A Tlili

V M Giordano

Y M Beltukov

P Desmarchelier

S Merabia

A Tanguy

Enhancement and anticipation of the Ioffe-Regel crossover in amorphous/nanocrystalline composites †

Nanocomposites made of crystalline nanoinclusions embedded in an amorphous matrix are at the forefront of current research for energy harvesting applications. However, the microscopic mechanisms leading alternatively to an effectively reduced or enhanced thermal transport still escape understanding. In this work, we present a molecular dynamics simulation study of model systems, where for the first time we combine a microscopic investigation of phonon dynamics with the macroscopic thermal conductivity calculation, for shedding light on thermal transport in these materials. We clearly show that crystalline nanoinclusions represent a novel scattering source for vibrational waves, modifying the nature of low energy vibrations and significantly anticipating the propagative to diffusive crossover (Ioffe-Regel), usually located at energies of few THz in amorphous materials. Moreover, this crossover position can be tuned by changing the elastic contrast between nanoinclusions and matrix, and anticipated by a factor as large as 10 for a harder inclusion. While the propagative contribution to thermal transport is drastically reduced, the calculated thermal conductivity is not significantly affected in the chosen system, as the diffusive contribution dominates heat transport when all phonons are thermally populated. These findings allow finally to understand the panoply of contradictory results reported on thermal transport in nanocomposites and give clear indications on the characteristics that the parent phases should have for efficiently reducing heat transport in a nanocomposite.

Introduction

Thermal management is one of the most urgent challenges arising in our modern technological society. Indeed, heat represents almost 66% of the energy waste in current processing and technologies, thus asking for concrete improvements aimed at reducing heat dissipation and waste, or at converting such heat in another form of energy [START_REF] Lawrence | Livermore National Laboratory, Data based on DOE/EIA MER[END_REF] . In the late years, with the aim of positively answering to this challenge, scientists have focused in understanding heat transport in different kinds of materials, for ultimately controlling it and engineering novel materials able to limit heat spread and dissipation while keeping other functional properties, such as optical or electrical.

Recently, the use of composites made of guest nanoparticles embedded in a uniform host matrix has arisen as a very promising approach [START_REF] Viskadourakis | [END_REF] . Indeed, many theoretical and experimental studies have evidenced a strong reduction of the vibrational contribution to thermal conductivity, which has been understood in terms of an enhanced phonon scattering due to inclusions, and cannot be accounted for by standard effective medium theories, although successful in describing macroscopic composites [3][4][5][6][7][8][9] . Modified effective medium theories have been recently proposed, able to account for phonon scattering processes arising on the nanometric scale both in the matrix and within the nanoinclusions, and have succeeded in reproducing the results of simulations on the thermal conductivity of nanocomposites and its dependence on interface density 10,11 . Still, even these models fail in reproducing exotic phenomena reported by numerical studies, such as the high energy phonon localization in the intergrain regions 12 or their filtering through nanometric inclusions in an amorphous matrix 13 , indicating thus that the microscopic mechanisms at play still escape understanding.

Intuitively, one of the key parameters ruling the vibrational en-ergy transport in such materials is the contrast of elastic properties between inclusion and matrix, similar to an impedance mismatch. The higher the contrast, the stronger the interface scattering on phonons traveling in the matrix. Indeed, it has been recently reported that no remarkable effect could be observed in thermal and electric transport properties in a nanocomposite when the elastic and electric properties of the crystalline inclusions are too similar to the ones of the matrix 14 . When the elastic contrast is significant, instead, contradictory results have been reported, such as a strong thermal conductivity reduction, as measured in composites made of Si nanocrystals embedded in a polystyrene matrix 9 or its enhancement beyond normal percolation as simulated in glassy silica with embedded nanocrystalline inclusions of GaN, both results being at odds with all effective medium predictions 15 . It is thus still not understood how the elastic contrast acts on thermal transport in such systems. To shed light on this question, it is essential to look directly at the effect of interfaces and elastic contrast onto the heat carriers, the phonons. However, until now only very few works have adopted a microscopic approach for understanding the mechanisms at play, the majority of the literature on the subject being focused on the simulation of the thermal conductivity or the interpretation of experimental data by means of effective medium models.

Here we report a molecular dynamics investigation aimed at unveiling the role of elastic contrast in amorphous/crystalline nanocomposites and getting a microscopic understanding of the effect of nanometric elastic heterogeneities and interfaces on thermal transport, through the study of phonon dynamics. Such composites, made of nanocrystalline particles embedded in an amorphous matrix, are here chosen for their high technological potential. They offer indeed the opportunity of realizing a material with a low glass-like, matrix-dominated, thermal conductivity, together with good electrical properties, as determined by the crystalline guest phase through a percolation phenomenon 2 . To this purpose, we have investigated a model 3D silicon amorphous system, with nanometric Si crystalline inclusions. The elastic contrast has been tuned by modifying the inter-atomic potential within the inclusion, in such a way that the Young modulus is the same as in the amorphous matrix in one case, and about five times larger in a second case.

It is worth reminding here that in amorphous materials, three dynamical regimes can be identified 16,17 : i) a low-energy planewave dominated regime, where phonons undergo weak scattering, insuring a propagative thermal transport ii) a strong scattering regime, above the Ioffe-Regel limit, which identifies the energy at which wave-packets lifetime has been so reduced that they cannot be considered propagative plane-waves anymore and the corresponding phonons are called "diffusons" 18 . This regime corresponds to a diffusive transport of vibrational energy. This definition is the one that we will use in the following when talking of "diffusive regime", "diffusive transport". The Ioffe-Regel crossover marks then the end of the propagative in favour of a diffusive thermal transport. Finally, iii) the Anderson localization regime near the mobility edge 16,19,20 . The question arises how the presence of interfaces and elastic heterogeneties at the nanoscale affects these dynamical regimes.

We show here that phonon dynamics is strongly modified in nanocomposites with and without elastic contrast, leading to a change in the nature of low energy phonons, significantly anticipating the propagative to diffusive crossover and the Ioffe-Regel limit with respect to the parent amorphous material. Surprisingly, the calculated thermal conductivity is apparently insensitive to the elastic contrast, which can be understood in terms of a dominant diffusive thermal transport already in the amorphous matrix.

Methods

The system is made of an amorphous silicon matrix where spherical nanoholes are scooped out and filled with fcc crystalline silicon of the same density. The amorphous matrix is obtained with molecular dynamics simulations by quenching very quickly a liquid sample of silicon atoms interacting with the Tersoff potential 21 and initially equilibrated at 3500K. Then a quench is performed with a quenching rate of 10 14 K/s in the NVT ensemble at a fixed density ρ = 2.303 g/cm 3 with a box size of L b = 60 Å (10 740 atoms), and periodic boundary conditions. An energy minimization is done using overdamped dynamics as soon as the temperature reaches 10K. As reported in 22 , the Tersoff potential allows to get a realistic structure, whose pair correlation function and static structure factor are in agreement with experimental data, which is not the case for the Stillinger-Weber potential 23 . However, this latter is better suited for the analysis of the mechanical properties 24 . For this reason, once the system generated, we replace the Tersoff potential by the Stillinger-Weber for our study. The system is first annealed at 100K for 10 ps, and then the total energy is minimized again to get an initial amorphous matrix at mechanical equilibrium. Nanoholes with spherical shapes are then cut in the matrix and filled with nanospheres cut in a fcc silicon configuration with the same density ρ = 2.303g/cm 3 . In order to prevent the large stresses at the boundaries of the inclusions, atoms with neighbors at distances below 1 Åare removed, and the nano-composite is again annealed at 100K for 10ps and quenched.This procedure allows atoms to readjust enough for avoiding instabilities initiated at the interface and getting a deeper mechanical equilibrium. Different inclusions sizes have been compared 25 . In this work, we focus on the case of an inclusion radius R = 25 Åthat was found to give rise to the largest effect on wave packets attenuation 25 . The corresponding crystalline volume fraction is thus φ = 30% and the total number of atoms is close to N = 10 740 in the basic cell. In order to tune the elastic contrast between inclusions and matrix, we have artificially increased the rigidity of the nanoinclusions by choosing a larger value for the prefactor Λ of the three-body term 23 in the Stillinger-Weber interaction, which is proportional to the shear modulus 16 , affecting the transverse sound velocity as √ Λ. For silicon it is Λ = 21. We have then kept this value in the matrix, and prepared an inclusion with the same value and another one with Λ = 100, doubling the transverse velocity with respect to the matrix: c T (100)/c T (21) = 2.20, while increasing the longitudinal only by 40% (c L (100)/c L (21) = 1.38 16,22 ). Since only the impedance ratio matters for sound propagation, we introduce the rigidity ratio χ = E(100)/E(21) = 4.6 to designate the nanocomposite with the more rigid inclusions and χ = 1 to designate the sample with Λ = 21 in the matrix and in the inclusions.

In our work, we analyze small vibrations around the equilibrium state. The properties of the near-equilibrium vibrations are determined by the properties of the dynamical matrix in the harmonic approximation

M iα, jβ = 1 √ m i m j ∂ 2 U ∂ r iα ∂ r jβ . ( 1
)
The numerical calculation of all second derivatives in Eq. ( 1) for a large atomic system is a time-consuming procedure. However, all interatomic potentials have a known analytical form, and the second derivatives can be calculated analytically. Due to the complicated structure of these latter, we use computer algebra system SymPy to find their analytical form. To generate the high-efficient numerical code from the analytical expression we use the Lambdify function. The dynamical matrix M was calculated for the equilibrium state obtained by molecular dynamics. In order to calculate the DOS for the given dynamical matrix, we use the Kernel Polynomial Method (KPM) 16,26 .

The dynamical structure factor S(q, ω) is obtained from the spatial and temporal Fourier Transform of the atomic displacements during a Molecular Dynamics run at constant total energy with random initial velocities on each atom. The method used to compute S(q, ω) is the same as in 13 . It allows discriminating the transverse and the longitudinal components of the dynamical structure factor. To perform the Spatial Fourier Transform, the minimum wavevector is q min = π/L and the maximum wavevector is q max = 2π/(1 Å ). The minimum frequency with a physical meaning is imposed by the size of the system ω min = c T /L. It allows defining the duration of the Molecular Dynamics run t run = 2π/ω min , while the maximum frequency is related to the inverse timestep ∆t = 10 f s as ω max = 2π/∆t.

