Dr Armelle Rancillac
email: armelle.rancillac@college-de-france.fr

Hélène Geoffroy

Jean Rossier

Impaired neurovascular coupling in the APPxPS1 mouse model of Alzheimer's disease

Keywords: Blood vessels, somatosensory cortex, DMPP, Nitric Oxide, interneurons, U46619, Neuropeptide Y, nicotinic cholinergic receptors

The tight coupling between neuronal activity and the local increase of blood flow termed neurovascular coupling is essential for normal brain function. This mechanism of regulation is compromised in Alzheimer's Disease (AD). In order to determine whether a purely vascular dysfunction or a neuronal alteration of blood vessels diameter control could be responsible for the impaired neurovascular coupling observed in AD, blood vessels reactivity in response to different pharmacological stimulations was examined in double transgenic APPxPS1 mice model of AD.

Blood vessels movements were monitored using infrared videomicroscopy ex vivo, in cortical slices of 8 monthold APPxPS1 and wild type (WT) mice. We quantified vasomotor responses induced either by direct blood vessel stimulation with a thromboxane A2 analogue, the U46619 (9,11-dideoxy-11a,9aepoxymethanoprostaglandin F2α) or via the stimulation of interneurons with the nicotinic acetylcholine receptor (nAChRs) agonist DMPP (1,1-Dimethyl-4-phenylpiperazinium iodide). Using both types of stimulation, no significant differences were detected for the amplitude of blood vessel diameter changes between the transgenic APPxPS1 mice model of AD and WT mice, although the kinetics of recovery were slower in APPxPS1 mice.

We find that activation of neocortical interneurons with DMPP induced both vasodilation via Nitric Oxide (NO) release and constriction via Neuropeptide Y (NPY) release. However, we observed a smaller proportion of reactive blood vessels following a neuronal activation in transgenic mice compared with WT mice. Altogether, these results suggest that in this mouse model of AD, deficiency in the cortical neurovascular coupling essentially results from a neuronal and not a vascular dysfunction.

Introduction

Alzheimer's Disease (AD) is a multifactorial and progressive neurodegenerative pathology associated with various vascular dysfunctions. During AD development, levels of beta amyloid (A) peptides increase and form depositions in neuritic plaques and in cerebral blood vessels, causing cerebral amyloid angiopathy (CAA).

Disturbances in neuronal circuits, such as loss of neurons and synapses [START_REF] Selkoe | Alzheimer's disease is a synaptic failure[END_REF][START_REF] Morrison | Selective vulnerability of corticocortical and hippocampal circuits in aging and Alzheimer's disease[END_REF][START_REF] Palop | Aberrant excitatory neuronal activity and compensatory remodeling of inhibitory hippocampal circuits in mouse models of Alzheimer's disease[END_REF] as well as progressive vascular dysregulation [START_REF] Iadecola | Neurovascular regulation in the normal brain and in Alzheimer's disease[END_REF] are well known features of this pathology.

Neurons and astrocytes are known to function in a close metabolic and structural interdependence with blood vessels forming the so called neurovascular unit [START_REF] Cauli | Revisiting the role of neurons in neurovascular coupling[END_REF][START_REF] Lecrux | The neurovascular unit in brain function and disease[END_REF]. The tight coupling between increased neuronal activity and local control of cerebral blood vessels diameter known as neurovascular coupling or functional hyperemia is precociously altered in AD patients [START_REF] Hock | Decrease in parietal cerebral hemoglobin oxygenation during performance of a verbal fluency task in patients with Alzheimer's disease monitored by means of near-infrared spectroscopy (NIRS)--correlation with simultaneous rCBF-PET measurements[END_REF][START_REF] Mentis | Increasing required neural response to expose abnormal brain function in mild versus moderate or severe Alzheimer's disease: PET study using parametric visual stimulation[END_REF][START_REF] Rombouts | Functional MR imaging in Alzheimer's disease during memory encoding[END_REF][START_REF] Zlokovic | Neurovascular pathways to neurodegeneration in Alzheimer's disease and other disorders[END_REF] and in various AD transgenic mouse models [START_REF] Iadecola | Neurovascular regulation in the normal brain and in Alzheimer's disease[END_REF][START_REF] Hamel | Oxidative stress and cerebrovascular dysfunction in mouse models of Alzheimer's disease[END_REF][START_REF] Iadecola | SOD1 rescues cerebral endothelial dysfunction in mice overexpressing amyloid precursor protein[END_REF][START_REF] Nicolakakis | Neurovascular function in Alzheimer's disease patients and experimental models[END_REF]. In AD, ultrastructural changes affecting diverse properties of the neurovascular unit have been observed, such as capillary distortions, blood-brain-barrier (BBB) dysfunction, losses of endothelial mitochondria, thickening of the basement membrane, endothelial degeneration, increase in degererative pericytes, perivascular neuroinflammation [START_REF] Shi | Vascular abnormalities: the insidious pathogenesis of Alzheimer's disease[END_REF]. Moreover, early in the physiopathological development of the disease, the AD brain is characterized by an altered perfusion to activated areas [START_REF] Brown | Review: Cerebral microvascular pathology in aging and neurodegeneration[END_REF]. These ultrastructural changes affecting the neurovascular unit result in disturbed regulatory mechanisms and contribute to cerebral damage. This neurovascular dysfunction observed in AD could result from purely vascular alterations or from a neurodegenerative process [START_REF] De La Torre | Vascular basis of Alzheimer's pathogenesis[END_REF][START_REF] Zlokovic | Neurovascular mechanisms of Alzheimer's neurodegeneration[END_REF][START_REF] Benarroch | Neurovascular unit dysfunction: A vascular component of Alzheimer disease?[END_REF] and a progressive loss of the neuronal control of blood vessels tonus.

Using pharmacological stimulation of blood vessels within cortical slices, the objective of this study was to determine ex vivo, the vascular or neuronal origin of the impaired neurovascular coupling occurring during AD.

To this aim, here we used 8 month-old double transgenic mice overexpressing human mutated genes of both human amyloid precursor protein (APP751 with both the Swedish and London mutations; SL) and humanpresenilin 1 (PS1M146L) [START_REF] Blanchard | Time sequence of maturation of dystrophic neurites associated with Abeta deposits in APP/PS1 transgenic mice[END_REF]. This APPxPS1 mouse model of AD develops early-onset brain amyloidosis (3-4 months), undergoes substantial cell loss and CAA could first be detected from about 20 weeks of age. We performed direct stimulation of blood vessels using a thromboxane A2 analogue, the U46619 (9,11dideoxy-11a,9a-epoxymethanoprostaglandin F2α) acting on G-protein coupled thromboxane receptor located on endothelial cells of blood vessels [START_REF] Narumiya | Structure and function of prostanoid receptors[END_REF], or indirect stimulation via neuronal activation using the nicotinic acetylcholine receptor (nAchRs) agonist DMPP (1,1-Dimethyl-4-phenylpiperazinium iodide). Indeed, in the neocortex, the neurovascular coupling is in part mediated by a direct interaction with interneurons [START_REF] Cauli | Revisiting the role of neurons in neurovascular coupling[END_REF][START_REF] Cauli | Cortical GABA interneurons in neurovascular coupling: relays for subcortical vasoactive pathways[END_REF][START_REF] Rancillac | Glutamatergic Control of Microvascular Tone by Distinct GABA Neurons in the Cerebellum[END_REF] via the release of vasoactive messengers. As functional postsynaptic nicotinic acetylcholine receptors (nAchRs) are only expressed by non Fast Spiking (FS) neocortical inhibitory interneurons [START_REF] Porter | Selective excitation of subtypes of neocortical interneurons by nicotinic receptors[END_REF][START_REF] Xiang | Cholinergic switching within neocortical inhibitory networks[END_REF][START_REF] Gulledge | Heterogeneity of phasic cholinergic signaling in neocortical neurons[END_REF], DMPP allows a specific neuronal activation.

Here, we observed that for both direct and indirect stimulation protocols, the amplitudes of evoked blood vessel movements were not significantly different in APPxPS1 versus WT mouse. We find that activation of neocortical interneurons with DMPP could dilate blood vessels via Nitric Oxide (NO) or constrict blood vessels via Neuropepide Y (NPY) release. However, theis neuronal stimulation using DMPP induced less vasomotor responses in APPxPS1 compared to WT mice. These findings suggest that a loss of neuronal control rather than a purely vascular impairment underlie the neurovascular alteration occurring during the development of AD.

