Sophie Lavalette

William Raoul

Marianne Houssier

Serge Camelo

Olivier Levy

Bertrand Calippe

Laurent Jonet

Francine Behar-Cohen

Sylvain Chemtob

Xavier Guillonneau

Christophe Combadière

M.D Florian Sennlaub
email: florian.sennlaub@inserm.fr

Interleukin-1β inhibition prevents choroidal neovascularization

Introduction :

The IL-1 family of cytokines plays a key role in the initiation of acute inflammatory responses 1 . Interleukin 1 beta (IL-1β) interacts with IL-1 receptor I (IL-1RI), composed of IL-1R and IL-1R accessory protein (IL-1RacP) subunits. IL-1 receptor antagonist (IL-1Ra) competes with IL-1β for its binding site 1 . IL-1β is a potent inflammatory mediator with chemotactic 2 and angiogenic 3,4 properties. It is a neurotoxic mediator in ischemic brain injury 5 but can attenuate glutamate neurotoxicity in the retina 6 and protect against lightinduced or hereditary photoreceptor degeneration 7,[START_REF] Whiteley | Photoreceptor rescue after low-dose intravitreal IL-1beta injection in the RCS rat[END_REF] .

Age-related macular degeneration (AMD) is the leading cause of vision loss in elderly people in industrialized countries [START_REF] Friedman | Prevalence of age-related macular degeneration in the United States[END_REF] . Its most prominent pathological features are photoreceptor degeneration and choroidal neovascularization (CNV) [START_REF] Sarks | Ageing and degeneration in the macular region: a clinico-pathological study[END_REF] . In AMD, IL-1β is secreted by RPE cells and CD68 positive cells in choroidal neovascular membranes [START_REF] Oh | The potential angiogenic role of macrophages in the formation of choroidal neovascular membranes[END_REF] and is therefore a possible proangiogenic and neuroprotective or neurotoxic mediator in AMD.

IL-1Ra is clinically used to treat juvenile idiopathic arthritis [START_REF] Pascual | Role of interleukin-1 (IL-1) in the pathogenesis of systemic onset juvenile idiopathic arthritis and clinical response to IL-1 blockade[END_REF] , it has been shown to inhibit neurotoxicity in ischemia in animal models [START_REF] Banwell | Systematic review and stratified meta-analysis of the efficacy of interleukin-1 receptor antagonist in animal models of stroke[END_REF] . Intravitreal human IL-1Ra injections have been shown to efficiently inhibit CNV in a rat model [START_REF] Olson | Intravitreal anakinra inhibits choroidal neovascular membrane growth in a rat model[END_REF] . To analyze the possible implication of IL-1β in CNV and retinal degeneration, we studied the expression of IL-1β in laser-induced CNV in mice and in light-induced retinal degeneration in rats and mice. We localized IL-1β expression by immunohistochemistry and inhibited IL-1β activity with IL-1Ra supplementation.

Materials and Methods

Animals

10 week old Wistar rats and C57B6j wild type mice were purchased from the Janvier breeding center (Le Genest-St-Isle, France). Cx3cr1 -/-C57B6j mice were backcrossed for six generations into the BALB/c background (Janvier) to obtain the Cx3cr1 -/-BALB/c strain and kept in specific pathogen-free conditions with food and water available ad libitum and housed in a 12/12 hr light/dark (100-500 lux) cycle. Animal experiments were approved by the Institutional Animal Care and Use Committee.

Laser coagulation

10-week-old C57Bl6 mice were anesthetized with an intramuscular injection of ketamine (50 mg/kg) and xylazine (10 mg/kg). Their pupils were fully dilated with 1% tropicamide.

Coverslips positioned on the mouse cornea were used as a contact glass. 4 laser coagulations were performed 4 to 5 disc diameters away from the papillae with an Argon laser (532 nm) mounted on a slit lamp (400 mW, 50 ms and 50 µm). Mice were treated with daily subcutaneous injections of PBS or human recombinant IL-1Ra (Kineret, Biovitrum, Stockholm, Sweden) at 1mg/d/kg until sacrifice.

Choroidal flatmounts, Immunohistochemistry and CNV quantifications

Eyes were enucleated, fixed in 4% PFA for 15 minutes at room temperature and sectioned at the limbus; the cornea and lens were discarded (for CNV quantification, mice were perfused with fluorescein dextran 10 6 before enucleation). Retinal and RPE/choroidal flatmounts were stained according to previously described standard immunohistochemical procedures [START_REF] Checchin | Potential role of microglia in retinal blood vessel formation[END_REF] . The primary antibodies and lectins used were Bandeirae simplicifolia lectin (Sigma Aldrich, Saint Quentin Fallavier, France), goat anti-mouse IL-1β (R&D, Lille, France), rabbit polyclonal anti-IL-1RI (Santa Cruz, Heildeberg, France), rabbit polyclonal anti-Iba1 (Wako, Neuss, Germany), rat anti-mouse Ly6G (Miltenyi Biotec, Paris, France), and goat anti-human Collagen IV (R&D, Lille, France). The corresponding Alexa secondary antibodies (Molecular Probes, Leiden, the Netherlands) were used to reveal the primary antibodies, and flatmounts were counterstained with 4-6-diamino-2-phenylindole (DAPI). The choroids and retinas were radially incised, flatmounted, and viewed with the same fluorescence microscope. Flatmounts were viewed with a fluorescence microscope (DM5500B, Leica, Nanterre, France). All immunostaining was repeated at least 3 times, and stains omitting the primary antibody served as negative controls. Ly6G positive and Iba1 positive cells were counted on stained whole RPE/choroidal flatmounts up to the ciliary body and on the outer segment side of the retina.

The surface covered by perfused CNV was measured on photographs of fluorescein dextran 10 6 perfused choroidal flatmounts with ImageJ analysis software (surface covered by fluorescein staining). The average CNV size was calculated per eye.

Microvascular Sprouting from Aortic Explants

Aortae from adult C57BL6 were cut into 1-mm-thick rings and covered with 30 µL of Matrigel (BD Biosciences, Le Pont de Claix, France) in 24-well tissue cultures plates. Aortic rings were cultured for 3 days in Dulbecco's Modified Eagle's Medium (DMEM, Invitrogen, Cergy Pontoise, France) containing 10% FCS, 1% penicillin/streptomycin, and 0.2% fungizone. Explants were exposed to IL-1β (5ng/mL; R&D system, Lille, France), IL-1Ra (10mg/mL) and VEGFR1s (150ng/mL; R&D) from day 3 to day 7 of culture. Photographs of individual explants were taken every day and the surface covered by the aortic ring and the vascular sprouts was measured daily from day 3 to 7. The surface of each individual aortic ring and pre-incubation sprouts at day 3 was subtracted from the surface at day 4 to calculate the vascular sprouting that occurred in the presence of the ligands and controls.

Reverse Transcription and Real Time Polymerase Chain Reaction (RT PCR)

Total RNA was isolated with NucleoSpin RNA II Kit (Macherey-Nagel, Hoerdt, France).

Single-stranded cDNA was synthesized from total RNA (pre-treated with DNaseI amplification grade) using oligodT as primer and superscript reverse transcriptase (Invitrogen, Cergy Pontoise, France). Subsequent real-time polymerase chain reaction (RT-PCR) was performed using cDNA, qPCR SuperMix-UDG Platinum SYBR Green (Invitrogen), and the following primers (0.5 pmol/µl): mm actin sense: 5'-AAGGCCAACCGTGAAAAGAT-3'; mm actin antisense : 5'-GTGGTACGACCAGAGGCATAC-3'; mm IL-1β sense : 5'-CATGGAATCCGTGTCTTCCT-3'; mm IL-1β antisense : 5'-GAGCTGTCTGCTCATTCACG-3'; mm VEGF sense : 5'-GTGAGCCAGGCTGCAGGAAG-3'; mm VEGF antisense : 5'-GAATGCGTCTGCCGGAGTCT-3'; rn actin sense : 5'-AAAGAAAGGGTGTAAAACGCAG-3'; rn actin antisense : 5'-AAAGACCTCTATGCCAACACAG-3'; rn IL-1β sense : 5'-GGAACCCGTGTCTTCCTAAA-3'; rn IL-1β antisense : 5'-CTGACTTGGCAGAGGACAAA-3' PCR reactions were performed in 40 cycles of 15 s at 95°C, 45 s at 60°C. Product was not generated in control reactions in which reverse transcriptase was omitted during cDNA synthesis.

Light-Induced Degeneration

10-week-old rats and 8-week-old Cx3cr1 -/-mice were were adapted to darkness for 12h and pupils were fully dilated with 1% Atropin (Novartis, Rueil Malmaison, France). Animals were then exposed to green LED light (4500 Lux) for 12 hours and subsequently kept in cyclic 12h/12h normal animal facility conditions. Control and light-exposed rats and mice were treated with daily subcutaneous injections of PBS or IL-1Ra (Kineret, Biovitrum, Stockholm, Sweden) at 1mg/d/kg until sacrifice and a volume of 5 (rats) or 2 µL (mice) of IL-1Ra at 150 mg/mL was injected intravitreally at day 0 and day 3 after illumination.