In order to study the wave packets propagation, we apply a quasi-monochromatic external pulse to a thin atomic layer between inclusions. Only the harmonic response is studied here. In this case, the equation of motion can be written using the dynam-

ical matrix M üiα (ω,t) + ∑ jβ M iα, jβ u jβ (ω,t) = f exc iα (ω,t), (2)
where

u i = √ m(r i -R i )
is a scaled displacement of the i-th atom from the equilibrium position R i and M iα, jβ is an element of the dynamical matrix M. The excitation force f exc iα (ω,t) is the α component of a complex excitation force

f exc i (ω,t) = f L/T exp iωt - t 2 2τ 2 exc - x 2 i 2w 2 . ( 3
)
The width of the excited layer is w = 1 Å and the duration of the excitation is τ exc = 0.36 ps. Such pulse duration is smaller than a typical phonon lifetime and gives a good enough frequency resolution δ ν ∼ 1/(2πτ exc ) = 0.4 THz. The direction of the applied force is defined by f L/T , which is common for all atoms in the excited layer. The subscript L (or T ) indicates the wave-packet polarization in case of coherent excitation. In this case, we use f L in x direction for the longitudinal polarization and f T in y direction for the transverse one (see Fig. 1).

In order to study the wave packets propagation for different frequencies we can use the linearity of Eq. 2 and impose an initial pulse containing all the frequencies. The deconvolution of the dynamics allows then studying all frequencies using only one simulation and the Fast Fourier Transform. The details of this procedure were described in Ref. 27 . The diffusivity can also be computed, by measuring the time dependence of the progressive enlargement of the wave packet with initial random polarizations. The diffusivity D(ω) is then obtained as the slope in the linear time dependence of the quantity

1 2 ∑ i (E k (i,t) × x 2 i ) ∑ i E k (i,t) - 1 2
∑ i x i E k (i,t) ∑ i E k (i,t) 2 = 1 2 < (x 2 -< x > 2 ) >∝ Dt (4
) where E k (i,t) is the kinetic energy averaged on the slice located at x i with width δ x = 2 Å.

For better averaging, we use first a system with 8 inclusions in a periodic box L × L × L (2 inclusions along each side of the cube), for a total number of atoms 8 times bigger than in the basic cell. There are 6 possible planes between inclusions (x = 0, x = L/2 = L b , y = 0, y = L/2 = L b , z = 0, and z = L/2 = L b ). We sequentially use all these planes as the excitation layers and average the results over them. Using periodic boundary conditions, this system is repeated a sufficient number of times in the direction perpendicular to the excitation plane in order to prevent the reflection of waves at the boundary during the duration of the simulation.

Finally, we computed the nanocomposite thermal conductivity k T using equilibrium simulations [28][29][30][31] . The thermal conductivity has been calculated by the Green-Kubo formula :

k T = 1 3V k B T +∞ 0 J α (t)J α (0) dt ( 5
)
where V is the nanocomposite volume, k B is Boltzmann's constant, J α (t) is the instantaneous heat flux vector 30 (Einstein summation convention has been implictly assumed), and the brackets indicate an ensemble average. NVE simulations (duration 0.25 ns) have been run to sample the heat flux vector every 10 time steps (time step = 0.5 fs). To obtain good convergence of the integral of Eq. 5, we have averaged the results over 10 indepen- dent simulations, for each studied structure. In this way, while the simulation time is large enough for the thermal conductivity calculation to converge, the error bar is reduced through the averaging over independent simulations rather than a longer simulation time, which could result in large numerical uncertainties 32,33 (see Supplementary Materials for more details). This calculation has been done using the same atomic configurations as for the microscopic dynamics simulations, on a single cell with a single inclusion and periodic conditions, with initial equilibrations at 10 K and also at 300 K. Before closing this section, it is worth commenting on the choice of considering only harmonic interactions. Indeed, anharmonicity has quite a relevant role in determining thermal transport properties at high temperature. Still, it was shown previously that it is negligible in amorphous samples at the phonon energies here concerned 34 , suggesting thus that the same is true in nanocomposites with a dominant amorphous component. Concerning all our results from the atomic structure equilibrated at 10 K, we can safely rule out an anharmonic contribution in the crystalline component as well, and thus in nanocomposites whatever the crystalline content. The question arises concerning the calculation of the thermal conductivity from the configuration equilibrated at 300 K. We will comment on this point later in the paper.

results

Density of States and Dynamical Structure Factor

We report in Fig. 2 panel (a) the total vibrational density of states (DOS) as calculated in the samples with and without elastic contrast, together with the DOS of a pure amorphous sample, from Ref. 16 . The χ = 1 system presents some differences from the fully amorphous sample, mostly localized at about 11 and 13.3 THz, where the DOS is sharper, reminiscent of sharper features in the crystalline system. As for the χ = 4.6 sample, new high energy modes appear in the density of states for ω ≥ 20 THz, characteristic of the harder inclusion. The number of modes in the ω ≤ 20 THz region obviously decreases because of the normalization of the DOS, still the relative weight of the modes at around 10 THz becomes more important. To better inspect the low energy acoustic modes, we report in panel (b) the reduced density of states, normalized to the Debye behavior, where the characteristic excess of low energy modes, the Boson Peak, clearly appears in the amorphous system at ≈4 THz. Interestingly, in both nanocomposites, this region is strongly affected, leading to the growing up of a low-frequency peak, at about 1.7 THz, already weakly present in the fully amorphous system. As such, the DOS of the nanocomposite reminds of the one calculated in 16 for a weak three-body interaction (parameter λ < 21 in the three-body term of the inter-atomic potential). This low energy peak was identified in that work as corresponding to the first strong minimum of the atomic participation ratio, i.e. the end of the plane-wave, Debye-like, regime, prior to the reaching of the Ioffe-Regel transition from propagative to fully scattered waves. In that work the low energy peak corresponds to a typical distance between scatterers of ≈ 20 Å , which is comparable here to the interparticle distance. Our results point thus to a larger number of modes implied by the breaking down of the Debye approximation in both nanocomposites.

To understand how the presence of the nanoinclusions modifies the individual modes, we have calculated the dynamical structure factor S(q, ω), as described in section 2. Similarly to what was done in Damart et al. 13 , to overcome the resolution limitation of the simulation, we have convoluted the S(q, ω) with a typical energy resolution function with a linewidth of ≈ 1.4 meV. The result of such data treatment is reported in Fig. 3 for modes with an exchanged momentum smaller than 0.7 Å -1 . Like in Damart's work, we also find a nice single peak at low-momentum transfers, which evolves into an envelope of many modes at higher q. However, while the behavior of the sample χ = 1 closely resembles the one of the amorphous matrix in Damart et al., in the sample χ = 4.6 this remains true only for the transverse polarization, while the breaking down of the single phonon in an envelope of modes takes place much earlier for the longitudinal polarization.

In order to get a more quantitative understanding, we have fitted these data using a Lorentzian function convoluted with the instrumental resolution. Where the peak was not single anymore, a fit with multiple modes has been done. In Fig. 4 we report the longitudinal and transverse acoustic dispersion obtained in this way for the nanocomposites, compared with the ones of amorphous silicon obtained by Damart et al., using the same simulation technique as well as analysis method 13 . Where the peak becomes an envelope, only the center of mass of the envelope has been reported. Here it can be seen that the acoustic dispersion is not affected by the presence of a nanocrystalline inclusion when there is no elastic contrast, in agreement with what previ- c)-(d) represent the longitudinal and transverse S(q,ω) calculated in the system without elastic contrast and with elastic contrast respectively. The intensity has been multiplied by the square of the phonon energy position for enhancing the high q modes. Different colors correspond to different wave-vectors: q = 0.1 (blue), 0.2 (black), 0.3 (orange), 0.4 (purple), 0.5 (green), 0.6 (red), 0.7 (violet) Å -1 . The breaking down of the phonon in an envelope of modes and loss of total intensity clearly appear with the presence of an elastic contrast.

Fig. 4 (a) and (b) report the phonons acoustic dispersion respectively for the longitudinal and transverse vibrations, up to the largest q at which an envelope was still visible for the systems χ = 1 (blue filled circles) and χ = 4.6 (red filled triangles). Data from Damart et al. 13 for the fully amorphous system are also reported as black squares. The vertical dotted line separates the low q region, where a single mode exists, from the high q region, where the mode breaks in an envelope of modes. In the single mode region, the symbols correspond to the phonon energy and their error bar to its broadening, inversely proportional to its lifetime. In the envelope-region, the symbols correspond to the envelope center of mass, and the error bar to its broadening, i.e. to the width of the distribution of modes in which the phonon breaks down.

ously reported and with our results on the DOS. Interestingly, in the χ = 4.6 sample, despite the different velocities of the modes within the envelope, the center of mass closely follows the acoustic dispersion in the pure amorphous, indicating that the dynamics is globally dominated by the matrix at these wavelengths. This result is at odds with a prediction based on the Reuss approximation for the homogenized calculation of the moduli (see Ref. 35,[START_REF] Torquato | Random heterogeneous materials: microstructure and macroscopic properties[END_REF] for the definition), which works well for macroscopic composites with spherical inclusions. This approximation leads in the long wavelength limit to an effective longitudinal or shear modulus M e f f = ( x M c + 1-x M a ) -1 , with subscript a or c for amorphous and crystalline phase, and x the crystalline fraction. Calling v a L/T the longitudinal/transverse velocity of the amorphous matrix, for χ = 4.6 and x=30%, we should find v L = 1.11v a L and v T = 1.14v a T . Such increase is obviously not reproduced by the center of mass of the envelop, pointing thus to a failure of the effective medium approach for the description of the dynamic elastic properties in our nanocomposites at lengthscales smaller than ≈ 6 nm.

In Fig. 4 we report as error bars the phonon broadening when a single peak exists, and the envelope full width at half maximum otherwise. From these results, we can safely conclude that the presence of a harder nanoinclusion hinders the phonon propagation, the phonon breaking down in an envelope of modes getting broader and broader as the average energy and wavevector increase.

It is worth reminding here that in pure amorphous silicon 16 , the Ioffe-Regel limit is located at 4.5 THz and 12.7 THz for transverse and longitudinal waves respectively, for an exchanged momentum of 0.73 and 1.0 Å -1 . The vibrational modes here reported lie thus all in the propagative regime, suggesting that the presence of hard nanoinclusions induces an enhanced diffusive character of phonons, with the anticipation of the Ioffe-Regel criterion validity to lower energies. Interestingly, the breaking down of the longitudinal phonon shown in Fig. 4(a), while starting at q ∼ 0.2 Å -1 , takes clearly place at q = 0.3 Å -1 , i.e. at a wavelength of ≈ 21 Å , comparable with the inter-particle distance.

