Shohini Bhattasali

Murielle Fabre

Wen-Ming Luh

Hazem Al Saied

Mathieu Constant

Christophe Pallier

Jonathan R Brennan

R Nathan Spreng

John Hale

Localising Memory Retrieval and Syntactic Composition: An fMRI Study of Naturalistic Language Comprehension

Keywords: memory retrieval, structural composition, syntax, lexical cohesion, naturalistic comprehension, fMRI

This study examines memory retrieval and syntactic composition using fMRI while participants listen to a book, The Little Prince. These two processes are quantified drawing on methods from computational linguistics. Memory retrieval is quantified via multi-word expressions that are likely to be stored as a unit, rather than built-up compositionally. Syntactic composition is quantified via bottom-up parsing that tracks tree-building work needed in composed syntactic phrases. Regression analyses localise these to spatially-distinct brain regions. Composition mainly correlates with bilateral activity in anterior temporal lobe and inferior frontal gyrus. Retrieval of stored expressions drives right-lateralised activation in the precuneus. Less cohesive expressions activate well-known nodes of the language network implicated in composition. These results help to detail the neuroanatomical bases of two widely-assumed cognitive operations in language processing.

Introduction

Our human ability to comprehend natural language relies on two fundamental of cognitive operations. One involves the retrieval of memorised elements, while the other composes those elements together into novel expressions. Despite a growing body of work on the brain's language network (see e.g. [START_REF] Dronkers | Lesion analysis of the brain areas involved in language comprehension[END_REF][START_REF] Fedorenko | Neural correlate of the construction of sentence meaning[END_REF][START_REF] Friederici | The language network[END_REF][START_REF] Hagoort | The neurobiology of language beyond single words[END_REF][START_REF] Pallier | Cortical representation of the constituent structure of sentences[END_REF] the neural bases of these two operations within this network remain imprecisely specified.

This study disentangles these two cognitive processes with an analysis of fMRI time-courses observed during an extended episode of naturalistic spoken story understanding. Like prior work, it suggests a localisation of memory retrieval and composition to largely distinct regions of the brain. But by linking these operations with quantitatively precise computational models, this study paves the way for more granular research into the relationship between these operations. Further, this spatial dissociation is based upon brain responses to a literary narrative, one that attests a variety of different linguistic phenomena in an ecologically valid setting rather than one or two constructions repeated out of context.

The processing of retrieving memorised elements is operationalised using a specific type of word sequence called "multiword expressions." This term (henceforth abbreviated MWE) comes from computational linguistics. Broadly, it refers to word sequences that are bettertreated as a single unit rather than as a structurally-composed combination [START_REF] Sag | Multiword expressions: A pain in the neck for NLP[END_REF]. MWE themselves raise a general question about the balance between productivity and reuse in language processing [START_REF] Goldberg | Constructions at work: The nature of generalization in language[END_REF][START_REF] Jackendoff | Foundation of language: Brain, meaning, grammar[END_REF][START_REF] O'donnell | Productivity and reuse in language: A theory of linguistic computation and storage[END_REF]. If they truly lack internal structure, then their processing might proceed via a single, unitary retrieval operation rather than step-by-step composition. Even if they are amenable to some sort of internal structure analysis, existing proposals hold that the entire item is stored and retrieved as a unit.

Adopting this idea as working hypothesis, we assume that MWE comprehension involves an additional memory retrieval, one that is not involved in non-MWE expressions. No assumptions regarding the nature of this retrieval are required, only that the stored element should correspond to the expression as a whole.

This distinction between retrieval and composition figures centrally in neurocognitive models of language processing [START_REF] Bornkessel-Schlesewsky | The role of prominence information in the realtime comprehension of transitive constructions: A cross-linguistic approach[END_REF][START_REF] Siyanova-Chanturia | Eye-tracking and ERPs in multi-word expression research: A state-ofthe-art review of the method and findings[END_REF][START_REF] Hagoort | On Broca, brain, and binding: A new framework[END_REF][START_REF] Hagoort | MUC (Memory,Unification, Control): A model on the neurobiology of language beyond single word processing[END_REF]; [START_REF] Ullman | A neurocognitive perspective on language: The declarative/procedural model[END_REF][START_REF] Ullman | Contributions of memory circuits to language: The declarative/procedural model[END_REF][START_REF] Ullman | The declarative/procedural model: A neurobiological model of language learning, knowledge, and use[END_REF]. These models all suppose, to some degree or another, that these two operations in language processing dissociate at the level of neurobiology. The neurobiological implications of this claim are discussed in more detail below in §2.5. The present study adds to prior work by instantiating these two operations in quantitatively precise computational models, and by probing their neural localisation during a naturalistic task.1

Present Study

We pursue our objective by investigating two specific research questions. The first is to localise these processes (cf. [START_REF] Yang | Syntactic complexity and frequency in the neurocognitive language system[END_REF]: Which brain structures support cognitive processes of structural composition and retrieval of stored linguistic elements, in naturalistic language comprehension? The second research question has to do with the retrieval operation specifically: Are some MWEs better candidates for comprehension via direct retrieval than others?

We investigate these questions using naturalistic spoken story-comprehension, with a view towards to maximising ecological validity. The stimulus, introduced below in §2.1, is a literary text annotated with numbers that quantify processing effort associated with each operation ( §2.2 and §2.4). This section finishes by deriving from the existing literature a broad hypothesis regarding the organisation of these two cognitive processes in the brain ( §2.5).

Naturalistic Stimulus

Participants heard de Saint-Exupéry's The Little Prince (1943) as translated into English by David Wilkinson and read out loud by Nadine Eckert-Boulet. The text constitutes a fairly lengthy exposure to naturalistic language, comprising 15,388 words and lasting over an hour and a half. This follows [START_REF] Willems | Cognitive neuroscience of natural language use[END_REF] encouragement to probe the neural bases of language comprehension with greater ecological validity.

The MWE that are identified in this text reflect contemporary usage of these expressions, and they are linguistically diverse (for details see §2.2 and §3.2). To get a sense of how these MWE attestation frequencies relate to distributional patterns in English more generally, Figure 1 compares attestation counts in the stimulus text to their corresponding frequencies in the 560million word Corpus of Contemporary American English (COCA; [START_REF] Davies | The Corpus of Contemporary American English (COCA): 560 million words, 1990present[END_REF]. Over half (56.54%) of the attestations are headed by a verb, and only two items are story-character names. These attestations are evenly distributed throughout the text with about 100 examples in each section.

MWEs as Indicators of Memory Retrieval

What are MWEs?

The designation 'MWE' is motivated in computational linguistics whenever a sequence of words is better treated as an indivisible whole, rather than a structure that is composed out of subparts.

This pragmatic criterion (see e.g. [START_REF] Calzolari | Towards best practice for Multiword Expressions in Computational Lexicons[END_REF] picks out a very wide class of linguistic phenomena. To get a sense of this diversity, Table 1 (after Siyanova-Chanturia 2013) cites some cases that might plausibly be analysed as MWEs. The specific MWE identification procedure used in this study is presented further below in §3.2.

fixed phrases

per se, by and large noun compounds black coffee, cable car verb compounds give a presentation, come along binomials heaven and hell, safe and sound complex prepositions in spite of idioms break the ice, spill the beans

MWE and Compositionality

What unifies cases of MWE is the absence of a wholly compositional linguistic analysis [START_REF] Sag | Multiword expressions: A pain in the neck for NLP[END_REF]. The naturalistic story used as a stimulus in this project attests nearly all of the types cited in Table 1; a few are shown below in Example (1).

