Odile Pr

Kellermann

Sophie Mouillet-Richard

Anne Baudry

Jean-Marie Launay

Odile Kellermann

MicroRNAs and depression

Keywords: microRNA, depression, antidepressant

With an estimated life-time prevalence of 15 to 17% and an incapacitating illness in 50% of cases, depressive spectrum disorders represent a heavy public health burden. Despite considerable efforts to underpin the molecular and cellular changes associated with depressive states, a global understanding of the pathophysiology of major depressive disorders (MDD) is still lacking. It is now acknowledged that deficits in synaptic plasticity, such as those resulting from chronic stress, can set the stage for the onset of depression. As a corollary, antidepressants balance neurostransmitter systems and help restore neuronal activity. In recent years, microRNAs have emerged as key protagonists in numerous physiopathological conditions including CNS function and disease. This review summarizes the current evidence for an involvement of microRNAs in the pathophysiology of depression and their contribution to the action of antidepressants.

Introduction

Depressive spectrum disorders are a major public health concern, affecting 121 million people worldwide. According to the World Health Organization, these diseases are predicted to become the second leading cause of disability by 2020. Despite considerable advances over the past decades, a clear understanding of the etiology of depression is still lacking. This notably relates to the complex interplay between genetic, immuno-inflammatory and environmental factors that dictate the relative susceptibility of individuals to affective disorders [START_REF] Blume | Immune suppression and immune activation in depression[END_REF][START_REF] Dudley | Epigenetic mechanisms mediating vulnerability and resilience to psychiatric disorders[END_REF][START_REF] Lesch | When the serotonin transporter gene meets adversity: the contribution of animal models to understanding epigenetic mechanisms in affective disorders and resilience[END_REF]. A widespread view is that neuronal plasticity is challenged in key areas of the depressed brain and that antidepressants trigger a set of adaptive changes that help restore these defects. In this review, we focus on the involvement of microRNAs in the pathophysiology of depression and on their contribution to the action of antidepressants.

Depressive disorders: pathophysiological and etiological aspects

Neurotransmitter systems involved in the biology of depression

In view of the complexity of the brain circuitry, grasping the overall changes associated with depression is obviously very challenging [START_REF] Benton | Biological alterations in depression. Psychiatric disorders -trends and development[END_REF][START_REF] Tanti | Open questions in current models of antidepressant action[END_REF]. Analyses carried out on post mortem brains of patients suffering from major depressive disorder (MDD) have allowed to characterize neuroanatomical alterations, including reduced neuronal densities in the prefrontal cortex, the raphe nuclei and the locus coeruleus, hypertrophy of the amygdala and reductions in hippocampal volume [START_REF] Hercher | Through the looking glass: examining neuroanatomical evidence for cellular alterations in major depression[END_REF]. The pathophysiology of depressive states appears to also include glial cells, since depressed patients exhibit a profound loss of oligodendrocytes and myelin as well as reductions in densities of astrocytes [START_REF] Allen | Neuroscience: Glia -more than just brain glue[END_REF][START_REF] Hercher | Through the looking glass: examining neuroanatomical evidence for cellular alterations in major depression[END_REF]. Brain imaging studies such as functional magnetic resonance imaging (fMRI) are providing further information on the functional disturbances associated with depressive states, leading to the identification of a brain-based endophenotype for MDD [START_REF] Peterson | A brain-based endophenotype for major depressive disorder[END_REF].

The involvement of various brain regions and neurotransmitter systems in the pathophysiology of depression is now well-established. Based on the observation over 50 years ago that the first classes of antidepressants, i.e. monoamine oxidases inhibitors (MAOIs) and tricyclic antidepressants (TCAs), target the monoaminergic systems, it was initially proposed that depression results from imbalances in the brain levels of monoamines [START_REF] Coppen | The biochemistry of affective disorders[END_REF][START_REF] Lapin | Intensification of the central serotoninergic processes as a possible determinant of the thymoleptic effect[END_REF][START_REF] Schildkraut | The catecholamine hypothesis of affective disorders: a review of supporting evidence[END_REF]. The "monoamine theory of affective disorders" was reinforced by the development of antidepressants targeting the serotonin transporter (SERT) (Serotonin Reuptake Inhibitors, SRIs), the norepinephrine transporter (NET) (Norepinephrine Reuptake Inhibitors, NRIs), or both (Serotonin Norepinephrine Reuptake Inhibitors, SNRIs). Incidentally, most antidepressants available to date increase serotonergic and/or noradrenergic neurotransmission [START_REF] Baudry | New views on antidepressant action[END_REF].

However, these drugs suffer several limitations: (i) relatively modest remission rates (symptom improvement in about 50% of depressed patients and remission in 30 to 40%) [START_REF] Rush | Bupropion-SR, sertraline, or venlafaxine-XR after failure of SSRIs for depression[END_REF], (ii) relatively slow onset of efficacy and delayed time to remission so that, of ultimate remitters, as many as half will not remit until 6 to 12 weeks of ongoing antidepressant therapy [START_REF] Trivedi | Evaluation of outcomes with citalopram for depression using measurement-based care in STAR*D: implications for clinical practice[END_REF], and (iii) lower effectiveness for certain depressive symptoms such as sleep disturbances and fatigue than for others [START_REF] Nierenberg | Residual symptoms in depressed patients who respond acutely to fluoxetine[END_REF]. These clinical observations have motivated the quest for a better knowledge of the molecular and cellular events sustaining depressive states. There is now clear evidence that depression is associated with deficits in neurotransmitter systems other than serotonergic or noradrenergic.

For instance, extensive analysis of the mesolimbic dopamine reward circuit in mouse models of depression has uncovered a detrimental overactivation of dopamine neurons of the ventral tegmental area (VTA), which is reversed upon long-term antidepressant treatment [START_REF] Cao | mTOR Signaling and Entrainment of the Mammalian Circadian Clock[END_REF][START_REF] Krishnan | Linking molecules to mood: new insight into the biology of depression[END_REF]. Dopaminergic neurons from the VTA project to the nucleus accumbens (NAc), a brain region located within the striatum, whose hypoactivity is associated with anhedonia, a prevalent manifestation of depression [START_REF] Krishnan | Linking molecules to mood: new insight into the biology of depression[END_REF][START_REF] Shirayama | Neurochemistry of the nucleus accumbens and its relevance to depression and antidepressant action in rodents[END_REF].

Most neurons within the NAc are medium spiny GABAergic projection neurons, whose activity is inhibited by dopamine [START_REF] Shirayama | Neurochemistry of the nucleus accumbens and its relevance to depression and antidepressant action in rodents[END_REF]. Dysfunction of the GABAergic system in depression extends beyond the NAc and deficits in GABAergic transmission may notably contribute to stress-induced impairment of hippocampal neurogenesis (see below) [START_REF] Luscher | The GABAergic deficit hypothesis of major depressive disorder[END_REF].

There is growing interest for the glutamatergic system, which is influenced by traditional antidepressants targeting the monoaminergic systems [START_REF] Mitchell | An update on the role of glutamate in the pathophysiology of depression[END_REF]. Increased glutamate levels have been reported in the plasma and the brains of patients suffering for MDD [START_REF] Hashimoto | The role of glutamate on the action of antidepressants[END_REF]. The most compelling evidence for a role of glutamate in depression relates to the antidepressant effect of antagonists targeting the NMDA receptor [START_REF] Li | Glutamate N-methyl-D-aspartate receptor antagonists rapidly reverse behavioral and synaptic deficits caused by chronic stress exposure[END_REF]. Among these drugs, the non-selective NMDA channel blocker ketamine was shown to operate by enhancing synaptic plasticity via the mTOR pathway [START_REF] Li | mTOR-dependent synapse formation underlies the rapid antidepressant effects of NMDA antagonists[END_REF]. Ketamine is now under trial and is expected to provide an alternative treatment for patients resistant to traditional therapies [START_REF] Cryan | Neuroscience. A glutamate pathway to faster-acting antidepressants[END_REF].

Genetic susceptibility to depression

In view of the significant heritability of MDD, estimated to range from 34% to 42% [START_REF] Ebmeier | Recent developments and current controversies in depression[END_REF], the identification of susceptibility genes has raised much interest. Candidate gene studies favour an association of several genes with the disease, including those encoding the brain-derived neurotrophic factor (BDNF), TPH2, and the SERT (SLC6A4 gene) [START_REF] Levinson | The genetics of depression: a review[END_REF]. More than 20 variants of the latter gene have been described [START_REF] Murphy | Human serotonin transporter gene (SLC6A4) variants: their contributions to understanding pharmacogenomic and other functional GxG and GxE differences in health and disease[END_REF]. Among these, variations in the SERT-linked polymorphic region (5-HTTPLR) have been associated with reduced SERT expression and function, comparable to those recorded in Slc6a4+/-mice [START_REF] Murphy | Targeting the murine serotonin transporter: insights into human neurobiology[END_REF]. This polymorphism may notably drive the vulnerability to depression in response to stressful life events [START_REF] Caspi | Influence of life stress on depression: moderation by a polymorphism in the 5-HTT gene[END_REF]. In the near future, genome-wide association studies, despite present poor replication [START_REF] Bosker | Poor replication of candidate genes for major depressive disorder using genome-wide association data[END_REF], might shed further light on the genetic susceptibility to MDD, as recently documented for polymorphisms in the sodium-dependent neutral amino acid transporter gene SLC6A15 [START_REF] Kohli | The Neuronal Transporter Gene SLC6A15 Confers Risk to Major Depression[END_REF] and the synaptic scaffolding protein HOMER1 [START_REF] Rietschel | Genome-wide association-, replication-, and neuroimaging study implicates HOMER1 in the etiology of major depression[END_REF].

Stress, impaired neurogenesis and disturbances in circadian rhythms: risk factors for depression

A major contributing factor in the etiology of MDD is stress. Indeed, major depression is associated with impairments in learning and memory, hippocampal atrophy as well as hyperactivity of the HPA axis, all of which are known to be direct consequences of acute and chronic stress [START_REF] Berton | New approaches to antidepressant drug discovery: beyond monoamines[END_REF][START_REF] Lee | Depression research: where are we now?[END_REF][START_REF] Pittenger | Stress, depression, and neuroplasticity: a convergence of mechanisms[END_REF].

Chronic stress triggers excess glucocorticoid levels, in turn leading to impaired neurogenesis, excitotoxic damage and reduced levels of neurotrophins in the hippocampus [START_REF] Anacker | The glucocorticoid receptor: pivot of depression and of antidepressant treatment?[END_REF]. Impaired hippocampal neurogenesis seems to represent a core pathological feature of major depression [START_REF] Duman | Depression: a case of neuronal life and death?[END_REF][START_REF] Macqueen | The hippocampus in major depression: evidence for the convergence of the bench and bedside in psychiatric research[END_REF][START_REF] Pittenger | Stress, depression, and neuroplasticity: a convergence of mechanisms[END_REF], although the "neurogenesis hypothesis of depression" is currently under debate [START_REF] Hanson | Depression, antidepressants, and neurogenesis: a critical reappraisal[END_REF]. Excess glucocorticoid receptor activation, deficient neurotrophic signals, altered CREB signalling have all been monitored in depressed patients [START_REF] Dwivedi | Evidence demonstrating role of microRNAs in the etiopathology of major depression[END_REF]. Basic and clinical studies provide some evidence for elevated secretion of the hypothalamic neuropeptides CRH and vasopressin in depression and anxiety [START_REF] Holsboer | Stress hormone regulation: biological role and translation into therapy[END_REF].

