Jean-Luc Adam

OPTICAL PROPERTIES AND APPLICATIONS OF NON-OXIDE GLASSES

Keywords: Chalcogenide, Fluoride, Optical Fiber, Infrared, Supercontinuum, Optical Sensor

The synthesis and fundamental properties of non-oxide glasses are overviewed. Applications are illustrated by the latest results in fields of technological or societal importance. The first one is the generation of supercontinuum of infrared light by using fluoride and/or chalcogenide optical fibers; the second one is the detection of green-house-effect gases, like CO2, by using optical sensors based on rare-earth-doped chalcogenide fibers.

INTRODUCTION

Vitreous materials based on fluorides or chalcogen elements (S, Se, Te) show large transparency windows in the infrared. Indeed, fluoride glasses are transparent from the UV to 7 µm in the infrared, while chalcogenides can be transparent from the visible up to 12-15 µm, depending on their compositions [START_REF] Sanghera | Optical Properties of Chalcogenide Glasses and Fibers[END_REF]. This is due to the lower phonon energies of non-oxide glasses, which are also responsible for enhanced luminescence of rare-earth ions embedded in such matrices, as compared to oxides. Thus, these glasses allow light emission at wavelengths not accessible with silica. In addition, chalcogenide glasses contain large polarisable atoms and external lone electron pairs that induce exceptional non-linear properties. The non-linear properties of chalcogenides can be 100 to 1000 times as high as the non-linearity of silica. Specific fluoride and chalcogenide glasses can be obtained in the form of optical fibers. Applications are directly related to the combination of unique optical properties and shaping abilities. The paper deals with an overview of the synthesis and fundamental properties of non-oxide glasses, supplemented by the latest results, in terms of applications, in two fields of technological or societal importance. The first one is the generation of supercontinuum of infrared light by using fluoride and/or chalcogenide optical fibers [START_REF] Martinez | [END_REF][3][4]; the second one is the detection of green-houseeffect gases, like CO2, by using optical sensors based on rare-earth-doped chalcogenide fibers [5].

GLASS COMPOSITIONS AND SYNTHESIS

Fluoride glasses

One of the most stable fluoride glass is ZBLAN glass, whose composition is shown in Table I. In this glass, ZrF4 is the glass former, BaF2 and NaF are defined as modifiers, and AlF3 and LaF3 are the stabilizers. Other compositions like fluoroaluminate (AlF3), fluoroindate (InF3), and fluorogallate (GaF3) glasses have been developed [START_REF] Poulain | Fluoride glasses : synthesis and properties[END_REF][START_REF] Miranday | New transition metal fluoride glasses isolated in the PbF2-M II F2-M III F3 systems[END_REF]. Fluoride glasses are usually synthesized either directly from a mixture of metallic fluorides or from a mixture of metal oxides and ammonium bifluoride in excess that react between 250-400 °C to form the fluorides. The fluoride mixture is then melted between 800 and 1000 °C during 30 minutes under inert atmosphere, and poured in a brass mold preheated at a temperature close to the glass temperature (Tg). Finally, the glass sample is annealed at nearly Tg and cooled down to room temperature. An indication of the glass stability towards crystallization is given by ΔT, the difference between the crystallization and the glass transition temperatures. Stable glasses, suitable for the preparation of optical fibers, show a stability parameter ΔT of nearly 100 °C and above, associated to critical cooling rates (Rc) of 1 °C/min or less. ZBLAN glass fulfills these requirements.

Chalcogenide glasses

Chalcogenide glasses refer to glassy materials based on S, Se, or Te, possibly associated to nearby elements of the periodic table such as Ge, As, in a lesser extent Ga, Sb, and also halides. Addition of these elements results in glasses with improved properties like stability against crystallization or optical properties [8]. Bondings between these elements, with close electronegativity, are essentially covalent in nature, contrary to more common glasses like silicates, where bondings are more polarized or even ionic. Chalcogenide glasses usually have relatively low Tg ranging typically from about 150 °C, for selenium-rich glasses, to a maximum of around 400 °C for glasses from the Ge-Ga-S (Se) system, as shown in Table II. The technique for synthesizing chalcogenide glasses should allow to avoid or more reasonably to minimize the presence of impurities like oxygen, hydrogen and water, which could altered the optical transparency. The raw materials are generally elements such as sulfur, selenium, germanium, antimony, arsenic, and tellurium typically with 4N -5N purity. Most of these elements are sensitive to oxidation. This is the reason why the contact with atmosphere should be avoided/minimized during transportation, storage and manipulation, even if the formed glasses are stable in normal air condition. Consequently, the synthesis of chalcogenide should be done in inert atmosphere or in vacuum. The commonly used procedure is the following: the raw materials are introduced in a pre-cleaned silica tube, which is then evacuated to vacuum with a residual pressure of about 10 -5 mbar. Then, the silica tube is hermetically sealed and put into a rocking furnace to ensure a thorough reaction and homogenization. The homogenization temperature is between 600 and 950 °C, depending on the glass composition during several hours. The quenching of the melt can be performed by putting the silica tube in water, ideally iced water, or in air. Similarly to fluorides, the glass sample is then annealed at nearly Tg and cooled down to room temperature. Other techniques like synthesis of As2S3 in open system under gas flow and synthesis in the Ge-Ga-Se system by ball-milling followed by spark plasma sintering have been also employed [START_REF] Guillevic | Fabrication of highly homogeneous As2Se3 glass under argon flow[END_REF][START_REF] Hubert | An innovative approach to develop highly performant chalcogenide glasses and glass-ceramics transparent in the infrared range[END_REF] OPTICAL PROPERTIES

Optical transmission

Optical transmission is probably the one optical parameter that best characterizes non-oxide glasses. Indeed, their ability to transmit mid-IR light has been known for decades. Figure 1 shows the infrared edge of some typical glass families. Silica glass transmits light up to 3 µm only. This limitation is related to the high frequency of intrinsic fundamental vibration modes due to Si  O stretching, which are around 1100 cm -1 .

In fluoride glasses, the infrared edge is located at longer wavelengths, around 7 µm. This is because vibrations modes are at much lower frequencies, as compared to oxides. Thus, in fluorozirconate glasses for instance, the fundamental Zr  F vibration in ZrFn polyhedra is equal to 570 cm -1 .

In chalcogenides, which contain heavier chemical elements, the main fundamental vibration modes are located at even lower frequencies, typically at 340 cm -1 in sulfide glasses where Ge and S are the main constituents, and at 220 cm -1 in Ge-Se selenide glasses. In Ge-Te glasses the main fundamental mode is around 130 cm -1 . These low frequencies, attributed to the "breathing" mode of Ge(S/Se/Te)4 tetrahedra, are responsible for the shift towards longer wavelengths of the infrared edge due to multiphonon absorption, in the series S < Se < Te.

Non-linear optical properties

Fluoride glasses like ZBLAN show standard non-liner properties with a non-linear refractive index n2 of 0.021 10 -18 m²W -1 , similar to that of silica [START_REF] Agger | Supercontinuum generation in ZBLAN fibersdetailed comparison between measurement and simulation[END_REF][START_REF] Milam | Measurement of nonlinear refractive-index coefficients using time-resolved interferometry: application to optical materials for high-power neodymium lasers[END_REF]. On the contrary, chalcogenide glasses are well known for exhibiting high third-order nonlinear properties in presence of a strong electric field, which results in phenomenon like optical Kerr effect, four-wave mixing, third-harmonic generation, or two-photon absorption. Applications related to third-order nonlinearities are, for example, all-optical switching, generation of continuum of light, optical limitation, etc. A complete review on optical nonlinearities in chalcogenide glasses can be found in Ref [START_REF] Barney | Correlating structure with nonlinear optical properties in xAs40Se60 -(1-x)As40S60 glasses[END_REF].

As shown in Table III, nonlinear refractive indices of chalcogenide glasses are 2 to 3 orders of magnitude higher than that of silica glass, because of the hyperpolarisability of chalcogen elements, as compared to oxygen. In the As-S-Se glass series, n2 increases gradually from 2.3 to 10.5 10 -18 m 2 W -1 when highly polarizable Se substitutes for S.

Luminescence

As described earlier, non-oxide glasses are characterized by intrinsic low-energy phonons responsible for the broad transparency window in the mid-IR. Low-energy phonons are also responsible for the low probabilities of non-radiative relaxations in non-oxide glasses, then the luminescence of rare-earth ions (RE) exhibits enhanced quantum efficiencies. Rare-earth-doped non-oxide glasses possess a high potential for generating light at wavelengths where rare-earthdoped oxide glasses cannot operate, especially in the mid-IR domain. Thus, with rare-earth-doped fluoride glasses, luminescence and lasing has been demonstrated at discrete wavelengths from the UV (Nd 3+ ) to 3.9 µm (Ho 3+ ) in the mid-IR [START_REF] Adam | Fluoride glass research in France : Fundamentals and Applications[END_REF][START_REF] Adam | Lanthanides in non-oxide glasses[END_REF][START_REF] Zhu | High-power ZBLAN glass fiber lasers: review and prospects[END_REF]. While luminescence can be observed in bulk samples, lasing is obtained usually with optical fibers, which provides better conditions for laser cavity pumping and heat evacuation. With lower phonon energies, rare-earth-doped chalcogenide glasses can generate luminescence at even longer wavelengths, up to 8 µm with Tb 3+ ions [START_REF] Abdellaoui | Tb 3+ doped Ga5Ge20Sb10Se65-xTex (x = 0-37.5) chalcogenide glasses and fibers for MWIR and LWIR emissions[END_REF]. Emissions at wavelengths nearly 3 µm and above have been achieved in both sulfide and selenide glasses, most of them with quantum efficiencies as high as 80 % and more. These transitions are of interest as sources for the 3-5 µm transparency window of the atmosphere, or for optical sensing of molecules that show specific fingerprints in that spectral region. Thus, the broad 4.4 µm emission (FWHM=300 nm) of Dy 3+ , displayed in Figure 2, has been successfully utilized to perform quantitative detection of CO2 gas [START_REF] Charpentier | Mid-IR luminescence of Dy 3+ and Pr 3+ doped Ga5Ge20Sb10S(Se)65 bulk glasses and fibers[END_REF][START_REF] Pelé | Dy 3+ doped GeGaSbS fluorescent fiber at 4.4 µm for optical gas sensing: comparison of simulation and experiment[END_REF], as described further. In that same spectral region, the combination of Pr 3+ transitions at 3.9 and 4.8 µm leads to a super-broad band spanning from 3.5 to 5.5 µm in Ge20Ga5Sb10S65 bulk glass and single-index fiber [START_REF] Chahal | Fiber evanescent wave spectroscopy based on IR fluorescent chalcogenide fibers[END_REF]. A similar band profile is recorded with Ge-As-Ga-Se small-core double-index fiber [START_REF] Tang | Mid-infrared photoluminescence in small-core fiber of praseodymium-ion doped selenide bases chalcogenide glass[END_REF] Figure 2 -Fluorescence spectrum around 4.4 µm of a Dy 3+ -doped Ge20Ga5Sb10S65 single-index glass fiber under excitation at 920 nm, modified by the presence of CO2 gas

PREPARATION OF OPTICAL FIBERS

Fibers are essential optical components for applications where light must be transported (power delivery, remote sensing) or confined (high non-linearity). Several types of architecture have been investigated from single-index fibers to single-mode core-clad fibers, not to mention photonic crystal fibers (PCF) with their extreme versatility in terms of cross-section design. Thanks to an appropriate viscosity-temperature behavior, fluoride and chalcogenide glasses can be obtained in the form of wires and fibers by heating a preform in a drawing tower, described elsewhere [START_REF] Shiryaev | Infrared fibers based on Te-As-Se glass system with low optical losses[END_REF].