Material and Methods

Animals

We used nine wild type C57Bl6J (WT) and six APPxPS1 transgenic 8 month-old male mice, as previously described [START_REF] Blanchard | Time sequence of maturation of dystrophic neurites associated with Abeta deposits in APP/PS1 transgenic mice[END_REF]. Transgenic mice (Tg) were generated by crossing homozygous PS1 (carrying presenilin 1 M146L mutation under the control of the HMG-CoA reductase promoter) with hemizygous APPSL mice (hAPP751 transgene with the Swedish K670N/M671L and London V717I mutations, under the control of the Thy1 promoter) to obtain APPxPS1 transgenics.

All animals were housed in a temperature-controlled (21-25°C) room under daylight conditions. They arrived in the laboratory 2 weeks before initiating experiments to acclimate to their new environment. Experimental procedures were conducted in strict compliance with approved institutional protocols and in accordance with the provisions for animal care and use described in the European Communities Council directive of 24 November 1986 (86-16-09/EEC). Experiments were performed in compliance and following approval of the Sanofi-Aventis Animal Care and use Committee.

Slice preparation

Mice were anesthetized using halothane and decapitated. Brains were quickly removed and placed into cold (~4 °C) cutting solution containing (in mM): 110 choline chloride, 11.6 Na-ascorbate, 7 MgCl2, 2.5 KCl, 1.25 NaH2PO4, and 0.5 CaCl2, continuously bubbled with 95% O2 -5% CO2. Coronal brain slices (300 µm thick) comprising the somatosensory cortex were cut with a vibratome (VT1200S; Leica, Nussloch, Germany), and transferred to a holding chamber filled with artificial cerebrospinal fluid (ACSF), containing (in mM): 126 NaCl, 2.5 KCl, 1.25 NaH2PO4, 2 CaCl2, 1 MgCl2, 26 NaHCO3, 20 glucose, and 1 kynurenic acid (a nonspecific glutamate receptor antagonist, Sigma, France), constantly oxygenated (95% O2 -5% CO2) and held at room temperature. Individual slices were next placed in a submerged recording chamber kept at 32°C and perfused (1.5 ml/min) with oxygenated ACSF (in the absence of kynurenic acid). Blood vessels were visualized in the slice using infrared videomicroscopy with Dodt gradient contrast optics.

Drugs

Thrombaxane A2 receptors and nicotinic acetylcholine receptors agonists (9,11-dideoxy-11a,9aepoxymethanoprostaglandin F2α, U46619, 50 nM and 1,1-Dimethyl-4-phenylpiperazinium iodide, DMPP, 100 µM, respectively, Sigma, France) were added in the ACSF perfusion of the cortical slices for 6 min. To block the NOS, slices were treated for at least 1 h with an irreversible inhibitor of constitutive nitric oxide synthase (nNOS) and a reversible inhibitor of inducible nitric oxide synthase (iNOS), Nω-Nitro-L-arginine (L-NNA; 100 µM; Sigma, France). NPY Y1 receptors were blocked with the selective antagonist N2-(Diphenylacetyl)-N-[(4hydroxyphenyl)methyl]-D-arginine amide (BIBP 3226; 100 µM; Sigma, France).

Vascular reactivity

Blood vessel movements were monitored at 32°C by infrared videomicroscopy on cortical slices. Blood vessels remaining in focal plane for more than 50 µm, exhibiting a well-defined luminal diameter (10-20 µm) and located in supragranular layers of the somatosensory cortex were selected for vascular reactivity. Images were acquired every 15 s using Image Pro Plus 7.0 (MediaCybernetics, San Diego, CA) and control baselines of blood vessel diameters were determined for 5 min. Blood vessels with unstable baseline were discarded from the analysis. U46619 or DMPP were applied during 6 minutes after stable baseline period. Luminal diameters were quantified off-line at different locations along the blood vessel using custom written routines running within IgorPro (Wavemetrics) to determine the most reactive part of the vessel. Maximal vasomotor responses were then expressed as percentages of the mean baseline diameter during the control period.

Statistical analyses

To determine the statistical significance of the vasomotor responses, changes in diameter were compared using a paired t-test at the time before the onset of the drug application where the mean diameter change is closest to 0% and at the time where the maximal response was observed. Vascular reactivities in control WT versus APPxPS1 mice were compared using t-test. To compare the proportions of reactive blood vessels in WT versus APPxPS1, we used a z test. The Area Under the Curve (AUC) for the vascular response of each blood vessel was calculated using the trapezoidal rule. For U46619 study, the AUC was calculated as the incremental or decremental values from the baseline obtained during the 27 minutes (from time 3 to 30 minutes after U46619 onset application).

For DMPP study, the AUC was calculated as the incremental or decremental values from the baseline obtained during the 12 minutes (from time 0 to 12 minutes after DMPP onset application). Comparisons of AUC among genotype groups were performed by Mann Whitney U test. The AUC was expressed in arbitrary units.

Histology and Immumohistochemistry

Following the recordings of blood vessel reactivity, slices were fixed over night. Amyloid deposits were labeled by standard Congo red staining [START_REF] Wilcock | Quantification of cerebral amyloid angiopathy and parenchymal amyloid plaques with Congo red histochemical stain[END_REF], and amyloid loads were manually delineated and quantified with Photoshop CS3. A total of 2-3 measures were performed and averaged for every mouse. Evaluation of amyloid loads was performed in the barrel somatosensory cortex and in the CA1 region of the hippocampus. Blood vessels in transgenic versus WT mice were assessed by immunostaining for basement membrane protein collagen-IV (goat anti-collagen type IV; 1: 400; Chemicon). Blood vessels density was measured using computer-based thresholding methods in Photoshop CS3. A total of 2-3 measures were performed and averaged for every mouse in the barrel somatosensory cortex.

Results

Amyloid deposition and vascular density in 8 month-old APPxPS1 mice

In order to examine the vascular reactivity in the somatosensory cortex of 8 months APPxPS1 versus WT mice, we first looked at the amyloid depositions and the vascular density in this region of interest. Neuropathological analyses revealed in APPxPS1 the presence of numerous extracellular aggregated (Congo red positive) A deposits (Fig. 1A-C) whereas none could be observed in WT mice. Interestingly, we observed that A deposits in the cortex were more numerous and bigger in deep layers, compared to supragranular layers (Fig. 1A andC). Some Congo red positive blood vessels were observed (Fig. 1B), indicating a cerebral amyloid angiopathy at 8 months in this APPxPS1 mice. We quantified Congo red positive aggregates in the cortex and the CA1 field of the hippocampus. Amyloid loads represented 4.53 ± 0.57% of the parenchymal tissue in the somatosensory cortex, and 5.41 ± 0.66% in the CA1 region (no significant difference between these regions) (Fig. 1E).

In the cortex, immunostaining of collagen-IV to label blood vessels revealed no difference in the mean vascular density in APPxPS1 (12.67 ± 1.48%) compared to the WT mice (15.15 ± 2.61%) (Fig. 1 D andF).

Direct stimulation of blood vessels using U46619

To determine if purely vascular dysfunctions or a loss of neural control underlay the neurovascular coupling impairment observed in AD, we examined blood vessels reactivity in 8 months APPxPS1 versus WT mice.

Using infrared videomicroscopy on cortical slice preparations, we first investigated a potential vascular dysfunction by using direct pharmacological stimulation of blood vessels. We applied a thromboxane A2 stable analogue, the U46619 (50 nM) for 6 minutes on acute WT or APPxPS1 cortical slices. We focused our study on penetrating arterioles that are known to be of prime importance in feeding deeply located microvessels and neurons [START_REF] Nishimura | Penetrating arterioles are a bottleneck in the perfusion of neocortex[END_REF]. Well defined arterioles of supragranular layers were selected for quantitative analyses. This direct vascular stimulation induced strong, significant and reversible vasoconstrictions in both WT (80.82 ± 6.59%, n = 8; p<0.05) and APPxPS1 mice (80.83 ± 3.79%, n = 10; p<0.001) (Fig. 2, Table 1). The diameter change measured either at 14 minutes after the onset of U46619 application, where the maximum of the response was reached, or at 30 minutes after the onset of U46619 application, did not differ significantly between APPxPS1 and WT mice. However, when the response of each arteriole to the U46619 application was expressed as the area under the response curve (AUC), the percent AUC for the U46619-induced vasoconstriction was significantly highest at 30 minutes in APPxPS1 mice compared to WT mice (325.15 ± 47.31 and 183.94 ± 37.91 respectively, P<0.05, Table 1). Finally, the proportion of blood vessels that constrict in response to U46619 in APPxPS1 mice (10 out of 18) compared to WT mice (8 out of 14) did not differ significantly (p>0.68, Table 1). These results suggest that, at 8 months of age, blood vessels recovery following a direct stimulation is slower but reactivity remains functional in these transgenic mice.