Wave-packets propagation

In order to better understand and confirm the possible change in the nature of the low energy vibrational modes induced by the presence of a harder inclusion, we have followed the propagation of longitudinal and transverse wave-packets generated in the center of the sample, as explained in section 2. Our results are directly comparable with the ones reported in 27 , for a fully amorphous silicon system. Following Ref. 27 , we calculate the kinetic energy density as a function of space and time, averaged over an atomic plane perpendicular to the wave-packet propagation, of dimensions L 2 , where L is the simulation box side:

E L/T (ω, x,t) = 1 2L 2 ∑ i | ui L/T (t)| 2 δ (x -x i ); (6)
where the sum runs over all the atoms i with longitudinal or transverse velocity ui Fig. 5 The Longitudinal (a,b) and Transverse (c,d) kinetic energy envelope, as calculated in Eq.7, is reported for the system with no elastic contrast (χ = 1) on the left, and with elastic contrast (χ = 4.6) on the right, for different frequencies. Each fifth curve is reported as a thicker black line, and its frequency is indicated (THz). Each n-th curve is multiplied by 0.9 n to make an offset along the y-axis for a better visualisation.

diffusive nature of the wave-packet of that frequency. A very convenient way for visualizing it consists in calculating the kinetic energy envelope, by taking the maxima of the kinetic energy density at all times, thus getting rid of the time dependence:

P L/T (ω, x) = max t [E L/T (ω,t, x)]; (7)
Depending on the regime (propagative or diffusive), the envelope has different spacial dependencies, ranging from an exponential dependence similar to the Beer-Lambert law for a propagative wave-packet [START_REF] Swinehart | [END_REF] to an hyperbolic dependence for a diffusive wave-packet:

P prop L/T = ε 0 πv 2 τ 2 exc e -x l ; (8)
P di f f L/T = ε 0 √ 2πe 1 x ; (9)
For the derivation of these dependencies, we address the reader to Ref. 27 . Here we just say that ε 0 represents the total kinetic energy delivered to the system per unit surface from the excitation force, τ exc is the excitation duration time, and v is the sound velocity with which the wave-packet moves in the propagative regime.

Figure 5 reports the envelopes for longitudinal and transverse polarization respectively, and for the two systems, for different frequencies ω. It is worth underlying that the curves should be considered only for x ≥ v L/T τ exc , due to the finite excitation time τ exc , i.e. x ≥ 2.8 nm and x ≥ 1.4 nm for the longitudinal and transverse polarizations respectively 13 . We have chosen here a representation in a logarithmic scale, where the Beer-Lambert law observed in the propagative regime is a straight line.

From the figure, a perfectly linear -and thus propagative -behavior is not easily found, because of the presence, in both systems, of periodic oscillations in the envelope, more evident in the case of a strong elastic contrast. These bumps present a period of 6 nm = L b i.e. the distance between inclusions. They can be associated to the enhancement of the local density of states in correspondence to the crystalline inclusions. Indeed, the oscillations are almost fully removed if we normalize the energy envelope by the spatially resolved density of states (see Supplementary Material), as reported in Figure 6. Only at low frequency and high elastic contrast, the oscillation remains after normalization, with a maximum energy located now inside the inclusions. The propagation of energy is thus inhomogeneous at low frequency, due to the different density of states in the matrix and in the inclusions, and an additional slight pinning of the normalized vibrational energy within the inclusions.

In Fig. 6 a red thick line identifies the x position at which the envelope changes from an exponential to a 1/x behavior, marking thus the traveled distance after which the wave-packet looses its propagative character and turns to a diffusive motion. Comparing with fully amorphous Silicon 27 , we find that already in the system with no contrast, χ = 1, the diffusive regime fully replaces the propagative one much earlier, pointing to a Ioffe-Regel cross-over located at 3 THz for the transverse mode and 4.2 THz for the Fig. 7 Thermal conductivity calculated with the Green Kubo method in the amorphous system and the nanocomposites as a function of the crystalline volume fraction φ at 300 K (a) and 10 K (b). Amorphous: green square, χ = 1: blue circles, χ = 4.6: red triangles. In (a) the EMT prediction is reported as a dashed line. In (b) we report also the thermal conductivity calculated starting from microscopic dynamics for the amorphous (green filled star) and the nanocomposites with φ = 30% and χ = 1 (blue filled star) and χ = 4.6 (red empty star).

longitudinal one, against 4.5 and 12.7 THz in the fully amorphous system.

Looking now to the χ = 4.6 system, the crossover is less distinguishable, so the transition line is much shorter. At low frequency the energy oscillations dominate the behavior: here the wave-packet is strongly scattered by the inclusions, which gives rise to an energy localization between them. It is important to specify that this is not a localization in the sense of the Anderson definition: by localization here we merely mean that there is a majority of kinetic energy between the inclusions and a minority within them, due to the lower density of vibrational states inside the hard inclusions at low frequencies. The propagative to diffusive regime crossover is now definitely anticipated, the diffusive regime taking it over at 2.2 and 3 THz for transverse and longitudinal wave-packets respectively. The phase space for propagative modes is thus reduced by a factor of 2 for transverse vibrations and 4 for the longitudinal ones, suggesting a strong thermal conductivity reduction in this system. This would indeed be the case if most of the heat were carried by the propagative acoustic modes, as in crystalline materials.

Thermal conductivity

In order to understand the effects of an anticipated Ioffe-Regel crossover on thermal transport, we have calculated the thermal conductivity at room temperature for both our nanocomposites as a function of the inclusion volume fraction, φ = V i L 3 , with V i the total volume filled by the inclusions and L 3 the simulation box size.

Results are reported in Fig. 7. First of all, we notice that the value obtained for the fully amorphous sample at 300 K, k am T = 1.7 ± 0.2 W/mK, is in good agreement with literature 38,39 for such system sizes 40 . As for the composites, two points are here to be commented: the volume fraction and the elastic contrast dependence. At 300 K, the thermal conductivity in the nanocomposites increases with the crystalline volume fraction, which can be expected due to the introduction of larger and larger well-conducting crystalline regions when the interface thermal resistance (Kapitza resitance) is negligible. The behaviour can be perfectly reproduced with an effective medium model, assuming for the fully nano-crystalline sample a thermal conductivity of 4 W/mK, as found in Ref 41 for a grain size of 5 nm, and for the amorphous phase k T = 1.5 W/mK. Our result is thus quite different from the case of nanocrystalline inclusions of GaN embedded in amorphous SiO 2 , where an enhanced percolation was found 15 . More intriguing is the elastic contrast dependence: surprisingly, we don't see any effect of the elastic contrast within our error bars, although we have shown that the vibrational dynamics much depends on it. A possible reason for this discrepancy is that our findings on the microscopic dynamics have been obtained at very low temperature. It is thus possible that the situation is different at high temperature. In order to understand thermal transport in such composites, we have thus calculated the thermal conductivity at 10 K, so that it is directly related to the observed microscopic dynamics. Results are also reported in Fig. 7. Here we see that now the thermal conductivity does not depend anymore on the crystalline volume fraction. Such a different behavior with respect to 300 K can be understood in terms of a temperature dependent equilibrium between the interface scattering process, which decreases the total thermal conductivity, and increases with the interface area, the addition of more conductive regions, which increases with the crystalline volume, and the anharmonic processes, which, if negligible at 10 K, are clearly important at 300 K. Indeed, the interface Kapitza resistance is a decreasing function of temperature 31 . Its larger importance at low temperature can be identified as the reason of the lack of a crystalline volume dependence of the thermal conductivity, due to a lucky compensation between interface scattering and the growing crystalline contribution. At high temperature, the latter seems to dominate on the former. Concerning the anharmonic contribution, one quick comparison of the thermal conductivity values at 10 and 300 K for a given crystalline volume fraction indicates that the thermal conductivity increases with temperature, which is not expected when anharmonic scattering dominates phonon lifetime. To clarify the role of anharmonicity, a thorough investigation of the thermal conductivity as a function of both temperature and elastic contrast in both amorphous and crystalline components of the nanocomposite is needed. This is however beyond the scope of this work, and makes the object of an on-going investigation. More intriguing is the fact that, at low temperature as well, the elastic contrast does not play any role, meaning that the strong reduction of the propagative regime with the elastic contrast does not reflect into a lower thermal conductivity. This could be understood in terms of a dominant role of diffusons in thermal transport already in the fully amorphous sample 39 , so that the reduction of the propagative regime would be negligible.

In presence of a diffusive transport mechanism, i.e. a transport assured by largely scattered vibrational modes (namely diffusons), the thermal conductivity can be calculated knowing the phonon diffusivity, which we report in Fig. 8 for the two systems with and without elastic contrast. First, we can see here that the high energy modes that appear for ω ≥ 20 THz have a negligible diffusivity, due to their localized nature 18,20,27 , thus do not contribute to thermal transport. Second, for the other modes, the elastic contrast introduces only tiny differences, mostly reducing the diffusivity of modes around 10 THz, which, in the amorphous, comes mainly from longitudinal modes 27 . Such a weak effect may thus explain why our simulated thermal conductivity is quite insensitive to the elastic contrast.

The situation would be different if diffusons were not the dominant heat carriers. To check this point, we have calculated the thermal conductivity of a crystal/crystal nanocomposite, where the crystalline inclusions, of same radius and volume content, are now embedded in a crystalline matrix, using the same potential and rigidity contrast as before. The nanoinclusions have been tilted by 45 • with respect to the crystalline matrix, for maximizing the interface effect. Indeed, the resulting thermal conductivity will depend on the transmission probability of phonons from the matrix to the inclusion and viceversa, which depends on the superposition of the DOS projected along the crystallographic directions of the contact surfaces 42,[START_REF] Eisenmenger | Phonon Scattering in Condensed Matter V[END_REF] . In such systems we find k T = 3.55 ± 0.15 W/mK and k T = 2.2 ± 0.4 W/mK at 300 K, for χ = 1 and χ = 4.6 respectively. We thus see a significant dependence on the elastic contrast, with a decrease by 40% of the thermal conductivity at room temperature in presence of a harder inclusion, compatible with a perturbation of the propagative acoustic modes.

