Laura Durieux1

Victor Mathis1

Karine Herbeaux1

Marc-Antoine Muller1

Alexandra Barbelivien1

Chantal Mathis1

Rémy Schlichter3

• Sylvain

Monique Majchrzak1

Lucas Lecourtier1

Involvement of the lateral habenula in fear memory

Keywords: Amygdala, Hippocampus, Prefrontal cortex, c-fos, Chemogenetic, Context

Increasing evidence points to the engagement of the lateral habenula (LHb) in the selection of appropriate behavioral responses in aversive situations. However, very few data have been gathered with respect to its role in fear memory formation, especially in learning paradigms in which brain areas involved in cognitive processes like the hippocampus (HPC) and the medial prefrontal cortex (mPFC) are required. A paradigm of this sort is trace fear conditioning, in which an aversive event is preceded by a discrete stimulus, generally a tone, but without the close temporal contiguity allowing for their association based on amygdala-dependent information processing. In a first experiment, we analyzed cellular activations (c-Fos expression) induced by trace fear conditioning in subregions of the habenular complex, HPC, mPFC and amygdala using a factorial analysis to unravel functional networks through correlational analysis of data. This analysis suggested that distinct LHb subregions engaged in different aspects of conditioning, e.g. associative processes and onset of fear responses. In a second experiment, we performed chemogenetic LHb inactivation during the conditioning phase of the trace fear conditioning paradigm and subsequently assessed contextual and tone fear memories. Whereas LHb inactivation did not modify rat's behavior during conditioning, it induced contextual memory deficits and enhanced fear to the tone. These results demonstrate the involvement of the LHb in fear memory. They further suggest that the LHb is engaged in learning about threatening environments through the selection of relevant information predictive of a danger.

Introduction

The lateral habenula (LHb) plays a key role in the integration of basal ganglia and forebrain limbic information. It is implicated in functions such as reward prediction error, coding of negative motivational value, spatial memory, and subjective decision biases [START_REF] Matsumoto | Representation of negative motivational value in the primate lateral habenula[END_REF][START_REF] Stamatakis | Activation of lateral habenula inputs to the ventral midbrain promotes behavioral avoidance[END_REF][START_REF] Stopper | What's better for me? Fundamental role for lateral habenula in promoting subjective decision biases[END_REF][START_REF] Mathis | Excitatory transmission to the lateral habenula is critical for encoding and retrieval of spatial memory[END_REF][START_REF] Mathis | The lateral habenula interacts with the hypothalamopituitary adrenal axis response upon stressful cognitive demand in rats[END_REF][START_REF] Baker | Control of behavioral flexibility by the lateral habenula[END_REF]. One of the main roles of the LHb is to process stressful situations [START_REF] Stamatakis | Activation of lateral habenula inputs to the ventral midbrain promotes behavioral avoidance[END_REF][START_REF] Amo | The habenulo-raphe serotonergic circuit encodes an aversive expectation value essential for adaptive active avoidance of danger[END_REF][START_REF] Hennigan | Distinct midbrain and habenula pathways are involved in processing aversive events in humans[END_REF][START_REF] Chou | Social conflict resolution regulated by two dorsal habenular subregions in zebrafish[END_REF]. LHb activation has been reported following stressful experiences such as electrical footshocks, immobilization, and tail pinch [START_REF] Chastrette | Effects of daytime and nighttime stress on Fos-like immunoreactivity in the paraventricular nucleus of the hypothalamus, the habenula, and the posterior paraventricular nucleus of the thalamus[END_REF][START_REF] Wirtshafter | Dopamine agonists and stress produce different patterns of Fos-like immunoreactivity in the lateral habenula[END_REF][START_REF] Cullinan | Pattern and time course of immediate early gene expression in rat brain following acute stress[END_REF]. Moreover, the LHb is a main afferent of the rostromedial tegmental nucleus (RMTg), a modulator of the activity of dopamine (DA) neurons, a structure also activated following repeated foot-shock delivery (Sanchez-Catalan et al. 2017;Li et al. 2019a, b). In addition, activation of afferent pathways to the LHb, from the globus pallidus, the ventral tegmental area (VTA) or the lateral hypothalamus (LH), promote avoidance behaviors [START_REF] Lammel | Input-specific control of reward and aversion in the ventral tegmental area[END_REF][START_REF] Root | Role of glutamatergic projections from ventral tegmental area to lateral habenula in aversive conditioning[END_REF][START_REF] Lecca | Aversive stimuli drive hypothalamus-to-habenula excitation to promote escape behavior[END_REF], whereas silencing of the LH-LHb pathway impairs escape behavior [START_REF] Lecca | Aversive stimuli drive hypothalamus-to-habenula excitation to promote escape behavior[END_REF]). Altogether, these data indicate that the LHb plays a prominent role in the behavioral adaptation to aversive situations. LHb neurons were also reported to progressively increase their activity during the presentation of a conditioned stimulus (CS) that precedes footshock delivery (unconditioned stimulus, US) during avoidance learning [START_REF] Trusel | Punishment-predictive cues guide avoidance through potentiation of hypothalamus-to-habenula synapses[END_REF] and Pavlovian fear conditioning [START_REF] Wang | Role of the lateral habenula in shaping context-dependent locomotor activity during cognitive tasks[END_REF], while maintaining US responding, suggesting it is involved in the encoding of the predictive relationship between the CS and the US. However, only few studies [START_REF] Wang | Role of the lateral habenula in shaping context-dependent locomotor activity during cognitive tasks[END_REF][START_REF] Song | Lesions of the lateral habenula facilitate active avoidance learning and threat extinction[END_REF][START_REF] Barrett | Prefrontal-limbic functional connectivity during acquisition and extinction of conditioned fear[END_REF] shed light on its contribution to Pavlovian fear memory. Such a contribution can also be expected given that in rodents the LHb is directly or indirectly connected with the main structures involved in fear conditioning. i.e., the medial prefrontal cortex (mPFC), the amygdala, and the hippocampus (HPC). If contextual fear learning is sensitive to hippocampal manipulation, fear learning to a discrete CS only requires the HPC during trace fear conditioning paradigms, when the CS and the US are separated by an empty temporal interval named the trace interval [START_REF] Misane | Time-dependent involvement of the dorsal hippocampus in trace fear conditioning in mice[END_REF][START_REF] Chowdhury | Dorsal hippocampus involvement in trace fear conditioning with long, but not short, trace intervals in mice[END_REF][START_REF] Esclassan | Differential contribution of dorsal and ventral hippocampus to trace and delay fear conditioning[END_REF]. Importantly, the mPFC is also required for trace conditioning [START_REF] Gilmartin | Trace and contextual fear conditioning require neural activity and NMDA receptor-dependent transmission in the medial prefrontal cortex[END_REF][START_REF] Guimarais | Time Determines the neural circuit underlying associative fear learning[END_REF][START_REF] Gilmartin | Prefrontal activity links nonoverlapping events in memory[END_REF] as well as for contextual fear learning [START_REF] Gilmartin | Trace and contextual fear conditioning require neural activity and NMDA receptor-dependent transmission in the medial prefrontal cortex[END_REF]. Anatomically, the LHb receives direct projections from several subregions of the mPFC [START_REF] Kim | Topography of descending projections from anterior insular and medial prefrontal regions to the lateral habenula of the epithalamus in the rat[END_REF]; recently, a connection with the amygdala has been described in mice [START_REF] Kim | Functional connectivity of basolateral amygdala neurons carrying orexin receptors and melanin-concentrating hormone receptors in regulating sociability and mood-related behaviors[END_REF][START_REF] Zhou | A neural circuit for comorbid depressive symptoms in chronic pain[END_REF], although other anatomical investigations fail to report it (e.g. [START_REF] Zahm | Review of the cytology and connections of the lateral habenula, an avatar of adaptive behaving[END_REF], so this needs to be confirmed in the Rat. Although the LHb and dorsal HPC (dHPC) are not directly connected, electrophysiological recordings in the head-restrained as well as in behaving rats, demonstrated coherent activity between these two structures [START_REF] Aizawa | Molecular characterization of the subnuclei in rat habenula[END_REF][START_REF] Goutagny | Interactions between the lateral habenula and the hippocampus: implication for spatial memory processes[END_REF], strongly suggesting they exchange information. These findings, along with the known role of the LHb in HPC-dependent spatial memory [START_REF] Goutagny | Interactions between the lateral habenula and the hippocampus: implication for spatial memory processes[END_REF][START_REF] Mathis | Excitatory transmission to the lateral habenula is critical for encoding and retrieval of spatial memory[END_REF][START_REF] Mathis | The lateral habenula interacts with the hypothalamopituitary adrenal axis response upon stressful cognitive demand in rats[END_REF] and in mPFC-dependent working memory [START_REF] Mathis | The lateral habenula as a relay of cortical information to process working memory[END_REF], suggest that the LHb could contribute to both CS trace and contextual fear learning. To test this hypothesis, we used a trace conditioning protocol with a trace interval (30 s) long enough to allow fear conditioning to both the context and the tone [START_REF] Detert | Differential effects of training intertrial interval on acquisition of trace and long-delay fear conditioning in rats[END_REF]. In a first experiment, we questioned whether the LHb was part of the network sustaining trace fear conditioning acquisition by quantifying the expression of the c-Fos protein in the habenular complex, the HPC, the mPFC, and the amygdala; we analyzed the presence of co-activation among these structures using a factorial analysis design. In a second experiment we used a chemogenetic approach, with a modified muscarinic hM4(Gi) receptor, to perform LHb inhibition during conditioning and assessed its effects on fear elicited by re-exposure to the context and to the CS. In addition, we studied in the same rats the effects of LHb chemogenetic inactivation on plus-maze behaviors and home cage locomotor activity, both previously shown to be sensitive to LHb inactivation [START_REF] Mathis | Excitatory transmission to the lateral habenula is critical for encoding and retrieval of spatial memory[END_REF].