These findings suggest that CRH1 receptors may be potential drug targets.

Depression may further be influenced by disturbances in the circadian clock [START_REF] Mendlewicz | Disruption of the circadian timing systems: molecular mechanisms in mood disorders[END_REF]. The integration of circadian rhythms operates in the suprachiasmatic nucleus (SCN), within the hypothalamus. This brain region receives dense serotonergic innervation from the raphe as well as glutamatergic input from the retina. Its major output is the paraventricular nucleus of the hypothalamus, which supports the connection between the circadian clock and the HPA axis [START_REF] Monteleone | Circadian rhythms and treatment implications in depression[END_REF].

Hence, the onset of depression appears to be governed by multiple etiological factors, which may fuel each other within pathophysiological interplays. Further expansions of the current armamentarium of drug treatments of depression will depend on the discovery of new pathways and targets for antidepressant treatment, but fortunately several other lines of research hold promise for making these discoveries.

Dysregulation of microRNAs in depression

Biogenesis and functions of microRNAs

Over the past few years, microRNAs (miRNAs) have gained center stage as key posttranscriptional regulators of gene expression. By fine-tuning the translation of 50% of proteincoding genes, miRNAs appear as robust determinants of cellular states [START_REF] Krol | The widespread regulation of microRNA biogenesis, function and decay[END_REF].

MiRNAs are transcribed as pri-miRNA by RNA-polII, exported to the cell cytoplasm after processing into pre-miRNA by the Drosha complex, then further cleaved by Dicer to generate double stranded mature miRNAs. One strand of the mature miRNA is loaded within the RISC complex and binds its target mRNA to impair its translation or promote its degradation, while the other strand of the duplex is degraded (reviewed in [START_REF] Krol | The widespread regulation of microRNA biogenesis, function and decay[END_REF]). The miRNAdependent regulation of gene expression may in fact be more complex since, in some instances, miRNAs may on the contrary act as activators of translation [START_REF] Krol | The widespread regulation of microRNA biogenesis, function and decay[END_REF].

Finally, recent evidence suggests that they may also impair transcription by binding to gene promoters [START_REF] Younger | Transcriptional gene silencing in mammalian cells by miRNA mimics that target gene promoters[END_REF].

microRNAs in the brain

A large number of miRNAs was shown to be either specifically expressed in the brain or enriched in the central nervous system [START_REF] Lagos-Quintana | Identification of tissue-specific microRNAs from mouse[END_REF]. A specificity of some neuronal miRNAs is their localization into the synaptodendritic compartment, associated with their ability to locally regulate the translation of target mRNAs (reviewed in [START_REF] Schratt | microRNAs at the synapse[END_REF]).

Interestingly, the turnover of miRNA in neurons appears to be faster than in other cell types, suggesting that the neuronal miRNA system may contribute to the rapid adaptation to neuronal activity [START_REF] Krol | The widespread regulation of microRNA biogenesis, function and decay[END_REF]. The production of miRNA at the synapse is indeed under the control of neuronal activity, and may involve calpain-dependent activation of Dicer [START_REF] Schratt | microRNAs at the synapse[END_REF].

microRNA dysfunction: genetic and epigenetic mechanisms

Disease-associated dysregulation of miRNAs may encompass a variety of aberrant pathways (Fig. 1). These include altered miRNA expression, which may itself relate to genomic deletion or amplification, epigenetic changes or single nucleotide polymorphisms (SNP) ( [START_REF] Sethupathy | MicroRNA target site polymorphisms and human disease[END_REF] and references therein). Alternatively, polymorphisms within target genes may abrogate the recognition site for relevant miRNAs and create seed sequence for inappropriate miRNAs [START_REF] Sethupathy | MicroRNA target site polymorphisms and human disease[END_REF][START_REF] Sun | SNPs in human miRNA genes affect biogenesis and function[END_REF]. While these overall pathogenic mechanisms have been largely documented in the cancer field [START_REF] Garzon | Targeting microRNAs in cancer: rationale, strategies and challenges[END_REF][START_REF] Ryan | Genetic variation in microRNA networks: the implications for cancer research[END_REF], only few of them have turned out to be involved in depression at present. To date, such examples are in fact limited to two polymorphisms located in pre-miR-30e [START_REF] Xu | A polymorphism in the microRNA-30e precursor associated with major depressive disorder risk and P300 waveform[END_REF] and pre-miR-182 [START_REF] Saus | Genetic variants and abnormal processing of pre-miR-182, a circadian clock modulator, in major depression patients with late insomnia[END_REF]) that are associated with major depressive disorder risk, in particular with late insomnia for the latter. In the case of pre-miR-182, the SNP leads to an up-regulation of miR-182 and a decrease in its targets -notably CLOCK-providing a molecular explanation for the clinical signs of circadian disturbance in patients. As for SNPs in the 3'UTR region of mRNAs encoding proteins involved in pathophysiology of depression, the only defect in miRNA recognition reported to date relates to the serotonergic 5-HT1B receptor [START_REF] Jensen | A common polymorphism in serotonin receptor 1B mRNA moderates regulation by miR-96 and associates with aggressive human behaviors[END_REF]. By lowering the affinity for miR-96, the rs13212041 SNP located in the 3'UTR of the 5-HT1B receptor mRNA leads to increased receptor expression and reduced aggressive behaviour [START_REF] Jensen | A common polymorphism in serotonin receptor 1B mRNA moderates regulation by miR-96 and associates with aggressive human behaviors[END_REF]. A very recent study further suggests that the rs27072 SNP in the 3'UTR of the dopamine transporter (DAT) may affect miRNA-mediated regulation [START_REF] Pinsonneault | Dopamine Transporter Gene Variant Affecting Expression in Human Brain is Associated with Bipolar Disorder[END_REF]. This SNP is associated with bipolar disorder, an affective disorder with partial overlap with MDD [START_REF] Pinsonneault | Dopamine Transporter Gene Variant Affecting Expression in Human Brain is Associated with Bipolar Disorder[END_REF]. In view of the multiplicity of SNPs in the genes encoding the SERT or the NET [START_REF] Baffa | Norepinephrine and serotonin transporter genes: impact on treatment response in depression[END_REF][START_REF] Murphy | Human serotonin transporter gene (SLC6A4) variants: their contributions to understanding pharmacogenomic and other functional GxG and GxE differences in health and disease[END_REF], one cannot exclude that some may impact miRNA recognition.

microRNAs and susceptibility to depression: facts and hypotheses

An appealing approach to probe for dysregulated miRNA in depression would be to carry out large-scale analyses on post-mortem brain material [START_REF] Dwivedi | Evidence demonstrating role of microRNAs in the etiopathology of major depression[END_REF]. Beyond technical hurdles, one difficulty relates to possible variations according to brain regions, as observed with BDNF levels [START_REF] Wang | The when and where of BDNF and the antidepressant response[END_REF]). An alternative approach would be to exploit so-called animal models of depression [START_REF] Nestler | Animal models of neuropsychiatric disorders[END_REF], with the possibility to further assess the impact of antidepressant treatment on global miRNA expression, as carried out for in transcriptome studies [START_REF] Surget | Corticolimbic transcriptome changes are state-dependent and regionspecific in a rodent model of depression and of antidepressant reversal[END_REF].

Theoretically, any dysregulation of miRNAs involved in the pathways associated with vulnerability to affective disorders may precipitate or aggravate depression (Fig. 1). As mentioned above, stress, impaired neurogenesis and defects in synaptic plasticity represent three interconnected factors that are associated with depression. The global miRNA response to acute and chronic stress is only beginning to be disclosed [START_REF] Meerson | Changes in brain MicroRNAs contribute to cholinergic stress reactions[END_REF][START_REF] Rinaldi | Stress induces region specific alterations in microRNAs expression in mice[END_REF][START_REF] Uchida | Early life stress enhances behavioral vulnerability to stress through the activation of REST4-mediated gene transcription in the medial prefrontal cortex of rodents[END_REF]. Along the glucocorticoid-stress axis, several candidate miRNAs promoting the downregulation of the glucocorticoid receptor (GR) have already been identified, and may have relevance as to depression. These include miR-18, which was shown to be increased in the paraventricular nucleus of the hypothalamus of stresshyperresponsive rats. Another GR-targeting miRNA is miR-124a, the most abundant miRNA in the brain [START_REF] Lagos-Quintana | Identification of tissue-specific microRNAs from mouse[END_REF]. Because GR activation may interfere with hippocampal neurogenesis [START_REF] Anacker | The glucocorticoid receptor: pivot of depression and of antidepressant treatment?[END_REF], miR-124 may be critical for maintaining GR expression at levels permissive to neurogenesis [START_REF] Cheng | miR-124 regulates adult neurogenesis in the subventricular zone stem cell niche[END_REF]. Beyond miR-124, adult hippocampal neurogenesis appear to be associated with changes of a wide array of miRNAs [START_REF] Luikart | Dentate gyrus neurogenesis, integration and microRNAs[END_REF]. miR-132 and miR-212 feature among the most highly upregulated miRNAs in response to neuronal activity in the dentate gyrus [START_REF] Luikart | Dentate gyrus neurogenesis, integration and microRNAs[END_REF]. In line with this observation, cre-mediated deletion of miR-132/212 results in decreased dendritic arborization of newborn neurons in this brain region [START_REF] Magill | microRNA-132 regulates dendritic growth and arborization of newborn neurons in the adult hippocampus[END_REF].

Much effort is definitely being made to assess the contribution of miRNAs to the control of neuronal maturation, synapse formation and functional integration of new neurons in the brain circuitry [START_REF] Shi | MicroRNA regulation of neural stem cells and neurogenesis[END_REF].

The involvement of miRNAs in synaptic plasticity is also being actively scrutinized (reviewed in [START_REF] Siegel | microRNAs in neurons: manifold regulatory roles at the synapse[END_REF]). The first report on the regulation of local protein synthesis in dendrites by microRNAs was by Schratt et al. [START_REF] Schratt | A brain-specific microRNA regulates dendritic spine development[END_REF]. In this work, the brainspecific miR-134 was shown to interfere with the growth of dendritic spines by targeting the Lim kinase 1 (LIMK-1) and to be downregulated when neurons are exposed to BDNF.