Protein Analysis

Eyes were enucleated, sectioned at the limbus and the cornea and lens were discarded. The complex retina/RPE/choroid/sclera was placed in 150 µL PBS 1x supplemented with a protease inhibitor cocktail (Calbiochem, Fontenay-sous-Bois, France) followed by homogenization with a plastic pestle. The lysate were cleared of debris by centrifugation at 2000 rcf for 10 minutes at 4°C. Total protein content of supernatant was determined by commercial assay (Bradford kit, Eragny-sur-Oise, France). Supernatant IL-1β level was determined with a sandwich enzyme-linked immunoabsorbent assay (ELISA), according to the manufacturer's instructions (R&D Systems, Lille, France) and normalized for total protein.

Histology

For histology, eyes were fixed in 0.5% glutaraldehyde, 4% PFA PBS for 2 hours, dehydrated and mounted in historesin. 5-µm oriented sections, crossing the centre of the laser injury (CNV model) or inferior pole, optic nerve and superior pole (light-induced model), were cut and stained with toluidin blue. Rows of nuclei in the outer nuclear layer were counted at different distances from the injury site (CNV model) or optic nerve (light-induced model).

Statistical Analysis

Graph Pad Prism 5 (GraphPad Software) was used for data analysis and graphic representation. All values are reported as mean ± SEM. Statistical analysis was performed by one-way and two-way analysis of variance followed by post-hoc Bonferroni's test for comparison among means. P < 0.05 was considered statistically significant.

Results

IL-1β is induced in laser-induced neovascularization and localizes to the injury site.

IL-1β is known for its proangiogenic properties 3,4 . It is secreted by CD68 positive cells in choroidal neovascular membranes in AMD [START_REF] Oh | The potential angiogenic role of macrophages in the formation of choroidal neovascular membranes[END_REF] . Its inhibition has recently been shown to prevent laser-induced subretinal neovascularization in rats, but its induction, expression pattern, and impact on inflammation in the laser model has not been analyzed [START_REF] Olson | Intravitreal anakinra inhibits choroidal neovascular membrane growth in a rat model[END_REF][START_REF] Zou | Effect of interleukin-1 blockers, CK112, and CK116 on rat experimental choroidal neovascularization in vivo and endothelial cell cultures in vitro[END_REF] . We first analyzed IL-1β expression at different time points in laser-induced CNV in the retina (Fig. 1A) and choroid (Fig. 1B) of C57Bl6 mice. A strong induction of IL-1β mRNA is observed as early as 1h in the retina (55fold) and at 6h (35fold) and 20 hours (18fold) in the choroid. To a lesser extent, a tendency for vascular endothelial growth factor (VEGF) mRNA induction was detected in the retina at 1h (Fig. 1C) but not in the choroid (Fig. 1D; control n=4, 1h n=8, 6h n=7, 20h n=5, 72h n=4 retinas/choroids). To identify IL-1β expressing cells, we performed immunohistochemistry of IL-1β (green staining Fig. 1 E, I andM), the neutrophil marker Ly6G (Fig. 1 F, J andN) and the monocyte/macrophage marker Iba1 (Fig. 1 G, K andO) on choroidal/RPE flatmounts of uninjured control eyes (Fig. 1E-H) and 10h (Fig. 1 I-L) and 72h (Fig. 1M-P) after laser impact. Uninjured control eyes did not stain positive for IL-1β, Ly6G or Iba1 (Fig. 1 E-H

IL-1Ra inhibits choroidal neovascularization independently of macrophage recruitment.

To evaluate the impact of IL-1β induction on CNV we treated one group of experimental animals with subcutaneous saline injections (n=4 mice) and one group with subcutaneous injections of 1mg/kg/day recombinant human IL-1Ra (n=5 mice). At day 14 after laser injury the CNV was visualized on isolectin-stained (red staining), fluorescein dextran perfused (green/yellow staining) RPE/choroidal flatmounts of saline (Fig. 2A) and Il-1Ra treated mice (Fig. 2B). Mature, perfused CNV (presenting a tubular structure with a lumen) was quantified as fluorescein positive surface (Fig. 2C). IL-1Ra treated animals showed a 60% reduction in choroidal neovascularization. Initial lesion size at 10h after laser impact, identified by the loss of the RPE cell layer, was quantified on choroidal/RPE flatmounts of saline and Il-1Ra treated mice and showed no significant difference (Fig. 2D), excluding a possible initial difference of the laser injury in the two groups.

IL-1β has chemotactic properties and can alter macrophage recruitment to the granulomateous tissue that forms after injury 2 . It might thereby indirectly influence angiogenesis. To test this hypothesis, we visualized neutrophils and macrophages/microglial cells 10h (Fig. 2 E andF) and 72h (Fig. 2 H andI) after laser injury, on Ly6G (red staining) and Iba1 (green staining) stained RPE/choroidal flatmounts of saline (Fig. 2 E andH; n=3 mice) and Il-1Ra treated (Fig. 2F andI; n=3 mice) mice. Neutrophil (Fig. 2 G) and macrophage/microglial cell (Fig. 2J) prevalence were quantified. Ly6G positive neutrophils were significantly recruited to the injury site, reaching their maximum at 10h. Neutrophil recruitment was stronger in IL-1Ra treated animals (Fig. 2G; n=3-4 mice per time point). IL-1Ra treatment did not alter Iba1 positive macrophage/microglial cell recruitment, which reached the maximum in both saline and IL-1Ra treated groups at 4d (Fig. 2 J).

IL-1β induces vascular sprouting in aortic rings.

IL-1β exerts its activity by acting on the IL-1 receptor I (IL-1RI) 1 . It induces angiogenesis in the cornea in vivo 3 , IL6 expression [START_REF] Boraschi | Endothelial cells express the interleukin-1 receptor type I[END_REF] , and microparticle release from vascular endothelial cells in vitro [START_REF] Leroyer | CD40 ligand+ microparticles from human atherosclerotic plaques stimulate endothelial proliferation and angiogenesis a potential mechanism for intraplaque neovascularization[END_REF] . IL-1RI is expressed on numerous cell types, notably it is constitutively expressed on vascular endothelium [START_REF] Boraschi | Endothelial cells express the interleukin-1 receptor type I[END_REF] . IL-1RI immunohistochemistry on choroidal/RPE flatmounts of C57Bl6 mice 5d after the injury reveals IL-1RI positive staining around the lesion site (Fig 3A) (n=3 mice). Double labeling using the vascular endothelial cell marker collagen IV (red fluorescence Fig. 3B) shows that IL-1RI is expressed in the vascular endothelium surrounding the lesion and in some unidentified cells (Fig. 3C merge).

To test the effect of IL-1β on vascular sprouting we used the aortic ring assay. Photographs at 4d, 24 hours after ligand incubation, show the vascular sprouting of PBS (Fig. 3D), IL-1β (Fig. 3E), IL-1Ra (Fig. 3F), or IL-1β and IL-1Ra incubated aortic rings (Fig. 3G).

Measurements of the area covered by the vascular sprouts revealed a significant increase of growth in IL-1β exposed aortic rings after 24 hours of IL-1β exposure at 4d that was inhibited by simultaneous IL-1Ra incubation (Fig. 3H). Vascular sprouting remained significantly increased up to 6d (72 hours of ligand exposure) before aortic rings started dying (data not shown). Simultaneous incubation of IL-1β with the soluble VEGFR1 did not significantly inhibit the effect of IL-1β incubation alone (Fig. 3I).

IL-1Ra does not affect photoreceptor survival laser and light-induced injury.

In AMD, CNV develops in the context of photoreceptor degeneration. IL-1β supplementation has been shown to inhibit photoreceptor degeneration in rat models of light-induced degeneration 7 and protect photoreceptor degeneration observed in Royal College of Surgeons rats [START_REF] Whiteley | Photoreceptor rescue after low-dose intravitreal IL-1beta injection in the RCS rat[END_REF] . IL-1β inhibition might have a deleterious effect on retinal degeneration. Laser-injury used to induce CNV also induces a local inflammatory reaction and retinal degeneration in the vicinity of the laser impact. To evaluate whether IL-1Ra influences the laser-induced photoreceptor degeneration, we treated laser-injured C57Bl6 mice with daily subcutaneous injections of saline (n=3 mice) or IL-1Ra (n=4 mice) and prepared histological sections crossing the lesion site at d14 (Fig. 4A andB respectively). Quantification of rows of nuclei at increasing distances from the injury site revealed no difference in photoreceptor cell loss in the vicinity of the laser impact (Fig. 3C).

To analyze the effect of IL-1β inhibition in a non-neovascular context, we exposed Wistar rats to light-induced retinal degeneration and analyzed IL-1β expression by RT-PCR in eye cups (Fig. 4D; control n=4 eyes, 1h n=5 eyes, 12h n=5 eyes). IL-1β transcript was significantly induced (x5) in the eye at 12 hours after the onset of light-induced stress. To evaluate the influence of the endogenously-produced IL-1β on photoreceptor survival we injected Wistar rats with either PBS (n=4 rats) or IL-1Ra (n=6 rats) (subcutaneously 1mg/kg/day and intravitrealy and 5µL of a 150 mg/mL solution at d1 and d3 to ensure penetration of a possibly intact blood retinal barrier in the eye). Control animals that were raised under normal light conditions revealed no signs of toxicity at this dosage (data not shown). At 7d, histology of control (Fig. 4E) light-exposed PBS treated (Fig. 4F) and light-exposed IL-1Ra treated rats (Fig. 4G) revealed significant thinning of the outer nuclear layer (ONL) in both groups.

