Ph.D Pothitos M Pitychoutis

Ph.D Arnauld Belmer

Imane Moutkine

Ph.D Joëlle Adrien

Ph.D Luc Maroteaux
email: luc.maroteaux@upmc.fr

Mice lacking the serotonin Htr 2B receptor gene present an antipsychotic-sensitive schizophrenic-like phenotype

Keywords: Serotonin, receptor, schizophrenia, antipyschotics, mice, endophenotype

Impulsivity and hyperactivity share common ground with numerous mental disorders, including schizophrenia. Recently, a population-specific serotonin 2B (5-HT 2B ) receptor stopcodon (i.e. HTR2B Q20*) was reported to segregate with severely impulsive individuals, while 5-HT 2B mutant (Htr 2B -/-) mice also showed high impulsivity. Interestingly, in the same cohort, early-onset schizophrenia was more prevalent in HTR2B Q*20 carriers. However, the putative role of 5-HT 2B receptor in the neurobiology of schizophrenia has never been investigated. We assessed the effects of the genetic and the pharmacological ablation of 5-HT 2B receptors in mice subjected to a comprehensive series of behavioral tests screening for schizophrenic-like symptoms and investigated relevant dopaminergic and glutamatergic neurochemical alterations in the cortex and the striatum. Domains related to the positive, negative and cognitive symptom-clusters of schizophrenia were affected in Htr 2B -/-mice, as shown by deficits in sensorimotor gating, in selective attention, in social interactions and in learning and memory processes. In addition, Htr 2B

-/-mice presented with enhanced locomotor response to the psychostimulants dizocilpine and amphetamine, and with robust alterations in sleep architecture. Moreover, ablation of 5-HT 2B receptors induced a region-selective decrease of dopamine and glutamate concentrations in the dorsal striatum. Importantly, selected schizophrenic-like phenotypes and endophenotypes were rescued by chronic haloperidol treatment. We report herein that 5-HT 2B receptor deficiency confers a wide spectrum of antipsychotic-sensitive schizophrenic-like behavioral and psychopharmacological phenotypes in mice and provide first evidence for a role of 5-HT 2B in the neurobiology of psychotic disorders.

INTRODUCTION

Schizophrenia is a devastating, complex and costly neuropsychiatric disorder that affects about 1% of the world's population [START_REF] Olesen | The economic cost of brain disorders in Europe[END_REF]. Schizophrenia is typically characterized by three symptom clusters: positive (e.g. hallucinations, delusions, disordered thoughts), negative (e.g. flattened affects, social withdrawal) and cognitive (e.g. attention and working memory deficits) [START_REF] Simpson | A possible role for the striatum in the pathogenesis of the cognitive symptoms of schizophrenia[END_REF]. This disease has traditionally been associated with a deregulated dopaminergic system, mainly due to the fact that for many years typical antipsychotics (i.e. strong dopamine (DA)-D 2 receptor antagonists, such as haloperidol) have served as the mainstream pharmacotherapy for psychotic patients [START_REF] Keshavan | Schizophrenia, "just the facts": what we know in 2008 Part 3: neurobiology[END_REF]. However, it has long been known that the serotonergic system is also implicated in the pathogenesis of schizophrenia [START_REF] Meltzer | The role of serotonin receptors in the action of atypical antipsychotic drugs[END_REF]. Indeed, atypical antipsychotics, of which clozapine is the prototype, consist of agents that are more potent antagonists of serotonin 2A (5-HT 2A ) receptors over D 2 receptors [START_REF] Gonzalez-Maeso | Identification of a serotonin/glutamate receptor complex implicated in psychosis[END_REF]. Many atypical antipsychotics show similar affinity for 5-HT 2B and 5-HT 2A receptors (e.g. clozapine, amisulpride, asenapine, aripiprazole or cariprazine) [START_REF] Abbas | Amisulpride is a potent 5-HT7 antagonist: relevance for antidepressant actions in vivo[END_REF][START_REF] Kiss | Cariprazine (RGH-188), a dopamine D(3) receptor-preferring, D(3)/D(2) dopamine receptor antagonist-partial agonist antipsychotic candidate: in vitro and neurochemical profile[END_REF][START_REF] Shahid | Asenapine: a novel psychopharmacologic agent with a unique human receptor signature[END_REF][START_REF] Shapiro | Aripiprazole, a novel atypical antipsychotic drug with a unique and robust pharmacology[END_REF], although the contribution of this 5-HT 2B receptor to the action of antipsychotic compounds has never been reported. Notably, the elucidation of the intricate mechanisms underlying the neurobiology of schizophrenia is imperative in order to develop novel drugs with improved therapeutic efficacy [START_REF] Kvajo | Avoiding mouse traps in schizophrenia genetics: lessons and promises from current and emerging mouse models[END_REF]. In this direction, models of genetic deletion in mice have been indispensable in characterizing the contribution of specific genes to disease pathophysiology [START_REF] Arguello | Modeling madness in mice: one piece at a time[END_REF].

Impulsivity, very broadly defined as action without foresight, novelty-seeking and hyperlocomotion share common ground with numerous mental disorders, including chronic substance abuse, attention deficit hyperactivity disorder and schizophrenia [START_REF] Fineberg | Probing compulsive and impulsive behaviors, from animal models to endophenotypes: a narrative review[END_REF][START_REF] Humby | Assaying dissociable elements of behavioural inhibition and impulsivity: translational utility of animal models[END_REF]. In a recent study conducted in a Finish cohort, we reported that a population-specific 5-HT 2B receptor stop-codon (HTR2B Q20*) segregates with severe impulsivity, psychosis and early-onset schizophrenia [START_REF] Bevilacqua | A populationspecific HTR2B stop codon predisposes to severe impulsivity[END_REF]. Moreover, mice in which the Htr 2B gene has been genetically inactivated (i.e. Htr 2B -/-) exhibit a hyperlocomotor phenotype as well as impulsivity, as assessed in the delay discounting task [START_REF] Bevilacqua | A populationspecific HTR2B stop codon predisposes to severe impulsivity[END_REF][START_REF] Doly | Serotonin 5-HT2B receptors are required for 3,4-methylenedioxymethamphetamine-induced hyperlocomotion and 5-HT release in vivo and in vitro[END_REF]. However, the extent to which 5-HT 2B receptor is implicated in the neurobiology of schizophrenia has never been investigated.