Fluoride glass fibers

Among the numerous vitreous compositions based on fluorides, only a few of them are sufficiently stable against crystallization upon heating to allow fiber drawing. Today, fluoride fibers based on zirconium, gallium or aluminum are commercial products [23,24]. The best performances, in terms of optical attenuation, are obtained with ZBLAN fibers. Figure 3 shows the attenuation curve of a single-mode ZBLAN fiber, with a minimum of attenuation of nearly 2 dB/Km at 2.6-µm wavelength. For comparison, the minimum loss of single-mode silica fibers, utilized for telecommunications, is less than 0.2 dB/Km at 1.55 µm. If fluoride fibers can not compete with silica for long distance applications, they are well adapted for short and medium ranges for applications like sensors, optical amplifiers, lasers, and transmission of infrared signals, where silica can not operate. The most efficient technique to prepare core-clad fluoride fibers is drawing from a solid prefom heated above Tg. Core-clad preforms can be elaborated by various methods, the most developed ones being the rotational casting technique and the rod-in-tube technique. In the first technique, the cladding glass in the liquid state is introduced in a cylindrical mold preheated at Tg. The mold is then rotated at high speed (ca 3000 rpm), which results in the formation of a glass tube on the inner wall. Then, the core glass is poured into the tube leading, after cooling to room temperature, to a preform with a good quality core-clad interface. The second technique necessitates, in a preliminary step, to elaborate separately a glass rod (core glass) and a glass tube (cladding glass). The two pieces are then inserted one in another, forming the preform. This technique is easier to implement and allows a broader combinations of glasses for the core and the cladding. But, the core-clad interface is usually of lower quality. At the end, singlemode ZBLAN optical fibers are obtained with core/clad diameters of 6.5/125 µm. The refractive index contrast between the core and the clad is 10 -2 typically, which gives a numerical aperture of 0.23.

Chalcogenide glass fibers

Single-index fibers are obtained usually from rods prepared by a classical melting-quenching procedure. Core-clad fibers are drawn from preforms elaborated by the rod-in-tube technique described above. Chalcogenide photonic crystal fibers or microstructured optical fibers (MOF) are drawn from specific preforms obtained by either the stack-and-draw process [START_REF] Brilland | Fabrication of complex structures of holey fibers in chalcogenide glass[END_REF], or the direct molding process [START_REF] Coulombier | Casting method for producing low-loss chalcogenide microstructured optical fibers[END_REF], or the drilling process [START_REF] Zhang | Fabrication of chalcogenide glass photonic crystal fibers with mechanical drilling[END_REF].

Several laboratories worlwide have developed intensive research on chalcogenide MOFs, with the aim to exploit the intrinsic third-order non-linearities of these glasses by controlling the design of the fiber, which is based on a periodic arrangement of air holes in a single-index matrix. Typical designs are portrayed in Figure 4 [START_REF] Trolès | Low-loss microstructured chalcogenide fibers for large non-linear effects at 1995 nm[END_REF][START_REF] Toupin | Small core Ge-As-Se microstructured optical fiber with single-mode propagation and low optical losses[END_REF][START_REF] Toupin | Comparison between chalcogenide glass single-index and microstructured exposed-core fibers for chemical sensing[END_REF][START_REF] Adam | Chalcogenide Glasses for Infrared Photonics[END_REF]. Provided that the ratio be smaller than 0.42 (d is the holes diameter, and Λ is the distance between two holes) and that the number of rings be large enough, the second mode is never confined in the core of the fibers. As a result, the fiber is singlemode whatever the wavelength. The optical losses of microstructured fibers prepared by the molding technique are less than 1 dB/m at the telecommunication wavelength, and less than 0.1 dB/m at 3.7 µm [START_REF] Trolès | Low-loss microstructured chalcogenide fibers for large non-linear effects at 1995 nm[END_REF].

Figure 4 -Cross-sections of chalcogenide glass microstructured optical fibers, (a) three rings large-core fiber [START_REF] Trolès | Low-loss microstructured chalcogenide fibers for large non-linear effects at 1995 nm[END_REF], small suspended-core fiber [START_REF] Trolès | Low-loss microstructured chalcogenide fibers for large non-linear effects at 1995 nm[END_REF], three-ring small-core fiber [START_REF] Toupin | Small core Ge-As-Se microstructured optical fiber with single-mode propagation and low optical losses[END_REF], and exposed-core fiber [START_REF] Toupin | Comparison between chalcogenide glass single-index and microstructured exposed-core fibers for chemical sensing[END_REF]

APPLICATIONS

Generation of infrared supercontinuum sources

First observed in silica optical fibers, the generation of supercontinuum (SC) with fluoride glasses became rapidly of high interest because of the intrinsic transparency of these materials in the infrared above 3 µm. One of the first SC, obtained with a ZBLAN core-clad single-mode fiber pumped at 1.55 µm, spanned from 0.8 to 4.5 µm with average output power of 23 mW [START_REF] Xia | Mid-infrared supercontinuum generation to 4.5 µm in ZBLAN fluoride fibers by nanosecond diode pumping[END_REF]. Then, thanks to optimized pumping, SCs with average power greater than 1 W have been obtained [START_REF] Xia | Power-scalable mid-infrared supercontinuum generation in ZBLAN fluoride fibers with up to 1.3-Watts time-averaged power[END_REF]. Fluoroindate fibers, whose infrared transparency is shifted towards longer wavelengths as compared to ZBLAN, permit the generation of SC up to 5.4 µm [START_REF] Gauthier | Mid-IR supercontinuum from 2.4 to 5.4 µm in a low-loss fluoroindate fiber[END_REF]. The combination of a ZBLAN fiber pumped at 1.55 µm with a sulfide glass fiber coupled itself to a selenide fiber lead to the generation of a SC with 417-mW average power from 1.6 to 11 µm [START_REF] Martinez | Mid-infrared supercontinuum generation from 1.6 to > 11 µm using concatenated step-index fluoride and chalcogenide fibers[END_REF]. Figure 5 shows the spectrum obtained at the output of a fluoride-glass-fiber supercontinuum source. This commercially-available device delivers an average power of 700 mw from 700 nm to 4 µm [23]. The nonlinear properties of chalcogenide glasses can be enhanced by using fibers with a very small core diameter, which can be achieved by tapering microstructured fibers. Tapered fibers permit to reach very small effective areas while maintaining good experimental conditions for light injection in non-tapered zone. Calculation of the nonlinear parameter γ (γ = 2πn2 / λAeff) allows a comparison between different fibers. In this equation, n2 is the nonlinear refractive index, Aeff is the effective area of the mode and λ is the wavelength. Thus, the nonlinear parameter is equal to 70 W -1 km -1 in silica MOF, while in chalcogenide MOF it can be as large as 46 000 W -1 km -1 for a tapered suspended-core As-Se fiber with an effective mode area of nearly 1 µm² [START_REF] Trolès | Novel designs of chalcogenide microstuctured optical fibers[END_REF]. Several nonlinear effects have been demonstrated in chalcogenide MOFS like self-phase modulation, fourwave mixing, and stimulated Brillouin scattering. But, the most promising application is probably the generation of infrared supercontinuum. Thus, a supercontinuum was observed from 1.7 to 8 µm, with an average output power of 15.6 mW, in a 18-cm long As-Se suspended-core fiber by pumping at 4.7 µm [3]. A broad continuum, spanning the 2 to 15.1 µm spectral range, was also demonstrated in a 3-cm long As-Se step-index fiber pumped at 9.8 µm [START_REF] Cheng | Mid-infrared supercontinuum generation spanning 2.0 to 15.1 μm in a chalcogenide step-index fiber[END_REF].

Optical detection of green-house-effect gases

Infrared spectroscopy is an efficient, non-destructive and highly selective technique for chemical and biological sensing. The potential applications include environmental monitoring, food security [START_REF] Lucas | Chalcogenide Glass Sensors for Bio-molecule Detection[END_REF][START_REF] Bureau | Chalcogenide optical fibers for mid-infrared sensing[END_REF] and medical diagnostics [START_REF] Rodolphe | NASH screening with a fast diagnostic tool using mid infrared spectroscopy: a bi-centric study[END_REF]. The spectral region of 2-15 µm, where chalcogenide glasses are transparent, is particularly interesting because it corresponds to the fingerprint absorptions of many chemical molecules/bonds and biological species. Fiber evanescent wave spectroscopy (FEWS), using single-index chalcogenide fibers, has been implemented successfully to detect various biological compounds like pathogen elements in food matrices or to establish metabolic profiling in human fluids [START_REF] Bureau | Chalcogenide optical fibers for mid-infrared sensing[END_REF][START_REF] Brandily | Identification of foodborne pathogens within food matrices by IR spectroscopy[END_REF][START_REF] Anne | Fibre Evanescent Wave Spectroscopy using the Mid-Infrared provides useful fingerprints for metabolic profiling in human[END_REF][START_REF] Jiang | Engineering of a Ge-Te-Se glass fibre evanescent wave spectroscopic (FEWS) mid-IR chemical sensor for the analysis of food and pharmaceutical products[END_REF].

A second category of optical sensors has been developed with rare-earth-doped chalcogenide fibers. The broad spectral overlap around 4.4 µm, shown in Figure 2, between the CO 2 absorption band and the 6 H 11/2  6 H 13/2 emission band of Dy 3+ in sulfide glasses, has been utilized to detect CO 2 . The sensor head, depicted in Figure 6, is based on a 20-cm long, 1000-ppm Dy 3+ -doped Ge-Ga-Sb-S fiber excited with 920-nm light from a laser diode transported through a silica fiber to the input of the sulfide fiber. The incoherent 4.4-µm light is collimated in the glass cell by using an aspheric chalcogenide lens. The cell length is controlled by the position of the flat mirror. The reflected beam is then focused in the undoped chalcogenide fiber, which brings the attenuated 4.4-µm signal to a pyroelectric detector [START_REF] Starecki | Mid-IR optical sensor for CO2 detection based on fluorescence absorbance of Dy 3+ :Ga5Ge20Sb10S65 fibers[END_REF]. The sensor is fast, reversible, and show a low detection limit of 500 ppm of CO 2 .