Materials and methods

Animals

This study, authorized by the French authorities (APAFIS#7114), required 66 male Long-Evans rats (250-350 g; Janvier Labs, France). They were housed in pairs on a 12 h light/dark cycle (lights on at 7:00 A.M.) with ad libitum access to food and water, controlled temperature (~ 23 °C), and a hygrometry of about 55%. Animals were distributed as follows: experiment 1: c-Fos study (n = 18); experiment 2: electrophysiological validation of the DREADD technique (n = 4) and behavioral study (n = 44). Testing took place between 9:30 am. and 2 pm.

Fear conditioning

Four conditioning chambers (25 cm x 27 cm × 18 cm) located in a lit-(6 lx) and sound-attenuating box (57 cm × 38 cm × 38 cm, Campden Instruments) were used. Chambers were made of transparent plastic with a loudspeaker fitted on one of the sidewalls, a transparent ceiling, and a grid floor (parallel 0.3 cm diameter stainless-steel bars spaced 0.8 cm apart) above a sawdust tray. A camera (MCT-210 MS, OptoVision, Toulouse, France) was fitted inside each box, above the center of the chamber, and monitored the entire chamber from the top through a 2.45 mm-wide angle lens. These chambers were used as conditioning context whereas triangular plexiglass boxes with one gray wall, one black and white striped wall, and a smooth white floor could be placed within the chambers and served as a new context. Chambers were cleaned with water between successive rats. Tones and electric shocks were delivered through a computerized interface (Med Associates Inc., St Albans, VT, USA). Fear conditioning (38 min-long session) was conducted as follows: rats were placed in the conditioning chamber and, after a 3 min baseline with no event, received six CS and US presentations (CS: 15 s tone, 4000 Hz, 10 dB above background; US: 0.5 mA before scrambling, 0.8 s, through the grid floor) with the US delivered 30 s after CS offset; the 6 presentations were made with a 4 min 20 ± 26 s intertrial interval. The automatic measurements of freezing behavior, defined as the suppression of all visible movements except those needed for breathing -and used as a measure of fear-, were performed as previously described [START_REF] Marchand | Evaluation of an improved automated analysis of freezing behaviour in rats and its use in trace fear conditioning[END_REF]. A set of procedures written under Excel® Visual Basic®, allowed the computation of the percentage of time spent freezing over blocks of selected duration.

Experiment 1. C-Fos expression related to fear conditioning A schematic representation of the experimental procedure is given in Fig. 1a. Tissue preparation and section processing Rats were conditioned as described above (Fear conditioning, FC, condition), or remained in their home cage (HC) to assess baseline c-Fos expression. An additional group of rats was exposed to the conditioning chambers with tones delivered exactly as during conditioning but without applying electric shocks (No shock, NS). Ninety minutes after the beginning of the session, rats were deeply anesthetized with pentobarbital overdose (120 mg/kg, i.p). Following intracardiac perfusion of phosphate-buffered saline (PBS,0.1 M) and then 4% paraformaldehyde (PFA)-PBS solution (pH 7.4; 4 °C), brains were removed, post-fixed in 4% PFAPBS (4 °C, 48 h), transferred into a 0.1 M PBS-20% sucrose solution (4° C, 48 h) and subsequently frozen (isopentane, -40 °C, 1 min). Serial 40 μm-thick free-floating sections were cut in the coronal plane at -20 °C and collected then stored in cryoprotectant at -20 °C. Immunohistochemistry Sections were first rinsed three times during 10 min in PBS before being soaked for 1 h in 5% normal horse serum in PBS/0.5% Triton X-100. They were subsequently transferred into the primary anti-Fos rabbit polyclonal antibody (1:750, polyclonal rabbit antibodies; SYSY; ref: 226 003, Synaptic System) solution for 20 h at room temperature, and then in a buffer solution containing biotinylated goat antirabbit secondary antibody for 1 h (1:500, Biotin-SP-conjugated affiniPure Goat anti-rabbit IgG, ref: BA1000, Vector). Staining was revealed with the avidin-biotin peroxidase method (Vectastain ABC kit, PK 6100; Vector Laboratories, Burlingame, CA, USA). After two PBS and one Tris (0,6%, pH 7,6) 10 min washes, sections were exposed for 8 min to a revelation solution (Kit DAB, SK 4100, Vector Laboratories) containing DAB (3,3′-diaminobenzidine) and H2O2 (hydrogen peroxide). Sections were then dropped on gelatinated slides, dried for 24 h, dehydrated by incrementally concentrated alcohol baths (70%, 90%, 95%, 100%, 100%), covered with Clearify (Americain MasterTech Scientific), fixed on microscopic slides with Diamount (Diapath S.P.A), and dried for 48 h, protected from light. Quantification Quantification of c-Fos + cells were performed (see Supplementary information and Supplementary Figs 123456) bilaterally in mPFC [prelimbic (PRL), infralimbic (IL), and anterior cingulate (ACC)], basolateral (BL) and lateral (LA) nuclei of the basolateral complex and central nucleus (CeA) of the amygdala, habenular complex, dorsal and ventral HPC (dHPC and vHPC respectively). The habenular complex was divided into a medial (MHb) subregion, and a lateral (LHb) subregion, the latter being further divided into a lateral (LHbL) and a medial (LHbM) part; in addition, we considered separately the LHb subregion in its most rostral part (rLHb) where LHbL and LHbM do not appear yet according to [START_REF] Paxinos | The rat brain in stereotaxic coordinates[END_REF]. Both the dHPC and vHPC were divided into ammonic fields 1 (CA1) and 3 (CA3), and dentate gyrus (DG). Countings were performed using a semi-automated method with ImageJ (Free License, Wayne Rasband, Research Services Branch, National Institute of Mental Health, Bethesda, Maryland, USA; for more details see Supplementary information and Supplementary Fig. 7) and expressed as mean number of c-Fos positive (c-Fos +) cells by mm2. The values of both hemispheres did not differ so that they were pooled (Supplementary Fig. 8). Factorial analysis Factorial analysis was conducted using the Statistica software. It was performed to unravel homogeneous functional networks, i.e. regional activities that are consistently correlated in FC and NS groups. The aim of this kind of analyses is to compress a large number of correlations into fewer factors. These factors are created through the computation of the correlation matrix of initial variables, without any prior (hypothesis-driven) input from the experimenter and represent ad hoc variables that show strong correlations with specific clusters of heavily intercorrelated observed variables. In addition to this ability to summarize an overwhelming number of covariations into fewer factors, this analysis is considered a way to unfold the underlying structure organizing these correlations. Applied to functional imaging studies, factorial analysis helps to unravel homogeneous functional networks (i.e. regional activities that are consistently correlated), as previously shown [START_REF] Veening | Activation of the septohippocampal system differentiates anxiety from fear in startle paradigms[END_REF][START_REF] Ali | Environmental enrichment enhances systems-level consolidation of a spatial memory after lesions of the ventral midline thalamus[END_REF]. Although this analysis initially extracts a high number of factors, only a set amount is classically described, as factors show diminishing relevance: they are determined in a series, each new one trying to bind residual correlations that had not yet been explained by preceding factors. The number of factors retained is determined by the observation of each factor's eigenvalue, describing the proportion of total variance that is explained by the adjunction of a supplementary factor relative to initial variables. For instance, an eigenvalue equal to 1 indicates that the corresponding factor explains exactly as much dispersion as one of these variables. To this aim, one of the most widely used criteria is the Guttman-Kaiser's: only factors with eigenvalues superior to 1 are conserved, to keep modeling of raw data from redundancy. When described, factors are associated with factor loadings of each observed variables, which correspond to correlations of the latter with the former, allowing for the description of the computed cluster, these correlations being considered strong above 0.6 or under -0.6. Computation of factorial analysis was conducted using varimax rotation, which increases discrepancies between these factor loadings on a given factor, thus clarifying the variables critically contributing to this factor's extraction.