Quantification of rows of nuclei at different distances from the optic nerve measured from the inferior pole to the superior pole revealed no differences in photoreceptor cell loss throughout the retina (Fig. 4H).

IL-1Ra does not affect photoreceptor survival in degeneration prone CX3CR1 -/-mice.

We have previously shown, that the T280M polymorphism of CX3CR1 leads to dysfunctional CX3CR1 and is associated with AMD. Furthermore, Cx3cr1 -/-mice develop spontaneous photoreceptor degeneration by the age of 18 months and exaggerated photoreceptor degeneration in light-induced models [START_REF] Combadiere | CX3CR1-dependent subretinal microglia cell accumulation is associated with cardinal features of age-related macular degeneration[END_REF][START_REF] Raoul | Role of the chemokine receptor CX3CR1 in the mobilization of phagocytic retinal microglial cells[END_REF] . Here, we analyzed the expression and impact of IL-1β in the Cx3cr1 -/-model of AMD. First, we analyzed IL-1β expression in C57Bl6 Cx3cr1 +/+ and C57Bl6 Cx3cr1 -/-mice at 3 and 18 months of age (Fig. 5A n=4 eyes/group). IL-1β protein was significantly increased in 18-month-old C57Bl6 Cx3cr1 -/-mice compared to Cx3cr1 +/+ mice. Similarly, 4500lux light-exposed BALB Cx3cr1 +/+ and BALB Cx3cr1 -/- animals revealed a significant increase of IL-1β protein 24h after onset of the light injury compared to normal light-raised Cx3cr1 +/+ and Cx3cr1 -/-mice (Fig. 5B; n=4 eyes/group).

Interestingly, the basal level of IL-1β expression was more important in young C57BL6 mice than in BALB mice. The increase of IL-1β protein levels in light-induced-injury in BALB Cx3cr1 +/+ and Cx3cr1 -/-mice were not different from each other and comparable to 18-month-old C57Bl6 Cx3cr1 -/-mice. To evaluate if IL-1β plays a role in photoreceptor degeneration in the context of CX3CR1 deficiency, we exposed two-month-old BALB Cx3cr1 -/-mice before significant spontaneous degeneration was observed to light-induced degeneration [START_REF] Combadiere | CX3CR1-dependent subretinal microglia cell accumulation is associated with cardinal features of age-related macular degeneration[END_REF] (Fig. 5C). PBS (n=4 mice) or IL-1Ra (n=6 mice) were injected subcutaneously 1mg/kg/day and intravitrealy (2µL of a 150 mg/mL solution) at d1 and d3 to ensure penetration of a possibly intact blood retinal barrier in the eye. Saline-treated controls (n=6 mice; Fig. 5D) degenerated similarly to IL-1Ra treated Cx3cr1 -/-mice (Fig. 5E; n=6 mice) and no significant difference could be observed throughout the retina (Fig. 5F). Control Cx3cr1 -/-animals that were raised under normal light conditions and treated with intravitreal and subcutaneous PBS or IL-1Ra revealed no signs of toxicity at this dosage (Fig. 5G).

Discussion

We demonstrate that IL-1β expression rapidly increases in laser-and light-induced injury.

The expression in laser-induced injury shows rapid induction that lasts for about two days (Fig. 1). Immunohistochemistry at 10h localizes IL-1β mainly to infiltrating neutrophils, as demonstrated by a positive Ly6G co-staining (specific for neutrophils and not expressed on monocytes [START_REF] Daley | Use of Ly6G-specific monoclonal antibody to deplete neutrophils in mice[END_REF] ) and negative Iba1 co-staining (specifically expressed on macrophages and microglial cells 22 ) (Fig. 1). The identity of IL-1β expressing cells was further confirmed by their multilobulated nuclei (data not shown) and is corroborated by the IL-1β expression kinetics that coincide with early neutrophil recruitment in CNV [START_REF] Tsutsumi-Miyahara | The relative contributions of each subset of ocular infiltrated cells in experimental choroidal neovascularisation[END_REF] . The majority of IL-1β expressing cells in our experiments were Ly6G positive neutrophils and the contribution of Iba1 positive macrophages/microglia was negligible. Nevertheless, occasional IL-1β staining was also detected in RPE cells adjacent to the laser injury. Activated polymorphonuclear neutrophils have been shown to express IL-1β [START_REF] Malyak | Peripheral blood neutrophil production of interleukin-1 receptor antagonist and interleukin-1 beta[END_REF] and our data shows that these cells are the predominant source of IL-1β after laser injury in the mouse. IL-1β is expressed in CD68 positive cells in human neovascular membranes which were interpreted to be macrophages [START_REF] Oh | The potential angiogenic role of macrophages in the formation of choroidal neovascular membranes[END_REF] .

Although macrophages can express CD68, it has recently been shown that the main CD68 expressing cells in the injured central nervous system are neutrophils [START_REF] Matsumoto | Antibodies to CD11b, CD68, and lectin label neutrophils rather than microglia in traumatic and ischemic brain lesions[END_REF] . It is possible that neutrophils participate in IL-1β in AMD. Studies using more specific markers of neutrophils and macrophages are needed to determine the cell types expressing IL-1β in the human disease.

IL-1β inhibition by recombinant IL-1Ra significantly reduced the development of subretinal neovascularization in the mouse model of laser-induced subretinal neovascularization (Fig. 2).

Our results corroborate an earlier report that IL-1Ra inhibits neovascularization in a rat model 14, 16 .

Neutrophils and macrophages promote CNV [START_REF] Tsutsumi-Miyahara | The relative contributions of each subset of ocular infiltrated cells in experimental choroidal neovascularisation[END_REF][START_REF] Espinosa-Heidmann | Macrophage depletion diminishes lesion size and severity in experimental choroidal neovascularization[END_REF][START_REF] Sakurai | Macrophage depletion inhibits experimental choroidal neovascularization[END_REF][START_REF] Zhou | Neutrophils promote experimental choroidal neovascularization[END_REF] ; IL-1Ra has been shown to inhibit macrophage recruitment to the atherosclerotic plaque in a mouse model 2 . We therefore tested the influence of IL-1Ra treatment on neutrophil and macrophage recruitment to the laserinjury site as a possible mechanism of CNV inhibition (Fig. 2). Interestingly, IL-1Ra treatment significantly increased the number of Ly6G positive neutrophils at 10h, suggesting a possible IL-1β dependent negative feedback in neutrophil recruitment. No inhibition of macrophage/microglial cell recruitment was observed in IL-1Ra treated mice after laser injury. Therefore IL-1Ra-dependent inhibition of CNV is unlikely to be mediated by an inhibitory effect on the macrophage recruitment.

IL-1β exerts its activity by acting on the IL-1 receptor I (IL-1RI) 1 . Vascular endothelium, which constitutively expresses IL-1RI [START_REF] Boraschi | Endothelial cells express the interleukin-1 receptor type I[END_REF] , releases IL6 [START_REF] Boraschi | Endothelial cells express the interleukin-1 receptor type I[END_REF] and microparticles [START_REF] Leroyer | CD40 ligand+ microparticles from human atherosclerotic plaques stimulate endothelial proliferation and angiogenesis a potential mechanism for intraplaque neovascularization[END_REF] upon IL-1β stimulation. We show that collagen IV expressing vascular endothelial cells in CNV express IL-1RI in vivo (Fig. 3). To evaluate a direct effect of IL-1β on vascular sprouting independently of inflammatory recruitment, we submitted aortic rings to IL-1β in vitro (Fig. 3). We show that IL-1β significantly induces vascular sprouting of aortic rings after 24 hours of IL-1β exposure. This difference was maintained for up to 72 hours of IL-1β incubation. A previous study has shown an inhibitory effect of IL-1β on vascular sprouting in the aortic ring assay at ten days [START_REF] Mountain | Interleukin-1beta-mediated inhibition of the processes of angiogenesis in cardiac microvascular endothelial cells[END_REF] . In our study, aortic rings became unviable after 7 days in culture and we could therefore not corroborate or contradict these findings. IL-1β has been suggested to induce angiogenesis via a sustained VEGF induction in fibroblasts [START_REF] Kvanta | Expression and regulation of vascular endothelial growth factor in choroidal fibroblasts[END_REF] . In our in vivo experiments we did not detect a sustained VEGF induction following the IL-1β induction in laser-injured eyes. Furthermore, vascular sprouting of aortic rings simultaneously incubated with IL-1β and the soluble VEGFR1 did not differ from rings incubated only with IL-1β, suggesting that the proangiogenic effect is independent of VEGF. Previous studies have shown that IL-1β directly induces IL-6 [START_REF] Boraschi | Endothelial cells express the interleukin-1 receptor type I[END_REF] and microparticles release from the vascular endothelium [START_REF] Leroyer | Microparticles from ischemic muscle promotes postnatal vasculogenesis[END_REF] and that IL-1β inhibitor CK112 and CK116 inhibit HUVEC cell proliferation [START_REF] Zou | Effect of interleukin-1 blockers, CK112, and CK116 on rat experimental choroidal neovascularization in vivo and endothelial cell cultures in vitro[END_REF] . The proangiogenic effect of IL-1β we observed in vivo and in vitro might therefore be a direct effect of endothelial cell proliferation.