Figure 1 -

 1 Figure 1-Infrared optical transmission of silica, fluoride and chalcogenide glasses

Figure 3 -

 3 Figure 3 -Attenuation of ZBLAN fluoride glass single-mode fiber, from [23].

Figure 5 -

 5 Figure 5 -Infrared supercontinnum spectrum obtained with single-mode fluoride glass fibers, commercially available from Le Verre Fluoré [23]

Figure 6 -

 6 Figure 6 -Optical sensor for CO2 gas, using a Dy 3+ -doped chalcogenide (Chg) fiber emitting at 4.4 µm.

Table I

 I Examples of fluoride glass compostions with their thermal characteristics and refractive index

	Acronym	Composition (mol %)	Tg (°C)	T (°C)	Rc (°C/min)	nD
	ZBLAN	53ZrF4 20BaF2 20NaF 4LaF3 3AlF3	262	90	1-3	1.498
	ZBLA	57ZrF4 34BaF2 5LaF3 4AlF3	305	80	-	1.512

Table II

 II

			Glass transition temperature of chalcogenide glasses
		Ge33As12Se55 Ge30As13Se32Te25 Ge10As40Se50 Ge28Sb12Se60	As40Se60
	Tg (°C)	368	275	225	285	185

Table III

 III

	Third-order nonlinear refractive index of silica, fluoride
		and chalcogenide glasses
	Glass composition	n2 (10 -18 m 2 W -1 )	Reference
	SiO2	0.027	[12]
	ZBLAN	0.021	[11]
	As2S3	2.3 (± 0.5)	[13]
	As40S45Se15	2.3 (± 0.5)	[13]
	As40S40Se20	3.4 (± 0.5)	[13]
	As40S30Se30	4.5 (± 0.5)	[13]
	As40S20Se40	6.0 (± 0.5)	[13]
	As40S15Se45	6.1 (± 0.5)	[13]
	As2Se3	10.5 (± 0.5)	[13]


Thomas D Pre´voˆt

Franc¸ois Gastambide

Ce´cile Viollet

Nadia Henkous1

Guillaume Martel

Jacques Epelbaum

Daniel Be´racoche´a

Jean-Louis Guillou
email: jean-louis.guillou@u-bordeaux.fr

Thomas D Prévôt

François Gastambide

Cécile Viollet

Nadia Henkous

Daniel Béracochéa

Lamotrigine in mood disorders and cocaine dependence Cortical glutamate in postpartum depression Roles of Hippocampal Somatostatin Receptor Subtypes in Stress Response and Emotionality

Keywords:

are to be accounted for in development of pathway-specific somatostatin receptor agents in the treatment of hypercortisolism (Cushing's disease) and stress-related neuropsychiatric disorders.

Abstract.

Altered brain somatostatin functions recently appeared as key elements for the pathogenesis of stress related neuropsychiatric disorders. The hippocampus exerts an inhibitory feedback on stress but the mechanisms involved remain unclear. We investigated herein the role of hippocampal somatostatin receptor subtypes in both stress response and behavioral emotionality using C57BL/6, wild type and sst 2 or sst 4 knockout mice. Inhibitory effects of hippocampal infusions of somatostatin agonists on stress-induced hypothalamo-pituitary-adrenal axis (HPA) activity were tested by monitoring peripheral blood and local hippocampus corticosterone levels, the latter by using microdialysis. Anxiolytic and antidepressant-like effects were determined in the elevated-plus maze, open field, forced swimming and stress-sensitive beam walking tests. Hippocampal injections of somatostatin analogs and sst 2 or sst 4, but not sst 1 nor sst 3 receptor agonists produced rapid and sustained inhibition of HPA axis. sst 2 agonists selectively produced anxiolytic-like behaviors whereas both sst 2 and sst 4 agonists had antidepressant-like effects. Consistent with these findings, high corticosterone levels and anxiety were found in sst 2 KO mice and depressive-like behaviors observed in both sst 2 KO and sst 4 KO strains. Both hippocampal sst 2 and sst 4 receptors selectively inhibit stress-induced HPA axis activation but mediate anxiolytic and antidepressive effects through distinct mechanisms. Such results

Introduction

Stress plays an important role in the pathogenesis of neuropsychiatric disorders such as major depression (Lin and Sibille, 2015a;[START_REF] Mcewen | Stress Effects on Neuronal Structure: Hippocampus, Amygdala, and Prefrontal Cortex[END_REF][START_REF] Pittenger | Stress, depression, and neuroplasticity: A convergence of mechanisms[END_REF]. The main effect of stress is to activate the hypothalamopituitary-adrenal axis (HPA) leading to release of glucocorticoids from adrenal glands into the blood. Because of their liposolubility, glucocorticoids reach the brain by crossing the blood-brain barrier and bind to glucocorticoid and mineralocorticoid receptors [START_REF] Joels | Functional actions of corticosteroids in the hippocampus[END_REF]. The hippocampus contains the highest concentration of glucocorticoid receptors and plays a key role for negative feedback regulations on the HPA axis [START_REF] Herman | Limbic system mechanisms of stress regulation: Hypothalamo-pituitary-adrenocortical axis[END_REF][START_REF] Mcewen | Stress Effects on Neuronal Structure: Hippocampus, Amygdala, and Prefrontal Cortex[END_REF].

High glucocorticoid levels contribute to hippocampal dysfunctions and may lead to hippocampal atrophy as observed in major depression or Cushing's disease [START_REF] Andela | MECHANISMS IN ENDOCRINOLOGY Cushing's syndrome causes irreversible effects on the human brain: a systematic review of structural and functional magnetic resonance imaging studies[END_REF][START_REF] Sheline | Hippocampal atrophy in major depression: A result of depression-induced neurotoxicity?[END_REF]. Ablation of the dorsal hippocampus or lateral fornix disrupts the diurnal corticosteroid rhythm and elevates resting corticosteroids in rodents [START_REF] Fendler | EFFECT OF HIPPOCAMPAL LESION ON PITUITARY-ADREENAL FUNCTION[END_REF]. The intrinsic hippocampal mechanisms responsible for this effect remain unclear, but transynaptic glutamatergic projections from GABA neurons in the peri-paraventricular region and the bed nucleus of the stria terminalis may be involved [START_REF] Herman | Role of the ventral subiculum in stress integration[END_REF][START_REF] Radley | A Common Substrate for Prefrontal and Hippocampal Inhibition of the Neuroendocrine Stress Response[END_REF]. As mineralocorticoid and glucocorticoid receptors are predominantly expressed in principal cells, previous studies concerning corticosteroid effects on neuronal activity in the hippocampus focused mainly on these neurons [START_REF] Joels | Functional actions of corticosteroids in the hippocampus[END_REF]. However, chronic stress also impairs GABAergic functions in the hippocampus by affecting the integrity of parvalbumin-expressing neurons [START_REF] Hu | Stress Impairs GABAergic Network Function in the Hippocampus by Activating Nongenomic Glucocorticoid Receptors and Affecting the Integrity of the Parvalbumin-Expressing Neuronal Network[END_REF] and somatostatin (SOM)expressing cells [START_REF] Czeh | Chronic Stress Reduces the Number of GABAergic Interneurons in the Adult Rat Hippocampus, Dorsal-Ventral and Region-Specific Differences[END_REF]. In vivo, rapid SOM release occurs in the dorsal dentate gyrus when acute stress or dexamethasone is applied to animals [START_REF] Arancibia | Acute stress and dexamethasone rapidly increase hippocampal somatostatin synthesis and release from the dentate gyrus hilus[END_REF], suggesting a role for SOM in the hippocampal regulatory feedback on stress. In addition, mice lacking SOM exhibit elevated behavioral emotionality, high basal plasma corticosterone (CORT) levels and reduced gene expression of BDNF, cortistatin and GAD67, together recapitulating behavioral, neuroendocrine and molecular features of human depression (Lin et al, 2015a).

Five somatostatin receptor subtypes have been described, sst 1 , sst 2 , sst 3 , sst 4 , and sst 5 . The development of selective SOM analogs [START_REF] Rohrer | Rapid identification of subtype-selective agonists of the somatostatin receptor through combinatorial chemistry[END_REF] and the availability of genetically modified animal models [START_REF] Zeyda | Null mutant mouse models of somatostatin and cortistatin, and their receptors[END_REF] has contributed to a better understanding of individual sst receptor characteristics and their roles in brain functions and disorders including anxiety and depression [START_REF] Engin | ANXIOLYTIC AND ANTIDEPRESSANT EFFECTS OF INTRACEREBROVENTRICULARLY ADMINISTERED SOMATOSTATIN: BEHAVIORAL AND NEUROPHYSIOLOGICAL EVIDENCE[END_REF][START_REF] Engin | Anxiolytic and antidepressant actions of somatostatin: the role of sst2 and sst3 receptors[END_REF][START_REF] Epelbaum | Somatostatin, Alzheimer's disease and cognition: An old story coming of age?[END_REF][START_REF] Viollet | Involvement of sst2 somatostatin receptor in locomotor, exploratory activity and emotional reactivity in mice[END_REF]. Recent studies identified a causal role for frontal and cingular SOM cells in modulating behavioral emotionality in mice (Lin et al, 2015a;Lin and Sibille, 2015b;[START_REF] Soumier | Opposing Effects of Acute versus Chronic Blockade of Frontal Cortex Somatostatin-Positive Inhibitory Neurons on Behavioral Emotionality in Mice[END_REF] and showed that SOM anxiolytic effects are mediated by sst 2 receptors expressed in the amygdala and the septum of the rat brain [START_REF] Yeung | Anxiolytic-like effects of somatostatin isoforms SST 14 and SST 28 in two animal models (Rattus norvegicus) after intra-amygdalar and intra-septal microinfusions[END_REF][START_REF] Yeung | The anxiolytic effects of somatostatin following intra-septal and intra-amygdalar microinfusions are reversed by the selective sst2 antagonist PRL2903[END_REF]. While all sst receptor subtypes (except for sst 5 ) are present in the hippocampus, SOM has been shown to modulate learning and memory formation through sst 2 and sst 4 receptors selectively [START_REF] Gastambide | Cooperation Between Hippocampal Somatostatin Receptor Subtypes 4 and 2: Functional Relevance in Interactive Memory Systems[END_REF][START_REF] Gastambide | Hippocampal SSTR4 somatostatin receptors control the selection of memory strategies[END_REF][START_REF] Guillou | EFFECTS OF INTRAHIPPOCAMPAL INJECTIONS OF SOMATOSTATIN AND CYSTEAMINE ON SPATIAL DISCRIMINATION-LEARNING IN MICE[END_REF]. The present study investigated the role of hippocampal SOM and its receptors in stress response and emotionality, an issue that has so far remained unexplored.

Materials / subjects and Methods

Experimental goals

The aims of the present study were first to determine whether any of the four sst receptor subtypes (sst 1-4 ) expressed in the hippocampus inhibits stress-induced activation of the HPA axis and then, to investigate whether such inhibitory control could have anxiolytic or antidepressant properties. Intrahippocampal injections of selective sst receptor agonists were given prior to a stress event and plasma CORT elevation was measured. Given that plasma levels do not always parallel brain concentrations of glucocorticoids, microdialysis was conducted to assess CORT levels after sst agonist infusions and acute stress directly in the hippocampus. Then, independent groups of mice received sst agonists intrahippocampally and anxiety or depressive-like behaviors were assessed. Finally, further hippocampal microdialysis assays and emotional evaluations were conducted in sst 2 or sst 4 knockout mice.