Indirect stimulation of blood vessels via neuronal activation using DMPP

Next, in order to investigate whether neuronal release of vasoactive compounds was altered in APPxPS1 mice, we used the nicotinic receptor agonist DMPP (100 µM) to stimulate interneurons. DMPP induced, both in WT or APPxPS1 mice, either vasodilatations (105.21 ± 1.21%, n = 12 and 106.08 ± 2.91%, n = 3, respectively), or vasoconstrictions (95.52 ± 1.08%, n = 7 and 93.42 ± 4.77%, n = 4, respectively; Table 1, Fig. 3). Here again, no significant differences (p>0.56) were observed for the mean vasodilatation or vasoconstriction amplitudes between both lines of mice. However, the proportion of blood vessels reactive to DMPP was smaller in APPxPS1 (7 out of 20) compared to WT (19 out of 27) mice (p<0.05, z-test). In particular, vasodilation were less frequently induced by DMPP in transgenic mice (3 out of 20; 15%) compared to WT mice (12 out of 27; 44%; Table 1). Following DMPP stimulation, proportion of vasoconstriction in transgenic mice (7 out of 27; 26%) was not significantly different compared to WT mice (4 out of 19, 20%; z-test, P>0.8; Table 1). However, no significant difference of AUC for DMPP-induced vasodilation or vasoconstriction were observed in APPxPS1 mice compared with WT mice (P>0.64, Table 1).

These results suggest that the impairment of the functional hyperemia observed in AD is primarily due to a defect in the neuronal control of blood vessels rather than to a pure vascular dysfunction.

Vasodilations induced by DMPP application are mediated by NO whereas vasoconstrictions are due to NPY release

The stimulation of GABAergic interneurons can induce vasomotricity via the production of vasoactive substances [START_REF] Cauli | Revisiting the role of neurons in neurovascular coupling[END_REF][START_REF] Cauli | Cortical GABA interneurons in neurovascular coupling: relays for subcortical vasoactive pathways[END_REF][START_REF] Rancillac | Glutamatergic Control of Microvascular Tone by Distinct GABA Neurons in the Cerebellum[END_REF]. DMPP depolarizes nicotinic cholinergic receptors (nAChR)-expressing interneurons that were shown to be also responsive to serotonergic via the serotonin 5-hydroxytryptamine 3A (5-HT3A) receptor [START_REF] Lee | The largest group of superficial neocortical GABAergic interneurons expresses ionotropic serotonin receptors[END_REF]. As the pharmacological stimulation of this receptor was recently reported to bi-directionally control arterioles diameter by the release of NO to dilate, or NPY to constrict [START_REF] Perrenoud | Activation of cortical 5-HT(3) receptor-expressing interneurons induces NO mediated vasodilatations and NPY mediated vasoconstrictions[END_REF], we hypothesized that DMPP induced vasomotor changes could also be due to NO or NPY release. Therefore, in order to determine the molecular events underlying vasomotor changes following DMPP stimulation, we successively blocked different possible mechanisms. Lowering basal NO levels by treatment with the constitutive nNOS inhibitor L-NNA (100 µM) prevent all vasodilations in response to DMPP applications. Only vasoconstrictions were recorded (94.65 ± 2.41 of baseline diameter; n = 7/16, P<0.01) (figure 4A). These results suggest that DMPP-induced vasodilations are mediated by NO release.

Then, to determine the molecular pathway underlying vasoconstrictions, treatment of DMPP was reproduced in the presence of NPY Y1 receptor antagonist (BIBP 3226, 1 µM). Indeed, vasoconstrictions mediated by NPY are known to be mediated by smooth muscle NPY Y1 vascular receptor [START_REF] Abounader | Expression of neuropeptide Y receptors mRNA and protein in human brain vessels and cerebromicrovascular cells in culture[END_REF]. Under BIBP 3226, constrictions where blocked and only dilations (106.24 ± 2.02%; n = 5/13; P<0.01) could be recorded (figure 4B). Amplitudes of vasomotor responses under these different conditions were not statistically different from control condition.

Altogether, these data strongly suggest that pharmacological stimulations of nAchRs-expressing interneurons induce vasodilations through NO release, whereas they induce vasoconstriction through NPY release and activation of its Y1 receptor.

Discussion

The alteration of the neurovascular coupling occurring during AD, could result either from a purely vascular disorder or from a neurodegenerative process leading to a loss of blood vessels control by neuronal activity. Indeed, increasing evidence suggests a causal role for vascular disorder in the development of AD. Vascular risk factors including hypertension, diabetes, and hypercholesterolemia increase the risk of incident AD dementia [START_REF] Li | Vascular risk factors promote conversion from mild cognitive impairment to Alzheimer disease[END_REF] and regional cerebral hypoperfusion is one of the earliest pathological features of AD. On the other hand, various modes of neurodegeneration can coexist in a same mouse model and lead to reduced blood vessels innervation [START_REF] Yang | Neuronal apoptosis and autophagy cross talk in aging PS/APP mice, a model of Alzheimer's disease[END_REF]. Previous findings already shown that A accumulates in neurons [START_REF] Cook | Alzheimer's A beta(1-42) is generated in the endoplasmic reticulum/intermediate compartment of NT2N cells[END_REF][START_REF] Hartmann | Distinct sites of intracellular production for Alzheimer's disease A beta40/42 amyloid peptides[END_REF][START_REF] Xu | Generation of Alzheimer beta-amyloid protein in the trans-Golgi network in the apparent absence of vesicle formation[END_REF][START_REF] Greenfield | Endoplasmic reticulum and trans-Golgi network generate distinct populations of Alzheimer beta-amyloid peptides[END_REF], as it was also observed at 2 months in this model [START_REF] Langui | Subcellular topography of neuronal Abeta peptide in APPxPS1 transgenic mice[END_REF], which might cause a synaptic degeneration [START_REF] Selkoe | Alzheimer's disease is a synaptic failure[END_REF] and lead to neuronal death [START_REF] Schmitz | Hippocampal neuron loss exceeds amyloid plaque load in a transgenic mouse model of Alzheimer's disease[END_REF]. Extracellular amyolid- induces significant entorhinal neuronal loss of principal and SOM/NPY neurons in these APPxPS1 mice [START_REF] Moreno-Gonzalez | Extracellular amyloid-beta and cytotoxic glial activation induce significant entorhinal neuron loss in young PS1(M146L)/APP(751SL) mice[END_REF]. Moreover, autophagy is induced but impaired in affected neurons in the AD brain [START_REF] Nixon | Extensive involvement of autophagy in Alzheimer disease: an immuno-electron microscopy study[END_REF][START_REF] Yu | Macroautophagy--a novel Beta-amyloid peptide-generating pathway activated in Alzheimer's disease[END_REF], causing autophagic vacuoles to accumulate profusely in dystrophic neuritis. This axonopathy disturbs neuronal trafficking [START_REF] Wirths | Axonopathy in an APP/PS1 transgenic mouse model of Alzheimer's disease[END_REF] and could perturb vasomotor peptidergic release.

Vascular dysfunction

In the present study, to determine if purely vascular or neuronal dysfunction were responsible for the decreased neurovascular coupling occurring during AD, we used either direct blood vessels stimulation, or neuronal activation, while recording vasomotor changes. Here, no differences could be observed in the peak amplitude of evoked blood vessels movements between WT and APPxPS1 mice using either type of stimulation. However, the AUC for the U46619-induced vasoconstriction was significantly highest in APPxPS1 mice compared to WT mice, accordingly to in vivo studies that reported in other models of AD that vasoconstrictions are usually exaggerated [START_REF] Niwa | Abeta 1-40-related reduction in functional hyperemia in mouse neocortex during somatosensory activation[END_REF][START_REF] Christie | Structural and functional disruption of vascular smooth muscle cells in a transgenic mouse model of amyloid angiopathy[END_REF][START_REF] Niwa | Alterations in cerebral blood flow and glucose utilization in mice overexpressing the amyloid precursor protein[END_REF]. This suggests reduced vessel elasticity, which could be due to CAA. Indeed, CAA was first detected from about 20 weeks of age and developed beyond the age of 40 weeks in this mouse line, with a high prevalenceof the A40 isoform forming amyloid deposits in the vascular system [START_REF] El Tayara | Detection of vascular alterations by in vivo magnetic resonance angiography and histology in APP/PS1 mouse model of Alzheimer's disease[END_REF]. The blood a vessel inelasticity could compromise mental function by preventing neural regions from reacting to new activity by adapting blood supply [START_REF] De La Torre | Can disturbed brain microcirculation cause Alzheimer's disease?[END_REF].

