Joseph D Buxbaum

Patrick R Hof

Catalina Betancur
email: catalina.betancur@inserm.fr

Mary Coleman

The Neuroscience of Autism Spectrum Disorders

Keywords: autism, genetic syndrome, intellectual disability, epilepsy, metabolic disorder, mutation, copy number variation, deletion, duplication intellectual

Autism spectrum disorders (ASD) encompass a group of behaviorally defined developmental disabilities characterized by marked clinical and etiological heterogeneity. There is increasing evidence that ASD can arise from rare highly penetrant mutations and genomic imbalances. There are at present over 100 disease genes and 50 recurrent genomic imbalances implicated in the etiology of ASD. These genes and loci have so far all been causally implicated in intellectual disability, indicating that these two neurodevelopmental disorders share common genetic bases. Similarly, many genes involved in epilepsy can also result in ASD. These observations indicate that these genes cause a continuum of neurodevelopmental disorders that manifest in different ways depending on other genetic, environmental or stochastic factors. Increased recognition of the etiological heterogeneity of ASD will expand the number of target genes for neurobiological investigations, reveal functional pathways and assist the development of novel therapeutic approaches.

Introduction

Autism spectrum disorders (ASD) encompass a group of behaviorally defined developmental disabilities characterized by marked clinical and etiological heterogeneity. ASD can be associated with the profound etiological heterogeneity underlying ASD.

The most obvious conclusion we can draw is that, when examined from an etiological perspective, ASD is not a single disease entity but a behavioral manifestation of many hundreds of single gene and genomic disorders. In addition, it is emerging that de novo variants are an important part of the architecture of ASD, consistent with purifying selection against deleterious genetic variants of major effect. One of the most important observations is that there is considerable overlap in high--risk genes and loci for ASD, ID, and epilepsy. Similarly, many of the rare recurrent CNVs identified recently have been found to confer risk for a broad range of neurologic and psychiatric phenotypes, including not only ID, ASD, and epilepsy, but also schizophrenia and attention deficit hyperactivity disorder (ADHD). This highlights how disruption of core neurodevelopmental processes can give rise to a wide range of clinical manifestations and that greater attention should be placed on the neurobiological processes of brain development and function rather than on the precise behavioral manifestation. Finally, we show how some of the genes implicate specific pathways, subcellular organelles, or systems in the pathophysiology of ASD, which can lead to biological and neurobiological insights into disease mechanisms. Genetic disorders strongly associated with ASD Table 1 shows genetic and genomic disorders in which ASD is a common manifestation. For some of these disorders, ASD is among the clinical hallmarks, including Phelan--McDermid syndrome (22q13 deletion syndrome/SHANK3 mutations), maternal 15q11--q13 duplications, Rett syndrome (MECP2) and MECP2 duplication syndrome, fragile X syndrome (FMR1), tuberous sclerosis (TSC1, TSC2), adenylosuccinate lyase deficiency (ADSL), Timothy syndrome (CACNA1C), cortical dysplasia--focal epilepsy syndrome (CNTNAP2), Smith--Lemli--Opitz syndrome (DHCR7), Smith--Magenis syndrome (17p11.2 deletion, RAI1 mutations), and Potocki--Lupski syndrome (17p11.2 duplication) (see Table 1 for references). Another disorder strongly associated with ASD is the recently described 2q23.1 microdeletion syndrome, caused by haploinsufficiency of the methyl--CpG--binding domain 5 (MBD5) gene. An analysis of 65 individuals with deletions or translocations involving MBD5 reported that all had "autistic--like" behaviors [START_REF] Talkowski | Assessment of 2q23.1 microdeletion syndrome implicates MBD5 as a single causal locus of intellectual disability, epilepsy, and autism spectrum disorder[END_REF]. If these findings were confirmed using standardized diagnostic assessments, this would constitute the first genetic disorder exhibiting fully-penetrant ASD. However, this appears unlikely, given that none of the disorders implicated in ASD to date are associated with ASD in 100% of cases, reflecting the variable expressivity of many genetic conditions.

Other disorders with common ASD manifestations are brain creatine deficiency (SLC6A8, GAMT, GATM), Cornelia de Lange syndrome (NIPBL, SMC1A), CHARGE syndrome (CHD7), Cohen syndrome (VPS13B), Joubert syndrome and related syndromes (AHI1, NPHP1, CEP290, RPGRIP1L), myotonic dystrophy type 1 (DMPK), X--linked female--limited epilepsy and ID (PCDH19), 2q37 deletion syndrome, Cri du Chat syndrome (5p deletion), Williams syndrome (7q11.23 deletion), 7q11.23 duplication syndrome, 8p23.1 deletion syndrome, WAGR syndrome (11p13 deletion), Angelman syndrome (maternal 15q11--q13 deletion), 16p11.2 microdeletion, and 22q11 deletion syndrome (velocardiofacial/DiGeorge syndrome).

In other disorders, ASD appear to be somewhat less frequent but still much higher than in the general population, such as in PTEN related syndromes, Kleefstra syndrome (9q subtelomeric deletion syndrome/EHMT1 mutations), Prader--Willi syndrome (paternal 15q11--q13 deletion), 15q24 microdeletion syndrome, and 16p11.2 microduplication. Finally, certain chromosomal aneuploidies are associated with an increased risk for ASD, including Down syndrome, Klinefelter syndrome (XXY), XYY syndrome, and XXYY syndrome.

Note that for most genetic disorders, no reliable estimates of the frequency of ASD among affected individuals or the frequency of the disorder among patients with ASD are available. Even in disorders for which such studies have been conducted, the samples are usually quite small and few are population--based. While it is assumed that these genetic syndromes are rare, some could be underdiagnosed, since only a minority of patients with ASD has been screened for most of these conditions. Several genetic disorders have been described only recently and their prevalence is unknown. Furthermore, the methods employed to diagnose ASD in these studies are very variable, and in some instances no standardized diagnostic assessments were used. Clearly, more data is needed on the prevalence of specific genetic disorders in ASD, and of ASD in genetic disorders, using reliable diagnostic assessment tools in large samples. The frequencies cited in Table 1 should serve to give an idea of the association between ASD and certain genetic disorders but should not be considered precise. Most of the disorders associated with a high risk for ASD are rare or very rare; apart from fragile X syndrome (∼2%), only a few account for at most ∼0.5%-1% of ASD cases (Table 1).

Genetic overlap between ASD and intellectual disability

Like ASD, ID is a common and highly heterogeneous neurodevelopmental disorder, affecting 2%-3% of the population. Like in ASD, chromosomal abnormalities detected with conventional karyotyping account for about 5% of cases of ID, while novel microarray--based methods have a diagnostic yield of 10%--15%, underscoring the major role of submicroscopic CNVs as causes of ID. Down syndrome (trisomy 21) is the most frequent chromosomal cause of ID, and has also been identified as a relatively frequent cause of autism in several epidemiological studies (Table 1). The most common single--gene defect in male patients with ID is fragile X syndrome, with full mutations identified in 2.6% of patients; the combined frequency in males and females with ID is 2% [START_REF] Michelson | Evidence report: Genetic and metabolic testing on children with global developmental delay: report of the Quality Standards Subcommittee of the American Academy of Neurology and the Practice Committee of the Child Neurology Society[END_REF], like in ASD. In females with moderate to severe ID, MECP2 testing is diagnostic in 1.5% [START_REF] Michelson | Evidence report: Genetic and metabolic testing on children with global developmental delay: report of the Quality Standards Subcommittee of the American Academy of Neurology and the Practice Committee of the Child Neurology Society[END_REF]. At least 93 genes have been identified that are implicated in X--linked ID; 52 are associated with syndromic ID, while 41 genes have been found to be associated with nonsyndromic ID (Figure 1) [START_REF] Gecz | The genetic landscape of intellectual disability arising from chromosome X[END_REF][START_REF] Ropers | Genetics of early onset cognitive impairment[END_REF]. The distinction between syndromic and nonsyndromic ID is not precise, and many genes, initially identified in syndromic conditions, were later reported in subjects with nonsyndromic forms. Among the 93 genes involved in X--linked ID, 45 have also been implicated in ASD (Figure 1), demonstrating the profound etiologic overlap between these phenotypes. In addition, numerous autosomal genes, either due to dominant, usually de novo mutations or to recessive gene defects, have been implicated in ID (and ASD), but many more remain unidentified.

Table 2 shows several recently identified recurrent microdeletions and microduplications reported in individuals with ID, ASD and other neurodevelopmental or neuropsychiatric disorders (Chapter 2.2). Some of these novel recurrent CNVs have a recognizable phenotype, such as the 17q21.31 microdeletion syndrome, with a distinctive facial dysmorphism. Others, such as CNVs at 1q21. 1, 15q13.3, 16p13.11 and 16p11.2, give rise to less consistent phenotypes (variable expressivity) and have been identified in cohorts of patients ascertained for ID, epilepsy, ASD, or schizophrenia, blurring the current nosological boundaries of these disorders. Several of these aberrations show incomplete penetrance, as demonstrated by their presence in clinically unaffected relatives and in controls. These CNVs have been studied in very large samples of subjects with various neurocognitive and neuropsychiatric conditions, and there appears to be a clear increased frequency in affecteds versus controls for some of them, suggesting that they act as risk factors; for other CNVs, particularly those that appear to be relatively more frequent in controls, the clinical significance is still uncertain (e.g., 15q13.3 and 16p13.11 duplications).

When reviewing these studies, it is clear that not all 'intellectual disability genes and loci' are necessarily associated with ID. As shown in Box 1, several genetic and genomic disorders have been reported in individuals with higher function ASD (Asperger syndrome). Similarly, not all genetic defects involved in the etiology of ID and ASD are identified in individuals presenting with marked dysmorphic features or other congenital malformations. In fact, many disease genes implicated in ASD can be associated with nonsyndromic presentations (Box 2).

It should be clear when looking at the genetic and genomic disorders for which ASD is a manifestation that variable expressivity is the rule rather than the exception, and none will invariably present with ASD. This point is important to consider from a neurobiological perspective. There is, for example, an emphasis on studying ASD--like behaviors in rodent and primate models of ASD; if mutations in the underlying genes do not reliably lead to ASD in humans, other intermediate neurobiological phenotypes are perhaps equally or even more relevant to understanding disease pathogenesis (Chapter 4).

Genetic overlap between ASD and epilepsy

Epilepsies are common and etiologically heterogeneous disorders, affecting up to 3% of the population. About 30% of children with epilepsy have ASD, and conversely, epilepsy is observed in about a third of ASD individuals. Many well--known genetic disorders share ID, ASD, and epilepsy as prominent phenotypic features, including fragile X syndrome, tuberous sclerosis, Rett syndrome and Angelman syndrome. In addition, monogenic forms involving mutations in genes encoding voltage-gated or ligand--gated ion channels, referred to as "channelopathies" have been identified in epilepsy, and increasingly in ASD, such as the neuronal voltage--gated sodium channel genes SCN1A and SCN2A (Table 3). Both genes have been implicated in various forms of epilepsy, including early--onset epileptic encephalopathies. This group of severe epilepsies is characterized by progressive intellectual deficits or regression, and includes West syndrome (infantile spasms), Dravet syndrome (severe myoclonic epilepsy of infancy), and Ohtahara syndrome (early infantile epileptic encephalopathy with burst-suppression) [START_REF] Mastrangelo | Genes of early--onset epileptic encephalopathies: from genotype to phenotype[END_REF]Paciorkowski et al., 2011). Table 3 shows several genes involved in early infantile epileptic encephalopathies that can also manifest with ASD (e.g., ARX, CDKL5, MECP2, MEF2C, FOXG1, STXBP1, and PCDH19).

