Elise Doveri
email: doveri@lma.cnrs-mrs.fr

Meysam Majnooni
email: meysam.majnooni@univ-amu.fr

Carine Guivier-Curien

Cécile Baron
email: cecile.baron@univ-amu.fr

Philippe Lasaygues
email: lasaygues@lma.cnrs-mrs.fr

Keywords: LIPUS stimulation, Computational model, Acoustic field aperture, Acoustic lens, Acoustic intensity

Low-Intensity Pulsed Ultrasound Stimulation (LIPUS) is a therapeutic modality used for bone tissue regeneration and healing. Its clinical efficacy is still debated, as the underlying physical phenomena remain poorly understood. The interaction between ultrasonic waves and cells, likely to trigger mechanotransduction inducing bone regeneration, is at the center of scientific concerns on the subject.

In order to get new insights into these phenomena, the development of in vitro experiments is a key step but special attentions should be paid concerning to the actual acoustic area covered that has to be sufficiently large and homogeneous. To address this issue, an acoustic lens can be placed on the transducer to improve the homogeneity of the acoustic field over the entire cell culture area. A computational model is developed to test several shapes and heights of acoustic lenses and compare their effectiveness in order to find a compromise between the surface covered, the homogeneity of the intensity distribution and the acoustic pressure loss.

All the lenses studied improve the enlargement of the field and its homogeneity

Introduction

Ultrasonic waves can be used for the diagnosis of bone pathologies [1] as well as for therapy, for the removal of kidney stones (lithotripsy) and for the ablation of cancerous tumors. They are also used for bone healing [START_REF] Romano | Low-Intensity Pulsed Ultrasound for the Treatment of Bone Delayed Union or Nonunion: A Review[END_REF], and this work focuses on the investigation of this application. The first clinical observations of the effect of Low Intensity Pulsed Ultrasound (LIPUS) for bone healing stimulation were reported in the early 1950s [START_REF] Buchtala | The present state of ultrasonic therapy[END_REF]. In 1994, the United States Food and Drug Administration (U.S. FDA) confirmed that LIPUS can be used for therapeutic clinical applications and approved clinical devices such as Exogen (Smith & Nephew Inc., Memphis, TN, USA), Osteotron IV (ITO Physiotherapy & Rehabilitation, Kawaguchi, Saitama, Japan), or Melmak (BTT Melmak Development & Production GmbH, Raisting, Germany).

In the literature, the clinical value of LIPUS treatment using this kind of devices is currently controversial. Some clinical studies show that the use of LIPUS can enhance the healing rate of nonunion fractures [START_REF] Zura | Treatment of chronic (>1 year) fracture nonunion: Heal rate in a cohort of 767 patients treated with low-intensity pulsed ultrasound (LIPUS)[END_REF][START_REF] Leighton | Healing of fracture nonunions treated with low-intensity pulsed ultrasound (LIPUS): A systematic review and meta-analysis[END_REF][START_REF] Romanò | Low-Intensity Pulsed Ultrasound in the Treatment of Nonunions and Fresh Fractures: A Case Series[END_REF] and metatarsal fractures [START_REF] Nolte | Heal rate of metatarsal fractures: A propensity-matching study of patients treated with low-intensity pulsed ultrasound (LIPUS) vs. surgical and other treatments[END_REF][START_REF] Teoh | The use of low-intensity pulsed ultrasound in treating delayed union of fifth metatarsal fractures[END_REF][START_REF] Anderson | Delayed Healing in Metatarsal Fractures: Role of Low-Intensity Pulsed Ultrasound Treatment[END_REF] whereas other studies show that LIPUS has no significant effect on fracture healing [START_REF] Busse | Low intensity pulsed ultrasonography for fractures: systematic review of randomised controlled trials[END_REF][START_REF] Schandelmaier | Low intensity pulsed ultrasound for bone healing: systematic review of randomized controlled trials[END_REF]. It has also been shown that the effectiveness of LI-PUS may depend on the patient [START_REF] Martinez De Albornoz | The evidence of low-intensity pulsed ultrasound for in vitro, animal and human fracture healing[END_REF]. One reason why the controversy cannot be resolved is that the underlying mechanisms triggering bone remodeling under LIPUS stimulation are still poorly understood [START_REF] Romano | Low-Intensity Pulsed Ultrasound for the Treatment of Bone Delayed Union or Nonunion: A Review[END_REF].

In order to make progress on this issue, in vitro setups are key tools to better understand the interaction between LIPUS and bone cells. A wide variety of in vitro setups with different LIPUS stimulation conditions are proposed in the literature [START_REF] Fontana | Development and validation of low-intensity pulsed ultrasound systems for highly controlled in vitro cell stimulation[END_REF][START_REF] Sun | In vitro effects of low-intensity ultrasound stimulation on the bone cells[END_REF][START_REF] Horne | LI-PUS far-field exposimetry system for uniform stimulation of tissues in-vitro : development and validation with bovine intervertebral disc cells[END_REF][START_REF] Tien | Effects of Pulsed Low-Intensity Ultrasound on Human Child Chondrocytes[END_REF]. In these papers, the ultrasound signals are in continuous or burst mode, with frequencies between 45 kHz and 3 MHz, with daily exposure times between 1 and 20 minutes, and acoustic intensities between 5 mW/cm 2 and 1 W/cm 2 [START_REF] Padilla | Stimulation of bone repair with ultrasound: A review of the possible mechanic effects[END_REF] which makes it difficult to rigorously compare the results obtained. In the present paper, only the frequency of 1 MHz is studied.

One of the challenges in achieving a relevant interpretation of the biological effects consists in the accurate monitoring of the acoustic dose received by the cells in a Petri dish. This monitoring is often mentioned in dedicated literature but rarely satisfactorily performed, because of multiple phenomena disturbing the propagation of the acoustic waves and the cell/wave interactions. Whatever the relative positions of the transducer and Petri dish, or the range of ultrasonic frequencies, one first disturbing effect is related to the standing waves that occur in the cell culture layer inside the Petri dish [START_REF] Secomski | In vitro ultrasound experiments: Standing wave and multiple reflections influence on the outcome[END_REF]. These standing waves can be constructive or destructive, and depend on the thickness of the fluid layer with a precision of a quarter of the wavelength (λ/4), i.e., a few hundred micrometers in water in the megahertz range [START_REF] Snehota | Therapeutic ultrasound experiments in vitro: Review of factors influencing outcomes and reproducibility[END_REF]. Consequently, it is very difficult to control these experimental conditions and to evaluate the consequences of these phenomena on the measurable acoustic intensity. A technical solution to this problem has been proposed recently [START_REF] Majnooni | Anti-reflection cover to control acoustic intensity in in vitro low-intensity ultrasound stimulation of cells[END_REF], consisting of an anti-reflection cover with high performance acoustic absorbing foam that reduces multiple reflections, traps transmitted waves and thus suppresses standing waves. However, even if these disturbing phenomena are neutralized, the acoustic field remains unsatisfactory in terms of spatial distribution (extent and homogeneity) in the Petri dish.