Animals

Male mice of the C57Bl/6J strain were obtained from Charles River Laboratories (L'Arbresle, France). sst 2 knockout (sst 2 KO) strain was originally generated by Zheng et al. (Zheng et al, 1997). sst 4 knockout (sst 4 KO) mice were produced by integration (knockin) of LacZ gene into the sstr4 gene [START_REF] Helyes | Impaired defense mechanism against inflammation, hyperalgesia, and airway hyperreactivity in somatostatin 4 receptor gene-deleted mice[END_REF]. All lines were backcrossed to the C57BL/6J background (N>11). KO and wild-type male littermates were obtained by intercrossing heterozygous +/-mice. Animals

Surgery

Mice were anesthetized with a ketamine (1mg/kg body weight)-xylazine (10mg/kg body weight) mix and placed on a stereotaxic frame. A microdialysis guide-cannulae (CMA/7 microdialysis probe, CMA microdialysis, Sweden) was implanted at the following coordinates from the bregma: AP: -2.2mm; L: ±1.35mm; V: 1mm. The laterality of the implantation was randomized between right and left side of the brain. Guide cannulae were fixed to the skull with dental cement (Palavit G, Promodentaire) and 3 screws (stainless steel, Ø 0.5mm, L=1mm; FOM2000). For behavioral experiments, mice were implanted at the same coordinates with stainless steel guide-cannulae (Le Guellec tubular components, France). Sterile stylets were inserted in the cannulae to maintain patency. Mice were allowed to recover from surgery for 1 week before experiments.

Drugs

Somatostatin analogs (RC160 and octreotide, TOCRIS, France), as well as selective agonists for sst 1 (L-797,591 Rahway, NJ, USA) sst 2 (L-054,264, TOCRIS, France), sst 3 (L-796,778 Rahway, NJ, USA) or sst 4 (L-803,087, TOCRIS, France) were all used at the dose of 5nmol. For microdialysis experiments, all drugs were diluted in artificial cerebrospinal fluid (aCSF) containing 10% of dimethylsulfoxide (Vehicle, Veh) and were delivered into the hippocampus via the microdialysis probe within 15 min. For other experiments, the drugs were prepared at the concentration of 5nmol/0.5µl based on previous experiments [START_REF] Gastambide | Cooperation Between Hippocampal Somatostatin Receptor Subtypes 4 and 2: Functional Relevance in Interactive Memory Systems[END_REF] and were delivered into the hippocampus within 5 minutes in freely-moving animals as previously described [START_REF] Martel | Stimulation of hippocampal adenylyl cyclase activity dissociates memory consolidation processes for response and place learning[END_REF].

Injections were given 15 min before testing sessions.

Microdialysis

Microdialysis was performed in freely moving animals to measure CORT levels in the dorsal hippocampus 1h before and 3h after acute stress as previously described [START_REF] Dorey | Stress-Induced Memory Retrieval Impairments: Different Time-Course Involvement of Corticosterone and Glucocorticoid Receptors in Dorsal and Ventral Hippocampus[END_REF]. The experiment was conducted by squad of 2 animals belonging to controls or experimental groups respectively. The day before the experiment, animals were individually placed in a 30cm-diameter chamber made of clear plexiglass walls and equipped with a grid floor. Animals had ad libitum access to their habitual chow and water. A microdialysis probe (CMA/7 membrane length 1mm; Ø=0.24mm; CMA Microdialysis, Sweden) was inserted into the guide-cannulae. The membrane was 1mm longer than the guide-cannulae to be directly in contact with the hippocampal tissue. Probe was continuously perfused with sterile, filtered Dulbecco's solution (mock CSF, Sigma, France) at a rate of 0.1µl/min for fluid equilibration during 1 night. After the equilibration phase, baseline dialysates were collected every 15 min with a flow rate of 1µl/min during 1h. After baseline establishment, three inescapable foot-shocks (0.9mA; 1sec) were applied from the grid floor within 1 min to provide stress. For pharmacological experiments, the perfusion was switched to another syringe containing the drug or the vehicle for 15 min after baseline. At the end of the injection, syringes were switched back to aCSF and the stress procedure was applied. The dialysates were collected every 15 min during 3h after stress. Samples were stored at -80°C until free CORT determination.

Plasma corticosterone collection

Mice received an intrahippocampal injection of either one sst agonist or Veh solution in their home-cage. Fifteen minutes after the injection, 3 foot-shocks were delivered as described above and 15 minutes later blood was collected from the retro-orbital vessel under anaesthesia (isofluran) by a capillary tube. After centrifugation at 3500 r.p.m. for 10 min at 4°C, the supernatant was stored at -80°C until ELISA assay. An enzyme immunoassay commercial kit (Correlate-EIA, Assay Designs, Ann Arbor, MI) was used to measure plasma or hippocampal CORT concentrations. The sensitivity of the ELISA assay is 18.6 pg/ml. Baseline sample concentration was more than 10-fold above the sensitivity threshold.

Data are expressed in pg/ml/15min for CSF and in ng/ml for plasma.

Behavior Motion recording during intrahippocampal microdialysis

Each mouse was video recorded and its activity was quantified by videotracking (Viewpoint, Champagne au Mont d'Or, France). Three displacement thresholds were determined from pilot trials to quantify resting or very small motions, small motions and large motions respectively as function of pixel displacements/16ms. Changes of movement types were counted as a motion unit and analyzed per period of 15 minutes to assess the evolution of the activity from the baseline to the end of the post-stress evaluation.

Elevated Plus-Maze

The elevated plus-maze (EPM) apparatus was composed of grey polyvinylcarbonate and consisted of four arms surrounding a central platform.

Each arm was 30cm long, 7cm wide, and 60cm above the ground. The four arms joined at the center on a 7cm square platform. Two opposite arms of the plus maze ("closed arms") were bordered by 24cm-high sidewalls opened on the top, and the two other arms ("open arms") were exposed to a 70lux light intensity. At the beginning of each test, the mouse was placed in the center of the maze in a cylinder (7cm in diameter, 17cm high) for 30 sec. Then, the cylinder was removed and the animal was allowed to explore all arms of the maze freely for 8 min. An entry was counted when the mouse entered an arm with all 4 feet. The number of entries, times in open and closed arms, latencies to entry were measured.

Open-Field

The open-field was a white circular arena (1m diameter) surrounded by a 25 cmhigh wall made of opaque plexiglass. One lamp, 2 meters above the apparatus, provided 70 lux of illumination distributed equally over the entire surface of the apparatus. Animals were placed facing to the wall and were allowed to freely explore the apparatus for 5 min. The floor was virtually divided into a periphery annulus 12cm wide and a central zone. Exploration parameters were quantified by videotracking.

Forced Swim Test

Mice were placed into a glass cylinder (diameter 15cm, height 29cm) containing 20cm of water at 25 ±1°C. The protocol used is that initially described by Porsolt et al. [START_REF] Porsolt | BEHAVIORAL DESPAIR IN MICE -PRIMARY SCREENING-TEST FOR ANTIDEPRESSANTS[END_REF]

Stress-sensitive Beam Walking Test

Mice were placed at the extremity of a 1-meter long beam, 80cm high from the floor. The start platform was narrow (3x4 cm) and exposed to intense light (120 lux) to encourage the animal to walk through the beam to reach the dark compartment. The floor was lined with foam to prevent hurting if animals fall down. The time to cross the beam and the number of hesitations (stop or slowing down) were recorded and quantified by the videotrack system. Animals were first trained for 3 days (3 trials/day) with beams of increased difficulties as a function of their shape and size: 10mm square beam, 15 mm round beam, 8mm square beam. A cut-off was applied if the mouse fell down from the beam or did not cross the beam in 2-min time. The next day, mice were injected with sst 2 or sst 4 agonists or vehicle 15 minutes before receiving a foot-shock stress as described above. Fifteen minutes later, mice were tested on the beams experienced during training and an additional beam (10 mm round) was introduced. Results are the mean of two successive trials for each beam.

Rationales and data regarding the stress sensitivity of the task are presented in Supp3.

Histology

At the end of the experiments, mice were anesthetized with 27mg/0.5ml of Sodium Pentobarbital and quickly decapitated. Entire heads were immersed in a solution of 10% formaldehyde at 4°C for two weeks. The brain was removed from the skull and was then cryoprotected and coronally sectioned into slices (60µm thickness). A Thionin acetate stain was used to check the exactness the implantations.

Statistical Analysis

Statistical analyses were performed using the Statview 5.0 software (Statistical Analysis System Institute Inc, NC, USA). The data were analyzed using ANOVA to determine main factor effects and followed by appropriate post-hoc tests.

Mean group comparisons were also performed using Student's t-tests.

Results

Plasma CORT levels increased by 4-fold following stress exposure relative to non-stressed conditions (Figure 1). Stress-induced CORT levels were lower following intrahippocampal injections of L-054,264 (t(11)=3.98; P=0.002) or L-803,087 (t(9)=2.94; P=0.016) but not of L-797,591 or L-796-778 agonists relative to stressed controls, indicating that hippocampal sst 2 and sst 4 , but not sst 1 or sst 3 receptors, contribute to inhibit stress-induced activation of the HPA axis. L-054,264 also reduced basal CORT levels significantly (t(8)=1.92; P=0.04) whereas L-803,087 was inefficient. Basal levels of plasma CORT were significantly higher in sst 2 KO mice (P=0.04) than in WT littermates and consistently, basal CORT levels in hippocampal dialysates were also higher in sst 2 KO mice (Figure 5). A significant genotype effect was observed both before (F(1.16)=9.91; P=0.006) and after stress exposure (F(1,16)=6.86 ; P=0.019). The time lag to produce a significant increase (45 min, P=0.009) was shorter and the maximal (P=0.02) and recovery (P=0.01) levels of CORT concentrations were significantly higher in sst 2 KO than in WT, indicating that the HPA axis is constitutively up-regulated and is more reactive to stress in the absence of sst 2 receptors. No change in CORT levels was observed in sst 4 KO mice (P=0.78). Finally, and to further investigate the inhibitory control induced by sst 4 receptor activation, intrahippocampal injections of L-803,087 were tested in sst 2 KO mice (Figure 5C). L-803,087 lowered stressinduced elevation of hippocampal CORT levels in sst 2 KO mice (F(1,12)=5.76; P=0.03), thus confirming that hippocampal sst 4 receptors contribute to the inhibition of the HPA axis. However, the inhibitory effect of L-803,087 occurred only later (+1h30) whereas it occurred immediately after stress onset in experiments with C57BL/6 mice (see Figure 2), indicating that the presence of hippocampal sst 2 receptors is required for the fast-acting inhibition of the HPA axis by somatostatinergic neurotransmission.