In the present study, we assessed the effects of the genetic ablation of Htr 2B in mice subjected to a comprehensive series of behavioral tests screening for positive, negative and cognitive schizophrenic-like symptoms. Following the demonstration that loss of function of the 5-HT 2B receptor in Htr 2B -/-mice confers a wide spectrum of schizophrenic-like behavioral phenotypes, we show that selected phenotypes are induced by acute pharmacological blockade of 5-HT 2B receptors with RS127445 (selective 5-HT 2B receptor antagonist) and rescued by chronic antipsychotic treatment with the typical antipsychotic drug haloperidol. The dorsal striatum (dSTR) and its connections with the prefrontal cortex (PFC) have been implicated in the pathogenesis of both the positive and the cognitive symptoms of schizophrenia [START_REF] Howes | Elevated striatal dopamine function linked to prodromal signs of schizophrenia[END_REF][START_REF] Simpson | A possible role for the striatum in the pathogenesis of the cognitive symptoms of schizophrenia[END_REF]. Indeed, neurochemical alterations in the dopaminergic and glutamatergic corticostriatal circuits underlie a substantial portion of the dysfunction seen in schizophrenic patients [START_REF] Gordon | Testing the glutamate hypothesis of schizophrenia[END_REF][START_REF] Howes | Elevated striatal dopamine function linked to prodromal signs of schizophrenia[END_REF][START_REF] Keshavan | Schizophrenia, "just the facts": what we know in 2008 Part 3: neurobiology[END_REF][START_REF] Simpson | A possible role for the striatum in the pathogenesis of the cognitive symptoms of schizophrenia[END_REF]. Thus, we further investigated relevant dopaminergic and glutamatergic neurochemical alterations in the PFC and the dSTR of Htr 2B -/-mice. Altogether, our findings present the first evidence for a role of 5-HT 2B receptors in the neurobiology of schizophrenia.

MATERIALS AND METHODS

Animals

Male Htr 2B

-/-mice used in these experiments were in a 129S2/SvPas (129S2) background and wild type (Htr 2B +/+ ) 129S2 mice (8-12 weeks old) used as controls were bred in our animal facility under standard 12 light/dark schedule, and housed in groups of 3-5 of the same genetic background and sex after weaning [START_REF] Diaz | 5-HT2B receptors are required for serotonin-selective antidepressant actions[END_REF]. Animal experiments were conducted in accordance with standard ethical guidelines (National Institutes of Health's "Guide for the Care and Use of Laboratory animals", and European Directive 2010/63/UE) and were approved by the Ethics Committee for Animal Experiments (N° 1170.01).

Drug treatments

The selective 5-HT 2B receptor antagonist RS127445 (RS; 0.5 mg/kg; Tocris, France) was administered acutely (i.e. 1 h before testing) and intraperitoneally (i.p.), as in previous studies from our group [START_REF] Diaz | 5-HT2B receptors are required for serotonin-selective antidepressant actions[END_REF][START_REF] Doly | Serotonin 5-HT2B receptors are required for 3,4-methylenedioxymethamphetamine-induced hyperlocomotion and 5-HT release in vivo and in vitro[END_REF]. The glutamatergic psychostimulant and Nmethyl-D-aspartic acid (NMDA) receptor antagonist dizocilpine (MK-801; 0.5 mg/kg; i.p.; Sigma-Aldrich, France) and the dopaminergic psychostimulant amphetamine (10 mg/kg; i.p.; Sigma-Aldrich, France) were dissolved in saline and administered at a rate 10 mL/kg. Different cohorts of Htr 2B +/+ and Htr 2B -/-mice were chronically treated (4 weeks) with haloperidol (Sigma-Aldrich, France) in drinking water (2.0 mg/kg/day) or vehicle, as described before [START_REF] Terry | Oral haloperidol or risperidone treatment in rats: temporal effects on nerve growth factor receptors, cholinergic neurons, and memory performance[END_REF]. Following chronic antipsychotic treatment, locomotor activity and prepulse inhibition (PPI) were tested, as described below. Separate mouse cohorts were used for the assessment of sleep architecture and for social interaction behavior and neurochemical estimations.

Locomotor response to novelty and to psychostimulants

Hyperactivity in response to novelty and locomotor hypersensitivity to acute administration of psychostimulants are considered useful correlates of positive schizophrenic-like symptoms in mice [START_REF] Arguello | Modeling madness in mice: one piece at a time[END_REF]. Locomotor activity was measured in a circular corridor with four infrared beams placed at every 90° (Imetronic, Passac, France), as previously described [START_REF] Blundell | Neuroligin-1 deletion results in impaired spatial memory and increased repetitive behavior[END_REF][START_REF] Doly | Serotonin 5-HT2B receptors are required for 3,4-methylenedioxymethamphetamine-induced hyperlocomotion and 5-HT release in vivo and in vitro[END_REF]. Mice were injected with saline and individually placed in the activity box for 2 h during four consecutive days in order to habituate to the apparatus. On the fifth day mice were placed in the activity box for 1 h and then treated with, dizocilpine or amphetamine. Locomotor activity was then screened for 2-6 h.

Prepulse Inhibition (PPI) of acoustic startle

PPI has been considered an operational measure of sensorimotor gating and pre-attentive processing, and its deficit has been consistently reported in schizophrenic patients and in relevant animal models [START_REF] Meyer | Towards an immuno-precipitated neurodevelopmental animal model of schizophrenia[END_REF]. Sensorimotor gating was assessed as previously described [START_REF] Yadav | The presynaptic component of the serotonergic system is required for clozapine's efficacy[END_REF]. PPI was indexed by the percentage inhibition of the startle response at each level of prepulse intensity by using the following formula: % PPI = [(mean reactivity on pulse-alone trials-mean reactivity on prepulse-pulse trials)/mean reactivity on pulse-alone trials] x 100%. % Average PPI values were calculated from all PPI values across all the four prepulse intensities.

Sociability and preference for social novelty

Decreased interactions with conspecifics are often used to model social withdrawal, a negative symptom of schizophrenia [START_REF] Arguello | Modeling madness in mice: one piece at a time[END_REF]. The three-chamber social test was performed as previously described [START_REF] Peca | Shank3 mutant mice display autistic-like behaviours and striatal dysfunction[END_REF]. Target subjects (Social I and Social II) were adult Htr 2B

+/+ males. For the sociability test, the test animal was introduced into the middle chamber and was left to habituate for 5 min, after what an unfamiliar mouse (Social I) was introduced into a wire cage in one of the side-chambers and an empty wire cage (EC) on the other sidechamber. The dividers were then opened and the test animal was allowed to freely explore all three chambers over a 10 min session. Preference for social novelty was assayed during a second 10 min session in which a novel stranger mouse (Social II) was inserted into the previously empty wire cage. The time spent in close interaction with the two cages was scored manually.

Novel Object Recognition (NOR)

Cognitive deficits, including working memory impairment, are typically observed in animal models of schizophrenia [START_REF] Arguello | Modeling madness in mice: one piece at a time[END_REF][START_REF] Meyer | Towards an immuno-precipitated neurodevelopmental animal model of schizophrenia[END_REF]. The NOR task represents a most useful method for studying cognitive impairment in schizophrenia [START_REF] Lyon | Spontaneous object recognition and its relevance to schizophrenia: a review of findings from pharmacological, genetic, lesion and developmental rodent models[END_REF]. The NOR test was performed as previously described [START_REF] Carlini | Decreased memory for novel object recognition in chronically food-restricted mice is reversed by acute ghrelin administration[END_REF] with retention tests performed at 1 h and at 24 h intervals in order to assess short-and long-term memory, respectively. Each trial lasted for 5 min. The time that the animals spent exploring the novel and the familiar objects were measured; results are expressed as % time of exploration of the novel object.