One of the solutions proposed in this work is to develop an acoustic lens that would broaden and homogenize the ultrasonic beam to obtain a targeted distribution of the acoustic intensity inside the Petri dish. For this purpose, a first step consists in the development of a computational model using Finite Element Method (FEM) to design the size and shape of the lenses, and to study their effects on the acoustic field delivered inside the Petri dish, in terms of broadening, inhomogeneity and acoustic pressure loss.

Materials & Methods

Preamble

The computational model used in this work was based on an experimental setup developed by our group [START_REF] Majnooni | Anti-reflection cover to control acoustic intensity in in vitro low-intensity ultrasound stimulation of cells[END_REF][START_REF] Majnooni | Monitoring of in-vitro ultrasonic 32 stimulation of cells by numerical modeling[END_REF]. A Petri dish was positioned on the surface of a water tank, in the far field of a transducer. Placed above the Petri dish, an absorbing cover was used to prevent the formation of standing waves in the water layer inside the Petri dish, as shown in Figure 1(a) (details can be found in [START_REF] Majnooni | Anti-reflection cover to control acoustic intensity in in vitro low-intensity ultrasound stimulation of cells[END_REF]).

The spatial distribution of the ultrasound stimulation inside the Petri dish was evaluated by measuring acoustic intensities (Appendix A) [START_REF]Output Measurements for Medical Ultrasound[END_REF]. The Temporal-Average Intensity (I TA ) is defined as follows:

I TA = 1/(ZT PRP ) T PRP 0 p(t) 2 dt , (1)
where p(t) is the total acoustic pressure, T PRP is the Pulse Repetition Period and Z is the acoustic impedance of the medium. The maximum intensity of I TA over the beam cross-sectional area S defines the Spatial-Peak Temporal-Average Intensity (I SPTA ), defined as follows:

I SPTA = max S (I TA ) , (2)
making it possible to define a normalized acoustic intensity ( ĨTA ) as follows:

ĨTA = I TA /I SPTA . (3)

Computational model

The computational model was developed in COMSOL Multiphysics ® v5.5 (Stockholm, Sweden) using the Pressure Acoustics module (transient mode) for the acoustic wave propagation in the fluid parts (water under and inside the dish) and the Solid Mechanics module for the elastic wave propagation inside the Petri dish.

Geometrical model and mechanical properties

The computational model was a 2D axisymmetric model, as shown in Figure 1(b). The Petri dish was made of polystyrene (Table 1) and had an external diameter of D dish = 2R dish = 55 mm, a depth of H dish = 15 mm and a wall and bottom thickness of t dish = 0.8 mm. The targeted area defined as the area intended for cell culture was assumed to be a disk with a diameter of D target = 2R target = 35 mm [START_REF] Majnooni | Monitoring of in-vitro ultrasonic 32 stimulation of cells by numerical modeling[END_REF]. The transducer was supposed to radiate as a plane piston of D td = 2R td = 13 mm diameter. The distance d f between the transducer and the Petri dish had been set at 25 mm. The horizontal red line in Figure 1(b) corresponds to the distance of observation z obs = 25.81 mm (which is the position of the cells inside the Petri dish), as in [START_REF] Majnooni | Monitoring of in-vitro ultrasonic 32 stimulation of cells by numerical modeling[END_REF]. The distance d f and the transducer diameter D td are constrained by the dimensions of the incubator provided in the associated experimental set-up [START_REF] Majnooni | Monitoring of in-vitro ultrasonic 32 stimulation of cells by numerical modeling[END_REF]. Previous results obtained by Majnooni et al. [START_REF] Majnooni | Monitoring of in-vitro ultrasonic 32 stimulation of cells by numerical modeling[END_REF] showed that, with this configuration, most of the targeted area is not insonified.

Acoustic lens geometry

The lens used to broaden the aperture angle and homogenize the ultrasonic beam was positioned above the transducer. Three shapes of lenses were tested. The first lens was a spherical cap made of epoxy resin (noted E-lens from now on) [Fig. 2(a)].

To cover the entire active front face of the transducer, the width of the epoxy resin cap was set such that L epoxy = D td + 2 mm = 15 mm. Its height H epoxy and its curvature radius R epoxy are related by the following equation:

R epoxy = H epoxy /2 + L 2 epoxy /(8H epoxy ) . (4)
In the present work, H epoxy is set as a variable, in order to study how it affects the ultrasonic fields and the acoustic intensity distribution inside the Petri dish.

Because the acoustic impedance in the epoxy resin was higher than in water (Table 2), a fraction of the wave energy was reflected from the lens/water interface. To reduce this effect, a second type of lens was designed (noted S-lens, for spherical lens, from now on): the epoxy resin cap was surrounded by a layer of polydimethylsiloxane (PDMS) with a constant thickness of 1 mm [Fig. 2(b)]. PDMS, with an acoustic impedance (Table 2) closer to that of water, acted as an acoustic impedance matching layer, allowing more energy to be transmitted through the lens.

The third lens design (noted C-lens, for cylindrical lens, from now on) was also a cap-shaped epoxy resin lens with a cylindrical top layer of PDMS forming a flat interface with water, as shown in Figure 2(c). This lens was similar to the one developed by Yang et al. [START_REF] Yang | Design and evaluation of a compound acoustic lens for photoacoustic computed tomography[END_REF]. The PDMS and the epoxy resin were assumed to be fluid because the attenuation coefficients of the shear waves α T inside them were much higher than the attenuation coefficients of the compression waves α L (Table 2). There was a small space of water (0.1 mm) between the transducer and the lens in order to mimic the fact that the lens was not directly in contact with the transducer.

Transmitted signal

The incident acoustic pressure P (t) was defined as follows:

P (t) = P 0 sin(2πf 0 t) , (5)
with P 0 = P max for 0 ≤ t ≤ t pulse = 200 µs, P 0 = 0 kPa for t pulse = 200 µs < t ≤ T PRP = 1 ms, where P 0 was the amplitude of the transmitted signal, f 0 = 1 MHz the nominal frequency of the transmitted wave, and t the time variable. Note that by definition ĨTA , in Equation (3), does not depend on the P max value.