Discussion

The hippocampus is important for episodic and spatial memory but is now also recognized for its role in mood regulation [START_REF] Mcewen | Stress Effects on Neuronal Structure: Hippocampus, Amygdala, and Prefrontal Cortex[END_REF]. There is evidence for a connection between hippocampal-HPA axis dysregulations and stress-related illnesses such as depression, generalized anxiety, post-traumatic stress disorders and memory deficits [START_REF] Lucassen | Neuropathology of stress[END_REF]. Chronic stress exposure [START_REF] Sapolsky | Glucocorticoids and hippocampal atrophy in neuropsychiatric disorders[END_REF], acute exposure to psychogenic stressors [START_REF] Popoli | The stressed synapse: the impact of stress and glucocorticoids on glutamate transmission[END_REF] or acute therapies with high-dose of corticoids [START_REF] Brown | Effects of Glucocorticoids on Mood, Memory, and the Hippocampus Treatment and Preventive Therapy. Glucocorticoids and Mood Clinical Manifestations, Risk Factors[END_REF] negatively impact hippocampal neuroplasticity, neurogenesis and neuronal or glial survival in relationship with depression [START_REF] Banasr | Regulation of neurogenesis and gliogenesis by stress and antidepressant treatment[END_REF][START_REF] Banasr | Cell atrophy and loss in depression: reversal by antidepressant treatment[END_REF][START_REF] Czeh | What causes the hippocampal volume decrease in depression?[END_REF][START_REF] Pittenger | Stress, depression, and neuroplasticity: A convergence of mechanisms[END_REF]. Studies have also linked depression to an increase in the excitatory-inhibitory ratio which may be attributed to a loss of GABAergic neurons. Parvalbumin and SOM subpopulations in both the dorsal and ventral hippocampus are particularly vulnerable to chronic stress exposure [START_REF] Czeh | Chronic Stress Reduces the Number of GABAergic Interneurons in the Adult Rat Hippocampus, Dorsal-Ventral and Region-Specific Differences[END_REF]. We observed a causal inhibitory effect of RC160, octreotide and selective sst 2 or sst 4 receptor agonists on the acute stress response, as measured by alterations in both plasma and hippocampal CORT levels, demonstrating the critical role played by SOM neurotransmission in inhibitory hippocampal feedback on the HPA axis. The levels and the time course of stress-induced increase in hippocampal CORT were comparable to previous experiments using the same protocol [START_REF] Chauveau | Rapid Stress-Induced Corticosterone Rise in the Hippocampus Reverses Serial Memory Retrieval Pattern[END_REF][START_REF] Dorey | Stress-Induced Memory Retrieval Impairments: Different Time-Course Involvement of Corticosterone and Glucocorticoid Receptors in Dorsal and Ventral Hippocampus[END_REF][START_REF] Tronche | Pharmacological modulation of contextual "episodic-like" memory in aged mice[END_REF]. Consistent with the finding that acute stress or dexamethasone produces rapid SOM release in the hippocampus [START_REF] Arancibia | Acute stress and dexamethasone rapidly increase hippocampal somatostatin synthesis and release from the dentate gyrus hilus[END_REF], SOM inhibitory effects were detectable within 15 minutes after infusion, indicating that activation of the hippocampal somatostatinergic network by glucocorticoids controls the stress response rapidly. sst 2 and sst 4 receptor agonists produced stronger effects than RC160 and octreotide, in agreement with better stability and longer half-life than peptidergic analogs [START_REF] Rohrer | Rapid identification of subtype-selective agonists of the somatostatin receptor through combinatorial chemistry[END_REF].

Mice reacted to footshocks by expressing a transient peak of locomotor activity which was comparable in all groups, indicating that sst receptor functions do not alter sensory thresholds in this context. Both sst 2 and sst 4 receptor agonists reduced locomotor activity, an inhibitory effect which occurred rapidly and persisted for the entire experiment. The time course of decreased activity in mice infused with sst 2 or sst 4 agonists suggests that the effect represents a more complete adaptation (habituation) to the aversive context than to be a motor deficiency. Accordingly, L-054,264 had no effect and L-803,087 improved performance in the beam walking test. Both agonists also had similar inhibitory effects on immobility in the FST. Interestingly, in this latter test, they produced different active behaviors, implying that distinct mechanisms underlie their behavioral effects. The sst 2 receptor agonist increased swimming behavior without affecting climbing behavior, a type of behavioral pattern which is characteristic of serotoninergic activity. In sharp contrast, the sst 4 receptor agonist increased climbing behavior without affecting swimming as produced by antidepressants with noradrenergic activity [START_REF] Detke | ACTIVE BEHAVIORS IN THE RAT FORCED SWIMMING TEST DIFFERENTIALLY PRODUCED BY SEROTONERGIC AND NORADRENERGIC ANTIDEPRESSANTS[END_REF][START_REF] Reneric | Idazoxan and 8-OH-DPAT modify the behavioral effects induced by either NA, or 5-HT, or dual NA/5-HT reuptake inhibition in the rat forced swimming test[END_REF]. Though this dissociation is less clear in mice than in rats, the behavioral profile produced by L-054,264 (decreased immobility/increased swimming) was similar to that previously observed in rats which received intracerebroventricular injections of SOM [START_REF] Engin | ANXIOLYTIC AND ANTIDEPRESSANT EFFECTS OF INTRACEREBROVENTRICULARLY ADMINISTERED SOMATOSTATIN: BEHAVIORAL AND NEUROPHYSIOLOGICAL EVIDENCE[END_REF] or L-779,976, another sst 2 selective agonist [START_REF] Engin | Anxiolytic and antidepressant actions of somatostatin: the role of sst2 and sst3 receptors[END_REF]. In addition, the hippocampal injection of sst 2 , but not the sst 4 agonist, also produced anxiolytic-like behavior in EPM and open field testing, corroborating the idea that hippocampal sst 2 and sst 4 regulate different emotional responses. These results indicate that the affinities for the sst receptor subtypes of L-054,264 (1250-fold superior for sst 2 than for sst 4 ) and L-803,087 (6700-fold superior for sst 4 than for sst 2 ) are sufficient to produce selective behavioral changes [START_REF] Rohrer | Rapid identification of subtype-selective agonists of the somatostatin receptor through combinatorial chemistry[END_REF][START_REF] Yang | Spiro 1H-indene-1,4 '-piperidine derivatives as potent and selective non-peptide human somatostatin receptor subtype 2 (sst(2)) agonists[END_REF].

Consistent with the pharmacological results and previous studies [START_REF] Viollet | Involvement of sst2 somatostatin receptor in locomotor, exploratory activity and emotional reactivity in mice[END_REF], sst 2 KO, but not sst 4 KO mice displayed increased anxiety-like behaviors.

CORT levels were also constitutively higher in sst 2 KO mice in agreement with the ACTH release previously described in these mice [START_REF] Viollet | Involvement of sst2 somatostatin receptor in locomotor, exploratory activity and emotional reactivity in mice[END_REF] and the increased basal level of CORT reported in SOMKO mice (Lin et al, 2015a), an effect undetected in sst 4 KO mice. In addition, increased immobility in the FST associated with decreased swimming in sst 2 KO mice whereas climbing was decreased in sst 4 KO mice corroborate the idea that sst 2 and sst 4 regulate emotionality by different mechanisms. Recent data showing increased immobility in the FST in sst 4 KO mice whereas intraperitoneal treatments with J-2156, another highly selective sst 4 agonist, decreased immobility in the tail suspension test in C57BL/6 mice, support the idea that the sst4 receptor subtype is involved in depression-like behaviors [START_REF] Scheich | Somatostatin receptor subtype 4 activation is involved in anxiety and depression-like behavior in mouse models[END_REF]. We previously showed that injections of sst 4 agonists in the dorsal hippocampus produced a shift from flexible cognitive to habit memory systems to rescue task performance [START_REF] Gastambide | Hippocampal SSTR4 somatostatin receptors control the selection of memory strategies[END_REF] as similarly produced by stress [START_REF] Schwabe | Stress and multiple memory systems: from 'thinking' to 'doing[END_REF]. Together with the present results, this suggests that dorsal hippocampal sst 4 receptors control both emotional and cognitive responses to cope with stressful situations.

Previous results showed that the functionality of hippocampal sst 4 receptors is lost following pharmacological blockade or knockout of sst 2 receptors. L-803,087-mediated increases in glutamatergic excitability and bursting frequency in hippocampal CA1 region, as well as L-803,087-mediated shifts toward the use of striatal memory systems are both blocked by sst 2 antagonists or sst 2 gene deletion [START_REF] Cammalleri | Compensatory changes in the hippocampus of somatostatin knockout mice: upregulation of somatostatin receptor 2 and its function in the control of bursting activity and synaptic transmission[END_REF][START_REF] Gastambide | Cooperation Between Hippocampal Somatostatin Receptor Subtypes 4 and 2: Functional Relevance in Interactive Memory Systems[END_REF][START_REF] Moneta | Somatostatin receptor subtypes 2 and 4 affect seizure susceptibility and hippocampal excitatory neurotransmission in mice[END_REF]. In line with these results we show here that in the absence of sst 2 receptors, the sst 4 -mediated inhibition on the HPA axis is lost during induction of the stress response while it remains effective during the stress recovery phase. Based on immunohistochemistry detections [START_REF] Gastambide | Cooperation Between Hippocampal Somatostatin Receptor Subtypes 4 and 2: Functional Relevance in Interactive Memory Systems[END_REF], binding experiments [START_REF] Viollet | Involvement of sst2 somatostatin receptor in locomotor, exploratory activity and emotional reactivity in mice[END_REF] and mRNA mapping [START_REF] Vanetti | mRNA distribution of two isoforms of somatostatin receptor 2 (mSSTR2A and mSSTR2B) in mouse brain[END_REF] in the mouse brain, the non-overlapping distribution of sst 2 and sst 4 respectively found in the dentate gyrus and in the CA1 subfield suggests that interactions between these sst receptor functions occur indirectly. The nonoverlapping distribution may also account for the differential effects of sst 2 and sst 4 on anxiety-like and depressive-like behaviors. Insofar as the role of the CA1 subfield in emotions remains unclear, the present results suggest that dentate gyrus sst 2 and CA1 sst 4 receptors conjointly control behavior in response to glucocorticoids by providing anxiolytic and antidepressant effects. Given the general consensus for involvement of the ventral hippocampus in regulation of stress responses and emotionality, it is interesting that the effects were produced by injections into the dorsal hippocampus. Whether the effects are relative to spread of injections in the ventral hippocampus and/or to intrahippocampal communication remains to be determined. Notwithstanding, a recent study demonstrated that optogenetic control of granule cells activity in the dorsal dentate gyrus selectively results in a dramatic increase in exploratory behavior in novel environments while affecting the ventral dentate gyrus causes an equally robust anxiolytic-like behavior [START_REF] Kheirbek | Differential Control of Learning and Anxiety along the Dorsoventral Axis of the Dentate Gyrus[END_REF].

In conclusion, SOM is a key regulator of the HPA axis activity not only at the pituitary level but also at the brain level and besides cortical, hypothalamic, amygdalar and septal involvements, hippocampal SOM neurotransmission critically controls both HPA function and emotionality through sst 2 and sst 4 receptors. This must be taken into account for the development and management of multiselective sst receptor ligands such as pasireotide which is currently used to control hypercortisolism in the Cushing's disease, a syndrome most often associated with anxiety, irritability, depression and memory troubles [START_REF] Colao | Managing Cushing's disease: the state of the art[END_REF].

Zheng H, Bailey A, Jiang MH, Honda K, Chen HY, Trumbauer ME, et al (1997). Somatostatin receptor subtype 2 knockout mice are refractory to growth hormone-negative feedback on arcuate neurons. Molecular Endocrinology 11(11): 1709-1717. mean CORT concentration (ng/ml for plasma levels or pg/ml/15min for intrahippocampal levels) ± s.e.m. *P<0.05, **P<0.01 vs wild-type groups or relative to the Veh group using Student's t-tests.

Legends to the Figures

  were bred in an animal facility equipped with an artificial 12h light/dark cycle (7:00AM on). Mice were 4-8-month-old at the beginning of the experiments. All experiments were conducted between 7h30 and 12h and in compliance with the © 2016 Macmillan Publishers Limited. All rights reserved.

  directive 2010/63/EU, Animal Care and Use Committee (Bordeaux) and approved under the number 5012098-A.

  in which the animal is maintained in water during 6 min. Given that mice are very active during the first two minutes and inactive during the last two minutes of FST, the times spent in swimming, climbing or immobility were analyzed during the last four minutes for pharmacology © 2016 Macmillan Publishers Limited. All rights reserved.

  experiments or during the first four minutes in knockout experiments, in order to unmask respectively antidepressant and depressive-like effects.