Contextual and cued fear conditioning

Cognitive deficits in associative fear memory have been reported in mice displaying schizophrenic-like phenotypes [START_REF] Arguello | Modeling madness in mice: one piece at a time[END_REF][START_REF] Gleason | Susceptibility of the adolescent brain to cannabinoids: long-term hippocampal effects and relevance to schizophrenia[END_REF]. Fear conditioning was assessed in computer-controlled operant chambers (Imetronic, Pessac, France), as previously described [START_REF] Blundell | Neuroligin-1 deletion results in impaired spatial memory and increased repetitive behavior[END_REF][START_REF] Cai | Postreactivation glucocorticoids impair recall of established fear memory[END_REF][START_REF] Powell | The presynaptic active zone protein RIM1alpha is critical for normal learning and memory[END_REF]. All results were expressed as % time freezing.

Latent Inhibition (LI)

Latent inhibition (LI) refers to the retardation in learning about the significance of a stimulus as a result of its prior repeated pre-exposures without consequence [START_REF] Lubow | Latent inhibition: the effect of nonreinforced pre-exposure to the conditional stimulus[END_REF]. LI was assessed in a conditioned freezing paradigm, as previously described [START_REF] Meyer | Towards an immuno-precipitated neurodevelopmental animal model of schizophrenia[END_REF][START_REF] Willi | Constitutive genetic deletion of the growth regulator Nogo-A induces schizophrenia-related endophenotypes[END_REF]. The test procedures consisted of four phases: pre-exposure, conditioning, context test, and tone test. Animals were randomly allocated to either the preexposed (PE) or non-pre-exposed (NPE) condition. The establishment of LI (i.e. difference in freezing between NPE and PE groups) reflects the ability to ignore stimuli that historically predict non-significant consequences. Thus, LI deficiency is a translational model that assesses attentional deficits in animal models of schizophrenia [START_REF] Arguello | Modeling madness in mice: one piece at a time[END_REF][START_REF] Meyer | Towards an immuno-precipitated neurodevelopmental animal model of schizophrenia[END_REF].

Sleep/wakefulness patterns

A battery of evidence shows that schizophrenia is associated with alterations in sleep architecture [START_REF] Keshavan | Schizophrenia, "just the facts": what we know in 2008 Part 3: neurobiology[END_REF][START_REF] Monti | Sleep in schizophrenia patients and the effects of antipsychotic drugs[END_REF]. Sleep architecture and assessment and spectral analysis of the electroencephalogram (EEG) recording were performed as previously described [START_REF] Boutrel | Key role of 5-HT1B receptors in the regulation of paradoxical sleep as evidenced in 5-HT1B knock-out mice[END_REF]. Polygraphic recordings were scored visually every 15 s epoch as wakefulness, non-rapid eye movement (NREM) sleep, or rapid eye movement (REM) sleep, following classical criteria [START_REF] Lena | Beta2-containing nicotinic receptors contribute to the organization of sleep and regulate putative micro-arousals in mice[END_REF][START_REF] Tobler | Sleep and sleep regulation in normal and prion proteindeficient mice[END_REF], using Somnologica®. For analysis of the spontaneous sleep-wakefulness patterns, the amounts of vigilance states for each animal were calculated for every hour throughout 48 h and summed over 24 h. REM sleep latency was calculated as the time elapsing from sleep onset after the animal had been awakened, to the first episode of REM sleep [START_REF] Popa | Contribution of 5-HT2 receptor subtypes to sleep-wakefulness and respiratory control, and functional adaptations in knock-out mice lacking 5-HT2A receptors[END_REF].

Neurochemical Assessments

Dopamine (DA), 3,4-dihydroxyphenylacetic acid (DOPAC) and glutamate tissue concentrations in the PFC and the dSTR of Htr 2B +/+ and Htr 2B -/-mice were assessed ex vivo by means of high-performance liquid chromatography (HPLC) as previously described [START_REF] Banas | Deconstructing antiobesity compound action: requirement of serotonin 5-HT2B receptors for dexfenfluramine anorectic effects[END_REF][START_REF] Pitychoutis | Individual differences in novelty-seeking predict differential responses to chronic antidepressant treatment through sex-and phenotype-dependent neurochemical signatures[END_REF]. A real-time qPCR-based approach was implemented to examine the mRNA expression of relevant key-receptor genes pertaining to the dopaminergic and the glutamatergic neurochemical systems. qPCR was performed as previously described [START_REF] Diaz | 5-HT2B receptors are required for serotonin-selective antidepressant actions[END_REF].

Statistical Analysis

Differences between experimental groups,were analyzed by unpaired t-tests, one-way or twoway analysis of variance (ANOVA) with genotype and treatment as main factors, depending on the experimental design. Bonferroni's post-hoc test was applied in order to elucidate specific differences between groups. Repeated measures ANOVAs were implemented in order to analyze locomotor activity and sleep architecture data. Statistical Analyses of the results are summarized in Supplementary Tables (Table S1-S2). In graphs/tables the values represent means±SEM and p < 0.05 was considered statistically significant.

RESULTS

Deficits in the PPI of the startle reflex

Genetic ablation of the 5-HT 2B receptor induced a global deficit in the PPI of the startle reflex (Fig. 1a) and a reduction in startle amplitude (Fig. S1a). Moreover, acute treatment of Htr 2B +/+ mice with the 5-HT 2B receptor antagonist RS127445, phenocopied the effects of genetic ablation of 5-HT 2B receptor in the PPI (Fig. 1b; Fig. S1b) Enhanced locomotor response to psychostimulants Administration of NMDA receptor antagonists (e.g. dizocilpine) and DA releasers (e.g. amphetamine) exacerbate existing symptoms in schizophrenic patients and cause psychoticlike symptoms in humans and in rodents [START_REF] Arguello | Modeling madness in mice: one piece at a time[END_REF]. As shown herein and before, Htr 2B

-/-mice exhibit enhanced locomotor response to novelty (Fig. 1c) [START_REF] Doly | Serotonin 5-HT2B receptors are required for 3,4-methylenedioxymethamphetamine-induced hyperlocomotion and 5-HT release in vivo and in vitro[END_REF]. The locomotor response of Htr 2B -/-mice to both dizocilpine (Fig. 1d) and amphetamine (Fig. 1e) was significantly enhanced, as compared to Htr 2B +/+ mice. Several other behaviors appear to be completely unaltered in Htr 2B -/-mice, including motor coordination on the Rotarod and nociception assessed in the hot-plate (Fig. S1c,d).

Defect in social interaction

According to our data, both Htr 2B -/-mice and RS-treated Htr 2B +/+ mice displayed dysfunctional social interaction behavior (Fig. 1f), as measured by observing the time mice spent in close interaction with the social partner vs. the inanimate stimulus (i.e. empty wire cage; EC). During the second session, preference for social novelty was evaluated by inserting a novel social stimulus (social II) in the previously empty cage and measuring the time spent interacting with Social I and Social II stimuli (Fig. 1g). All three groups showed preference for social novelty as revealed by the greater % time interacting with Social II vs. Social I (Fig. S1e). Social memory was assessed by comparing the time mice spent in close interaction with Social I stimulus during the two different trials. This analysis showed that genetic ablation of 5-HT 2B receptor impaired social memory, since Htr 2B -/-mice spent similar time interacting with Social I partner during both trials (Fig. 1h). Social memory in both Htr 2B +/+ and Htr 2B +/+ -RS-treated mice appeared to be intact. In addition, social interaction with a juvenile conspecific was also impaired in Htr 2B -/-mice (Fig. S1f). Of note, the differences observed in social interaction could not be attributed to altered olfactory function due to the genetic ablation of 5-HT 2B receptor in mice (Fig. S1g,h).