Experiment 2. Effects of chemogenetic LHb inactivation on fear memory, anxiety, and locomotor activity A schematic representation of the experimental procedure is given in Fig. 1b. Surgery Rats underwent surgery under isoflurane anesthesia [4% for induction in an induction box (3 min), 1.5% throughout surgery] delivered in O2. Prior to surgery, they received a painkiller (meloxicam, 1 mg/kg, s.c.). After their head was shaved, rats were secured into a stereotactic apparatus, covered with an aluminum blanket to prevent hypothermia, and lidocaine (0.02 mg in 0.1 ml, s.c.) was injected at the incision location. Following incision, burr holes were drilled above the LHb. AAV8-CamKII-hM4(Gi)-mCherry [Viral Production Unit, Universitat Autonoma de Barcelona (VPU); [START_REF] Piedra | Development of a rapid, robust, and universal picogreen-based method to titer adeno-associated vectors[END_REF]] or phosphatebuffered saline (PBS) as a Sham operation was bilaterally injected into the LHb, by means of a 33gauge Hamilton syringe, at the two following locations and volumes: (1) anteroposterior (AP) = -3.3 mm from Bregma, mediolateral (ML) = ± 0.7 mm from the midline of the sagittal sinus, dorsoventral (DV) = -4.5 mm from dura (0.2 μL); (2) AP = -3.5 mm from Bregma, ML = ± 0.7 mm from the midline of the sagittal sinus, DV = -4.4 mm from dura (0.15 μL). Among the 44 rats of the behavioral study, 22 were injected with the viral vector and 22 with PBS. Once the injections terminated the syringe was left in place for 7 min and then slowly removed from the brain, before the skin was stitched. Animals were then placed in cages under a heating lamp until complete awakening. Rats received a second injection of painkiller (meloxicam, 1 mg/kg, s.c.) the following day. After a one week recovery period in individual cages, the same pre-surgery pairs of rats were housed together again. Experiments started three weeks after surgery (Fig. 1b). In vitro electrophysiological recordings The efficacy of the chemogenetic strategy was first tested using patch-clamp recordings. Three rats underwent the surgical procedure described above, to express hM4(Gi) receptors within the LHb, and one rat was used to assess the consequences of Clozapine N-Oxyde (CNO) administration on the excitability of LHb neurons not expressing the hM4(Gi) receptor. Acute slice preparation Rats were deeply anesthetized with an i.p. injection of a mixture of ketamine (82.5 mg/kg, Imalgène 1000, Merial) and xylasine (11 mg/kg, Rompun 2%, Bayer). Intracardiac perfusion was performed with 300 ml of cold (0-4 °C) sucrose-based artificial cerebrospinal fluid containing: 248 mM sucrose, 11 mM glucose, 26 mM NaHCO3, 2 mM KCl, 1.25 mM KH2PO4, 2 mM CaCl2, 1.3 mM MgSO4, and 2.5 mM kynurenic acid (bubbled with 95% O2 and 5% CO2). After 15 min, rats were decapitated, their brain removed and placed in a matrix allowing to split it in two at the level of the longitudinal sulcus and to remove the posterior part using a razor blade. Each brain half was mounted on the vibratome stage (VT1200S; Leica, Nussloch, Germany) by gluing the posterior end, the median plan facing the blade. This allowed us to perform transverse slices (thickness: 300 μm) and to start slicing from the midplane of each brain half. Slices were stored at room temperature in a chamber filled with artificial cerebrospinal fluid (aCSF) containing: 126 mM NaCl, 26 mM NaHCO3, 2.5 mM KCl, 1.25 mM NaH2PO4, 2 mM CaCl2, 2 mM MgCl2, and 10 mM glucose (bubbled with 95% O2 and 5% CO2; pH = 7.3; 310 mOsm measured). Patch-clamp recordings Slices were transferred to a recording chamber and continuously superfused with oxygenated aCSF at 34 °C. LHb neurons, selected based on the presence of mCherry fluorescence (hM4 neurons) or on its absence (no hM4 neurons), were recorded in the whole-cell configuration of the patch-clamp technique. Patch pipettes were pulled from borosilicate glass capillaries (Harvard Apparatus, Edenbridge, UK) with a P-1000 puller (Sutter Instruments, Novato, CA, USA). They were filled with a solution containing 140 mM KCl, 2 mM MgCl2, 10 mM HEPES, 2 mM MgATP, (pH = 7.3, adjusted with KOH; osmolarity, 310 mOsm, adjusted with sucrose) and had final tip resistances of 3.5-4.5 MΩ. Current-clamp experiments were performed with a Multiclamp 700 A amplifier (Molecular Devices, Union City, CA, USA) and recorded with Win-WCP or WinEDR softwares (John Dempster, University of Strathclyde). Recordings were performed at a holding current allowing to maintain the resting potential at ca. -60 mV. In the first set of experiments we investigated, on LHb neurons including hM4(Gi) receptors (n = 2 rats), the consequences of bath perfusion of CNO (5 μM) on firing frequency in response to intracellular injection of 1 s-lasting current pulses of increasing amplitude (from 0 to 240 pA, increments of 20 pA). Responses were recorded twice, once before and once following CNO bath application. In the second set of experiments, we investigated the consequences of local application of 1 μl CNO (500 μM) on spontaneous firing frequency of LHb neurons, with (n = 1 rat), and without (n = 1 rat) hM4(Gi) receptor expression, by means of puff application through a second pipette positioned just above the recorded LHb neurons. Spike frequency was analyzed with Clampfit 10.2 (Molecular Devices, Union City, CA, USA). This was done to be more precise and selective of the recorded neurons and to increase the chance to observe a recovery of neuronal firing to baseline levels, through the local administration of a very small amount of CNO solution. Behavioral study Drug treatments CNO (1 mg/kg; freshly prepared before each experiment and dissolved in 0.9% NaCl-0.5% DMSO), or vehicle (Veh, 0.9% NaCl-0.5% DMSO) were administered intraperitoneally. Sixteen hM4 animals and 16 Sham animals were administered CNO (hM4-CNO group and Sham-CNO respectively). Six rats of each surgery condition (hM4-Veh group and Sham-Veh group) were administered vehicle, hM4-Veh animals serving to control for possible adverse effects of the administration of the viral construct and subsequent inclusion of hM4 receptors at the cellular membrane. The Sham-CNO group was used to control for potential adverse effects of CNO [START_REF] Ilg | Behavioral effects of acute systemic low-dose clozapine in wild-type rats: implications for the use of DREADDs in behavioral neuroscience[END_REF][START_REF] Manvich | The DREADD agonist clozapine N-oxide (CNO) is reverse-metabolized to clozapine and produces clozapine-like interoceptive stimulus effects in rats and mice[END_REF][START_REF] Campbell | The use of chemogenetics in behavioural neuroscience: receptor variants, targeting approaches and caveats[END_REF] in our conditions. Fear conditioning The protocol took place over four consecutive days. Rats were subjected to each session in the same chamber. On day 1 (Conditioning), rats were conditioned as described above. On day 2 (Context test; 15 min-long), they were placed in the conditioning chamber to assess contextual fear. On day 3 (New context exposure; 15 min-long), rats were placed in the new context, the CS test context, to reduce baseline fear before CS fear assessment [START_REF] Jacobs | The accurate measurement of fear memory in pavlovian conditioning: resolving the baseline issue[END_REF]). On day 4 (CS test; 26 min-long), after a 2 min baseline in this context with no event, five CS were delivered with a 4 min interval. Rats received CNO or Veh on day 1, 30 min before the start of conditioning, and were administered Veh on each subsequent testing day, 30 min before the start of the session. Freezing scores were averaged on periods of interest on day 1 (the 3 min before the first tone presentation serving to assess baseline freezing), day 2 and day 3 (the 5 first min to capture the primary reaction to the conditioning context, and to the new context, respectively), and day 4 (the 15 s preceding the first tone being used to assess baseline, and the five 15 s tone presentations being used to assess fear conditioned to the CS). Elevated plus-maze It was made of black Plexiglas, elevated 73 cm above the floor, and consisted of four arms (50 cm × 10 cm), two comprising 40 cm-high walls (closed arms) and two comprising 1.5 cm-high borders (open arms). Light intensity was 10 lx in open arms, 7 lx at the center of the maze, and 2.5 lx in closed arms. The maze was cleaned with water and 70% ethanol between each rat. Thirty min after CNO or vehicle injections, rat were put in the maze for 5 min. The data analyzed were the total number of visits in the four arms, the number of visits in the open arms (in the percentage of the total number of visits), and the time spent in the open arms (in the percentage of the total time spent in the four arms). Locomotor activity Locomotor activity was assessed in the home cage (HC) by means of two infrared light-beams perpendicular to the width of the cage, placed 4.5 cm above floor level and 28 cm apart along the length of the cage. The consecutive interruptions of both light beams were counted as longitudinal crossings, whose numbers were monitored and saved in 15 min bins. Following a 1 h baseline, rats were administered CNO or Veh, and activity was recorded for an additional 2 h. Histology Tissue preparation and section processing were conducted as described in experiment 1, and sections were mounted on gelatin-coated slides with a DAPI-fluoromount medium. Observation of the spread of hM4(Gi) receptor expression (visualization of mCherry fluorescence) was performed with a Zeiss Apotome microscope (Zeiss, Muenchen, Germany). Statistical analyses For Experiment 1, freezing scores during each 15 s tone presentation and c-Fos + cells densities in the habenular complex were analyzed using one-way ANOVAs with repeated factor (Trial and Subregion, respectively). For the analyses of electrophysiological data from Experiment 2, one-and two-way ANOVAs (with Group as the factor and Current pulse amplitude or Time as the repeated measures) was used. For the analyses of all behavioral data from experiment 2, Group was used as the factor. For fear conditioning, freezing scores during baseline (day 1), reexposure to the conditioning context (day 2), and exposure to the new context (day 3) were analyzed using one-way ANOVAs. Twoway ANOVAs were used to compare groups through conditioning (factor Trial, freezing scores during each 15 s tone presentation, day 1) and during the test of conditioned fear to the CS (factor Tone, 15 s before the first CS presentation vs during CS, day 4). Data collected in the plusmaze and in the home cage were analyzed using one-and two-ways ANOVAs (with Time as the repeated measure), respectively. Post hoc tests used the Newman Keuls multiple range test (except when specified) when appropriate. Values of p < 0.05 were considered significant.

Results

Experiment 1

The increase in freezing scores to the tone during the session (Trial, F5,15 = 18.85, p < 0.0001, first tone vs second tone, p < 0.05, second tone vs third tone, p < 0.01, and asymptotic score-90% ± 6.3-thereafter) indicated that FC rats were successfully conditioned. Although a significant Trial effect was found in NS condition (F7,35 = 4.09, p < 0.01), the increase in behaviors coded as freezing by our automated system only reached its low (26 ± 9.4%) asymptotic level during the fifth tone presentation (p < 0.05 vs the previous ones), very unlikely to be indicative of the onset of unconditioned fear responses to this stimulus. As compared to the home cage, NS and FC conditions both induced low to marked c-Fos expression in all the investigated structures (Supplemental Table 1). In the habenular complex (Fig. 2 a,b), c-Fos + cells densities differed according to the subregion of both NS and FC rats (HC, F3,15 = 1.56, p > 0.2; NS, F3,21 = 31.17, p < 0.0001; FC, F3,9 = 17.38, p < 0.001). Post hoc comparisons indicated that the density of c-Fos + cells were higher in rLHb and LHbM than in LHbL and MHb in NS (p < 0.01 at least), and FC (p < 0.01 at least) conditions, and higher in the rLHb than in LHbM only in NS condition (p < 0.01). Factorial analyses of c-Fos data are shown in Table 1. In the NS condition, analysis of eigenvalues led to the conservation of a 3 factor-model accounting for 87.85% of the total variance observed. Factor 1 explained 65.34% of this variance and included positive loadings of the whole mPFC, the rLHb and LHbM, the whole CA3 region (ventral and dorsal), and the vDG. Factor 2 explained 15.8% of the total variance and included positive loadings of the CeA and LA, MHb, dDG, and vCA3. Factor 3 explained 6.73% of the total variance and included positive loadings of BL, LHbL, and vCA1. In the FC condition, analysis of eigenvalues led to the conservation of a 3 factor-model accounting for nearly 100% of the variance observed. Factor 1 explained 51% of the total variance and included positive loadings of the entire mPFC, the MHb, and the entire dHPC, and negative loadings of the BL and rLHb. Factor 2 explained 29.66% of the total variance and included positive loadings of CeA, LA, and LHbL, and negative loadings of LHbM and MHb. Factor 3 explained 19.34% of the total variance and included positive loadings of dCA3 and vCA3, and negative loadings of vCA1 and vDG. As illustrated in Fig. 2c, NS, and FC seem to engage the structures of interest but within distinct networks.

