], we investigated Network Portrait Divergence [START_REF] Bagrow | [END_REF] and Laplacian Spectra Descriptor [START_REF] Tsitsulin | Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining[END_REF] to compare the similarity between movie stories. We extracted for each movie a multilayer network [START_REF] Mourchid | International Conference on Complex Networks and their Applications[END_REF] composed of three layer entities (character, keyword, and location). We ignored inter-relationships and compared monolayers of the same entities. This research aims to quantify the similarity between movies. There have been multiple approaches to quantifying visual content [9-12, 11, 13-16]. Here, we consider multilayer network movies with inter-relationships between layers. To the best of our knowledge, there is currently no approach for measuring the similarity between multilayer networks considering inter-relationships. Analyzing the structure of interlayer relationships provides extra information and a comprehensive overview of the multilayer network. Moreover, previous studies have focused on comparing layers with the same number of nodes. Multilayer movie networks consist of layers of different sizes, making finding an appropriate measure challenging. We propose an approach that captures intralayer and intralayer relationships in the multilayer network, considering networks of various sizes.

Methodology

A graph G is a set of nodes N connected by edges E. Based on this property, we consider nodes of the same entity linked by intra-relationships as graphs G intra and nodes of different entities connected by inter-relationships as graphs G inter . We work on multilayer network movie scripts with three entities (character, keyword, and location). So, the multilayer network includes six types of networks: G intra CC is the character graph, G intra KK is the keyword graph,G intra LL is the location graph, G inter CK consists of inter-relationships connecting character and keywords, G inter KL consists of inter-relationships connecting keyword and location nodes, and G inter KL consists of inter-relationships connecting keyword and location nodes.

The proposed algorithm (Algorithm 1) maps G intra , G inter , and network features into one matrix P as follows.

-(i) Six rows, where the first three rows represent the three intralayers (G intra CC , G intra KK , and G intra LL ), and the last three rows represent the three interlayers (G inter CK , G inter KL , and G inter CL ). -(ii) Six columns, each one represents a network features: max degree, max centrality, density, adjacency, Laplacian, and network portrait. -(iii) Each cell c ij encodes a network feature j of the network type i.

Algorithm 1 Matrix property extraction

input: G intra CC , G intra KK , G intra LL , G inter CK , G inter KL , G inter CL output: matrix property P 1: for i in G intra CC , G intra KK , G intra LL , G inter CK , G inter KL , G
: P ← [v G intra CC , v G intra KK , v G intra LL , v G inter CK , v G inter KL , v G inter CL ]
Consider a pair of multilayer network movies M and M'. In the first step, we extract property matrices P from M and P ′ from M'. Second, we flatten matrices P and P ′ to vectors v and v′ . Then, we compute the distance between v and v′ using the Euclidean Distance.

D = G inter CL i=G intra CC (λ vi -λ ′ v′ i ) 2
(1)

Experimental Results

We performed experiments using movie scripts from various categories. In our previous work, it appears that the romance films were the more challenging. Therefore we concentrate on these movies. We compare: Titanic (1997), episode I of Twilight (2008), and episode II of Twilight (2009). For each movie, we extracted three layers (character, keyword, location), intra-relationships and inter-relationships. We collected ground-truth data by inviting a group of individuals to rank the similarity between romance movies. Based on the evaluation, we obtained the following ranking: Episodes I and II of Twilight are in the first rank, Titanic and I of Twilight in the second, and episodes II and three also in the second.

To illustrate the efficiency of the proposed method in quantifying the similarity between romance movies (Titanic, episodes I and II of Twilight) we compare the obtained results (Table 2) to those of previous studies (Table 1). In a previous investigation (Table 1), the Network Portrait Divergence revealed the similarity between character layers, and the Network Laplacian Spectra detected the similarity between location layers. But, no measure indicated the similarity between keyword layers.

✗ ✗ ✗ NetMF ✗ ✗ ✗ D-measure ✗ ✗ ✗ Network Portrait Divergence ✗ ✗ Laplacien Spectra ✗ ✗
According to Table 2, the distance between episodes I and II of Twilight (272.93) is the smallest. That means episodes I and II are the most similar. Indeed, the ground-truth data shows episodes I and II of Twilight are in the first rank. On the other hand, Titanic is closer to episode II of Twilight (403.52) than Episode I (450.24). However, according to the ground-truth data, both pairs of movies are second, which reveals how Titanic is far from episodes I and II of Twilight. Table 2 Distance between romance movies using the proposed method In brief, the proposed method revealed the high similarity between episodes I and II of Twilight and the distance between Titanic compared to both movies. In contrast, in the previous research, no measure revealed the similarity between keyword layers. Furthermore, the time complexity of the proposed technique is much smaller than the prior one. That is because we compare the overall multilayers at one time.