Boundary conditions

In order to model an infinite medium of water around the Petri dish, a Perfect Matching Layer (PML) was placed on the right side of the model [blue honeycomb in Figure 1(b)]. To simulate the effect of the absorbing cover on top of the Petri dish, there was also a PML at this interface.

To mimic the air above the top right-end water layer, the boundary condition was the impedance of air (Z air ) [top-right green line in Fig. 1(b)].

Because the experimental set-up [Fig. 1(a)] was immersed in a water tank, the bottom boundary condition of the computational model was supposed to be water layers with an acoustic impedance (Z water ) [blue lines in Fig. 1(b)].

Mesh

The mesh size was homogeneous in all the materials of the model. It was such that there were at least 8 elements per smallest wavelength.

In the water and Petri dish, the mesh was made with free quads and in the lens it was made with free triangles. The time step of the simulation was defined as follows:

∆t = ∆x min /( √ 2V max ) , (6)
with ∆x min was the smaller mesh size and V max was the maximum wave velocity. This led to a Courant-Friedrichs-Lewy (CFL) condition equal to 1/ √ 2 ≈ 0.7, which was smaller than 1, ensuring the stability of the numerical scheme.

Wave attenuation

The Petri dish was considered as an elastic solid. However, the wave attenuation inside the walls was not taken into account. In fact, the attenuation coefficient of the compression wave in polystyrene being 1.36 dB/cm [START_REF] Waters | Differential forms of the Kramers-Kronig dispersion relations[END_REF] and the thickness of the walls being 800 µm, the attenuation due to the Petri dish was assumed negligible compared with the one due to the lens. Only the wave attenuation in the fluid parts (acoustic lens and water) was taken into account using General dissipation for the fluid model [START_REF]Nonlinear Acoustics -Modeling of the 1D Westervelt Equation, COMSOL Multiphysics v.5.5[END_REF], which requires to define the sound diffusivity δ as [START_REF] Jordan | On the sound attenuation in fluid due to the thermal diffusion and viscous dissipation[END_REF]:

δ = ν[φ/µ s + (γ -1)/P r ] , (7) with:
• ν = µ s /ρ, the kinematic viscosity [m 2 /s],

• µ s , the shear viscosity [Pa.s],

• ρ, the mass density [kg/m 3 ],

• φ = 4µ s /3 + µ b , the longitudinal viscosity [Pa.s],

• µ b , the bulk viscosity [Pa.s],

• γ, the ratio of specific heats,

• P r = ν/a, the Prandtl number,

• a, the thermal diffusivity [m 2 /s].

The coefficient of sound attenuation α Np L (expressed in Np/m) is defined as [29]:

α Np L = [φ + ρa(γ -1)]ω 2 /(2ρV 3 L ) , (8) with:
• V L , the compression wave velocity in the fluid [m/s],

• ω = 2πf 0 , the pulsation [rad/s],

• f 0 , the working frequency [Hz].

Finally, when combining Eq. ( 7) with Eq. ( 8), the sound diffusivity was defined in terms of the compression wave attenuation coefficient:

δ = 2V 3 L α Np L /ω 2 = 2V 3 L α L /[20 log(e 1 )ω 2 ] ≈ 0.23V 3 L α L /ω 2 , (9)
with α L the compression wave attenuation coefficient expressed in dB/m. Table 3 gives the sound diffusivity at f 0 = 1 MHz, for each fluid material. In order to characterize the broadening of the field aperture, three different areas were defined [Fig. 3] inside the Petri dish:

Material

• The targeted area [Fig. 3(a)], which corresponds to the area intended for cell culture in the in vitro set-up [START_REF] Majnooni | Monitoring of in-vitro ultrasonic 32 stimulation of cells by numerical modeling[END_REF]. It is a disk of radius R target = D target /2 = 17.5 mm.

• The insonification area [Fig. 3(b)], which corresponds to the area where the acoustic intensity is not zero. It is a disk of radius R ins .

• The beam area [Fig. 3(c)], which corresponds to the area where ĨTA is greater than 0.25 (details can be found in Appendix A). Finally, an aperture ratio (expressed in %) was introduced as the ratio of the beam area over the targeted area. One of the aims of this study was then to design an acoustic lens that provides a beam area equal to the targeted area, in order to obtain an aperture ratio of 100%.

Inhomogeneity

In order to quantify the inhomogeneity of the intensity distribution inside the Petri dish, the mean value and the standard deviation of the absolute value of the gradient of ĨTA were calculated [START_REF] Horne | LI-PUS far-field exposimetry system for uniform stimulation of tissues in-vitro : development and validation with bovine intervertebral disc cells[END_REF]. The absolute value of the gradient of the ĨTA was defined (more details in Appendix A) as:

∇ ĨTA = ∂ ĨTA ∂x + ∂ ĨTA ∂y . (10)
The mean value and the standard deviation of |∇ ĨTA | give information about, respectively, the number and the amplitude of oscillations of the intensity distribution. An homogeneous acoustic intensity field means that the mean value and the standard deviation of |∇ ĨTA | are equal to 0.

Pressure acoustic loss

In order to estimate the wave attenuation induced by the different shapes of lens, the pressure acoustic loss (P AL, expressed in dB) due to each lens was calculated by multiplying the attenuation coefficient of each lens material (epoxy or PDMS) of the lens (Table 2) by the distance traveled by the wave inside each corresponding part of the lens : 2), and where d epoxy and d PDMS are respectively the distances traveled inside epoxy and PDMS by a ray starting at the end of the transducer (at x = D td /2 = 6.5 mm) drawn in red on Figure 4.

P AL = α epoxy

Analytical modeling of the aperture angle

The computational model allowed us to quantify the effect of lenses on the broadening of the aperture angle, the acoustic field inhomogeneity, and the acoustic pressure loss. The aperture angle can be assessed analytically. Such assessment can thus participate in partially validating the computational model and rapidly quantify the influence of the lens geometry on the aperture angle, especially the optimal value of H epoxy . An analytical modeling (details can be found in [START_REF] Engholm | Increasing the field-of-view of row-column-addressed ultrasound transducers: implementation of a diverging compound lens[END_REF]) was implemented to compare the aperture of the acoustic field with and without the shaped lenses.