In AMD, subretinal neovascularization occurs simultaneously with retinal degeneration. To make anti-IL-1β therapy a viable option, the influence of IL-1β on photoreceptors has to be evaluated in vivo. Previous reports have shown that IL-1β develops neurotoxic properties in the brain 5, [START_REF] Banwell | Systematic review and stratified meta-analysis of the efficacy of interleukin-1 receptor antagonist in animal models of stroke[END_REF][START_REF] Cardona | Control of microglial neurotoxicity by the fractalkine receptor[END_REF] . However in the retina IL-1β was shown to inhibit ganglion cell death 6 and IL-1β substitution saved photoreceptors in light-induced models 7 and genetic retinal degeneration [START_REF] Whiteley | Photoreceptor rescue after low-dose intravitreal IL-1beta injection in the RCS rat[END_REF] . Here we show that IL-1β is significantly induced in the light-induced model in albino rats (Fig. 4D) and in albino mice (Fig. 5B). To test the effect of IL-1β inhibition on retinal degeneration, we treated the laser-injured C57Bl6 mice and light-exposed rats with IL-1Ra at dosages that were sufficient to significantly inhibit CNV in mice (see above) and rats [START_REF] Olson | Intravitreal anakinra inhibits choroidal neovascular membrane growth in a rat model[END_REF] . Interestingly, IL-1Ra at a dose that significantly inhibits CNV does not exacerbate photoreceptor degeneration in vivo (Fig. 4).

We have previously shown that a dysfunctional CX3CR1 is associated with AMD. Cx3cr1 -/- mice develop spontaneous subretinal macrophage accumulation that is associated with photoreceptor degeneration by the age of 18 months [START_REF] Combadiere | CX3CR1-dependent subretinal microglia cell accumulation is associated with cardinal features of age-related macular degeneration[END_REF] . Similarly, these mice develop exaggerated photoreceptor degeneration in light-induced models [START_REF] Raoul | Role of the chemokine receptor CX3CR1 in the mobilization of phagocytic retinal microglial cells[END_REF] . We analyzed the expression and impact of IL-1β in the Cx3cr1 -/-model of AMD. IL-1β significantly increases with age in Cx3cr1 -/-animals. This increase coincides with the observed subretinal inflammation in these animals [START_REF] Combadiere | CX3CR1-dependent subretinal microglia cell accumulation is associated with cardinal features of age-related macular degeneration[END_REF] and might reflect IL-1β produced by subretinal inflammatory cells. In light-induced injury BALB Cx3cr1 +/+ and Cx3cr1 -/-mice, IL-1β was rapidly induced, similar to the rat and to laser-induced CNV. As shown in the mouse CNV model and rat models (Fig. 4), IL-1Ra substitution in control and light-exposed Cx3cr1 -/-mice in dosages sufficient to significantly inhibit CNV had no deleterious effect on photoreceptor survival (Fig. 5). Although exogenous IL-1β substitution has been shown to protect photoreceptor degeneration in a light-induced model 7 , its inhibition does not exacerbate degeneration in wild type mice (CNV model) or light-exposed rats, nor in degeneration-prone Cx3cr1 -/-mice.

The apparent contradiction of IL-1β mediated photoreceptor protection and the absence of an exacerbating effect of IL-1Ra might be explained by different IL-1β concentrations. While the neuroprotective effect of LaVail et al. 7 and Whiteley et al. [START_REF] Whiteley | Photoreceptor rescue after low-dose intravitreal IL-1beta injection in the RCS rat[END_REF] are found at concentrations of 0,5-5µg of IL-1β per eye, the maximum IL-1β concentration we measured with age-and light-induced injury correspond to approximately 50 pg/eye. The endogeneously-produced amount of IL-1β might therefore be well below neuroprotective concentrations and therefore have no effect on photoreceptor survival.

In summary, IL-1β is induced by laser-and light-induced injury. We confirm that IL-1Ra substitution efficiently inhibits the development of CNV in the mouse and show that it has no deleterious effect on photoreceptor survival. Taken together, our data suggests that IL-1Ra is a safe alternative for the treatment of neovascular AMD.

  ). At 10h after the injury small IL-1β positive cells are visible at the site of impact (Fig1I) (n=4 eyes). Triple labeling at 10h using the neutrophil marker Ly6G (red fluorescence Fig.1J) and the monocyte/macrophage marker Iba1 (blue fluorescence Fig.1K), reveals that the IL-1β positive cells are Ly6G positive and Iba1 negative (Fig.1L merge). The vast majority of detected IL-1β positive cells were Ly6G positive. 72 hours after the impact IL-1β (Fig.1M) and Ly6G (Fig.1N) was faint and diffuse, while numerous Iba1 positive cells were detected at the injury site (blue staining Fig.1O, P merge). Immunohistochemistry at 10h and 72h omitting the primary antibodies resulted in no staining (data not shown) (n=4 eyes per time point).

Fig. 1 :

 1 Fig.1 : IL-1β expression in laser-induced neovascularization.

Fig. 2 :Fig. 5 :

 25 Fig.2: IL-1Ra inhibits choroidal neovascularization independently of macrophage

Acknowledgement :

The authors wish to thank Christopher Brent Murray for critical review of the manuscript and Bernadette Lescure and Nadège Brunel for technical assistance. This work was supported by grants from INSERM, ANR "blanc" (AO5120DD), European Grant "Innochem" (LSHB-CT-2005-518167), ANR Maladies Neurologiques et Psychiatriques (ANR-08-MNPS -003) and ERC starting Grant (ERC-2007 St.G. 210345) F.S. is a recipient of a contract « Interface » from Assistance Publique-Hopitaux de Paris.


Eugene Asarin

Nicolas Basset

Aldric Degorre

Dominique Perrin

Generating functions of timed languages ⋆

In order to study precisely the growth of timed languages, we associate to such a language a generating function. These functions (tightly related to volume and entropy of timed languages) satisfy compositionality properties and, for deterministic timed regular languages, can be characterized by integral equations. We provide procedures for closed-form computation of generating functions for some classes of timed automata and regular expressions.

Introduction

Since the introduction of timed automata in [START_REF] Alur | A theory of timed automata[END_REF], these automata and their languages are extensively studied both in theoretical perspective and in applications to verification of real-time systems. However, the natural question of measuring the size of timed languages was addressed only recently in [START_REF] Asarin | Volume and entropy of regular timed languages: Discretization approach[END_REF][START_REF] Asarin | Volume and entropy of regular timed languages: Analytic approach[END_REF] and a couple of subsequent works. In these articles we explored the asymptotic behavior of the volume of a timed language when the number of events tends to ∞. We showed that for most deterministic timed automata this volume grows (or decreases) exponentially, defined entropy as its growth rate, characterized this entropy as a logarithm of the spectral radius of an integral operator Ψ and showed how to compute the entropy symbolically or numerically.

We believe that size analysis can be useful in several aspects: entropy is a measure of information content in timed words [START_REF] Asarin | Volume and entropy of regular timed languages: Discretization approach[END_REF] and a key to a timed code theory (work in progress). Whenever the entropy is not too small, timed automata have nice robustness properties [START_REF] Basset | Thin and thick timed regular languages[END_REF]. As a practical perspective, we are exploring applications of size analysis to random generation and compression of timed words. We also find the study of the size of timed languages a natural and mathematically appealing generalization of classical results on regular languages and formal series.

In this article, we make a much more precise size analysis of timed languages accepted by deterministic timed automata. We associate to such a language L the sequence of its volumes Vol(L n ), and the generating function f (z) = ⋆ The support of Agence Nationale de la Recherche under the project EQINOCS (ANR-11-BS02-004) is gratefully acknowledged. ⋆⋆ A part of the work of this author was done during his stay at the Institute of Mathematical Sciences (National University of Singapore) within its Programme "Automata Theory and Applications". n Vol(L n )z n . Thus the function f (z) contains a complete information on the "size profile" of Vol(L n ) as a function of n. To relate it to the previous work, we show that f (z) can be expressed in terms of the resolvent of the operator Ψ , and that the entropy of a timed language depends only on the convergence radius of f (z).

Throughout the paper we use examples in Fig. 1, to illustrate the notions of volume, generating function and techniques for computing the latter. Thus, the timed language recognized by automaton A 1 is L = {t 1 , a, t 2 , b, t 3 , a, . . . |∀i (t i + t i+1 ≤ 1)}.

For any number of events n we have a polytope in IR n :

L n = {t 1 , t 2 , t 3 , . . . , t n |∀i (t i + t i+1 ≤ 1)}, the sequence of volumes V n of these polytopes is and it was shown in [START_REF] Asarin | Volume and entropy of regular timed languages: Analytic approach[END_REF] that this sequence behaves asymptotically like (2/π) n . The methods developed in this paper yield a closed-form expression for the generating function of volumes: tan z + sec z. The convergence radius of the series, π/2, is the inverse of the growth rate of the sequence V n . This series describes precisely the sequence of volumes, and a closed-form formula for V n can be deduced:

V 2n-1 = B 2n (-4) n (1 -4 n )/(2n)! ; V 2n = (-1) n E 2n /(2n)! ,
where B s stand for Bernoulli numbers and E s for Euler numbers. Generating functions behave in a natural way with respect to simple operations on timed languages (disjoint union, unambiguous concatenation, unambiguous star). However in order to obtain an exact characterization and eventually closed-form expressions for generating function of timed regular languages a more involved analysis is needed. Such an analysis constitutes the main contribution of the article.