To go further, and better link macroscopic and microscopic information, we have used our phonon dispersions, mean free path and diffusivity results for directly calculating the thermal conductivity. Two different regimes are considered: the propagative regime only below the Ioffe-Regel energy, and the diffusive regime, corresponding to a multiple scattering of plane waves, which remains the only one above the Ioffe-Regel energy. In the propagative regime, the thermal conductivity can be calculated as

k prop T = 1 3 ω IR 0 C(T, ω)v(ω)l(ω)g(ω) dω (10)
Here v(ω) is the group velocity calculated from the simulated phonon dispersions fitted to a sinusoidal q dependence, in the energy region where an individual phonon mode is well defined, C(T, ω) is the quantum heat capacity, calculated from phonon dispersions and density of states, l(ω) is the mean-free path obtained by the Beer-Lambert fit of the kinetic energy envelope in the wave-packet propagation study (see Eqq. 7 and 8), reported in the Supplementary Material, and g(ω) the simulated density of vibrational states. In this calculation, we have taken a lower energy limit for the integral of 0.5 THz, for convergence issues, which implies that the propagative contribution is slightly underestimated. Such approximation remains reasonable as the neglected modes, despite a very long mean free path, have low heat capacity and density of states. In the diffusive regime, the thermal conductivity expression becomes:

k di f f T = ω max ω IR C(T, ω)D(ω)g(ω) dω (11)
with D(ω) the diffusivity of vibrational modes with energy ω. Eq.10 has been calculated as the sum of the longitudinal and transverse contributions, each having a different Ioffe-Regel limit and different mean free path. To this aim, we have calculated the projected longitudinal and transverse density of states using Voronoi cells (see Supplementary Material). The total thermal conductivity is then k

T = k prop T + k di f f T .
In this calculation, assuming that the S(q, ω) does not change with temperature, the temperature dependence of the calculated k T only results from the one of the Bose factor, i.e. from the increased phonon population. As such, the thermal conductivity calculated this way at high temperature, when all phonons are thermally populated, corresponds to the thermal conductivity as calculated with the Green-Kubo method, where all phonons are always populated. We can then directly compare the Green-Kubo results obtained from an atomic configuration stabilized at 10 K, with the results of our calculation from the same atomic configuration, but taken at high temperature for having the same classical phonon population. This comparison is reported in Fig. 7, and shows a very good agreement for amorphous and nanocomposites with a volume fraction of 30%. We can clearly see the effect of the elastic contrast on the calculated thermal conductivity, however it is smaller than the error bars of the Green-Kubo calculations, explaining thus the apparent insensitivity of these latter.

To get more insight on the effect of the elastic contrast on the two different thermal transport regimes, we report in Fig. 9 the two contributions as a function of temperature. The high temperature saturation of the propagative contribution is clearly due to the absence of contributing modes above Ioffe-Regel: the saturation temperature corresponds to T IR ≈ hω IR /k B and is clearly reduced in nanocomposites with respect to the amorphous. It is evident that nanostructuration causes a reduction of the propagative part, stronger in presence of a harder inclusion. As for the diffusive part, this is enhanced by the presence of the nanoinclusions, with almost no effect of the elastic contrast. Such enhancement is due to a major interface scattering and the increased number of diffusive modes, as indicated by the drastic reduction of the Ioffe-Regel crossover. At the end, due to the reduction of the propagative phase space, if at low temperature propagons dominate heat transport, at room temperature they only account for about 10% of it in the harder nanocomposite, against 30% for the amorphous sample (see inset of the figure).

Conclusions

In this work we have reported a microscopic investigation of phonon dynamics in amorphous/nanocrystalline composites, with the aim of evidencing and understanding the role of elastic contrast between matrix and nanoinclusions. As cited above, the elastic contrast is expected to play a major role in determining thermal transport properties in such nanocomposites. Still, contradictory experimental results have been reported.

Here we have shown that the presence of a crystalline nanoinclusion modifies phonon dynamics even in absence of an elastic contrast. Still, the effect is much stronger when the nanoinclusion is harder than the matrix. The presence of a disorder/order interface, coupled with the elastic heterogeneity, is at the origin of a modification in acoustic wave-packets, which loose their propagative character, leading to a significant anticipation of the Ioffe-Regel crossover between a propagative and a diffusive regime. Such anticipation translates in several modifications of the microscopic dynamics, from the breaking down of the individual phonon in an envelope of modes, to the rising up of a low energy peak in the reduced phonon density of states, at energies below the Boson Peak. The inspection of the time evolution of longitudinal and transverse wave-packets generated in the sample clearly confirms such anticipation and the arising of a strongly inhomogeneous energy propagation in presence of elastic contrast, with the pinning of vibrational energy within the harder inclusions. An anticipated Ioffe-Regel crossover means that the transition from a ballistic to a diffusive thermal transport takes place at lower frequencies, reducing the phase space for propagons to about half the one in the fully amorphous system for the high contrast nanocomposite. Despite such a strong effect on acoustic modes, the thermal conductivity, as calculated with the Green-Kubo formula, is almost unaffected. This is essentially due to the major diffusive nature of thermal transport already in the amorphous matrix when all phonons are thermally populated. The dramatic reduction of the propagative contribution affects thus only partially the total thermal conductivity, being balanced by the increased diffusive contribution in the nanocomposites, which, moreover, is quite insensitive to the elastic contrast. Interestingly, the interface scattering dominates thermal conductivity only for the atomic configuration equilibrated at low temperature, where it counterbalances the larger crystalline contribution. At room temperature however, this is not true anymore: the higher conductivity in the crystalline nanoinclusions takes it over the interface scattering, whose strength decreases indeed with temperature, assuring a gradual increase of the thermal conductivity with crystalline volume fraction, resulting in a behavior in agreement with effective medium predictions.

Our findings, obtained on a model system, allow to clarify the panoply of experimental and simulation results on thermal conductivity in systems with and without elastic contrast. We find indeed that this latter is a key ingredient, dramatically perturbing phonon dynamics, and reducing the propagative contribution to thermal transport. Such reduction is due to the strong phonon scattering introduced with the nanostructuration, which leads to two consequences: the reduction of the phonon mean free path, and the anticipation of the Ioffe-Regel crossover, reducing thus the phase space of propagative phonons and then the number of modes available for a propagative thermal transport. However, the effect of such modifications of the microscopic dynamics on the macroscopic thermal conductivity is not straightforward. The result will depend on the relative weight of propagative and diffusive contribution to thermal transport in the matrix material. If propagons are the main heat carriers, a strong reduction of the thermal conductivity can be expected. If diffusive transport dominates already in the matrix, then no reduction is expected, and the thermal conductivity could even increase, as the diffusive contribution increases. Depending on this crucial point -which will depend on the material and on temperature -the sensitivity of thermal conductivity to elastic contrast will strongly change as well. Our findings clearly indicate that an amorphous matrix where propagons carry most of the heat at room temperature would be better suited for enhancing the interface scattering and elastic contrast effect and ultimately engineering nanocomposites with even lower thermal conductivity than the matrix. Our study illustrates as well the relevance of coupling a microscopic dynamics investigation to thermal conductivity simulations to get a real insight onto the mechanisms at play and the key parameters for thermal transport in heterogeneous systems.

Fig. 1

 1 Fig. 1 Atomistic model of a nanocomposite made of an amorphous matrix (blue) and crystalline inclusions (green) with radius R = 25 Å and distance between inclusions L b = 60 Å (L = 2L b = 120 Å ). The excitation of the wave packet is imposed along the red layer.

Fig. 2

 2 Fig. 2 Panel (a): vibrational density of states for the nanocomposite with χ = 1 (thick blue line), χ = 4.6 (thin red line) with 30% crystalline volume fraction, and for a fully amorphous sample from Ref 16 (dotted black line). While in case of no contrast the DOS is only weakly affected by the presence of the inclusion, the harder inclusion introduces high energy vibrational modes. Panel (b): reduced density of states, normalized by the Debye behavior (ω 2 ) to evidence the Boson Peak at low energy. Same colors as in panel (a) for the three systems. The presence of the inclusion causes the growing up of a peak at about 1.7 THz.

Fig. 3

 3 Fig. 3 Panels (a)-(b) and (c)-(d) represent the longitudinal and transverse S(q,ω) calculated in the system without elastic contrast and with elastic contrast respectively. The intensity has been multiplied by the square of the phonon energy position for enhancing the high q modes. Different colors correspond to different wave-vectors: q = 0.1 (blue), 0.2 (black), 0.3 (orange), 0.4 (purple), 0.5 (green), 0.6 (red), 0.7 (violet) Å -1 . The breaking down of the phonon in an envelope of modes and loss of total intensity clearly appear with the presence of an elastic contrast.

  L/T . The space and time behavior of E L/T for a given frequency ω allows to understand the propagative or o u r n a l N a me , [ y e a r ] , [ v o l . ] , 1-10 | 5

Fig. 6

 6 Fig. 6 Same as in Fig. 5, but after normalization by the local density of states. Thick red (grey) lines show the transition from a propagative to a diffusive regime.

Fig. 8

 8 Fig.8The phonon diffusivity is reported between 0 and 20 THz for the χ = 1 system (thick blue line) and the χ = 4.6 system (thin red line), compared with a pure amorphous sample (black dotted line).The high frequency modes in the χ = 4.6 composite, with ω ≥ 20 THz have negligible diffusivity.

Fig. 9

 9 Fig. 9 Propagative and Diffusive contributions to the thermal conductivity calculated using Eqs. 10 and 11 as a function of temperature for the amorphous system (black dashed line), and two composite systems with χ = 1 (blue solid line) and χ = 4.6 (red dot-dashed line) at a volume fraction φ = 30%. Inset: Relative weight of the propagative contribution, same colors and line styles as the main figure.

[START_REF] Lawrence | Livermore National Laboratory, Data based on DOE/EIA MER[END_REF][START_REF] Viskadourakis | [END_REF][3][4][5][6][7][8][9][10]

J o u r n a l N a me , [ y e a r ] , [ v o l . ] ,

1-10 J o u r n a l N a me , [ y e a r ] , [ v o l . ] ,

Acknowledgement

The authors are grateful to Région Rhône-Alpes who has funded this work through the ARC Energie PhD funding program. V. M. G. thanks the Lyon IDEX Scientific Breakthrough program for funding of the project IPPON. The work of Y.M.B. was supported by the Russian Federation President Grant no. MK-3052.2019.2 and the 2018 Metchnikov fellowship of the French Embassy in Russia.