  Hippocampal CORT concentrations sampled by microdialysis rose within one hour following stress onset in control mice (Figure2) and progressively returned to baseline levels within the next two hours. SOM analogs RC160 and octreotide, known to target sst 2-3-5 receptor subtypes attenuated stress-induced increase in hippocampal CORT levels (Supp2; Treatment effects Fs≥4.10; Ps≤0.05 for both analogs). The non-peptidic agonists L-054,264 and L-803,087 (Figure3) targeting sst 2 and sst 4 respectively, fully blocked the elevation of CORT levels in the hippocampus following acute stress (treatment effects:© 2016 Macmillan Publishers Limited. All rights reserved.

Fs≥7. 61 ;

 61 Ps≤0.01 for both agonists). Inhibitory effects started immediately after infusion (Ps≤0.01 for both agonists) and lasted for at least 3 hours post stress onset (Ps≤0.05 for both agonists). As indicated on Figure2B, stress induced a transient peak of hyperactivity in all groups. Then, locomotor activity in stressed mice infused with either the sst2 or sst4 agonist rapidly recovered the level of non-stressed controls and was significantly lower relative to stressed controls (treatment effects: Fs≥6.18; Ps≤0.02 for both agonists), indicating that both sst 2 and sst 4 receptors attenuate stress-induced reactivity. However, further behavioral investigations revealed that each of these agonists alleviates stressinduced emotional responses differently (Figure3). L-054,264 selectively reduced anxiety-like behaviors as assessed by exploration levels of open places either in the EPM (P≤0.009 for both entries and time in open arms) or in the open-field (P=0.01). In the EPM, L-054,264 also shortened the latency to enter an open arm (P=0.001). Both L-054,264 and L-803,087 modified behaviors in the FST but the latter increased the attempts to escape by climbing on the wall of the glass cylinder (P=0.04) whereas the former increased the time spent in swimming (P=0.02). Finally, only L-803,087 improved performance in the beam walking test. The sst4 agonist prevented both stress-induced hesitations (P=0.007, data not shown) and increases in time (P<0.01) to cross the beams as a function of difficulty and novelty, further highlighting that hippocampal sst 2 and sst 4 receptors control different emotional behaviors. Consistent with such findings, behaviors of sst 2 KO and sst 4 KO mice in the EPM and in the FST were altered in an opposite manner (Figure 4). Relative to wild type littermates, sst 2 KO mice displayed increased latency to enter an open arm (P=0.04) and lower exploration time in open arms (P=0.001) whereas no significant effect was detected between sst 4 KO and wild type. In the FST, both sst 2 KO and sst 4 KO mice displayed increased immobility relative to their wild type littermates (P=0.001 and P=0.004, respectively). However, swimming decreased selectively in sst 2 KO mice (P=0.002) whereas climbing decreased selectively in sst 4 KO mice (P=0.02) indicating that active behaviors are qualitatively altered depending on genotypes.

Figure 1 :

 1 Figure 1: Hippocampal infusions of sst 2 and sst 4 receptor agonists

Figure 2 :

 2 Figure 2: Hippocampal concentrations of CORT and general motion

Figure 3 :

 3 Figure 3: Hippocampal sst 2 and sst 4 receptors regulate emotional

Figure 4 :

 4 Figure 4: sst 2 and sst 4 receptor gene deletion differently modulate

© 2016 Macmillan Publishers Limited. All rights reserved.

© 2016 Macmillan Publishers Limited. All rights reserved.© 2016 Macmillan Publishers Limited. All rights reserved.

Acknowledgements

We thank the CPN and the INCIA small animal facility staffs for technical assistance and animal care. We thank Christophe Tronche, Una Avdic (Erasmus fellow), Marie Mennesson and Bertrand Beauchoux for their assistance during experiments. We also thank Dr. Gary Gilmour for helpful comments on the manuscript.

Funding and Disclosures

This study was supported by the Région Aquitaine and by the Programme Interdisciplinaire du CNRS "Longévité et Vieillissement". The authors declare no

competing financial interests or conflict of interest in relation to the work described.


Paolo Tagliolato

MUSIC STRUCTURE REPRESENTATION: A FORMAL MODEL

In the present work we introduce a general formal (Object Oriented) model for the representation of musical structure information, taking into account some common feature of the information vehiculed by music analysis. The model is suitable to represent many different kinds of musical analytical entities. As an example of both musical and mathematical-computational relevance, we introduce the D.Lewin's GIS theory. A GIS is equivalent to a particular kind of group action over a set: we show that other related structures can be treated in a similar manner. We conclude with the prototype implementation of these ideas in MX, an XML format for music information.

INTRODUCTION

Music representation in XML is a field in evolution. There are many proposals for a possible standard. MX's peculiarity is its multilayered structure(cfr. [START_REF] Poli | Ingegneria del software e informatica musicale[END_REF], [START_REF] Haus | Towards a symbolic/time-based music language based on xml[END_REF], [START_REF] Haus | Music segmentation: an xml-oriented approach[END_REF], [START_REF]Draft Trial-Use Recommended Practice for Definition of a Commonly Acceptable Musical Application Using the XML Language[END_REF]). One of these layers is devoted to music structure information, i.e. compositional and analytical aspects of musical information. The present work constitues a theoretical survey to individuate formal entities for possible representations in the format and contains the sketch of a general model which could suggest some idea to extend its current implementation. Analytical information encoding into a format containing music like mx, could open some different possibilities. We will imagine here some of these application. A musicanalysis tool based on mx format would give an analyst a graphical interface to edit his own analysis, not only by visual, but also by aural means. The same capability (to aurally show the elements of the analysis), could make an mx-based tool a strongly popularizing tool, not only from an educational point of view. A software based on the mx format would allow to listen (in several audio versions) to parts of pieces described in the analyses, to directly follow the comments of the analyst on the score while music involved by these is executed. To have more than one analysis of the same piece would provide an effective instrument for comparative analysis. Furthermore, the possibility to compare different analyses could have still wider effects. Let's imagine as an example to have access to sources in a big database, containing varied analyses that could also be generated automatically, as an example, at the submission of the file in the database. By means of searches on analysis-data in the files, one could recognize similarities on structural level between pieces analyzed by common features. An opportune agent could lead surveyings on a specific feature, searching recurrences in order to determine affinities. The way in which analytical information could be stored is the subject of the next section.

MUSICAL STRUCTURE

The first problem is to define what is musical structure. If we consider the approaches to music analysis, we'll find each one to have its own tools, methods and results. Each analysis reveals a specific structure. The problem could become then: how to model music analyses. We are not interested here in each motivation for a specific methodology. We'll limit ourselves to observe some common features.

Basic common structure of analyses

A very basic structure (of an analysis, not of a piece!), common to almost all anaytical approaches (e.g. Forte and set-theory [START_REF] Forte | The Structure of Atonal Music[END_REF], Nattiez and Paradigmatic analysis [START_REF] Nattiez | Analyse musicale et sémiologie: à propos du prélude de 'pelléas[END_REF], automated analysis conducted in ways like Meudic's [START_REF] Meudic | musical pattern extraction: from repetition to musical structure[END_REF]) is the segmentation of a piece. In each school of music analysis, segmentation has its own specificities in term of when and how it is conducted. Very interesting theoretical problems arises around segmentation, intended as the process by which a piece is segmented, but a discussion here is beyond the scope of the present work. We will call here segmentation the result of the process. So we will define a segmentation of a piece, in a very general sense, as a collection of segments. A segment will be a set of references to events in a music piece.

When the process of segmentation is done and a collection of segments is obtained, the next step we want to concentrate on is the relating the segments one another. This is done accordingly the specific methodology in action. Now we simply notice the presence of relations between segments.

Segmentation, segments and relations between segments are the very basic structure contained in MX format to represent music analyses. We will intend this structure as the formal shadow of an analysis of a piece of music.

A formal (object oriented) model for specific analysis representation

To discover similarities between segments of music we need a particular musical sensibility. We need to concentrate to particular music features. In order to define relation between segments we need a way to define these features. First of all we must decide which musical feature we are interested in. Many approaches associate feature vectors to segments (like in [START_REF] Anagnostopoulou | Classification in music: a computational model for paradigmatic analysis[END_REF]). We think that the more general concept of object could offer a more powerful possibility: in an object oriented perspective, one will be able to define his own feature object class in order to model his own music attitude. In a limit case a feature object could be just a textual description (not so naturally formalizable within a vector). So, if we have a segmentation σ we can imagine an arrow into an abstract FeatureObject Ω class:

f : Σ -→ Ω σ -→ ω (1)
Depending on the particular decisions for a concrete derived class of objects, let's call it S (Space of objects), a part of an analysis will be based on a new association:

f my : Σ -→ S σ -→ s (2)
Now we can predicate something on the segments σ ∈ Σ in our particular point of view (one could say frame) by means of the featureObjects s ∈ S. In this perspective we can relate segments by defining relations between our objects associated to them. In certain cases there will be formal rules to establish such relations. In certain cases there will be methods to say if a segment is related to another and in which way it is.

The next two sections show a formal mathematical model (of music analytical relevance) which could constitue an example of a possible instantiation for feature objects and relations. There will be shown that in these models relations and objects have particular algebraic structures. This suggests the possibility (beyond the scope of this paper, but constituing a trace for further developements) to take advantage from these structures for further investigation.

GENERALIZED INTERVAL SYSTEMS

We're going to introduce a structure proposed by the american music-theorist D. Lewin ([9]). This structure is based on the concept of interval in an extended manner. With GIS it's possible to represent not only pitch-related, but also rythmic musical objects and their relationships. We want to emphasize the idea that the choice of a particular GIS lets the modelling of particular features of music. A GIS will be a candidate model for a class of objects to put in the S-level introduced in the previous section. First of all we proceed with the formal definition of GIS as given by Lewin. Then we will give an alternative characterization of a GIS which makes use of the concept of a simply transitive action over a set1 . This will permit us to generalize the model in order to represent a more general family of algebraic musical structures. Definition 3.1. A GIS is defined as (S, IVLS , int ), being:

• S ('space'), a set • IVLS (intervals), a group

• int an application int : S × S -→ IVLS satisfying the following conditions:

(A) ∀r, s, t ∈ S int (r, s) int (s, t) = int (r, t) (3) (B) ∀s ∈ S, i ∈ IVLS ∃ ! t ∈ S : int (s, t) = i (4)
Definition 3.2. We will say commutative a GIS which intervals group is abelian.

Given a GIS , we'll be able to describe particular musical features: those represented by the 'space' of musical objects (fixing a 'space' S is a sort of concentrating on a particular musical dimension). We'll also be able to calculate -via IVLS -an interval between two musical objects (elements of S). In a certain way choosing a GIS is like modelling a particular analythical attitude. Some very simple examples.