(1) a. So I thought a lot about the adventures of the jungle and in turn, I managed with a coloured pencil to make my first drawing.

b. My little fellow, I don't know how to draw anything except boa constrictors, closed and open. c. 'What are you doing there?', he said to the drinker who he found sitting in silence in front of a number of empty bottles and a number of full bottles d. You must see to it that you regularly pull out the baobabs as soon as they can be told apart from the rose bushes to which they look very similar to when they are young.

The syntactic and semantic properties of the boldfaced expressions in (1) indeed resist analysis by parts; they could be considered in some way as lexicalised.

MWE Processing

To the extent that MWEs are not compositional, their processing may depend more on memory retrieval than composition. This idea is most clear in the case of idioms where it has been dubbed the Configuration Hypothesis (CH) by [START_REF] Cacciari | The comprehension of idioms[END_REF].

For the CH, idioms are processed word-by-word, just like any other piece of language, until enough information has accumulated to render the sequence of words identifiable as -or highly expected to be -a memorised chunk. Only at this point the idiomatic meaning is retrieved. [START_REF] Cacciari | Processing multiword idiomatic strings: Many words in one?[END_REF] Here, the CH is generalised in supposing that all categories of MWE exhibit this greater dependence on memory retrieval. Section 2.5.1, below, reviews experimental evidence regarding the processing of MWEs. Of course, whether or not retrieval is actually used on a particular occasion may reflect gradations in compositionality. These gradations, as the next section details, can be quantified precisely.

Lexical Cohesion of MWEs

The crisp theoretical distinction of the CH between compositional and non-compositional expressions obscures the somewhat messier reality: expressions fall along a graded spectrum of compositionality (see Table 2). To quantify this gradient of MWE cohesion, we use a measure called Pointwise Mutual Information (PMI; [START_REF] Church | Word association norms, mutual information, and lexicography[END_REF]. PMI is commonly used in corpus linguistics to identify collocations.2 Intuitively, PMI is high when the word sequence under consideration occurs more often together than one would have expected, based on the frequencies of the individual words (Manning & Schütze, 1999, §5.4). More formally, PMI is a log-ratio of observed and expected probabilities:

PMI = log 2 O E (1)
where

O = count(whole expression) corpus size and E = count(w 1 ) * count(w 2 ) * • • • * count(w n ) corpus size n
In definition 1 the numerator O is a probability for the n-word collocation, w 1 w 2 ...w n-1 w n . The denominator E is what one would expect if the occurrences of each word in the collocation were probabilistically independent.

MWEs that receive a higher PMI score are lexically more cohesive, suggesting less compositionality, at least at the superficial level of word co-occurrence. To the extent that their greater cohesion results in a kind of word-sequence memorisation, these highly-cohesive MWEs are excellent candidates for a processing account based on memory retrieval. Conversely, low PMI scores signal a kind of "incohesion." These expressions might be better accounted-for by a compositional analysis.

It is important to appreciate that PMI is different from the raw attestation counts that go into it. This is because PMI reflects the ratio of the attestations of the whole relative to the attestations of the parts. It was raw counts, rather than the ratio PMI, that were used in a fMRI study by [START_REF] Yang | Syntactic complexity and frequency in the neurocognitive language system[END_REF] that examined memory retrieval in comparison to syntactic complexity. This study found no effects of frequency in the brain beyond the single-word level. [START_REF] Yang | Syntactic complexity and frequency in the neurocognitive language system[END_REF] did observe, however, an effect of syntactic complexity by parametrically varying six types of nominal, verbal and prepositional phrases. The next section presents an alternative way to investigate this same factor using naturalistic text.

Bottom-up Parsing as Composition

To study composition itself, some independent characterisation of compositional processes themselves are needed. In this study, bottom-up parsing plays that role. The intermediate states of this parsing strategy quantify the amount of compositional work that an idealised system would do, in the course of processing the naturalistic stimulus text introduced above in §2.1. While there exists a large literature on bottom-up parsing within computational linguistics (see e.g. [START_REF] Hale | Automaton theories of human sentence comprehension[END_REF] its essential character is easy to grasp.

Bottom-up parsing amounts to a repeated cycle of choice: whether to shift to the next word or reduce a sequence of transient elements held in memory. As shown in Figure 2, reduce actions are individuated by particular grammar rules. The number of parser actions required at each word defines an incremental complexity metric. We use this complexity metric to quantify composition effort in the brain, following prior work in neurolinguistics [START_REF] Brennan | Syntactic structure building in the anterior temporal lobe during natural story listening[END_REF][START_REF] Brennan | MEG evidence for incremental sentence composition in the anterior temporal lobe[END_REF][START_REF] Brennan | Abstract linguistic structure correlates with temporal activity during naturalistic comprehension[END_REF][START_REF] Nelson | Neurophysiological dynamics of phrase-structure building during sentence processing[END_REF] and psycholinguistics [START_REF] Frazier | Syntactic complexity[END_REF]. shift and reduce) that would build the colour-coded tree nodes during bottom-up parsing.

Hypotheses regarding Retrieval of MWEs and Composition

With operational definitions sketched-out for both memory retrieval ( §2.2) and composition ( §2.4), this section turns to what is already known about their processing in the brain. Synthesising several different literatures, a general prediction emerges that these two processes are subserved by different brain areas.

MWE Processing is Different: Experimental Evidence

Prior work supports the claim that MWE comprehension is distinct from other language processing such as structural composition and involves additional memory retrieval. For instance, it is well-established at the behavioural level that MWEs are produced and understood faster due to their frequency, familiarity, and predictability [START_REF] Siyanova-Chanturia | The idiom principle revisited[END_REF]. This would follow if MWEs were remembered as chunks, in the sense of [START_REF] Miller | The magical number seven, plus or minus two: Some limits on our capacity for processing information[END_REF] that was later formalised by [START_REF] Laird | Chunking in Soar, anatomy of a general learning mechanism[END_REF][START_REF] Rosenbloom | Learning by chunking: A production-system model of practice[END_REF].

Eye-tracking and EEG work further documents this processing advantage across a wide range of MWE sub-types, e.g. binomials [START_REF] Siyanova-Chanturia | Adding more fuel to the fire: An eyetracking study of idiom processing by native and non-native speakers[END_REF], phrasal verbs [START_REF] Yaneva | Cognitive processing of multiword expressions in native and non-native speakers of English: Evidence from gaze data[END_REF]), complex prepositions (Molinaro, Canal, Vespignani, Pesciarelli, & Cacciari, 2013;[START_REF] Molinaro | Cloze probability does not only affect N400 amplitude: The case of complex prepositions[END_REF], nominal compounds [START_REF] Molinaro | Electrophysiological evidence of interaction between contextual expectation and semantic integration during the processing of collocations[END_REF][START_REF] Molinaro | Semantic combinatorial processing of nonanomalous expressions[END_REF], lexical bundles [START_REF] Tremblay | Holistic processing of regular four-word sequences: A behavioral and ERP study of the effects of structure, frequency, and probability on immediate free recall[END_REF][START_REF] Tremblay | Processing advantages of lexical bundles: Evidence from self-paced reading and sentence recall tasks[END_REF], and idioms [START_REF] Laurent | On understanding idiomatic language: The salience hypothesis assessed by ERPs[END_REF][START_REF] Rommers | Context-dependent Semantic Processing in the Human Brain: Evidence from Idiom Comprehension[END_REF][START_REF] Siyanova-Chanturia | Seeing a phrase time and again matters: The role of phrasal frequency in the processing of multiword sequences[END_REF][START_REF] Strandburg | Eventrelated potentials in high-functioning adult autistics: Linguistic and nonlinguistic visual information processing tasks[END_REF][START_REF] Underwood | The eyes have it[END_REF][START_REF] Vespignani | Predictive Mechanisms in Idiom Comprehension[END_REF].