Neuronal dysfunction

Here, we observed that following a neuronal stimulation, reactive vessels were less numerous in transgenic mice compared to WT mice. Therefore, it is likely that neurodegeneration rather than a purely vascular impairment could be responsible for the alteration of neurovascular coupling in AD. Indeed, a neuronal loss in the entorhinal cortex [START_REF] Moreno-Gonzalez | Extracellular amyloid-beta and cytotoxic glial activation induce significant entorhinal neuron loss in young PS1(M146L)/APP(751SL) mice[END_REF] and an extensive, selective and early neurodegeneration of the dendritic inhibitory interneurons of the hippocampus have been observed in this model. At 6 months of age, a diminution of 50-60% of SOM-immunopositive neurons was noted [START_REF] Ramos | Early neuropathology of somatostatin/NPY GABAergic cells in the hippocampus of a PS1xAPP transgenic model of Alzheimer's disease[END_REF], in accordance with the loss of SOM and/or NPY neurons frequently reported in AD patients and the linear correlation between SOM and/or NPY deficiency and A content which have been reported [START_REF] Ramos | Early neuropathology of somatostatin/NPY GABAergic cells in the hippocampus of a PS1xAPP transgenic model of Alzheimer's disease[END_REF]. As extracellular A deposits are observed in the hippocampus and in the neocortex (Fig. 1) and [START_REF] Langui | Subcellular topography of neuronal Abeta peptide in APPxPS1 transgenic mice[END_REF], this suggest that a neurodegeneration of cortical interneurons could be responsible for the decreased proportion of reactive blood vessels following a neuronal stimulation observed in this study.

Alternatively, Alzheimer's disease is characterized by large losses of nicotinic cholinergic receptors [START_REF] Whitehouse | Cholinergic receptors in aging and Alzheimer's disease[END_REF][START_REF] James | Genetic and environmental aspects of the role of nicotinic receptors in neurodegenerative disorders: emphasis on Alzheimer's disease and Parkinson's disease[END_REF], that could also contribute to the lack of reactivity following DMPP application.

Dual role of nAChR-expressing interneurons

Our results indicate that activation of nAChR-expressing interneurons induces a complex vascular response. Indeed the selective nAChR agonist DMPP induced either constrictions (37%) or dilations (63%) of penetrating arterioles within supragranular layers. All vasoconstrictions were abolished in the presence of the NPY receptor antagonist (BIBP 3226), suggesting that they were elicited by NPY release. This result confirm and extend prior studies implicating NPY in vasoconstrictions [START_REF] Cauli | Cortical GABA interneurons in neurovascular coupling: relays for subcortical vasoactive pathways[END_REF][START_REF] Abounader | Characterization of neuropeptide Y (NPY) receptors in human cerebral arteries with selective agonists and the new Y1 antagonist BIBP 3226[END_REF][START_REF] Dacey | Vasomotor responses of rat intracerebral arterioles to vasoactive intestinal peptide, substance P, neuropeptide Y, and bradykinin[END_REF].

Conversely, DMPP-induced dilations were blocked in the presence of a nNOS inhibitor suggesting that NO is the predominant messenger inducing the vasodilation, in line previous studies performed in the cerebellum [START_REF] Rancillac | Glutamatergic Control of Microvascular Tone by Distinct GABA Neurons in the Cerebellum[END_REF][START_REF] Yang | Nitric oxide is the predominant mediator of cerebellar hyperemia during somatosensory activation in rats[END_REF][START_REF] Yang | Stellate neurons mediate functional hyperemia in the cerebellar molecular layer[END_REF] and in the cortex [START_REF] Cauli | Cortical GABA interneurons in neurovascular coupling: relays for subcortical vasoactive pathways[END_REF][START_REF] Brown | Fluorescent imaging of nitric oxide production in neuronal varicosities associated with intraparenchymal arterioles in rat hippocampal slices[END_REF][START_REF] Lovick | Neurovascular relationships in hippocampal slices: physiological and anatomical studies of mechanisms underlying flow-metabolism coupling in intraparenchymal microvessels[END_REF][START_REF] Estrada | Nitric oxide-producing neurons in the neocortex: morphological and functional relationship with intraparenchymal microvasculature[END_REF][START_REF] Tong | Basal forebrain nitric oxide synthase (NOS)-containing neurons project to microvessels and NOS neurons in the rat neocortex: cellular basis for cortical blood flow regulation[END_REF][START_REF] Liu | Interaction of nitric oxide, 20-HETE, and EETs during functional hyperemia in whisker barrel cortex[END_REF]. This result confirm the previous observation that nAChRs activation increased levels of the metabolites of NO, and therefore presumably of NO [START_REF] Pogun | Nicotine modulates nitric oxide in rat brain[END_REF]. Together, our observations that in the somatosensory cortex a selective nAChR agonist induce NO mediated vasodilations confirms and extends prior studies reinforcing the central role of NO in the neurovascular coupling.

Role of astrocytes in the vascular response to DMPP

Astrocytes are known as cellular intermediaries that couple neuronal activity to local blood flow changes through the phospholipase A2 (PLA2)-mediated synthesis of arachidonic acid, which leads to production of prostaglandins and epoxyeicosatrienoic acids [START_REF] Petzold | Role of astrocytes in neurovascular coupling[END_REF].

In the present study, we demonstrated that following DMPP stimulation, vasoconstrictions were mediated by NPY release. This neuropeptide is released by interneurons and directly activates NPY Y1 receptors located on vascular smooth muscle cells. Otherwise, we demonstrated that following DMPPP stimulation, vasodilations were mediated by NO release. NO is known to either directly acts on vascular smooth muscle cells through cGMP-PKG [START_REF] Garthwaite | Nitric oxide signaling in the central nervous system[END_REF][START_REF] Jaffrey | Nitric oxide: a neural messenger[END_REF] or indirectly interacts with numerous signaling pathways also involved in the vascular control [START_REF] Fleming | Cytochrome p450 and vascular homeostasis[END_REF][START_REF] Fujimoto | Effects of reactive oxygen and nitrogen species on cyclooxygenase-1 and -2 activities[END_REF][START_REF] Salvemini | Regulation of cyclooxygenase enzymes by nitric oxide[END_REF]. In particular, NO could induce astrocytic cytosolic Ca 2+ increased [START_REF] Willmott | A fundamental role for the nitric oxide-G-kinase signaling pathway in mediating intercellular Ca(2+) waves in glia[END_REF]. However, in vivo data strongly suggest that the involvement of astrocytes is likely to occur in the late phase of the neurovascular coupling [START_REF] Calcinaghi | Metabotropic glutamate receptor mGluR5 is not involved in the early hemodynamic response[END_REF][START_REF] Wang | Astrocytic Ca2+ signaling evoked by sensory stimulation in vivo[END_REF]. Even if further experiments would be necessary to fully characterize the NO mediated vasodilations following DMPP application, it appears that in our stimulation conditions, no other components possibly released by astrocytes are involved in the neurovascular coupling.

Conclusion

In this study, we observed by infrared videomicroscopy, the cortical vascular response induced by either a vascular activator, the thomboxane A2 analogue, or by a neuronal activator, the nicotinic agonist DMPP in an APPxPS1 transgenic mouse model of AD. No differences were observed in the amplitude of vascular responses between WT and transgenic mice following both types of activators. Therefore, in this mouse model of AD a neuronal alteration rather than vascular changes seems responsible for the impaired neurovascular coupling.

Figure 1 :

 1 Figure 1: Neuropathological profile of 8 month-old APPxPS1 mice. A-C, APPxPS1 develop brain amyloidosis as evidenced by numerous Congo red positive plaques in the parenchyma and blood vessels. B and C are magnifications of a blood vessel positive for Congo red and a parenchymal amyloid deposit, respectively, as outlined in A. D, Immunolabbeling of collagen IV revealed blood vessels architecture within the cortex. E, Regional amyloid loads (mean ± sem) were calculated from Congo red sections in different brain region (somatosensory cortex and CA1). Note that the relative volume occupied by amyloid loads is not significantly different in CA1 versus Cortex of APPxPS1 mice. F, Vascular density (mean ± sem) was calculated from collagen IV immunostaining in the somatosensory cortex.