Experiment 2 CNO reduced the excitability of hM4(Gi)-containing LHb neurons CNO reduced action potentials (AP) firing in response to pulses up to 100 pA (F12,48 = 4.25, p < 0.001; Fig. 3a). For example, the AP frequency in response to 100 pA current pulses in the presence of CNO represented 46 ± 14% of the frequency recorded in the control condition. Note that under strong current pulses injection (> 100 pA) the inhibitory effect of CNO was overcome (see traces with 240 pA injection in Fig. 3b). To control for possible nonspecific effects of CNO, we examined the consequences of local puff applications of high CNO concentration (500 μM) on the spontaneous firing of LHb neurons with (hM4; n = 1 rat;n = 4 neurons) and without (no hM4; n = 1 rat; n = 5 neurons) hM4(Gi) receptors. Whereas CNO significantly reduced the firing frequency of hM4 neurons, it had no effects on the firing of no hM4 neurons (Time X Group, F7,49 = 3.83, p < 0.01; p < 0.05 at least, vs no hM4, Duncan post-hoc test; Fig. 3 c,d). Altogether, these results indicate that the effect of CNO was specific to neurons expressing hM4Gi receptors and strongly suggest that such an effect accounts for the behavioral alterations observed in hM4-CNO animals. Histology and groups size for the behavioral study Only rats in which mCherry expression was bilaterally mainly present in the LHb were kept. In these rats, hM4(Gi) receptors were expressed within the LHb in its entire rostrocaudal extent in both its lateral and medial parts (Fig. 4). In few of those rats, the expression of hM4 receptors also slightly impinges on the underlying paraventricular thalamic nucleus (PVT), particularly in its posterior part. Five hM4 rats were discarded because hM4(Gi) receptor expression spread to other surrounding structures. In addition, the two groups of rats which were administered Veh (hM4-Veh and Sham-Veh) were pooled into a single Ctl-Veh group after verification that their performances did not differ (Supplementary Information). Therefore, groups were composed as follows: hM4-CNO (n = 11), Sham-CNO (n = 16), Ctl-Veh (n = 12) and indicated as such in the figures. LHb inactivation altered conditioned fear memory Conditioning. All groups showed freezing scores close to 0% during baseline (F2,36 = 0.90, p > 0.4), and a similar increase of freezing to the tone during the session (Fig. 5a). Indeed, the analysis indicated no significant effect of Group (F2,36 = 1.44, p > 0.2), a significant effect of Trial (F5,180 = 20.32, p < 0.0001), and no significant Group x Trial interaction (F10,180 = 1.32, p > 0.2). This suggests that LHb inactivation did not affect the rate of fear learning. Context test (Fig. 5b). Reexposure to the training context-induced freezing behavior, but hM4-CNO group showed lower freezing score than the two other groups, the latter displaying similar freezing scores (Group, F2,36 = 3.86, p < 0.05; hM4-CNO vs Crt-Veh and Sham-CNO, p < 0.05 for each comparison). This suggests that LHb inactivation during conditioning impaired conditioned fear of the training context. New context exposure (Fig. 5c). Freezing scores were low in all groups and of a similar level (Group, F2,36 = 0.65, p > 0.5), suggesting that exposure to this new context did not induce generalized contextual fear, whatever the group. CS test (Fig. 5d). All groups showed similarly low freezing scores during baseline and enhanced freezing during CS presentation (Tone, F1,36 = 54.64, p < 0.0001), especially in the hM4-CNO group (Group, F2,36 = 5.88, p < 0.01; Tone x Group, F2,36 = 5.18, p < 0.05). Post hoc comparisons indicated that hM4-CNO rats showed significantly more freezing than the two control groups only during CS presentation (p < 0.05). This suggests that, if all groups displayed fear to the CS, LHb inactivation during conditioning enhanced such fear response. LHb inactivation induced mild anxiety and increased locomotor activity In the elevated plus-maze, LHb inactivation did not affect the total number of arm visits (Group, F2,36 = 1.29, p > 0.2; Fig. 6a), nor did it impact the percent time spent in the open arms, despite a tendency (Group, F2,36 = 2.13, p = 0.13; Fig. 6b right). However, the percent of open arms entries were lower in the LHb inactivated group than in the two other groups (Group, F2,36 = 4.21, p < 0.05 for each comparison; Fig. 6b left). With regard to home cage locomotor activity, there was no difference during baseline (Group, F2,36 = 0.89, p > 0.4; Time, F3,108 = 12.09, p < 0.0001; Group x Time interaction, F6,108 = 0.22, p > 0.9; Fig. 6c). LHb inactivation induced marked hyperactivity (bins 5-12; Group: F2,36 = 38.70, p < 0.0001; Time: F7,252 = 16.28, p < 0.0001; Group x Time interaction: F14,252 = 11.36, p < 0.0001). Noteworthy, this hyperactivity peaked 30 min following CNO administration, further validating our strategy to perform behavioral testing 30 min following CNO administration, and lasted approximately 75 min (Fig. 6c). Together, these results indicate that chemogenetic LHb inactivation induced only mild anxiety and a marked behavioral activation in a familiar (i.e. home cage), but not an unfamiliar -and anxiogenic-(i.e. the elevated plus maze) environment. These behavioral alterations have already been reported following LHb inactivation [START_REF] Mathis | Excitatory transmission to the lateral habenula is critical for encoding and retrieval of spatial memory[END_REF].