Memory deficits in the NOR and the fear conditioning 5-HT 2B receptor deficiency induced a significant impairment of both short-(1 h; Fig. 2a) and long-term NOR memory (24 h; Fig. 2b). Similarly, Htr 2B -/-mice exhibited impaired fear learning and memory, as evidenced by the reduced time they spent freezing in response to the context (Fig. 2c) and the cue in a fear conditioning paradigm (Fig. 2d). These cognitive deficits appear to be independent of object exploration behavior or sensitivity to foot shocks. Indeed, Htr 2B -/-and Htr 2B +/+ mice spent similar time exploring the objects during both NOR trials (Fig. S1i) and showed comparable freezing behavior in response to foot shock during training in the fear conditioning paradigm (Fig. S1j).

Attention deficits in the LI paradigm

In the test of the conditioned response to the tone, the presence of LI in Htr 2B

+/+ mice was demonstrated by a clear reduction in the amount of freezing to the tone in pre-exposed (PE) mice relative to the non-pre-exposed (NPE) mice. However, LI was not established in Htr 2B -/mice (Fig. 2e).

Alterations in sleep architecture Htr 2B

-/-mice exhibited increased wakefulness duration at the expense of NREM sleep (Fig. 3a), as well as decreased latency to REM sleep (Fig. 3b). Under baseline conditions, Htr 2B +/+ and Htr 2B -/-mice exhibited the typical polyphasic structure of vigilance states found in rodents and a diurnal rhythm of sleep and wakefulness, with larger amounts of sleep during the light period classically observed in nocturnal species (Fig. 3c-e). However, Htr 2B -/-mice exhibited significantly increased amounts of wakefulness, at the expense of NREM sleep, throughout the dark period and at the onset of light period (Fig. 3c,d) while REM sleep was not affected (Fig. 3e).

Chronic haloperidol treatment rescued selected schizophrenic-like phenotypes in Htr 2B

-/mice Haloperidol was selected herein on the basis of its minor binding capacity to 5-HT 2 receptors including murine 5-HT 2B receptors unlike many other antipsychotics [START_REF] Meltzer | Update on typical and atypical antipsychotic drugs[END_REF]) (Fig. S1k). Chronic oral intake of haloperidol (2mg/kg/day) for 4 weeks restored the balance between wakefulness and NREM sleep (Fig. 3c-e) in Htr 2B -/-mice. Moreover, chronic haloperidol treatment normalized the PPI deficits (Fig. 3f) and the psychomotor agitation (Fig. 3g) conferred by 5-HT 2B receptor deletion. However, haloperidol treatment did not ameliorate the social deficit observed in Htr 2B -/-mice in the 3-chamber social interaction test (Fig. 3h,i).

Neurochemical alterations in Htr 2B

-/-mice In our experimental setup, DA, and glutamate tissue levels were found decreased in the dSTR of Htr 2B -/-mice, while in the PFC no major neurochemical alterations were observed (Fig. 4a,b). Moreover, our results showed that the genetic ablation of the 5-HT 2B receptor resulted in a differential regulation of mRNA expression of DA and glutamate receptors in the PFC and the dSTR. In particular in the dSTR, D 2 R mRNA expression was decreased (Fig. 4c), while GluR2 was upregulated in the PFC (Fig. S2f). It is noteworthy, that chronic haloperidol treatment normalized the observed neurochemical alterations in DA and glutamate concentrations (Fig. 4a,b) and D 2 R receptor expression in the dSTR (Fig. 4c).

DISCUSSION

Herein we investigated the effects of the genetic ablation of 5-HT 2B receptor across a battery of translational behavioral paradigms relevant for assessing face validity of animal models of schizophrenia [START_REF] Arguello | Modeling madness in mice: one piece at a time[END_REF][START_REF] Meyer | Towards an immuno-precipitated neurodevelopmental animal model of schizophrenia[END_REF][START_REF] Willi | Constitutive genetic deletion of the growth regulator Nogo-A induces schizophrenia-related endophenotypes[END_REF]. We report herein that loss of function of 5-HT 2B receptor confers a wide spectrum of schizophrenic-like behavioral and psychopharmacological phenotypes in mice. Importantly, domains related to the positive, negative and cognitive symptom-clusters of schizophrenia appear to be affected upon 5-HT 2B receptor gene ablation (Table 1). Htr 2B -/-mice display a global deficit in sensorimotor gating (i.e. impaired PPI performance) as well as psychomotor agitation (i.e. novelty-induced hyperlocomotion), psychostimulant hypersensitivity (i.e. enhanced locomotor response to psychostimulants), all phenotypes that have been related to the positive symptoms of schizophrenia [START_REF] Gunduz-Bruce | The acute effects of NMDA antagonism: from the rodent to the human brain[END_REF][START_REF] Meyer | Towards an immuno-precipitated neurodevelopmental animal model of schizophrenia[END_REF][START_REF] Pratt | Advancing schizophrenia drug discovery: optimizing rodent models to bridge the translational gap[END_REF]. Moreover, 5-HT 2B receptor gene ablation impaired social interaction behavior with conspecifics, a trait used commonly to model negative symptoms of schizophrenia in rodents [START_REF] Arguello | Modeling madness in mice: one piece at a time[END_REF][START_REF] Pratt | Advancing schizophrenia drug discovery: optimizing rodent models to bridge the translational gap[END_REF]. Most importantly, Htr 2B -/-mice present with selective attention deficits (i.e. lack of LI establishment) and learning and memory impairments (i.e. poor NOR and fear conditioning performance and social memory deficit) that closely resemble the cognitive deficits observed in schizophrenic patients [START_REF] Keshavan | Schizophrenia, "just the facts": what we know in 2008 Part 3: neurobiology[END_REF]. Remarkably, the alterations in sleep architecture observed in Htr 2B -/-mice (i.e. reduced duration of NREM sleep and decreased latency to REM sleep) mimic the sleep abnormalities often seen in schizophrenic patients [START_REF] Keshavan | Schizophrenia, "just the facts": what we know in 2008 Part 3: neurobiology[END_REF]. The list of behavioral paradigms implemented herein, although not exhaustive, is consistent with schizophrenia-like behavioral abnormalities. Moreover, selected schizophrenic-like phenotypes (i.e. PPI and social interaction deficits) were phenocopied in Htr 2B +/+ mice by acute pharmacological ablation of 5-HT 2B receptors with the selective receptor antagonist RS127445.

We next assessed whether selected schizophrenic-like phenotypes could be rescued by chronic antipsychotic treatment. For this reason Htr 2B -/-and Htr 2B +/+ were chronically treated with the typical antipsychotic haloperidol. Of note, chronic antipsychotic treatment completely abolished the observed PPI deficits and alterations in sleep architecture. As expected, haloperidol treatment suppressed psychomotor agitation in Htr 2B -/-mice, but was ineffective against social withdrawal, a negative symptom of schizophrenia. Moreover, chronic haloperidol treatment also had an effect in Htr 2B +/+ mice, normalizing the differences observed between the two genotypes during locomotor activity and social interaction testing.