Figure 2 :

 2 Figure 2: Vasoconstrictions induced by direct stimulation of blood vessels using U46619 were not significantly different in APPxPS1 compared to WT mice. A, Mean vascular constrictions ± sem induced by U46619 (50 nM) in 8 month-old APPxPS1 (n = 10, open circles) and WT (n = 8, filled lozenges) mice. B, Infrared images of an intracortical blood vessel that reversibly constricted in response to bath application of U46619 (50 nM) in an APPxPS1 mouse. C, At 14 min (T14),vasoconstrictions shown in (A) were significant in WT (p<0.05) as in APPxPS1 (p<0.001), compared to values at T0. No significant differences were detected at 14 min, or at 30 min, between these two lines of mice (p>0.3).

Figure 3 :

 3 Figure 3: Vasodilatations and vasoconstrictions induced by indirect blood vessels stimulation, via DMPP induced interneurons activation, were not significantly different in APPxPS1 compared to WT mice. A, Mean vascular dilatations (n = 12) and constrictions (n = 7) ± sem induced by DMPP (100 µM) in 8 month-old WT mice. B, Mean vascular dilatations (n = 3) and constrictions (n = 4) ± sem induced by DMPP (100 µM) in 8 month-old transgenic APPxPS1 mice. C, Mean maximal or minimal responses for vasodilatation and vasoconstriction, respectively, were compared to initial values (at T0) for WT and APPxPS1 mice and between WT and APPxPS1 mice.

Figure 4 :

 4 Figure 4: DMPP induced vasodilations are mediated by NO while constrictions are mediated by NPY in 8 month-old WT mice. A, Mean vasoconstriction (n = 7) ± sem induced by DMPP (100 µM) in the presence of nNOS inhibitor L-NNA (100 µM). B, Mean vascular dilation (n = 5) ± sem induced by DMPP (100 µM) in the presence of the NPY Y1 receptor antagonist BIBP 3226 (1 µM).

Table 1

 1 Vascular reactivity induced by U46619 (50 nM) or DMPP (100 µM) applications in WT and APPxPS1 mice cortical slices. Mean vVascular dilation (white line) or constriction (grey lines) are indicated for each condition as mean diameter change or mean AUC ± sem. Proportions of reactive of blood vessels following by U46619 (50 nM) or DMPP (100 µM) applications were determined in WT and APPxPS1 mice cortical slices.

		Diameter change (%)		AUC	Reactivity (%)
		WT	APPxPS1	WT	APPxPS1	WT	APPxPS1
	U46619	80.18 ± 4.79	80.83 ± 3.79	183.94 ± 37.91	325.15 ± 47.31	57	56
	DMPP	105.21 ± 1.21 95.52 ± 1.08	106.08 ± 2.91 94.71 ± 0.88	39.40 ± 8.49 27.77 ± 9.64	39.22 ± 21.23 35.86 ± 14.89	44 26	15 20

Table 2

 2 Vascular reactivity induced by DMPP applications in WT mice cortical slices in the presence of L-NNA or BIBP 3226. Vascular dilation (white line) or constriction (grey lines) are indicated for each condition as mean diameter change or mean AUC ± sem. Proportions of reactive of blood vessels following DMPP (100 µM) applications were determined in WT mice.

		Diameter change (%)	AUC	Reactivity (%)
		WT	WT	WT
	DMPP + BIBP 3226	106.24 ± 2.02	39.75 ± 6.21	38
	DMPP + L-NNA	94.65 ± 2.41	26.76 ± 6.00	44

Acknowledgments

We thank the Sanofi-Aventis Neurodegenerative Disease Group for the generous gift of the animals involved in this study and Marcel Leopoldie for animal husbandry. We are grateful to Dr. Isabelle Férézou for helpful comments reading the manuscript and to Marion Daenens and Thierry Dendele for technical support. This work was supported by the French National Research Agency (ANR-06-NEURO-033-01grant).


J Zhang
email: zhang@irphe.univ-mrs.fr

M Benoit
email: benoit@irphe.univ-mrs.fr

RÉSONANCE DE FABRY-PÉROT APPLIQUÉE AUX VAGUES CÔTIÈRES : COMPARAISON D'APPROCHES DE MODÉLISATION LINÉAIRES FABRY-PÉROT RESONANCE OF COASTAL WATER WAVES: COMPARISON OF LINEAR MODELING APPROACHES

 [6], lorsque la longueur d'onde des vagues est exactement le double de la longueur d'onde des ondulations du fond. Récemment, Couston et al. (2015) [1] ont étendu cette théorie au cas de deux zones de rides séparées par une zone de profondeur uniforme et montré que sous certaines conditions une amplication importante des vagues peut se produire dans la zone entre les rides (résonateur) : ce phénomène correspond à la résonance de Fabry-Pérot initialement étudiée en optique. Dans ce travail, nous étudions par le biais de simulations numériques (réalisées avec la version linéaire du code potentiel de vagues whispers3D), les eets de certaines des hypothèses de la théorie ALAT, notamment pour ce qui est de la formulation de la condition à la limite sur le fond ondulé. Nous montrons en particulier que lorsque l'amplitude relative des rides augmente, la résonance de Fabry-Pérot se produit pour un nombre d'onde des vagues incidentes légèrement plus faible que celui prédit par ALAT ('wave number downshift'). La forme de la courbe de résonance est également étudiée, et il apparaît que la gamme de nombres d'onde susceptibles d'entrer en résonance est réduite, et ce d'autant plus que l'amplitude relative des rides augmente.

Summary

When periodic waves propagate in a medium of uniform water depth except on an area having a series of regular ripples, signicant reection of the incident waves can occur in some cases. This phenomenon, called Bragg resonance, occurs from the asymptotic linear

theory established by [START_REF] Mei | Resonant reection of surface water waves by periodic sandbars[END_REF] [START_REF] Mei | Resonant reection of surface water waves by periodic sandbars[END_REF] when the wavelength of the surface waves is exactly twice the wavelength of the bottom ripples. Recently, Couston et al. (2015) [START_REF] Couston | Fabry-Perot resonance of water waves[END_REF] extended this theory to the case of two patches of ripples separated by a zone of uniform depth and showed that under certain conditions a signicant amplication of the waves can occur in the zone between the patches (resonator): this phenomenon corresponds to the Fabry-Pérot resonance initially studied in optics. In this work, we study, through numerical simulations (carried out with the linear version of the potential wave code whispers3D), the eects of some of the assumptions of the ALAT, in particular with regard to the formulation of the bottom boundary condition. We show in particular that when the relative amplitude of the ripples increases, the Fabry-Pérot resonance occurs for a slightly lower incident wave number than that predicted by ALAT ('wave number downshift'). The shape of the resonance curve is also studied, and it appears that the range of wave numbers likely to come into resonance is reduced, especially as the relative amplitude of the ripples is increased.

I Introduction

Since the 1970s, it has been observed that when water waves propagate over the a patch of sinusoidal corrugations on an otherwise at bottom, signicant reection of the incident waves can be observed under certain conditions. When the incident wavelength is twice than that of the bottom ripples, the so-called 'Bragg resonance' may take place. This phenomenon has been conrmed experimentally in wave ume experiments by [START_REF] Heathershaw | Seabed-wave resonance and sand bar growth[END_REF] [START_REF] Heathershaw | Seabed-wave resonance and sand bar growth[END_REF] and [START_REF] Davies | Surface-wave propagation over sinusoidally varying topography[END_REF] [START_REF] Davies | Surface-wave propagation over sinusoidally varying topography[END_REF].

In the linear framework, [START_REF] Davies | On the interaction between surface-waves and undulations on the seabed[END_REF] [START_REF] Davies | On the interaction between surface-waves and undulations on the seabed[END_REF] and [START_REF] Mei | Resonant reection of surface water waves by periodic sandbars[END_REF] [START_REF] Mei | Resonant reection of surface water waves by periodic sandbars[END_REF] have developed mathematical models to describe linear incident wave propagating over a series of ripples on an otherwise at bottom. In both models, the amplitude of the free surface elevation as well as the amplitude of the bottom ripples are assumed to be small. It is known that Davies model can only predict non-resonance cases with a small number of bottom corrugations, whereas Mei's asymptotic linear analytical theory (ALAT) [START_REF] Mei | Resonant reection of surface water waves by periodic sandbars[END_REF] is capable to describe the resonance.