Further, excessive levels of miR-134 impair memory and plasticity by repressing the expression of the cAMP response element binding protein (CREB) [START_REF] Gao | A novel pathway regulates memory and plasticity via SIRT1 and miR-134[END_REF]. The downregulation of this transcription factor in turn decreases the expression of BDNF [START_REF] Gao | A novel pathway regulates memory and plasticity via SIRT1 and miR-134[END_REF], highlighting reciprocal antagonism between miR-134 and BDNF [START_REF] Siegel | microRNAs in neurons: manifold regulatory roles at the synapse[END_REF]. Another feedback loop contributing to neuronal plasticity involves the miR-124/CREB duo (reviewed in [START_REF] Siegel | microRNAs in neurons: manifold regulatory roles at the synapse[END_REF]). In Aplysia, miR-124 controls serotonin-induced synaptic facilitation through post-transcriptional repression of CREB [START_REF] Rajasethupathy | Characterization of small RNAs in Aplysia reveals a role for miR-124 in constraining synaptic plasticity through CREB[END_REF] and, conversely, CREB can regulate the expression of miR-124 [START_REF] Rajasethupathy | Characterization of small RNAs in Aplysia reveals a role for miR-124 in constraining synaptic plasticity through CREB[END_REF]. The network of synaptic miRNAs also includes miR-132, which is increased in response to BDNF and regulates neuronal morphogenesis by targeting the GTPase activating protein p250GAP [START_REF] Vo | A cAMP-response element binding protein-induced microRNA regulates neuronal morphogenesis[END_REF]. Interestingly, excess glucocorticoids were recently shown to interfere with the BDNF/miR-132 cascade [START_REF] Kawashima | Glucocorticoid attenuates brain-derived neurotrophic factordependent upregulation of glutamate receptors via the suppression of microRNA-132 expression[END_REF]. Recent data actually posit miR-132 as a key player in the integration of neuronal functions. For instance, a wellcharacterized target of miR-132 is the transcriptional regulator MeCP2 [START_REF] Klein | Homeostatic regulation of MeCP2 expression by a CREB-induced microRNA[END_REF], involved in neuronal activity [START_REF] Chahrour | MeCP2, a key contributor to neurological disease, activates and represses transcription[END_REF][START_REF] Diaz De Leon-Guerrero | In sickness and in health: the role of methyl-CpG binding protein 2 in the central nervous system[END_REF][START_REF] Greer | From synapse to nucleus: calcium-dependent gene transcription in the control of synapse development and function[END_REF]. Finally, miR-132 may play a role in circadian rhythms since it is induced by light in the suprachiasmatic nucleus and regulates the expression of Period genes [START_REF] Alvarez-Saavedra | miRNA-132 orchestrates chromatin remodeling and translational control of the circadian clock[END_REF].

Involvement of microRNAs in the action of antidepressants

A role for miR-16 in the action of SRI antidepressants

The involvement of microRNAs in the response to antidepressants is only beginning to be explored. Recently, we have unveiled a key role for miR-16 in the antidepressant action of the SRI fluoxetine [START_REF] Baudry | MiR-16 Targets the Serotonin Transporter: A New Facet for Adaptive Responses to Antidepressants[END_REF]. miR-16 acts as a posttranscriptional repressor of the SERT, the pharmacological target of SRIs. miR-16 is expressed at higher levels in noradrenergic than serotonergic cells, both in vitro and in vivo. In noradrenergic neurons, neutralization of miR-16 unlocks SERT expression and renders the cells competent to bind SRI antidepressants. Based on the observation that SRIs reduce SERT protein while not altering its mRNA levels [START_REF] Benmansour | Serotonin clearance in vivo is altered to a greater extent by antidepressant-induced downregulation of the serotonin transporter than by acute blockade of this transporter[END_REF], we postulated that antidepressants may promote changes in miR-16. Indeed, in mice, chronic fluoxetine treatment increases the level of miR-16 in raphe, which decreases the protein level of SERT. From a mechanistic point of view, the raise in miR-16 results from an increased maturation from pri/pre-miR-16. This process depends on the activation of GSK3, an effector of the Wnt canonical pathway.

Serotonergic neurons additionally respond to fluoxetine by releasing the neurotrophic factor S100. S100 secreted by serotonergic neurons acts on noradrenergic neurons by promoting a decrease in miR-16, which turns on the expression of SERT. Other serotonergic functions are also implemented following the reduction of miR-16, notably serotonin synthesis. This observation is in keeping with the monoamine theory of affective disorders, according to which increasing brain serotonin levels is beneficial to depressive states. Building upon these observations, we explored whether miR-16 may relay the action of SRIs in other brain regions. This is indeed the case in the hippocampus: fluoxetine promotes a decrease in miR-16, which in turn triggers a raise in the protein levels of SERT as well as those of the antiapoptotic factor bcl-2, a well-established target of miR-16 in cancer [START_REF] Aqeilan | miR-15a and miR-16-1 in cancer: discovery, function and future perspectives[END_REF], thereby sustaining hippocampal neurogenesis [START_REF] Launay | Raphe-mediated signals control the hippocampal response to SRI antidepressants via miR-16[END_REF]. These results allow to define miR-16 as a "missing link" between SRIs and adult hippocampal neurogenesis.

Further, in view of its impact on multiple targets, miR-16 behaves as a micromanager of the hippocampal response to fluoxetine. Taken together, our observations support the view that miR-16 is widely involved in the brain response to fluoxetine. Importantly, the fluoxetineinduced changes in miR-16 actually sustain the antidepressant action of the drug. Indeed, using mouse models of depression, we observed that the upregulation of miR-16 in raphe or the downregulation of miR-16 in the locus coeruleus [START_REF] Baudry | MiR-16 Targets the Serotonin Transporter: A New Facet for Adaptive Responses to Antidepressants[END_REF] or in the hippocampus [START_REF] Launay | Raphe-mediated signals control the hippocampal response to SRI antidepressants via miR-16[END_REF] induce behavioural responses that compare that induced by fluoxetine.

As rightly suggested by Millan [START_REF] Millan | MicroRNA in the regulation and expression of serotonergic transmission in the brain and other tissues[END_REF], whether fluoxetine influences the expression of miRNAs other miR-16, notably the closely related miR-15 and miR-107 will have to be assessed. Another related issue concerns the potential miRNA-mediated post-transcriptional regulation of other key monoaminergic functional molecules. For instance, can miRNAs regulate the expression of the NET and take part to the response to antidepressants? In direct support of this hypothesis, long-term treatments with various antidepressants reduce the protein levels of NET while having no effect on NET mRNA [START_REF] Zhao | Association of changes in norepinephrine and serotonin transporter expression with the long-term behavioral effects of antidepressant drugs[END_REF]. The regulation of NET expression by miRNAs would notably provide some explanation for the perplexing lack of alteration in the expression of genes associated with monoaminergic systems in MDD [START_REF] Fiori | Broadening our horizons: Gene expression profiling to help better understand the neurobiology of suicide and depression[END_REF].

Antidepressant responses: established pathways, candidate microRNAs

Our understanding of the pathways sustaining the response to antidepressant drugs has greatly progressed over the past few years [START_REF] Baudry | New views on antidepressant action[END_REF]. The current knowledge of the effectors at play further allows considering the implication of miRNAs (Fig. 2). For instance, a large body of evidence supports a contribution of the Wnt-GSK3 pathway to the action of antidepressants ( [START_REF] Baudry | New views on antidepressant action[END_REF][START_REF] Wada | Lithium and neuropsychiatric therapeutics: neuroplasticity via glycogen synthase kinase-3beta, beta-catenin, and neurotrophin cascades[END_REF] and references therein). The GSK3 kinase is the target of the mood stabilizer lithium [START_REF] Beaulieu | Looking at lithium: molecular moods and complex behaviour[END_REF]. Large scale analysis of the hippocampal miRNA changes induced by lithium and valproate, another mood stabilizer, has revealed decreases in let-7b, let-7c, miR-128a, miR-24a, miR-30c, miR-34a, miR-221 and a raise in miR-144 [START_REF] Zhou | Evidence for selective microRNAs and their effectors as common longterm targets for the actions of mood stabilizers[END_REF]. Except for an increase in the mGluR7 receptor as a result of reduced miR-34a levels, the mechanisms through which these changes in miRNA contribute to the therapeutic effect of lithium remain to be investigated [START_REF] Zhou | Evidence for selective microRNAs and their effectors as common longterm targets for the actions of mood stabilizers[END_REF]. Further, from a mechanistic point of view, how GSK3 may regulate the levels of specific miRNAs is elusive. It may notably relate to the blockade of pri/pre-miRNA maturation, as we observed for miR-16 in the locus coeruleus [START_REF] Baudry | MiR-16 Targets the Serotonin Transporter: A New Facet for Adaptive Responses to Antidepressants[END_REF]. That GSK3 may regulate pri-miR maturation is notably supported by the recent observation that GSK3 controls the nuclear localization of the RNAase III enzyme Drosha through phosphorylation [START_REF] Tang | Glycogen Synthase Kinase 3 Beta (GSK3beta) Phosphorylates the RNAase III Enzyme Drosha at S300 and S302[END_REF] The therapeutic action of antidepressants further involves the restoration of neuronal plasticity [START_REF] Duman | Depression: a case of neuronal life and death?[END_REF][START_REF] Macqueen | The hippocampus in major depression: evidence for the convergence of the bench and bedside in psychiatric research[END_REF][START_REF] Pittenger | Stress, depression, and neuroplasticity: a convergence of mechanisms[END_REF]. This process may in part rely on neurotrophic signals, able to counteract stress-induced impairment of hippocampal neurogenesis [START_REF] Duman | Depression: a case of neuronal life and death?[END_REF]. In direct support of this hypothesis is our recent demonstration that fluoxetine operates on the hippocampus by mobilizing BDNF, Wnt2, and 15-Deoxy-delta12,14-prostaglandin J2 [START_REF] Launay | Raphe-mediated signals control the hippocampal response to SRI antidepressants via miR-16[END_REF]. These signalling molecules emanating from the raphe synergize to downregulate miR-16, and increase SERT and bcl-2 protein levels in the hippocampus [START_REF] Launay | Raphe-mediated signals control the hippocampal response to SRI antidepressants via miR-16[END_REF]. Notably, the normalization of BDNF levels in response to antidepressants [START_REF] Castren | The role of BDNF and its receptors in depression and antidepressant drug action: Reactivation of developmental plasticity[END_REF][START_REF] Launay | Raphe-mediated signals control the hippocampal response to SRI antidepressants via miR-16[END_REF] is likely to sustain a cascade of beneficial events leading to appropriate control of miRNA expression, beyond the regulation of miR-16. Of note, BDNF has the capacity to regulate gene expression by activating the mTOR pathway [START_REF] Schratt | BDNF regulates the translation of a select group of mRNAs by a mammalian target of rapamycin-phosphatidylinositol 3-kinase-dependent pathway during neuronal development[END_REF]. Whether the BDNFdependent activation of mTOR also regulates miRNA expression deserves further consideration.