For all the lenses, the first enlargement of the field was due to the epoxy part. Noting θ 1 the incident angle and θ 2 the refracted angle at the epoxy/PDMS interface (or at the epoxy/water interface in the case of the E-lens) :

θ 1 = arcsin[D td /(2R epoxy )] , (12)
θ 2 = arcsin(n 1 sin θ 1 ) , (13)
with n 1 = V PDMS /V epoxy in the cases of the C-lens and S-lens, and n 1 = V water /V epoxy in the case of the E-lens, and with V PDMS , V epoxy and V water respectively the compression wave velocities in the PDMS layers, in the epoxy resin cap and in the water.These values are given in Table 2.

For the E-lens the aperture is characterized by the angle θ 1 -θ 2 .

For the PDMS/water interface, the angle of incidence θ 3 (represented in Figure 4) was calculated using Snell's law. Assuming that the PDMS layer was thin, this angle was written as follows:

θ 3 ≈ θ 2 , for S-lens, ( 14
) θ 3 = θ 1 -θ 2 , for C-lens, (15)
Finally the refraction angle at the PDMS/water interface θ 4 is given by:

θ 4 = arcsin(n 2 sin θ 3 ) with n 2 = V water /V PDMS , (16)
with V PDMS and V water the compression wave velocities in the PDMS layers and water, respectively. These values are given in Table 2.

The aperture angle is θ 1 -θ 4 for the S-lens, and θ 4 for the C-lens.

The algorithm was implemented using Matlab ® (The MathWorks, Inc., Natick, MA, USA).

Results and discussion

Optimal design of the lens

The analytical modeling presented in Section 2.4 was used to determine the appropriate range of H epoxy values for each shape of lens. The minimum and maximum values of H epoxy were determined as follows:

• The minimum value of the H epoxy range was set such that an aperture of 30% of the targeted area (S ins = 30%S target = 289 mm 2 , which corresponds to R 30% = 10 mm) was reached.

• The maximum value of the H epoxy range was set such that an aperture of 95% of the Petri dish area (R max = 26.8 mm) was reached, or when the maximum of the insonification radius R ins [Fig. 3(b)] was reached.

For the three lens shapes, Figure 5 shows:

1. the insonification radius R ins as a function of H epoxy between 0 and 7.5 mm in thick black line, 2. the minimum and maximum thresholds R 30% and R max in black dashed lines, 3. the radius of the targeted area R target in black dotted line.

The minimum (blue vertical lines) and maximum (red vertical lines) values for the H epoxy range are shown for each lens in Figure 5. On each side of a graphic are represented the insonification areas along the z-axis for each lens: in blue for R 30% (the minimum value of the range), and in red for R max (the maximum value of the range).

Table 4 shows the H epoxy values that were chosen to be equi-distributed over the previous range. They were used in the computational model to explore the influence of H epoxy on the aperture angle of the lens, the inhomogeneity of the intensity distribution inside the Petri dish, and the pressure acoustic loss due to the lens.

The selected H epoxy for the E-lens and the S-lens are similar, while the selected ones for the C-lens are different. This means that the PDMS layer of the S-lens has a smaller effect on the increase in the insonification radius than the PDMS layer of the C-lens.

Shape of lens

H The blue vertical lines correspond to the values of H epoxy that match an insonification radius of R 30% . The red vertical lines correspond to the values of H epoxy that match an insonification radius of R max or the maximum value of the insonification radius. On both sides of each graphic are represented the insonification areas along the z-axis, for the corresponding lenses. The blue ones correspond to a H epoxy that matches an insonification radius of R 30% . The red ones correspond to a H epoxy that matches an insonification radius of R max or the maximum value of the insonification radius.

Acoustic intensity distribution along the z-axis

To better understand the influence of the lenses on the acoustic intensity distribution, the results of the analytical model (presented in 2.4) and those of the computational model were overlaid for ĨTA along the z-axis (Figure 6). The comparisons are performed for the minimum and maximum of the H epoxy range (Table 4). The vertical black line at x = 17.5 mm corresponds to R target .

In order to see small details in the ĨTA distributions, the colormap of Figure 6 was shifted, so that when the ĨTA is higher than 1%, pixels are coloured in black, so the maximum value of the colormap is set to 1% and the pixels corresponding to an ĨTA between 1% and 100% are all coloured in black. This figure validates the analytical model, since the simulated insonification area (in black) corresponds to the analytical insonification area (delimited by the thick red lines).

On the bottom line graphs (d to f) of Figure 6, oscillations can be seen around the Petri dish. These oscillations appear more clearly in the zoom part of Figure 6(f), where a maximum of intensity can be found at the center of the Petri dish and where oscillations can be seen on the interior surface of the dish bottom. This type of ĨTA distribution can be related to the work of Hensel et al. [START_REF] Hensel | Analysis of Ultrasound Fields in Cell Culture Wells for In Vitro Ultrasound Therapy Experiments[END_REF] assuming that when the transducer diameter is close to the Petri dish diameter, "a ring-shaped pattern with maximum acoustic pressure at the well center can be observed". According to them, when the transducer diameter is larger than the Petri dish diameter, acoustic pressure waves are coupled into the Petri dish walls with compression and shear wave components. Then, these waves are transmitted through the liquid and are reflected by the Petri dish walls several times, resulting in constructive superposition at the Petri dish center. In our case, as the lens enlarges the acoustic field sufficiently to insonify almost all the Petri dish bottom [Fig. 6(f)], this effect may occur, which could explain the peak of ĨTA at the center of the Petri dish, and the small oscillations aorund it. This central peak is also increased by the axi-symmetry of the Petri dish, which "favors the formation of a symmetric and tapered focus along the central vertical axis" [START_REF] Tamboia | A comparative analysis of low intensity ultrasound effects on living cells: from simulation to experiments[END_REF].

Acoustic intensity along the radius of the Petri dish

In order to understand how the acoustic intensity was distributed inside the Petri dish according to the different designs of the lens, the ĨTA was calculated on a plane perpendicular to the z-axis at z obs = 25.81 mm, for each size and each shape of lens. This height corresponds to the place where the cells will be during the in vitro stimulation. The area where ĨTA is plotted corresponds to the targeted area [as defined in Fig. 3(a)].

Figure 7 shows ĨTA along the radius of the Petri dish, for different H epoxy , in the case of the C-lens. On all curves, there is a main peak that far exceeds all other peaks in the center of the dish. But, as this peak represents only 0.04 % of the targeted area (corresponding to the blue area of 0.35 mm radius in Fig. 7), and as the purpose of this study is to investigate the inhomogeneity of the acoustic intensity distribution inside the Petri dish on all the targeted area (and not only on the peak area), we have chosen to remove this peak area from the following results. Omitting this peak in the ĨTA distribution thus leads to define a larger beam area than the one that would otherwise be assessed.