1 Example. Let the music space be the any musical scale steps. Given two steps s, t ∈ S, let int (s, t) be the number of steps between the two. For instance let's take into account the diatonic scale. We have

int (C 4 , D 4 ) = 1 int (C 4 , A 3 ) = -2
If we move by two steps (e.g. from C 4 to E 4 ), then by two more steps (e.g. from E 4 to G 4 ) we'll have done 2+2=4 degrees of the scale:

int (C 4 , E 4 ) = 2 int (E 4 , G 4 ) = 2 int (C 4 , G 4 ) = 4
The 'space' S so described, with the additive group of relative integers Z and the application int is a first, trivial example of a GIS . In fact: int satisfies (A). It satisfies also (B): given any step s of the scale S and any n ∈ Z one can find a step moving by n steps of the scale such that int An interval will be the ratio between two spans (we perceive a relation between two couples of spans if their ratios coincides). We will then consider

(f irstnote, f oundnote) = n 2 Example.
S = Q IVLS = ( Q * , •) int : (x, y) → y x
As we said at the beginning of this section, it is possible to define the GIS structure in the following alternative way. Definition 3.5 (of GIS, alternative). A GIS is (S, IVLS, τ ), where S is a set, IVLS a group and τ a simply transitive action of IVLS over S.

HARMONIC OBJECTS: KLANG AND THEIR TRANSFORMATIONS

In this section we will describe a new family of objects that is interesting from many points of view. The theory discussed here is introduced by D.Lewin in his book. He was inspired by the work of the music-theorist H.Riemann. The original purpose was to find a valid alternative to music analysis with roman numbers2 . We found very interesting relations between the model presented here and the entities automatically recognized by the algorithm suggested in [START_REF] Stuart | Visual hierarchical key analysis[END_REF]. In particular we think that the objects we're going to present, are a suitable formal model for those entities.

At the end of this section we will demonstrate that this Riemannian model is an example of a generalization of the GIS structure, not only an example of an exception. This suggested us to the possibility to take into account a more general family of structures. We are going to show how these are again well defined mathematical structures. We can now define some particular transformation of the set of Klangs K. Consider the following example: a C major harmony followed by a F Major one. We will where the pith class 'C' is represented by 0 ∈ Z 12 and 'F' by 5. In this way one can formalize what in perceived music is the variation of a tonal reference. It's interesting to observe that with this formula we do represent the same information contained in the expression 'V-I' but not only: we also have the explicit information about which are the tonics and modalities of the considered harmonies. We will consider bijectives transformations of K in K and we will call them Klang transformations. In general, given T ransf : K -→ K we will write (p, ρ)T ransf = (q, σ) (we use right notation for historical motivations and to simplify intuition). The meaning is: the harmony (p, ρ) gets the role given by T ransf in the next harmony (q, σ).

Transformation examples

We described informally the DOM transformation. We can give a formal definition. Following the musical semantic we can say that with DOM if we start with a major (minor) tonality, we obtain a new major (minor) tonality having its tonic 5 semitones upper with respect to the original tonic. It's possible to formalize this in the following way:

Definition 4.4 (formal, of DOM transf ). DOM : K -→ K (p, σ) -→ (p + 5, σ) (5)
Remark 4.5. Here and in this section we will use + and -followed by numbers to mean the result of the action τ of on Z 12 (we can consider the cartesian product introduced in the definition of Klang as a particular GIS). We do not enter in more formal details here, but we want to emphasize that this is not only a stenography and that the calculations that will follow are formally justifiable.

Remark 4.6. DOM is a bijection of K in itself. 3 What kind of musical sensibility do we model with this kind of transformation? What do we understand of a composition if we can remark DOM transformations in the harmonic texture? A composition ruled only by DOM could contain any tonality of a fixed mode. We can model by DOM harmonic movements to neighbour tonality (in tonal music). We can also think that a only-DOM-sensible ear could be able to model macro-areas of tonal influence, without going into deeper details about every single chord structure in a single area. Let's now consider a finer grained harmonic sensibility, introducing a new transformation of Klang: We will call M ED a Klang transformation that describes an harmony (p, ρ) becoming mediant of a new one (q, σ) (III -I or III -I). Starting from a major harmony we get to a minor one having its tonic 3 semitones 'lower'.

M ED : K -→ K (p, ρ) -→ p -3 if ρ = +1 p -4 if ρ = -1 , ρ * (-1)
Remark 4.8. M ED is a bijection 3 It's injective:

∀k 1 = (p 1 , σ 1 ), k 2 = (p 2 , σ 2 ) ∈ K we have (k 1 )DOM = (k 2 )DOM ⇔ (p 1 + 5, σ 1 ) = (p 2 + 5, σ 2 ) ⇔ k 1 = k 2 . It's surjective: ∀k 1 ∈ K ∃k 2 ∈ K so that (k 1 )DOM = k 2 . One can consider p 2 := p 1 -5 and σ 2 := σ 1 .
Let's now go back to the concept of a harmony ruled by repeated applications of DOM . We can formalize this concept considering < DOM >, the group generated by the DOM transformation4 . We will write DOM n to indicate repeated compositions of DOM with itself n times. DOM -1 will be the inverse of DOM . We said that it's possible to reach with DOM any tonality of given modality.To formalize this concept we need some more precise definition. It's trivial that Proposition 4.9. < DOM > acts on K.

Let's remind the following definitions and propositions (cfr. [START_REF] Kosniovski | A first course in algebraic topology[END_REF])

Definition. Let G be a group acting on a set X (that's equivalent to say that X is a G-set). We define orbit of a an element x ∈ X the subset of X

G • x = {g • x|g ∈ G} Proposition.
The orbits of a G-set coincide or are disjointed. So a G-set is decomposable as the union of disjointed subsets.

Proposition. Given a G-set X, the relation ∼ on X

x ∼ y ⇔ ∃g ∈ G : g • x = y is an equivalence relation.

Definition. X/G is the set of the equivalence classes of ∼ (quotient of X mod G).

We can now enunciate the following Proposition 4.10. the orbit of an element (p, +1) (major harmony) of the < DOM >-set K, is the set of all and only those elements of the form (_, +1), i.e. all the major harmonies. (the same happens for the minor harmonies). The set {major harmonies, minor harmonies } is the quotient set K/ < DOM > that is:

K/ < DOM > Z 2
Consider now the following musical fact: the mediant of the mediant of any given harmony (major or minor) is it's dominant. Algebrically we should have M ED 2 = DOM . In fact:

((p, -1)M ED)M ED = (p -3, +1)M ED = (p -3 -4, -1) = (p -7, -1) = (p + 5, -1) ((p, +1)M ED)M ED = (p -4, -1)M ED = (p -4 -3, +1) = (p -7, +1) = (p + 5, +1)
(we used the fact that 5 ∼ = 7(mod12)). And < DOM > is a subgroup of < M ED >. What about the behavior of M ED? We have the following Proposition 4.11. We have

5 ∀k ∈ K orbit <M ED> (k) = K i.e. K/ < M ED > {•} Proof.
First of all let's try to define M ED n . Given k = (p, σ) ∈ K, we'll call (q, ρ) the Klang (p, σ)M ED

The modality (second component) of a Klang change sign for odd values of n, it doesn't for even values. That

is ρ = σ • (-1) n .
What happens to the tonic (first component of the Klang)?

Let n be even. For σ = +1 we have q = p-3-4-3-...-3-4 (mod12) For σ = -1 We have q = p-4-3-4-...-4-3 (mod12) Hence, in both cases,

q = p - n 2 • 7 (mod12)
Let n be odd. For σ = +1 we have q = p -3 -4 -... -4 -3 (mod12).

So in this case

q = p - n -1 2 • 7 -3 (mod12)
For σ = -1 we have q = p -4 -3 -... -3 -4 (mod12).

So in this case

q = p - n -1 2 • 7 -4 (mod12)
How then acts < M ED > on the set K? Musically reasoning we should expect the orbit of any k ∈ K to be the whole K. So, we wonder if the following proposition is true

Proposition. Given (p, σ) ∈ K ∀(q, ρ) ∈ K ∃n ∈ N sothat(p, σ)M ED n = (q, ρ)
Remark that calculating n correspond to the musical question: how many mediant harmony modulations do we need to go from any given harmony to another? This question is answered solving the last equation with respect to n.

We can proceed in the following way (solution algorithm).

The parity of n is determined by the condition σ•(-1) n = ρ.

σ, ρ of the same sign: n is even. Depending on the parity of n and on σ we will consider one of the three pieces of M ED n . We'll have three cases:

n σ equation in Z 12 even +1 or -1 7h = r (h := n 2 , r := p -q) odd +1 7h = r (h := n-1 2 , r := p -q -3 odd -1 7h = r (h := n-1 2 , r := p -q -4
Every equation has only one solution in Z 12 because 7 and 12 are coprime. The proposition follows.

Let's now introduce a third transformation of particular musical interest. We are taking into account the harmonic movement from an harmony to the one having the same tonic, but opposed modality (an example is the wellknown phenomenon of the 'terza piccarda': when a piece in minor mode ends with a chord of the same tonic, but major mode).

P AR : K -→ K (p, σ) -→ (p, -σ) (6)
Remark 4.13. P AR is a bijection. (It's trivial)

As usual let < P AR > be the group generated by P AR. Remark 4.14. P AR 2 = 1.

Consider now the group generated by M ED and P AR.

We immediately note that 1 Exercise. the group < M ED, P AR > is not abelian. < M ED, P AR > contains M ED, hence it's clear that the orbit of an element will coincide with the whole K, and that < M ED >⊆< M ED, P AR >. It's not difficult to prove that Proposition 4.16. < M ED > =< M ED, P AR >.

Proof. We have to show that P AR ∈< M ED >, i.e. there is no value of n such that M ED n = P AR. If there were a solution what would it be? For instance let's pose the following harmonic question: is there a value of n ∈ N for which C Maj becomes M ED n of C min? Formally:

(0, +1)M ED n = (0, -1) (

Let's follow the procedure of the proof in the previous note. Modes of harmonies in 7 are opposed, so n will be odd. The starting mode is +1, hence 7h = r (where h := n-1 2 and r := p -q -3 = 0 -0 -3 = -3) i.e. 7h = -3 which (unique) solution is for h = 3, so n = 7.

Then we have (0, +1)M ED 7 = (0, -1). Let's consider the detailed definition of M ED 7 :

(p, σ)M ED 7 = p if σ = +1 p -1 if σ = -1 , σ • (-1)
Hence M ED 7 = P AR. But it's the only element of < M ED > that maps C Maj to C min, and so we proved that P AR ∈< M ED >, and so < M ED >⊂< M ED, P AR > .

Conclusions: generalizing the GIS structure

Summarizing. We exhibited a family of musical entities, which reduce harmonic information to single element of a set. We showed how harmonic facts can be algebrically interpreted. We showed that Klangs and particular transformation groups (of musical meaning), constitue more general structures than GIS. In which way, exactly? Let's go into deeper details. We note that Proposition 4.17. The action (application of transformations) of the group < M ED > on K is simply transitive.

Proof.

In fact < M ED > •k = K (4.11). Hence Proposition 4.18. K, < M ED >, action is a GIS (com- mutative).
We discovered in 4.16 that also the orbit < M ED, P AR > •k is the whole K. But is the action of < M ED, P AR > simply transitive? The existence in the axiom ?? is guaranteed by the orbit coincident to K. On the other hand the solution of 7 in proposition 4.16 falsifies the unicity in the same axiom.

Let's remember at this point the following Definition. The action of a group on a set X is transitive iff ∀x, y ∈ X ∃g ∈ G : x • g = y

Then we can say that Proposition 4.19. The action of < M ED, P AR > on K is transitive.