Neural Basis for Retrieving MWEs and Composition

The hypothesis that MWEs involve additional memory retrieval reflects trends in both Construction Grammar and Connectionism (see e.g. [START_REF] Ellis | Constructions, chunking, and connectionism: The emergence of second language structure[END_REF]. In these quarters, however, the relevant 'memory' does not receive a precise psychological or neural characterisation. It may be that memory for MWE fall into a distinct theoretical category "between" episodic memory and general semantic memory [START_REF] Renoult | Personal semantics: at the crossroads of semantic and episodic memory[END_REF]. Memories in this category are encoded on the basis of repeated personal experience, an idea that is certainly plausible for MWEs. Neuroanatomical sites for the category of memory that [START_REF] Renoult | Personal semantics: at the crossroads of semantic and episodic memory[END_REF] discuss include medial temporal regions and the Precuneus. This last suggestion converges well with PET results on single word retrieval [START_REF] Halsband | PET studies on the memory processing of word pairs in bilingual Finnish-English subjects[END_REF].

Regarding composition operations, one prominent line of work implicates frontal regions, such as the pars triangularis and pars opercularis of the inferior frontal gyrus [START_REF] Friederici | The language network[END_REF][START_REF] Hagoort | MUC (Memory,Unification, Control): A model on the neurobiology of language beyond single word processing[END_REF][START_REF] Snijders | Retrieval and unification of syntactic structure in sentence comprehension: An fMRI study using word-category ambiguity[END_REF][START_REF] Zaccarella | The neurobiological nature of syntactic hierarchies[END_REF]. Other lines of investigation implicate anterior temporal regions, based on deficit-lesion data [START_REF] Dronkers | Lesion analysis of the brain areas involved in language comprehension[END_REF]) and text comprehension tasks [START_REF] Ferstl | The extended language network: A meta-analysis of neuroimaging studies on text comprehension[END_REF]. These anterior temporal areas are sensitive to parametric variation of phrase size, which would follow if they were somehow representing composed structures (Bemis & Pylkkänen, 2011a;[START_REF] Brennan | Syntactic structure building in the anterior temporal lobe during natural story listening[END_REF][START_REF] Pallier | Cortical representation of the constituent structure of sentences[END_REF][START_REF] Pylkkänen | Composition of complex meaning: Interdisciplinary perspectives on the left anterior temporal lobe[END_REF]. These latter findings, although, conflict with patient studies showing that atrophy to anterior temporal regions does not systematically impair processing of syntactically complex sentences (Wilson et al., 2014a). This tension between functional and deficit/lesion results remains to be resolved.

This array of empirical work, in combination with the neurocognitive models of language processing discussed below ( §2.5.3) serves to motivate the present investigation into the brain bases for MWE processing and compositional processing.

Neurocognitive Models of Language and Localisation

Current neurobiological models for language suggest that retrieval and composition should be sub-served by different brain areas. These models do however differ both in their conceptualisation of these operations, and on their anatomical localisation.

The Declarative-Procedural model [START_REF] Ullman | A neurocognitive perspective on language: The declarative/procedural model[END_REF] for instance, is founded upon a distinction that contrasts memory-related with non-memory-related processing. [START_REF] Ullman | The declarative/procedural model: A neurobiological model of language learning, knowledge, and use[END_REF] links rulebased mechanisms to frontal regions and sub-cortical structures, while memory for words is supported by medial temporal regions.

In Hagoort's Memory, Unification and Control framework (2016), composition falls under the scope of the Unification operation and is assigned to inferior frontal areas. While operating under different assumptions, this localisation is in general agreement with [START_REF] Zaccarella | The neurobiological nature of syntactic hierarchies[END_REF] who propose that hierarchical processing is subserved by a sub-part of the left inferior frontal gyrus. Regarding the Memory aspect of their model, Hagoort and colleagues agree partly with Ullman, associating that function (among others) to posterior temporal regions [START_REF] Hagoort | Reflections on the neurobiology of syntax[END_REF][START_REF] Hagoort | The neurobiology of language beyond single words[END_REF].

The Dual Streams Model [START_REF] Hickok | The cortical organization of speech processing[END_REF] similarly locates the Lexical Interface, where individual words would be processed, to posterior middle temporal gyrus. Syntactic phrases would be composed, part-by-part, by a combinatorial network within the anterior temporal lobes.

Another perspective is offered by the Extended Argument Dependency Model (eADM; [START_REF] Bornkessel-Schlesewsky | The role of prominence information in the realtime comprehension of transitive constructions: A cross-linguistic approach[END_REF][START_REF] Siyanova-Chanturia | Eye-tracking and ERPs in multi-word expression research: A state-ofthe-art review of the method and findings[END_REF], that divides up language processing in a different way. In this model sequential information (for instance about word order) is handled by a dorsal stream, while dependency information (as expressed through case-marking) is handled by a ventral stream. If MWE comprehension is sequential processing in this sense, then structures along this dorsal stream, including the inferior parietal lobule, should be involved. On the other hand, composition should activate temporal regions along the ventral stream.

These models' localisation claims contrast in detail with each other, and with an important body of evidence from patient work (e.g. [START_REF] Dronkers | Lesion analysis of the brain areas involved in language comprehension[END_REF]Wilson et al., 2014b;Wilson, Galantucci, Tartaglia, & Gorno-Tempini, 2012, inter alia). Despite this tension, they agree on the common proposal that retrieval and composition should each be independently localisable. It is this point of agreement that motivates the present study.

Methods

A spoken narrative serves as the stimulus. Participants hear this narrative over headphones while they are in the scanner (e.g. [START_REF] Brennan | Syntactic structure building in the anterior temporal lobe during natural story listening[END_REF][START_REF] Willems | Prediction during natural language comprehension[END_REF]. The sequence of neuroimages collected during their session becomes the dependent variable in a regression against word-by-word predictors, derived from the text of the story. The overall approach to deriving time series predictions regarding the comprehension of this auditory stimulus is shown in Figure 3.

Subjects and Design

Participants were fifty-one volunteers (32 women and 19 men, 18-37 years old) with no history of psychiatric, neurological, or other medical illness or history of drug or alcohol abuse that might compromise cognitive functions. All strictly qualified as right-handed on the Edinburgh handedness inventory [START_REF] Oldfield | The assessment and analysis of handedness: the Edinburgh inventory[END_REF]. They self-identified as native English speakers and gave their written informed consent prior to participation, in accordance with Cornell University IRB guidelines.

Stimulus and MWE Identification

The audio stimulus was a literary text, Antoine de Saint-Exupéry's The Little Prince. We applied two different approaches to MWE identification, which together yield 669 distinct MWE types for a total of 1292 attestations in the stimulus text. This section explains the two identification approaches.

The first approach uses lgtagger [START_REF] Constant | MWU-aware part-of-speech tagging with a crf model and lexical resources[END_REF], a freely-available program that combines two submethods:

(1) string-based matching to look up MWE in external dictionaries (2) a tagging model, based on conditional random fields and trained on hand-checked examples Regarding submethod 1, the external dictionaries are: the Unitex lexicon [START_REF] Paumier | Unitex, a Corpus Processing System with Multi-Lingual Linguistic Resources[END_REF], the SAID corpus [START_REF] Kuiper | Syntactically Annotated Idiom Dataset (SAID) LDC2003T10[END_REF], the Cambridge International Dictionary of Idioms [START_REF] White | Cambridge International Dictionary of Idioms[END_REF], and the Dictionary of American Idioms [START_REF] Makkai | A Dictionary of American Idioms[END_REF].