  Twilight & episode II of Twilight 272.93 Titanic & episode I of Twilight 450.24 Titanic & episode II of Twilight 403.52

  B ← sum(B) 11: v i ← [D, BC, Dens, s A , s L , s B ] 12: end for 13

					inter CL do
	2: D ← max((deg(1), deg(2), ..., deg(N )))	//return the max node degree of i.
	3: BC ← max( s̸ =t∈V	σ st (N ) σ st	)	//return the max node betweeness centrality of i
	//σst: total shortest paths passing from a node s to a node t
	//σst(N ): total number of σst that passing through a node n
	4: Dens ← E/(N * (N -1))		//return density of i
	5: A ← Extract Adjacency matrix(i)
	6: L ← Extract Laplacian matrix(i)
	7: B ← Extract N etworkP ortrait matrix(i)
	8: s A ← sum(eigenvalues(A))
	9: s L ← sum(eigenvalues(L))
	10: s

Table 1

 1 Checklist table for similarity between romance movies

	Measures/Layers	Character	Keyword	Location
	NetLSD


Gilles Burel
email: gilles.burel@univ-brest.fr

Stéphane Azou

A New Receiver for Chaotic Digital Transmissions: The Symbolic Matching Approach

Keywords: Chaos, Digital Transmissions, Communications, Receiver, Symbolic Matching

Chaotic digital transmissions have recently emerged as a way to improve the security and privacy of digital transmissions. In this paper, we propose a new approach to estimate the transmitted symbols on the receiver side. This approach, that we call Symbolic Matching, consists in matching vectors built from the received signal with a symbolic model of the chaotic oscillator trajectories. Simulation results are also provided and they show that low error rates are obtained.

Introduction

In the last few years, chaos-based modulation techniques have emerged as an interesting alternative to standard spread spectrum techniques [START_REF] Hasler | Synchronization of chaotic systems and transmission of information[END_REF][7] [START_REF] Rulkov | Chaosbased Communication over Turbulent Channel[END_REF]. Indeed, chaotic transmissions provide many advantages, among which we can mention:

Their wideband nature which provides robustness against frequency selective fading in multipath channels and against narrowband interference. The intricate dynamic of chaos which signi�cantly increases the privacy of communications in comparison with standard pseudo-noise codes used in spreadspectrum. Without knowledge about the chaotic oscillator on which the transmission is based, it is extremely dif�cult for the unauthorized user aware of the transmission to access the information. Indeed, standard spread spectrum transmissions are not so secure. For instance, in [START_REF] Burel | Detection of Spread Spectrum Transmissions using �uctuations of correlation estimators[END_REF] [START_REF] Burel | Blind estimation of the pseudo-random sequence of a direct-sequence spread spectrum signal[END_REF] we proposed approaches for interception of standard spread spectrum transmissions. On the contrary, no ef�cient approach is known today for interception of chaotic signals.

Chaos can be used in multiple ways in a digital communication system. In this paper we will focus on one of the most ef�cient technique, which is called CD3S (Chaotic Direct-Sequence Spread Spectrum) [START_REF] Heidari-Bateni | A chaotic direct-sequence spread spectrum system[END_REF]. In order to get reliable communication for realistic propagation conditions, a robust synchronization procedure and gain control has to be developed at the receiver side. Readers interested by these aspects can refer to our previous papers [START_REF] Azou | A chaotic direct sequence spread-spectrum system for underwater communication[END_REF] [START_REF] Azou | Sea trial results of a chaotic direct sequence spread spectrum underwater communication system[END_REF], in which we report experiments on real-world application and signals in the context of a chaotic underwater acoustic network.

In this paper, we assume synchronization and gain control done and we focus on the estimation of the transmitted symbols. A simple and ef�cient approach was proposed by Milanovic et al. [START_REF] Milanovic | Combating noise and other channel distorsions in chaotic communications[END_REF]. In the present paper, we propose a new approach, which we call Symbolic Matching. From the received signal, we build N -dimensional vectors composed of N successive received samples, and we match these vectors with a symbolic model. A criterion, based on the result of the matching, is then used to estimate the transmitted symbols. Simulation results show that this approach provides a lower error rate than the approach described in [START_REF] Milanovic | Combating noise and other channel distorsions in chaotic communications[END_REF].