Interestingly, the mTOR pathway is also involved in the antidepressant action of ketamine [START_REF] Li | mTOR-dependent synapse formation underlies the rapid antidepressant effects of NMDA antagonists[END_REF]. Whether the ketamine-mTOR cascade mobilizes miRNAs to enhance synaptic plasticity has yet to be investigated. mTOR further exerts a regulatory role in the entrainment of the mammalian circadian clock [START_REF] Cao | mTOR Signaling and Entrainment of the Mammalian Circadian Clock[END_REF]. Again, whether mTOR signalling regulates miRNA expression in the suprachiasmatic nucleus has not been addressed. Beyond mTOR-dependent mechanisms, the restoration of the BDNF/CREB/miR-132 cascade in this brain region following antidepressant treatment is certainly critical to the resetting of the circadian clock [START_REF] Cheng | Revealing a role of microRNAs in the regulation of the biological clock[END_REF].

Adding another layer of complexity to the picture, there is now clear-cut evidence that the action of antidepressants may involve chromatin remodeling [START_REF] Baudry | New views on antidepressant action[END_REF][START_REF] Covington | From synapse to nucleus: novel targets for treating depression[END_REF]. Epigenetic mechanisms distinct from miRNA changes notably operate at the level of BDNF gene transcription [START_REF] Tsankova | Sustained hippocampal chromatin regulation in a mouse model of depression and antidepressant action[END_REF]. Their contribution to adult hippocampal neurogenesis is also largely documented [START_REF] Hsieh | Epigenetics, hippocampal neurogenesis, and neuropsychiatric disorders: unraveling the genome to understand the mind[END_REF][START_REF] Ma | Epigenetic choreographers of neurogenesis in the adult mammalian brain[END_REF]. The regulation of miRNA expression is intimately connected with the epigenetic machinery [START_REF] Iorio | Interplay between microRNAs and the epigenetic machinery: an intricate network[END_REF]. A dramatic example is the association between aberrant methylation of defined miRNAs and cancer [START_REF] Iorio | Interplay between microRNAs and the epigenetic machinery: an intricate network[END_REF]. Cleary, more work is needed to assess the potential implication of epigenetic regulators in the miRNAs response to antidepressants.

Lessons from resilience

Our ability to adapt to adverse events is highly promiscuous and may notably depend on early life stress [START_REF] Feder | Psychobiology and molecular genetics of resilience[END_REF]. The molecular mechanisms associated with resilience, i.e. the capacity to respond positively to stressfull events, are still very elusive [START_REF] Feder | Psychobiology and molecular genetics of resilience[END_REF].

Recent works suggest that resilience is associated not only with an absence of changes in gene expression seen in vulnerable individuals but also with the induction of distinct beneficial changes, in response to stress [START_REF] Feder | Psychobiology and molecular genetics of resilience[END_REF][START_REF] Uchida | Epigenetic status of Gdnf in the ventral striatum determines susceptibility and adaptation to daily stressful events[END_REF]. The possibility that these changes include the regulation of specific miRNAs deserves consideration. Analysing the profile of miRNA expression in the brain of stress-resilient versus stress-sensitive mice may help shed some light on this issue. Another approach would be to exploit the few transgenic mouse models exhibiting a depression-resistant phenotype, such as animals rendered insensitive to regulators of GPCR signalling (RGS), one of the first example being a negative regulator of the serotonin 5-HT1A receptor function [START_REF] Talbot | RGS inhibition at G(alpha)i2 selectively potentiates 5-HT1Amediated antidepressant effects[END_REF]. Profiling studies may alternatively benefit from mice deficient in NET, which behave like antidepressant-treated wild-type mice [START_REF] Haenisch | Depression and antidepressants: insights from knockout of dopamine, serotonin or noradrenaline re-uptake transporters[END_REF].

Conclusion and future directions

To date, the actual experimental evidence for an involvement of miRNAs in the pathophysiology and treatment of depression remains scarce. Nevertheless, the involvement of miRNAs in the diverse pathways known to be associated with depression argues for a dysregulation of miRNA-related homeostatic loops in MDD. On another hand, the restoration of efficient neurogenesis, neurotransmission, structural and functional brain plasticity as well as circadian rhythms upon antidepressant treatment is likely to require a concerted miRNA response. The challenge now is to reach an integrated view of the network of miRNAs that are mobilized by antidepressants as a function of the brain regions at play. Further, the identification of functionally relevant targets of these miRNAs should help unravel the global molecular and cellular changes sustaining the therapeutic action of antidepressants. Another important issue to be addressed relates to the reversibility of these changes. Indeed, the rapid turnover of neuronal miRNAs allows for a dynamic adaptation to external inputs [START_REF] Rajewsky | microRNAs and the Operon Paper[END_REF]. Deciphering the step-wise changes in the expression of appropriate miRNAs should eventually help identify the molecular events that trigger recovery from depression.

Legends to Figures

Figure 1 :

 1 Figure 1: Potential dysregulated pathways involving miRNAs that may contribute to

Figure 2 :

 2 Figure 2: Schematic representation of the pathways mobilized by antidepressants that may

Acknowledgements

We apologize to all researchers whose work could not be cited owing to space limitation. We thank Vincent Mutel, Rolf Kettler and Andrea Cesura for critical reading of the manuscript. O.K. is a professor at Paris XI University. Work at the Kellermann's lab is funded by ANR and INSERM.


C Piotrowicz

B Mohamad

B Rrustemi

N Malbert

M.-A A Jaud

W Vandendaele

M Charles

R Gwoziecki

N Malbert+

A Comprehensive Analysis of AlN spacer and AlGaN n-doping effects on the 2DEG Resistance in AlGaN/AlN/GaN Heterostructures

Keywords: AlGaN/AlN/GaN heterostructure, 2DEG, AlN spacer, n-doped AlGaN, Schrödinger-Poisson simulations, Measurements

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

GaN HEMT technology based on AlGaN/AlN/GaN heterostructures on 200 mm diameter Si (111) substrate has aroused a lot of interest for power electronics. Thanks to their crystal and polarization properties, III-N materials are very promising due to high breakdown field and excellent transport properties. To further increase the output power in HEMTs, the resistance of the bi-dimensional electron gas channel (R2DEG) needs to be reduced as far as possible. This improvement can be achieved through either ns or µs enhancement. The AlN spacer between the AlGaN and GaN layer is known to increase the ns, but its physical root cause is not clearly identified [START_REF] Ambacher | Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N-and Ga-face AlGaN/GaN heterostructures[END_REF][START_REF] Shrestha | Simulation study on electrical characteristic of AlGaN/GaN high electron mobility transistors with AlN spacer layer[END_REF]. In addition, the effect of n-doped AlGaN on the ns is investigated [START_REF] Ambacher | Two dimensional electron gases induced by spontaneous and piezoelectric polarization in undoped and doped AlGaN/GaN heterostructures[END_REF][START_REF] Sonmez | The effect of barrier layers on 2D electron effective mass in Al0.3Ga0.7N/AlN/GaN heterostructures[END_REF][START_REF] Mondal | Effect of Doped AlGaN Width Variation on Analog Performance of Dual Gate Underlap MOS-HEMT[END_REF][START_REF] Wośko | Influence of growth process scheme on the properties of AlGaN/AlN/GaN heterostructures[END_REF].

Experiments

In this work, four different epitaxial-variations grown by MOCVD are investigated. In this work, ID(VG) and C(VG) measurements are performed and reported for all the splits (A,B,C,D) in Figure 2. The Vth1 in this article is defined as the gate voltage at which the 2DEG channel is depleted and Vth2 is the gate voltage required to form a channel inside the AlGaN barrier. ID(VG) characteristics show firstly a Vth1 variation toward negative values when tAlN is increased. Moreover, an n-doped AlGaN barrier leads to an even more negative Vth1 value and to a higher current density at VG=0V compared to the undoped cases. Comparison with C(VG) measurements confirms these electrostatic effects observed on ID(VG) curves. Table 1(a) reports the extractions of R2DEG, ns and µs, performed from the ID(VG) and C(VG) according to equations 1 to 4. For samples A, B and C, extractions at VG=0V refer directly to 2DEG parameters, where ns is calculated by Eq.1 from the C(VG) and R2DEG with Eq.2 from the ID(VG). Finally, µs is calculated with Eq.3. As reported in Table 1(a), nS increases with larger AlN spacer but R2DEG enhancement is limited by mobility degradation, likely due to the rise of surface roughness.

Regarding the n-doped AlGaN sample, the extraction of the 2DEG parameters are indirect because ndoping leads to an earlier formation of the second channel in AlGaN (for VG<<0V, see Figure 4(d)).

Consequently, the extraction at VG=0V accounts for the 2DEG and the AlGaN channel as details in Table 1(b). ns(total) refers to the value extracted at VG=0V corresponding to the sum of both channels (Eq.4). R2DEG is extracted at Vth2 before the formation of the n-AlGaN channel and RAlGaN is deduced using Eq.5, where Rtot is the resistance at VG=0V. Surprisingly, the ns(2DEG) barely increases with doping and the R2DEG reduction is caused by a mobility degradation, likely induced by the high level of doping near the 2DEG channel. Depicted in Figure 4, the simulations of the electron density as a function of the depth for the four-studied samples explain and validate this weak rise of nS for the doped AlGaN. Indeed, since the onset of charge density in the AlGaN barrier occurs near the AlGaN/AlN interface and thus near the 2DEG channel, a variation of the gate voltage will induce a simultaneous variation of both channels. We attribute the difference between the doped and undoped sample (1)

𝑅 2𝐷𝐸𝐺,𝑡𝑜𝑡 = 𝜋 ln(2) . ∆𝑉 𝐼 (2)
µ S,𝑡𝑜𝑡 = 1 q.R 2DEG,tot .n S,𝑡𝑜𝑡 (3)
n S (total) = 1 q ∫ C. dV 0 Vth1 (4)
R AlGaN = R tot .R 2DEG R 2DEG -R tot (5)
to the enhance electric field in the AlGaN near the interface induced by the high level of donors. Hence, the overall resistance improvement is only attributed to the high ns in the AlGaN barrier.