Like MECP2, mutations in the X--linked cyclin--dependent kinase--like 5 (CDKL5) gene are more common in girls and are associated with a Rett--like phenotype with infantile spasms and ID; several cases have been described with autism (Table 3). Another X--linked gene, protocadherin 19 (PCDH19), was recently implicated in "epilepsy and mental retardation limited to females", a familial disorder with an unusual mode of inheritance, since only heterozygous females are affected and transmitting males are asymptomatic. PCDH19 mutations, mostly occurring de novo, have also been shown to be a frequent cause of sporadic infantile--onset epileptic encephalopathy in females, and have been reported in females with epilepsy without cognitive impairment [START_REF] Depienne | PCDH19--related infantile epileptic encephalopathy: an unusual X--linked inheritance disorder[END_REF]. ASD or autistic features appear to be frequent among patients with PCDH19 mutations, with rates varying between 22%--38% (Table 1). Interestingly, a PCDH19 mutation was reported in a female with Asperger syndrome and normal IQ, with a history of infantile onset seizures [START_REF] Hynes | Epilepsy and mental retardation limited to females with PCDH19 mutations can present de novo or in single generation families[END_REF]. The female-limited expression is explained by a phenomenon called cellular interference; random X inactivation in mutated females leads to tissue mosaicism, with PCDH19--positive and PCDH19--negative cells, with altered interactions between the two populations [START_REF] Depienne | PCDH19--related infantile epileptic encephalopathy: an unusual X--linked inheritance disorder[END_REF]. In contrast, complete absence of the protein, as seen in mutated males, is not deleterious. The only affected male reported to date was shown to be mosaic for the PCDH19 deletion in skin fibroblasts [START_REF] Depienne | PCDH19--related infantile epileptic encephalopathy: an unusual X--linked inheritance disorder[END_REF].

In addition to the genes involved in early onset epilepsy and ASD listed in Table 3, many other genes implicated in ASD and ID are associated with epilepsy, including those involved in metabolic disorders (Table 3), Joubert syndrome and related disorders (Table 3), and disorders of the RAS/mitogen activated protein kinase (MAPK) pathway (Table 4). Moreover, several recently discovered recurrent CNVs associated with ID and ASD, such as 15q13.3 and 16p13.11 deletions, increase risk for various forms of epilepsy (Table 2). Large, rare non--recurrent CNVs also play a role in the genetic etiology of epilepsy [START_REF] Mulley | Epilepsy and the new cytogenetics[END_REF], similar to what has been observed in ID, ASD and other neuropsychiatric disorders.

The strong association between ASD and epilepsy suggests that they share common mechanisms of synaptic dysfunction. From the neurobiological perspective, understanding this shared vulnerability is an important direction and the model of excitatory/inhibitory imbalance, first developed in epilepsy, is now being considered in forms of ASD (Chapter 3.9). Metabolic disorders associated with ASD Several metabolic disorders have been associated with an autistic phenotype (Table 3). Although inborn errors of metabolism are rare and probably account for a small proportion of individuals with ASD, their diagnosis is important because some are potentially treatable. Metabolic disorders may be suspected on the basis of parental consanguinity, affected family members, early seizures, episodic decompensation, developmental regression, and coarse facial features. However, many recently described disorders can present as nonsyndromic ID and/or ASD and should therefore be considered in the etiological diagnosis of ASD [START_REF] Kayser | Inherited metabolic diseases in neurodevelopmental and neurobehavioral disorders[END_REF].

Phenylketonuria was identified as a relatively common cause of ASD in older studies, but since the introduction of newborn screening programs and with early dietary intervention, affected children can now expect to lead relatively normal lives [START_REF] Baieli | Autism and phenylketonuria[END_REF]. Unfortunately, phenylketonuria is still identified among patients with ASD in emerging countries without neonatal testing or among subjects born before these screening programs were started [START_REF] Steiner | Genotype and natural history in unrelated individuals with phenylketonuria and autistic behavior[END_REF].

Cerebral creatine deficiency syndromes may be due to two disorders of creatine synthesis, arginine:glycine amidinotransferase deficiency (GATM) and guanidinoacetate methyltransferase deficiency (GAMT), inherited as autosomal recessive traits, or to creatine transporter deficiency (SLC6A8), an X--linked disorder [START_REF] Longo | Disorders of creatine transport and metabolism[END_REF]. All three deficiencies are characterized by ID, severe speech impairment, epilepsy and autistic behavior (Table 3). Although GATM and GAMT mutations are very rare, creatine transporter deficiency could account for up to 1% of unexplained ID in males [START_REF] Clark | X--linked creatine transporter (SLC6A8) mutations in about 1% of males with mental retardation of unknown etiology[END_REF]. Because the presentation is nonsyndromic and autistic behavior is common, this condition could be underdiagnosed in populations of lower functioning males with ASD.

Autism may also occur in the context of mitochondrial disorders, resulting either from mutations in mitochondrial DNA or, more commonly, in nuclear DNA genes encoding mitochondrial--targeted proteins (see Chapter 2.5). Mitochondrial disorders can present with a vast range of symptoms, severity, age of onset and outcome, with a minimum prevalence estimated at 1:5000.

Understanding how metabolic disorders affect brain development and function can lead to a better understanding of the pathophysiology of ASD. At the same time, the frequently indirect nature of this relationship may make such studies more challenging than, for example, studying how synaptic genes alter brain functioning. However, because many metabolic disorders are treatable, understanding the range of metabolic disorders associated with ASD and testing for them can provide immediate clinical benefits, and allow for genetic counseling.

Other examples of etiological subgroups associated with ASD Joubert syndrome is a clinically and genetically heterogeneous group of disorders characterized by a distinctive cerebellar and brainstem malformation, cerebellar ataxia, ID and breathing abnormalities, sometimes including retinal dystrophy and renal disease. ASD is a relatively frequent finding in individuals with Joubert syndrome, present in 13%--36% of patients (Table 1). Sixteen genes have been implicated in Joubert syndrome, the majority very recently; thus, it is not surprising that so far only 4 of these genes have been reported to be mutated in subjects with ASD/autistic traits (Table 3). Joubert syndrome and related disorders arise from ciliary dysfunction and are collectively termed ciliopathies. Other ciliopathies reported in subjects with ASD include Leber congenital amaurosis and Bardet--Biedl syndrome, both of which exhibit phenotypic overlap with Joubert syndrome (Table 3). The means by which cilia are involved in neurodevelopmental processes, and by which ciliopathies lead to neurodevelopmental disorders, are areas of active research. One exciting emerging finding is that primary (or nonmotile) cilia, found on most neurons and astrocytes, play roles as modulators of signal transduction during both brain development and homeostasis [START_REF] Lee | Cilia in the nervous system: linking cilia function and neurodevelopmental disorders[END_REF]. The primary cilia can mediate signaling through sonic hedgehog, wingless, planar cell polarity and fibroblast growth factor pathways.

Another group of disorders that can be associated with ASD is muscular dystrophies (Table 3). Duchenne and Becker muscular dystrophies are caused by deficient expression of the cytoskeletal protein dystrophin, coded by the DMD gene on chromosome Xp21.2--p21.1. One--third of the children with Duchenne muscular dystrophy and about 12% of those with the Becker type also have ID. A small subgroup of these boys with both of these disorders also have ASD, with frequencies varying between 3% and 19% [START_REF] Hinton | Association of autistic spectrum disorders with dystrophinopathies[END_REF][START_REF] Kumagai | Central nervous system involvements in Duchenne/Becker muscular dystrophy[END_REF]Wu et al., 2005). Several maternally--inherited exonic duplications of DMD have been identified in males ascertained for ASD, with no documented muscle disease [START_REF] Pagnamenta | A family with autism and rare copy number variants disrupting the Duchenne/Becker muscular dystrophy gene DMD and TRPM3[END_REF]Pinto et al., 2010), suggestive of the mild end of the spectrum of dystrophinopathies seen in Becker muscular dystrophy, with later onset or subclinical muscle involvement. Another form of muscular dystrophy that includes cases with autistic features is myotonic dystrophy type 1, also known as Steinert disease, caused by expansion of a CTG trinucleotide repeat in the 3'--untranslated region in the DMPK gene (Table 3). The clinical findings span a continuum from mild to severe. In a study of 57 children with myotonic dystrophy type 1, 49% were found to have ASD; the more clinically severe the myotonic dystrophy, the higher the frequency of children with autistic features [START_REF] Ekstrom | Autism spectrum conditions in myotonic dystrophy type 1: a study on 57 individuals with congenital and childhood forms[END_REF]. This may be an underdiagnosed disease entity in autistic populations [START_REF] Coleman | The Autisms[END_REF].

Dysregulation of the RAS/MAPK cascade is the common molecular basis for multiple congenital anomaly syndromes known as neuro--cardio--facio--cutaneous syndromes and characterized by a distinctive facial appearance, heart defects, musculocutaneous abnormalities, and ID, including Noonan syndrome, LEOPARD syndrome (Lentigines, Electrocardiogram abnormalities, Ocular hypertelorism, Pulmonic valvular stenosis, Abnormalities of genitalia, Retardation of growth, and Deafness), cardio--facio--cutaneous syndrome, and Costello syndrome (Table 4) [START_REF] Samuels | MAP'ing CNS development and cognition: an ERKsome process[END_REF].

These overlapping phenotypes can arise from heterozygous mutations in many genes, including PTPN11, BRAF, RAF1, KRAS, HRAS, MAP2K1, MAP2K2, SOS1 and SHOC2. Neurofibromatosis type I and neurofibromatosis type I--like syndrome, which are caused by loss--of--function mutations of NF1 and SPRED1, respectively, can also be included in the same disease entity [START_REF] Aoki | The RAS/MAPK syndromes: novel roles of the RAS pathway in human genetic disorders[END_REF]. As shown in Table 3, all these disorders have been reported in subjects with ASD. In particular, ASD was observed in 8% of 65 children with Noonan syndrome [START_REF] Pierpont | Genotype differences in cognitive functioning in Noonan syndrome[END_REF], as well as in several patients with cardio--facio--cutaneous syndrome or Noonan syndrome with BRAF, KRAS or MAP2K1 mutations [START_REF] Nava | Cardio--facio--cutaneous and Noonan syndromes due to mutations in the RAS/MAPK signalling pathway: genotype--phenotype relationships and overlap with Costello syndrome[END_REF][START_REF] Nystrom | Noonan and cardio--facio-cutaneous syndromes: two clinically and genetically overlapping disorders[END_REF].

Myriad biological pathways

When considering genes involved in autism, neurobiologists usually think about synaptic genes such as those coding for the postsynaptic cell adhesion molecules neuroligins 3 and 4 (NLGN3, NLGN4X), their presynaptic partner neurexin 1 (NRXN1), and the postsynaptic scaffolding proteins SHANK2 and SHANK3 [START_REF] Betancur | The emerging role of synaptic cell--adhesion pathways in the pathogenesis of autism spectrum disorders[END_REF]. In addition to this pathway (described in Chapter 4.7), further evidence implicating synaptic dysfunction in the pathogenesis of ASD has come from the study of genetic disorders with increased rates of ASD, such as fragile X syndrome (FRM1), Rett syndrome (MECP2), tuberous sclerosis (TSC1 and TSC2) and Angelman syndrome (UBE3A). Rare mutations in numerous other genes encoding pre--and postsynaptic proteins have also been reported in ID and ASD, including STXBP1, SYNGAP, as well as the X--linked genes AP1S2, ARHGEF6, CASK, GRIA3, FGD1, IQSEC2, IL1RAPL1, OPHN1, RAB39B, and SYN1 (Figure 1) (for references implicating these genes in ASD, see [START_REF] Betancur | Etiological heterogeneity in autism spectrum disorders: more than 100 genetic and genomic disorders and still counting[END_REF]; for a general review, see [START_REF] Van Bokhoven | Genetic and epigenetic networks in intellectual disabilities[END_REF].