Conflicts of interest

There are no conflicts to declare.


Elodie Despouy
email: elodiedespouy@cnrs.fr

Jonathan Curot

Leila Reddy

Lionel G Nowak

Martin Deudon

Jean-Christophe Sol

Jean-Albert Lotterie

Marie Denuelle

Ali Maziz

Christian Bergaud

Simon J Thorpe

Luc Valton

Emmanuel J Barbeau

Recording local field potential and neuronal activity with tetrodes in epileptic patients

Keywords: Hybrid electrode, depth electrode, tetrode, intracranial EEG, single unit, epilepsy ACSF: artificial cerebrospinal fluid, EIS: electrochemical impedance spectroscopy, MRI: magnetic resonance imaging, RMS: root mean square, SEEG: stereo-electroencephalography

Background: Recordings with tetrodes have proven to be more effective in isolating single neuron spiking activity than with single microwires. However, tetrodes have never been used in humans. We report on the characteristics, safety, compatibility with clinical intracranial recordings in epileptic patients, and performance, of a new type of hybrid electrode equipped with tetrodes.

New Method: 240 standard clinical macroelectrodes and 102 hybrid electrodes were implanted in 28 patients. Hybrids (diameter 800 µm) are made of 6 or 9 macro-contacts and 2 or 3 tetrodes (diameter 70-80 µm).

Results: No clinical complication or adverse event was associated with the hybrids. Impedance and noise of recordings were stable over time. The design enabled multiscale spatial analyses that revealed physiopathological events which were sometimes specific to one tetrode, but could not be recorded on the macro-contacts. After spike sorting, the single-unit yield was similar to other hybrid electrodes and was sometimes as high as > 10 neurons per tetrode.

Comparison with Existing Method(s): This new hybrid electrode has a smaller diameter than other available hybrid electrodes. It provides novel spatial information due to the configuration of the tetrodes. The single-unit yield appears promising.

Conclusions: This new hybrid electrode is safe, easy to use, and works satisfactorily for conducting multi-scale seizure and physiological analyses.

HIGHLIGHTS

• First in vivo tetrode recordings in humans

• No clinical complications or adverse events • On average 2.08 ± 2.86 single units per tetrode (sometimes >10)

INTRODUCTION

About 30% to 40% of epileptic patients suffer from drug refractory epilepsy [START_REF] Kwan | Drug-Resistant Epilepsy[END_REF][START_REF] Laxer | The consequences of refractory epilepsy and its treatment[END_REF]Dalic and Cook, 2016). Epilepsy surgery in these patients provides a good rate of seizure control, with 58-73% of patients seizure-free two years after surgery [START_REF] Capraz | Surgical outcome in patients with MRI-negative, PET-positive temporal lobe epilepsy[END_REF]. Implantation of subdural and/or depth electrodes is often required prior to epilepsy surgery to delineate the seizure onset zone [START_REF] Behrens | Subdural and depth electrodes in the presurgical evaluation of epilepsy[END_REF][START_REF] Yang | Intracranial electroencephalography with subdural and/or depth electrodes in children with epilepsy: Techniques, complications, and outcomes[END_REF][START_REF] Podkorytova | Stereo-Encephalography Versus Subdural Electrodes for Seizure Localization[END_REF][START_REF] Isnard | French guidelines on stereoelectroencephalography (SEEG)[END_REF].

Hybrid depth electrodes that combine both macro-contacts and microwires have been used in the investigation of drug-refractory epilepsy for approximately 45 years [START_REF] Babb | Analysis of extracellular firing patterns of deep temporal lobe structures in man[END_REF][START_REF] Fried | Cerebral microdialysis combined with single-neuron and electroencephalographic recording in neurosurgical patients: Technical note[END_REF], although this has only been in a few centers worldwide. They provide a unique opportunity to study neuronal activity in the human brain for a better understanding of the mechanisms of epileptogenesis [START_REF] Keller | Heterogeneous neuronal firing patterns during interictal epileptiform discharges in the human cortex[END_REF][START_REF] Alvarado-Rojas | Single-unit activities during epileptic discharges in the human hippocampal formation[END_REF][START_REF] Lambrecq | Single-unit activities during the transition to seizures in deep mesial structures: Seizures and Single-Unit Activities[END_REF][START_REF] Schevon | Multiscale recordings reveal the dynamic spatial structure of human seizures[END_REF], seizure propagation [START_REF] Eissa | Cross-scale effects of neural interactions during human neocortical seizure activity[END_REF] and seizure prediction [START_REF] Bower | Changes in granule cell firing rates precede locally recorded spontaneous seizures by minutes in an animal model of temporal lobe epilepsy[END_REF][START_REF] Park | Early detection of human focal seizures based on cortical multiunit activity[END_REF]Misra et al., 2017), and of the neurobiological basis of cognition [START_REF] Fried | Single Neuron Activity in Human Hippocampus and Amygdala during Recognition of Faces and Objects[END_REF][START_REF] Kreiman | Single-neuron correlates of subjective vision in the human medial temporal lobe[END_REF][START_REF] Quiroga | Invariant visual representation by single neurons in the human brain[END_REF][START_REF] Mormann | Latency and Selectivity of Single Neurons Indicate Hierarchical Processing in the Human Medial Temporal Lobe[END_REF][START_REF] Rutishauser | Human memory strength is predicted by theta-frequency phase-locking of single neurons[END_REF][START_REF] Reddy | Learning of anticipatory responses in single neurons of the human medial temporal lobe[END_REF][START_REF] Staresina | Recollection in the human hippocampal-entorhinal cell circuitry[END_REF]. They are also more effective in detecting pathological activities such as high-frequency oscillations [START_REF] Zijlmans | How to record high-frequency oscillations in epilepsy: A practical guideline[END_REF][START_REF] Despouy | Neuronal spiking activity highlights a gradient of epileptogenicity in human tuberous sclerosis lesions[END_REF] and microseizures [START_REF] Schevon | J Microphysiology of Epileptiform Activity in Human Neocortex[END_REF][START_REF] Stead | Microseizures and the spatiotemporal scales of human partial epilepsy[END_REF]. They are as safe and effective as standard clinical macroelectrodes for intracranial monitoring [START_REF] Hefft | Safety of Hybrid Electrodes for Single-Neuron Recordings in Humans[END_REF][START_REF] Carlson | Safety and utility of hybrid depth electrodes for seizure localization and single-unit neuronal recording[END_REF].

Since the end of the 1990s, the most commonly used hybrid electrode has been a bundle of microwires (up to 9, each 40 µm in diameter) inserted inside the shaft of the electrode and which exit at the tip of the macroelectrode [START_REF] Babb | Analysis of extracellular firing patterns of deep temporal lobe structures in man[END_REF][START_REF] Fried | Cerebral microdialysis combined with single-neuron and electroencephalographic recording in neurosurgical patients: Technical note[END_REF]. Despite their success, such hybrid electrodes suffer from several drawbacks: 1-the diameter of the macroelectrode (1,300 µm) is large compared to other standard clinical macroelectrodes used for intracranial monitoring (800 µm). 2-there is an imperfect spatial correspondence between the macrocontact and microelectrode recording sites due to the extension of the microwires by approximately 5 mm off the tip of the macroelectrode, i.e. macro-and microelectrodes may not record the same neuronal populations; 3-the length of this extension limits the number of brain regions in which these hybrid electrodes can be implanted; 4-It is difficult to localize the microelectrodes as they spread out of the shaft like an umbrella, i.e. the spatial relationship between the different microwires is unknown which might present a difficulty during spike sorting.

The use of tetrodes in animal studies has led to a major improvement in single-unit isolation [START_REF] Buzsáki | Large-scale recording of neuronal ensembles[END_REF]. There are two independent challenges: (1) some areas have high cell densities in which cell bodies are closely packed together and cells can fire in a complex spike pattern; (2) during burst firing, the height of the action potential can vary in amplitude by as much as 50% [START_REF] Kandel | Electrophysiology of hippocampal neurons: II. After-potentials and repetitive firing[END_REF][START_REF] Ranck | Studies on Single Neurons in Dorsal Hippocampal Formation and Septum in Unrestrained Rats[END_REF]. The use of a stereotrode (two microwires twisted together) improves the detection of single units compared to single wire recording: cells with different ratios of distances from the two electrode tips have different spike amplitude ratios when recorded on two channels [START_REF] Mcnaughton | The stereotrode: A new technique for simultaneous isolation of several single units in the central nervous system from multiple unit records[END_REF]. According to the same logic, tetrode configurations (four microwires bundled together) have been shown to further improve the yield of single units at single recording sites [START_REF] O'keefe | Phase relationship between hippocampal place units and the EEG theta rhythm[END_REF][START_REF] Wilson | Dynamics of the hippocampal ensemble code for space[END_REF][START_REF] Gray | Tetrodes markedly improve the reliability and yield of multiple single-unit isolation from multi-unit recordings in cat striate cortex[END_REF][START_REF] Buzsáki | Large-scale recording of neuronal ensembles[END_REF]. Therefore, up to nine neurons at a time could be recorded from a single tetrode [START_REF] Gray | Tetrodes markedly improve the reliability and yield of multiple single-unit isolation from multi-unit recordings in cat striate cortex[END_REF], although in principle, many more could be recorded [START_REF] Henze | Intracellular Features Predicted by Extracellular Recordings in the Hippocampus In Vivo[END_REF]. Tetrode configuration has been highly valuable to record the neuronal dynamics of interneurons and principal cells during ictogenesis in in vitro analyses of animal brain slices [START_REF] Lévesque | Interneurons spark seizure-like activity in the entorhinal cortex[END_REF][START_REF] Lévesque | Dynamic interneuron-principal cell interplay leads to a specific pattern of in vitro ictogenesis[END_REF].

In this study, we describe a new hybrid electrode which has been under clinical investigation in our epilepsy center since 2015. It contains two or three tetrodes inserted in a macroelectrode of the same diameter as standard clinical macroelectrodes (800 µm). Once the hybrid electrode is inserted in the brain, the tetrodes can be extended up to 2 mm with a micrometer screw. Because the tetrodes extend between two macrocontacts, they allow a true multiscale recording of the same brain region. To our knowledge, this is the first time that tetrodes have been used in a clinical trial in humans. Therefore, the effectiveness in humans was unknown. The purpose of this study was to report on the characteristics of this new hybrid electrode, its safety and its ability to record EEG signals and neuronal activity at several spatial scales simultaneously. Our results suggest that tetrodes may be as useful in humans as in animal studies, both from a clinical and an electrophysiological perspective.