Recently, Couston et al. (2015) [START_REF] Couston | Fabry-Perot resonance of water waves[END_REF] considered water wave trapping by using the analogous mechanism to the Fabry-Pérot (F-P) in optics. Using the same assumptions and extending the ALAT approach of Mei, they developed the mathematical model for linear incident waves propagating over two separated patches of ripples on an otherwise at bottom. The at bottom region between two patches is called 'resonator'. The F-P resonance condition is composed of two parts: as Bragg resonance, the incident wavelength must be twice that of the bottom ripples; in addition, the resonator must have a particular length with respect to bottom wavelength. In this condition, the wave energy could be trapped within the resonator, resulting in large amplication of the incident waves and forming standing waves with large amplitude.

The present work aims at studying some of the assumptions of the ALAT, in particular focusing on the wave-bottom interaction. Through numerical simulations we examine in which manner higher-order bottom eects associated with the expression of the bottom boundary condition (BBC) adopted in the mathematical models of Mei [START_REF] Mei | Resonant reection of surface water waves by periodic sandbars[END_REF] and Couston et al. [START_REF] Couston | Fabry-Perot resonance of water waves[END_REF] may aect the occurrence of the F-P resonance on a particular conguration. The eect of the relative ripple amplitude is also addressed through these numerical simulations, and comparisons with ALAT predictions are analyzed.

II Mathematical modeling of Fabry-Pérot resonance

The wave problem is expressed in a 2D vertical Cartesian coordinate system (x, z), with x-axis located at the mean free surface level, and z-axis positive upward. We assume sandbars to be parallel to the y-axis and waves propagate along the x direction only (i.e. with normal incidence with respect to the sand ripples). Assuming (i) the uid is incompressible and homogeneous, (ii) the ow is irrotational, and (iii) surface waves are of innitesimal amplitude, the linear water wave problem (labeled as system A in the following) can be formulated in terms of velocity potential φ(x, z, t):

φ xx + φ zz = 0, -h(x) ≤ z ≤ 0, (1a)
φ tt + gφ z = 0, z = 0, (1b) hx φ x + φ z = 0, z = -h(x), (1c)
in which subscripts denote partial derivatives, and g is the gravitational acceleration. The free surface elevation η(x, t) is related to the velocity potential through the linearized free surface boundary condition (FSBC) η = -φ t | z=0 /g. The water depth is considered to be of the form

h(x) = h -ζ(x)
, where h is the (constant) mean water depth and ζ(x) is the shape of ripples superimposed on the at bottom:

ζ =    d sin [k b (x -x s 1 ) -θ 1 ], x ∈ [x s 1 , x e 1 ], d sin [k b (x -x s 2 ) -θ 2 ], x ∈ [x s 2 , x e 2 ], 0, elsewhere.
(

where x s j and x e j represent the start and the end of patch j (j = 1, 2), L j = x e j -x s j = N j L b (N j ∈ N) is used to denote the length of the patch j. k b is the wave number of the bottom corrugations, the corresponding wavelength is L b = 2π/L b . The resonator length is denoted by L r = x s 2 -x e 1 . θ j is the initial phase of the j-th patch (chosen to be 0 or π in order to have a continuous bottom shape). Considering the form of Eq. ( 2), the assumption of bottom ripples of small amplitude (i.e. k b d 1 and d/h 1) allows to write Eq. (1c) as an approximate BBC, by retaining only the rst order terms of its Taylor expansion:

-

(ζφ x ) x + φ z = 0, z = -h. (3)
Note that Eq. ( 3) is now applied at the constant elevation z = -h. The system composed of Eqs. (1a, 1b, 3) is called system B in the following. We know from ALAT that the Bragg resonance happens when the incident wave number is k B = k b /2 (to be clear, the subscript 'B' stands for 'Bragg' and 'b' for 'bottom'). Now the monochromatic incident water waves come from x = -∞ with a wave number k = k B + κ, κ k b . This means that the incident wave number lies in the vicinity of the Bragg wave number k B . For the incident wave number k, the corresponding wave angular frequency is obtained by the dispersion relation ω = gk tanh(kh), Bragg resonance condition in frequency is ω B = gk B tanh(k B h), the detuning frequency corresponding to κ is Ω = C g κ, where C g is the group celerity at the Bragg wave number:

C g = 1 2 1 + 2k B h sinh 2k B h ω B k B . (4)
Due to the wave-bottom interaction, the amplitude is slow varying. The general solution of F-P resonance over the j-th patch of the linearized system reads

φ = f (z)[A j (x , t )e -ikx + B j (x , t )e ikx ]e iωt + c.c., (5)
where c.c. denotes the complex conjugate, i = √ -1, x and t are the slow variables. A j , B j denote the slowly varying complex amplitudes of the incident and reected waves over the patch j, the time variation for both of them is simple harmonic motion. Then the time variation can be written explicitly A j = A j (x )e iΩt , B j = B j (x )e iΩt . f (z) is the vertical structure of the velocity potential:

f (z) = - ig 2ω cosh k(h + z) cosh kh . (6)
By using the multiple-scale method, and keeping the leading order terms of the solution, the following system can be obtained:

iΩA j + C g ∂A ∂x = -Ω c e iθ j B j , (7a)
iΩB j -C g ∂B ∂x = Ω c e -iθ j A j , (7b)
where

Ω c = ω B k B d 2 sinh 2k B h . ( 7c
)
It can be seen that for the at bottom parts (ζ(x) = 0), Ω c = 0: the incident and reected waves propagate at the speed of group velocity without coupling. If only one patch j is considered, the model returns to Mei's theory for Bragg resonance [START_REF] Mei | Resonant reection of surface water waves by periodic sandbars[END_REF]. The (complex) Bragg reection and transmission coecients are then obtained analytically as functions of normalized detuning frequency Ω/Ω c :

R B j (P)| x s j = B(x s j , t ) A(x s j , t ) = e -iθ j sinh S j Q Q cosh S j Q + iP sinh S j Q , ( 8a
)
T B j (P)| x e j = A(x e j , t ) A(x s j , t ) = Q Q cosh S j Q + iP sinh S j Q (8b)
where

P = Ω/Ω c , Q = √ 1 -P 2 , S j = Ω c L j C g . (8c)
If the following, R B j and T B j denote the modulus of these coecients, and α B j denotes the complex argument of R B j . If two patches are considered, and the system is solvable thanks to the continuity of the free surface elevation over the resonator area, i.e. the amplitude of the incident waves of the second patch equals that of the transmitted wave of the rst patch. Thus, the reection and transmission coecients for F-P resonance read (see more details in [START_REF] Couston | Fabry-Perot resonance of water waves[END_REF]):

R F P | x s 1 = (R B 1 ) 2 + (R B 2 ) 2 -2R B 1 R B 2 cos γ 1 + (R B 1 R B 2 ) 2 -2R B 1 R B 2 cos γ 1/2 , (9a)
T F P | x e 2 = 1 -(R B 1 ) 2 1 -(R B 2 ) 2 1 + (R B 1 R B 2 ) 2 -2R B 1 R B 2 cos γ 1/2 , (9b)
where

γ = π -2θ 1 + 2k B L r -α B 1 -α B 2 . ( 10
)
In addition to Bragg resonance condition, the wave energy will be trapped within the resonator when γ = 2mπ (m ∈ N). This is denoted as F-P resonance condition in which the standing waves with highest achievable amplitude are expected between two patches. The non-dimensional amplitude of the standing waves within the resonator is dened as enhancement factor E F P and can be computed as follows [START_REF] Couston | Fabry-Perot resonance of water waves[END_REF]:

E F P = |A(x e 1 , t )| + |B(x e 1 , t )| |A(x s 1 , t )| = 1 + R B 2 T F P T B 2 . ( 11
)
The water depth aects the results through entering S in Eqs. [START_REF] Tian | A numerical model on the interaction between nearshore nonlinear waves and strong currents[END_REF]. C g is related to the mean water depth h. Ω c is not only aected by the water depth but also by the amplitude of the ripples d. The relation between the relative ripple amplitude d/h and the enhancement factor E F P is plotted in Fig. 1. In this asymptotic linear analytical theory (ALAT), the enhancement factor E F P increases exponentially with the relative ripple amplitude d/h, as shown in Fig. 1, achieving values of about 4 around d/h = 0.15 and exceeding 10 for d/h = 0.3. Such local amplication of incident waves might be of interest for coastal engineering applications in certain cases, for example harnessing wave energy.