Although the focus on the synaptic pathway in recent years has contributed to our understanding of the pathophysiology of autism, there are dozens of other non--synaptic genes that have been implicated in ASD and which encompass a wide range of biological functions and cellular processes. Mechanisms by which such genes disrupt brain and neuronal development and function will provide a deeper understanding of ASD pathogenesis. Some examples of biological pathways and organelles recurrently implicated in ASD are highlighted in Tables 3 and4. In addition to the genes involved in ciliopathies (Table 3) and the RAS/MAPK signaling pathway (Table 4) mentioned above, Table 4 shows genes involved in channelopathies, genes coding for cell--adhesion molecules and genes implicated in the protein kinase mammalian target of rapamycin (mTOR) signaling pathway. Hyperactivation of mTOR as a consequence of loss--of--function mutations in the genes TSC1, TSC2, and PTEN is responsible for the development of tuberous sclerosis, PTEN hamartoma--tumor syndrome (including Cowden syndrome, Bannayan--Riley--Ruvalcaba syndrome, and Proteus syndrome), and macrocephaly/autism syndrome (for review, see (de Vries, 2010)). Molecularly--targeted treatments using mTOR inhibitors (such as rapamycin) are currently in clinical trials, providing great promise and hope.

Another emerging pathway involves ASD/ID genes that encode regulators of chromatin structure and of chromatin--mediated transcription (for review, see van Bokhoven and Kramer, 2010). Table 4 shows the genes mutated in ASD involved in epigenetic regulation of neuronal gene expression. Prominent examples of epigenetic ASD/ID genes include MECP2, CHD7 (CHARGE sydrome), EHMT1 (Kleefstra syndrome), and the recently implicated gene MBD5 (2q23.1 microdeletion syndrome), all listed in Table 1 as being frequently associated with ASD.

Conclusion

The findings discussed in this review clearly indicate that autism represents the final common pathway for hundreds of genetic and genomic disorders. Despite the abundant evidence, this etiological heterogeneity is still not widely recognized by autism researchers, and most studies fail to take it into account. The genetic overlap and the frequent comorbidity of ASD, ID and epilepsy indicate that the disruption of essential neurodevelopmental processes can give rise to a wide range of manifestations, where the final outcome is likely modulated by the genetic background of each individual as well as other factors including possibly environmental and stochastic factors. Increased understanding of the common genetic, molecular, and cellular mechanisms underlying these neurodevelopmental disorders may provide a framework for novel therapeutic interventions.

Chromosome microarray analysis has revolutionized the molecular diagnostic process in ASD and other neurodevelopmental conditions and is now recommended as a first--line test in the genetic workup of these children, providing an etiological diagnosis in 10 to 15% of cases. Novel high-throughput whole--exome and whole--genome sequencing technologies have hugely accelerated the mutation finding process for Mendelian disorders in the past two years, and hopefully will soon become a first--line approach in the etiological exploration of patients with ASD, replacing targeted sequencing of candidate disease genes (Chapter 2.4).

Currently the most applicable benefit of genetic testing is family planning. A prospective longitudinal study of 664 infants with an older biological sibling with ASD found that 18.7% developed ASD [START_REF] Ozonoff | Recurrence risk for autism spectrum disorders: a baby siblings research consortium study[END_REF]. Although many of the mutations associated with autism so far identified are de novo, future siblings are at risk in the cases where the variant is inherited from a parent, such as in autosomal dominant disorders with variable expressivity inherited from mildly affected parents (e.g., tuberous sclerosis, PTEN related syndromes, 22q11 deletion syndrome), autosomal recessive disorders or maternally--transmitted X--linked disorders (or even a paternally--transmitted X--linked disorder, as for PCDH19). Germinal mosaicism in one of the parents can also explain rare instances of familial recurrence. This mechanism has been implicated in a surprising number of cases of siblings with ASD carrying apparently de novo mutations, not found in the parents' DNA (e.g., SHANK3 mutations and deletions, NRXN1 deletions, NLGN4X mutation, 16p11.2 deletion, 2q23.1 deletion, Rett syndrome and tuberous sclerosis) and may remain unrecognized in sporadic cases in small families.

An etiologic diagnosis has important benefits for the patients with ASD and their families. For the patients, it can help anticipate and manage associated medical and behavioral comorbidities. For the parents, the benefits include relieving anxiety and uncertainty, limiting further costly or invasive diagnostic testing, improving understanding of treatment and prognosis, genetic counseling regarding recurrence risk as well as preventing recurrence through screening for carriers and prenatal testing. A specific disease diagnosis can be empowering to parents who wish to become involved in more targeted support and research groups. For the medical and research community, each child who is accurately diagnosed adds to our presently limited understanding of the pathological cascades which result in autistic features; undoubtedly new findings will include previously unrecognized disease mechanisms. For the neurobiologist especially, the myriad genetic findings in ASD offer a rich source of targets for further study, providing a window into brain and neuronal development and function. The deeper understanding of these brain and neuronal processes will ultimately lead to better outcomes in ASD and other neurodevelopmental disorders. Genes implicated in syndromic and/or nonsyndromic forms of X--linked intellectual disability (XLID) and their localization on the X chromosome. Genes reported to be mutated in ASD are highlighted in red. Genes that cause syndromic forms of XLID are shown on the left; those that can cause nonsyndromic forms are on the right. The distinction between syndromic and nonsyndromic genes is not always clear--cut, and several genes on the right have been involved in syndromic as well as nonsyndromic XLID; the syndromic presentation is indicated in parentheses. Abbreviations: ATRX (alpha thalassemia, mental retardation syndrome, X--linked) syndrome; MASA (mental retardation, aphasia, shuffling gait, and adducted thumbs) syndrome; MHBD (2--methyl--3--hydroxybutyryl--CoA dehydrogenase) deficiency; PRS (phosphoribosylpyrophosphate synthetase) superactivity; VACTERL (vertebral anomalies, anal atresia, cardiac malformations, tracheoesophageal fistula, renal anomalies, and limb anomalies); XLAG (X--linked lissencephaly and abnormal genitalia) syndrome. This figure is an updated version of the one originally published in [START_REF] Betancur | Etiological heterogeneity in autism spectrum disorders: more than 100 genetic and genomic disorders and still counting[END_REF], Copyright 2011, with permission from Elsevier. [START_REF] Antshel | Autistic spectrum disorders in velo--cardio facial syndrome (22q11.2 deletion)[END_REF][START_REF] Fine | Autism spectrum disorders and symptoms in children with molecularly confirmed 22q11.2 deletion syndrome[END_REF][START_REF] Niklasson | Autism, ADHD, mental retardation and behavior problems in 100 individuals with 22q11 deletion syndrome[END_REF][START_REF] Vorstman | The 22q11.2 deletion in children: high rate of autistic disorders and early onset of psychotic symptoms[END_REF] Joubert syndrome (1:100,000)

AHI1 (6q23.3), NPHP1 (2q13), CEP290 (12q21.32), RPGRIP1L (16q12.2); recessive

Mutation, deletion

Very rare 13%--36% ASD ID, distinctive cerebellar and brainstem malformation (molar tooth sign on MRI), ataxia, breathing abnormalities, some times retinal dystrophy and renal disease. Only 4 of 16 genes implicated in Joubert syndrome so far have been reported to be mutated in subjects with ASD [START_REF] Ozonoff | Autism and autistic behavior in Joubert syndrome[END_REF][START_REF] Takahashi | Joubert syndrome is not a cause of classical autism[END_REF] Female--limited epilepsy and ID (unknown) ID, facial dysmorphism, hypotonia, joint laxity, short stature, heart defects [START_REF] Fombonne | Autism and associated medical disorders in a French epidemiological survey[END_REF][START_REF] Kielinen | Associated medical disorders and disabilities in children with autistic disorder: a population--based study[END_REF][START_REF] Lowenthal | Prevalence of pervasive developmental disorder in Down's syndrome[END_REF][START_REF] Oliveira | Epidemiology of autism spectrum disorder in Portugal: prevalence, clinical characterization, and medical conditions[END_REF] a The prevalence of many of these disorders is likely underestimated, due to lack of systematic surveillance and because individuals with atypical, milder features are not diagnosed. b Additional references implicating these disorders in ASD can be found in [START_REF] Betancur | Etiological heterogeneity in autism spectrum disorders: more than 100 genetic and genomic disorders and still counting[END_REF]. Abbreviations: ADHD, attention deficit hyperactivity disorder; ASD, autism spectrum disorder; ID, intellectual disability; MRI, magnetic resonance imaging; OCD, obsessive--compulsive disorder; PDD--NOS, pervasive developmental disorder not otherwise specified; SCZ, schizophrenia. [START_REF] Brunetti--Pierri | Recurrent reciprocal 1q21.1 deletions and duplications associated with microcephaly or macrocephaly and developmental and behavioral abnormalities[END_REF][START_REF] Mefford | Recurrent rearrangements of chromosome 1q21.1 and variable pediatric phenotypes[END_REF]Pinto et al., 2010;[START_REF] Moessner | Contribution of SHANK3 mutations to autism spectrum disorder[END_REF] 2p15--p16.1 microdeletion syndrome 2p15--p16.1 57.7--61.7 ID, growth retardation, microcephaly, dysmorphic features, congenital abnormalities; 4 of 6 subjects reported with the microdeletion have ASD/autistic behavior [START_REF] Liang | A newly recognised microdeletion syndrome of 2p15--16.1 manifesting moderate developmental delay, autistic behaviour, short stature, microcephaly, and dysmorphic features: a new patient with 3.2 Mb deletion[END_REF][START_REF] Rajcan--Separovic | Clinical and molecular cytogenetic characterisation of a newly recognised microdeletion syndrome involving 2p15--16.1[END_REF] 3q29 microdeletion/ microduplication syndrome 3q29 195.7--197.5 3q29 deletions are associated with reduced head size, mild dysmorphic features, multiple congenital anomalies and have been reported in patients with ID, ASD (including one with Asperger syndrome and another one with autism and normal IQ) and schizophrenia. No microduplications have been described thus far in ASD [START_REF] Ballif | Expanding the clinical phenotype of the 3q29 microdeletion syndrome and characterization of the reciprocal microduplication[END_REF][START_REF] Quintero--Rivera | Autistic and psychiatric findings associated with the 3q29 microdeletion syndrome: case report and review[END_REF][START_REF] Willatt | 3q29 microdeletion syndrome: clinical and molecular characterization of a new syndrome[END_REF] 10q22--q23 deletion 10q22.3--q23.2 81.7--88.9 Recurrent 10q22--q23 deletions of varying sizes have been associated with cognitive and behavioral abnormalities including ASD and hyperactivity [START_REF] Alliman | Clinical and molecular characterization of individuals with recurrent genomic disorder at 10q22.3q23.2[END_REF][START_REF] Balciuniene | Recurrent 10q22--q23 deletions: a genomic disorder on 10q associated with cognitive and behavioral abnormalities[END_REF] 15q13 ( [START_REF] Ben--Shachar | Microdeletion 15q13.3: a locus with incomplete penetrance for autism, mental retardation, and psychiatric disorders[END_REF][START_REF] Miller | Microdeletion/duplication at 15q13.2q13.3 among individuals with features of autism and other neuropsychiatric disorders[END_REF]Pinto et al., 2010;[START_REF] Sharp | A recurrent 15q13.3 microdeletion syndrome associated with mental retardation and seizures[END_REF]van Bon et al., 2009) 15q24 microdeletion syndrome 15q24.1--q24.2 74.4--76.2 Microdeletion syndrome characterized by ID, typical facial characteristics, and mild hand and genital anomalies. 23% (8/35) of reported cases have ASD [START_REF] Marshall | Structural variation of chromosomes in autism spectrum disorder[END_REF][START_REF] Mcinnes | A large--scale survey of the novel 15q24 microdeletion syndrome in autism spectrum disorders identifies an atypical deletion that narrows the critical region[END_REF][START_REF] Mefford | Further clinical and molecular delineation of the 15q24 microdeletion syndrome[END_REF] 16p13 Recessive [START_REF] Haliloglu | Clinical spectrum of muscle--eye--brain disease: from the typical presentation to severe autistic features[END_REF][START_REF] Hehr | Novel POMGnT1 mutations define broader phenotypic spectrum of muscle--eye--brain disease[END_REF] POMT1 protein--O--mannosyltransferase 1 9q34.13 Limb--girdle muscular dystrophy with ID; Walker--Warburg syndrome Recessive [START_REF] D'amico | Expanding the clinical spectrum of POMT1 phenotype[END_REF] Abbreviations: ID, intellectual disability

PCDH19 (Xq22.