METHODS

Patients

The patients included in this study suffered from drug-resistant epilepsy. They were included between 2015 and 2018. They had a detailed pre-surgical evaluation including medical history and neurological examination, neuropsychological testing, cerebral MRI, and long-term video-EEG monitoring. All patients had functional imagery such as 18-FDG positron emission tomography and some patients had ictal single photon emission computer tomography. Due to the fact that noninvasive assessments failed to localize the seizure onset zone with certainty, patients underwent stereo-electroencephalography (SEEG) on the Toulouse epilepsy unit so that the seizure onset zone could be more precisely delineated. SEEG was performed as part of the patients' regular clinical care. Intracranial EEG electrodes were implanted stereotaxically according to Talairach's method [START_REF] Talairach | Stereotaxic exploration in frontal epilepsy[END_REF]. Each patient received detailed information concerning the objectives of the SEEG technique before intracerebral electrode implantation. They were also given the details and informed about the use of these new hybrid electrodes, after which they agreed to participate in the study (ClinicalTrials Identifier: NCT02491476). Implantation of the hybrid electrodes and the use of the data were approved by the local ethics committee and the French Drug and Health Product Safety Agency (CPP Sud-Ouest et Outre-Mer I, no.1-14-23 and ANSM 2014-A00747-40).

Electrodes

Up to 12 semi-rigid multi-contact standard clinical depth macroelectrodes and up to 4 new hybrid electrodes were implanted in each patient. The macroelectrodes (Microdeep, DIXI Medical, France) were 0.8 mm in diameter and contained 5-18 contacts (platinum/iridium).

The contacts were 2 mm long and were regularly spaced 1.5 mm apart.

The hybrid electrodes (DIXI Medical, France) consisted of a standard macroelectrode (diameter: 0.8 mm) equipped with two or three tetrodes that protruded up to 2 mm from the shaft between the first and second most distal macrocontacts (Fig. 1). A micrometer screw was used to extend the Tetrodes from the electrode shaft after implantation. The tetrodes protruded from the shaft at a 30° angle. Each tetrode was made of 4 tungsten wires, 20 µm in diameter. The theoretical surface of each microcontact was 6.28x10 -4 mm 2 . Each tetrode was 70-80 µm in diameter. For hybrid electrodes equipped with two tetrodes, tetrodes were at a 180° angle in relation to each other. They also had nine 2 mm-long macrocontacts regularly spaced at intervals along the shaft according to the overall length of the electrode. Three models with 2 tetrodes were available in a total exploration length / space between contacts (in mm) of: 33.2/1.9; 40.4/2.8; 50.8/4.1. The length of the electrodes was chosen according to the depth of the cerebral target. For hybrid electrodes equipped with three tetrodes, the tetrodes were at a 120° angle in relation to each other. They had six 2 mm-long macrocontacts, three distal and three lateral, all spaced 2 mm apart (3 models were available with exploration lengths of: 34, 42 and 50 mm). These three tetrodes cover a theoretical triangular surface of 2 mm 2 . The three-tetrode configuration was generally chosen when the electrode passed through white matter between the lateral and distal cerebral targets, where it is usually not useful to record EEG signal.

These new hybrid electrodes have been under clinical investigation in our epilepsy center since 2015, with an agreement to implant up to four hybrid electrodes per patient. ensure that the wires have an optimal curvature radius, and therefore, correct functioning of the input/output mechanism. (D) The design of these electrodes enables the multiscale recording of EEG, LFP, single and multi-unit activity. In blue: an interictal epileptiform discharge (i.e. an epileptic spike) recorded by a macrocontact (raw signal). Note the difference in morphology between the interictal epileptic spikes recorded simultaneously on the micro-and macroelectrodes. In orange: the same interictal epileptiform discharge recorded by the four microwires of a tetrode (raw signal) and the corresponding neuronal activity (high pass 500 Hz).

Neurosurgery

Considering that the hybrid electrodes had the same diameter as the clinical electrodes, they did not require any change in the neurosurgical procedure. The rigidity of the clinical and hybrid electrodes in particular, remained the same. After the implantation of each hybrid electrode, a sheath support was placed on the guide screw to ensure that the cables had an optimal radius of curvature, resulting in correct functioning of the extension or retraction of the tetrodes (Fig. 1C). The sheath support slightly increased the safety distance to 5 mm instead of 3 mm between the entry point of each hybrid electrode and the other electrodes. In our center, the tetrodes were generally extended by 2 mm immediately after implantation in the neurosurgery room, or the patient's room, as long as the micrometer screw is not buried under the bandage.

Electrode location

The choice of electrode locations was based solely on pre-SEEG clinical observations and on hypotheses concerning the location of the seizure onset zone based on a non-invasive assessment, i.e. tailored implantation. The electrodes were implanted in brain regions chosen for clinical purposes only.

Recordings

Macrocontact signals were recorded using two SystemPLUS EVOLUTION 64-channel acquisition units (Micromed, France) at a sampling rate of 2,048 Hz (anti-aliasing filter: 926.7

Hz; high-pass filter: 0.15 Hz; low-pass filter: 1,000 Hz). Microelectrode signals were recorded using a 64-channel Cerebus System (Blackrock Microsystems, Salt Lake City, UT, USA) at a sampling rate of 30 kHz (0.3-7.5 kHz bandwidth). Line noise cancellation at 50 Hz was applied.

For patient 22, macroelectrode and microelectrode signals were both recorded using a 256channel Atlas System (Neuralynx, Bozeman, MT, USA). A macrocontact located in white matter was used as a reference for both systems.

Macrocontacts were recorded 24/day. Microelectrodes were recorded for 1 hour in five morning sessions while patients watched episodes of a TV show.

Neuroimagery control

The location of each electrode contact was based on a pre-operative 3D T1-weighted MRI/postoperative CT-scan data fusion. The resolution of this fusion allowed visual verification of the anatomical location of each contact and whether it was located in gray or white matter. This visual inspection was completed by normalization in the MNI anatomical space (Fig. 2).

In order to investigate potential subclinical complications of electrode implantation such as tissue damage or bleeding along the electrode trajectory, a postoperative MRI was performed 3 to 6 months after the end of the SEEG and the explantation of the electrodes (Siemens/Philipps 3 Tesla, T2, FLAIR, and T2 gradient-echo based sequences (T2*)). A neurologist (JC) verified each postoperative MRI for subclinical injuries (i.e. not inducing any symptom) caused by electrode insertion (easily visible as a linear hypersignal on T2 and FLAIR sequences) and subclinical bleeding (hyposignal on T2 gradient echo-based sequences). We distinguished ( 1) clearly visible linear hyperintensities on T2 and FLAIR sequences that follow a part or the entire trajectory of the electrode; (2) hematomas, which correspond to a relatively well circumscribed pool of blood (with a mass effect at the acute phase, but small enough to remain subclinical); (3) "linear bleeding" which generally corresponds to a pseudolinear hyposignal on T2* that follows a part or the entire trajectory of the electrode. To our knowledge, this type of "linear bleeding" is not well described in the literature. It corresponds to the asymptomatic sequelae of electrode paths, which are well known and expected, but which, until now, have mainly been taken into account on T2 or Flair sequences. These sequences show only a linear cavity and gliosis but not possible hemorrhage. The magnetic inhomogeneity of iron and deoxyhemoglobin causes sustainably faster T2* relaxation, leading to signal intensity loss on T2* MRI [START_REF] Chavhan | Principles, Techniques, and Applications of T2*-based MR Imaging and Its Special Applications[END_REF][START_REF] Ko | Characteristic Signs on T2*-Based Imaging and Their Relationship with Results of Reperfusion Therapy for Acute Ischemic Stroke: A Systematic Review and Evidence to Date[END_REF]. Therefore, T2* MRI is highly sensitive in the detection of recent or old cerebral hemorrhages and T2 hyperintense foci most frequently highlight lesions involving white matter [START_REF] Bekiesińska-Figatowska | T2-hyperintense foci on brain MR imaging[END_REF].

Please note that tetrodes could not be accurately localized using brain imagery, neither with a variety of MRI sequences and machines nor with X-ray imaging. This means that we do not know exactly where the tetrodes spread radially although we know the distance between the tetrodes and the adjacent macrocontacts as well as the precise distance between each tetrode (see above and Figure 1A for more information).

Impedance

Electrochemical Impedance spectroscopy (EIS) measurements were taken using a VMP3 Biologic potentiostat coupled with EClab Software. The impedance spectra were recorded by applying a 5 mV root mean square (RMS) for the sine wave with frequencies varying from 1 Hz to 10 kHz. Measurements were taken by immersing tetrodes in artificial cerebrospinal fluid (ACSF) at 37°C for 24 days [START_REF] Lecomte | In vitro and in vivo biostability assessment of chronically-implanted Parylene C neural sensors[END_REF]. Impedances at 1 kHz were recorded every day. Due to recording failures, the impedance of microelectrodes 1, 5 and 7

were not collected respectively on days 0, 1 and 2.

Noise level

We used a MATLAB toolbox (MicMac) developed in our lab, which allows simultaneous and synchronized macro and micro EEG signal visualization, to analyze EEG signals on different scales. We also measured the noise level and the changes in noise level throughout the days of recording. The noise level in an electronic system can be measured as the root mean square (RMS) voltage, which provides an overall measure of the magnitude of the recorded signal.

We computed the RMS voltage on filtered signals (300-3,000 Hz) for each electrode for a 10min period (bin duration: 60 s).

Single-unit isolation

Spike sorting was performed on one-hour daily recordings (maximum of five days per patients). We considered only the electrodes located in gray matter. Each microwire signal was bandpass filtered using a Butterworth filter (300-3,000 Hz) and whitened. We detected action potentials offline on each tetrode. Spike sorting was performed using SpyKING CIRCUS, which relies on a template matching-based algorithm [START_REF] Yger | A spike sorting toolbox for up to thousands of electrodes validated with ground truth recordings in vitro and in vivo[END_REF]). An action potential was detected if the amplitude was above a threshold of six times the median of the absolute deviation of the voltage. We took into account different quality metrics (interspike interval histograms and refractory period violations, scatterplots of the different clusters, action potential amplitude over time, autocorrelogram, crosscorrelogram and density plots) to optimize spike sorting [START_REF] Hill | Quality Metrics to Accompany Spike Sorting of Extracellular Signals[END_REF]. The resulting data were inspected manually and refined using the MATLAB graphical interface SpyKING CIRCUS.