Related work. Our generating functions generalize those of regular languages, thoroughly studied and applied, [START_REF] Berstel | Noncommutative rational series with applications[END_REF][START_REF] Flajolet | Analytic combinatorics[END_REF][START_REF] Salomaa | Automata-theoretic aspects of formal power series[END_REF]. We are not aware of any work on generating functions of timed languages. Techniques and ideas used in this article build on our previous works on volumes and entropy of timed languages, especially on [START_REF] Asarin | Volume and entropy of regular timed languages: Analytic approach[END_REF] (however the current article is self-contained). As for automata and languages under study, we investigate timed regular languages of [START_REF] Alur | A theory of timed automata[END_REF], clock languages and expressions as in [START_REF] Bouyer | A Kleene/Büchi-like theorem for clock languages[END_REF], and subclasses of timed automata: regenerating automata from [START_REF] Sassoli | Close form derivation of state-density functions over DBM domains in the analysis of non-Markovian models[END_REF] , their subclass known as real-time automata [START_REF] Dima | Real-time automata[END_REF] and 1 3 4 -clocks automata, that extend both regenerating automata and 1 1 2 -clocks automata from [START_REF] Asarin | Volume and entropy of regular timed languages: Analytic approach[END_REF].

Whenever the alphabet of a timed automaton contains only one letter, its language can be seen as a sequence of polytopes P n ⊆ IR n . Some of such sequences, known as Fibonacci polytopes, have been studied (independently of timed automata) in combinatorics (see [START_REF] Stanley | A survey of alternating permutations[END_REF] and references therein). In particular, [START_REF] Stanley | A survey of alternating permutations[END_REF] provides a full analysis of the sequence of polytopes produced by automaton A 1 on Fig. 1. Thus our work points at a connection between timed automata and enumerative combinatorics.

Article structure. In Sect. 2 we introduce a formalism (inspired by [START_REF] Bouyer | A Kleene/Büchi-like theorem for clock languages[END_REF]) for timed and clock languages, introduce volume functions of such languages, and investigate the properties of these functions. In Sect. 3 we introduce generating functions of timed languages and investigate their general properties. In Sect. 4 we explain how to compute generating functions for several subclasses of timed automata. We summarize the contributions and discuss the directions of future work in Sect. 5.

Preliminaries

Clock languages and timed languages

In this paper, we study timed languages (mostly regular) using an approach based on clock languages introduced in [START_REF] Bouyer | A Kleene/Büchi-like theorem for clock languages[END_REF]. We present this approach in a slightly different form along with a multi-stage semantics. The general idea is as follows: we are interested in timed languages. Timed languages are obtained as projections of clock languages. Clock languages are homomorphic images of discrete "triplet languages". Triplet languages, in turn, can be generated by automata, regular expressions or grammars. Below we define formally all these notions and illustrate them on a running example.

An alphabet of timed events is the product IR + × Σ where Σ is a finite alphabet. The meaning of a timed event (t, a) is that t is the time delay before the event a. A timed word is a sequence of timed events and a timed language is just a set of timed words.

Inspired by [START_REF] Bouyer | A Kleene/Büchi-like theorem for clock languages[END_REF] we enrich timed words and languages with d-dimensional clock vectors. A clock is a variable which takes values in IR + . In our setting, values of clocks will be bounded by a positive integer M . A clock word is a timed word together with an initial and a final clock vector, i.e. an element of IR d × (IR + × Σ) * × IR d . Two clock words [x w y] and [x ′ w ′ y ′ ] are said to be compatible if y = x ′ , in this case we define their product by [x w y]•[y w ′ y ′ ] = [x ww ′ y ′ ]. A clock language is a set of clock words. The product of two clock languages L and

L ′ is L • L ′ = {c • c ′ | c ∈ L, c ′ ∈ L ′ , c and c ′ compatible}. The neutral element E is {[x ǫ x] | x ∈ IR d }
and the Kleene star of a language L is as usual

L * = k L k with L 0 = E.
A clock language L is said to be deterministic whenever for each clock word the final clock vector is uniquely determined by the initial clock vector and the timed word, in other words there exists a function σ L : IR d × (IR + × Σ) * → IR d such that for any clock word [x w y] of L, we have that y = σ L (x, w). In the following, we work with deterministic clock languages.

To a clock language we associate its timed projections. Given L, we define L(x, x ′ ) as the timed language leading from x to an element lower than x ′ :

L(x, x ′ ) = {w | ∃y [x w y] ∈ L ∧ y ≤ x ′ }.
We also define the timed language

L(x) = {w | ∃y [x w y] ∈ L}
as the language starting from x. Note that L(x) = L(x, M ) where M = (M, . . . , M ) is the greatest clock vector possible.

Triplet, clock and timed languages

Triplets. Following [START_REF] Bouyer | A Kleene/Büchi-like theorem for clock languages[END_REF] we define a triplet as a tuple a, g, r with: a a letter in Σ; g a conjunction of constraints, x i ⊲⊳ c (i ∈ {1, . . . , d}, c ∈ {0, . . . , M }, ⊲⊳∈ {<, >, ≤, ≥}), called guard, and r ⊆ {1 . . . d} a set of indices of clocks to be reset. We suppose moreover that guards are such that all the clocks remain bounded by M. We denote by T the finite alphabet of such triplets.

Clock semantics of triplets. The clock language of a triplet a, g, r is L( a, g, r

) = {[x (t, a) x ′ ] | x+t |= g∧r(x+t) = x ′ }.
Here, for a clock vector x = (x 1 , . . . , x d ), we denote by x+t the vector (x 1 +t, . . . , x d +t). Clock vectors are updated as follows: r(y 1 , . . . , y d ) = (y ′ 1 , . . . , y ′ d ) with y ′ i = 0 if i ∈ r and y ′ i = y i otherwise. The reader acquainted with timed automata will notice that a triplet corresponds to a transition of such an automaton.

This definition can be extended to all triplet words by: L

(ǫ) = E and L(π 1 . . . π n ) = L(π 1 ) . . . L(π n ). Finally for a language L ⊆ T * , we define L(L) = {L(π) | π ∈ L}.
In fact, L is a morphism between the two Kleene algebras (P(T * ), ∪, •, ∅, ǫ) and (P(

IR d × (IR + × Σ) * × IR d ), ∪, •, ∅, E).
Timed automata and their languages. A timed automaton A is a finite automaton with alphabet T . Its discrete semantics L is the language of triplet words accepted by A seen as a finite automaton; its clock semantics is L A = L(L) and its timed semantics is L A (0). The timed automata considered in this article are assumed to be deterministic in the sense of [START_REF] Alur | A theory of timed automata[END_REF], i.e. outgoing transitions from the same state with the same label must have pairwise incompatible guards. Clock languages of deterministic automata are also deterministic.

A timed regular expression is defined as expression over the finite alphabet T . Its discrete, clock and timed semantics are defined similarly to the automata.

This multi-stage timed semantics is equivalent to the usual semantics of timed automata, timed regular expressions, etc.

Example 1 (Our running example). Automata A 2 and A 3 on Fig. 1 have the same discrete semantics3 which is captured by a regular expression:

( a, x ≤ 1, {y} + b, y ≤ 1, {x} ) * .
An example of clock words recognized by the automaton is

[(0.5, 0.8) (0.3, a)(0.1, a)(0.9, b) (0, 0.9)].
The timed language recognized is:

{(t 1 , a) • • • (t n1 , a)(t n1+1 , b) • • • (t n2 , b)(t n2+1 , a) • • • (t n3 , a) • • • | ∀j ≥ 0, nj+1 i=nj +1 t i ≤ 1},
with n 0 = 0 and possibly n 1 = 0.

A convenient way to see automata is the matrix form. A timed automaton A over a set of control states Q and an alphabet of transitions T is uniquely described by three ingredients:

a Q × Q-matrix Ì whose element Ì qq ′ is the set of triplets labelling transi- tions from q to q ′ ; -a row vector I describing initial states: for each control state p, its element I p = {ǫ} iff p is initial, and ∅ otherwise; -a column vector F describing final states: for each control state q, its element F q = {ǫ} iff q is final, and ∅ otherwise.

The coefficient (Ì n ) p,q of Ì n contains the language of all the triplet words of length n from p to q. The p th coordinates of the column vector Ì n F contains the language recognized from state p and IÌ n F contains the language of triplet words of length n recognized by A. For instance the matrices for A 3 are:

I = {ǫ} ∅ ; Ì = { a, x ≤ 1, {y} } { b, y ≤ 1, {x} } { a, x ≤ 1, {y} } { b, y ≤ 1, {x} } ; F = {ǫ} {ǫ} .