Acoustic intensity distribution inside the Petri dish in the absence of a lens

Figure 8 shows the normalized intensity ĨTA inside the Petri dish (z obs = 25.81 mm), which represents the initial configuration without lens (i.e., the experimental set-up developed by Majnooni et al. [START_REF] Majnooni | Monitoring of in-vitro ultrasonic 32 stimulation of cells by numerical modeling[END_REF]). Here, the aperture ratio (defined in Section 2.3.1) is equal to 2.5 %, which means that the beam area represents 2.5 % of the targeted area. The inhomogeneity [defined as the mean value and the standard deviation of |∇ ĨTA | (Eq. 10)] is equal to 0.024 ± 0.069 mm -1 .

Figure 8: ĨTA inside the Petri dish on a plane perpendicular to the z-axis at z obs = 25.81 mm, without lens. The white area in the middle corresponds to the removed peak area (corresponing to the blue area in Fig. 7), where the ĨTA is set to 0.

Acoustic intensity distribution inside the Petri dish in the presence of the shaped lens 3.5.1. With the E-lens

Figure 9 shows the distribution of ĨTA along the radius of the targeted area (z obs = 25.81 mm, 0 < x < R target = 17.5 mm), for the H epoxy values reported in Table 4 for the E-lens.

Figures 12(a) and (d) show the aperture ratio values and the inhomogeneity for the E-lens, as a function of H epoxy values (Table 4).

For the E-lens, the maximum aperture ratio is achieved (43.5 %) for H epoxy = 4.25 mm [Fig. 9 and12(a) and(d)]. The inhomogeneity is higher (0.245 ± 0.330 mm -1 for H epoxy = 4.25 mm) when the aperture ratio is higher (43.5 % for H epoxy = 4.25 mm). 4). The white area (for x between 0 and 0.35 mm) corresponds to the removed peak area (corresponding to the blue area in Fig. 7), where the ĨTA is set to 0.

With the S-lens

Figure 10 shows the distribution of the normalized intensity ĨTA along the radius of the targeted area (z obs = 25.81 mm, 0 < x < R target = 17.5 mm), for the H epoxy values reported in Table 4 for the S-lens.

Figures 12(b) and (e) show the aperture ratio values and the inhomogeneity for the S-lens, as a function of H epoxy values (Table 4).

For the S-lens, the maximum aperture ratio is achieved (32.7 %) for H epoxy = 3.70 mm [Fig. 10 and 12(b) and (e)]. The corresponding inhomogeneity is equal to 0.269 ± 0.337 mm -1 . The layer of PDMS around the epoxy lens does not improve the aperture ratio obtained with the E-lens (-11%) and weakly increases the inhomogeneity of the acoustic intensity distribution (+ 0.012 mm -1 ).

Figure 10: Distribution of the normalized intensity ĨTA along the radius of the targeted area (R target = 17.5 mm) for the fixed distance of z obs = 25.81 mm, using the S-lens. In the center (respectively on the left and the right) are shown the half-profile views (respectively the top views of the total areas) for the selected heights H epoxy of the epoxy resin cap. The white area (for x between 0 and 0.35 mm) corresponds to the removed peak area (corresponding to the blue area in Fig. 7), where the ĨTA is set to 0.

With the C-lens

Figure 11 shows the distribution of the normalized intensity ĨTA along the radius of the targeted area (z obs = 25.81 mm, 0 < x < R target = 17.5 mm), for the H epoxy values reported in Table 4 for the C-lens.

Figures 12(c) and (f) shows the aperture ratio values and the inhomogeneity for the C-lens, as a function of H epoxy values (Table 4).

The C-lens achieves coverage of the largest targeted area (67.9%) for H epoxy = 2.45 mm [Fig. 11 and 12(c) and 12(f)]. The corresponding inhomogeneity is equal to 0.445 ± 0.452 mm -1 .

The C-lens can cover most of the targeted area (67.9 %) whereas the E-lens (43.5%) and S-lens (32.7%) cannot even cover half of the targeted area. Nevertheless, the inhomogeneity of the acoustic intensity field inside the Petri dish is higher with the C-lens (0.445 ± 0.452 mm -1 ) than for the E-lens (0.245 ± 0.330 mm -1 ) and the S-lens (0.269 ± 0.337 mm -1 ). This non-homogeneity can also be seen more clearly in Fig. 11 where, for the highest H epoxy value (4.10 mm), oscillations of ĨTA along the radius appear. Those oscillations form rings of different intensities inside the Petri dish. In the center (respectively on the left and the right) are show the half-profile views (respectively the top views of the total areas) for the selected heights H epoxy of the epoxy resin cap. The white area (for x between 0 and 0.35 mm) corresponds to the removed peak area (corresponding to the blue area in Fig. 7), where the ĨTA is set to 0. 4).

Acoustic pressure loss

Figure 13 shows the acoustic pressure loss for each shape of lens, along a path followed by a ray starting at the edge of the transducer x = D td /2 = 6.5 mm. This ray passes through the two components of the S-lens and C-lens, which allows us to investigate the attenuation effect of the material each component is made of (epoxy and PDMS).

Inside the E-lens, multiple wave reflections, due to the high impedance contrast between epoxy resin and water, occur at the boundaries. Moreover, the attenuation of the compression waves is high in the epoxy resin. The combination of these two phenomena results in an acoustic pressure loss that increases as a function of H epoxy , from 0.3 dB for a height of 1.70 mm to 1.7 dB for a height of 6.80 mm. To counteract these effects, the epoxy resin cap was surrounded by a PDMS layer, allowing the acoustic impedance between the epoxy resin cap and the water to be better matched.

For the S-lens, Figure 13(b) shows that the acoustic pressure loss is the same as for the E-lens, for a slightly lower H epoxy height. In addition, most of the acoustic pressure loss is due to the epoxy resin cap (88%), and not to the PDMS layer which attenuates the waves less (11%).

For the C-lens, the PDMS layer is thicker at the extremities of the lens. This cylindrical shape allows the increase of the aperture ratio, but the most diffracted ultrasonic waves on the edges of the epoxy resin cap are propagated through a greater thickness of PDMS, and are thus more attenuated [Figure 13(c)]. Indeed, the loss due to the PDMS layer of the lens is close to the one due to the epoxy resin cap of the lens (≈ 50%). 4). The blue bars represent the acoustic pressure loss due to the epoxy resin cap and the red bars the one due to the PDMS layer.