Electrical brain stimulation (EBS)

EBS is a standard clinical procedure that is used to stimulate the brain areas sampled with the intracerebral electrodes in order to assess the local propensity of these brain areas to induce seizures, and to assess functional mapping before neurosurgery. In our center high frequency EBS at 50 Hz or low frequency EBS at 1 Hz (pulse duration: 1 msec, duration of EBS:

from 5 seconds for high frequency EBS to 10-30 seconds for low frequency EBS) is applied in a bipolar manner to each contact in grey matter, with current intensity gradually increasing from 0.5 to 3 mA according to French SEEG guidelines [START_REF] Isnard | French guidelines on stereoelectroencephalography (SEEG)[END_REF]. The question of whether hybrid electrodes are compatible with this standard clinical procedure might be asked. We analyzed the stimulation procedures of all patients included in this study.

RESULTS

A total of 240 clinical macroelectrodes (mean per patients ± SD: 8.57 ± 1.85, range [5][6][7][8][9][10][11][12]) and 102 hybrid electrodes (mean per patient ± SD: 3.6 ± 0.62, range [2-4]) were implanted in 28 patients (median age= 33.5 years old; range [14-63]). Of the 102 hybrid electrodes, 69 had two tetrodes and 33, three tetrodes. The tetrodes were mainly placed in the medial temporal lobes but also in various other brain structures (Fig. 2).

Figure 2 -Hybrid electrode locations in all patients. A total of 102 hybrid electrodes were implanted in 28 patients. The most distal contact of all the hybrid electrodes is depicted in a normalized glass brain after normalization in the MNI anatomical space [START_REF] Xia | BrainNet Viewer: a network visualization tool for human brain connectomics[END_REF]. The tetrodes are adjacent to each of these contacts.

Safety of the new hybrid electrode

The use of the hybrid electrode caused no adverse events or clinical complications such as hemorrhage, infection, or clinical deficit during or after this clinical trial in any of the 28 patients. A routine CT scan was performed several hours after implantation and after the recovery room to verify the absence of acute brain lesion (such as brain hemorrhage, subdural or extradural hematoma or brain edema). No specific lesion was linked to the hybrid electrodes on these CT scans.

In order to investigate subclinical complications caused by the electrodes (i.e., the percentage of hematomas, linear hyperintensities or linear bleeding on the ancient trajectories of the electrodes), fourteen patients (50%) had a T2 and a T2*-weighted MRI scan at least 2 months after the removal of the hybrid electrodes (Fig. 3A). This corresponds to 116

Impedances

Impedance was measured for four tetrodes (12 microelectrodes) immersed in ACSF for 24 days at 37°C (Fig. 4). The median impedance was 103 kΩ (range: 67-179) for the first ten days after which it started to increase slightly until the twentieth day, with a median impedance of 224 kΩ (range: 129-285), (Friedman rank sum test, df = 14, P = 8.91 -16 ). Of note, tetrodes are usually retracted around the 8 th day on our epilepsy unit. Each point represents a single wire.

Noise levels

The noise level was measured on all hybrid electrodes from patients 10 to 28 during the 5 days of recording (the days were not necessarily consecutive because of the week-ends) (Fig. 5). No difference from the days after implantation was observed (median level = 22.25 µV; range: 14.87-52.19 µV; Friedman rank sum test, df = 7, P = 0.1386).

Please note that the noise level may vary considerably from one setting to another as shown in Fig. 6, where comparisons were made between different amplifiers or with the same amplifier but different settings (a clinical room and a Faraday cage).

Figure 5 -Noise level. Change in noise amplitude (RMS voltage, filtered signal 300-3,000 Hz) on the recording days following implantation. Each point represents an electrode and the total number of electrodes is indicated below each day. It is important to note that the choice of recording system and the clinical conditions under which the recordings take place can have a significant impact on the noise level as shown in Fig. 6.

Multiscale events

The fact that the tetrodes extended between two macrocontacts enabled simultaneous multiscale recordings of the same brain region (Fig. 1C). Signals can be recorded on different scales (such as between the adjacent macrocontacts and tetrodes, but also between the tetrodes themselves), and may reveal different information. An example is that interictal epileptiform discharges may only be detected by microelectrodes (Fig. 7A) or have a different temporal and morphological structure between micro-and macroelectrodes (Fig. 1D). Microelectrodes also enabled the detection of additional focal events such as fast ripples which are considered to be new biomarkers of the seizure onset zone [START_REF] Zijlmans | How to record high-frequency oscillations in epilepsy: A practical guideline[END_REF][START_REF] Roehri | High-frequency oscillations are not better biomarkers of epileptogenic tissues than spikes: HFOs Are Not Better Than Spikes[END_REF] (Fig. 7B). It is interesting to note that even if the three tetrodes of an electrode are localized in a confined space, they may not detect the same events, or not at the same latency. For example, a fast ripple may only be seen on one or two tetrodes or sometimes can be delayed on one tetrode in relation to the others (Fig. 8).

Electrical brain stimulation (EBS)

The implantation of the hybrid electrodes did not impact EBS, neither on the two macrocontacts surrounding the tetrodes nor on remote contacts. It was possible to use the tetrodes to record LFP and single-unit activity dynamics induced by EBS, before and after EBS artifacts (corresponding to the train of pulses) (Fig. 9).

Neuronal yield

Patients 1-11 were excluded from the analysis due to interference between the microwire and macrowire acquisition systems that created an artifact at 2,048 Hz (the microelectrodes were recorded on the Blackrock system while the macroelectrodes were recorded in parallel on our Micromed clinical system, (see Methods). This problem was solved when the Micromed amplifiers were run on batteries). Patients 17 and 22 were also excluded because the SEEG ended prematurely and because the signal was acquired with another acquisition system (patient 22).

We analyzed the signal recorded by 30 electrodes (68 tetrodes) in five days. On average, we were able to record 2.08 ± 2.86 single neurons per tetrode per day on the tetrodes that recorded neuronal activity. In most cases, the tetrodes that did not record activity were located in white matter. Single neurons were recorded on 60% of the tetrodes. Neuronal activity could be recorded on 71% of the tetrodes when multi-unit activity was taken into account. We were able to detect a maximum of 13 single neurons with a single tetrode located in the amygdala of patient 14. The number of neurons per tetrode was > 6 neurons in 32% of the tetrodes, and it is of note that most were in the amygdala (Fig. 10). We also assessed whether the neuronal yield depended on the regions under investigation for the regions that were the most often recorded. On average, it was in the amygdala that we recorded the most neurons (3.68, SD=3.20), followed by the entorhinal region (3.45, SD = 3.98). In contrast, we recorded comparatively fewer neurons in the hippocampus (head of the hippocampus: 0.95, SD = 1.61, tail of the hippocampus: 1.75, SD = 2.69). In our study, the global neuronal yield remained fairly constant over the days of recording (up to 8 days post-implantation). However, we have no way of knowing whether it was the same units that were recorded as recordings took place during different sessions rather than continuously.

DISCUSSION

In this study, we present a new hybrid electrode equipped with tetrodes which enables simultaneous analysis of intracerebral activity on macro-and micro-scales in epileptic patients.

In comparison to standard clinical macroelectrodes, the use of new hybrids electrodes caused no particular or additional clinical complications or adverse events. They also did not cause more subclinical brain damage (i.e. hematomas, linear bleeding or more tissue damage at electrode insertion sites) than clinical macroelectrodes. The same conclusions were reached with another type of hybrid electrodes with a bundle of wires, which appeared to be as safe as standard depth electrodes [START_REF] Fried | Cerebral microdialysis combined with single-neuron and electroencephalographic recording in neurosurgical patients: Technical note[END_REF][START_REF] Hefft | Safety of Hybrid Electrodes for Single-Neuron Recordings in Humans[END_REF][START_REF] Carlson | Safety and utility of hybrid depth electrodes for seizure localization and single-unit neuronal recording[END_REF].

Therefore, in general, hybrid electrodes, including the one presented in this study, appear to be as safe as clinical electrodes.

Please note that we report on "linear bleeding" in 50% of our patients. To our knowledge, this type of linear bleeding is not well described in the literature. It corresponds to the sequelae of electrode paths which are well known, inevitable and expected, but which, until now, have mainly been taken into account on T2 or Flair sequences. These sequences only show a linear cavity and gliosis but not the possibility of hemorrhage. The main reason for this is probably a lack of focus by clinicians and neuroimagers on this type of anomaly, because they are perfectly asymptomatic and do not evolve with time. Furthermore, T2 * sequences are not systematically performed if no research protocol and surgery are scheduled. Finally, 3T MRI scans have been performed in most of our patients with a much better resolution than has been reported in studies of only a few years ago. Therefore, overall, we highlight asymptomatic sequelae which have probably been largely overlooked until now.

Several types of device now make it possible to record brain electrophysiological activity in humans, including needle hybrid electrodes such as the one used in this study, grids and multi-electrode arrays. An adequate comparison of the safety of the different types of electrodes and models remains to be done. One pending question in particular is whether the smaller diameter of the Dixi electrode (0.8 mm) is of any consequence compared to the larger diameter of other needle electrodes, which can reach up to 1.3 mm.

In our clinical study, these hybrid electrodes also had a minimal impact on the daily routine of SEEG exploration. However, we found that special care had to be taken not to bend the cable running from the electrodes to the connectors with the bandage. Special attention must also be paid to the electromagnetic environment of the room and the way in which the different cables are arranged in order to avoid electronic interference. In this article, we also show that the general noise level can vary significantly depending on the amplifiers and setup.

This means that special care and effort should be devoted to attaining a suitable noise level.

The most significant change for us was to decide which electrode insertion path plan should be used for the hybrid electrodes. This was because the spacing between the macrocontacts of hybrid electrodes is different from that of clinical electrodes and the number of macrocontacts is limited according to the model (see Methods section). In addition, these new hybrid electrodes are not MRI-compatible as they are made of tungsten. However, that did not affect our clinical routine because we use a CT scan after implantation to verify the absence of bleeding. Overall, these hybrid electrodes were easily adopted by our team of epileptologists and our neurosurgeon, with no increase in the duration of the implantation.