However, it remains to be tested whether such large amplication can be really achieved in practice, especially the eects associated with the assumptions used in ALAT, i.e. the assumptions on the smallness of surface waves and on the smallness of the bottom ripples, needs to be studied. In the current study, we focus on the assumption on the bottom ripples amplitude by using an ecient potential linear wave model (presented in the next section). The test conditions to be simulated are identical to the ones used in the gure 6 of [START_REF] Couston | Fabry-Perot resonance of water waves[END_REF], except for the relative amplitude of bottom ripples d/h which varies in our study from 0.1 to 0.2.

III The numerical model: whispers3D

The numerical simulations are performed with a highly accurate code, called whispers-3D, currently developed at Ecole Centrale Marseille and Irphé lab. It solves the fully nonlinear potential wave problem with variable bottom conditions, in the form of two coupled nonlinear equations, corresponding to the two nonlinear free surface boundary conditions (FSBC). In dimensional form for the case of a single horizontal dimension [START_REF] Zakharov | Stability of periodic waves of nite amplitude on the surface of a deep uid[END_REF], they are expressed as:

η t = -η x φx + w 1 + (η x ) 2 , ( 12a
) φt = -gη - 1 2 ( φx ) 2 + 1 2 w2 1 + (η x ) 2 , (12b)
where φ(x, t) ≡ φ(x, z = η(x, t), t) is the free surface velocity potential and w(x, t) ≡ φ z (x, z = η(x, t), t) is the vertical velocity at the free surface.

In order to march Eqs. (12) in time, the vertical velocity w(x, t) has to be determined as a function of (η(x, t), φ(x, t)), which corresponds to a so-called 'Dirichlet-to-Neumann (DtN)' problem. The modeling approach used is presented in previous works [START_REF] Yates | Accuracy and eciency of two numerical methods of solving the potential ow problem for highly nonlinear and dispersive water waves[END_REF][START_REF] Raoult | Validation of a fully nonlinear and dispersive wave model with laboratory non-breaking experiments[END_REF] and summarized hereafter. Following Tian and Sato [START_REF] Tian | A numerical model on the interaction between nearshore nonlinear waves and strong currents[END_REF], a spectral approach is used in the vertical to approximate the velocity potential. Using the set of orthogonal Chebyshev polynomials of the rst kind, denoted T n (s), n = 0, 1, ..., N T , with s ∈ [-1, 1], as an expansion basis, the potential is approximated at any given time t (omitted for brevity hereafter) as

φ(x, z) = ϕ(x, s) ≈ N T n=0 a n (x)T n (s), (13)
where s(x, z, t) is the scaled vertical coordinate allowing to map the water column z ∈ [-h(x), η(x, t)] into the xed range s ∈ [-1, 1], and the a n coecients (n = 0, 1, ..., N T ) depend upon the local abscissa x (and time t).

The main steps involved in solving the DtN problem and marching Eqs. (12) in time are summarized as follows: (i) rst, the system of governing equations composed of the Laplace equation, a Dirichlet FSBC on the potential and the BBC is expressed in the (x, s) coordinate system, (ii) then, the approximation given in Eq. ( 13) is inserted into those equations, (iii) the so-called Chebyshev-tau method, a variant of the Galerkin method, is used to project the Laplace equation onto the T n polynomials for n = 0, 1, ..., N T -2 eliminating the s coordinate and giving a set on N T -1 equations on the a n coecients at each location x, (iv) two additional equations are obtained by considering the Dirichlet FSBC and the BBC so that a system of N T + 1 linear equations with N T + 1 unknowns (a n , n = 0, ..., N T ) at each abscissa is formed, (v) once this linear system is solved for the a n coecients, the free surface vertical velocity at t is obtained as

w(x, t) = 2 h(x) + η(x, t) N T n=1 a n (x, t)n 2 , ( 14
)
allowing Eqs. (12) to be integrated in time.

In the current study, the linearized version of whispers3D is used. The linearization is achieved by (i) ignoring the nonlinear terms in Eqs. ( 12), (ii) solving the DtN problem on a domain bounded by the level z = 0 (instead of z = η(x, t)), and (iii) replacing the total water depth h(x) + η(x, t) in Eq. ( 14) by the still water depth h(x). Then, two dierent variants are considered: one with the exact BBC (system A), and one with the approximate BBC (system B). In all cases horizontal derivatives are approximated using fourth-order nite dierence formulas using stencils of 5 nodes on a regular grid, and an explicit third-order Runge-Kutta scheme (SSP-RK3) is used for time marching.

IV Numerical simulations of Fabry-Pérot resonance IV 1 General settings of the numerical simulations

In the present work, we simulate the F-P resonance case as was chosen in the gure 6 of [START_REF] Couston | Fabry-Perot resonance of water waves[END_REF] with two the systems A and B. The original set-up is recalled here: N 1 = 11 ripples, N 2 = 15 ripples, θ 1 = θ 2 = 0, k b h = 1.64, d/h = 0.1, a/h = 10 -5 and resonator length L r /L b = 5.5 (corresponding to m = 5 in Eq. ( 10)). To investigate the eect associated with the assumption of small ripple amplitude, we further studied the case d/h = 0.2. In practice, we choose L b = 0.5 m for the wavelength of the ripples (giving a theoretical Bragg wavelength

L B = 2L b = 1 m).
In terms of the numerical parameters, two relaxation zones with 3L B in length are set to generate regular waves on one side and absorb them on the other side. The numerical basin is uniformly meshed with ∆x = L b /64 (= L B /128), the time step is ∆t = T B /256 resulting in a Courant-Friedriechs-Lewy number CF L = 0.5. These discretization parameters were selected after a convergence study. Meanwhile, the time required to reach an established periodic regime depends on the relative ripple amplitude.

In the following, the theoretical results A and B, corresponding to the solution of ALAT shown in Eqs. [START_REF] Raoult | Validation of a fully nonlinear and dispersive wave model with laboratory non-breaking experiments[END_REF], are the complex wave amplitudes, the modulus of the solutions |A| and |B| correspond to the envelopes of the amplitudes, excluding time variations as well as fast spatial variations. On the at bottom zones, standing waves are expected resulting in the amplitude envelopes to be constant. In addition, because of the reection from the ripples downwave, the value of normalized amplitude envelope falls in the range [START_REF] Couston | Fabry-Perot resonance of water waves[END_REF][START_REF] Davies | On the interaction between surface-waves and undulations on the seabed[END_REF] before the rst patch. Meanwhile, due to the conservation of energy, the amplitude envelope of transmitted waves is less than 1 after the second patch. Between the two patches, the wave energy is trapped in the resonator area, thus a high constant level is expected from the ALAT results over this zone.

IV 2 Perfectly tuned cases with dierent ripple amplitudes

We start with the same case as in the gure 6 of [START_REF] Couston | Fabry-Perot resonance of water waves[END_REF] with d/h = 0.1, in order to validate our numerical model. Based on ALAT, for the incident wave number k = k B , we nd: global reection coecient R F P ≈ 0.245, enhancement factor E F P ≈ 2.476 (resonator), and transmission coecient is T F P ≈ 0.970. Note that (R F P ) 2 + (T F P ) 2 = 1. Considering two patches separately, the results of ALAT for Bragg resonance in Eqs. [START_REF] Tian | A numerical model on the interaction between nearshore nonlinear waves and strong currents[END_REF] give: reection coecient of the rst patch (N 1 = 11) R B 1 ≈ 0.597 and reection coecient of the second patch (N 2 = 15) R B 2 ≈ 0.734. Both are larger than the reection coecient in F-P resonance, due to the fact that the energy is trapped within the resonator.

The comparison between the theoretical result and simulated results (with systems A and B) is shown in Fig. 2a. The fast spatial variation of the envelope is kept in the simulated results, corresponding to the envelope of the free surface elevation (or to the local amplitude of the free surface oscillation).

It is seen that in the case d/h = 0.1 the agreement between the simulated results and the theoretical result is reasonably good. It is also observed that the two numerical systems give almost the same results. Only in very local scale can some dierences be seen. This justies that the approximation on the BBC in the current case is acceptable. The reection of the patches is stronger in the simulations than predicted by ALAT. The simulated reection coecient is R F P ≈ 0.329 with System A, whereas the enhancement factor E F P ≈ 2.385 and the transmission coecient T F P ≈ 0.944 are slightly smaller than ALAT predictions.