The energy transmitted through the lens depends also of the reflection and transmission coefficients (given in Appendix B under normal incident conditions) at each material interface. For the E-lens, the only interface is between epoxy and water, which gives a transmission coefficient of T epoxy, water = 0.93 and a reflection coefficient of R epoxy, water = 0.07 (using the material properties given in Table 2). For the S-lens and the C-lens, there are two interfaces: the first one between epoxy and PDMS gives a transmission coefficient of T epoxy, PDMS = 0.84 and a reflection coefficient of R epoxy, PDMS = 0.16, and the second one between PDMS and water gives transmission coefficient of T PDMS, water = 0.97 and a reflection coefficient of R PDMS, water = 0.03, which leads to a total transmission coefficient of T epoxy, PDMS × T PDMS, water = 0.81. As the transmission coefficient of the Elens is higher than the one of the S-lens and C-lens, it means that more energy is transmitted through the E-lens. However, those coefficients are calculated under normal incident conditions, which is the case only at the center of each lens. Anywhere else along the radius of the lens, the incident angle varies, which varies the reflection and transmission coefficients at each interface [START_REF] Ainslie | Plane-wave reflection and transmission coefficients for a three-layered elastic medium[END_REF].

Summary

In order to summarize all those results, let us compare the results obtained for different lenses that give the same aperture ratios.

The E-lens and the C-lens

Table 5 summarizes the different results obtained for an E-lens and a C-lens that give an aperture ratio of 41%. In this case, the best choice would be the Elens because it reduces the inhomogeneity (-30%) ant it leads to a smaller acoustic pressure loss than the C-lens. It is also a lens easier to make because it is made of epoxy only.

Shape of lens

The E-lens and the S-lens

Table 6 summarizes the different results obtained for an E-lens and a S-lens that give an aperture ratio of 33%. In this case, the E-lens would still be a better choice because it reduces the inhomogeneity a little more than the S-lens (-8%). The drawback is that the acoustic pressure loss is a little higher (+0. If the purpose of using the lens is only to increase the aperture ratio as much as possible, the best choice would be the C-lens. However it also increases the inhomogeneity. By contrast, if an aperture ratio of 40% is enough, the E-lens would provide a better homogeneity of the acoustic intensity distribution inside the Petri dish. Finally, the use of a S-lens does not improve anything: the aperture ratio is smaller than with an E-lens and the inhomogeneity is higher.

Conclusion

This work proposes a computational FEM-model to guide the design of acoustic lenses in terms of size and shape, with the objective of ensuring a homogeneous ultrasound stimulation on the whole cell culture area inside a Petri dish, within the framework of LIPUS stimulation of bone cells. Three different lens shapes based on an epoxy resin part with various heights covered with a PDMS layer were studied. Lens efficiency was quantified in terms of field aperture, homogeneity of the acoustic intensity distribution inside the Petri dish and attenuation due to the lens.

There is no ideal lens configuration that homogenizes the field and distributes the acoustic intensity to the whole targeted area. The design of the lens should be a compromise which depends on the priority given to the 2 criteria related to the biological effects induced. For example, if it is only used to enlarge the acoustic field, the C-lens is the best choice, but it will lead to a less homogeneous acoustic intensity distribution and to more attenuation. Then, more attenuation means increased voltage input, which may damage the transducer.

The computational model we developed allowed us to access parameters the experimental measurements of which are complicated, such as the acoustic intensity at the cells level. It also allowed us to test different configurations, such as the geometry of the lens, its composition, the excitation frequency, etc..., in order to eliminate the irrelevant configurations that will not be experimentally tested. The next step will be to make the lens and to measure its effect on the acoustic field, in order to compare it with the corresponding simulated one to validate the computational model.

[W/m 2 ], is a key parameter [START_REF]Guide for Measuring and Reporting Acoustic Output of Diagnostic Ultrasound Medical Devices[END_REF]. In a fluid, it is given by the time average of the product between the acoustic pressure p(t) and the particle velocity u(t) over the period of oscillations T [START_REF] Royer | Elastic waves in solids I: Free and Guided Propagation[END_REF]: For a harmonic wave, the acoustic intensity can be written as follows:

I = lim
I = 1 2
[p(t)u(t) * ], with p(t) = P 0 e -i(ωt-φ) u(t) = U 0 e -i(ωt-φ) (A.2)

where * denotes the complex conjugate and , the real part. Different parameters can be calculated, in order to characterize the acoustic field, such as:

• the Temporal Average Intensity (I TA );

• the Spatial Peak Temporal Average Intensity (I SPTA ) corresponding to the maximum value of I TA over the beam cross-sectional area S;

• the Spatial Average Time Average Intensity (I SATA ), which corresponds to the I TA averaged over the surface S;

• the ratio of surface covered by at least 25% I SPTA over the surface S [START_REF]Output Measurements for Medical Ultrasound[END_REF]. It corresponds to a level of attenuation of -6dB, which is the threshold defining the active area of a plane transducer;

• the mean value and standard deviation of the absolute value of the gradient of the normalized I TA ( ĨTA ) [START_REF] Horne | LI-PUS far-field exposimetry system for uniform stimulation of tissues in-vitro : development and validation with bovine intervertebral disc cells[END_REF]. They represent the variation of the intensity field (the inhomogeneity of the field): the smaller the mean gradient, the flatter the intensity field.

In the case of a pulsed signal, the Pulse-Intensity Integral I PI is written as follows ( [START_REF]Output Measurements for Medical Ultrasound[END_REF], p.96):

I PI = P I /Z , (A.3) with P I , the Pulse-Pressure-Squared Integral, defined as follows ( [START_REF]Output Measurements for Medical Ultrasound[END_REF], p.93):

P I = T PRP 0 p(t) 2 dt , (A.4)
where T PRP is the Pulse Repetition Period, which corresponds to the total duration of the signal, including the pulse duration. Then, the Temporal-Average Intensity I TA is written as follows ( [START_REF]Output Measurements for Medical Ultrasound[END_REF], p.97):

I TA = F PRF I PI , (A.5)
where F PRF = 1/T PRP is the Pulse Repetition Frequency.

Finally, when reporting Eq. (A.4) into Eq. (A.5), we obtain: where S corresponds to the beam cross-sectional area.