Discussion

This study aimed to assess whether the LHb contributes to fear memory formation using a long trace fear conditioning procedure. Our results indicate that it participates to both contextual and CS memories formation. Notably, whereas LHb inactivation did not affect fear expression during learning, demonstrating that it did not impair the ability of the rats to perceive the footshocks, it resulted in poor freezing levels upon re-exposure to the context, consistent with contextual memory deficit, whereas it exacerbated fear response to the CS, which is evocative of CS memory enhancement. It is unlikely that those effects resulted from partial inactivation of the adjoining paraventricular thalamic nucleus (PVT); in rats, PVT inactivation before conditioning did not impact fear retention [START_REF] Padilla-Coreano | A time-dependent role of midline thalamic nuclei in the retrieval of fear memory[END_REF]; also, PVT inactivation during a conditioning phase similar to ours induced in mice a decreased freezing during the CS retention test [START_REF] Penzo | The paraventricular thalamus controls a central amygdala fear circuit[END_REF], whereas here hM4-CNO rats showed increased freezing during the CS test. The contextual memory deficits induced by LHb chemogenetic inhibition can be discussed according to the two main processes leading to contextual fear conditioning: context encoding per se, and the association between the context and the aversive event. Context encoding rapidly takes place during conditioning through the elaboration of a conjunctive representation of the various cues of the conditioning chamber, a process mainly dependent on the dHPC [START_REF] Rudy | Hippocampal formation supports conditioning to memory of a context[END_REF]). An interaction between the LHb and dHPC might be critical to the proper encoding of contextual information. Such a view is strengthened by the presence of highly correlated metabolic activations in these structures during contextual fear conditioning [START_REF] González-Pardo | Different brain networks underlying the acquisition and expression of contextual fear conditioning: a metabolic mapping study[END_REF], as well as the depiction of electrophysiological communication between them [START_REF] Goutagny | Interactions between the lateral habenula and the hippocampus: implication for spatial memory processes[END_REF][START_REF] Aizawa | The synchronous activity of lateral habenular neurons is essential for regulating hippocampal theta oscillation[END_REF], including in a task involving spatial information processing [START_REF] Goutagny | Interactions between the lateral habenula and the hippocampus: implication for spatial memory processes[END_REF]. Strikingly, in parallel with contextual memory deficits, hM4-CNO rats were able to perform and memorize the CS-US association. LHb neurons were recently shown to increase CS activity while continuing to be activated by the US during aversive classical conditioning [START_REF] Wang | Role of the lateral habenula in shaping context-dependent locomotor activity during cognitive tasks[END_REF]) and avoidance learning [START_REF] Trusel | Punishment-predictive cues guide avoidance through potentiation of hypothalamus-to-habenula synapses[END_REF] suggesting that the LHb may be involved in the formation of CS-US association supporting predictive learning, at least when the CS is a perfect predictor of the US according to both its temporal and contingent relation with the US. In those conditions and based on this assumption, an alteration of LHb function may be expected to alter learning about the predictive value of the CS. However, LHb lesions were shown to facilitate avoidance learning [START_REF] Song | Lesions of the lateral habenula facilitate active avoidance learning and threat extinction[END_REF]) and contrasting effects of its lesion were reported on CS fear memory (unaffected, [START_REF] Heldt | Lesions of the habenula produce stressand dopamine-dependent alterations in prepulse inhibition and locomotion[END_REF][START_REF] Song | Lesions of the lateral habenula facilitate active avoidance learning and threat extinction[END_REF]; altered fear memory to both the CS and the context, [START_REF] Wang | Role of the lateral habenula in shaping context-dependent locomotor activity during cognitive tasks[END_REF]. The recent demonstration of coexistence of two distinct populations of LHb neurons displaying opposite responses to footshocks indicates that LHb encoding of aversive cues involves a complex excitation/ inhibition process [START_REF] Congiu | Opposite responses to aversive stimuli in lateral habenula neurons[END_REF], and our current results showing enhanced fear conditioned to a trace CS in hM4-CNO rats suggest a more complex role of the LHb in predictive learning. Conditioning protocols where the Pavlovian law of temporal contiguity is manipulated -from the introduction of a long time interval to a complete lack of contingency between the CS and the UStypically lead to an increased fear to the context at the expense of the CS. This phenomenon indicates the onset of overshadowing of the CS by the context, a process commonly thought to rely on competition between potential predictors at the time of conditioning [START_REF] Rescorla | Probability of shock in the presence and absence of CS in fear conditioning[END_REF][START_REF] Rescorla | A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and nonreinforcement[END_REF]. The fact that control animals (Ctl-Veh and Sham-CNO) displayed higher levels of freezing to the context than to the CS confirms the overshadowing effect in our long trace protocol. It is likely that an impaired ability to encode the context has led to a lack of competition with the CS, resulting in the ability of the latter to elicit increased fear responses in hM4-CNO rats. The Rescorla-Wagner model [START_REF] Rescorla | A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and nonreinforcement[END_REF] posits that differences in the emotional value acquired by the different cues is based on differences in the perceived salience of the US and of those cues. The lack of alteration of the rate of fear learning in hM4-CNO rats suggests unaltered stimuli salience processing; therefore, the role of the LHb may be to compute the relative pertinence of potential predictors (to shed light on the strongest one) and favor -or inhibit-their association with the aversive event. The reduced contextual-and enhanced CS-conditioned fear observed in hM4-CNO rats might thus be the consequence of an alteration of this computation. Interestingly, opposite changes in the acquired emotional value of contextual and CS cues were previously reported by Calandreau and collaborators (2005). Indeed, those authors have shown that LA inactivation during conditioning using a delay procedure (i.e. no trace interval between the CS and the US, leading to a strong overshadowing of contextual cues by the CS), reduced fear to the more predictive CS and enhanced fear to the less predictive context. Our results suggest that the LHb might modulate, along with the LA, fear learning according to the predictive value of competing cues. The c-Fos experiment results, rather than opposing two potential roles of the LHb in context encoding vs. association between stimuli, seem to be congruent with both, although with some anatomical peculiarities. First, the factorial analysis conducted in the NS and FC conditions led to different distributions of the investigated structure among factors. This indicates that even though neuronal activations were observed in the same structures in both conditions, as expected by the fact that all rats were exposed to novelty (see [START_REF] Milanovic | Production of the Fos protein after contextual fear conditioning of C57BL/6N mice[END_REF][START_REF] Radulovic | Relationship between Fos production and classical fear conditioning: effects of novelty, latent inhibition, and unconditioned stimulus preexposure[END_REF][START_REF] Cho | Brain-wide maps of Fos expression during fear learning and recall[END_REF], the onset of functional associations in these networks can be unraveled through correlational analysis of data. Specifically, the analysis of the FC group led to the extraction of 3 factors gathering neuronal ensembles classically described as sustaining different facets of information processing during fear conditioning. The first factor encompassed the whole dHPC and mPFC subregions, as well as the BL; it may represent the network implicated in contextual information processing (see for review [START_REF] Maren | The contextual brain: implications for fear conditioning, extinction and psychopathology[END_REF]). However, the BL shows a negative loading on this factor, even though it has been repeatedly reported to be the main locus of association between the context representation and the US in contextual fear conditioning (see for review Duvarci and Pare 2014). Thus, this factor likely underlines cognitive facets of information processing in contrast with associative and/or emotional ones. Such an interpretation is congruent with the involvement of both the mPFC and the dorsal HPC in broad cognitive functions (see for review [START_REF] Fanselow | Are the dorsal and ventral hippocampus functionally distinct structures?[END_REF][START_REF] Connor | The role of working memory and declarative memory in trace conditioning[END_REF]. The second factor, characterized by heavy loadings of the LA and the CeA, may sustain the associative process as well as the onset of fear responses (see for review [START_REF] Duvarci | Amygdala microcircuits controlling learned fear[END_REF]. Indeed, if CS-US association mainly takes place in the LA, the latter also supports context-US association [START_REF] Calandreau | A different recruitment of the lateral and basolateral amygdala promotes contextual or elemental conditioned association in Pavlovian fear conditioning[END_REF]. The mild positive loading of the BL on this factor might reflect associations between the US and its predictors. The third factor, characterized by heavy loadings on the whole ventral HPC, likely reflects emotional processes [START_REF] Fanselow | Are the dorsal and ventral hippocampus functionally distinct structures?[END_REF]. However, this factor puts forward CA3 activations in both dorsal and ventral portions of the HPC, in opposition to ventral DG and CA1 activations. This subfield of the HPC has historically been proposed as a relevant substratum to the encoding of polymodal representations such as a context [START_REF] Marr | Simple memory: a theory for archicortex[END_REF]). Thus, this factor might also underline the processing of contextual information. The fact that the BL (along with the CeA) also displays a mild positive loading on this factor suggests that it might represent contextual computations relevant to fear conditioning, in contrast with factor 1. Of utmost interest is the presence of habenular subregions within these ensembles. The association of the MHb with factors 1 and 2 suggests it contributes to both cognitive facets of information processing, i.e. associative learning and the onset of fear responses. The implication of the MHb in trace fear conditioning has not yet been investigated. This region in rodents [START_REF] Yamaguchi | Distinct roles of segregated transmission of the septohabenular pathway in anxiety and fear[END_REF][START_REF] Zhang | Presynaptic excitation via GABAB receptors in habenula cholinergic neurons regulates fear memory expression[END_REF][START_REF] Geng | ErbB4 receptors in the medial habenula regulate contextual fear memory[END_REF], or its equivalent in zebrafish (the dorsal habenula; [START_REF] Agetsuma | The habenula is crucial for experience-dependent modification of fear responses in zebrafish[END_REF], modulate fear responses, although [START_REF] Hsu | Dorsal Medial Habenula Regulation of mood-related behaviors and primary reinforcement by tachykinin-expressing habenula neurons[END_REF] found no contextual fear conditioning deficits following its lesion. The LHb is also included in these two factors, but with regional specificity. If the inclusion of its rostral part in factor 1 suggests its contribution to the cognitive aspects of fear conditioning, the inclusion in factor 2 of the LHbM and the LHbL, along with the LA and CeA, suggests they contribute to the associative process and/or to the onset of fear responses. The memory deficits observed following LHb inactivation appear in accordance with the position of the different LHb subdivisions within the clusters of structures revealed by the factorial analysis. Contextual memory deficits are in accordance with rLHb inactivation and the consecutive disturbance of the network including the dHPC (factor 1), whereas inappropriate CS-US association is in accordance with LHbM/ LHbL inactivation and the consecutive disturbance of the network encompassing the CeA and the LA (factor 2). The c-Fos pattern in the LHb is also interesting regarding its potential role as a modulator of DA transmission during fear conditioning. Indeed, the part of the LHb that we defined as "rostral" includes the LHbMA subdivision described by [START_REF] Andres | Subnuclear organization of the rat habenular complexes[END_REF] which neurons send excitatory projection to VTA DA neurons [START_REF] Metzger | Habenular connections with the dopaminergic and serotonergic system and their role in stressrelated psychiatric disorders[END_REF]). In addition, neurons situated in the more caudallateral region of the LHb send excitatory projection on GABA neurons of the RMTg [START_REF] Petzel | Anterior and posterior parts of the rat ventral tegmental area and the rostromedial tegmental nucleus receive topographically distinct afferents from the lateral habenular complex[END_REF] which in turn inhibit VTA DA neurons activity [START_REF] Brown | Habenula-induced inhibition of midbrain dopamine neurons is diminished by lesions of the rostromedial tegmental nucleus[END_REF]. The strong activation of these LHb subregions, where hM4(Gi) receptors were abundantly expressed, suggests that the observed deficits could be the consequence of the disturbance of the fine-tuning of DA transmission exert by the LHb during conditioning. Interestingly, infusion of D1-like receptor antagonist during training was shown to reduce trace CS fear memory when infused in the prelimbic region of the mPFC [START_REF] Runyan | Intra-medial prefrontal administration of SCH-23390 attenuates ERK phosphorylation and long-term memory for trace fear conditioning in rats[END_REF], and to increase contextual fear when infused in the shell part of the nucleus accumbens [START_REF] Albrechet-Souza | D(1)-like receptors in the nucleus accumbens shell regulate the expression of contextual fear conditioning and activity of the anterior cingulate cortex in rats[END_REF]. One can, therefore, postulate that the enhanced CS-and reduced contextual memory seen in hM4-CNO rats is consecutive to an increased DA flow within these two brain regions, due to the disinhibition of DA transmission by LHb inactivation, as suggested by previous work [START_REF] Lecourtier | Differential tonic influence of lateral habenula on prefrontal cortex and nucleus accumbens dopamine release[END_REF]. However, the fact that D1-like antagonist infusion during conditioning in other VTA DA targets, i.e. the dorsomedial PFC [START_REF] Stubbendorff | Dopamine D1-like receptors in the dorsomedial prefrontal cortex regulate contextual fear conditioning[END_REF], dHPC and BL [START_REF] Heath | Dopamine D1-like receptor signalling in the hippocampus and amygdala modulates the acquisition of contextual fear conditioning[END_REF] reduces contextual fear rather than increasing it, renders difficult to explain our results just by global disinhibition of the DA transmission. Moreover, D1-like antagonism and agonism were both shown to decrease contextual fear when infused in the LHb during conditioning [START_REF] Chan | Dopamine D1-like receptor in lateral habenula nucleus affects contextual fear memory and long-term potentiation in hippocampal CA1 in rats[END_REF]. Thus, if numerous evidence pointed to the involvement of DA in various target regions of VTA neurons during fear conditioning where and how perturbations of DA signaling upon LHb inactivation modulates fear learning remains to be explored. Besides DA, another key implication of the LHb during conditioning could be to contribute to the proper modulation of the hypothalamicpituitary-adrenal axis response. [START_REF] Kaouane | Glucocorticoids can induce PTSD-like memory impairments in mice[END_REF] have shown that acute restraint stress and postconditioning intrahippocampal, or systemic, corticosterone (CORT) injections, induce memory impairments reminiscent with those observed in the current study in hM4-CNO rats. Indeed, animals conditioned with an explicitly CS-US unpaired training procedure disregarded the context as the correct predictor of a footshock and showed fear of the unpredictive cue. We have previously shown that the blood CORT response to a stressful experience was higher in rats with inactivated LHb [START_REF] Mathis | The lateral habenula interacts with the hypothalamopituitary adrenal axis response upon stressful cognitive demand in rats[END_REF]; one can, therefore, hypothesize that a too large CORT release during conditioning has contributed to the memory impairments observed in hM4-CNO rats. Such a CORTmediated effect would be congruent with the general role of the LHb in the modulation of fear learning. In summary, we have shown that LHb inactivation during the conditioning phase of a trace fear conditioning paradigm led to contextual memory deficits along with an enhanced response to a discrete cue. This study improves our understanding of the neuroanatomical bases of fear memory by showing that the LHb is crucially involved in the selection of the more relevant cue predicting a danger.