Figure 1 .

 1 Figure1. Genes implicated in syndromic and/or nonsyndromic forms of X--linked intellectual disability (XLID) and their localization on the X chromosome. Genes reported to be mutated in ASD are highlighted in red. Genes that cause syndromic forms of XLID are shown on the left; those that can cause nonsyndromic forms are on the right. The distinction between syndromic and nonsyndromic genes is not always clear--cut, and several genes on the right have been involved in syndromic as well as nonsyndromic XLID; the syndromic presentation is indicated in parentheses. Abbreviations: ATRX (alpha thalassemia, mental retardation syndrome, X--linked) syndrome; MASA (mental retardation, aphasia, shuffling gait, and adducted thumbs) syndrome; MHBD (2--methyl--3--hydroxybutyryl--CoA dehydrogenase) deficiency; PRS (phosphoribosylpyrophosphate synthetase) superactivity; VACTERL (vertebral anomalies, anal atresia, cardiac malformations, tracheoesophageal fistula, renal anomalies, and limb anomalies); XLAG (X--linked lissencephaly and abnormal genitalia) syndrome. This figure is an updated version of the one originally published in[START_REF] Betancur | Etiological heterogeneity in autism spectrum disorders: more than 100 genetic and genomic disorders and still counting[END_REF], Copyright 2011, with permission from Elsevier.

Table 1 . Genetic disorders strongly associated with ASD Disorder (prevalence) a Gene (locus); inheritance Mutations Prevalence in ASD Proportion with ASD Clinical features Selected references b

 1

	Fragile X syndrome (1:4,000	FMR1 (Xq27.3); X--	Trinucleotide	~2%	~60% males and ~20%	ID, ASD, ADHD, characteristic facial	(Clifford et al., 2007;
	males; 1:6,000 females)	linked	repeat expansion		females with the full	appearance, macroorchidism. Females are	Hagerman et al.,
					mutation have ASD.	generally less affected than males.	2010; Kielinen et al.,
					Among premutation		2004)
					carriers, 15% males and
					5% females have ASD
	22q13 deletion	SHANK3	22q13 deletion,	~0.5%	55% (6/11) individuals	ID, absent or severely delayed speech,	(Durand et al., 2007;
	syndrome/Phelan--McDermid	(22q13.33);	mutation		with 22q13 deletions had	autistic behavior, seizures, hypotonia,	Jeffries et al., 2005;
	syndrome (>800 cases	dominant			autistic behavior; among	decreased sensitivity to pain,	Manning et al., 2004)
	diagnosed)				subjects with ring	mouthing/chewing, dysplastic toenails
					chromosome 22 including
					a 22q13 deletion, 44%
					(12/27) had a clinical
					diagnosis of ASD and 85%
					(23/27) had autistic traits
	Rett syndrome (1:8,500	MECP2 (Xq28); X--	Mutation,	~1% in females,	ASD/autistic features are	MECP2 mutations or deletions cause Rett	(Carney et al., 2003;
	females); MECP2 duplication	linked	deletion,	rare in males	frequent in girls with Rett	syndrome in females (severe ID and speech	Ramocki et al., 2010)
	syndrome (~1% in males		duplication		syndrome; 76% (13/17)	impairment, loss of purposeful hand use,
	with ID)				males with MECP2	ataxia, hyperventilation), and are often fatal
					duplication have	in males; MECP2 duplication syndrome
					autism/autistic features	occurs mostly in males
	15q11--q13 duplication	UBE3A (15q11.2);	Interstitial	~1%	81% (44/54) with	ID, language impairment, seizures, mild	(Depienne et al.,
	syndrome (1:20,000--30,000)	dominant;	duplication or		isodicentric chromosome	dysmorphism, infantile hypotonia.	2009; Hogart et al.,
		imprinted	isodicentric		15 met criteria for autism	Maternally derived duplications confer a	2010)
			chromosome 15,		and 92% (50/54) for ASD	high risk of ASD, whereas duplications of
			usually of			paternal origin usually remain phenotypically
			maternal origin			silent but can lead to ASD/ID
	Angelman syndrome	UBE3A (15q11.2);	Maternal 15q11--	Rare	63% (38/60) ASD (range	ID, lack of speech, inappropriate laughter,	(Sahoo et al., 2006;
	(1:12,000--20,000)	dominant;	q13 deletion,		50%--81%)	seizures, microcephaly, ataxia	Trillingsgaard and
		imprinted	paternal				Østergaard, 2004)
			uniparental
			disomy, mutation,
			imprinting defect
	Prader--Willi syndrome	HBII--85 snoRNA	Paternal 15q11--	Rare	23% (49/209) ASD (range	ID, obsessive compulsive behavior, skin	(Descheemaeker et
	(1:10,000--25,000)	cluster (15q11.2);	q13 deletion,		19%--25%)	picking, psychosis, hypotonia, obesity,	al., 2006; Veltman et
		dominant;	maternal			hypogonadism, short stature	al., 2005)
		imprinted	uniparental
			disomy,
			imprinting defect
	Smith--Magenis syndrome	RAI1 (17p11.2);	17p11.2 deletion,	Rare	90% (18/20) ASD	ID, hyperactivity, sleep disorder, seizures,	(Laje et al., 2010)
	(1:15,000)	dominant	mutation			self--mutilation, hoarse voice, brachydactyly,
						hypotonia
	Potocki--Lupski syndrome	RAI1 (17p11.2);	17p11.2	Rare	Autistic features are	ID, ASD, ADHD, infantile hypotonia, failure to	(Treadwell--Deering et
	(1:20,000)	dominant	duplication		present in the majority;	thrive, sleep apnea, cardiovascular	al., 2010)
					67% (10/15) meet criteria	abnormalities
					for ASD
	Tuberous sclerosis (1:5,800)	TSC1 (9q34.13),	Mutation,	~1%	40% ASD (20%--60%)	ID, non--malignant tumors in the brain,	(Numis et al., 2011)
		TSC2 (16p13.3);	deletion			kidneys, heart, eyes, lungs, and skin, seizures
		dominant
	Adenylosuccinate lyase	ADSL (22q13.1);	Mutation	Extremely rare	~50% autism/autistic	Disorder of purine metabolism characterized	(Spiegel et al., 2006)
	deficiency (<100 cases	recessive			features	by ID, epilepsy and autistic features
	reported)
	Smith--Lemli--Opitz syndrome	DHCR7 (11q13.4);	Mutation	Rare	53% (9/17) autism, 71%	Inborn error of metabolism affecting	(Sikora et al., 2006;
	(1:20,000--40,000)	recessive			(10/14) ASD	cholesterol biosynthesis characterized by	Tierney et al., 2006)
						growth retardation, microcephaly, ID, and
						multiple malformations of variable severity
	CHARGE syndrome	CHD7 (8q12.2);	Mutation,	Rare	68% (17/25) ASD/autistic	ID, coloboma, heart anomaly, choanal	(Johansson et al.,
	(1:10,000)	dominant	deletion (rare)		traits (including 48% with	atresia, genital and ear anomalies	2006)
					ASD)
	Timothy syndrome (<20	CACNA1C	Mutation	Extremely rare	80% (4/5) ASD	ID, ASD, cardiac abnormalities (long QT	(Splawski et al., 2004)
	individuals reported)	(12p13.33);				syndrome, malformations), hand/foot
		dominant				syndactyly, facial dysmorphism, seizures
	Cortical dysplasia--focal	CNTNAP2 (7q35);	Mutation	Extremely rare	67% (6/9) autism or ASD	Severe intractable seizures, ID, ASD, and	(Strauss et al., 2006)
	epilepsy syndrome (10	recessive				focal brain malformations in Amish children
	individuals reported)
	Brain creatine transporter	SLC6A8 (Xq28); X--	Mutation	Very rare	Frequent ASD/autistic	Inborn error of creatine metabolism	(Longo et al., 2011)
	deficiency syndrome (>150	linked			features	characterized by ID, speech delay, autistic
	individuals diagnosed; 45					behavior and seizures
	families reported)
	2q23.1 microdeletion	MBD5 (2q23.1);	2q23.1 deletion	0.17% (7/4061)	100% (65/65) subjects	Angelman--like phenotype including ID,	(Talkowski et al.,
	syndrome (65 individuals	dominant			with deletions or	severe speech impairment, seizures,	2011)
	reported)				translocations involving	behavioral problems, microcephaly, mild
					MBD5 had autistic	dysmorphism, short stature, ataxic gait
					features
	8p23.1 deletion syndrome	? (8p23.1);	8p23.1 deletion	Rare	57% (4/7) autism	Congenital heart defects, congenital	(Fisch et al., 2010)
	(1:10,000--30,000)	dominant				diaphragmatic hernia, mild facial
						dysmorphism, ID, hyperactivity
	Cohen syndrome (500--1000	VPS13B (8q22.2);	Mutation,	Very rare	49% (22/45) autism	ID, typical facial dysmorphism, retinal	(Howlin et al., 2005)
	individuals diagnosed)	recessive	deletion			dystrophy, neutropenia, obesity,
						microcephaly
	Cornelia de Lange syndrome	NIPBL (5p13.2),	Mutation,	Very rare	47%--67% autism	ID, facial dysmorphism, upper limb	(Bhuiyan et al., 2006;
	(1:50,000)	dominant; SMC1A	deletion (rare)			malformations, growth retardation	Moss et al., 2008;
		(Xp11.22), X--linked					Oliver et al., 2008)

Table 2 . Novel microdeletion and microduplication syndromes reported in individuals with ASD and other neurodevelopmental disorders

 2

	Disorder	Cytoband	Position (Mb) a	Comment	References reporting ASD
	1q21.1 microdeletion/	1q21.1	146.5--147.7 Neurodevelopmental disorders (ID, learning disability, ASD,	(Brunetti
	microduplication syndrome			schizophrenia, ADHD, epilepsy), dysmorphic features, congenital
				abnormalities, microcephaly (deletions) or macrocephaly (duplications).
				Both deletions and duplications exhibit incomplete penetrance
				(reported in unaffected parents and controls)