The paper is organized as follows. In Section 2, we recall the principle of CD3S chaotic digital transmissions. Then, in Section 3, the approach described in [START_REF] Milanovic | Combating noise and other channel distorsions in chaotic communications[END_REF] is summarized. Our Symbolic Matching method is explained in Section 4, and illustrated by simulation results in Section 5. Finally, a conclusion is drawn in Section 6.

Principle of CD3S chaotic digital transmissions 2.1 Overview

A chaotic oscillator is a system which is extremely dependent on the initial conditions. If we consider two identical chaotic oscillators, an extremely small difference of their initial state causes the signals they gener-WSEAS Int. Conf. On Telecommunications and Informatics, New York, USA, Sept. [15][16][17]2003 ate to quickly diverge. A chaotic signal is therefore unpredictable in the long term. An n-dimensional chaotic system can be described by state space equations:

c k = g (c k-1 ) (1)
where c k ∈ R m is called the state, and nonlinear function g maps state c k-1 to the next state c k .

A chaotic dynamical system is one that is deterministic but appears not to be so, as a consequence to its extreme sensitivity to initial conditions. This can be observed even for very simple (one dimensional discrete time) chaotic dynamical system. In this paper, for clarity purpose, we will focus on one dimensional discrete time chaotic dynamical system (another reason is that most chaotic oscillators in use in actual transmission applications are one dimensional).

In most chaotic transmission systems, a BPSK (Binary Phase Shift Keying) is used, hence the symbols a n belong to {-1, +1}. If we note P the spreading factor (i.e. the number of chaotic samples per symbol), the transmitted signal is:

x k = a �k/P� c k (2)
where �k/P� stands for k/P rounded to the nearest integer towards minus in�nity. The received signal is then:

y k = x k + n k (3)
where n k stands for the noise. Usually, the nonlinear function g is such that:

g (-x) = g (x) (4)
Let us note g a (x) = ag(x). We can write:

x k = a �k/P� c k (5) = a �k/P� g (c k-1 ) (6)
= a �k/P� g � a �k/P� c k-1 � (7) = a �k/P� g (x k-1 ) (8) = g a �k/P� (x k-1 ) (9)
This result will be used for estimation of the symbol.

A simple example of chaotic oscillator

Let us consider the nonlinear function below:

g (x) = 1 -2x 2 (10)
This map generates intricate chaotic trajectories. The corresponding state equation is:

c k = 1 -2c 2 k-1 (11)

Milanovic receiver

Principle

In [START_REF] Milanovic | Combating noise and other channel distorsions in chaotic communications[END_REF], Milanovic et al. proposed a simple and ef�cient chaotic receiver. Synchronization is assumed, hence we know where begins and ends each block of P samples corresponding to a symbol. For simplicity, let us explain the approach for the estimation of the �rst symbol (generalization to other symbols is trivial, since it only requires an offset on index k). The index �k/P� of a �k/P� is also omitted.

First, from the received samples {y k }, the values below are computed:

� y k = g (y k-1 ) (12)
for k ∈ {1, 2, . . . , P -1}. Then, the following criterion is computed:

C = P-1 � k=1 � y k y k (13) If C < 0 then the estimated symbol is � a = -1 else it is � a = +1.

Justi�cation

If there is no noise, we have y k = x k and

� y k = g (y k-1 ) (14) = g (x k-1 ) (15)
Therefore:

C = P-1 � k=1 � y k y k (16) = P-1 � k=1 g (x k-1 ) x k (17) = P-1 � k=1 g (x k-1 ) ag (x k-1 ) (18)
= a P-1 � k=1 g 2 (x k-1 ) (19)
Since the sum is positive, the sign of C is equal to the sign of the transmitted symbol a.

Proposed Symbolic Matching receiver 4.1 Overview

Let us consider P successive received samples corresponding to a symbol. For simplicity, as in the previous Section, let us explain the approach for the estimation of the �rst symbol (generalization to other symbols is trivial, since it only requires an offset on index k). The index �k/P� of a �k/P� is also omitted. Let us consider N successive received samples (N � P ) and note:

y k = � y k , y k+1 , ..., y k+N -1 � T (20)
and de�ne a symbolic vector z a (x):

z a (x) = � x, g a (x), g (2) a (x), . . . , g (N -1) a (x) � T ( 21
)
where x is a symbolic variable, a ∈ {-1, +1}, and g (n) (x) is de�ned as:

g (n) (x) = g(g(. . . (g � �� � n times (x)))) (22)
z a (x) is a symbolic model of chaotic trajectories. When x varies, z a (x) moves along a one-dimensional curve into an N -dimensional vector space.