Noël Brunetière
email: noel.brunetiere@univ-poitiers.fr

Michał Wodtke
email: mwodtke@pg.edu.pl

Considerations about the applicability of the Reynolds equation for analyzing high-speed near field levitation phenomena

Keywords: Near field levitation, Reynolds equation, CFD, High frequencies

This paper presents the results of investigations into the applicability of the

Reynolds equation for analyzing near field levitation (NFL) phenomena. Two separate approaches were developed, experimentally verified, and applied to meet the research objective. One was based on the Reynolds equation and the other was based on general conservation equations for fluid flow solved using computational fluid dynamic (CFD). Comparing the calculation results revealed that, for certain operating conditions, differences in the predicted system parameters appear. It was proven that both the Reynolds and Helmholtz numbers define border operating conditions for the applicability of the Reynolds equation to analyze NFL problems.

Introduction

The possibility to levitate objects offers an interesting alternative to eliminate friction and adhesive forces in micro-systems and allow accurate and controlled displacements of micro-objects [START_REF] Vandaele | Non-contact handling in microassembly: Acoustical levitation[END_REF][START_REF] Stolarski | Self-lifting contacts: From physical fundamentals to practical applications[END_REF]. Among all the possible technical solutions, Vandaele et al. [START_REF] Vandaele | Non-contact handling in microassembly: Acoustical levitation[END_REF] indicated that acoustic levitation is the most interesting solution because it is more versatile than other methods. The levitation is provided by a vibrating plate that modifies the air pressure distribution between the plate and the object, leading to a net force able to balance the weight of the object. Depending on the ratio of the air gap, h, to the acoustic wavelength, λ, two levitation modes can be identified [START_REF] Vandaele | Non-contact handling in microassembly: Acoustical levitation[END_REF]. If h is higher than λ, this is standing wave levitation; otherwise, it is near field levitation (NFL) or squeeze film levitation [START_REF] Stolarski | Self-lifting contacts: From physical fundamentals to practical applications[END_REF].

NFL, which provides much higher force [START_REF] Vandaele | Non-contact handling in microassembly: Acoustical levitation[END_REF], is the topic of this paper. One of the first experimental studies of squeeze film levitation was performed by Salbu in 1964 [START_REF] Salbu | Compressible Squeeze Films and Squeeze Bearings[END_REF]. A piston-like motion of the bottom plate was imposed by an electromagnetic actuator, allowing Salbu to measure the net force value as a function of the average distance between the plates. The piston-like motion, which can be easily measured and theoretically modeled, was adopted in several later studies where the electromagnetic actuator was replaced by a more efficient piezoelectric actuator [START_REF] Beck | Experiment and Analysis of a Flat Disk Squeeze-Film Bearing Including Effects of Supported Mass Motion[END_REF][START_REF] Hashimoto | Acoustic levitation of planar objects using a longitudinal vibration mode[END_REF][START_REF] Ueha | Non-contact transportation using near-field acoustic levitation[END_REF][START_REF] Ilssar | On the slow dynamics of near-field acoustically levitated objects under High excitation frequencies[END_REF][START_REF] Brunetière | Conditions of Lift-Off and Film Thickness in Squeeze Film Levitation[END_REF]. This solution can provide high levels of levitation force, estimated as up to 70 kNm -² by Ueha et al. [START_REF] Ueha | Non-contact transportation using near-field acoustic levitation[END_REF]. Even if, in many studies, the system was operated at its natural oscillation frequency (generally corresponding to several tenths of 1 kHz) to reduce the energy needed for actuating the bottom plate, a very stiff and massive plate is usually needed to ensure perfect motion of the top surface. To reduce the thickness of the bottom plate and allow lateral size increases, it was necessary to operate the system at its structural flexural mode. In this case, the bottom plate motion is more complex and controlled by structural dynamic behavior. This solution was adopted in many papers [START_REF] Ueha | Non-contact transportation using near-field acoustic levitation[END_REF][START_REF] Hashimoto | Near-field acoustic levitation of planar specimens using flexural vibration[END_REF][START_REF] Minikes | Coupled dynamics of a squeeze-film levitated mass and a vibrating piezoelectric disc: Numerical analysis and experimental study[END_REF][START_REF] Li | Modelling and experimental study on the influence of surface grooves on near-field acoustic levitation[END_REF]. This approach has been extended to create journal bearings [START_REF] Stolarski | Air journal bearing utilizing near-field acoustic levitation stationary shaft case[END_REF][START_REF] Atherton | Some fundamental aspects of selflevitating sliding contact bearings and their practical implementations[END_REF][START_REF] Stolarski | Acoustic journal bearing -A search for adequate configuration[END_REF][START_REF] Stolarski | Acoustic journal bearing-Performance under various load and speed conditions[END_REF][START_REF] Zhao | An ultrasonic levitation journal bearing able to control spindle center position[END_REF][START_REF] Stolarski | Acoustic Journal Bearing With Changeable Geometry and Built-in Flexibility[END_REF] or methods to move the levitated objects [START_REF] Hashimoto | Transporting objects without contact using flexural traveling waves[END_REF][START_REF] Matsuo | Holding characteristics of planar objects suspended by near-field acoustic levitation[END_REF][START_REF] Gabai | A rotational traveling wave based levitation device -Modelling, design, and control[END_REF][START_REF] Wei | Theoretical investigation and prototype design for non-parallel squeeze film movement platform driven by standing waves[END_REF][START_REF] Wei | Tangential motion mechanism and reverse hydrodynamic effects of acoustic platform with nonparallel squeeze film[END_REF]. In addition, the vibrating motion of the bottom plate can lead to the flexural vibration of the top plate, which makes the problem more complex [START_REF] Nomura | Theoretical and experimental examination of near-field acoustic levitation[END_REF].

The airflow between the vibrating plate and the levitated object is governed by the general Navier-Stokes set of equations. Depending on the configuration, some physical contributions vanish, and the equations can be simplified. When the levitation distance, h, is small compared to the characteristic width of the object, L, viscous effects are dominant and inertia effects in the fluid are negligible. It is thus possible to use the Reynolds equation to describe the flow in the air gap [START_REF] Reynolds | On the Theory of Lubrication and Its Application to Mr. Beauchamp Tower's Experiments[END_REF]. This approach has been successfully used by many authors [START_REF] Salbu | Compressible Squeeze Films and Squeeze Bearings[END_REF][START_REF] Brunetière | Conditions of Lift-Off and Film Thickness in Squeeze Film Levitation[END_REF][START_REF] Crandall | The Air-Damped Vibrating System: Theoretical Calibration of the Condenser Transmitter[END_REF][START_REF] Langlois | Isothermal squeeze films[END_REF][START_REF] Andrews | A comparison of squeeze-film theory with measurements on a microstructure[END_REF][START_REF] Stolarski | Numerical modeling and experimental verification of compressible squeeze film pressure[END_REF][START_REF] Garratt | A compressible flow model for the air-rotor-stator dynamics of a high-speed, squeeze-film thrust bearing[END_REF]. When the configuration is axisymmetrical and the air gap is a sine function of time, the Reynolds equation has an analytical solution involving Kelvin functions, as demonstrated by Crandall [START_REF] Crandall | The Air-Damped Vibrating System: Theoretical Calibration of the Condenser Transmitter[END_REF].

Crandall also showed that two regimes exist, depending on the value of what is currently named the squeeze number: σ = 12 µ ω R d ²/(p a h²) where µ is the viscosity of air, ω is the vibration pulsation, R d is the outer radius of the system, h is the average gap, and p a is the atmospheric pressure. At low squeeze numbers, the flow is mainly viscous, leading to a high damping coefficient for the film and low stiffness. However, at high squeeze numbers, the compressibility effect becomes dominant, damping is low, and the stiffness is maximal [START_REF] Crandall | The Air-Damped Vibrating System: Theoretical Calibration of the Condenser Transmitter[END_REF][START_REF] Andrews | A comparison of squeeze-film theory with measurements on a microstructure[END_REF]. In this last regime, the fluid behaves as an enclosed mass of air that is compressed between the plates without any radial flow.

The gas film can be described by Boyle's law [START_REF] Salbu | Compressible Squeeze Films and Squeeze Bearings[END_REF][START_REF] Garratt | A compressible flow model for the air-rotor-stator dynamics of a high-speed, squeeze-film thrust bearing[END_REF].

If it is assumed that the oscillating motion is fast enough so that no viscous fluid motion is observed, the Navier-Stokes equations can be reduced to the Euler set of equations.

The pressure distribution can thus be obtained from acoustic theory [START_REF] Stolarski | Self-lifting contacts: From physical fundamentals to practical applications[END_REF][START_REF] Hashimoto | Acoustic levitation of planar objects using a longitudinal vibration mode[END_REF][START_REF] Ueha | Non-contact transportation using near-field acoustic levitation[END_REF]. [START_REF] Brunetière | Conditions of Lift-Off and Film Thickness in Squeeze Film Levitation[END_REF].