Table 3 . Disease genes and genetic disorders reported in individuals with ASD: examples of clinical and etiological subgroups

 3 Xq28Brain creatine deficiency can be caused by mutation in the creatine transporter gene SLC6A8, or by defects in the biosynthesis of creatine (GAMT and GATM genes)

	Gene	Gene name	Cytoband Disorder	Inheritance	References reporting ASD
					pattern
	Early--onset epilepsy
	SCN1A	sodium channel, voltage--gated, type I,	2q24.3	Severe myoclonic epilepsy of infancy (Dravet syndrome) Dominant	(Marini et al., 2009; O'Roak et
		alpha subunit				al., 2011)
	SCN2A	sodium channel, voltage--gated, type II,	2q24.3	Early infantile epileptic encephalopathy; benign familial	Dominant	(Neale et al., 2012; Sanders et
		alpha subunit		infantile seizures		al., 2012)
	MEF2C	myocyte enhancer factor 2C	5q14.3	5q14.3 microdeletion syndrome	Dominant	(Novara et al., 2010)
	ALDH7A1	aldehyde dehydrogenase 7 family,	5q23.2	Pyridoxine--dependent epilepsy	Recessive	(Burd et al., 2000; Mills et al.,
		member A1				2010)
	CNTNAP2	contactin associated protein--like 2	7q35--	Cortical dysplasia--focal epilepsy syndrome; Pitt--Hopkins--	Recessive	(Strauss et al., 2006; Zweier et
			q36.1	like syndrome--1		al., 2009)
	STXBP1	syntaxin binding protein 1	9q34.11	Early infantile epileptic encephalopathy	Dominant	(Milh et al., 2011; Neale et al.,
						2012)
	FOXG1	forkhead box G1	14q12	Congenital variant of Rett syndrome	Dominant	(Brunetti--Pierri et al., 2011;
						Philippe et al., 2010)
	PAFAH1B1 platelet--activating factor	17p13.3	Isolated lissencephaly	Dominant	(Saillour et al., 2009)
		acetylhydrolase 1b, regulatory subunit
		1
	CDKL5	cyclin--dependent kinase--like 5	Xp22.13 Early infantile epileptic encephalopathy	X linked	(Archer et al., 2006; Russo et
						al., 2009)
	ARX	aristaless related homeobox	Xp21.3	Early infantile epileptic encephalopathy; brain	X linked	(Partington et al., 2004; Turner
				malformations with abnormal genitalia; nonsyndromic ID		et al., 2002)
	SYN1	synapsin I	Xp11.23 X--linked epilepsy with variable learning disabilities and	X linked	(Fassio et al., 2011; Garcia et al.,
				behavior disorders		2004)
	PCDH19	protocadherin 19	Xq22.1	X--linked female--limited epilepsy and cognitive	X linked	(Hynes et al., 2010; Marini et
				impairment; early infantile epileptic encephalopathy		al., 2010)
	DCX	doublecortin	Xq22.3	X--linked lissencephaly	X linked	(Leger et al., 2008)
	SLC9A6	solute carrier family 9 (sodium/	Xq26.3	Syndromic X--linked ID, Christianson type (ID,	X linked	(Garbern et al., 2010)
		hydrogen exchanger), member 6		microcephaly, epilepsy, and ataxia)
	RAB39B	RAB39B, member RAS oncogene family Xq28	X--linked ID associated with autism, epilepsy, and	X linked	(Giannandrea et al., 2010)
				macrocephaly
	Metabolic disorders
	Disorders of purine and pyrimidine metabolism
	ADSL	adenylosuccinate lyase	22q13.1	Adenylosuccinate lyase deficiency (purine synthesis	Recessive	(Spiegel et al., 2006)
				disorder)
	DPYD	dihydropyrimidine dehydrogenase	1p21.3	Dihydropyrimidine dehydrogenase deficiency (disorder	Recessive	(van Kuilenburg et al., 2009;
				of the pyrimidine degradation pathway)		Van Kuilenburg et al., 1999)
	Creatine deficiency syndromes
	SLC6A8	solute carrier family 6			X linked	(Poo--Arguelles et al., 2006;
		(neurotransmitter transporter,				Sempere et al., 2009a)
		creatine), member 8
	GATM	glycine amidinotransferase	15q21.1	Arginine:glycine amidinotransferase (AGAT) deficiency	Recessive	(Battini et al., 2002)
	GAMT	guanidinoacetate N--methyltransferase 19p13.3	Guanidinoacetate methyltransferase (GAMT) deficiency Recessive	(Sempere et al., 2009b)
	Disorder of cholesterol synthesis
	DHCR7	7--dehydrocholesterol reductase	11q13.4	Smith--Lemli--Opitz syndrome	Recessive	(Sikora et al., 2006; Tierney et
						al., 2006)
	Defect of phenylalanine metabolism
	PAH	phenylalanine hydroxylase	12q23.2	Phenylketonuria	Recessive	(Baieli et al., 2003; Steiner et
						al., 2007)
	Disorders of vitamin metabolism
	BTD	biotinidase	3p24.3	Biotinidase deficiency (defect in the recycling of the	Recessive	(Zaffanello et al., 2003)
				vitamin biotin)
	ALDH7A1	aldehyde dehydrogenase 7 family,	5q23.2	Pyridoxine--dependent epilepsy (vitamin B6)	Recessive	(Burd et al., 2000; Mills et al.,
		member A1				2010)
	Defect of GABA catabolism
	ALDH5A1	aldehyde dehydrogenase 5 family,	6p22.2	Succinic semialdehyde dehydrogenase deficiency	Recessive	(Knerr et al., 2008; Pearl et al.,
		member A1		(gamma--hydroxybutyric aciduria)		2003)
	Lysosomal storage disorders
	SGSH	N--sulfoglucosamine sulfohydrolase	17q25.3	Mucopolysaccharidosis IIIA (Sanfilippo syndrome A)	Recessive	(Petit et al., 1996; Wolanczyk et
						al., 2000)
	NAGLU	N--acetylglucosaminidase, alpha	17q21.31 Mucopolysaccharidosis IIIB (Sanfilippo syndrome B)	Recessive	(Heron et al., 2011)
	HGSNAT	heparan--alpha--glucosaminide N--	8p11.21	Mucopolysaccharidosis IIIC (Sanfilippo syndrome C)	Recessive	(Heron et al., 2011)
		acetyltransferase
	GNS	glucosamine (N--acetyl)--6--sulfatase	12q14.3	Mucopolysaccharidosis IIID (Sanfilippo syndrome D)	Recessive	(Heron et al., 2011)
	Folate deficiency syndrome
	FOLR1	folate receptor 1	11q13.4	Cerebral folate transport deficiency	Recessive	(Cario et al., 2009)
	Urea cycle disorder
	OTC	ornithine carbamoyltransferase	Xp11.4	Ornithine transcarbamylase deficiency	X linked	(Gorker and Tuzun, 2005)
	Defect of amino acid metabolism
	MUT	methylmalonyl CoA mutase	6p12.3	Methylmalonic aciduria	Recessive	(de Baulny et al., 2005)
	Joubert syndrome and related ciliopathies
	AHI1	Abelson helper integration site 1	6q23.3	Joubert syndrome 3	Recessive	(Ozonoff et al., 1999; Takahashi
						et al., 2005)
	NPHP1	nephronophthisis 1	2q13	Joubert syndrome 4; nephronophthisis 1, juvenile	Recessive	(Tory et al., 2007)
	CEP290	centrosomal protein 290kDa	12q21.32 Joubert syndrome 5; Leber congenital amaurosis 10;	Recessive	(Coppieters et al., 2010;
				Bardet--Biedl syndrome 14		Perrault et al., 2007; Tory et al.,
						2007)
	RPGRIP1L	RPGRIP1--like (retinitis pigmentosa	16q12.2	Joubert syndrome 7	Recessive	(Doherty et al., 2010)
		GTPase regulator--like)
	GUCY2D	guanylate cyclase 2D, membrane	17p13.1	Leber congenital amaurosis 1	Recessive	(Coppieters et al., 2010)
	RPE65	retinal pigment epithelium--specific	1p31.3	Leber congenital amaurosis 2	Recessive	(Coppieters et al., 2010; Yzer et
		protein 65kDa				al., 2003)
	MKKS	McKusick--Kaufman syndrome	20p12.2	Bardet--Biedl syndrome 6; McKusick--Kaufman syndrome Recessive	(Barnett et al., 2002; Moore et
						al., 2005)
	BBS10	Bardet--Biedl syndrome 10	12q21.2	Bardet--Biedl syndrome 10	Recessive	(Deveault et al., 2011)
	Muscular dystrophies
	DMD	dystrophin	Xp21.2--	Muscular dystrophy, Duchenne and Becker types	X linked	(Hinton et al., 2009; Young et
			p21.1			al., 2008)
	DMPK	dystrophia myotonica--protein kinase	19q13.32 Myotonic dystrophy 1 (Steinert disease)	Dominant	(Ekstrom et al., 2008)
	POMGNT1 protein O--linked mannose beta1,2--N--	1p34.1	Muscle--eye--brain disease (congenital muscular
		acetylglucosaminyltransferase		dystrophy, structural eye abnormalities and
				lissencephaly)

a The protein name and its function are indicated after the gene name.b The protein function is indicated after the gene name.c The epigenetic function is indicated after the gene name. Abbreviations: ASD, autism spectrum disorder; ID, intellectual disability

Cardio--facio--cutaneous syndrome, Noonan syndrome, LEOPARD syndrome Dominant [START_REF] Nava | Cardio--facio--cutaneous and Noonan syndromes due to mutations in the RAS/MAPK signalling pathway: genotype--phenotype relationships and overlap with Costello syndrome[END_REF][START_REF] Nystrom | Noonan and cardio--facio-cutaneous syndromes: two clinically and genetically overlapping disorders[END_REF]) KRAS v--Ki--ras2 Kirsten rat sarcoma viral oncogene homolog (KRAS, small G--protein)

12p12.1 Cardio--facio--cutaneous syndrome, Noonan syndrome Dominant [START_REF] Nava | Cardio--facio--cutaneous and Noonan syndromes due to mutations in the RAS/MAPK signalling pathway: genotype--phenotype relationships and overlap with Costello syndrome[END_REF][START_REF] Nystrom | Noonan and cardio--facio-cutaneous syndromes: two clinically and genetically overlapping disorders[END_REF]) MAP2K1 mitogen--activated protein kinase kinase 1 (MEK1, tyrosine/serine/threonine kinase)

15q22.31 Cardio--facio--cutaneous syndrome Dominant [START_REF] Nava | Cardio--facio--cutaneous and Noonan syndromes due to mutations in the RAS/MAPK signalling pathway: genotype--phenotype relationships and overlap with Costello syndrome[END_REF] PTPN11 protein tyrosine phosphatase, non--receptor type 11 (SHP2, tyrosine phosphatase) 12q24.13 Noonan syndrome, LEOPARD syndrome Dominant [START_REF] Pierpont | Genotype differences in cognitive functioning in Noonan syndrome[END_REF][START_REF] Watanabe | A familial case of LEOPARD syndrome associated with a high--functioning autism spectrum disorder[END_REF]) HRAS v--Ha--ras Harvey rat sarcoma viral oncogene homolog (HRAS, small G--protein)

11p15.5 Costello syndrome Dominant [START_REF] Kerr | Genotype--phenotype correlation in Costello syndrome: HRAS mutation analysis in 43 cases[END_REF] NF1 neurofibromin 1 (neurofibromin, RAS--GTPase activator protein) [START_REF] Gillberg | Asperger syndrome in 23 Swedish children[END_REF]) 45,X/46,XY mosaicism [START_REF] Fontenelle | Asperger Syndrome, obsessive--compulsive disorder, and major depression in a patient with 45,X/46,XY mosaicism[END_REF]) Mitochondrial disorder (A3243G mutation) [START_REF] Pons | Mitochondrial DNA abnormalities and autistic spectrum disorders[END_REF] Box

Genes involved in nonsyndromic ASD

There is a widely spread misconception that genetic disorders are only identified in individuals with ASD that have a syndromic presentation, i.e., that exhibit facial dysmorphism, congenital malformations, and/or neurologic abnormalities such as microcephaly or structural brain malformations. However, the evidence is clear that many genetic defects can be observed in children that present only with ASD, with no apparent physical abnormalities.