Regarding submethod 2, conditional random fields are used to guess MWE marks like those shown in the final column of Table 3. N. Smith (2011, §3.5) offers a general introduction to the technique. The key idea here is to construe a word's MWE status as a latent variable that is related to the surface string via a collection of weak clues, the "features." Here the training examples come from MWE annotations in the English Universal Dependencies English Web Treebank [START_REF] Silveira | Universal dependencies english web treebank[END_REF]. This annotated data comprises several web-related genres including weblogs, newsgroups, emails, reviews, and Yahoo! answers. The conditional random field makes its marking decisions based on feature templates that specify, for example, particular lexical items, capitalisation, or part of speech tags. A complete table of these templates is presented in [START_REF] Constant | Evaluating the impact of external lexical resources into a crf-based multiword segmenter and part-of-speech tagger[END_REF]. The approach is limited to contiguous MWEs and identifies mainly fixed expressions such as complex function words and nominal MWEs. Crucially, this way of identifying MWEs is blind to hierarchical syntactic structure.

The second approach uses a transition based system (Al [START_REF] Saied | The ATILF-LLF system for the PARSEME Shared Task: a Transition-based Verbal Multiword Expression Tagger[END_REF]. This system is a variant of the well-known Nivre parser [START_REF] Constant | A transition-based system for joint lexical and syntactic analysis[END_REF], in which abstract "actions" update an abstract computational state that moves through the text, emitting MWE marks as a side-effect. The choice of which action to take is made by a classifier, here a support vector machine (for textbook introductions see e.g. Abney 2007, §6.4 or Murphy 2012, §14.5). There are actions that Add or Remove words from the computational state, as well as actions which Mark candidates as MWEs. This last type is actually divided into two cases, one of which composes candidate MWEs in a binary fashion, and another which marks these binary elements in a way that produces output. Figure 4 shows how this transition-based system would apply to the example "see to it."

The input to this classifier includes word form, lemma and part-of-speech information as shown in Table 3, as well as dictionary-based features and features that relate to previous actions ("history-based"). The classifier's output is a selection of one particular action. This second way of identifying MWEs is similarly denied access to any hierarchical syntactic information.

The training data for this transition-based system come from the Children's Book Test [START_REF] Hill | The Goldilocks principle: Reading children's books with explicit memory representations[END_REF], which is close to the genre of the stimulus narrative. Training examples were generated by string-matching based on the external dictionaries cited above. Compared to the conditional random field-based approach, this second approach seems betterable to find verbal MWE.

The MWEs used in the fMRI study reported here were identified by applying both of the approaches described above to the stimulus text. The union of their outputs yielded a time series of indicator variables as shown in Panel C of Figure 5. This time series has the value 1 on words that are final in a MWE, and 0 otherwise.

PMI scores for each MWE define a gradient metric and were determined using equation 1. The probabilities used in this calculation were estimated using the Corpus of Contemporary English (COCA; [START_REF] Davies | The Corpus of Contemporary American English (COCA): 560 million words, 1990present[END_REF]. The observed probability O of entire MWEs was estimated by querying the 2018 version of COCA, which contains 560 million words. The expected probabilities E are based on an earlier COCA release that comprises 450 million words. These estimates are based on counts of lemmas, rather than inflected words. This use of lemmas serves to focus the analysis on the cohesion of the expression rather than its morphology.

Annotating compositional structure

Compositional processing was modeled as bottom-up parsing, as introduced above in §2.4. The tree structures were obtained using Stanford parser (D. [START_REF] Klein | Accurate unlexicalized parsing[END_REF] and follow the conventions of the Penn Treebank [START_REF] Marcus | Building a large annotated corpus of english: The penn treebank[END_REF]. They capture constituency relationships, including phrase type and embedding, but do not explicitly mark long-distance dependencies. The complexity metric is defined as the number of reduce actions that would be taken during bottom-up parsing of these trees. Since narrative stimulus does not readily lead to mis-understandings or "garden-path" effects in the sense of [START_REF] Bever | The cognitive basis for linguistic structures[END_REF] we restrict ourselves to just the globally-correct parser action sequence.

Figure 5 illustrates the two retrieval estimators and the one compositional estimator for a single sentence from The Little Prince. Correlation matrices for all estimators are given in the Appendix.

Data Presentation

After giving their informed consent, participants were familiarised with the MRI facility and assumed a supine position on the scanner gurney. The presentation script was written in Psy-choPy [START_REF] Peirce | Psychopy: Psychophysics software in Python[END_REF]. Auditory stimuli were delivered through MRI-safe, high-fidelity headphones (Confon HP-VS01, MR Confon, Magdeburg, Germany) inside the head coil. The headphones were secured against the plastic frame of the coil using foam blocks. Using a spoken recitation of the US Constitution, an experimenter increased the volume until participants reported that they could hear clearly. Participants then listened passively to the audio storybook for 1 hour 38 minutes. The story was divided into nine sections and at the end of each section the participants were presented with a multiple-choice questionnaire with four questions (36 questions in total), concerning events and situations described in the story. These questions served to confirm participants' comprehension. They were viewed via a mirror attached to the head coil and answered through a button box with their right hand. The entire session lasted around 2.5 hours.

Data Acquisition

Imaging was performed using a 3T MRI scanner (Discovery MR750, GE Healthcare, Milwaukee, WI) with a 32-channel head coil at the Cornell MRI Facility. Blood Oxygen Level Dependent (BOLD) signals were collected using a T2 -weighted echo planar imaging (EPI) sequence (repetition time: 2000 ms, echo time: 27 ms, flip angle: 77deg, image acceleration: 2X, field of view: 216 x 216 mm, matrix size 72 x 72, and 44 oblique slices, yielding 3 mm isotropic voxels). Anatomical images were collected with a high resolution T1-weighted (1 x 1 x 1 mm 3 voxel) with a Magnetisation-Prepared RApid Gradient-Echo (MP-RAGE) pulse sequence.

Data Analyses

Preprocessing

FSL's Brain Extraction Tool (Jenkinson, Beckmann, Behrens, Woolrich, & Smith, 2012; S. M. [START_REF] Smith | Advances in functional and structural MR image analysis and implementation as FSL[END_REF] was used for skull-stripping with a fractional intensity threshold setting of 0.5. Subsequent preprocessing steps were carried out using AFNI version 16 [START_REF] Cox | AFNI: software for analysis and visualization of functional magnetic resonance neuroimages[END_REF]. Anatomical and functional images were co-registered using the in-built AFNI function 3dseg, images were normalised to the MNI-152 template, and images were resampled to 2mm isotropic voxels. We used multi-echo independent components analysis (ME-ICA) [START_REF] Kundu | Integrated strategy for improving functional connectivity mapping using multiecho fMRI[END_REF][START_REF] Kundu | Differentiating BOLD and non-BOLD signals in fMRI time series using multi-echo EPI[END_REF] to improve the signal-to-noise ratio in these data. ME-ICA splits the T 2 * signal into BOLD-like and non BOLD-like components. Removing these non-BOLD components mitigates noise due to participants' head motion, physiology and scanner conditions such as thermal changes [START_REF] Kundu | Multiecho fMRI: A review of applications in fMRI denoising and analysis of BOLD signals[END_REF]. Indeed, there were no exclusions based on degree of head movement. Nor was any high-pass filtering or smoothing applied at this stage.

Statistical Analysis

The research questions layed out above in §2 motivate two statistical analyses. The first analysis localises composition and memory-retrieval operations during naturalistic listening. The second analysis investigates multiword expressions along a quantitative gradient of cohesion. Both analyses employ the General Linear Model, and were carried out using SPM12 (Friston, Ashburner, Kiebel, Nichols, & Penny, 2007). The predictors were convolved using SPM's canonical HRF.