For the case d/h = 0.2, we doubled the ripple amplitude while holding the water depth unchanged. As is shown in Fig. 1, the enhancement factor increases exponentially as the relative ripple amplitude increases. The wave-bottom interaction is more signicant in this case. When the resonance takes place, according to ALAT, the reection coecient R F P ≈ 0.462, the enhancement factor E F P ≈ 5.785 and the transmission coecient T F P ≈ 0.887. However, Fig. 2b shows that the resonance does not take place in this case. In addition, system A with exact BBC and system B with approximate BBC show dierent results. The system B with the approximate BBC (which is also adopted in ALAT) give even smaller enhancement factor within the resonator. Neither system A nor system B is close to the ALAT prediction (bold line). Incident waves are almost fully reected by the ripples resulting in R F P ≈ 1. The enhancement factor is E F P ≈ 1.711 with system A. IV 3 Slightly detuned incident waves for the case d/h = 0.2 Do our results indicate that, when the amplitude of the bottom ripples is relatively large, the F-P resonance completely disappears and that incident waves will be strongly reected by the bottom ripples? This question is interesting because of the signicant dierence between the expected value from ALAT E F P ≈ 5.785 and the simulated result E F P ≈ 1.711 (system A) in the case d/h = 0.2, and the dierence is greater when d/h is larger.

In seeking an explanation of this behavior, the so-called 'wave number downshift' drew our attention. As was observed in the early experiments [START_REF] Davies | Surface-wave propagation over sinusoidally varying topography[END_REF], in the case with large ripple amplitude d/h = 0.16, the strongest Bragg reection of incident waves did not take place when the theoretical Bragg resonance condition was met, i.e. the incident wave number k = k B , but it happened when the incident waves were of slightly smaller wave number. This phenomenon was also observed in the numerical study of [START_REF] Liu | On generalized Bragg scattering of surface waves by bottom ripples[END_REF] who simulated the experiments of Davies & Heathershaw [START_REF] Davies | Surface-wave propagation over sinusoidally varying topography[END_REF] using a highly accurate model. In [START_REF] Liu | On generalized Bragg scattering of surface waves by bottom ripples[END_REF] a high-order spectral (HOS) method was adopted to solve the nonlinear water wave problem up to an arbitrary order M (in their study, M = 4 was used). In their work the simulated results showed limited but clear downshift of wave number when compared to ALAT predictions, while in terms of reection coecients R B , their simulated results were in good agreement with ALAT predictions. This motivated us to perform a series of runs by slightly perturbating the incident wave number around the theoretical value k B . The results shown in Fig. 3 illustrate that, with relatively large bottom ripples, the incident waves are strongly amplied within the resonator and the reection coecient reaches its minimum value when the incident wave number is slightly smaller than k B . The downshift of the wave number is limited, as was observed in the simulations of Bragg resonance [START_REF] Liu | On generalized Bragg scattering of surface waves by bottom ripples[END_REF], whereas it largely aects the result in the present F-P resonance. This can be explained by the fact that the F-P resonance condition is more 'strict' than Bragg resonance condition: the F-P resonance only manifests for a relatively narrow range of wave numbers. When the incident wave number falls outside this range, the wave energy will be largely reected by the ripples. The slight deviation from the F-P resonance condition due to 'wave number downshift' is thus of signicant importance. The duration of the simulations is t = 900T .

In system A with exact BBC, the maximum enhancement factor E F P ≈ 5.805 is observed at 2k/k b ≈ 0.99263, and the corresponding reection coecient is R F P ≈ 0.5065. These values are in better agreement with the ALAT prediction than those of system B. Regarding system B with approximate BBC, the minimum reection happens for 2k/k b ≈ 0.992, the enhancement is overestimated E F P ≈ 6.363 and the reection coecient R F P ≈ 0.512 is smaller than expected in ALAT. The degree of wave number downshift is dierent only by 1% for k/k B , but the dierences it makes is obvious. Both systems A and B show a similar wave number downshift but with small dierences in the magnitude of downshift. This is due to the dierence in the formulation of the BBC (the unique dierence between the two systems) and implies, as the resonance curves are rather sharp, that for a same incident wave number the amplication of wave from system A and B can be signicant dierent (in particular in the vicinity of the resonance condition).

As a further illustration of the occurrence of F-P resonance with slightly detuned incident wave number, the comparison between the envelope of wave amplitude from ALAT and the envelopes of the free surface elevation from systems A and B is given in Fig. 4 for the case d/h = 0.2. The simulations are done with an incident wave number tuned to the resonating wave number for system A, namely k/k B = 0.99263. Again, the dierences between system A and system B mainly result from the dierent degree of frequency downshift between the two systems (as this wave number is close to, but dierent from, the resonating wave number of system B).

V Conclusions

Recently, Couston et al. (2015) [START_REF] Couston | Fabry-Perot resonance of water waves[END_REF] studied the resonance of water waves using two distinct sets of corrugations, in analogy with Fabry-Pérot resonance in optics by using an asymptotic linear analytical theory (ALAT). The ALAT solution is derived based on the assumptions that incident monochromatic water waves are of small amplitudes and that the bottom ripple amplitudes are small too. In the present study, we examined the higher-order eects of BBC by using the linearized version of a highly accurate code (whispers3D) solving the fully nonlinear potential and dispersive wave problem. The motivation is that, in the coastal areas as the water depth decreases, the nonlinear eect of wave-bottom interaction becomes more and more important. According to ALAT prediction, the enhancement factor should increase exponentially as the relative ripple amplitude d/h increases, which may result in very large amplication of incident waves between the two patches when the resonance conditions are met. Two cases following the work of Couston et al. (2015) [START_REF] Couston | Fabry-Perot resonance of water waves[END_REF] were analyzed and simulated, with dierent relative ripple amplitudes d/h = 0.1 and 0.2. The rst case with d/h = 0.1 is identical to the one used in [START_REF] Couston | Fabry-Perot resonance of water waves[END_REF], it validates our numerical model to the corrugated bottom condition. Two systems with exact BBC and approximate BBC are tested. They show very similar results in the case d/h = 0.1, the resonance is indeed realized, even though in simulations E F P ≈ 2.385 is smaller than expected under the F-P condition. However, in the case d/h = 0.2 the resonance does not manifest when the incident wave number k = k B , and two systems show dierences. This can be explained by the so-called 'wave number downshift' in Bragg resonance, meaning that the resonance takes place for an incident wave number slightly lower than k B . For the F-P resonance this eect makes a signicant dierence because the resonance condition is 'strict', especially when the d/h is large enough. A number of simulations have been conducted in order to demonstrate this wave number downshift and to nd the shifted F-P resonance condition. The dierences between the system A (with exact BBC) and system B (with approximate BBC) become considerable, and result (at least partially) from the dierent degree of wave number downshift between the two systems.

With the shifted incident wave number, the F-P resonance can be realized in the numerical model with a satisfying agreement with ALAT results regarding the enhancement factor of wave amplitude. In system A, the enhancement factor E F P ≈ 5.805 agrees well with the predicted value from ALAT E F P ≈ 5.785. In system B, although the enhancement factor is overestimated, the wave number downshift is also well described. This means that the wave number downshift eect is already present when the BBC is approximated at rst order. Last but not least, given the good agreement in the case with large ripple amplitude obtained by taking the downshift into consideration, the analysis and understanding of the rst case with d/h = 0.1 can be improved in the same way.

Future work will address the study the higher relative ripples' amplitude, as well as the eects introduced by taking into consideration waves of nite amplitude (free surface nonlinearity), and how these eects aect the occurrence of F-P resonance.

Figure 1 :

 1 Figure 1: Enhancement factor E F P as a function of the relative amplitude of the bottom ripples d/h, other parameters such as N 1 , N 2 , k b L r , k b h and k b d are chosen to be identical to the ones used in the gure 6 of [1].

Figure 2 :

 2 Figure 2: Computed envelope of the free surface elevation at the end of the simulations of system A and B, t = 200T for the case d/h = 0.1 (left panel), t = 900T for the case d/h = 0.2 (right panel). The results of ALAT which are the envelopes of the amplitude are also superimposed.

Figure 3 :

 3 Figure 3: The enhancement factor (left panel) and the reection coecient (right panel) as functions of incident wave number k, normalized by k B = k b /2 for the case d/h = 0.2. The duration of the simulations is t = 900T .

Figure 4 :

 4 Figure 4: Computed envelope of the free surface elevation at the end of the simulations of systems A and B, t = 900T for the case d/h = 0.2 and k/k B = 0.99263. The result of ALAT which is the envelope of the amplitude is also superimposed.