How 5-HT 2B receptor gene deficiency may lead to the observed schizophrenic-like phenotype can only be speculated at present since the function of this receptor in the brain remains largely uninvestigated. It is noteworthy that 5-HT 2B receptors have been shown to modulate aspects of both serotonergic and dopaminergic neurotransmission [START_REF] Bevilacqua | A populationspecific HTR2B stop codon predisposes to severe impulsivity[END_REF][START_REF] Diaz | Implication of 5-HT2B receptors in the serotonin syndrome[END_REF][START_REF] Doly | Serotonin 5-HT2B receptors are required for 3,4-methylenedioxymethamphetamine-induced hyperlocomotion and 5-HT release in vivo and in vitro[END_REF]. In particular, ex vivo studies have indicated that 5-HT 2B receptors may serve as positive autoregulator of the serotonergic tone through modulating the 5-HT transporter (SERT) in raphe neurons [START_REF] Launay | Serotonin transport and serotonin transporter-mediated antidepressant recognition are controlled by 5-HT2B receptor signaling in serotonergic neuronal cells[END_REF], while in vivo studies in mice further confirmed that 5-HT 2B receptors contribute to the behavioral effects of the SERT-targeting 5-HT releasers, MDMA (i.e. ecstasy) and dexfenfluramine [START_REF] Banas | Deconstructing antiobesity compound action: requirement of serotonin 5-HT2B receptors for dexfenfluramine anorectic effects[END_REF][START_REF] Doly | Role of serotonin via 5-HT2B receptors in the reinforcing effects of MDMA in mice[END_REF][START_REF] Doly | Serotonin 5-HT2B receptors are required for 3,4-methylenedioxymethamphetamine-induced hyperlocomotion and 5-HT release in vivo and in vitro[END_REF]. Recently, Auclair et al. reported that 5-HT 2B receptors control the mesoaccumbal DA pathway activity, as an acute systemic administration of the 5-HT 2B subtype-selective antagonist RS127445 in rats induced a regionselective reduction of extracellular DA levels in the NAC in response to amphetamine challenge [START_REF] Auclair | The central serotonin 2B receptor: a new pharmacological target to modulate the mesoaccumbens dopaminergic pathway activity[END_REF].

In light of the schizophrenic-like phenotype observed herein, the relevant neurochemical alterations found are not surprising. Indeed, the enhanced psychomotor response of Htr 2B -/-mice to dizocilpine and to amphetamine indicates that the glutamatergic and/or the dopaminergic systems must, at least to some extent, be altered in Htr 2B -/-mice. Schizophrenia has been largely associated with dopaminergic hypofunction in the PFC and hyperfunction in the basal ganglia; indeed numerous post-mortem studies report increased D 2 R densities in the dSTR [START_REF] Hirvonen | Increased caudate dopamine D2 receptor availability as a genetic marker for schizophrenia[END_REF][START_REF] Seeman | Schizophrenia: more dopamine, more D2 receptors[END_REF] and decreased D 2 R levels in the PFC of schizophrenic patients [START_REF] Takahashi | The role of extrastriatal dopamine D2 receptors in schizophrenia[END_REF]. Moreover, aberrations in glutamate-mediated neurotransmission through NMDA receptors in schizophrenia is supported by the reduced glutamate levels in the cerebrospinal fluid (CSF) and NMDA receptor expression in the PFC of schizophrenic patients [START_REF] Gordon | Testing the glutamate hypothesis of schizophrenia[END_REF][START_REF] Keshavan | Schizophrenia, "just the facts": what we know in 2008 Part 3: neurobiology[END_REF]. In our experimental setup, DA and glutamate tissue concentrations and D 2 R mRNA expression were found decreased in the dSTR of Htr 2B -/-mice. In accordance with our findings, DA tissue concentrations were also found decreased in the dSTR of Nogo-A -/-mice that also present with a schizophrenic-like phenotype [START_REF] Willi | Constitutive genetic deletion of the growth regulator Nogo-A induces schizophrenia-related endophenotypes[END_REF]. Intriguingly, the observed dopaminergic and glutamatergic alterations in the dSTR were corrected upon chronic antipsychotic treatment. The latter finding indicates that both D 2 R-related and unrelated defects in Htr 2B -/-mice are reversed by haloperidol treatment and provide evidence that the effects of 5-HT 2B receptors extend beyond the serotonergic and the dopaminergic systems.

Acute pharmacological ablation of 5-HT 2B receptors (i.e. RS127445 treatment) phenocopied both positive (i.e. PPI deficit) and negative (i.e. impaired social interaction) symptoms observed in Htr 2B -/-mice. Moreover, in an earlier study, administration of another 5-HT 2B receptor-selective antagonist (i.e. SB-215505) was shown to increase wakefulness at the expense of NREM and REM sleep in rats [START_REF] Kantor | Increased wakefulness, motor activity and decreased theta activity after blockade of the 5-HT2B receptor by the subtypeselective antagonist SB-215505[END_REF] and in another study it was concluded that 5-HT exerts a 5-HT 2B receptor-mediated facilitation of NREM sleep [START_REF] Popa | Contribution of 5-HT2 receptor subtypes to sleep-wakefulness and respiratory control, and functional adaptations in knock-out mice lacking 5-HT2A receptors[END_REF], which are in accordance with our findings in Htr 2B -/-mice. Thus, based on these observations, we propose that the schizophrenic-like phenotype of Htr 2B -/-mice results from a combination of both the direct absence of 5-HT 2B receptor signaling and the neural adaptations triggered by the permanent lack of this receptor.

Antipsychotic drugs ameliorate hallucinations and delusions in patients with neuropsychiatric disorders, particularly schizophrenia and bipolar disorders. The two main classes are known as typical and atypical antipsychotics. Many atypical antipsychotics show similar affinity for 5-HT 2B and 5-HT 2A receptors (e.g. clozapine, amisulpride, asenapine) [START_REF] Abbas | Amisulpride is a potent 5-HT7 antagonist: relevance for antidepressant actions in vivo[END_REF][START_REF] Shahid | Asenapine: a novel psychopharmacologic agent with a unique human receptor signature[END_REF] or even higher affinity for 5-HT 2B receptors (e.g. aripiprazole, cariprazine) [START_REF] Kiss | Cariprazine (RGH-188), a dopamine D(3) receptor-preferring, D(3)/D(2) dopamine receptor antagonist-partial agonist antipsychotic candidate: in vitro and neurochemical profile[END_REF][START_REF] Shapiro | Aripiprazole, a novel atypical antipsychotic drug with a unique and robust pharmacology[END_REF]. Notably, it was recently shown that the efficacy of clozapine, but not haloperidol, is diminished in Pet1 -/-mice, that lack 5-HT neurons, and thus depends on an intact presynaptic serotonergic system [START_REF] Yadav | The presynaptic component of the serotonergic system is required for clozapine's efficacy[END_REF]. Given that many marketed atypical antipsychotic drugs present high affinity for 5-HT 2B receptors [START_REF] Shapiro | Aripiprazole, a novel atypical antipsychotic drug with a unique and robust pharmacology[END_REF] and that these receptors are expressed in 5-HT neurons [START_REF] Diaz | 5-HT2B receptors are required for serotonin-selective antidepressant actions[END_REF], our findings bear broader significance for the elucidation of the effects of these drugs in the treatment of psychotic disorders.