Impedance can greatly affect the quality of neuronal recordings as the ability to record action potentials from individual neurons is dependent on a tradeoff between the geometric area of the tip of the recording instrument and the impedance of the tip [START_REF] Ludwig | Chronic neural recordings using silicon microelectrode arrays electrochemically deposited with a poly(3,4-ethylenedioxythiophene) (PEDOT) film[END_REF].

A small geometric area enables isolation of action potentials from distant neural sources [START_REF] Drake | Performance of planar multisite microprobes in recording extracellular single-unit intracortical activity[END_REF][START_REF] Humphrey | Extracellular Single-Unit Recording Methods[END_REF][START_REF] Paik | Roughened polysilicon for low impedance microelectrodes in neural probes[END_REF]. Usually, a decrease in geometric area causes an increase in impedance [START_REF] Humphrey | Extracellular Single-Unit Recording Methods[END_REF][START_REF] Paik | Roughened polysilicon for low impedance microelectrodes in neural probes[END_REF]. Lowering the impedance could improve neuronal recording quality by increasing the signal-to-noise ratio [START_REF] Ferguson | Creating low-impedance tetrodes by electroplating with additives[END_REF]. The expected impedance for the microwires of a tetrode between 10 and 30 µm in diameter is 100 -500 kOhm (at 1 kHz) [START_REF] Ferguson | Creating low-impedance tetrodes by electroplating with additives[END_REF]. Similarly, the impedance of multi-wire electrodes with a diameter of 12 to 25 µm is usually approximately 100 kOhm [START_REF] Hill | Quality Metrics to Accompany Spike Sorting of Extracellular Signals[END_REF]. In our study, the diameter of each microwire was 20 µm and the impedance was approximately 100 kOhm during the first ten days. The impedance slightly increased to 220 kOhm after 3 weeks in vitro. Overall, the impedances we measured remained within reported standards.

It is crucial to obtain a minimal noise level in order to detect a larger number of units in the recordings, as only units with amplitudes above the noise level can be detected. Our median noise level was 22.25 µV. This noise level appears to be satisfactory as we were able to record single units. However, our results were acquired in suboptimal recording conditions and the quality of the signal could be improved. In fact, the noise level observed in other centers is sometimes lower. For example, [START_REF] Rutishauser | Representation of retrieval confidence by single neurons in the human medial temporal lobe[END_REF] were able to detect units with an amplitude of 15 µV, whereas we had difficulty detecting units with amplitudes below 20-25 µV. There are several possible explanations for the difference between centers: the quality of the electromagnetic environment, the quality of the cables between the tetrodes and the headstage and the quality of the amplifier, all of which should be optimized in the future.

In a model of hybrid electrodes that is already commercialized, single neurons were observed on 40% of the microwires, and a bundle of 8 microwires recorded an average of 2.1 ± 0.9 single neurons [START_REF] Misra | Methods for implantation of micro-wire bundles and optimization of single/multi-unit recordings from human mesial temporal lobe[END_REF]. With more experience, 70-80% of the bundles of microwires could record neurons [START_REF] Carlson | Safety and utility of hybrid depth electrodes for seizure localization and single-unit neuronal recording[END_REF]. Carlson et al. also reported that the number of neurons per bundle varies widely, from 0 to 16, possibly according to the brain region explored. For example, the amygdala appears to be a brain region where it is relatively easy to record a large number of neurons.

The tetrode configuration was introduced in animal research in the early nineties and has been widely used since [START_REF] Gray | Tetrodes markedly improve the reliability and yield of multiple single-unit isolation from multi-unit recordings in cat striate cortex[END_REF][START_REF] Lévesque | Interneurons spark seizure-like activity in the entorhinal cortex[END_REF]2018;[START_REF] O'keefe | Phase relationship between hippocampal place units and the EEG theta rhythm[END_REF][START_REF] Wilson | Dynamics of the hippocampal ensemble code for space[END_REF]. It facilitates the isolation of single units. To our knowledge, the maximum number of single neurons that have been isolated with one tetrode in animals was 9 [START_REF] Gray | Tetrodes markedly improve the reliability and yield of multiple single-unit isolation from multi-unit recordings in cat striate cortex[END_REF], although tetrodes have the potential to record many more neurons simultaneously [START_REF] Buzsáki | Large-scale recording of neuronal ensembles[END_REF].

The number of neurons recorded on each tetrode in the current study varied widely, from 0 to >10. We observed single neurons on 60% of the tetrodes and we were able to record 2.08 ± 2.86 single neurons per tetrode, per day. Taking into account multiple unit activity, up to 70% of the tetrodes showed some level of activity. This level of activity persisted with no obvious difference over time (in days) after implantation (from day + 1 -day + 8). With the new hybrid electrode, we were able to isolate up to 13 neurons with one tetrode, which seems promising and demonstrates the potential of this new hybrid electrode. Given our suboptimal recording conditions mentioned above, we think it will be possible to improve the neuronal yield once the signal-to-noise ratio of the recordings is improved.

With regard to this, there may still be more work to do to validate and compare different spike-sorting software as it is unclear whether they can be optimized in comparison to older software and whether they use the spatial information provided by the tetrode configuration optimally [START_REF] Chaure | A novel and fully automatic spike-sorting implementation with variable number of features[END_REF]. Spike sorting remains a field in which some level of subjectivity remains despite useful guidelines [START_REF] Hill | Quality Metrics to Accompany Spike Sorting of Extracellular Signals[END_REF]. New spike sorting software is being developed (e.g. the one we are using -SpyKing Circus -by [START_REF] Yger | A spike sorting toolbox for up to thousands of electrodes validated with ground truth recordings in vitro and in vivo[END_REF] and progress can be expected in the years to come.

Another important specificity of hybrid electrodes, which is unique to our 2 or 3 tetrodes per electrode configuration, is that the tetrodes can be extended from the shaft between the most distal macrocontacts by up to 2 mm, and the distance between the macrocontacts and the tetrode tips is known with relative precision. This is particularly suitable for multiscale analysis, as both macro and microelectrodes are located in the same target brain region.

Therefore, it is possible to record different signals within the same area such as: [START_REF] Isnard | French guidelines on stereoelectroencephalography (SEEG)[END_REF] the local field potential between the two macrocontacts adjacent to the tetrodes; (2) the local field potential of each tetrode in relation to either a distant contact (suitable for the recording of "larger" events) or to another tetrode (for "smaller", more local, events). Theoretically, the surface covered by or the distance between tetrodes can be determined and can therefore be used to approximate the size of the patch of cortex from which the electrophysiological events of interest originate. Of course, multiple and single-unit activities are recorded at the same time. Using such schemes, it is possible to uncover neurophysiopathological events that are recorded on the tetrodes but not on the macrocontacts, such as local interictal epileptic discharge (Figure 7A) and fast ripples (Figure 7B). Events can also be recorded on a smaller scale, such as fast ripples occurring on two tetrodes but not on the third [START_REF] Despouy | Neuronal spiking activity highlights a gradient of epileptogenicity in human tuberous sclerosis lesions[END_REF], or different dynamics on each tetrode (Figure 8).

In conclusion, the design of this new hybrid electrode is different from that of other available hybrid electrodes implanted in humans. These electrodes are especially promising for conducting clinical multiscale analyses (of interictal epileptic discharges, high frequency oscillations and seizure onset or propagation), without changing the implantation procedure [START_REF] Schevon | Multiscale recordings reveal the dynamic spatial structure of human seizures[END_REF]. They offer the possibility to provide unique information regarding neurophysiopathological aspects of epilepsy or the physiological substrates of cognition. We showed that these electrodes are safe and work satisfactorily. At present, two or three tetrodes can be extended by up to 2 mm. Further improvements are planned, such as extending tetrodes further to 3 mm. The possibility of moving tetrodes over such a range with the micrometer screw will improve the neuronal search, which in turn may considerably improve the neuronal yield. Another possibility is to have tetrodes protruding from different levels of the macroelectrode, which would allow recordings from lateral parts of the cortex and the possibility to study brain regions that, so far, have been poorly explored in humans.

Figure 1 :

 1 Figure 1: The new hybrid electrode. (A) The electrode contains two connectors, one for the

  clinical macroelectrodes and 47 hybrid electrodes. The remaining patients have not yet had a control MRI. The percentage of hematomas, linear hypersignals on the ancient trajectories of the electrodes, and linear bleeding[START_REF] Hefft | Safety of Hybrid Electrodes for Single-Neuron Recordings in Humans[END_REF] was assessed by a neurologist (JC) who was blinded to the type of electrode (hybrid or clinical). Subclinical complication rates were similar in clinical and hybrid electrodes (Fig.3B, Fisher's Exact Test for Count Data, T2/FLAIR linear hyperintensities, P = 1, 95% confidence interval = 0.07 -4.78; hypersignal, P = 1, 95% confidence interval =0.25 -2.47; linear bleeding, P = 0.49, 95% confidence interval = 0.61 -2.67).

Figure 3 -

 3 Figure 3 -The new hybrid electrode does not increase the number of subclinical

Figure 4 -

 4 Figure 4 -Electrochemical impedances. Notched boxplot representing the changes in

Figure 6 -

 6 Figure 6 -Comparison of different acquisition systems and clinical conditions. A) Filtered

Figure 7 -

 7 Figure 7 -Contributions of microelectrode recordings. (A) An interictal epileptic discharge

Figure 8

 8 Figure 8 -A) Different features and time of onset between macroelectrodes and tetrodes of

Figure 9 -

 9 Figure 9 -Example of neuronal activity recorded after EBS. (A) Diagram of the electrodes

Figure 10 -

 10 Figure 10 -Neuronal spiking activity. (A) Raw and filtered signal (high pass: 300 Hz) recorded

ACKNOWLEDGEMENTS

We would like to thank Pierre Yger for helping us to understand the technical features of SpyKING CIRCUS, Karine Bouyer for managing the electrodes, and Sophie Muratot and Simona

FUNDING This work received funding from the European Research Council, under the European Union Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 323711 (the M4 project), the Fondation Française pour la Recherche sur l'Epilepsie (Prix Marion Clignet), the Fédération Française sur le Cerveau / Rotary Club, and a call for research projects in technological and organizational innovation (Toulouse University Hospital).

Celebrini for their help in acquiring the data. We are thankful to the team from the Hospices Civils de Lyon (Lyon University Hospital Center) for their help with Figure 6 (Prof. S. Rheimsfunding from the IHU CESAME in the "Investissements d'Avenir" program -ANR-10-IBHU-0003). We would also like to thank all the people involved, at one point or another, in the EPIFAR project.