Simulations

To understand the physical root causes leading to the enhancement of the 2DEG properties, the previous C(VG) measurements are compared to 1DSP simulations and reported in Figure 3. The Poisson and Schrödinger equations are solved self-consistently under the parabolic mass approximation for the Γ-valley. The material parameters used in the simulations are based on Ref.7. The AlGaN polarization charge at the SiN/AlGaN interface is assumed to be completely compensated by a positive charge [START_REF] Rrustemi | Investigation on interface charges in SiN/AlxGa1-xN/GaN heterostructures by analyzing the gate-to-channel capacitance and the drain current behaviors[END_REF]. Polarization charges at the AlGaN/AlN and AlN/GaN interfaces (σAlGaN/AlN and σAlN/GaN) are determined to account for the experimental Vth1 and Vth2 and reported in Table 2. These charges are found to be lower than the theoretical values induced by AlN with complete polarization [START_REF] Ambacher | Pyroelectric properties of Al(In)GaN/GaN hetero-and quantum well structures[END_REF]. Indeed, theoretical charges are calculated according to the Ambacher's works [START_REF] Ambacher | Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N-and Ga-face AlGaN/GaN heterostructures[END_REF][START_REF] Ambacher | Two dimensional electron gases induced by spontaneous and piezoelectric polarization in undoped and doped AlGaN/GaN heterostructures[END_REF][START_REF] Ambacher | Pyroelectric properties of Al(In)GaN/GaN hetero-and quantum well structures[END_REF], by considering two heterojunctions: the AlN/GaN heterojunction resulting in a positive polarization interface charge responsible for the well-known 2DEG, and the AlGaN/AlN heterojunction resulting in a negative polarization interface charge. The interface charges calculations take into account for the spontaneous (Psp) and the piezoelectric polarization (Ppz). GaN is considered as a fully relaxed material. The theoretical charges values are very high (σAlGaN/AlN = -5.42x10 13 cm -2 and σAlN/GaN=6.56x10 13 cm - 2 ) and they induce a much more large shift on both Vth1 (towards more negative voltage values) and Vth2 (towards more positive voltage values) compared to the experimental results. In addition, not enough Vth1 dependency with the AlN thickness is observed by considering these theoretical charges. Thereby, these results suggest that the polarization of AlN is attenuated for very thin layers. Figure 4 illustrates the decrease of ns with thinner AlN spacer. To take into account this reduction, we introduce a factor PR (polarization rate) in our calculations of polarization charges see Eq.6 and Eq.7. Interestingly, as depicted in Figure 5, using the same PR for the top and bottom charges, these calculated charges are similar to the ones extracted in Table 2, for the different tAlN.

These results support our assumption regarding the attenuation of the AlN polarization for very thin layer. Indeed, the theoretical charges calculation was demonstrated for thickness from 30nm [START_REF] Ambacher | Pyroelectric properties of Al(In)GaN/GaN hetero-and quantum well structures[END_REF]; in this work the studied thicknesses are approximately a few atomic layers considering that the lattice parameter c of the AlN is around 4.98Ả. Another hypothesis is that this effect can also, in part, be due to a reduction of the Al fraction in AlN, which becomes an Al-rich AlGaN for very thin layers. However, this assumption seems not sufficient to explain the experimental observations. Indeed, both charges are interdependent and by decreasing the Al content and assuming a complete polarization, the simulated capacitance well reproduces the effect on the Vth2 but not on Vth1. In addition, the Al molar fraction to account for the AlN/GaN polarization interface charge (Tab.2) are found to be too low, about 0.305, 0.35 and 0.46 respectively for the AlN 0.7nm, 1.1nm and 1.5nm and do not reproduce the Vth1 variations.

Going back at Table 2, we can notice that for samples D and A, which differ only by AlGaN doping, the same interface charges are determined despite the very different Vth1 and Vth2 values obtained on the two samples. This is consistent and corroborates our estimations of σAlGaN/AlN and σAlN/GaN. Figure 6 outlines the polarization rate reduction with thinner tAlN through an empirical model. This simple model gives good approximation for simulations and it seems in accordance with Berdalovic's work, which supposes a 65% of polarization in a 3.5nm AlN layer.

σ Al x Ga 1-x N/AlN (x) = {P tot (AlN) × PR -[ P sp (Al x Ga 1-x N) + P pz (Al x Ga 1-x N)]}/q (6) σ AlN/GaN = { P sp (GaN) -P tot (AlN) × PR}/q (7)

Conclusions

In summary, we have demonstrated that n-doped AlGaN improves the on state resistance due to the highly n-doped channel formation in AlGaN for negative voltages. We also showed that the 2DEG density was barely not affected by AlGaN n-doping. Then, we assumed that AlN polarization is attenuated for very thin layers, and increases with thicker AlN layers, resulting in a reduction of R2DEG. This assumption seems the most suitable to explain experimental results. Nevertheless, we find out that R2DEG enhancement with thicker AlN spacer is limited despite the nS improvement, by mobility degradation for tAlN larger than 1.1nm.

Figure 1 (

 1 a) depicts a cross-section of the studied structure, where the AlN spacer thickness (tAlN) and the n-type AlGaN doping (ND) are the key parameters of this study. Samples variations are summarized in Figure 1(b) and Figure 1(c) is a top view of the tested normally-ON Van der Pauw structure.

Figure 1 :

 1 Figure 1: (a) Cross-section of the studied structure (b) Summary table of epitaxial variations (c) Top view of gated Van der Pauw test structure (200µm x 200µm).

|

  1-x N 21nm N D AlN t AlN GaN:UID 100nm p-GaN 150nm GaN:C + transitions layers Si(111) Substrate Metal 𝐭 𝐀𝐥𝐍 (nm) 𝐍 𝐃 (cm -3 ) V={ 0 if 𝑉 𝑡ℎ2 ≥0V 𝑉 𝑡ℎ2 if 𝑉 𝑡ℎ2 ≪0V

Figure 2 :

 2 Figure 2: ID(VG) and C(VG) measurements at 1kHz respectively on B1500 device analyzer and HP4284 LCR meter.

Figure 3 :

 3 Figure 3: C(VG) simulations (lines) and comparison with the experimental C(VG) (dots) for samples A, B, C and D.

Figure 4 :

 4 Figure 4: Electron density versus Detph at various VG obtained from 1DSP simulations, (a) for tAlN=0.7nm, (b) for tAlN=1.1nm, (c) for tAlN=1.5nm, (d) for the n-doped AlGaN, tAlN=0.7nm.

Figure 5 :

 5 Figure 5: Lines: σ AlGaN/AlN and σ AlN/GaN vs PR (Eq.6-7). Dots: σ AlGaN/AlN and σ AlN/GaN extracted from measurements.

Table 1 :

 1 (a) Extracted 2DEG parameters R2DEG, ns and µs at VG=0V for samples A, B, C and at Vth2 for sample D, (b) AlGaN and 2DEG detailed channel parameters for sample D.

		𝑛 𝑆 (𝑐𝑚 -2 ) R 2DEG (Ω/□) µs (cm².V -1 .s -1 )
	A	7.9x10 12	371		2123
	B	8.2x10 12	344		2145
	C	8.4x10 12	351		1981
	D	8.2x10 12	441		1728
		D	𝑛 𝑆 (𝑐𝑚 -2 ) 𝑅 (Ω/□) µ (cm².V -1 .s -1 )
	AlGaN channel	2.1x10 13	1207	216
	2DEG (VG~-43V) 8.2x10 12	441	1728
	Total (VG=0V)	2.9x10 13	323	663
		t SiN (nm)	σ AlGaN /AlN (cm -2 )	σ AlN /GaN (cm -2 )
	A	73.5	-6.00x10 12	1.57x10 13
	B	74.5	-8.00x10 12	1.82x10 13
	C	75.5	-1.42x10 13	2.5x10 13
	D	73.5	-6.00x10 12	1.57x10 13

Table 2 :

 2 1DSP simulated parameters for all samples. The fitting parameters are the interface charges σAlGaN/AlN and σAlN/GaN and the fitting references are Vth1 and Vth2.

Acknowledgements

The authors would like to acknowledge the funding support from the PSPC French national program « G-Mobility ».


Laurence Carassus
email: carassus@math.jussieu.fr

Miklós Rásonyi
email: rasonyi@sztaki.hu

Risk-averse asymptotics for reservation prices

Keywords: Utility indifference price, Superreplication price, Convergence, Utility maximization, Risk aversion. MSC 2000 Subject Classification: Primary: 91B16, 91B28, Secondary: 93E20, 49L20 OR/MS Subject Classification: Primary: Utility / Value theory, Secondary: Finance / Asset Pricing JEL classification: C61, C62, G11, G12

An investor's risk aversion is assumed to tend to infinity. In a fairly general setting, we present conditions ensuring that the respective utility indifference prices of a given contingent claim converge to its superreplication price.

Introduction

In this article we investigate the effect of increasing risk aversion on utilitybased prices. We are dealing with the utility indifference price (or reservation price), defined in [START_REF] Hodges | Optimal replication of contingent claims under transaction costs[END_REF] for the first time. This is the minimal amount added to an option seller's initial capital which allows her to attain the same utility that she would have attained from her initial capital without selling the option, see Definition 4.2 below. Intuitively, when risk aversion tends to infinity, reservation price should tend to the superreplication price (i.e. the price of hedging the option without any risk).

This result was shown in [START_REF] Rouge | Pricing via utility maximization and entropy[END_REF] for Brownian models and in [START_REF] Delbaen | Exponential hedging and entropic penalties[END_REF] in a semimartingale setting when the agent has constant absolute risk aversion (i.e. for exponential utility functions). Certain other classes of utility functions were treated in [START_REF] Bouchard | Stochastic control and applications in finance[END_REF], models with transaction costs were considered in [START_REF] Bouchard | Option pricing by large risk aversion utility under transaction costs[END_REF].

However, an extension of this result to general utility functions was lacking. In [START_REF] Carassus | Convergence of utility indifference prices to the superreplication price[END_REF] and [START_REF] Carassus | Convergence of utility indifference prices to the superreplication price: the whole real line case[END_REF] the case of discrete-time markets was treated for utilities on the positive axis as well as on the real line. Now we prove this result in a continuoustime semimartingale framework, under suitable hypotheses.

1

In section 2 we model the agent's preferences and introduce a growth condition (related to the elasticity of utility functions), in section 3 the market model and a compactness assumption are discussed. In section 4 the concept of utility indifference price is formally defined and the two main theorems are proved.

Risk averse agents

We consider agents trading in the market with initial endowment z ∈ R. We assume that Assumption 2.1 U n , n ∈ N are twice continuously differentiable, strictly concave and increasing functions on R such that for each x ∈ R,

r n (x) := -U ′′ n (x) U ′ n (x) -→ n→+∞ ∞. (1)
The function r n is called the (absolute) risk aversion of an agent with utility function U n . This concept was introduced in [START_REF] Arrow | Essays in the Theory of Risk-Bearing[END_REF] and [START_REF] Pratt | Risk aversion in the small and in the large[END_REF]. Here we are interested in the case where this measure of risk aversion tends to infinity.

We take the Fenchel conjugates of U n :

V n (y) := sup x∈R {U n (x) -xy}, y ∈ (0, ∞). (2)
As easily checked, V n is a finite convex function.

We stipulate a growth condition on the conjugates of the utility functions we consider. Such assumptions are often referred to as "elasticity conditions". Assumption 2.2 For each [λ 0 , λ 1 ] ⊂ (0, ∞) there exist positive constants C 1 , C 2 , C 3 such that for all n and for all y > 0,

V n (λy) ≤ C 1 V n (y) + C 2 y + C 3 .
(3)

holds for each λ ∈ [λ 0 , λ 1 ]. Remark 2.3 For n fixed, condition (3) is equivalent to lim sup x→∞ xU ′ n (x) U n (x) < 1, lim inf x→-∞ xU ′ n (x) U n (x) > 1, (4)
as shown in [START_REF] Frittelli | Equivalent formulations of Reasonable Asymptotic Elasticity[END_REF].