It should be borne in mind that the behavioral phenotype of Htr 2B -/-mice portrayed herein is not exclusive of a putative role of the 5-HT 2B receptor in the neurobiology of a broader range of neuropsychiatric diseases that share common domains of dysfunction with psychotic disorders. Interestingly, it was recently reported that two polymorphisms of the HTR2B gene were associated with intelligence quotation, intellectual disability and language onset delay in a cohort of children and young adults suffering from autistic spectrum disorders [START_REF] Hervas | The involvement of serotonin polymorphisms in autistic spectrum symptomatology[END_REF]. In agreement, present data strongly support a novel role for 5-HT 2B in the regulation of cognitive function and social behavior. Despite the fact that Htr 2B -/-mice display deficits in social interaction and in learning and memory processes that closely resemble autistic-like behavior, they also present with deficits in sensorimotor gating and locomotor hypersensitivity to psychostimulants which are core endophenotypes of schizophrenia [START_REF] Arguello | Modeling madness in mice: one piece at a time[END_REF][START_REF] Powell | Prepulse inhibition and genetic mouse models of schizophrenia[END_REF].

Overall, the present study revealed that 5-HT 2B receptor deficiency induces a wide spectrum of antipsychotic-sensitive schizophrenic-like phenotypes in mice. Most importantly, further pharmacological validations with selective 5-HT 2B receptor antagonists (given acutely or chronically) that extend present findings to other behavioral tests, mouse strains and species, or other brain regions and neurochemical systems are clearly warranted to further dissect the role of the 5-HT 2B receptor in the neurobiology of psychotic spectrum disorders. This genetic mouse model holds additional value in further elucidating the schizophrenia-relevant neurodevelopmental, epigenetic and physiological mechanisms that may be sensitive to 5-HT 2B receptor polymorphisms. +/+ mice as determined by unpaired t-test, one way ANOVA or 2--way ANOVA; full statistical analysis is presented in Table S1. +/+ mice following unpaired t-test or 2--way ANOVA; #p < 0.05 significantly different from NPE in 2--way ANOVA; full statistical analysis is presented in Table S1.

Figure 3: In Htr 2B

-/-mice across 24 h, the total amount of wakefulness is increased while that of NREM sleep is decreased (a), as is the latency to REM sleep, calculated as the time elapsing from sleep onset after the animal had been awakened to the first episode of REM sleep (b) (N=4-5 per group). Htr 2B -/-mice exhibit significantly increased amounts of wakefulness at the expense of NREM sleep throughout the dark period and at the onset of the light period as assessed by the amount of vigilance states, expressed as minutes per 3 h (c,d) but REM sleep is not affected (e). Chronic oral haloperidol treatment (Hal, 2 mg/kg/day during 4 weeks) normalizes sleep-wakefulness states across the light/dark cycle (N=4-5 per group) (c-e). The effects of chronic oral haloperidol treatment were also tested in the prepulse inhibition (f) and locomotor activity (g) (N=8-10 per group); social interaction was assessed in the 3-chamber sociability test by comparing the time Htr 2B +/+ (h) and Htr 2B -/-mice (i) spent interacting with a novel mouse (Soc I) vs. an empty wire cage (EC; N=7-9 per group). *p < 0.05; **p < 0.01; ***p < 0.001 significantly different from the Veh-treated Htr 2B +/+ mice; # p < 0.05; #### p < 0.0001 significantly different from the Veh-treated Htr 2B -/-mice; full statistical analysis is presented in Table S1. -/-mice in 2--way ANOVA; full statistical analysis is presented in Table S1.

Figure 1 :

 1 Figure 1: Prepulse inhibition (PPI) is impaired in Htr 2B-/-mice across prepulse intensities over 66db and in average of all intensities (N=27 per group) (a). Acute injection of RS127445 (0.5 mg/kg), a selective 5-HT 2B receptor antagonist (RS) also impaired PPI in Htr 2B +/+ mice (N=8 per group) (b). Genetic ablation of the 5-HT 2B receptor enhanced novelty-induced locomotion during the first 10 min of testing (N=8 per group) (c). Genetic blockade of the 5-HT 2B receptor enhanced locomotor responses to the glutamatergic psychostimulant and non-competitive NMDA receptor antagonist dizocilpine (MK-801; 0.5 mg/kg; i.p.) (N=8 per group) (d) and to the dopaminergic psychostimulant amphetamine (10 mg/kg; i.p.) (N=8 per group) (e). Genetic and pharmacological ablation of the 5-HT 2B receptor impaired social interactions, as evidenced by the lack of preference for the social partner (social I vs. the empty wire cage; EC) (N=7-9 per group) (f). Htr 2B +/+ mice spent significantly more time interacting with the novel social stimulus (Social II vs. Social I), but this was not the case for Htr 2B -/-mice or Htr 2B +/+ -RStreated mice (N=7-9 per group) (g). Genetic ablation of 5-HT 2B receptor impaired social memory, since Htr 2B -/-mice spent similar time interacting with Social I partner during both trials, whereas both Htr 2B +/+ and Htr 2B +/+ -RS-treated mice spent significantly less time interacting with Social I partner in the second trial (N=7-9 per group) (h). *p < 0.05, **p < 0.01, ***p < 0.001, ****p < 0.0001 significantly different from vehicle-treated Htr 2B +/+ mice as determined by unpaired t-test, one way ANOVA or 2--way ANOVA; full statistical analysis is presented in TableS1.

Figure 2 :

 2 Figure 2: Htr 2B -/-mice present with impaired short-(a) and long-term (b) memory for a novel object, as assessed in the NOR test (N=8 per group). Contextual (c) and cued (d) fear memory is impaired in Htr 2B -/-mice (N=6-7 per group). In the latent inhibition (LI) paradigm, LI was established in Htr 2B +/+ but not in Htr 2B -/-mice as indicated by the absence of difference in freezing between NPE and PE groups (N=7 per group) *p < 0.05, **p < 0.01, ***p < 0.001, ****p < 0.0001 significantly different from Htr 2B+/+ mice following unpaired t-test or 2--way ANOVA; #p < 0.05 significantly different from NPE in 2--way ANOVA; full statistical analysis is presented in TableS1.

Figure 4 :

 4 Figure 4: Region-distinctive neurochemical profile defects in the prefrontal cortex (PFC-left) and the dorsal striatum (dSTR-right) of Htr 2B +/+ (black circle) and Htr 2B -/-(open square) mice. Htr 2B -/-mice present lower tissue concentrations of DA (a) and glutamate (b) in the dSTR (N=6-8 per group). Genetic ablation of the 5-HT 2B receptor was associated with a downregulation of D 2 R mRNA expression in dSTR (c) (N=4-5 per group). Neurochemical alterations in the dSTR are effectively reversed upon chronic haloperidol treatment (2 mg/kg/day during 4 weeks) (Hal) (a-c). *p < 0.05; **p < 0.01; significantly different from Veh-treated Htr 2B +/+ mice in unpaired t-test or 2--way ANOVA; #p < 0.05, ###p < 0.001 significant effect from Veh-treated Htr 2B-/-mice in 2--way ANOVA; full statistical analysis is presented in TableS1.