I TA = 1/(ZT PRP )
The Spatial Average Time Average Intensity I SATA can then be defined as [START_REF] Acevedo | The measurement of the spatial average temporal average intensity Isata and ultrasonic power W in composite ultrasonic transducers for medical application[END_REF]: where I TA (r) is the I TA at the point defined by the coordinate vector r on the surface S. The integral limit S 6 represents the integration over the surface where the I TA is greater than 25% I SPTA (-6 dB), and A 6 is the -6 dB beam cross-sectional area (which corresponds to the active beam diameter).

I SATA = 1/A 6
The normalized I TA is written as follows:

ĨTA = I TA /I SPTA , (A.9) and its gradient is given by:

∇ ĨTA = ∂ ĨTA ∂x x + ∂ ĨTA ∂y y . (A.10)
The absolute value of the gradient of the Ĩ TA is:

∇ ĨTA = ∂ ĨTA ∂x + ∂ ĨTA ∂y . (A.11)
Finally, the inhomogeneity is characterized by the mean value and the standard deviation of ∇ ĨTA .

Figure 1 :

 1 Figure 1: Schematic view of (a) the experimental set-up developed by Majnooni et al. [21] and (b) the 2D axisymmetric (around the z-axis) computational model, based on the experimental set-up.

Figure 2 :

 2 Figure 2: Schematic view of (a) the E-lens, (b) the S-lens and (c) the C-lens

Figure 3 :

 3 Figure 3: Representations of (a) the targeted area, (b) the insonification area and (c) the beam area, based on the normalized intensity ĨTA calculated in the absence of lens, at a distance of z obs = 25.81 mm, on the x/y plane

L

  d epoxy , for the E-lens, P AL = α epoxy L d epoxy + α PDMS L d PDMS , for the S-lens and the C-lens, (11) where α epoxy L and α PDMS L are respectively the compression attenuation coefficients of epoxy and PDMS (given in Table

Figure 4 :

 4 Figure 4: Schematic view of the angular orientation of the beam inside (a) the C-lens and (b) the S-lens.

Figure 5 :

 5 Figure 5: Insonification radius for (a) the E-lens, (b) the S-lens and (c) the C-lens, as a function of the height of the epoxy lens H epoxy . The black dashed horizontal lines correspond to the two thresholds (R 30% = 10 mm and R max = 26.8 mm). The black dotted horizontal line corresponds to the radius of the targeted area R target = 17.5 mm.The blue vertical lines correspond to the values of H epoxy that match an insonification radius of R 30% . The red vertical lines correspond to the values of H epoxy that match an insonification radius of R max or the maximum value of the insonification radius. On both sides of each graphic are represented the insonification areas along the z-axis, for the corresponding lenses. The blue ones correspond to a H epoxy that matches an insonification radius of R 30% . The red ones correspond to a H epoxy that matches an insonification radius of R max or the maximum value of the insonification radius.

Figure 6 :

 6 Figure 6: Ray traces obtained by the analytical modeling (thick red lines) overlayed over the ĨTA obtained by the computational model along the z-axis, for the minimum and the maximum of the H epoxy range (Table 4), for (a and d) the E-lens, (b and e) the S-lens, and (c and f) the C-lens. The corresponding lenses are drawn in red. On top of the bottom line graphics are plotted zooms of the ĨTA close to the bottom of the Petri dish.

Figure 7 :

 7 Figure 7: ĨTA along the radius for different H epoxy , in the case of the C-lens. The blue area corresponds to the peak area.

Figure 9 :

 9 Figure 9: Distribution of the normalized intensity ĨTA along the radius of the targeted area (R target = 17.5 mm) for the fixed distance of z obs = 25.81 mm, using the Elens. In the center (respectively on the left and the right) are shown the half-profile views (respectively the top views of the total areas) for the selected heights H epoxy of the epoxy resin cap (defined in Table4). The white area (for x between 0 and 0.35 mm) corresponds to the removed peak area (corresponding to the blue area in Fig.7), where the ĨTA is set to 0.

Figure 11 :

 11 Figure 11: Distribution of the normalized intensity ĨTA along the radius of the targeted area (R target = 17.5 mm) for the fixed distance of z obs = 25.81 mm, using the Clens.In the center (respectively on the left and the right) are show the half-profile views (respectively the top views of the total areas) for the selected heights H epoxy of the epoxy resin cap. The white area (for x between 0 and 0.35 mm) corresponds to the removed peak area (corresponding to the blue area in Fig.7), where the ĨTA is set to 0.

Figure 12 :

 12 Figure 12: Aperture ratio (top diagrams), and inhomogeneity (bottom diagrams) for (a) and (d) the E-lens, (b) and (e) the S-lens and (c) and (f) the C-lens, as a function of the values selected for the height H epoxy of the epoxy resin cap (Table4).

Figure 13 :

 13 Figure 13: Acoustic pressure loss at the distance x = D td /2 = 6.5 mm from the center of the transducer, along the transducer radius, in (a) an E-lens, (b) a S-lens and (c) a Clens, as a function of the values selected for the height H epoxy of the epoxy resin cap (Table4). The blue bars represent the acoustic pressure loss due to the epoxy resin cap and the red bars the one due to the PDMS layer.

6 )

 6 From the I TA , the Spatial Peak Temporal Average Intensity I SPTA is given by ([START_REF]Output Measurements for Medical Ultrasound[END_REF], p.98):I SPTA = max S (I TA ) , (A.7)

S 6 I

 6 TA (r)dS , (A.8)

Table 1 :

 1 Characteristics of the polystyrene which composes the Petri dish[START_REF] Waters | Differential forms of the Kramers-Kronig dispersion relations[END_REF]

	Material	Young Modulus E [GPa]	Poisson's ratio ν	Mass density ρ [kg/m 3 ]	Compression wave velocity V L [m/s]	Shear wave velocity V T [m/s]	Impedance Z [MRayls]
	Polystyrene	3.6	0.34	1050	2367	1131	2.49

Table 2 .

 2

		Mass	Compression	Impedance	Compression	Shear attenuation
	Material	density ρ	wave velocity	Z	attenuation coefficient	coefficient α T at
		[kg/m 3 ]	V L [m/s]	[MRayls]	α L at 1 MHz [dB/cm]	1 MHz [dB/cm]
	Water (17.8 o C) [20]	1000	1474	1.47	2.2 10 -3	-
	Sylgard 184 [25]	1031	1030	1.06	1.91	1499
	Epoxy [26]	1080	2300	2.48	5.32	30

Table 2 :

 2 Acoustic properties of water, PDMS and epoxy resin. The given values are mean values obtained from the literature. The values for water are the ones of a previous experiment[START_REF] Majnooni | Anti-reflection cover to control acoustic intensity in in vitro low-intensity ultrasound stimulation of cells[END_REF]. The values for epoxy are for an Araldite 2011 epoxy material (Huntsman Advanced Materials, Everberg, Belgium)[START_REF] Wang | Bulk shear wave propagation in an epoxy: attenuation and phase velocity over five decades of frequency[END_REF].