Proof. The existence of an element of the group which maps an element of the set in another one, for any choice of the elements, is ensured by the presence of M ED and by the proposition 4.11.

Finally, let's consider the actions of < P AR > and < DOM > on K. These are not even transitive (4. 15 and 4.15).

In conclusion, we characterized some musical entities, considered in conjunction with some transformations, which are or not GIS, depending on the family of transformations. By way of compensation we can say that when the axioms of GIS are not respected, we can anyhow find very similar algebrical structure. Weakening the axioms we can still model these musical structure with group actions over sets.

We want to conclude this formal part of the work emphasizing that the characterization of the entities described in terms of group actions over sets will let us to define, in a uniform way, verification rules for binary relations between elements of a (finite) G-set, relations that emerge from the G-set structure. Given a G-set X, being g ∈ G, two elements x, y ∈ X will be g-related iff action(x, g) = y.

IMPLEMENTATION

We conclude with the sketch of the implementation in the structural layer of MX. For a general description of MX format see [START_REF]Draft Trial-Use Recommended Practice for Definition of a Commonly Acceptable Musical Application Using the XML Language[END_REF], [START_REF] Haus | Towards a symbolic/time-based music language based on xml[END_REF].

Representing more analyses

<!ELEMENT structural (themes?, CPN * , analysis * )>

We added to the structural element the capacity to contain zero or more analyses as sub-elements. In this way it will be possible to contain more different analytical viewpoints. We want to emphasize that this is in line with the general phylosophy of MX: many different points of view on the same represented entity (the same happens in the audio, performance and notational levels).

<!ELEMENT analysis (segmentation, relationships?, featureObjectRelationships?) > <!ATTLIST analysis id ID #REQUIRED author CDATA #IMPLIED description CDATA #IMPLIED > An analysis is made up by a segmentation, (optionally) a set of relations between segments and (optionally) a set of relations between featureObects that one will be able to use when defining relations between segments.

Representing the segmentation

<!ELEMENT segmentation (segment+)> <!ATTLIST segmentation id ID #REQUIRED description CDATA "null" method CDATA "null" > A segmentation is made up of one or more segments. A segmentation is identified by an unique ID in the MX document. A segmentation is described by a generic parameter ('null' by default). A segmentation is described by an indication of which method has been used to generate it ('null' by default).

<!ELEMENT segment (segmentevent+, featureObject * )> <!ATTLIST segment id ID #REQUIRED >

A segment is made up of a set of events in the music piece. A segment is identified by an unique ID in the MX document. A segment contains zero or more feature objects, so one can predicate in many ways on the same segment of an analysis.

<!ELEMENT segmentevent EMPTY> <!ATTLIST segmentevent id_ref IDREF #REQUIRED >

An event of the piece of music is identified by a reference to an event in the element of the spine of the MX document (cfr. [START_REF]Draft Trial-Use Recommended Practice for Definition of a Commonly Acceptable Musical Application Using the XML Language[END_REF]).

We want to emphasize that a segment can be any collection of musical events (in one or more voices) and that segments can overlap (as in the analyses of Lewin) or not (as in other analyses). In our model there is no limitation, so we can support any approach to segmentation.

FeatureObject implementation

Feature objects will be assigned to a segment. Now the problem is to let custom implementation of featureObject classes. To address this problem we follow an extensible content model pattern (see [START_REF] Roland | Design patterns in xml music representation[END_REF]). So, we have a basic structure of the element featureObject, defined with some parameter entities that will allow the extensions.

<!ENTITY \%addedFeatureObjectClasses ""> <!ENTITY \%supportedFeatureObjectClasses "simpleDescription \%addedFeatureObjectClasses;" > <!ELEMENT featureObject (\%supportedFeatureObjectClasses)> <!ATTLIST featureObject id ID #IMPLIED name CDATA #REQUIRED > <!ELEMENT simpleDescription (#PCDATA)>
A feature object is identified by an unique ID in the MX document and will have (optionally) a name. A feature object will contain an object of one of the supported classes.

The standard model contains a trivial featureObject class simpleDescription which content is a simple text.

Adding a custom featureObject class

If we want to extend the model to represent our custom feature object we have to create our own definition of a class and introduce the name of the new class in the parameter entity defined in the dtd. For instance what follows is a Klang implementation 6 . We chose to represent pitch class and modality of a Klang as attributes (attribute models part of the internal state of an object

<!DOCTYPE mx SYSTEM "mx.dtd"[ <!ENTITY \%addedFeatureObjectClasses "|Klang"> <!ELEMENT Klang EMPTY> <!ATTLIST Klang pc (0|1|2|3|4|5|6|7|8|9|10|11) #REQUIRED mod (-1|+1) "+1" #REQUIRED implementingClass CDATA #IMPLIED > ]> .
.. <featureObject id="497" name="klang c-"> <Klang pc="0" mod="-1"/> </featureObject> ...

Representing relations between segments

As we said in 2.2, we want to relate segments by means of relations between feature objects associated to them.

In our model a relationship is a sub-element of the container relationships, which is part of an analysis.

<!ELEMENT relationship EMPTY> <!ATTLIST relationship id ID #REQUIRED description CDATA #IMPLIED segmentA_id IDREF #REQUIRED segmentB_id IDREF #REQUIRED featureObjectA_id IDREF #IMPLIED featureObjectB_id IDREF #IMPLIED featureObjectRelationship IDREF #IMPLIED >
A relation between segments is identified by an unique identifier in the mx document.

A relation between segments is defined by the references to the two segments A and B involved.

One can also specify which featureObject to chose from the feature objects associated to segments A and B. When the objects are specified, it is possible to specify also in which kind of relation A and B are 7 .

Now we introduce a prototypical definition of feature-Object relations.

<!ELEMENT featureObjectRelationship (verRule)> <!ATTLIST featureObjectRelationship id ID #IMPLIED > <!ELEMENT verRule (#PCDATA)>
A relation of featureObjects is identified by an unique identifier in the MX document (that can be referenced in the definition of a relationship between segments). It is defined by a sub-element which can contain the rule to check if the relation is verified. The idea is to let the user to describe this rule in terms of a syntax that a parser will understand. We make an example in a java-jsp expression language-like syntax [START_REF]The j2ee(tm) 1.4 tutorial[END_REF].

If we want to introduce an implementation of the DOM transformation of Klang we will add the following lines to the document. <!DOCTYPE mx SYSTEM "mx.dtd"[ ... <!ENTITY verRule_KlangMED 7 Note that it is possible to have more than one relation between A and B. For instance, if we have a G-set X with an action that is not simply transitive, we can have the same objects (i.e. set elements x 1 , x 2 ∈ X) in more than one relation (if there exist two distinct group-elements g, h ∈ G such that action(x 1 , g) = x 2 and action(x 1 , h) = x 2 ).

"((k1.mod eq 1)&&(k1.pc-3 eq k2.pc)) || ((k1.mod eq(-1))&&(k1.pc-4 eq k2.pc))" > ]> ... <featureObjectRelationship id="1045"> <verRule>&verRule_KlangMED;</verRule> </featurefeatureObjectRelationship/> ...

CONCLUSIONS

Let's try to sketch a possible not so sci-fi implementation of the ideas here exposed.

Let's imagine a musical database containing mx-encoded files. Now, the system could proceed to examine the harmonic successions of the pieces and to write in the file the obtained analysis, defining at the same time, automatically, the tie between the recognized chords and the segments of the piece which such chords are attributed. Imagine segmentations like those obtained by the algorithm described by S.C.Sapp(cfr. [START_REF] Stuart | Visual hierarchical key analysis[END_REF]) 8 . We will have segments with keys associated to them (key regions). But keys are representable by Klang objects. At the end of the process, chain of relations between successive harmonies would be exposed and saved in the xml file. An agent which we asked to carry out a search of pieces harmonically similar to others could try (e.g. using approximate string matching techniques) to search pieces containing chains of relations with a sufficient number of correspondences. The presence of an algebraic structure could inform an intelligent agent to use algorithms which could make use of this structure. The encoding of the analytical information obtained by the work of complex algorithms, and the possibility to take advantage of it later, through more simple algorithms, would remarkably speed up some different tasks of information retrieval. Future developements will be the refinement of the implementation here proposed and the examination of other existing models from the perspective of objects here described.

Figure 1 .

 1 Figure 1. C Major, F Major express what one could write V-I using roman numbers notation (the first chord has the role of the dominant with respect to the second one thought as the tonic), by (C, Major)DOM = (F Major)The meaning is that C Major becomes the dominant of F Major and, using the suggested formalism, we will write (0, +1)DOM =(5, +1)

Figure 2 .

 2 Figure 2. A minor, F Major

Figure 3 .

 3 Figure 3. D minor, D Major Proceeding formally: Definition 4.12 (of P AR transformation).

  Counter-example: ((0, +1)M ED)P AR = (9, -1)P AR = (9, +1) (i.e. C Major becomes MEDiant of A minor that becomes PARallel of A major) ((0, +1)P AR)M ED = (0, -1)M ED = (8, +1) (i.e. C Major becomes PARallel of C minor that becomes ME-Diant of A flat Major) What about the orbit of an element k ∈ K? We have: Proposition 4.15. ∀k := (p, σ) ∈ K orbita <P AR> (k) = {(p, σ), (p, -σ)} i.e. K/ < P AR > Z 12

  Another example of GIS is the space of picth classes of equal temperament. We forget the octave of a pitch and concentrate uniquely on its class. We can represent the pitch classes disposed around a 'clock', a well known representation. int(s, t) will be the number of 'minutes' in clockwise sense to from the 'minute' s to the 'minute' t. It's easy to see that

Proposition 3.3. (S, ( Z 12 , +), int ) is a GIS. Remark 3.4. S is the model of set-theoretical analysis.

Now some examples of rythmic GIS 3 Example. Let S be a succession of equidistant temporal points (imagine a pulsing uniform 'tactus'). We can represent it with Z . The interval between to of such points will be the number of 'tactus' between them. Hence we'll have:

IVLS = ( Z , +) int : (s,

t) → number of temporal units between s and t i.e. int : (s, t) → t -s 4 Example. Another rythmic space is the set of time spans.

See[START_REF] Vuza | Some mathematical aspects of david lewin's book generalized musical intervals and transformations[END_REF] for a punctual formalization of the equivalence between the two definitions

For an interesting introductive overview about the evolution of harmony notation see the book of Nicolas Cook[START_REF] Cook | A Guide to Musical Analysis[END_REF]

Note that this is a group with respect to composition of transformations (bijections) of K to K, being a subgroup of all the bijections of K in itself.

We intend with orbit <M ED> (k) the orbit of k as an element of the< M ED > -set K, that is < M ED > •k

The optional attribute implementingClass could contain the name of an implementing class written in some programming language. This could be useful in an environment involving an application.

The algorithm operates by different granularities. Segmentations of different granularity could be separated from the others in the file.

Acknowledgements

Special thanks are due to prof. Goffredo Haus for his fundamental role in the definition and discussion of the research topics of this paper.

Furthermore, this work has been made possible beacuse of the previous efforts of many researchers and graduate students of LIM.

The author wishes to acknowledge the members of the IEEE MX WG (PAR1599) for their cooperation and interest in our work, and the partial support of this project by