Volume(s) of timed and clock languages

Measurable timed languages and clock languages. A timed language L is measurable if, for any word w ∈ Σ * , the projection

L w = {t ∈ IR |w| | (t, w) ∈ L} 4 is a Lebesgue-measurable subset of IR |w| . A clock language L is measurable if it is deterministic and for every w ∈ Σ * , σ L (•, (•, w)) is a Lebesgue-measurable function of IR d × IR |w| → IR d .
We remark that timed languages and deterministic clock languages obtained from triplet languages are measurable because their timed projections are polytopes.

Volumes of a timed language [START_REF] Asarin | Volume and entropy of regular timed languages: Analytic approach[END_REF]. The sequence of volumes (V n (L)) n∈IN associated to a measurable timed language is V n (L) = w∈Σ n Vol(L w ), where Vol is the hyper-volume (i.e. Lebesgue measure) in IR n . For dimension 0 we define V 0 (L) = 1 if ǫ ∈ L, and V 0 (L) = 0 otherwise. Now, for a clock language L and a word w ∈ Σ * of length n ≥ 0, we define the clock language L(w

) = {[x (t, v) x ′ ] | v = w}.
Volumes constrained by initial and final clock vectors. Timed regular languages considered below come from clock languages (which themselves come from triplet languages). The information about clock vectors is crucial to compute the volume of timed languages in a compositional manner.

Thus we define parametric volumes depending on initial and final clock vectors as follows

V 2 n (x, x ′ ) = V n (L(x, x ′ )).
We call this function the cumulative volume function (CVF) 5 of L. We also allow the following notations: for a clock language L and a discrete events word w, V 2

L(w) (x, x ′ ) = V 2 |w| (L(w)(x, x ′ )); and for a triplets word π, V 2 π (x, x ′ ) = V |π| (L(π)(x, x ′ )).
The notion of parametric volumes can be also applied to the clock language constrained only by initial clock vector L(x):

V 1 n (x) = V n (L(x)). Clearly V 1 n (x) = V 2 n (x, ∞) = V 2 n (x, M ).
CVFs for a triplet word. According to the following result, a CVF is easy to compute for a triplet word, and hence for a finite triplet language.

Proposition 1. For a triplet word π the CVF V 2 π is piecewise polynomial with rational coefficients of degree ≤ |π|. The pieces are polytopes, and an expression of this function is computable.

Proof. The clock language L(π) is a polytope in IR d ×IR n ×IR d . Its timed section L(π)(x, x ′ ) is a polytope in IR n whose coefficients are linear functions of x and x ′ . Volumes of such polytopes are computable and have the required form.

Composing CVFs. In order to define a composition for CVF corresponding to the concatenation of triplet words and languages, we proceed as follows. We define composition of two functions of IR d × IR d → IR as:

V 2 1 ⋆ V 2 2 (x, x ′ ) = y V 2 2 (y, x ′ )V 2 1 (x, dy),
where the integral is the Lebesgue-Stieltjes integral 6 . We also define

V 2 1 ⋆ v(x) = y v(y)V 2 1 (x, dy),
when v is defined on IR d . Then we can state the key lemma (to transpose concatenation of words to the CVFs world):

Proposition 2. For any measurable clock languages L 1 and L 2 and discrete words w 1 and w 2 , V 2 L1(w1) ⋆V 2 L2(w2) is well defined and satisfies:

V 2 L1(w1) ⋆V 2 L2(w2) = V 2 L1(w1)•L2(w2) .
Proof. First recall that

V 2 L1(w1)•L2(w2) (x, x ′ ) = Vol{t | σ L1•L2 (x, (t, w 1 w 2 )) ≤ x ′ } = t dt • 1l σ L 1 •L 2 (x,(t,w1w2))≤x ′ , which gives V 2 L1(w1)•L2(w2) (x, x ′ ) = t1,t2 dt 1 dt 2 • 1l σ L 2 (σL 1 (x,(t1,w1)),(t2,w2))≤x ′ .
By Fubini's theorem this can be rewritten as

V 2 L1(w1)•L2(w2) (x, x ′ ) = t1 dt 1 t2 dt 2 1l σ L 2 (σL 1 (x,(t1,w1)),(t2,w2))≤x ′ .
Applying again the formula for V 2 L2(w2) , we get that

V 2 L1(w1)•L2(w2) (x, x ′ ) = t1 dt 1 • V 2 L2(w2) (σ L1 (x, (t 1 , w 1 )), x ′ ) = x1 V 2 L2(w2) (x 1 , x ′ ) V 2 L1(w1) (x, dx 1 ),
as required. The last change of variables can be justified as follows. It has the form:

t1 dt 1 • u(σ L1 (x, (t 1 , w 1 ))) = x1 u(x 1 ) V 2 L1(w1) (x, dx 1 ), (1)
and we have to prove that it holds for any measurable u. Indeed, whenever u(y) is an indicator 1l y≤a , both left-hand and right-hand sides equal V 2 L1(w1) (x, a), thus (1) holds. Any other function u can be obtained as a limit of linear combinations of such indicators.

Volume functions in timed automata. As we did for languages, we introduce a Q-vector V n (x) of volumes of clock languages and a Q × Q-matrix Î(x,x ′ ) of cumulative volume functions of elements of the transition matrix Ì: formally

V n,q (x) = V n (L q (x))
and

Î qq ′ (x, x ′ ) = V 1 (L(Ì qq ′ )(x, x ′ )),
with L q = L((Ì n F) q ). It follows from the proposition above that the matrix element (Î ⋆n ) pq (of the matrix power wrt ⋆) contains the CVF of L ((Ì n ) pq ), that is of the language of all the clock words of length n leading from p to q. Finally, we get the formula for volumes:

V n = Î ⋆n ⋆ V F , (2)
with V F a column vector with V F,p = 1 if p is final, and V F,p = 0 otherwise.

The following not so obvious property of volume functions will be used in the sequel. Proposition 3. In a timed automaton A, for n ≥ 1, the volume functions V n (x) and Î ⋆n (x, x ′ ) are continuous wrt the initial clock vector x.

Proof. We prove the result for V 2 π for a triplet word π. It is then straightforward to lift the result to the cumulative and non-cumulative volume functions of the n-language of an automaton (finite sum of triplet words).

For a triplet word π, L(π)(x, x ′ ) is actually a polytope, intersection of halfspaces H of equations of one of the following forms:

-either x i + s j ⊲⊳ c, -x i + s |π| ⊲⊳ x ′ i , -s j -s l ⊲⊳ c -or s |π| -s l ⊲⊳ x ′ i
Choices of i, j, ⊲⊳ and c depend on H. Here s j is the time since the beginning of the word after j transitions, i.e. s j = j i=1 t i , in particular s 0 is the constant 0. Note that the Jacobian of the change of variables t = (t 1 , . . . , t |π| ) to s = (s 1 , . . . , s |π| ) is 1. So we can write:

V 2 π (x, x ′ ) = 1l L(π)(x,x ′ ) (t)dt = H:half-space 1l H(s,x,x ′ ) ds,
thus, for any i ∈ 1..d:

∂V 2 π (x, x ′ ) ∂x i = H l :xi appears ±δ(x i + s ji -c l or x i + s |π| -x ′ i ) H =H l 1l H ds = H l :xi appears ± H =H l 1l H[xi ← x ′ i -s |π| or xi ← c l -sj l ] ds.
where s is the vector of coordinates of s different from j l for all l such that H l : x i + s j l ⊲⊳ k l c k l . We note that the Dirac's δs were all eliminated after the integration by s j l and that the expression that remains under the integral is a proper function of x, x ′ and s. This implies that V 2 π (x, x ′ ) was actually continuous with respect to x i for all i and thus continuous.

Generating functions

Definitions

To study volume sequences associated to timed and clock languages we define their generating functions. As usual for generating functions, they allow recovering the sequence, its growth rate, momenta etc; and they have nice compositional properties. Given a timed language L its generating function is defined as follows:

f L (z) = k z k V k (L).
Given a clock language L, we define a (parametric) generating function with a given initial clock vector

f 1 L (z, x) = k z k V k (L(x)) = f L (z), with L = L(x).
For a clock language L we also define another cumulative generating function with a given initial clock vector and a bound on the final clock vector:

f 2 (z, x, x ′ ) = k z k V 2 k (x, x ′ ) = f L (z), with L = L(x, x ′ ).
To summarize, we are interested in computing f (z), but this computation will be based on f 1 (z, x), and sometimes on f 2 (z, x, x ′ ). Given a timed automaton, timed and clock languages, and thus generating functions are naturally associated to its states, for example

f 1 q (z, x) = k z k V k (L q (x)) = f L (z), with L = L q (x).
Taken for all states, functions f q and f 1 q form |Q|-dimensional vector functions f (z, x), f 1 (z, x), while functions f 2 q,q ′ form a Q × Q-matrix function 2 (z, x, x ′ ).

Analytic characterization

Elementary properties. First, let us state the relations between the three kinds of generating functions:

Proposition 4. The functions f, f 1 , f 2 are related as follows:

f (z) = f 1 (z, 0); f 1 (z, x) = f 2 (z, x, M ).
By definition, f 2 , f 1 and f are analytic functions of z. Since we consider timed automata with guards bounded by some constant M , all the volumes V k (with any initial or final conditions) can be upper bounded by (M |Σ|) k . This implies that convergence radius of series for f 2 , f 1 and f is at least (M |Σ|) -1 > 0. More precisely, the radius of convergence of f is

1/ lim sup k→∞ (V k (L)) 1/k = 2 -H(L) ,
where H(L) is called the volumetric entropy of L (see [START_REF] Asarin | Volume and entropy of regular timed languages: Analytic approach[END_REF]).