Based on this last finding, it can be postulated that the Reynolds solution, which has the advantage of being solved at low computation cost, can be us ed to solve all near field problems. However, depending on the operating conditions, many authors found that additional effects not considered in the Reynolds equation must be accounted for.

Andrews et al. [START_REF] Andrews | A comparison of squeeze-film theory with measurements on a microstructure[END_REF] performed some tests at low ambient pressure (80 Torr) and showed that the rarefaction effect, inducing slip of the air on the plates, must be considered. Indeed, in this case, the mean free path of the gas, λg, is increased, leading to high values of the Knudsen number, Kn = λg/h. Based on the Knudsen definition, rarefaction effects can also be observed at low film thickness values, h [START_REF] Mohite | A compact squeeze-film model including inertia, compressibility, and rarefaction effects for perforated 3-D MEMS structures[END_REF].

Mohite et al. [START_REF] Mohite | A compact squeeze-film model including inertia, compressibility, and rarefaction effects for perforated 3-D MEMS structures[END_REF] also showed that fluid inertia can affect the behavior of the squeeze film when the magnitude of the Reynolds number, Re = h² is higher than 1. For small Re values, Stolarski and Chai [START_REF] Stolarski | Inertia effect in squeeze film air contact[END_REF] verified that the Reynolds equation is relevant.

Li et al. [START_REF] Li | Influence of gas inertia and edge effect on squeeze film in near field acoustic levitation[END_REF] also included the effect of convective inertia in a modified Reynolds equation. In addition, they considered the pressure drop occurring during fluid suction in the contact due to the sudden section reduction at the inlet. They showed that this effect becomes significant when the squeeze number increased. However, Minikes

and Bucher [START_REF] Minikes | Comparing numerical and analytical solutions for squeeze-film levitation force[END_REF] showed that edge effects, manifested as a pressure difference at the inlet, can be observed at low squeeze numbers due to flow interaction with the oscillating plate when the plate is bigger than the levitated object. Finally, in the case when one of the plates rotates, centrifugal inertia effects can be experienced [START_REF] Garratt | A compressible flow model for the air-rotor-stator dynamics of a high-speed, squeeze-film thrust bearing[END_REF]. This last point is beyond the scope of this work. These additional effects can make the Reynolds equation ineffective for solving the levitation problem, particularly when high oscillating frequencies are considered. As previously discussed, high operating frequencies are encountered in many applications where the system is used at its natural frequency. However, the limit of using Reynolds has not been clearly identified for this type of squeeze problem.

In this work, simulations were carried out with Navier-Stokes equations and the Reynolds equation. The results were compared to find the limit of validity for the Reynolds equation. Some deviations are observed at high oscillating frequencies and are related to acoustic wave propagation in the contact. It was possible to define a dimensionless number equivalent to the Helmholtz number to identify the limit of validity for the Reynolds equation.

Goal and scope of the research

The Reynolds equation was derived from general fluid dynamics conservation equations using a number of simplifying assumptions, one of them is neglecting fluid inertia. The goal of this work is to determine the range in which the assumptions of the Reynolds equation are acceptable for analyzing NFL problems. For this purpose, calculations of the piston-like motion system were carried out using two separate computational models that differ in their description of airflow in the air gap: a) a model Selected results of the experimental tests were also used to verify the correctness of developed models.

The calculations were carried out using both theoretical models for the same operating conditions and oscillation frequencies of the plate, assumed in a wide range. A comparison of the calculated levitation parameters showed differences between the results and determined the validity range of the results obtained from solving the Reynolds equation for analyzing an NFL system. The research was carried out assuming that oscillating mass and oscillating plate remain rigid during operation. That simplification could influences the whole system behavior in the case when relatively elastic elements were applied (thin and/or with a large diameter). However, despite of this assumption, conclusions about applicability of different approaches for investigations of the air flow in the gap, which is the goal of this paper, will be still valid.

Problem description

The principle of obtaining load capacity in an aerodynamic gas film, as a result of forced oscillations using a piston-like motion system, is illustrated schematically in Fig. 1. An oscillating plate, as a result of actuator operation, is nominally introduced into uniaxial oscillations. Above the plate, the levitating mass (disk-shaped) is placed. Between those elements, due to forced sinusoidal oscillations, an aerodynamic lubricating film is formed (on the principle of squeeze film effect). During system operation, pressure in the film exerts force on the levitated mass and causes its lift. As a result, a levitated mass is also introduced into oscillations, the amplitude of which depends on the frequency of the oscillating plate.

Methodology

Taking into account the operating principle of NFL, the aerodynamic film parameters (e.g. pressure field or film thickness) should be analyzed in the time domain. This applies both to the flow in the air gap and the displacement of the surfaces limiting it:

the oscillating plate and levitated mass. In the described problem, oscillating plate displacement is imposed and known while the mass elevation distance and its oscillations are the result of forces acting on the object and could be calculated using the fundamental dynamics law of motion.

CFD model

The model was developed using the CFX package [START_REF]ANSYS CFX -Pre Theory Guide[END_REF], which uses the finite volume method for discretizing and solving 3D conservation equations for transient flow of compressible fluids:

the continuity equation:

∂𝜌 ∂𝑡 + ∇ • (𝜌𝐔) = 0 (1)
the momentum equations:

∂(𝜌𝐔) ∂𝑡 + ∇ • (𝜌𝐔 𝐔) = -∇𝑝 + ∇ • 𝛕 (2)
which, together with the ideal gas equation of state:

𝜌 = 𝑀𝑝 𝑅𝑇 (3)
allows obtaining a solution to the air gap flow problem. Symbols in equations ( 1) -( 3) mean: ρ -air density, t -time, U -velocity vector, p -pressure, τ -stress tensor, M -air molar mass (28.96x10 -3 kg mol -1 ), Ttemperature, R -gas constant (8.314 J mol -1 K -1 ).

Since air is Newtonian fluid, stress tensor τ (in equations ( 2)) is proportional to velocity gradients ∇U and dynamic viscosity μ, according to Equation ( 4):

𝛕 = 𝜇(∇𝐔 + (∇𝐔) T - 2 3 𝛿∇ • 𝐔) (4)
where: δ is the Identity matrix. Mathematical operators in equations ( 1) to ( 4) are: ′∇ •' means divergence, ′𝐔 𝐔′ is the dyadic operator of velocity vectors U and ∇ means gradient.

Taking advantage of axial symmetry, the developed discrete model was simplified so that it represents only a part of the air gap with symmetry boundary conditions at the side walls (Fig. 2). At the outer wall of the model, an opening boundary condition was defined with the value of total pressure equal to ambient pressure. At the bottom and top walls, a 'no-slip' boundary condition was imposed. The bottom wall represents the surface of the oscillating plate and its displacement was imposed on the vertical axis (z in Fig. 2) using 'mesh motion' capability, according to Equation ( 5):

ℎ1 = -𝑒 • sin(2 • 𝜋 • 𝑓 • 𝑡) (5)
where: h1 is imposed position of the oscillation plate in vertical direction as a function of time t, e is oscillation amplitude, f is oscillation frequency. Consequently, the top wall movement was obtained with the Rigid Body solver using the solution of the dynamic equation of motion for one direction (using the Newmark integration scheme):

𝑚𝑧̈= 𝐹 aero + 𝑚𝑔 [START_REF] Ueha | Non-contact transportation using near-field acoustic levitation[END_REF] where m is the mass of the levitated object, kg; 𝑧̈ is the object's acceleration, ms -2 ; Faero is the force in the air gap due to the squeeze effect, N; and g is gravitational acceleration, ms -2 .

Analyses were carried out using a mesh divided into 500 finite volumes in the air gap radial direction, 20 in the film thickness direction and one in the circumferential direction. Mesh parameters were determined using the results of a mesh independence study. Angular size of the grid was assumed αg = 1, to get its adequate quality. Calculations were started with the assumed initial air gap thickness (usually 5 μm). During calculations, the thickness of the air gap changed as a result of interaction between structural and aerodynamic loads. Calculations were stopped after obtaining a stable and repeatable oscillating movement of the upper surface (levitation mass).

Each oscillation period was divided into 400 time steps. To achieve stable elevation of the levitated object and repeatable oscillations, one calculation task required, depending on the case, from several to hundreds of analyzed periods of enforced oscillation.

Reynolds model

The model using the solution of the Reynolds equation to describe flow in the air gap of an NFL problem, which was applied in this study, was presented in detail in work [START_REF] Brunetière | Conditions of Lift-Off and Film Thickness in Squeeze Film Levitation[END_REF]. The equations are solved in a home-made code written in Fortran 2003 and based on the Finite Volume Method. As discussed in the introduction, the Reynolds equation is a simplification of the Navier-Stokes equations due to the thin film configuration where only a part of the viscous contributions are preserved [START_REF] Reynolds | On the Theory of Lubrication and Its Application to Mr. Beauchamp Tower's Experiments[END_REF]. Therefore, the inertia terms are neglected and the pressure is constant in the through film direction. The fluid velocity profiles are thus parabolic and proportional to the radial pressure gradient. The Reynolds model, similar to the CFD model, used the axial symmetry of the analyzed system (for simplifying calculations) and the same set of boundary conditions and assumptions for air behavior in the gap (ideal gas). In addition, movement of the oscillation plate was imposed using Equation ( 5) and the levitated mass had one degree of freedom, with position calculated using the dynamic law of motion with Equation ( 6), as for the CFD model.

Input data

The geometrical and physical parameters of the system, as well as operating conditions used in analyses, are shown in Table 1. The air dynamic viscosity was assumed to be μ = 1.831x10 -5 Pas, gravitational acceleration was g = 9.81 ms -2 and temperature was T = 300 K.