For example, many genes involved in X--linked or autosomal nonsyndromic ID have also been implicated in nonsyndromic ASD (see table below and Figure 1). In addition, many other genes have been implicated in both syndromic as well as nonsyndromic forms of ASD/ID. The table below summarizes examples of genes for which nonsyndromic cases have been reported. For genes for which both nonsyndromic and syndromic cases have been reported, this is noted. [START_REF] Giannandrea | Mutations in the small GTPase gene RAB39B are responsible for X--linked mental retardation associated with autism, epilepsy, and macrocephaly[END_REF]


Paola Goatin
email: paola.goatin@inria.fr

Matthias Mimault
email: matthias.mimault@inria.fr

A MIXED SYSTEM MODELING TWO-DIRECTIONAL PEDESTRIAN FLOWS

Keywords: 2010 Mathematics Subject Classification. Primary: 35L65 -35M30; Secondary: 90B20 Systems of conservation laws, mixed hyperbolic-elliptic systems, macroscopic models for pedestrian flows, finite volume schemes, Young measures

In this article, we present a simplified model to describe the dynamics of two groups of pedestrians moving in opposite directions in a corridor. The model consists of a 2 × 2 system of conservation laws of mixed hyperbolicelliptic type. We study the basic properties of the system to understand why and how bounded oscillations in numerical simulations arise. We show that Lax-Friedrichs scheme ensures the invariance of the domain and we investigate the existence of measure-valued solutions as limit of a subsequence of approximate solutions.

1. Introduction. Systems of conservation laws that are not everywhere hyperbolic in the phase space arise naturally in the modeling of physical phenomena. Two wellknown examples are the two-fluid single-pressure model for two-phase flow [START_REF] Stewart | Two-phase flow: models and methods[END_REF], and a model for three-phase porous medium flow that has been widely used in petroleum reservoir simulation [START_REF] Bell | Conservation laws of mixed type describing three-phase flow in porous media[END_REF]. Another model arises in modeling two-directional traffic flows [START_REF] Berres | An adaptive finite-volume method for a model of two-phase pedestrian flow[END_REF][START_REF] Bick | A continuum model for two-directional traffic flow[END_REF]. These models display an elliptic region in the phase space. An elliptic state in the solution space is a state where the Jacobian matrix of the vectorvalued flux function has complex eigenvalues. The set of all elliptic points forms the elliptic region. This type of systems has been addressed since several decades now, see [START_REF] Fitt | The numerical and analytical solution of ill-posed systems of conservation laws[END_REF][START_REF] Keyfitz | A geometric theory of conservation laws which change type[END_REF][START_REF] Keyfitz | Singular shocks: retrospective and prospective[END_REF] for a general overview. Nevertheless, their solutions have not been completely understood yet. Rapidly oscillating but bounded numerical approximations suggest that solutions could be defined in the framework of Young measures [START_REF] Frid | Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type[END_REF].

In this article, we consider a mixed type system of conservation laws describing two populations of pedestrians moving in opposite directions, adapted from [START_REF] Benzoni-Gavage | An n-populations model for traffic flow[END_REF][START_REF] Bick | A continuum model for two-directional traffic flow[END_REF].

Let u t + f (u, v) x = 0, v t -f (v, u) x = 0, (1) 2
PAOLA GOATIN AND MATTHIAS MIMAULT be the governing equations, together with the initial conditions u(0, x) = u 0 (x),

v(0, x) = v 0 (x), (2)
where t ∈ R + , x ∈ R. The flux function is therefore given by

F(U ) = f (u, v) -f (v, u) , (3)
where f (u, v) = u(1 -u -v), and U = (u, v) are densities of the two groups of pedestrians that take values in

Ω = {(u, v) ∈ R 2 : u ≥ 0, v ≥ 0 and u + v ≤ 1}.
As announced, the main feature of system (1) lies in the hyperbolicity loss for certain density values. Indeed, the Jacobian of the flux exhibits complex eigenvalues in an elliptic region of the phase space Ω. It was suggested in [START_REF] Berres | An adaptive finite-volume method for a model of two-phase pedestrian flow[END_REF] that oscillations arising in the elliptic region could be related to the lane formation phenomenon observed in groups of pedestrians moving in opposite directions [START_REF] Helbing | Self-organizing pedestrian movement[END_REF][START_REF] Moussaïd | Traffic instabilities in self-organized pedestrian crowds[END_REF]. We aim to investigate solutions properties in relation with the modeled pedestrian dynamics, also relying on numerical simulations. In particular, we will study the solutions of ( 1), (2) corresponding Riemann-like initial data U 0 = (u 0 , v 0 ) of the form

U 0 (x) = U L = (u L , v L ), if x < 0, U R = (u R , v R ), if x > 0. (4)
A similar problem was addressed by Vinod [START_REF] Vinod | Structural stability of Riemann solutions for a multiple kinematic conservation law model that changes type[END_REF], who considered a slightly different version of model (1) including a parameter β > 0, which sensibly changes the solutions behavior. The paper is organized as follows. Section 2 contains the basic analytical study of the models properties. In Section 3, we introduce a Lax-Friedrichs finite volume scheme and we prove an L ∞ bound on the corresponding approximate solutions, ensuring the convergence towards Young measures. In Section 4 we give examples of weak solutions in distributional sense. The study two examples of initial data generating persistent oscillations is presented in Section 5 and we give conclusions in Section 6.

2. Basic analytical study. This section is devoted to the study of the basic properties of system (1), and to the identification of the wave types appearing in the solutions of the corresponding Riemann problem (1)-( 4). This study does not pretend to be exhaustive, the problem being non classical and still not completely understood. In particular, we cannot give any global existence result for weak solutions, and their uniqueness is not expected. Some examples of solutions displaying the described features will be showed through numerical computations in Section 4.

First of all, we compute the Jacobian of the flux (3):

J(u, v) = 1 -2u -v -u v -1 + u + 2v
and its characteristic polynomial

p(λ) = λ 2 + (u -v)λ -2u 2 -2v 2 + 3u + 3v -4uv -1. ( 5
)
The discriminant of ( 5) is which is negative in the region

∆(u, v) = 4 + 14uv -12u -12v + 9u 2 + 9v 2 ,
E = (u, v) ∈ R 2 : 4 + 14uv -12u -12v + 9u 2 + 9v 2 ≤ 0 ⊂ Ω,
see Fig. 1. Therefore the eigenvalues can take complex values :

λ 1 (u, v) = 1 2 v -u -4 + 14uv -12u -12v + 9u 2 + 9v 2 , λ 2 (u, v) = 1 2 v -u + 4 + 14uv -12u -12v + 9u 2 + 9v 2 ,
and corresponding eigenvectors are :

r 1 (u, v) = 2 u 1 -2u -v -λ 1 = 2u 2 -3u -3v + ∆(u, v) = 2 1 -u -2v + λ 1 v = 2 -3u -3v -∆(u, v) 2v , r 2 (u, v) = 2 u 1 -2u -v -λ 2 = 2u 2 -3u -3v -∆(u, v) = 2 1 -u -2v + λ 2 v = 2 -3u -3v + ∆(u, v) 2v .
The gradient of the eigenvalues are:

∇λ 1 (u, v) = 1 2     1 - 9u + 7v -6 ∆(u, v) 1 - 7u + 9v -6 ∆(u, v)     , ∇λ 2 (u, v) = 1 2     -1 + 9u + 7v -6 ∆(u, v) 1 + 7u + 9v -6 ∆(u, v)     .
For seek of clarity, we recall here the notion of distributional solution.

Definition 2.1. A function U ∈ L 1 (R + × R, Ω) is a weak solution of (1)-(2) if for all φ ∈ C 1 c (R 2 ; R) we have R + R (U φ t + F(U )φ x ) (t, x) dxdt + R U 0 (x)φ(0, x) dx = 0. ( 6
)
Weak solutions in distributional sense, when they exist, consist of a combination of rarefactions and shock waves, which are described below.

2.1. Rarefaction waves. The solution of a Riemann problem (1)-( 4) can be a rarefaction wave U = U (x, t) of the i-th family if it reads

U (x, t) =    U L , if x < λ i (U L )t, V (x/t), if λ i (U L )t ≤ x ≤ λ i (U R )t U R , if x > λ i (U R )t, (7)
where

V (ξ) satisfies V = r i (V (ξ)), V (λ i (U L )) = U L , V (λ i (U R )) = U R , i = 1 or i = 2. ( 8
)
Let R i (U L ) denote the solution of [START_REF] Diperna | Measure-valued solutions to conservation laws[END_REF]. The integral curves of r i (U ) are illustrated on Fig. 2. The arrows indicate directions of increase of the corresponding eigenvalue. The direction is reversed across the fognals F i , see Sec. 2.3.

Vector fields r i (U ), i = 1, 2, oriented so that ∇λ i • r i (U ) ≥ 0, are given in Fig. 3, which give an idea of the orientation of rarefaction curves.

Shocks. The discontinuous function

U (x, t) = U L , if x < st, U R , if x > st.
is a weak solution of (1)-( 4) called shock wave if and only if it satisfies the Rankine-Hugoniot relation

s(U R -U L ) = F(U R ) -F(U L ) (9
) for some speed s = s(U L , U R ) ∈ R. Given any point U L ∈ Ω, the Hugoniot locus [START_REF] Bressan | Hyperbolic systems of conservation laws[END_REF] is defined by

H(U L ) = U ∈ R 2 : ∃ s = s(U, U R ) ∈ R s. t. s(U -U L ) = F(U ) -F(U L ) . (10)
A Hugoniot locus is composed of three disjoint branches. When U L lies in the hyperbolic region, one of them has a loop closing at U L , which crosses the elliptic region. Another branch can also cross a different hyperbolic region but the three branches can never be in Ω at the same time. Those informations are visible on Fig. 4. Since the system is not genuinely non-linear (and not even hyperbolic), the entropy admissible branch can be selected using the Liu-Oleinik condition [START_REF] Liu | The Riemann problem for general 2 × 2 conservation laws[END_REF]:

Definition 2.2. U R ∈ H(U L ) is joined to U L by an entropy admissible shock if and only if s(U L , U R ) ≤ s(U L , U ) (11)
for each U ∈ H(U L ) between U L and U R Anyway, each branch section of H(U L ) belonging to Ω lies in a region were the corresponding field is genuinely non-linear, as shown in Fig. 4. Therefore, if U R belongs to the same branch than U L , condition [START_REF] Fjordholm | High-order accurate entropy stable numerical schemes for hyperbolic conservation laws[END_REF] coincides with the usual Lax

λ i (U R ) ≤ s(U L , U R ) ≤ λ i (U L ) for i = 1 or i = 2.
2.3. Fognals and umbilic points. Due to the mixed character of system (1), weak solutions can display other type of discontinuities. Combinations of contact discontinuities moving with zero speed appear along fognals :