The first of the two conditions in (4) was introduced in [START_REF] Kramkov | The condition on the Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets[END_REF], the second one in [START_REF] Schachermayer | Optimal Investment in Incomplete Markets when Wealth may Become Negative[END_REF]. A utility function U n satisfying (4) is said to have reasonable asymptotic elasticity (terminology of [START_REF] Schachermayer | Optimal Investment in Incomplete Markets when Wealth may Become Negative[END_REF]). Thus condition (3) is a (dually formulated) uniform reasonable asymptotic elasticity condition. Another uniform asymptotic elasticity condition appears as Assumption 2.3 of [START_REF] Carassus | Optimal strategies and utility-based price converge when agents' preferences do[END_REF]. About the derivation of equivalances like that of (4) and (3) consult section 6 of [START_REF] Kramkov | The condition on the Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets[END_REF], section 4 of [START_REF] Schachermayer | Optimal Investment in Incomplete Markets when Wealth may Become Negative[END_REF] and [START_REF] Frittelli | Equivalent formulations of Reasonable Asymptotic Elasticity[END_REF].

Market Model

Our market is modelled by an adapted d-dimensional stochastic process S on a given continuous-time stochastic basis (Ω, F , (F t ) t≥0 , P ). We think that S represents the evolution of the (discounted) prices of d assets. For simplicity we assume S locally bounded (to avoid technical complications related to σmartingales). Absence of arbitrage is stipulated by

M = ∅,
where M denotes the set of measures Q ∼ P such that S is a local martingale under Q.

We make the following compactness assumption.

Assumption 3.1 There exists Q 0 ∈ M such that the sequence V n (dQ 0 /dP ) is uniformly integrable (with respect to P ). We denote by M v the set of such Q 0 s.

Remark 3.2 Similar conditions have already appeared in investigations on the stability of optimal strategies with respect to perturbations of utility functions, see [START_REF] Larsen | Continuity of utility-maximization with respect to preferences[END_REF] and [START_REF] Kardaras | Stability of the utility maximization problem with random endowment in incomplete markets[END_REF]. Both papers consider a sequence U n of utility functions converging to a limiting utility U and show convergence of the corresponding optimal strategies/utility prices.

In [START_REF] Larsen | Continuity of utility-maximization with respect to preferences[END_REF] a sequence U n is assumed to be dominated by some U with conjugate function V such that V (Z) is integrable, where Z is (the candidate for) the density of the minimal martingale measure. In Assumption (UI) of [START_REF] Kardaras | Stability of the utility maximization problem with random endowment in incomplete markets[END_REF] we find essentially the uniform integrability of V n (ydQ n /dP ) for each y > 0, where Q n is a suitable sequence in M. Note that [START_REF] Kardaras | Stability of the utility maximization problem with random endowment in incomplete markets[END_REF] also considers perturbations of the underlying probability measure and that in both mentioned papers the U n are defined on the positive axis. This elasticity hypothesis 2.2 allows us to prove that the set M v is in fact large.

Lemma 3.3 Assume that there is x 0 ∈ R such that U n (x 0 ) is bounded from below. Under Assumptions 2.2 and 3.1, the set M v is dense in M with respect to the total variation norm topology.

Proof. This result was essentially reported in Proposition 6 of [START_REF] Biagini | On the super-replication price of unbounded claims[END_REF], without proof. Fix Q 0 ∈ M v . We know from [START_REF] Yu | On equivalent martingale measures with bounded densities[END_REF] that the set

M b := {Q ∈ M : dQ/dQ 0 is bounded } is dense in M.
It follows by concavity of M that also

M bb := {αQ 0 + (1 -α) Q : 0 < α < 1, Q ∈ M b }
is a dense subset of M. We shall show that M bb ⊂ M v . Take an arbitrary Q ∈ M b such that dQ/dQ 0 ≤ K and any fixed 0 < α < 1. We have

V n (αdQ 0 /dP + (1 -α)dQ/dP ) ≥ U n (x 0 ) -x 0 (αdQ 0 /dP + (1 -α)dQ/dP ).
Moreover,

V n (αdQ 0 /dP + (1 -α)dQ/dP ) ≤ C 1 V n (dQ 0 /dP ) + C 2 (dQ 0 /dP ) + C 3 , by Assumption 2.2 (choose C 1 , C 2 , C 3 corresponding to the interval [α, α + (1 - α)K] and note that α + (1 -α)dQ 0 /dQ falls into this interval).
Uniform integrability of V n (αdQ 0 /dP + (1α)dQ/dP ), n ∈ N now follows from Assumption 3.1. 2

Utility indifference prices

It follows from M = ∅ that S is a semimartingale and we may define the set of admissible trading strategies as predictable S-integrable processes with a finite credit line (to avoid doubling strategies).

Definition 4.1

A := {φ : φ predictable, S-integrable and for some w > 0, V 0,φ ≥ -w}, where we write V x,φ t := x + (φ • S) t for the value process of strategy φ starting from initial endowment x.

Fix T > 0 and a bounded random variable G, interpreted as a contingent claim to be delivered at the end of the period [0, T ]. By optimal trading an agent with initial capital x ∈ R and utility U n delivering the claim G may attain

u n (x, G) := sup φ∈A EU n (V x,φ T -G), (5)
this is well-defined as admissible strategies have bounded from below value processes.

Definition 4.2

The utility indifference price for U n and initial endowment z is

p n (z, G) := inf{p ∈ R : u n (z + p, G) ≥ u n (z, 0)}, (6)
the minimal extra capital that allows for delivering G while attaining the same utility as without claim delivery.

The utility-free concept of the superhedging price of a bounded contingent claim (random variable) G is defined as

π(G) := inf{x ∈ R : there is φ ∈ A such that V x,φ T ≥ G a.s.}.
The following dual characterization is a fundamental result of mathematical finance, see e.g. [START_REF] Föllmer | Optional decomposition and Lagrange multipliers[END_REF] and the references therein.

Theorem 4.3 Under M = ∅, we have π(G) = sup Q∈M E Q G.
We are now ready to state our main result i.e. the convergence of utility indifference prices to the superreplication price. We will present this result under two types of assumptions. The first one (see Assumption 4.4) refers to the existence of a initial wealth for which all the investors have (asymptotically) a common preference for x 0 and also a common non zero growth rate for their preferences near x 0 .

The second kind of assumptions impose the elasticity Assumption 2.2 and the compactness Assumption 3.1 on a normalized family of utility functions (see Theorem 4.13).

Assumption 4.4 There exists some x 0 , α, β ∈ R with α ∈ (0, ∞) and |β| < ∞ such that :

U ′ n (x 0 ) -→ n→+∞ α U n (x 0 ) -→ n→+∞ β.
Remark 4.5 We will see in the examples below that this assumption is satisfied for power and exponential utility functions. In those examples, we have that U n (0) = 0 and U ′ n (0) = 1. This means that all investors consider that utility of nothing is zero and that for very small wealth their utility functions are approximatively linear with slope one.

The following proposition is the first step to Theorem 4.7 : Proposition 4.6 Under Assumptions 2.1 and 4.4, for each y > 0,

V n (y) -→ n→+∞ β -x 0 y.
Proof. First, the argument of Lemma 4 in [START_REF] Carassus | Convergence of utility indifference prices to the superreplication price[END_REF] shows that as n → ∞,

U ′ n (x) → ∞, x < x 0 , U ′ n (x) → 0, x > x 0 . (7)
Let I n be the inverse of U ′ n (which exists by strict concavity of U n ). We claim

I n (y) → x 0 , for y > 0.
Indeed, let y > 0. If we had I n k (y) ≥ x 0 + ε for some ε > 0 and a subsequence

n k then U ′ n k (I n k (y)) = y ≤ U ′ n k (x 0 + ε)
, but this latter tends to 0 by [START_REF] Carassus | Convergence of utility indifference prices to the superreplication price: the whole real line case[END_REF], a contradiction. The proof of the reverse inequality is similar and I n (y) → x 0 .

First by definition V n (y) ≥ U n (x 0 )x 0 y. Since V n can be calculated as Remark 4.8 The convergence in Theorem 4.7 holds only for x 0 . We will see in Theorem 4.13 how the convergence for all initial capital z can be checked under a second type of assumptions.

V n (y) = U n (I n (y)) -I n (y)y; concavity of U n implies that V n (y) ≤ U n (x 0 ) + (I n (y) -x 0 )U ′ n (x 0 ) -I n (y)y. Thus 0 ≤ V n (y) -(U n (x 0 ) -x 0 y) ≤ (I n (y) -x 0 )(U ′ n (x 0 ) -y) → 0, n → ∞,
Proof. It is a standard fact that that p n (x 0 , G) ≤ π(G), see page 152 of [START_REF] Carassus | Convergence of utility indifference prices to the superreplication price[END_REF]. For the reverse inequality, we argue by contradiction. Suppose that for some ε > 0 and a subsequence n k one has p k := p n k (x 0 , G) ≤ π(G)ε for each k. We may suppose that

n k = k and p k → π(G) -ε, k → ∞. Take arbitrary φ ∈ A and Q ∈ M v . Then E Q (V x0+p k ,φ T -G) ≤ x 0 +p k -E Q G (V 0,
φ is a supermartingale by results of [START_REF] Ansel | Couverture des actifs contingents et prix maximum[END_REF]). Proposition 4.6 and

Q ∈ M v imply lim sup k→∞ E(V k (dQ/dP ) + (dQ/dP )(V x0+p k ,φ T -G)) (8)
≤ β -x 0 + (x 0 + lim sup k→∞ p k -E Q G). Introduce v φ k := inf Q∈Mv E(V k (dQ/dP ) + (dQ/dP )(V x0+p k ,φ T -G)). It follows from Lemma 3.3 that sup Q∈Mv E Q G = sup Q∈M E Q G, thus (
v φ k ≤ β + lim k→∞ p k -π(G) = β -ε.
By the definition of conjugate functions we have,

lim sup k→∞ EU k (V x0+p k ,φ T -G) ≤ lim sup k→∞ v φ k ≤ β -ε.
It follows that u k are finite and lim sup k→∞ u

k (x 0 + p k , G) ≤ β -ε. But u k (x 0 + p k , G) ≥ u k (x 0 , 0) ≥ U k (x 0 ), thus lim sup k→∞ u k (x 0 + p k , G) ≥ β a contradiction. 2
Remark 4.9 Let us consider the condition that there exists Q 0 ∈ M such that dQ 0 /dP and dP/dQ 0 are both bounded (by some K > 0). Note that a convex function attains its maximum on an interval at one of the endpoints. Hence

U n (x 0 ) -|x 0 |K ≤ V n (dQ 0 /dP ) ≤ |V n (K)| + |V n (1/K)|
which is bounded by Proposition 4.6, showing that this condition is stronger that Assumption 3.1.

We claim that, replacing Assumption 3.1 by this condition, one may drop Assumption 2.2 from the hypotheses of Theorem 4.7. Indeed, the result of [START_REF] Yu | On equivalent martingale measures with bounded densities[END_REF] directly implies that measures Q 0 with the above property are dense in M, without appeal to Assumption 2.2. The rest of the proof is identical.