Analysis 1: With Categorical Predictors

We regressed the word-by-word predictors described below against fMRI timecourses recorded during passive story-listening in a whole-brain analysis. Along with the parser action count and MWE indicators of theoretical interest, four "nuisance" variables of non-interest were entered into the GLM analysis, including a unigram word frequency predictor based on attestations in movie subtitles [START_REF] Brysbaert | Moving beyond Kučera and Francis: A critical evaluation of current word frequency norms and the introduction of a new and improved word frequency measure for American English[END_REF]. This enables the effects of MWE, Parser-action count, and MWE cohesion to be assessed in way that is distinct from single-word frequency effects. We also included two variables that account for prosodic effects: the RMS amplitude and the fundamental frequency of the auditory stimulus. These control variables serve to improve the sensitivity, specificity and validity of activation maps [START_REF] Bullmore | Methods for diagnosis and treatment of stimulus-correlated motion in generic brain activation studies using fMRI[END_REF][START_REF] Lund | Non-white noise in fMRI: does modelling have an impact?[END_REF]. The predictors entered into Analysis 1 are summarised in Table 4. Figure 5 shows the theoretical predictors' time series for a particular stimulus sentence.

Bottom-up parser action count

Analysis 2: With Gradient Predictors

Analysis 2 uses the same predictors as in Analysis 1, except that the categorical indicator of MWE presence is replaced with a gradient predictor. As explained above in §2.3, this predictor rates the propensity of a MWE to be memorised or not using a standard measure of collocation strength called pointwise mutual information (PMI). The same MWEs that were annotated with a value of "1" in Analysis 1 are in Analysis 2 marked with their PMI score. This is shown in the bottom row of Figure 5. Note that parser action count is not correlated with the categorical nor with the gradient MWE predictor (r = 0.09 in both cases). Correlation matrices for all terms entered into the regression analyses are given in the Appendix.

Group-level Analysis

In the second-level group analysis, each contrast was analysed separately at the group-level. An 8 mm FWHM Gaussian smoothing kernel was applied on the contrast images from the first-level analysis to counteract inter-subject anatomical variation. All the group-level results reported in the next section underwent FWE voxel correction for multiple comparisons which resulted in T-scores > 5.3.

Results

Behavioural results of the comprehension task showed attentive listening to the auditory story presentation. Across 51 participants, average accuracy on the comprehension questions was 90% (SD = 3.7%). All whole-brain effects reported survived a p < 0.05 Family-Wise-Error threshold at the voxel level. Tables introduced below use brain region labels from the Harvard-Oxford Cortical Structure Atlas.

Analysis 1: Results with Categorical Predictors

Results for Composition

Bottom-up parser action count shows a broad activation pattern both in right and left hemisphere. The peak activation is right lateralised in the anterior temporal lobe within a main cluster of activation which extends through the middle and superior temporal gyri. While anterior temporal activation is bilateral, both middle temporal gyrus and posterior superior temporal gyrus are only right lateralised. The second strongest cluster of increased activation is observed in the left inferior frontal gyrus stretching over pars orbitalis and triangularis and extending to the anterior insula and the putamen. A similar increased activation is observed in the right inferior frontal gyrus. Additional activation clusters were seen in areas including the prefrontal cortex, supplementary motor areas, the cerebellum, the left temporal-parietal junction, as well as inferior temporal lobes. These are all detailed in Table 5 andFigure

Results for MWE Presence

The categorical MWE predictor gives rise to two clusters of activation both in the right precuneus cortex, as presented below in Figure 6 and Table 6.

Analysis 2: Results with Gradient Predictors

Analysis 2 investigates memory retrieval further by rating how cohesive each MWE is. This is done using pointwise mutual information, introduced above in §2.3. Increasing MWE cohesion, as seen through the positive correlation with PMI, yields a single cluster in the right precuneus.

Left-lateralised activity in superior frontal gyrus, angular gyrus, pars triangularis, posterior middle temporal gyrus, and frontal pole was observed in proportion to decreasing lexical cohesion, as seen through the negative correlation with PMI scores. These are detailed in Table 7 and in Figure 7.

Discussion

This study investigated the neural substrates of two processes in language comprehension: retrieval and composition of stored linguistic elements. Departing from prior work, we distinguish these operations using computational methods and a naturalistic stimulus. Multiword expressions identified in The Little Prince using the methods of Al [START_REF] Saied | The ATILF-LLF system for the PARSEME Shared Task: a Transition-based Verbal Multiword Expression Tagger[END_REF] and [START_REF] Constant | MWU-aware part-of-speech tagging with a crf model and lexical resources[END_REF] serve as a hypothesis about points in the narrative where extra memory retrievals would occur.

Lexical Retrieval and MWEs

Significant activation in the precuneus for MWEs in both Analysis 1 and Analysis 2 offers initial support for the idea that this region subserves memory retrieval of these stored elements.

Precuneus and Retrieval of MWEs

Both the categorical Analysis 1 and the gradient Analysis 2 indicate the centrality of the precuneus for MWE-related memory processes. This area features a graded effect such that it is more active for increasingly cohesive MWEs, as estimated by the PMI metric. As discussed above in §2.3, highly-cohesive MWEs are excellent candidates for a processing account based on direct retrieval from memory.

The involvement of the precuneus in the sort of memory retrieval prompted by MWEs is consistent with studies attesting its participation in larger memory networks, e.g. for verbal material [START_REF] Halsband | PET studies on the memory processing of word pairs in bilingual Finnish-English subjects[END_REF]. The functional characterisation of the precuneus as part of a network sub-serving memory tasks, has been reported for different memory-based processes, such as verbal memory [START_REF] Halsband | PET studies on the memory processing of word pairs in bilingual Finnish-English subjects[END_REF], spatial memory [START_REF] Wallentin | Accessing the mental spacespatial working memory processes for language and vision overlap in precuneus[END_REF], episodic memory [START_REF] Andreasen | Remembering the past: two facets of episodic memory explored with positron emission tomography[END_REF], memory-related imagery [START_REF] Fletcher | The mind's eyeprecuneus activation in memory-related imagery[END_REF][START_REF] Mashal | The role of the precuneus in metaphor comprehension: evidence from an fMRI study in people with schizophrenia and healthy participants[END_REF]. Notably, some studies found that the Precuneus is active in non-imagery related episodic memory for musical sequences or abstract words (e.g. [START_REF] Platel | The structural components of music perception. a functional anatomical study[END_REF].

The precuneus has also been identified as a part of the Default Network, perhaps playing a role in a Dorsal-Medial subsystem of that network [START_REF] Andrews-Hanna | The default network and self-generated thought: Component processes, dynamic control, and clinical relevance[END_REF]. This subsystem is said to support story comprehension among other aspects of selfgenerated thought. Although the precuneus has been designated as part of the Protagonists Perspective Interpreter Network [START_REF] Mason | Neuroimaging contributions to the understanding of discourse processes[END_REF] and attested in naturalistic reading by [START_REF] Wehbe | Simultaneously uncovering the patterns of brain regions involved in different story reading subprocesses[END_REF], an interpretation of the results of this particular study in terms of reference to story characters is implausible. This is because less than 2% of the MWEs in the stimulus narrative are names of story characters.