Results

Experimental validation

The correctness of the developed theoretical models was evaluated using the results of experimental measurements obtained using a test stand in the piston-like-motion NFL problem configuration. Details regarding the construction of the test stand were presented in [START_REF] Brunetière | Conditions of Lift-Off and Film Thickness in Squeeze Film Levitation[END_REF]. Due to the limitations of the test stand, it was only possible to carry out measurements for oscillation frequencies up to 4 kHz. Test parameters were the same as shown in Table 1. Each test was repeated three times. The maximum difference in average mass distance between the different tests is 0.2 µm indicating an excellent reproducibility of the experiments. In the case of the RMS value of levitated mass displacement, which is a measure of similarity of calculated and experimentally observed oscillations of the levitated object, very good agreement was noticed for the whole analyzed range of frequencies. Experimental validation of the theoretical results for high frequencies was not possible with the use of available equipment. Good agreement for low frequencies confirmed, that both models work well, and predicted parameters of NFAL system were very close to measured data. However, when try to expand that statement also for high frequencies, it could be supposed, that simplifications introduced to the Reynolds approach (e.g. assumption of parabolic velocity profile across gap or no inertia effects) can lead to not adequate results, because flow inertia has presumably strong influence on operating parameters of the system in such conditions.

Parametric study

Parametric studies were carried out for the frequency range of the oscillating plate, from 2 to 60 kHz. The calculated average mass distance is compared in Fig. 4. The highest mass levitation height was obtained for the lowest analyzed frequency.

With increased oscillation frequency up to 10 kHz, mass lift distance decreased. At higher excitation frequencies (> 10 kHz), the lifting height did not depend strongly on the oscillation frequency. It is worth noting that results obtained from both theoretical models showed the same trend of changes, whereas average mass lift distances calculated using CFD were slightly higher than those obtained from the Reynolds model. However, differences in results for frequencies < 6 kHz were relatively low (about 3%). For higher oscillation frequencies, differences between the results of CFD and Reynolds models were slightly higher (up to 8%).

In Fig. 5, the calculated maximum air flow rate at the gap border (inlet/outlet) (a) and maximum air gap pressure (b) are compared. Similar to the average mass distance, the trend of changes for air flow rate as a function of frequency, predicted using both models, was the same (Fig. 5a). Except for the initial frequency range (f < 4 kHz), the flow rate increased with increasing oscillation frequency; while for frequencies above 20 kHz, the calculated flow rate using the Reynolds model was higher than for CFD.

The trends of maximum pressure changes in the film (Fig. 5b) calculated by the CFD and Reynolds models look slightly different. In the initial frequency range for both models (<20 kHz), the maximum pressure in the film decreased with increasing frequency. For higher frequencies, this trend is also visible in the results obtained from the Reynolds model; however, the calculated change in the maximum pressure value was relatively small. The opposite trend was noticed for results obtained from the CFD model, which for frequencies > 20 kHz, the calculated maximum pressure in the air film increased with increases in the oscillation frequency. In the case of analyzed quantities at frequencies up to 20 kHz, both models predicted nearly identical parameter values (Fig. 5). For higher frequencies (> 20kHz), a comparison of the results obtained from the CFD and Reynolds models shows visible differences, which increase with increasing oscillation plate frequency.

Discussion

The derivation of the Reynolds equation assumes that the inertia effects in the flow are negligible. Consequently, one of the main reasons for the deviation between the Reynolds and CFD solutions is the appearance of a significant inertia effect. Based on the work of Mohite et al. [START_REF] Mohite | A compact squeeze-film model including inertia, compressibility, and rarefaction effects for perforated 3-D MEMS structures[END_REF], the Reynolds equation for the problem can be defined as:

𝑅𝑒 = 𝜌𝜔ℎ 2 𝜇 ( 7
)
where h is the averaged levitation height in one period. Fig. 4 shows the levitation height calculated by both methods as a function of the oscillation frequency. It has been replotted as a function of Reynolds number in Fig. 6. It is clear from this graph that the deviation occurs when the Re becomes higher than 1, confirming that the deviation is due to inertia effects in the gas flow.

Doing the same analysis for the maximum pressure and air-flow presented in Fig. 5 does not give satisfactory results, as can be seen in Fig. 7. Indeed, the deviation starts at Re much higher than 1. Another physical phenomenon should explain this difference. This point is discussed in the following part of this section. To analyze this assumption, the Helmholtz number of the flow, defined as the ratio of the time necessary for acoustic waves to travel across the contact divided by the oscillation period, is used. It is:

𝐻 = 𝑅 𝑑 𝑓 𝑎 ( 8
)
where a is the speed of sound. When Fig. 5 is modified to be presented as a function of H, it appears that the deviation occurs when the Helmholtz number becomes higher than 1, confirming our assumption (see Fig. 10). When the oscillation period is lower than the time necessary for the pressure information to travel across the contact, a To analyze more deeply the effect of the acoustic waves propagation on the simulation results, air velocity profiles across the film thickness at r = 14 mm calculated with the use of Reynolds and CFD solutions were compared in Fig. 11. First of all, it can be seen that at H values much lower than 1, the velocity profiles obtained with both approaches were very close and with a parabolic shape (2 kHz case, Fig. 11a). When the frequency was increased to get H values higher than 1 (Fig. 11b andc), some significant differences can be observed. CFD solution showed that the velocity profiles are no more parabolic certainly because of the fluid acceleration which can be significant at high frequency. Secondly, the velocity calculated with the Reynolds solution is, at some particular time period, inverted when compared to the exact solution. This is due to the pressure gradients that are not correctly calculated with the Reynolds equation. Indeed, the pressure waves can provided pressure gradient inversions not captured with the Reynolds equation (see Fig. 8). This last part clearly confirms, that when the Helmholtz number is higher than 1, air flow in the NFAL system should be analyzed with the use of Navier-Stokes set of equations.

Summary and conclusion

The gas flow in an NFL system can be simulated by the Navier-Stokes approach, which is more general, and the Reynolds equation, which is a simplified approach. The latter has the advantage of a very low computation time.

These two methods were satisfactory compared to experiments performed on a levitated disk at low-frequency values (a few kHz). However, many near field levitation systems work at high frequency (several 10 kHz) for which the Reynolds equation is not necessarily valid. A comparison of the results obtained with the two methods was performed on a range of frequencies.

It was shown that, as expected, the Reynolds method deviates from the NS solution when the Reynolds number (Eq. 6) is higher than 1. A second deviation was observed at higher oscillation frequency. It appears when the plate oscillation period is shorter than the time necessary for pressure information to radially travel through the gas film.

It exactly corresponds to a Helmholtz number (Eq. 7) of 1. Above this value, some acoustic pressure waves, which cannot be captured by the Reynolds equation, appear in the gas film.

It is thus necessary to check these two conditions before using the Reynolds equation to simulate the gas flow during near field acoustic levitation.

  frequency (pulsation of the vibration)

  using general fluid dynamics conservation equations, b) a model using the Reynolds equation. The developed models considered both time-varying flow in the air gap and the movement of the levitated mass caused by aerodynamic force. Investigations were carried out for object geometry and oscillation parameters accessible at the test stand.

Fig. 1 .

 1 Fig. 1. Configuration of the problem.

  It was assumed that airflow was isothermal, laminar, without external forces acting on the fluid and air satisfies the ideal gas assumptions. Within the scope of this work, two different approaches for describing airflow were implemented: a) The computational fluid dynamics model (CFD model) based on the solution of the set of general conservation equations of the fluid flow, which for a Newtonian fluid, reduces to the Navier-Stokes set of equations, b) Reynolds model based on the solution of the Reynolds equation.

Fig. 2 .

 2 Fig. 2. Schematic view of the CFD mesh, (a) side view; (b) top view.

Fig. 3

 3 Fig.3shows a comparison of the measured and calculated values of average levitated mass distance (a) and mass RMS (Root Mean Square) displacement (b). Similar average levitated mass distances were calculated using both models (the maximum relative difference was not larger than 3.5%). Measured values differ from calculated ones, especially for low frequencies; however, for frequencies over 3 kHz, good agreement was observed. Slightly better agreement observed for the Reynolds approach results is rather accidental and is probably caused by uncertainty of experimental data. Surfaces planarity flatness errors or deviations from assumed oscillating plate motions are possible examples of measurement inaccuracies sources.Two initial points from experimental research show measured film thickness under conditions of non-fully developed squeeze film lubrication. Direct contact between cooperating surfaces took place for lowest tested frequencies. Such situation was not possible to consider with the use of developed methodology for investigations of NFAL system operation.

Fig. 3 .

 3 Fig. 3. Comparison of the results of calculations and measurements; (a) average mass distance, μm; (b) mass RMS displacement, μm.

Fig. 4 .

 4 Fig. 4. Average mass distance, µm, as a function of oscillation frequency, comparison between CFD and Reynolds model results.

Fig. 5 .

 5 Fig. 5. Air flow rate, x10 -5 kg s -1 ; (a) and maximum film pressure, Pa (b) as a function of oscillation frequency, comparison between CFD and Reynolds model results.

Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .

 6789 Fig. 6. Average mass distance, µm, as a function of Reynolds number, comparison between CFD and Reynolds model results.

Fig. 10 .

 10 Fig. 10. Air flow, x10 -5 kg s -1 ; (a) and maximum pressure, Pa, (b) as a function of Helmholtz number, comparison between CFD and Reynolds model results.

Fig. 11 .

 11 Fig. 11. Comparison of the calculated velocity profiles across air gap (at r = 14 mm) for selected time points of the oscillation period obtained from CFD and Reynolds models; (a) 2 kHz; (b) 30 kHz; (c) 60 kHz.

Table 1 .

 1 Problem input data.

	parameter	value
	mass, m	19.28x10 -3 kg
	mass radius, R d	15 mm
	oscillation amplitude, e	1 μm
	oscillation frequency, f	2 -60 kHz

Acknowledgments

This work was partially carried out as a part of research grant no. 2016/23/B/ST8/00210 entitled "Study on the use of Poisson's acoustic levitation effect for non-contact transportation of light objects" financed by the Polish National Science Centre. CFD calculations were carried out at the Academic Computer Centre (TASK) in Gdańsk. This work pertains to the French Government program "Investissements d'Avenir" (LABEX INTERACTIFS, reference ANR-11-LABX-0017-01).