Unfortunately, in continuous-time models one rarely finds such a Q 0 . In discrete-time models, however, such measures often exist, see [START_REF] Rokhlin | Lower bounds of martingale measure densities in the Dalang-Morton-Willinger theorem[END_REF] for an extensive discussion.

We now turn to our second Theorem. To state its hypothesis, we first need to introduce some normalization of the functions U n . Fix some initial wealth z ∈ R and set :

Ũn (x) := U n (x) -U n (z) U ′ n (z) , n ∈ N, x ∈ R. (9)
We now restate Assumptions 2.2 and 3.1 for the family ( Ũn ) n .

showing the claim. 2 Theorem 4 . 7

 247 If Assumptions 2.1, 2.2, 3.1 and 4.4 hold then for each bounded random variable G the quantities u n (x 0 , G), n ∈ N are finite and the corresponding utility indifference prices p n (x 0 , G) tend to π(G) as n → ∞.

Assumption 4 . 10

 410 For each [λ 0 , λ 1 ] ⊂ (0, ∞) there exist constants C 1 , C 2 , C 3 such that for all n and for all y > 0,Ṽn (λy) ≤ C 1 Ṽn (y) + C 2 y + C 3 . (10)holds for each λ ∈ [λ 0 , λ 1 ].

The second author acknowledges financial support from the Hungarian Science Foundation (OTKA) grant F 049094 and dedicates this paper to Annamária Brecz.

 Assumption 4.11There exists Q 0 ∈ M such that the sequence Ṽn (dQ 0 /dP ) is uniformly integrable (with respect to P ). Remark 4.12 We think that Assumptions 4.10 and 4.11 should be considered as technical assumptions to be checked on Ũn . We will see below that they hold true in our examples. Proof. Obviously, Ũn (z) = 0 and Ũ ′ n (z) = 1, thus Assumption 4.4 is satisfied for the sequence Ũn at x 0 = z. Let ũn be defined analogously as in [START_REF] Bouchard | Option pricing by large risk aversion utility under transaction costs[END_REF], with Ũn replacing U n . Then since the Ũn are affine transforms of the U n , we may alternatively write

It is obvious that if Assumption 2.1 hold for U n then it holds also for Ũn , so Theorem 4.7 applied for Ũn and x 0 = z allow us to conclude. 2

Example 4.14 (The exponential case) Let

α n and thus U n satisfy Assumption 2.1 and 4.4 for x 0 = 0. Moreover, calculation gives V n (y) = (1/α n )[y ln y + 1y] showing that Assumption 3.1 holds provided that there is Q 0 ∼ P with E Q0 ln(dQ 0 /dP ) < ∞, i.e. whenever a finite-entropy martingale measure exists. It is enough to show (3) for one function, namely V (y) = y ln y. This is trivial since λy ln(λy) ≤ λy ln y + λ| ln λ|y for λ, y > 0.

So Theorem 4.7 applies and the respective reservation prices p n (0, G) converge to π(G).

In fact, one can check that Theorem 4.13 applies for each z ∈ R. We have thus retrieved the result in [START_REF] Delbaen | Exponential hedging and entropic penalties[END_REF]. (They make a weaker moment assumption on G; this extension follows by our method, too.) Example 4.15 (The power case)

with α n > 0, β n > 1 tending to ∞. These functions are continuously differentiable with strictly monotone derivatives, hence they are strictly concave. They are also twice continuously differentiable on R \ {0} and satisfy (1) on R \ {0}. Take now β n := α n + 2 then U ′′ n exists and (1) holds in 0, too.

1-x ) and U n satisfy Assumptions 2.1 and 4.4 for x 0 = 0.

It remains to check (3) and Assumption 3.1. We have, for y ∈ (0, ∞),

Thus if there exists Q 0 ∈ M such that dQ 0 /dP ∈ L 1+ε for some ε > 0 then Assumption 3.1 will be satisfied for n large enough since (α n + 2)/(α n + 1) converges to 1. Finally, (3) is trivially satisfied by the scaling properties of the power functions. We may conclude by Theorem 4.7 that the reservation prices p n (0, G) tend to π(G) as n → ∞.

Tedious calculations show that we get the same conclusion for any initial capital z ∈ R and we may apply Theorem 4.13.


Anna Teledahl

Linda Marie Ahl

Ola Helenius
email: ola.helenius@ncm.gu.se

Cecilia Kilhamn
email: cecilia.kilhamn@ped.gu.se

Quality in students' writing in mathematics

Keywords: Writing, quality, progression

We present a literature review with the aim to identify and characterise research frameworks that propose standards for assessing the quality of students' mathematical writing. Eleven studies that investigated students' solutions to problems, journal writing, essays, or instructions, and proposed frameworks for analysing such writing, were identified. The analysis sought to characterise what the frameworks attended to in students' mathematical writing and whether they described a progression with a reference to quality of the writing. The main result is that the frameworks mainly attend to the content and that there is little attention to form. Further, quality is most often regarded as a question of content, if the text contains for example an explanation, then the explanation is seen as a sign of quality with no reference the form. Our review concludes that research rarely describes levels of quality in students' mathematical writing, something we believe further research should do.

Introduction

The universal practice of having students present their solutions to mathematical problems in writing and using this writing to draw conclusions about students' proficiency, has been common for centuries [START_REF] Morgan | Writing mathematically: The discourse of investigation[END_REF]. Reporting mathematical work in writing requires knowledge on how to represent mathematical investigations, conclusions, justifications, and arguments. In many curricula communication is presented as a competency and as such it can develop so that students over time could exhibit a progression, for example in the quality of their mathematical writing. In this paper we have investigated the question of quality in mathematical writing through a literature review where we set out to answer the questions: Does research provide standards for assessing the quality of students' mathematical writing (MW) and if so, what are these standards?

A theoretical point of departure is the idea that competence in mathematical communication, in our case in writing, is a skill that can be analytically separated from other skills. This idea is captured by the concept of communicative competence [START_REF] Hymes | On communicative competence[END_REF], which suggests that in a particular social situation all things said are not communicated with equal competence (Craig & Muller, 2007;[START_REF] Rickheit | The concept of communicative competence[END_REF]. The simplest criteria for what can be considered competent communication are efficiency and appropriateness [START_REF] Rickheit | The concept of communicative competence[END_REF]. Efficiency relates to the idea that communication takes place to fulfil a goal and if you achieve your goal then the communication is effective. Appropriateness connects with the social aspect of communication: to communicate well in a social situation is to be familiar with all the ways of making meaning within that situation. Wittgenstein referred to this social localisation using the metaphor of a language game. He suggested that competent communicators are those who know and abide by the rules of the language game that the social situation constitutes (Wittgenstein, 1953(Wittgenstein, /1986)). It can thus be argued that what is considered good communication changes with the situation. This idea is fundamental to the movement labelled Writing Across Disciplines which has advocated the differentiation of writing skills in relation to different academic disciplines of which mathematics is one [START_REF] Bazerman | Reference guide to writing across the curriculum[END_REF].

In line with the ideas presented above it can be noted that modern communication in mathematics is a special kind of communication which has developed over two centuries (Solomon & O'Neill, 1998) and is defined by its multi-semiotic nature [START_REF] O'halloran | Mathematical discourse: Language, symbolism and visual images[END_REF]. Because mathematical concepts are often difficult to articulate in ordinary language, mathematical symbolism has developed to express meaning beyond what can be expressed by ordinary language [START_REF] Schleppegrell | The linguistic challenges of mathematics teaching and learning: A research review[END_REF]. It has been proposed that MW can be decomposed into a formal logical structure in which we find definitions, theorems and proofs, and a complementary informal structure in which motivation, analogies, examples, and explanations have their place [START_REF] Steenrod | How to write mathematics[END_REF]. The logical structure relies on mathematical notation and symbolism which owe their strengths to their universality and independence of context [START_REF] Sfard | Learning mathematics through conversation: Is it as good as they say? For the Learning of Mathematics[END_REF]. The complementary structure relies on natural language, which owes its strength to its flexibility, offering a nearly endless number of ways of conveying meaning through multiple modes.

Mathematical writing takes place in relation to different objectives and prompts. Students use writing to personally make sense of, record or explore mathematical ideas, but also to document processes, explain, describe, and make arguments in social situations with peers or teachers [START_REF] Casa | Types of and purposes for elementary mathematical writing: Task force recommendations[END_REF][START_REF] Morgan | Mathematics and human activity: Representation in mathematical writing[END_REF][START_REF] Stylianou | An examination of middle school students' representation practices in mathematical problem solving through the lens of expert work: towards an organizing scheme[END_REF]. Within these different categories there is a difference between writing in mathematics and writing about mathematics [START_REF] Bosse | Learning and assessing mathematics through reading and writing[END_REF]. Writing to personally make sense of mathematics and writing for others such as teachers and peers represent two very different processes, but commonly, they are concurrent in school mathematics [START_REF] Morgan | Writing mathematically: The discourse of investigation[END_REF]. The definition for MW used in this paper is borrowed from [START_REF] Casa | Types of and purposes for elementary mathematical writing: Task force recommendations[END_REF] and encompasses both these categories. MW is understood to be writing to reason (personal) and to communicate (with others) in an educational and mathematical context. We assume that standards for assessing the quality of students' mathematical writing apply to either one specific form of writing or several forms.

Review methodology

We conducted a literature search in ERIC (EBSCO) where we searched abstracts using the following search words: [writing AND (mathematical or mathematics) AND (pupil* or student* or children*)]. The search words were chosen with the intent to capture research in which students' mathematical writing is treated as an object separate from communication in general [START_REF] Schleppegrell | The linguistic challenges of mathematics teaching and learning: A research review[END_REF][START_REF] Steenrod | How to write mathematics[END_REF]. The search was performed on the 24th of May 2021. We found 806 peer-reviewed articles fitting the inclusion criteria (see table 1).

Using the software Covidence to manage our literature review, each paper was screened by two reviewers. The screening was made in two steps. First, we screened the title and abstract to forward relevant studies to full-text review. In cases where we were hesitant, we chose to forward the paper to full-text screening. In the full-text screening, papers were included if they contained frameworks or guidelines on how to write, or explicit descriptions of progression or levels of writing.

Of a total of 141 papers, 50 remained after the full-text screening (see table 1). These were categorized according to what was investigated in the paper in terms of: Studies that compare different frameworks for analysing mathematical writing (1); Studies that create a framework for analysing mathematical writing, but where the research question concerns something else (the framework created is a means to study for example the way digital technologies influence students' writing or how writing affords learning) (26); Studies that use a pre-existing framework for analysing mathematical writing (the framework is a means) (12); and finally, Studies that result in a framework for analysing mathematical writing (11). In this study we have chosen to focus on the latter category to investigate only research that focuses exclusively on students' mathematical writing as an object. The literature review is a configurative review [START_REF] Gough | Clarifying differences between review designs and methods[END_REF] aimed at identifying, interpreting, and characterising research frameworks that propose standards for assessing the quality of students' mathematical writing. After an initial review of overarching data such as age groups and types of writing [START_REF] Bosse | Learning and assessing mathematics through reading and writing[END_REF][START_REF] Casa | Types of and purposes for elementary mathematical writing: Task force recommendations[END_REF], the analysis focused on the proposed frameworks. The first part of this analysis sought to characterise the frameworks in relation to what they attended to, and the second part was concerned with investigating hierarchies or descriptions of progression in relation to quality, within the identified frameworks.