For generating functions associated to timed automata, the following result is a straightforward corollary of Prop. 3: Proposition 5. Within its convergence radius, the generating function f 1 (z, x) associated to a timed automaton A is continuous wrt the initial clock vector x.

Integral equation for generating functions. Consider a timed automaton. Using formula (2), its generating function can be computed as follows:

f 1 (z, x) = k z k V k (x) = k z k Î ⋆k ⋆ V F ,
which implies our first main result.

Theorem 1 (Integral equation). In the interior of its convergence circle, the generating function f 1 is the unique solution of the integral equation

f 1 -zÎ ⋆ f 1 = V F . (3)
Example (1, continued). For the automaton A 3 (using the notation x -y for max(xy, 0)):

Î = min(x ′ , 1) -x1l y ′ ≥0 min(y ′ , 1) -y1l x ′ ≥0 min(x ′ , 1) -x1l y ′ ≥0 min(y ′ , 1) -y1l x ′ ≥0 ; V F = 1 1 .
Equation (3) gives:

f 1 p (z, x, y) = f 1 q (z, x, y) = 1 + z 1 x f 1 p (z, x ′ , 0)dx ′ + z 1 y f 1 q (z, 0, y ′ )dy ′ .
In Section 4.1 below we develop a technique for solving such equations (for a subclass of automata including this one), and compute the generating function for this language.

Volumes, generating functions and functional analysis

In this section (which can be skipped by a reader not interested in functional analysis), similarly to [START_REF] Asarin | Volume and entropy of regular timed languages: Analytic approach[END_REF], we rephrase previous results in terms of the spectral theory of linear operators.

Given a timed automaton, consider the Banach space F of Q-vectors of continuous functions on clock valuations. Thus an element of F is a vector v whose components are continuous v q : [0; M ] d → IR, and F = C([0; M ] d ) Q . The matrix Î corresponds to an operator Ψ : F → F defined by

Ψ (v) = Î ⋆ v
(a variant of this operator plays the central role in [START_REF] Asarin | Volume and entropy of regular timed languages: Analytic approach[END_REF]). In terms of this operator, Prop. 3 and equation ( 2) can be rephrased as follows:

Proposition 6 ([2]
). Ψ is a bounded linear operator on F (represented by a matrix of integral operators). The volume vector can be obtained by iteration of this operator:

V n = Ψ n (V F ).
Recall that, by definition, the resolvent of an operator A is R(λ, A) = (A -λI) -1 ; it is well defined when λ does not belong to the spectrum of A, in particular for |λ| > ρ(A), where ρ denotes the spectral radius. We obtain as a consequence of Thm. 1 another characterization of the generating function:

Proposition 7 (Generating function and resolvent). The generating function f 1 satisfies the formula:

f 1 = -z -1 R(z -1 , Ψ )V F , which holds in the interior of the circle |z| < ρ(Ψ ) -1 .

Inductive characterization of generating functions

The form of generating functions of finite triplet languages follows from Prop. 1: Proposition 8. For a finite triplet language L with maximal word length ℓ, the generating functions f 2 , f 1 are piecewise polynomial in z, x, x ′ (pieces are polytopes in x, x ′ ) of degree ≤ ℓ wrt z and wrt x and x ′ .

More complex languages can be obtained from finite ones using Kleene algebra operations. As usual in the context of generating functions, we suppose that the operations are unambiguous. A language operation is ambiguous if a word of the resulting language can be obtained in several ways by composing different words from the operands. We consider first the simple case of timed languages. Proposition 9. Generating functions behave well for unambiguous operations on measurable timed languages:

-f L1∪L2 = f L1 + f L2 ; -f L1•L2 = f L1 f L2 ; -f L * = 1 + f L f L * provided ǫ ∈ L.
However, in order to obtain general timed regular languages we need operations on clock languages, which are more involved. Proposition 10. Generating functions f 2 behave well for unambiguous operations on deterministic measurable clock languages:

-f 2 L1∪L2 = f 2 L1 + f 2 L2 ; -f 2 L1•L2 = f 2 L1 ⋆ f 2 L2 ; -f 2 L * = 1l x≤x ′ + f 2 L ⋆ f 2 L * provided E ∩ L = ∅. Proof. -Union: f 2 L1∪L2 (x, x ′ , z) = w∈Σ * z |w| V 2 L1∪L2(w) (x, x ′ ) Since the union is unambiguous, ∀w ∈ Σ * , L 1 (w) ∩ L 2 (w) = ∅ and thus ∀w ∈ Σ * , V 2 L1∪L2(w) = V 2 L1(w) + V 2 L2(w)
. Finally :

f 2 L1∪L2 (x, x ′ , z) = f 2 L1 (x, x ′ , z) + f 2 L2 (x, x ′ , z) -Product:
Recall that:

f 2 L1•L2 (x, x ′ , z) = w∈Σ * z |w| V 2 L1•L2(w) (x, x ′ ).
Since the product is unambiguous,

(L 1 • L 2 )(w) = w1,w2∈Σ * w=w1w2 L 1 (w 1 ) • L 2 (w 2 ),
and thus

V 2 L1•L2(w) = w1,w2∈Σ * w=w1w2 V 2 L1(w1)•L2(w2) = w1,w2∈Σ * w=w1w2 V 2 L1(w1) ⋆ V 2 L2(w2) ,
with the last equality given by proposition 2. We deduce:

w∈Σ * z |w| V 2 L1•L2(w) = w∈Σ * w1,w2∈Σ * w=w1w2 z |w1| V 2 L1(w1) ⋆ z |w2| V 2 L2(w2) = w1∈Σ * w2∈Σ * z |w1| V 2 L1(w1) ⋆ z |w2| V 2 L2(w2) = w1∈Σ * z |w1| V 2 L1(w1) ⋆ w2∈Σ * z |w2| V 2 L2(w2) ,
and finally:

f 2 L1•L2 = f 2 L1 ⋆ f 2 L2 . -Kleene star:
We have L * = E ∪ L • L * , the union is empty since E ∩ L = ∅ and the product is unambiguous since the star is unambiguous thus using the two first items:

f 2 L * = f 2 E + f 2 L ⋆ f 2 L * . To complete the proof it remains to prove that f 2 E (x, x ′ , z) = 1l x≤x ′ . Recall that: E = {[x ǫ x] | x ∈ IR d } thus E(x, x ′ ) = {w ∈ (R + × Σ) * | ∃y ≤ x ′ , (x, w, y) ∈ E} = {w ∈ (R + × Σ) * | x ≤ x ′ , w = ǫ} this set is equal to the 0-dimensional set {ǫ} whose volume is 1 iff x ≤ x ′ ,
otherwise it is empty and thus of volume 0.

p q r b, x < 7, {x} a, 2 < x < 3, ∅ b, x < 5, {x} b, x < 8, {x} p q r b, [1; 9] a, [2; 3] a, [1; 5] b, [2; 8]
Fig. 2. A regenerating automaton (left) and a real-time automaton (right)

Corollary 1. Generating function f 1 for unambiguous compositions of clock languages can be computed as follows:

-

f 1 L1+L2 = f 1 L1 + f 1 L2 ; -f 1 L1•L2 = f 2 L1 ⋆ f 1 L2 ; -f 1 L * = 1 + f 2 L ⋆ f 1 L * provided E ∩ L = ∅.

Computing generating functions

The generating function of a timed language represented by an automaton is characterized by a system of integral equations (3). The generating function of a timed language represented by a regular expression can be found recursively from piecewise polynomial functions using operations +, ⋆ and solving fixpoint integral equations of Prop. 10 and Cor. 1. Unfortunately, both procedures involve computation of integrals, and solution of integral equations, for this reason, the result cannot be always presented by an explicit formula. Below we consider several subclasses of timed automata, for which generating functions can be obtained in closed form, or at least admit a simpler characterization.

Generating functions for particular classes of automata

System of equations. Our closed-form solutions for subclasses of timed automata will be obtained using a variant of language equations. Let Q = G ∪ B be a disjoint partition of the states of a timed automaton A into good and bad. We want to describe the vector L of triplet languages L q recognized from good states q ∈ G only. This vector satisfies the equation:

L = Ì • L + F, (4)
where Ì is a G × G-matrix and F is a G-vector of triplet languages. Their elements are defined as follows: T pq consists of all words leading from p to q via bad states only; F p consists of all words leading from p to a final state via bad states only 7 .

Automata with regeneration. Following [START_REF] Sassoli | Close form derivation of state-density functions over DBM domains in the analysis of non-Markovian models[END_REF], we call an automaton regenerating if there exists a partition Q = G ∪ B having two properties:

(a) every cycle in the automaton contains a state in G (good); (b) all the transitions coming into a good state reset all clocks. W.l.o.g. we suppose that the initial state is good (this can be achieved by adding a new initial state). Condition (a) implies that no cycle is possible within bad states, and thus all the elements of Ì and F are finite triplet languages (with maximal word length ≤ |B| + 1). Condition (b) means that (4) can be rewritten in timed languages (instead of clock languages), since when entering in a good state all clocks are reset. This gives

L timed = Ì timed • L timed + F timed . (5)
Applying simple compositionality conditions for generating functions for timed languages (Prop. 9) we obtain that

f = f + f F .
Due to Prop. 8 all the coefficients (elements of matrix and vector f F ) are polynomials of z. Solving this linear |G|-dimensional system we express f as a vector of rational functions of z:

f = I - -1 f F . (6)
The generating function f of the timed language accepted by the automaton is just one element of this vector f . We conclude.