More broadly, the postero-medial portion of the parietal lobe has been widely linked to processing of complex lexical information by previous studies. Its sensitivity to the number of complements of a verb was reported by [START_REF] Shetreet | Cortical representation of verb processing in sentence comprehension: Number of complements, subcategorization, and thematic frames[END_REF] in the medial precuneus and the anterior cingulate cortex (see also den [START_REF] Den Ouden | Argument structure effects in action verb naming in static and dynamic conditions[END_REF][START_REF] Shetreet | Cortical representation of verbs with optional complements: The theoretical contribution of fMRI[END_REF]. The precuneus has also been linked to lexical processing of information of a relatively high complexity [START_REF] Shetreet | An fMRI study of syntactic layers: Sentential and lexical aspects of embedding[END_REF]. These results converge with a strand of the fMRI literature over the past decade that has characterised precuneus as a language-relevant region.

Incohesive MWEs and Broca's Area

While highly cohesive expressions evoke a focal pattern of activation that is distinct from classical left-hemisphere language areas, the cortical sites that showed a sensitivity to a decrease in MWEs' cohesion strength include the left superior frontal gyrus and also encompass several areas of the perisylvian language network. Notably the left inferior frontal gyrus, or Broca's area (pars triangularis and pars orbitalis), and also anterior and posterior regions of the left temporal gyrus. These perisylvian areas have been connected with compositional processes (e.g. [START_REF] Pallier | Cortical representation of the constituent structure of sentences[END_REF], and this is consistent with the leading idea that less-cohesive expressions are more likely to require compositional processes. Indeed a similar set of regions are activated in response to the Bottom-up parsing predictor, as discussed further in the next sub-section.

Parser Action Count as Composition

Word-by-word composition effort, quantified in terms of bottom-up parser actions, correlates in Analysis 1 with a highly bilateral pattern across several areas in the language network. Notably, both anterior-frontal and anterior-temporal regions are involved bilaterally.

Anterior Frontal and Anterior Temporal regions

Inferior frontal regions encompassing different sub-parts of the inferior frontal gyrus and anterior insula are commonly attributed a role in compositional processes [START_REF] Friederici | The language network[END_REF][START_REF] Hagoort | MUC (Memory,Unification, Control): A model on the neurobiology of language beyond single word processing[END_REF][START_REF] Snijders | Retrieval and unification of syntactic structure in sentence comprehension: An fMRI study using word-category ambiguity[END_REF][START_REF] Zaccarella | Merge in the human brain: A sub-region based functional investigation in the left Pars Opercularis[END_REF], and our findings are consistent with that work. Interestingly, prior work using methods most similar to ours (bottomup parsing, a naturalistic stimulus) have not highlighted the inferior frontal gyrus [START_REF] Brennan | Syntactic structure building in the anterior temporal lobe during natural story listening[END_REF][START_REF] Brennan | Naturalistic sentence comprehension in the brain[END_REF]. We hesitate to draw strong inferences from comparing "significant" to "nonsignificant" results, but this discrepancy may reflect on the increase in statistical power afforded by our current study, which uses a stimulus that is over seven times longer that used by Brennan and colleagues, and also reports data from almost twice as many participants.

The balance of activation that we observe in response to increasing parser action count is consistent with Ullman's Declarative-Procedural model. It shows a predominately anterior distribution in left hemisphere, alongside smaller activation clusters in the inferior parietal lobe and temporo-occipital-parietal junction. Further, the observed activation of putamen within the large bilateral cluster in inferior frontal gyrus and anterior insula conforms to Ullman's prediction that rule-governed language use is instantiated in a Fronto-basal ganglia network [START_REF] Ullman | Contributions of memory circuits to language: The declarative/procedural model[END_REF][START_REF] Ullman | The declarative/procedural model: A neurobiological model of language learning, knowledge, and use[END_REF].

The composition effect observed bilaterally in anterior temporal lobe is consistent with previous work on compositional in naturalistic narrative [START_REF] Brennan | Syntactic structure building in the anterior temporal lobe during natural story listening[END_REF]. It confirms and extends a broad range of prior findings, including those from simple two-word phrases (Bemis & Pylkkänen, 2011b), from parametric variation of constituent size [START_REF] Pallier | Cortical representation of the constituent structure of sentences[END_REF], and comparisons of simple sentences to unstructured lists of words [START_REF] Humphries | Syntactic and semantic modulation of neural activity during auditory sentence comprehension[END_REF][START_REF] Rogalsky | Selective attention to semantic and syntactic features modulates sentence processing networks in anterior temporal cortex[END_REF][START_REF] Stowe | Localizing components of a complex task: Sentence processing and working memory[END_REF][START_REF] Vandenberghe | The response of left temporal cortex to sentences[END_REF][START_REF] Xu | Language in context: Emergent features of word, sentence, and narrative comprehension[END_REF]. Our result also conforms with evidence that anterior temporal lesions impair "basic levels of constituent-structure processing" (Dronkers et al., 2004, p. 161).

Recent neuroimaging work correlating brain activity with processing-complexity metrics across grammar types and parsing strategies has convergently pointed to the anterior temporal complex [START_REF] Brennan | Abstract linguistic structure correlates with temporal activity during naturalistic comprehension[END_REF][START_REF] Li | Grammatical predictors for fMRI timecourses[END_REF] Appear; for a review see [START_REF] Brennan | Naturalistic sentence comprehension in the brain[END_REF]. The precision of this localisation has also been supported by results using intracranial recording [START_REF] Nelson | Neurophysiological dynamics of phrase-structure building during sentence processing[END_REF].

The results reported here, along with previous findings, underline the involvement of anterior temporal lobe in basic composition processes [START_REF] Friederici | The language network[END_REF][START_REF] Hickok | The cortical organization of speech processing[END_REF]). Yet, along-side many of the findings just reviewed, our results stand in contrast to the apparent stability of compositional processes following anterior temporal atrophy (e.g. Wilson et al., 2014a[START_REF] Wilson | Variable disruption of a syntactic processing network in primary progressive aphasia[END_REF]. The present study does not resolve the tension between these two literatures. We speculate that such a resolution might involve more detailed consideration of the specific mechanisms that contribute to observed neural activation patterns, and perhaps how compensatory processes respond to neuronal atrophy.

The unexpected right lateralisation of posterior temporal activation may suggest that the way the human brain processes linguistic stimuli within a contextually rich setting -one more similar to the everyday language environment -shows a strongly bilateral involvement of the language network. Future work that parametrically varies contextual richness, from more isolated to more naturalistic stimuli, may shed light on this speculation.

Left Posterior Temporal regions and Parser Action Count

The operationalisation of compositional processing in terms of bottom-up parser actions highlights the language network to a large extent. However, we did not find a significant correlation between this metric and activity in left posterior temporal areas. Our result contrasts with posterior temporal effects of composition that are reported in some prior imaging studies (e.g. [START_REF] Brennan | Abstract linguistic structure correlates with temporal activity during naturalistic comprehension[END_REF][START_REF] Pallier | Cortical representation of the constituent structure of sentences[END_REF][START_REF] Snijders | Retrieval and unification of syntactic structure in sentence comprehension: An fMRI study using word-category ambiguity[END_REF][START_REF] Vandenberghe | The response of left temporal cortex to sentences[END_REF] and are also highlighted in studies of primary progressive aphasia [START_REF] Wilson | The neural basis of syntactic deficits in primary progressive aphasia[END_REF]. This absence of evidence should be treated carefully; caution is especially warranted as different studies take different perspectives on composition itself.