Let us compute:

� x a,k = arg min x �y k -z a (x)� 2 (23)
� x a,k is the value of x which moves z a (x) as close as possible to y k (hence, which provides the best correspondence between the symbolic model and the actual received data). Details about this computation are given in next subsection.

Then, compute the vector below:

� x a,k = z a � � x a,k � (24)
that is:

� x a,k = � � x a,k , g a (� x a,k ), g (2) a (� x a,k ), . . . , g (N -1) a (� x a,k ) � T (25)
The criterion below is computed for a ∈ {-1, +1}:

C a = P-N � k=0 � � y k -� x a,k � � 2 (26)
This criterion represents the sum of the distances between vectors y k built from the received signal and their closest neighbors on the symbolic model.

Finally, if

C -1 < C +1 then the estimated symbol is � a = -1 else it is � a = +1. The reason is obvious: if C -1 < C +1 ,
this means that the symbolic model corresponding to a = -1 is closer to the received signal than the symbolic model corresponding to a = +1.

Computation of � x a,k

Since the problem is similar whichever the values of k and a are, these indexes are omitted below. We have:

� x = arg min x �y -z(x)� 2 (27) = arg min x N -1 � n=0 � y n -g (n) (x) � 2 (28)
Let us note

d(x) = �y -z(x)� 2 (29) = N -1 � n=0 � y n -g (n) (x) � 2 ( 30
)
Its derivative is:

d � (x) = 2 N -1 � n=0 � g (n) (x) -y n � ∂g (n) ∂x (x) (31)
Functions d (x) and d � (x) can be easily precomputed using symbolic computation software, such as Maple or Matlab symbolic toolbox. For instance, when g (x) = 1 -2x 2 , d(x) and d � (x) are polynomials.

The numerical value of x is then obtained by solving

d � (x) = 0 (32)
and keeping the solution which produces the lowest value of d(x) when there is more than one root. This solution is � x a,k .

Simulation Results

In this section, we show simulation results obtained with the map g(x) = 1 -2x 2 .

Let us take N = 3 and consider a signal to noise ratio equal to 15d B on the receiver side. Figure 2 shows the symbolic models z +1 (x) and z -1 (x), which are 1D curves in the 3D space. Vectors y k = � y k , y k+1 , y k+2 � T , represented by symbols +, and built from a block of the received signal corresponding to a transmitted symbol a = +1, are shown. It is clear that these vectors are (on average) closer to the symbolic model z +1 (x) than to z -1 (x). Hence, since criterion C a represents the sum of the distances between the vectors y k and the symbolic model z a (x), we will have C +1 < C -1 and the receiver will (correctly) estimate that the transmitted symbol is +1. If the signal to noise ratio on the receiver side is only 3d B, things are less obvious, as shown on �gure 3, and estimation errors may occur. However, on average, vectors y k are closer to the right model than to the wrong one. The table below shows simulation results using a CD3S transmission system with a spreading factor P = 32. On the receiver side, the signal to noise ratio was S N R = 3d B and 10 4 symbols were transmitted (i.e. 32×10 4 signal samples were received). The table shows the number of symbol errors with respect to the method used for symbol estimation. The table shows that Symbolic Matching provides a lower error rate than Milanovic et al. approach. However, except for N = 2, the Symbolic Matching requires more computational power. Hence, due to its simplicity, Milanovic et al. approach is a very interesting method when low computational power is available, while when reasonable computational power is available Symbolic Matching may be preferred.

Conclusion

In this paper, we have proposed a new approach for symbol estimation in a digital chaotic transmission receiver. The originality of the approach is to match vectors built from the received signal with a symbolic model of the chaotic oscillator trajectories. Experimental results show that good error rates are obtained. Further work will include a more theoretical study of the performances of this receiver.

Figure 1 Figure 1 :

 11 Figure 1 illustrates two chaotic sequences: a low cross-correlation is clearly seen, although the initial states for the two sequences differ only by 10 -3 .

Figure 2 :

 2 Figure 2: Representation of vectors y k and symbolic models z +1 (x) and z -1 (x) in a 3-dimensional space (N = 3). The map is g(x) = 1 -2x 2 and the signal to noise ratio is S N R = 15d B.

Figure 3 :

 3 Figure 3: Representation of vectors y k and symbolic models z +1 (x) and z -1 (x) in a 3-dimensional space (N = 3). The map is g(x) = 1 -2x 2 and the signal to noise ratio is S N R = 3d B.

  method nb. errors Milanovic et al. 290 Symbolic Matching (N = 2) 179 Symbolic Matching (N = 3) 151 Symbolic Matching (N = 4) 148