Table 1 Schizophrenia-like behavioral phenotypes in Htr 2B -/-mice Functional domain Factor tested Htr 2B -/-

 1

	Reference

Acknowledgements

The authors would like to thank Silvina L. Diaz for help with mice colony, Véronique Fabre for support in sleep experiments, and Cédric Mombereau for advices in prepulse inhibition experiments.

Funding and Disclosure

This work has been supported by funds from the Centre National de la Recherche Scientifique, the Institut National de la Santé et de la Recherche Médicale, the Université Pierre et Marie Curie, and by grants from the Fondation de France, the Fondation pour la Recherche Médicale "Equipe FRM DEQ2014039529", the French Ministry of Research (Agence Nationale pour la Recherche ANR-12-BSV1-0015-01 and the Investissements d'Avenir program managed by the ANR under reference ANR-11-IDEX-0004-02. LM's team is part of the École des Neurosciences de Paris Ile-de-France network and of the Bio-Psy Laboratory of Excellence. P.M.P. was supported by a DIM NERF fellowship from the "Région Ile-de-France" and from start-up funding from the University of Dayton. Authors report no biomedical financial interests or potential conflicts of interest.


Pascal Giorgi
email: pascal.giorgi@lirmm.fr

Bastien Vialla
email: bastien.vialla@lirmm.fr

Generating Optimized Sparse Matrix Vector Product over Finite Fields

Keywords: sparse linear algebra, finite fields, SpMV

Sparse Matrix Vector multiplication (SpMV) is one of the most important operation for exact sparse linear algebra. A lot of research has been done by the numerical community to provide efficient sparse matrix formats. However, when computing over finite fields, one need to deal with multi-precision values and more complex operations. In order to provide highly efficient SpMV kernel over finite field, we propose a code generation tool that uses heuristics to automatically choose the underlying matrix representation and the corresponding arithmetic.

Introduction

Modern sparse linear algebra is fundamentally relying on iterative approaches such as Wiedemann or Lanczos. The main idea is to replace the direct manipulation of a sparse matrix with its Krylov subspace. In such approach, the cost is therefore dominated by the computation of the Krylov subspace, which is done by successive matrix-vector products. Let A ∈ F n×n be a sparse matrix with O(n log o (1) n) non zero entries where F is a finite field. The matrix-vector product y = Ax where y, x ∈ F n costs O(n log o(1) n) operations in F. We call this operation SpMV in the rest of this paper. SpMV is a particular operation in the linear algebra framework, since it requires as much memory accesses as arithmetic operations. Basically, one entry in the matrix contributes to the SpMV computation only once. Therefore, on modern processor where memory hierarchy has a larger impact then arithmetical operations, data access is a major challenge to reach good performances. This challenge has been widely studied by the numerical community, and led to many different matrix storage for floating point numbers.

Over finite fields the situation is slightly different. Basically, the underlying arithmetic is more complex. Indeed, modern processor does not provide efficient support for modular operations. Furthermore, finite fields can be large and then requiring multiple precision arithmetic. Our main concern is weather the numerical formats are still satisfying when computing over finite fields and which arithmetic strategy is the most suited to the particularity of SpMV. This question has been already addressed in many papers, as in [START_REF] Boyer | Exact sparse matrix-vector multiplication on gpu's and multicore architectures[END_REF][START_REF] Jeljeli | Accelerating iterative spmv for discrete logarithm problem using gpus[END_REF], but no general optimization approach has been designed. In this paper, we propose a general framework which incorporates most of the optimization techniques for SpMV over finite field. We provide a software tool, available at www.lirmm.fr/ ~vialla/spmv.html, that emphasizes our approach for prime fields at any precision.

General optimization approach

It is well know that SpMV performance is limited by memory accesses. Indeed, the irregularity access of the x[i]'s during SpMV does not allow the processor to prefetch data to the cache memory in advance. In order to minimize cache misses, one need to minimize the memory footprint of the matrix while preserving a cache aware structure. To further speed-up SpMV over finite field, one need to minimize the number of clock cycle per arithmetic operation. This could be done by minimizing the modular reductions or taking care of particular entries in the matrix, i.e. ones and minus ones. One can evaluate a priori the impact of these optimizations by using the roofline model of [START_REF] Williams | Roofline: An insightful visual performance model for multicore architectures[END_REF].

Preprocessing the matrix in advance is a key tool to detect the most suited optimization. This can be done at runtime, as in the OSKI library [START_REF] Vuduc | OSKI: A library of automatically tuned sparse matrix kernels[END_REF]. Our proposed approach is to do this at compile time through two steps: 1) preprocessing the matrix to provide an optimization profile.; 2) generating an optimized SpMV with this profile.

Optimized SpMV generator

The workflow of our generator is given in Figure 1. It receives a matrix A and a prime number p such that the SpMV with A is performed over Fp. Depending on the prime p and some characteristics of A, such as the number of ±1 or the dimension, the generator choose the best suited matrix format and an arithmetic strategy. Then, it generates an optimization profile that can be used to compile a SpMV implementation for A over Fp.

Matrix formats

A lot of matrix format have been proposed by the numerical community for SpMV: e.g. CSR, COO, BlockCSR [START_REF] Vuduc | OSKI: A library of automatically tuned sparse matrix kernels[END_REF], Compress Sparse Block [START_REF] Buluç | Reduced-bandwidth multithreaded algorithms for sparse matrix-vector multiplication[END_REF]. In general, matrices over finite fields do not have any structural properties than can be used to improve performances. Therefore, we choose to focus on the CSR and SELL-C format [START_REF] Kreutzer | A unified sparse matrix data format for modern processors with wide simd units[END_REF], and some adaptation avoiding the storage of ±1. However, our approach is generic and more format can be added if necessary.

The CSR format compress data according to the row indices. It needs three arrays: val[] for the matrix entries; idx[] for the column indices; ptr[] for the number of non-zero entries per row. The SELL-C format is a variant of CSR which is designed to incorporate SIMD operations. It sorts the rows according to their sparsity and split the matrix by chunk of size C. Each chunk is padded with zeros such that each row in a same chunck has exactly the same sparsity. The parameter C is a chosen to be a multiple of the SIMD unit's width. In order to minimize the memory footprint, our generator adapt the data types of every arrays, e.g. 4 bytes for idx[] when column dimension is < 65 536. It also choose a data type related to ||A|| ∞ rather than p for val[].

Delayed modular reduction

As demonstrated in [START_REF] Dumas | Dense Linear Algebra over Finite Fields: the FFLAS and FFPACK package[END_REF], performing modular reductions only when necessary leads to better performances. Hence, the computation is relaxed over the integer and needs that no overflow occurs.

Depending on the finite field, our generator will compute a priori the maximum value of k such that k(p -1)||A|| ∞ does not overflow. Note that knowing k at compile time will allow the compiler to perform loop unrolling. As delayed modular reduction is fundamentally tied with the underlying data type, our approach is to use the best suited one to reduce the number of modular reductions. Nevertheless, some compromises must be done between the cost of the standard operations (+, ×) vs the number of reductions.