Table 3 :

 3 Sound

diffusivity at f 0 = 1 MHz, for each fluid material 2.3. Criteria of effectiveness 2.3.1. Aperture ratio

Table 4 :

 4 Five selected H epoxy values for each shape of lens

Table 5 :

 5 H epoxy , inhomogeneity and acoustic pressure loss for an E-lens and a C-lens that achieve an aperture ratio of 41%

Table 6 :

 6 3 dB). H epoxy , inhomogeneity and acoustic pressure loss for an E-lens and a S-lens that achieve an aperture ratio of 33%3.7.3. The S-lens and the C-lensTable7summarizes the different results obtained for a S-lens and a C-lens that give an aperture ratio around 25%. For this particular aperture ratio, both the smallest (H Epoxy = 0.80 mm) and the highest (H Epoxy = 4.10 mm) C-lenses work, but they give different results. The smallest C-lens gives the smallest inhomogeneity (-66%) and the smallest acoustic pressure loss (-1 dB) in comparison with the highest C-lens. The inhomogeneity is also reduced in comparison with the S-lens (-40%).

	Shape of lens	E-lens	S-lens
	H epoxy [mm]	5.53	3.70
	Aperture ratio [%]	33.3	32.7
	Inhomogeneity [mm -1 ]	0.248 ± 0.271 0.269 ± 0.337
	Acoustic pressure loss [dB]	1.2	0.9

Table 7 :

 7 H epoxy , inhomogeneity and acoustic pressure loss for an E-lens and a S-lens that achieve an aperture ratio around 25%

Acknowledgments

The project leading to this publication has received funding from the CNRS project AAP2017-OSEZ-BARON, and the Excellence Initiative of Aix-Marseille University -A*Midex, a French "Investissements d'Avenir programme" -Institut Mécanique et Ingénierie (IMI, AMX-19-IET-010). The authors would like to thank Julien Deschamps and Eric Bertrand of the Institute for Research on Non-Equilibrium Phenomena (IRPHE) for their help with PDMS, and Marie-Hélène Mignard for English language revision.

Appendix A. Acoustic intensities and gradient

In the design and development of US medical applications, the acoustic intensity, which is defined as the rate of energy transfer per unit time per unit area

Appendix B. Reflection and transmission coefficients

The reflection coefficient of a wave on a medium 1/medium 2 interface is defined under normal incident conditions as:

and the transmission coefficient is defined as:

where Z 1 = ρ 1 c 1 and Z 2 = ρ 2 c 2 are the acoustic impedance (expressed in MRayls) of medium 1 and of medium 2 respectively. ρ is the mass density of the medium and c is the compression wave velocity of the medium.


Olivier Faugeras

Etienne Tanré

A new twist on the large size limit behaviour of networks of Hopfield-like neurons

dX t = g(X t )dt + λdW t Law of X 0 = µ 0 , (1)
we couple N such neurons with independent initial distribution (Law of (X 1 0 , • • • , X N 0 ) = µ ⊗N 0 ) as

dX i t = g(X i t ) + Σ j J ij f (X j t ) dt + λdW i t 1 ≤ i ≤ N (2
) f is a regular bounded function (typically a sigmoid) that converts membrane potential to activities. The W i s are independent Brownians. The synaptic weights J ij are i.i.d. Gaussian random variables

J ij ≃ N ( J N , σ 2 N ), i, j = 1, • • • , N (3)
The theory of large deviations allows us, inspired by [START_REF] Benarous | Large deviations for Langevin spin glass dynamics[END_REF], to write the mean-field equations:

   X t = X 0 + t 0 g(X s ) ds + λB t B t = t 0 m Q (s) ds + B 0 t B 0 t = W t + t 0 s 0 Ks Q (s, u) dB 0 u ds m Q (t) = JE Q [ f (X t )], J = J/λ K Q (t, s) = σ2 E Q [f (X t )f (X s )], σ = σ/λ Q is the law of X t , W t is a Brownian under Q (4) Kt Q is a simple function of K Q .
The theory of Volterra equations allows us to compute B 0 as a function of the Brownian W and the covariance K Q :

B 0 t = W t + t 0 W s ds + t 0 s 0 H s Q (s, u) W u du ds where W t = t 0 Kt Q (t, s) dW s , (5)
and H t Q (t, s) is the resolvent kernel of the corresponding Volterra equation in the variable B 0 t which is a simple function of KQ and hence of K Q . Equations ( 4) and ( 5) are a new description of the mean-field dynamics as a set of non-Markov stochastic equations.

The functions K Q and m Q are unknown functions of g, f , J and σ. We have proved that a certain map defined on the set of pairs of functions (m(t), K(t, s)) has a unique fixed point, i.e. (m Q (t), K Q (t, s)) which defines the dynamics ( 4) and characterized its rate of convergence.

We have implemented the corresponding algorithm in the Julia language using Monte Carlo methods and found experimentally that it converged in just a few iterations, allowing us to simulate the mean-field dynamics [START_REF] Alexander Van Meegen | Large-Deviation Approach to Random Recurrent Neuronal Networks: Parameter Inference and Fluctuation-Induced Transitions[END_REF].

We show some of our numerical results in the Figure below for a network of Hopfield neurons, i.e. g(x) = -x, f a sigmoid. From left to right we use J = -6 (inhibitory network), J = 0 and J = 2 (excitatory network). In all cases we have σ = λ = 1. The simulations run over 10 seconds. Due to lack of space we do not show any results for the spin-glass network.

To summarize we have generalized and set on a firm mathematical basis the work of Sompolinsky et al., e.g. [START_REF] Crisanti | Path integral approach to random neural networks[END_REF], Helias et al., e.g. [START_REF] Alexander Van Meegen | Large-Deviation Approach to Random Recurrent Neuronal Networks: Parameter Inference and Fluctuation-Induced Transitions[END_REF]. We have also extended the work of Ben Arous and Guionnet, e.g. [START_REF] Benarous | Large deviations for Langevin spin glass dynamics[END_REF], and clarified and complemented the work of Faugeras et al., e.g. [START_REF] Faugeras | A constructive mean-field analysis of multi population neural networks with random synaptic weights and stochastic inputs[END_REF].

Figure 1 :

 1 Figure 1: Results of the Hopfield network mean-field estimation