Results

The types of mathematical writing that was investigated varied across the studies in the sense that the writing samples were collected through different prompts. They were, however, all examples of accounts of writing that communicated mathematics, i.e., the texts were created with the understanding that they were supposed to be read by someone other than the student herself. The most common were prompts related to documenting and reporting solutions to mathematical problems. This type of writing was found in six of the eleven studies that were part of the review [START_REF] Hughes | Evaluating various undergraduate perspectives of elementary-level mathematical writing[END_REF][START_REF] Holt | Promoting student buy-in: Using writing to develop mathematical understanding[END_REF][START_REF] Kosko | Emergence of argument in children's mathematical writing[END_REF][START_REF] Morgan | What does social semiotics have to offer mathematics education research[END_REF][START_REF] Pugalee | A comparison of verbal and written descriptions of students' problem solving processes[END_REF][START_REF] Teledahl | How young students communicate their mathematical problem solving in writing[END_REF]. The remaining prompts concerned proofs [START_REF] Lew | Linguistic conventions of mathematical proof writing at the undergraduate level: Mathematicians' and students' perspectives[END_REF], journals [START_REF] Clarke | Probing the structure of mathematical writing[END_REF], essays [START_REF] Stonewater | The mathematics writer's checklist: The development of a preliminary assessment tool for writing in mathematics[END_REF], and instructions [START_REF] Kline | Procedural explanations in mathematics writing: A framework for understanding college students' effective communication practices[END_REF][START_REF] Shield | The analysis of student expository writing in mathematics[END_REF]. The writing samples were collected from different age groups, three from students in higher education, two from students aged 17-19 and the remaining from different constellations of students aged 6 -16.

In our characterisation of what the frameworks attended to, we were originally interested in questions of form and content, inspired by previous research in which content and form in students' MW emerged as contrasting qualities. In our analysis, content and form proved to be useful categories although they failed to capture everything the frameworks attended to. Beyond these categories, we identified attention to the meta-functions of language as the focus of one of the frameworks. Our final characterisation therefore consists of the categories meta-functions of language, content, and form, which will be presented below.

Meta-functions of language

Only one of the proposed frameworks attended to the meta-functions of language and communication [START_REF] Morgan | What does social semiotics have to offer mathematics education research[END_REF]. Morgan suggested using tools from systemic functional linguistics (SFL) to analyse students' MW. Such an approach would enable us to understand how students represent their understanding of who does mathematics, what kind of objects are involved in mathematics and what relationships are constructed (the ideational aspect); who the participants are in the written mathematical communication including their relationship to each other and to mathematics (the interpersonal aspect); and what the written text does (the textual aspect).

Content

The most common question that the frameworks attended to was what?, i.e., there was a concern with content in all but one of the studies. A typical framework lists several textual elements that should appear in the MW. In the studies investigating problem-solving accounts or solutions, such elements could include recounts of the steps of the problem-solving process, explanations, accounts of reasoning, and referrals to context [START_REF] Hughes | Evaluating various undergraduate perspectives of elementary-level mathematical writing[END_REF][START_REF] Holt | Promoting student buy-in: Using writing to develop mathematical understanding[END_REF][START_REF] Kosko | Emergence of argument in children's mathematical writing[END_REF]. Mathematical language or the use of mathematical terms can also be viewed as elements of MW that are either present or not in the text, hence it becomes a question of content [START_REF] Holt | Promoting student buy-in: Using writing to develop mathematical understanding[END_REF][START_REF] Kline | Procedural explanations in mathematics writing: A framework for understanding college students' effective communication practices[END_REF]. One of the studies characterises the content of the MW as descriptions of sub-processes in the problem-solving process [START_REF] Pugalee | A comparison of verbal and written descriptions of students' problem solving processes[END_REF]. There is an orientation element, an organizational element, an execution element, and a verification element, all of which correspond to processes in students' problem-solving behaviour. The studies that describe types of writing other than solutions to problems also list content. These include descriptions of context, descriptions of steps, exemplars, kernels, goal statements, justifications, links to prior knowledge, dialogue, summaries, and recounts [START_REF] Clarke | Probing the structure of mathematical writing[END_REF][START_REF] Kline | Procedural explanations in mathematics writing: A framework for understanding college students' effective communication practices[END_REF][START_REF] Lew | Linguistic conventions of mathematical proof writing at the undergraduate level: Mathematicians' and students' perspectives[END_REF][START_REF] Shield | The analysis of student expository writing in mathematics[END_REF][START_REF] Stonewater | The mathematics writer's checklist: The development of a preliminary assessment tool for writing in mathematics[END_REF].

Some of the content elements described in these frameworks are described with a reference to quality.

The most common quality articulated is mathematical. The term mathematical is attached to textual elements such as steps, descriptions, explanations, language, and reasoning. There are, however, no descriptions of how a textual element qualifies as being mathematical.

Form

Three of the proposed frameworks address the question of mode or form. One of the studies [START_REF] Teledahl | How young students communicate their mathematical problem solving in writing[END_REF] lists empirically identified modes of communication in MW together with their apparent function in the MW. Two of the studies investigating proofs [START_REF] Lew | Linguistic conventions of mathematical proof writing at the undergraduate level: Mathematicians' and students' perspectives[END_REF] and essay writing [START_REF] Stonewater | The mathematics writer's checklist: The development of a preliminary assessment tool for writing in mathematics[END_REF] attend to students' use of mathematical notation and in the case of poofs, and in particular their adherence to rules of academic language and the formality of the proof.

Hierarchies or progression

Only four of the eleven studies address the question of hierarchies or progression in students' MW.

The remaining seven either do not mention any differences in quality or treat quality as a question of whether the MW contains a desirable element or not or how often it appears. The two studies that address quality describe three [START_REF] Clarke | Probing the structure of mathematical writing[END_REF] or six [START_REF] Kosko | Emergence of argument in children's mathematical writing[END_REF]) levels of writing and details the differences between these. In the case of Kosko & Zimmerman the analysis, and subsequently the quality criteria, details the difference in quality between six different ways of expressing an argument in students' MW. Two other studies describe elements of MW that were highly valued by respondents in relation to writing an instruction [START_REF] Kline | Procedural explanations in mathematics writing: A framework for understanding college students' effective communication practices[END_REF] or an essay [START_REF] Stonewater | The mathematics writer's checklist: The development of a preliminary assessment tool for writing in mathematics[END_REF]. The latter is the only study that mentions form as something that affects the quality of the MW. Stonewater suggest that "Highest scoring writers use both a greater number of algebraic, numeric, and graphic representations […] and a greater variety of these representations to augment their written work than do the lowest scoring writers" (2002, pp. 330-331).

Discussion

The review aimed to identify research-based frameworks that propose standards for assessing the quality of students' mathematical writing. The results indicate that while there were several frameworks that in different ways describe students' mathematical writing there are very few that do so with a focus on different levels of quality. Of the four frameworks that describe quality beyond the inclusion of different elements in the MW, only one investigated accounts of problem solving. This framework however focused on a specific part of such accounts: the expressing of an argument.

The other three studies investigated other types of mathematical writing.

If quality is thought of as something that can be decided based on which elements are included in MW, then there are more examples of frameworks. The choice whether to include specific features in MW can be related to what prompted a student to write the text or to her idea of the discourse in which the text is written. The simplest of the criteria for communicative competence are appropriateness and effectiveness [START_REF] Rickheit | The concept of communicative competence[END_REF] both of which depend on knowledge about the discourse or, to use Wittgenstein's words, the rules of the language game (1953/1986). Adherence to rules and conventions regarding the form and content of MW is connected to the appropriateness as well as the effectiveness of the communication. If you communicate in ways that let readers recognise the writing as mathematical discourse, then this is appropriate and if readers understand what is communicated then the writing is effective. In the written discourse of mathematics there are conventions and expectations [START_REF] Schleppegrell | The linguistic challenges of mathematics teaching and learning: A research review[END_REF][START_REF] Sfard | Learning mathematics through conversation: Is it as good as they say? For the Learning of Mathematics[END_REF] particularly in mathematical domains such as proofs that rely heavily on mathematical notation [START_REF] Steenrod | How to write mathematics[END_REF]. The result of this review, however, indicates that in school mathematics such conventions and expectations are, if not missing, then at least not described in research, neither regarding content nor form. It is possible that this is connected to the fact that communication in school mathematics, because the students are young and/or still learning, rely more on natural language than other modes of communication [START_REF] Sfard | Learning mathematics through conversation: Is it as good as they say? For the Learning of Mathematics[END_REF].

In preparing for the review, we assumed that we would find research that had taken an interest in how the use of different representations or forms of communication could contribute to an increased quality of MW. Not only did we encounter very few frameworks that addressed the issue of different levels of quality at all, but we only found one that mentioned form. The study that mentioned form, however, proposed that a greater number of different representations together with a greater variety of forms of representations was a sign of high-quality writing without addressing the appropriateness of the representations in relation to the context. How students communicate in writing is, from what we can conclude, not an issue connected to the quality of the writing. If we instead turn to content, we still encounter a lack of agreement on the features that signal quality in accounts of problem solving. The idea that the quality of MW increases if it contains arguments and/or justifications is only visible in frameworks that investigate other types of writing. Considering the small number of papers in our study it is likely that our scope was too narrow and that our study will yield different results when we, in the future as part of our longitudinal study, include also studies that propose frameworks for analysing MW while investigating something other than the quality of the writing itself. But this issue pinpoints our concern regarding the separation of MW as a means from MW as the primary object of interest. Our aim was to identify studies in which the focus was on the MW itself, as an object worth investigating in detail.

Considering that the studies investigated different types of writing from different age groups, one might ask if we should expect agreement or universal ideas on quality in student MW? Is writing not intimately connected to different curricula, different contexts, and different levels of mathematics?

We believe there could be universal ideas on some elements of MW, for example the inclusion of arguments and/or justifications in students' accounts of problem solving. It is our hope that future research will attend to such ideas.

Table 1 : Screening process

 1

		Inclusion Criteria	Exclusion Criteria	Excluded
	Title and	Mathematical writing	Journal writing without guidance for	665 studies
	abstract	Students	assessing writing	irrelevant
	screening	Frameworks for assessing writing	No instruction or identification of
	806 studies	or identifying aspects of writing	aspects of writing or frameworks for
	imported		assessment
	Full-text	Frameworks for how to write	Not a research article	57
	screening	Guidelines for how to write	No criteria for analysing or assessing	21
	141 studies	Explicit ideas for progression	mathematical writing	8
	imported	Descriptions for progression or	Full text not available.	5
	50 remained for	levels of writing	Not about mathematical writing
	data extraction
		Meta-reflections on the benefits of
		writing

Acknowledgment

This research was conducted with funding from the Swedish Institute for Educational Research under grant 2020-00066.