Hybrid

In most applications over finite fields, many matrix entries are ±1. Thus, one can avoid superfluous multiplication within SpMV and further reduce the memory footprint of the matrix. This approach have been developed in [START_REF] Boyer | Exact sparse matrix-vector multiplication on gpu's and multicore architectures[END_REF] using a splitting of the matrix A in 3 matrices: A 1 storing only 1's , A -1 storing -1's and finally, A λ store the rest of the entries. SpMV is then computed independently for each matrices and the results are sum up, i.e.

y = Ax = A 1 x + A -1 x + A λ x.
The drawback of this method is to amplify the number of cache misses arising during the reading of the vector x. Indeed, most of the matrices have a spatial locality in their row entries which is useful to avoid cache misses.

Our proposed hybrid approach is to keep this spatial locality such that SpMV still can be performed row by row. CSR format is well designed for this approach. Indeed, for each row we can store in idx[], the column indices of the entries different ±1, then the indices of ones and the minus ones. We can do exactly the same for ptr[], and val[] only stores the entries different from ±1. We call this hybrid format CSRHyb. Note this approach, cannot be directly applied to SELL-C since the zero padding may introduce too much memory overhead.

To circumvent this, one must store entries different from ±1 in SELL-C format and the ±1 in CSR format, but these two formats must be interleaved by row.

Benchmarks

Our benchmarks have been done on matrices arising in mathematical applications. The Table 1 gives the characteristic of such matrices (available at http://hpac.imag.fr). In this table, nnz is the number of non zero entries, nnz row is the average number of non zero entries in a row, and k max is the maximum number of non zero entries in a row. We used g++ 4.8.2 and an Intel bi-Xeon E5-2620, 16GB of RAM for our benchmarks. We performed a comparison with the best SpMV available at this time in the LinBox library1 (rev 4901) based on the work of [START_REF] Boyer | Exact sparse matrix-vector multiplication on gpu's and multicore architectures[END_REF].

Prime field F p with small p

In this section we consider the case where (p -1) 2 fits the mantissa of a double floating point number, e.g. 53 bits. In this case, the modular reduction is costly compare to the standard operations. However, it does not worth it to extend the precision beyond 53 bits to avoid most of the reductions. Our strategy is then to use double and to find the largest k such that k||A|| ∞ (p -1) < 2 53 and perform reduction at least every k entries in a row. If the matrix does not have too many ±1, the SELL-C format will be chosen to better exploit the SIMD vectorization, otherwise the CSRHyb format will be preferred.

The Figure 2 gives the relative performances of our optimized SpMV against the one of LinBox for the prime field F 1048583 . One can see that our code is always faster than the CSR implementation of LinBox, up to a speed-up of 2.2.

Indeed, this can be explain by the fact that most matrices have a many ±1 entries and that the CSR of LinBox is not handling such particularity. The implementation of the hybrid format from [START_REF] Boyer | Exact sparse matrix-vector multiplication on gpu's and multicore architectures[END_REF] is not yet fully operational in LinBox and we did not get the chance yet to compare to it. However, following the speed-up of the hybrid format vs the CRS one given in [3, Figure 3], which is less than 1.5, we are confident in the performance of our optimized SpMV.

Prime field F p with p multiple precision

Our motivations come primarily from the computation of discrete logarithms over finite fields [START_REF] Barbulescu | Discrete logarithm in GF(2 809 ) with FFS[END_REF]. We focus only on matrices which have small entries compare to the prime p, e.g. less than a machine word, since it is mostly the case in mathematical applications.

In order to compute with multiple precision integers, one can use the well known GMP library2 which is the fastest one for each single arithmetic operations. However, when dealing with vectors of small integers, e.g. ≈ 1024 bits, the GMP representation through pointer makes it difficult to exploit cache locality. In such a case, one should prefer to use a fixed precision representation through a residue number system [START_REF] Bernstein | Multidigit modular multiplication with the explicit chinese remainder theorem[END_REF], called RNS for short. Such approach provides intrinsic data locality and parallelism which are good for SpMV and its SIMD vectorization. The difficulty is then transferred to the reduction modulo p that cannot be done in the RNS basis. However, one can use the explicit chinese remainder theorem [START_REF] Bernstein | Multidigit modular multiplication with the explicit chinese remainder theorem[END_REF][START_REF] Jeljeli | Accelerating iterative spmv for discrete logarithm problem using gpus[END_REF] to provide a reduction that can use SIMD instructions. Furthermore, one can use matrix multiplication to perform modular reduction of a vector of RNS values and then better exploit data locality and SIMD.

In order to minimize the memory footprint of the matrix, we propose to store the matrix entries in double and convert them to RNS on the fly. The m i 's are chosen so that (m i -1) 2 fits in 53-bits to allow floating point SIMD in the RNS arithmetic. Larger m i 's could be chosen to reduce the RNS basis but this would induce a more complex SIMD vectorization which makes it harder to reach sustainable performances. In our multiple precision SpMV, the m i 's are chosen so that k max (m i -1) 2 < 2 53 and M = i m i > (p -1) • k max • ||A|| ∞ . This both ensures that the modular reduction mod m i and mod p can be done only once per row. The matrix format is CSRHyb since the SIMD vectorization is done over the arithmetic rather then being on SpMV operations.

Fig. 1 .

 1 Fig. 1. Generator workflow

Table 1 .

 1 List of matrices arising in mathematical applications

	Name	Dimensions	nnz nnzrow kmax ±1 = ±1 Problems
	cis.mk8-8.b5 564 480 × 376 320 3.3M	6	6 3.3M 0	(A)
	GL7d17 1 548 650 × 955 128 25M	16	69 25M 382K	(B)
	GL7d19 1 911 130 × 1 955 309 37M	19	121 36M 491K	(B)
	GL7d22	349 443 × 822 922 8.2M 23	403 7M 307K	(B)
	M06-D9 1 274 688 × 1 395 840 9.2M	7	10 9.2M 0	(E)
	rel9	5 921 785 × 274 669 23M	3	4 23M 19K	(C)
	relat9	9 746 231 × 549 336 38M	3	4 23M 29K	(C)
	wheel 601	902 103 × 723 605 2.1M	2	602 38M 29K	(F)
	ffs619	653 365 × 653 365 65M 100 413 60M 5M	(D)
	ffs809	3 602 667 × 3 602 667 360M 100 452 335M 25M	(D)

(A) Simplicial complexes from homology; (B) Differentials of the Voronoï complex of the perfect forms (C) Relations;(D) Function field sieve ; (E) Homology of the moduli space of smooth algebraic curves Mg,n ; (F) Combinatorial optimization problems.

This work has been supported by the Agence Nationale pour la Recherche under Grants ANR-11-BS02-013 HPAC, ANR-12-BS02-001 CATREL.

www.linalg.org

https://gmplib.org/

The Figure 3 gives the relative performances of our optimized SpMV against the one of LinBox for prime fields of bitsize, 64, 128, 256, 512, and 1024. One can see that our code is always faster than LinBox one, up to a speed-up of 67. Indeed, this can be explain by the fact that most matrices have many ±1 entries. But, mainly because LinBox stores matrix entries as GMP integers while our SpMV stores them as double. This has two consequences on LinBox SpMV performances. First, matrix entries are not store contiguously and then many cache misses are done. Secondly, LinBox use the arithmetic of GMP which is not using any SIMD vectorizations for mixed precision arithmetic.