The phrase-structures used in our analysis only capture some aspects of composition, and so a more comprehensive view of this aspect of language processing may shed light on discrepancies between studies. For example, the LPTL composition effect reported by [START_REF] Brennan | Abstract linguistic structure correlates with temporal activity during naturalistic comprehension[END_REF] did not appear when analyzing the same types of tree structures used in the present analysis, but only from tree structures that explicitly encode long-distance dependencies and other more abstract aspects of grammar. More broadly, one can speculate that the absence of activation in our study might be a consequence of the unlexicalised character of the bottom-up parsing metric. It counts distances, for instance along the spine of a tree as in Figure 3 (C), but it treats all nodes as equal, ignoring lexical information except at the preterminal level. Subjects and direct objects receive the same score if they close off the same number of constituents. This conception of composition leaves out many classical distinctions that are known to affect cerebral activity, such as semantic selection, argument structure [START_REF] Frankland | An architecture for encoding sentence meaning in left mid-superior temporal cortex[END_REF][START_REF] Thompson | Neural correlates of verb argument structure processing[END_REF], long-distance dependencies [START_REF] Ben-Shachar | Neural correlates of syntactic movement: Converging evidence from two fMRI experiments[END_REF][START_REF] Santi | fMRI adaptation dissociates syntactic complexity dimensions[END_REF][START_REF] Shetreet | The processing of different syntactic structures: fMRI investigation of the linguistic distinction between wh-movement and verb movement[END_REF], and case-checking [START_REF] Nieuwland | Brain regions that process case: Evidence from Basque[END_REF].

Such a more comprehensive perspective may also shed light on another aspect of the present results. While parser action count does not correlate with left posterior temporal activation, we do see such a correlation for decreasing MWE cohesion (see Section 6.1.2). Such a correlation is consistent with the hypothesis that less coherent expressions load more heavily onto certain compositional operations, and such compositional operations evoke, at least in some cases, posterior temporal activity.

Overall the results support the idea that composition and retrieval of stored linguistic elements like MWEs involve different parts of the language network. They also confirm that bilateral anterior frontal regions play an important role in linguistic composition during naturalistic spoken story comprehension.

Composition and the Cerebellum

An often reported but less-discussed area in language processing is the Cerebellum. In this study its relationship to parser action count in Crus I suggests its involvement in compositional processes bilaterally. A growing number of studies have provided evidence for cerebellar involvement in non-motor language functions (see [START_REF] Stoodley | Functional topography in the human cerebellum: A meta-analysis of neuroimaging studies[END_REF] for a meta-analysis), revealing interesting patterns of co-activation with classical left prefrontal language areas and right cerebellar hemisphere during language tasks and comprehension [START_REF] Fedorenko | Language-selective and domain-general regions lie side by side within Brocas area[END_REF], or during the processing of complex syntactic structures [START_REF] Christensen | Interfaces, syntactic movement, and neural activation: A new perspective on the implementation of language in the brain[END_REF][START_REF] Fabre | The sentence as cognitive object -the neural underpinnings of syntactic complexity in Chinese and French[END_REF][START_REF] Shetreet | The processing of different syntactic structures: fMRI investigation of the linguistic distinction between wh-movement and verb movement[END_REF]. Specifically, Crus I and Lobule VII are typically proposed as members of Prefrontal-cerebellar connectivity loops [START_REF] Stoodley | Functional topography in the human cerebellum: A meta-analysis of neuroimaging studies[END_REF].

Moreover, an emerging view of perceptual and cognitive processing in the Cerebellum puts special emphasis on sequencing and predictive processes [START_REF] Leggio | Cognitive sequencing impairment in patients with focal or atrophic cerebellar damage[END_REF][START_REF] Molinari | Cerebellum and detection of sequences, from perception to cognition[END_REF], which are presumably also required in syntactic sentence parsing operations. Although no special modelling of predictive processing was undertaken in this study, bottom-up approaches such as the LR parser do include predictive information in their control state (see e.g. [START_REF] Shieber | Variations on incremental interpretation[END_REF][START_REF] Stabler | Avoid the pedestrian's paradox[END_REF].

Composition of Form and of Meaning

The foregoing discussion has remained neutral as to whether activation within the anterior temporal lobe and other brain areas that correlate with bottom-up parsing reflects syntactic processing, or compositional semantic processing. Syntactic structure-building is transparently associated with bottom-up parser action count (panel B of Figure 5) that is itself defined in terms of syntactic phrases such as NP, VP and S. However, our approach is not able to rule out a semantic explanation. A semantic role would follow within approaches to natural language grammar that adopt the "rule-to-rule" assumption [START_REF] Bach | An extension of classical transformational grammar[END_REF]. Following the seminal work of [START_REF] Montague | Formal philosophy: selected paper of Richard Montague[END_REF] these theories of linguistic competence suppose that syntactic rules generally are paired with corresponding semantic rules. This organisation of grammar, which is introduced briefly in [START_REF] Barker | Introduction: Direct compositionality[END_REF], typifies Combinatorial Category Grammar [START_REF] Steedman | The syntactic process[END_REF], Generalized/Head-driven Phrase-Structure Grammars (E. [START_REF] Klein | Type-driven translation[END_REF] and certain forms of Minimalism (see e.g. Kobele, 2006, §2.2). A consensus from these conceptions of grammar is that aspects of linguistic performance that correlate with syntactic processing will necessarily also correlate with semantic processing.

Of course, the particular rules being processed do matter, and in this case the rules are plainly syntactic in nature. To tease apart semantic from syntactic explanations, future work could develop an alternative grammar, unlike the Penn Treebank phrase structures used here, and evaluate whether this alternative semantically-based theory offers a better explanation of the observed fMRI timecourses. Such an approach may be particularly fruitful in constructions where an additional semantic rule of "coercion" applies, unaccompanied by a corresponding syntactic rule [START_REF] Pylkkänen | Mismatching meanings in brain and behavior[END_REF]. These considerations, regarding the kinds of rules that are used during comprehension, are in any event quite orthogonal to questions about their order application [START_REF] Hale | Automaton theories of human sentence comprehension[END_REF]

Conclusion

Analysis of MWEs and parser action counts in naturalistic spoken story comprehension supports the localisation of memory retrieval to the precuneus in a way that trades-off with the other wellknown language regions such as inferior frontal gyrus and anterior temporal lobe. This trade-off seems to reflect the degree to which particular MWE form a cohesive unit. These findings are broadly consistent with several contemporary neurocognitive models of language processing, such as the proposals of [START_REF] Hagoort | MUC (Memory,Unification, Control): A model on the neurobiology of language beyond single word processing[END_REF], [START_REF] Friederici | The language network[END_REF] and [START_REF] Ullman | The declarative/procedural model: A neurobiological model of language learning, knowledge, and use[END_REF].

Apart from the light that they shed on the neural bases of these two language-related cognitive processes, these results also demonstrate the benefits of using combining computational methods for the automatic annotation of narrative texts. Such methods can be leveraged to tease out distinct sub-processes of complex cognitive processes, like language understanding, using experimental setups with greater ecological validity. As such they pave the way for increasing synergy between computational linguistics and the cognitive neuroscience of language.

Figure 1 .

 1 Figure 1. MWE frequency comparison. The vertical axis shows counts of multiword expression attestations within the natural spoken story stimulus text, The Little Prince (LPP). The horizontal axis is organised by log-frequency in the Corpus of Contemporary English (COCA; Davies, 2008).

Figure 2 .

 2 Figure 2. Panel (A) depicts hierarchical structure for John loves Mary to be recognised via processes of syntactic composition with the word-by-word parser action counts given in orange. Panel (B) shows the sequences of parser actions (i.e. shift and reduce) that would build the colour-coded tree nodes during bottom-up parsing.

Figure 3 .