Theorem 2. For a regenerating automaton the generating function f (z) is a rational function.

Example 2. Consider a regenerating automaton on Fig. 2 (left). We choose good and bad states as follows: G = {p, q}; B = {r}. The system of equations on timed languages of good states takes the form

L p L q = ∅ T pq T qp ∅ • L p L q + F p ∅ with (7)
T pq ={(t 1 , a)(t 2 , b)|2 < t 1 < 3 ∧ t 1 + t 2 < 5} ∪ {(t, b)|t < 8}; T qp ={(t, b)|t < 7}; F p ={(t, a)|2 < t < 3}.
For generating functions this yields:

f p f q = 0 2.5z 2 + 8z 7z 0 • f p f q + z 0 .
Solving this linear system we find the required

f p (z) = 2z 2 -35z 3 -112z 2 .
It converges for |z| < 0.1309, its Taylor coefficients (i.e. volumes V n for n = 0.. Real-time automata. We consider here transition-labeled real-time automata (t-RTA) from [START_REF] Dima | Real-time automata[END_REF]. They are automata, in which to any transition p a → q is associated a time interval [l, u]. This transition can be taken after spending between l and u time units in p. Equivalently, a real-time automaton can be seen as a timed automaton with only one clock, which is reset on transition.

It is easy to see that real-time automata are regenerating (all their states are good). Thus equation ( 6) applies. Its coefficients can be found in a more explicit form:

f = (I -z ) -1 V F , (8)
where matrix A pq is the sum of lengths of all time intervals associated to transitions from p to q, and, as before, V F q = 1 iff q is final, and 0 otherwise. This can be seen as a simplified version of the resolvent equation ( 3) for real-time automata: instead of a matrix of integral operators, a matrix of polynomials is inverted.

Example 3. For the real-time automaton on the right of Fig. 2 equation ( 8) takes the form:

  f p f q f r   =   I -z   0 6 1 8 0 0 0 4 0     -1 •   0 0 1   ,
which gives the required generating function:

f p = z 1 -48z 2 -
4 -clocks automata. We call an automaton 1 3 4 -clocks if there exists a partition of Q = G ∪ B into good and bad states having three properties:

every cycle in the automaton contains a good state; -the initial state is a good one; -for each good state p there is at most one clock x i(p) not reset by incoming transitions.

Similarly to regenerating automata, we apply equations (4), and observe that all the coefficients are finite triplet languages. Unfortunately, since some clocks are not reset, we cannot write an equation on timed languages similar to [START_REF] Berstel | Noncommutative rational series with applications[END_REF].

Instead, we pass to clock languages and their generating functions, as in the general case. This gives:

f 1 = 2 ⋆ f 1 + f 1 F , (9)
an integral equation with piecewise polynomial coefficients. We notice that functions in the last equation depend on the clock vector x ∈ IR d (or on two clock vectors x,x ′ ), but in fact for any good state p ∈ G only one clock x i(p) matters. This allows extracting simpler integral equations from (9), involving only functions of scalar argument. We proceed as follows: given a G-vector v whose elements v p are functions on IR d , we define reduced functions on IR: v p (x) = v p (0, . . . , 0, x, 0, . . . , 0), with the argument x at position i(p). Reduced G-vector v consists of reduced elements v p . Reduced version of matrices are defined similarly.

The following identity is based on the requirement of clock resets:

Lemma 1. For a 1 3 4 -clocks automaton the following holds: 2 ⋆ f 1 = 2 ⋆ f 1 . Equation ( 9), reduced to

f 1 = 2 ⋆ f 1 + f 1 F ,
implies that the reduced vector of generating functions is a solution of equations of the form:

f (z, x) = ( ⋆ f )(z, x) + b(z, x), (10)
where all the coefficients are piecewise polynomial functions of z and a scalar argument x.

Lemma 2. An integral equation of the form [START_REF] Sassoli | Close form derivation of state-density functions over DBM domains in the analysis of non-Markovian models[END_REF] can be transformed into a system of linear ordinary differential equation with piecewise polynomial coefficients (depending on x and z).

Theorem 3. For a 1 3 4 -clocks automaton the generating function f can be obtained by solving a system of linear ordinary differential equations with piecewise polynomial coefficients.

We notice that the theorem gives a rather explicit characterization of f , but not always a closed-form expression.

Example (1, completed). A 3 is 1 We use equation [START_REF] Sassoli | Close form derivation of state-density functions over DBM domains in the analysis of non-Markovian models[END_REF] and remark that by symmetry of , the two generating functions f 1 p and f 1 q are equal to a unique function f 1 which satisfies

f 1 (z, x) = z 1 x f 1 (z, x ′ ) dx ′ + z 1 0 f 1 (z, x ′ ) dx ′ + 1.
Differentiating it one time w.r.t x we obtain: ∂f 1 ∂x (z, x) = -zf 1 (z, x). The solution has the form f 1 (z, x) = A(z)e -zx . We remark that f 1 (z, 0) -1 = 2z ,

where Ai and Bi stand for Airy functions (see Appendix for detailed computations).

Conclusions

In this article, we have introduced generating functions of timed languages, explored their properties and characterized them by integral equations. For subclasses of timed regular languages we have presented closed-form expressions or simpler characterization of generating functions. Generating functions describe with a high precision the quantitative behaviour of timed languages. At the current stage of research, the computation of generating functions is a semi-manual task and restrictions are imposed to the automata. We are planning to explore theoretical and practical algorithmics of timed generating functions, and to implement the algorithm. On the other hand, we want to see whether closed form solutions are possible beyond the class of 1 3 4 -clocks languages. We hope that this approach will lead to new combinatorial results for timed regular languages and sequences of polytopes, better quantitative characterization of such languages with applications to information theory and verification of real-time systems. Also, the approach can be extended to timed formal series, non-regular timed languages, or to richer models such as hybrid automata.

Fig. 1 .

 1 Fig. 1. Timed automata. First line: A1, A2; second line: A3, A4.

3 4 -

 4 clocks 8 with good states G = {p, q} and no bad state B = ∅. The matrix and the vector b are = z(min(x ′ , 1) -x) z min(x ′ , 1) z min(x ′ , 1) z(min(x ′ , 1) -x)

1 0f 1

 11 (z, x ′ ) dx ′ = 2(f 1 (z, 1) -1)ande are done sincef (z) = f 1 (z, 0) = A(z) = 1 2e -z -1 .

Example 4 .

 4 Automata A 1 and A 4 are also 1 3 4 -clocks, and their generating functions are tan z + sec z and 4 π Bi ′ (0)Ai(-z) -Ai ′ (0)Bi(-z)

The difference will appear in section

4.1 since A3 is 1 3 4 -clocks and A2 is not.

by a slight abuse of notation, since (IR × Σ) n is isomorphic to IR n × Σ n

similarly to cumulative distribution functions in probability theory.

By definition, the Lebesgue-Stieltjes integral f (x)g(dx) is the Lebesgue integral of f wrt the measure µ having cumulative distribution function g.

if this final state is good the word should be ǫ.

A3 can be seen as A2 whose state is split to make it a 1 3 4 -clock automaton.

A Some details for the examples

The matrices defining A 1 are:

The matrices of CVF are:

The automaton A 1 is 1

Equations ( 10) on generating functions take the form:

By symmetry the two generating functions f 1 p and f 1 q are equal to a unique function f 1 which satisfies

Differentiating it twice w.r.t x we obtain:

The solution has the form

The matrices of A 2 are 1 × 1-dimensional since it has only one state:

The matrices of CVF are:

The equation on generating function is

This automaton is not 1 3 4 -clocks, but language equivalent to its split form A 3 treated in the article.

A.3 Example A 4

The automaton A 4 is 1 3 4 -clocks, it suffices to choose G = {p, q}, B = {r}. As in the previous example we have pq (x, x ′ ) = min(1 -x, x ′ )1l x ′ ≥0 . For qp , we must compute the volume of the language

. This is the area of the right triangle defined by equations t 1 , t 2 ≥ 0,

We can now give the matrix and the vector b:

Equations ( 10) on generating functions take the form:

Differentiating w.r.t x we obtain:

Differentiate once again the former equation gives:

∂x (z, 1x), combining with the latter one this gives:

The solution has the form f 1 (z, x) = α(z)Ai(-zx) + β(z)Bi(-zx) where Ai and Bi are the Airy's functions and α(z), β(z) two functions to be determined with the border conditions f 1 (z, 1) = 1, ∂f 1 ∂x (z, 0) = 0. We obtain the following equations: α(z)Ai(-z) + β(z)Bi(-z) = 1; α(z)Ai ′ (0) + β(z)Bi ′ (0) = 0.

Solving this system and simplifying using classical formulae for Ai(0), Ai ′ (0), Bi(0), Bi ′ (0) and Euler's reflection formula, we obtain the final result:

.