F i = U ∈ R 2 : ∇λ i (U ) • r i (U ) = 0 , i = 1, 2. ( 12
)
We define

F = F 1 ∪ F 2 = (u, v) ∈ R 2 : u + v = 1 ∪ (u, v) ∈ R 2 : u + v = 2 3 , (u, v) ∈ E .
(13) F coincides with the line u + v = 1 and crosses ∂E and ∂Ω at points called umbilic points. One can see orientation's change on the eigenvector field in Fig. 3 and observe that the wave type on the u and the v-axis changes at points 2 3 , 0 , 0, 2 3 and 1 2 , 1 2 . In particular, we will call crossing shocks the discontinuities satisfying

Σ(U L ) = U ∈ H(U L ) : λ 1 (U L ) ≤s(U L , U ) ≤ λ 2 (U L ) λ 1 (U ) ≤s(U L , U ) ≤ λ 2 (U ) , (14)
see [START_REF] Isaacson | The Riemann problem near a hyperbolic singularity: the classification of solutions of quadratic Riemann problems. I[END_REF]. For example, if

U L = (u L , 0) for some u L ∈ [0, 2/3[, we have Σ(U L ) = U = (u, 0) : 1 - u L 2 ≤ u ≤ 2 -2u L ,
and for

U L = (0, v L ) with v L ∈ ]2/3, 1] Σ(U L ) = U = (0, v) : 2 -2v L ≤ v ≤ 1 - v L 2 .
3. Numerical scheme. We take a space step ∆x and a time step ∆t subject to a CFL condition which will be specified later in [START_REF] Keyfitz | A geometric theory of conservation laws which change type[END_REF]. For j ∈ Z and n ∈ N, let x j+1/2 = j∆x be the cells interfaces, x j = (j-1/2)∆x the cells centers and t n = n∆t the time mesh. We want to construct a finite volume approximate solution of ( 1)-( 2) of the form

U ∆x (t, x) = U n j = (u n j , v n j ) for (t, x) ∈ C n j = [t n , t n+1 [×[x j-1/2 , x j+1/2
[. We use the following Lax-Friedrichs scheme:

u n+1 i = u n i - ∆t ∆x [F 1 (u n i , v n i ; u n i+1 , v n i+1 ) -F 1 (u n i-1 , v n i-1 ; u n i , v n i )], v n+1 i = v n i - ∆t ∆x [F 2 (u n i , v n i ; u n i+1 , v n i+1 ) -F 2 (u n i-1 , v n i-1 ; u n i , v n i )], (15)
with the numerical flux F = (F 1 , F 2 ) defined by

F 1 (u 1 , v 1 ; u 2 , v 2 ) = f (u 1 , v 1 ) + f (u 2 , v 2 ) 2 + α 2 (u 1 -u 2 ), F 2 (u 1 , v 1 ; u 2 , v 2 ) = - f (v 1 , u 1 ) + f (v 2 , u 2 ) 2 + α 2 (v 1 -v 2 ), (16)
for α ≥ 1. We prove by induction that the domain Ω is invariant for ( 15)-( 16)

Lemma 3.1. Under the Courant-Friedrichs-Lewy (CFL) stability condition ∆t ≤ ∆x α , α ≥ 1, (17)
for any initial data U 0 ∈ Ω the approximate solutions computed by scheme (15)-( 16) satisfy the following uniform bounds:

U n j = (u n j , v n j ) ∈ Ω ∀j ∈ Z, n ∈ N.
Proof. We proceed by induction: assuming that u n j ≥ 0, v n j ≥ 0 and u n j + v n j ≤ 1 for all j ∈ Z, we show that the same holds for u n+1 j and v n+1 j .

To prove positiveness, we focus on the u component, the procedure being similar for v. Dropping the index n, we compute

u n+1 i = u i - ∆t 2∆x [f (u i+1 , v i+1 ) -f (u i-1 , v i-1 )] - α∆t 2∆x (-u i-1 + 2u i -u i+1 ) = 1 -α ∆t ∆x u i + α∆t 2∆x (u i-1 + u i+1 ) + ∆t 2∆x [f (u i-1 , v i-1 ) -f (u i+1 , v i+1 )] = 1 -α ∆t ∆x u i + ∆t 2∆x [u i-1 (α + 1 -u i-1 -v i-1 ) + u i+1 (α -1 + u i+1 + v i+1 )] .
By assumption 1

-u i-1 -v i-1 ≥ 0, α ≥ 1 -u i+1 -v i+1 and 1 -α∆t/∆x ≥ 0 by (17), ensuring u n+1 i ≥ 0. To prove u n+1 i + v n+1 i ≤ 1, we observe that f (a, b) -f (b, a) = (a -b)(1 -a -b) = a(1 -a) -b(1 -b),
and we compute (dropping again the index n)

u n+1 i + v n+1 i = = u i + v i + α∆t 2∆x (-u i-1 + 2u i -u i+1 -v i-1 + 2v i -v i+1 ) - ∆t 2∆x [f (u i+1 , v i+1 ) -f (v i+1 , u i+1 ) -f (u i-1 , v i-1 ) + f (v i-1 , u i-1 )] = 1 -α ∆t ∆x (u i + v i ) + α∆t 2∆x (u i-1 + u i+1 + v i-1 + v i+1 ) - ∆t 2∆x [(u i+1 -v i+1 )(1 -u i+1 -v i+1 ) -(u i-1 -v i-1 )(1 -u i-1 -v i-1 )] = 1 -α ∆t ∆x (u i + v i ) + ∆t 2∆x [α(u i-1 + v i-1 ) + (u i-1 -v i-1 )(1 -u i-1 -v i-1 )] + ∆t 2∆x [α(u i+1 + v i+1 ) -(u i+1 -v i+1 )(1 -u i+1 -v i+1 )] = 1 -α ∆t ∆x (u i + v i ) + ∆t 2∆x [u i-1 (α + 1 -u i-1 ) + v i-1 (α -1 + v i-1 )] + ∆t 2∆x [u i+1 (α -1 + u i+1 ) + v i+1 (α + 1 -v i+1 )] .

Using the hypothesis that v

i-1 ≤ 1 -u i-1 , u i+1 ≤ 1 -v i+1 and u i + v i ≤ 1, we get u n+1 i + v n+1 i ≤ 1 -α ∆t ∆x + α ∆t 2∆x + α ∆t 2∆x = 1,
therefore concluding the proof.

The uniform L ∞ -bound provided by Lemma 3.1 ensures the convergence towards Young measures, which are weak- * measurable maps ν : R + × R → P(R 2 ), where P(R 2 ) denote the space of probability measures on R 2 .

Theorem 3.2 ([23]

). Let U ∆x be a sequence of approximate solutions of (1)-(2) constructed by the scheme (15)- [START_REF] Isaacson | The Riemann problem near a hyperbolic singularity: the classification of solutions of quadratic Riemann problems. I[END_REF]. Then there exists a subsequence, still denoted by U ∆x , and a Young measure ν with supp(ν t,x ) ⊂ Ω such that

h(U ∆x ) * ν t,x , h(λ) := R 2 h(λ) dν t,x (λ) in L ∞ (R + × R; R 2 ) ( 18
)
for all h ∈ C(R 2 ; R).

Frid and Liu [START_REF] Frid | Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type[END_REF] provide an explicit formula for computing the probability measure ν satisfying [START_REF] Keyfitz | Singular shocks: retrospective and prospective[END_REF] in the case of Riemann-type initial data (4). For any space step ∆x fixed, let U k = U ∆x/k , k ∈ N, be the sequence of approximate solutions obtained dividing ∆x by k. Then, for any h ∈ C(R 2 ; R) it holds

ν t,x , h(λ) = lim T →∞ 2 T 2 T 0 h(U ∆x (τ, (x/t)τ ))τ dτ (19)
for almost every x/t ∈ R. (For the proof, see [13, Appendix A.2].) For numerical purposes, we will use the following discretized version of formula [START_REF] Lax | Systems of conservation laws[END_REF] to compute the moments of interest:

ν t,x , h(λ) = lim N →∞ 2 N (N + 1) N k=1 k h(U k j k ), j k = k ∆t ∆x x t , (20)
where [•] denotes the integer part.

Relying on Young measures, DiPerna [START_REF] Diperna | Measure-valued solutions to conservation laws[END_REF] introduced the concept of measurevalued solutions. Definition 3.3. Let P(R 2 ) denote the space of probability measures on R 2 . A measure-valued solution of ( 1) is a measurable map ν : R + × R → P(R 2 ) such that for all φ ∈ C 1 c (R 2 ; R) we have

R + R ( ν t,x , λ φ t + ν t,x , F(λ) φ x ) dxdt + R U 0 (x)φ(0, x) dx = 0. ( 21
)
In order to prove that the Young measure obtained as limit of finite volume approximations is indeed a measure-valued solution, one needs some weak BV estimates that allow to pass to the limit in the discrete conservation equation, replacing the strong convergence argument used in the proof of Lax-Wendroff theorem [START_REF] Lax | Systems of conservation laws[END_REF]. Weak BV estimates have been derived in the case of monotone schemes for scalar conservation laws in several space-dimensions [START_REF] Coquel | Convergence of finite difference schemes for conservation laws in several space dimensions: a general theory[END_REF][START_REF] Eymard | Finite volume methods[END_REF], but also for entropy stable methods applied to hyperbolic systems [START_REF] Fjordholm | High-order accurate entropy stable numerical schemes for hyperbolic conservation laws[END_REF][START_REF] Fjordholm | Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws[END_REF]. Unfortunately, in the present case the system's entropy is not convex, and these techniques don't apply.

4. Distributional solutions. In this section, we present some numerical computations illustrating the principal features of weak (distributional) solutions of problem (1)-(4). From [START_REF] Holden | Some qualitative properties of 2 × 2 systems of conservation laws of mixed type[END_REF] we know that, if U L , U R ∈ Ω \ Ē, then corresponding weak solutions must satisfy U (x, t) ∈ Ω \ E a.e. In particular, U consists of a combination of rarefactions and shock waves, as illustrated by the following numerical tests, where we have taken ∆x = 0.001, α = 1, CF L = 0.9, and we display the approximate solution U ∆x at time t = 1, both in the x-u, v plane (Figures 56789 Test 1. We consider initial data U L = (0.2, 0.1) and U R = (0.1, 0.2). The solution showed in Fig. 5 consists of a rarefaction of the first family joining U L with an intermediate state on ∂E, followed by a rarefaction of the second family to U R . This is a limit situation, since if the Lax curves do not intersect in the same connected region, the structure of the solution becomes more complex, as illustrated by the following examples.

Test 2. We consider initial data U L = (0.2, 0.1) and U R = (0.1, 0.3). The solution showed in Fig. 6 consists of a shock of the first family joining U L with a state U 1 = (u 1 , 0) on the u-axis, followed by a crossing shock between U 1 and the state (1, 0), a contact discontinuity form (1, 0) to (0, 1) with zero speed, another crossing shock from (0, 1) to a point U 2 = (0, v 2 ) and a 2-shock from U 2 to U R .

Test 3. We consider initial data U L = (0.2, 0.1) and U R = (0.1, 0.8). The solution showed in Fig. 7 consists of a shock of the first family joining U L with a state U 1 = (u 1 , 0) on the u-axis, followed by a crossing shock between U 1 and the state (1, 0), a standing contact discontinuity form (1, 0) to a point U 2 = (u 2 , 1 -u 2 ) ∈ ∂Ω and a 2-shock from U 2 to U R .

Test 4. We consider initial data U L = (0.2, 0.1) and U R = (0.85, 0.1). The solution showed in Fig. 8 consists of a shock of the first family joining U L with a 4) with U L = (0.2, 0.1) and U R = (0.1, 0.3). We observe a 1-shock, a crossing shock, a contact discontinuity, another crossing shock and a 2-shock. 4) with U L = (0.2, 0.1) and U R = (0.1, 0.8). We observe a 1-shock, a crossing shock, a contact discontinuity and a 2-shock.

state U 1 = (u 1 , 0) on the u-axis, followed by a crossing shock between U 1 and a state U 2 = (u 2 , 0) and a 2-shock from U 2 to U R . We remark that the crossing shock is sharply captured. 4) with U L = (0.2, 0.1) and U R = (0.85, 0.1). We observe a 1-shock, a crossing shock and a 2-shock.

Test 5. We consider initial data U L = (0.2, 0.1) and U R = (0.75, 0.1). The solution showed in Fig. 9 consists of a shock of the first family joining U L with a state U 1 in the interior of the domain, followed by a crossing shock between U 1 and a state U 2 superposed to a 2-shock from U 2 to U R . Note that this composite wave could be replaced by a 2-shock joining directly U 1 to U R , see [START_REF] Isaacson | The Riemann problem near a hyperbolic singularity: the classification of solutions of quadratic Riemann problems. I[END_REF]. Indeed, we have

U 2 ∈ H(U 1 ), U R ∈ H(U 2 ), U R ∈ H(U 1 ), and s(U 1 , U 2 ) = s(U 2 , U R ) = s(U 1 , U R ).
Anyway, the two solutions are identical as L 1 functions. 4) with U L = (0.2, 0.1) and U R = (0.75, 0.1). We observe a 1-shock, a crossing shock and a 2-shock.

Remark 1. The configurations displayed in Tests 2-3 are unrealistic from the modeling point of view, because they result in a complete blocking of one or both groups of pedestrians, represented by the vacuum regions delimited by the standing contact discontinuities. In reality, such stuck situations never occur in normal conditions, and the flows always organize so that few people manage to pass, even if the resulting capacity can be very reduced [START_REF] Buchmüller | Parameters of pedestrians, pedestrian traffic and walking facilities[END_REF]. System (1) must be seen as a toy model, whose understanding can give some insight for more realistic approaches.

If one or both values of the Riemann initial data U 0 = (U L , U R ) belong to E, we can still observe distributional solutions in some cases. In accordance to [START_REF] Holden | Some qualitative properties of 2 × 2 systems of conservation laws of mixed type[END_REF], since E is convex, if U L ∈ E or U R ∈ E and U is a weak solutions, then if U (t, x) ∈ E for some t, x, than U (t, x) ∈ {U L , U R }. Indeed, the initial state belonging to E will be connected through a shock to some U M ∈ Ω \ E. In particular, given any point Examples of weak (distributional) solutions are given in Figures 111213. In these cases, the Hugoniot loci of the states belonging to E intersect the Lax curves of the following state, and no oscillations appear in the numerical approximation.

U L ∈ E, H(U L ) ∩ E = {U L }, see
5. Measure-valued solutions. In general, if the Riemann initial data take values in the elliptic region E, the approximate solutions computed through the numerical scheme ( 15)-( 16) display persistent oscillations. As examples, in this section we analyze the behavior of the approximate solutions corresponding to the initial data U L = (0.1, 0.2), U R = (0.4, 0.5) and U L = (0.4, 0.5), U R = (0.1, 0.2). Figures 14,15 show the corresponding approximate solutions for ∆x = 0.001 and ∆x = 0.0002 at time t = 1. It appears that oscillations joining the state in E with the hyperbolic region increase in number as the mesh size decreases.

To get further information on the Young measures associated to the above initial data, we compute their means (or expected values) and variance relying on formula [START_REF] Liu | The Riemann problem for general 2 × 2 conservation laws[END_REF]. In particular, we get 4) with U L = (0.2, 0.1) and U R = (0.1, 0.6) ∈ E. We observe a 1-shock, a crossing shock and a contact discontinuity in the hyperbolic region, followed by a shock joining directly U R (the last state in the hyperbolic region belongs to H((1, 0)) ∩ H(U R )).

U (t, x) := R 2 λ dν t,x (λ) = lim N →∞ 2 N (N + 1) N k=1 k U k j k (22)
for the mean densities,

F(U )(t, x) := R 2 F(λ) dν t,x (λ) = lim N →∞ 2 N (N + 1) N k=1 k F(U k j k ) (23)
for the mean fluxes, and

Var(U )(t, x) := R 2 (λ -U (t, x)) 2 dν t,x (λ) = lim N →∞ 2 N (N + 1) N k=1 k (U k j k -U (t, x)) 2 , = lim N →∞ 2 N (N + 1) N k=1 k (U k j k ) 2 -U (t, x) 2
(24) for the density variance. As expected, the means coincide with the densities and the variance is zero for values in the hyperbolic region, see Figures 16,17. and the variance computed for ∆x = 5 • 10 -6 , that is N = 2 • 10 6 . Note that we still observe oscillations for values in the elliptic region.

Aiming at giving an estimation of the accuracy of the results, we also compute the conservation errors and the convergence rates, which are given in Tables 1,2. The conservation error is given by the formula

E U cons (N, t) = E u cons (t) E v cons (t) (25) = 1 -1 U N (t, x) dx - 1 -1 U 0 (x) dx + t 0 F(U ) N (s, 1) ds - t 0 F(U ) N (s, -1) ds,
where

U N (t, x) := 2 N (N + 1) N k=1 k U k j k and F(U ) N (t, x) := 2 N (N + 1) N k=1 k F(U k j k ).
The L 1 -convergence rate is defined by

γ(N ) = γ u (N ) γ v (N ) = log 2 (e u (N )/e u (2N )) log 2 (e v (N )/e v (2N )) , (26)
where the L 1 -error is computed at final time t = 1 as

e(N ) = e u (N ) e v (N ) = U N (t, •) -U 2N (t, •) L 1 ([-1,1])
. (In the above expressions, the integrals are intended component by component.) We observe that conservation errors remain quite stable, while the L 1 -convergence rate is lower than one. Indeed, the very long computing times prevented us to reach sharper approximations. 6. Conclusions. The 2 × 2 system (1) is a simplified model for the motion of two groups of people walking in opposite directions along a corridor. This situation is known for displaying characteristic patterns as lane formation [START_REF] Helbing | Self-organizing pedestrian movement[END_REF][START_REF] Moussaïd | Traffic instabilities in self-organized pedestrian crowds[END_REF]. The system consists of two conservation laws of mixed hyperbolic-elliptic type. Though the solution configurations are not always realistic, the instabilities observed for densities in the elliptic region could be related to these auto-organization phenomena that result in a transition from a mixture to separate phases.

Following [START_REF] Frid | Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type[END_REF], we suggest that the corresponding Cauchy problem should be recast in the framework of Young measures. This allows to compute the expected values and the variance for densities and fluxes, and to estimate the flow characteristics.

Figure 1 .

 1 Figure 1. The domain Ω and the elliptic region E delimited by the curve 4 + 14uv -12u -12v + 9u 2 + 9v 2 = 0.

Figure 2 .

 2 Figure 2. 1-rarefaction curves (blue lines), 2-rarefaction curves (green lines) and fognals (dashed lines).

Figure 3 .

 3 Figure 3. Eigenvectors fields of r 1 (left) and r 2 (right) outside the elliptic region E, oriented so that ∇λ i • r i ≥ 0.

Figure 4 .

 4 Figure 4. Hugoniot locus for U L = (0.2, 0.1) in blue, U L = (0.8, 0.1) in red and U L = (0.1, 0.8) in green.

  and 11-15, left) and in the phase-plane u-v (Figures 5-9 and 11-15, right).

Figure 5 .

 5 Figure 5. Solution to (1)-(4) with U L = (0.2, 0.1) and U R = (0.1, 0.2). We observe a classical configuration consisting of a 1rarefaction and a 2-rarefaction separated by an intermediate state, here belonging to ∂E.

Figure 6 .

 6 Figure 6. Solution to (1)-(4) with U L = (0.2, 0.1) and U R = (0.1, 0.3). We observe a 1-shock, a crossing shock, a contact discontinuity, another crossing shock and a 2-shock.

Figure 7 .

 7 Figure 7. Solution to (1)-(4) with U L = (0.2, 0.1) and U R = (0.1, 0.8). We observe a 1-shock, a crossing shock, a contact discontinuity and a 2-shock.

Figure 8 .

 8 Figure 8. Solution to (1)-(4) with U L = (0.2, 0.1) and U R = (0.85, 0.1). We observe a 1-shock, a crossing shock and a 2-shock.

Figure 9 .

 9 Figure 9. Solution to (1)-(4) with U L = (0.2, 0.1) and U R = (0.75, 0.1). We observe a 1-shock, a crossing shock and a 2-shock.

  Fig.10.

Figure 10 .

 10 Figure 10. The Hugoniot locus H(U L ) for U L = (0.4, 0.5) ∈ E. Remark that H(U L ) ∩ E = U L .

Figure 11 .

 11 Figure 11. Distributional solution to (1)-(4) with U L = (0.2, 0.1) and U R = (0.6, 0.1) ∈ E. We observe a 1-shock in the hyperbolic region, followed by a shock connecting the state U R in the elliptic region (the intermediate state belongs to H(U L ) ∩ H(U R )).

Figure 12 .

 12 Figure 12. Distributional solution to (1)-(4) with U L = (0.2, 0.1) and U R = (0.1, 0.6) ∈ E. We observe a 1-shock, a crossing shock and a contact discontinuity in the hyperbolic region, followed by a shock joining directly U R (the last state in the hyperbolic region belongs to H((1, 0)) ∩ H(U R )).

Figure 13 .

 13 Figure 13. Distributional solution to (1)-(4) with initial data in the elliptic region: U L = (0.2, 0.5) ∈ E and U R = (0.6, 0.1) ∈ E. We observe two shocks joining U L and U R with an intermediate state in the hyperbolic region (the intermediate state belongs to H(U L )∩ H(U R )).

Figure 14 .

 14 Figure 14. Solution to (1)-(4) with initial data U L = (0.1, 0.2) and U R = (0.4, 0.5) ∈ E for ∆x = 0.001 (top) and ∆x = 0.0002 (bottom).

Figure 15 .

 15 Figure 15. Solution to (1)-(4) with initial data U L = (0.4, 0.5) ∈ E and U R = (0.1, 0.2) for ∆x = 0.001 (top) and ∆x = 0.0002 (bottom).

Figure 16 . 1 .

 161 Figure 16. Means (top for density values (22) and bottom right for flux (23)) and variance (24) (bottom left) corresponding to initial data U L = (0.1, 0.2) and U R = (0.4, 0.5) and N = 4 • 10 6 iterations. U L = (0.1, 0.2) U R = (0.4, 0.5) U L = (0.4, 0.5) U R = (0.1, 0.2) N E u cons

Figure 17 .

 17 Figure 17. Means (top for density values (22) and bottom right for flux (23)) and variance (24) (bottom left) corresponding to initial data U L = (0.4, 0.5) and U R = (0.1, 0.2) and N = 4 • 10 6 iterations.

Table 2 .

 2 Convergence rates (26) at time t = 1.

	1, 0.2)

This research was supported by the European Research Council under the European Union's Seventh Framework Program (FP/2007-2013) / ERC Grant Agreement n. 257661. Received xxxx 20xx; revised xxxx 20xx.