 3 Figure 3. Deriving an expected blood-oxygen level dependent (BOLD) signal from a naturalistic text. (A) shows a segment of the spoken stimulus, with word boundaries in light blue. (B) highlights multiword expressions (MWEs), which may be retrieved directly from memory during comprehension and (C) depicts the phrase structure of the example sentence, which is composed part-by-part during comprehension. In (D) the value of these word-by-word predictors are shown together. The orange values reflect steps taken between each word by a bottom-up (BU) parser. The purple values reflect PMI scores, which quantify the cohesion of MWEs like has to and look after. Panel (E) shows the expected BOLD response, after these predictors are convolved with a haemodynamic response function (HRF).

Figure 4 .

 4 Figure4. A transition-based approach to finding the multiword expression see to it from example (1-d). Identification involves adding see, to and it to consecutive computational states, creating a binary element and marking this element as a multiword expression. For each state, the classifier must choose the appropriate action among the four available actions. Panel (A) shows the effect of some features on choosing the action Add instead of the action Remove for the word "to". Panel (B) shows the effect of choosing Mark instead of Remove for the composed element "see to it". Panel (C) show the effect of some other features on choosing the action Remove instead of Add for the word "that". These feature templates help the classifier choose the right action. For display purposes we show feature weights as they would be used in a simpler Logistic Classifier, rather than the actual support vector-machine (SVM) that is used in the model. In this simplified setup, decisions are made by passing the summed feature weights -the coloured bars -through a logistic function. The SVM decision rule is more complicated. For details, see Al[START_REF] Saied | The ATILF-LLF system for the PARSEME Shared Task: a Transition-based Verbal Multiword Expression Tagger[END_REF].

Figure 5 .

 5 Figure5. Comparing the word-by-word predictors on a single sentence: (A) depicts the syntactic structure of a single sentence of the naturalistic stimulus text, as recovered by the Stanford parser, (B) gives the bottom-up parser action count with respect to this tree, represented in orange and annotated at the offset of each word in the story, (C) is the categorical multiword expression predictor, represented in blue, where 0 or 1 is tagged at the offset of a word if it is the final word of a MWE, (D) is the gradient multiword expression predictor, represented in dark blue, and similar to the previous predictor the offset of the final word in a MWE is tagged with the corresponding PMI score, quantifying its lexical cohesion.

Figure 6 .

 6 Figure 6. Whole brain contrast images with significant clusters are projected onto a template brain (Holmes et al., 1998): Panel (A) shows us the significant clusters Bottom-up parser action count in orange; Panel (B) shows the significant clusters for Multiword expressions in blue. All images are underwent FWE voxel correction for multiple comparisons with p < 0.05. A cluster-extent threshold k > 50 is applied for display purposes.

Figure 7 .

 7 Figure 7. Significant clusters for the increasing and decreasing cohesion measure of MWEs after FWE voxel correction for multiple comparisons with p < 0.05. Increasing cohesion measures are represented in blue and decreasing cohesion measures are represented in orange.

Table 1 .

 1 A wide variety of linguistic phenomena are plausibly treated as MWEs.

Table 2 .

 2 Graded spectrum of compositionality revealed by pointwise-mutual information.

	PMI multiword expression receiving this score
	26.59 heart skipped a beat
	23.80 have nothing to do with
	21.26 forehead with a handkerchief
	21.18 burst into tear
	20.17 once upon a time
	20.15 boa constrictor
	18.85 peal of laughter
	-2.34 be order
	-2.49 do calculation
	-2.72 be object
	-2.98 be hundred
	-3.15 a well
	-3.50 drink anything
	-3.63 have plan

Table 3 .

 3 Training data for MWE identification systems. Each word of the sentence is on a separate line with its lemma and its POS tag. + and -indicate whether the word is part of a MWE or not.

  Number of reduce actions taken since the last word ( §2.4)

	Categorical MWE predictor	1 at the last word of a MWE, 0 otherwise ( §2.2)
	Word rate	Indicator for spoken word offset
	Word frequency	log-frequency in movie subtitles (Brysbaert & New, 2009)
	f0	fundamental frequency of the narrator's voice, reflects pitch
	RMS amplitude	intensity, an acoustic correlate of volume

Table 4 .

 4 Predictors used in Analysis 1

Table 5 .

 5 Significant clusters of increasing activation for bottom-up parser action count after FWE voxel correction for multiple comparisons with p < 0.05 and cluster-extent threshold (k > 50). Peak activation is given in MNI Coordinates, and brain region labels come from the Harvard-Oxford Cortical Structure Atlas.

	6.

Table 6 .

 6 Significant clusters of increasing activation for multiword expressions after FWE voxel correction for multiple comparisons with p < 0.05. Peak activation is given in MNI Coordinates.

Table 7 .

 7 Significant clusters for the increasing and decreasing cohesion measure of MWEs after FWE voxel correction for multiple comparisons with p < 0.05. Peak activation is given in MNI Coordinates.

	Regions for PMI	Cluster size MNI Coordinates	p-value	T-score
		(in voxels)	x	y	z	(corrected) (peak level)
	Correlated with increasing MWE cohesion
	R Precuneus	244	6 -68	56	0.000	7.33
	Correlated with decreasing MWE cohesion
	L Superior Frontal Gyrus	2039 -18 32	52	0.000	8.39
	L Precentral Gyrus (BA9)		-44	8	40	0.000	7.26
	L Middle Frontal Gyrus		-38 22	46	0.000	6.89
	L Angular Gyrus	688 -42 -58	34	0.000	7.27
	L Inferior Parietal Lobule		-48 -46	50	0.000	5.76
	L Inferior Temporal Gyrus	320 -60 -44	-4	0.000	7.25
	L Inferior Frontal Gyrus Pars Triangularis	211 -46 30	18	0.000	6.41
	L Middle Temporal Gyrus	152 -56	0	-32	0.000	6.49
	L Frontal Pole (BA10)	50	-6 64	20	0.001	5.88
	R Superior Frontal Gyrus	35 14 52	28	0.001	6.15
	L Inferior Frontal Gyrus orbitalis	33 -38 48	18	0.001	5.55
	R Inferior Temporal Gyrus	21 58 -10	-32	0.001	5.78
	R Superior Frontal Gyrus/ SMA (BA6)	15 12 24	58	0.001	5.94

While the experience of listening to an audiobook is quite a natural one e.g. for daily commuters, it is not quite conversational. We put aside the question of which form of language is most basic, contenting ourselves with a comparative sense of the term "naturalistic." The implicit comparison is between story-listening and lists of unrelated sentence stimuli, devoid of literary content.

PMI is one of several cohesion metrics that have been advanced in both computational linguistics and corpus linguistics;[START_REF] Evert | Corpora and collocations[END_REF] offers a tutorial introduction.

Acknowledgements

The authors would like to thank Adam Mahar, Skyler Yeatman, Jaelyn Moore, Jacob Collard, Katey Huddleston, and Marissa Holl at Cornell University for assisting us with data collection.

Funding

This material is based upon work supported by the National Science Foundation under Grant Number 1607441 (USA). We also gratefully acknowledge support from the French National Research Agency (ANR) under grant ANR-14-CERA-0001.

Disclosure statement

We have reported all measures, conditions and data exclusions. The sample size was enlarged by a factor of 2 compared to previous work with similar regressors. Appendix C. Full cluster list for bottom-up parsing regressor

Significant clusters for Bottom-up parser action count (abbreviated table provided in §5.1):


Ahmed Yousfi
email: a.yousfi@mail.lagh-univ.dz

Sylvain Fréour

Mohamed Rahmani

Hocine Osmani

Frédéric Jacquemin

Djelloul Gueribiz

Yousfi Ahmed

Fréour Sylvain

Rahmani Mohamed

Osmani Hocine

Jacquemin Frédéric

Gueribiz Djelloul

Mechanical effects of oxidation induced shrinkage on organic matrix composites

Keywords: thermo-oxidation, shrinkage, polymer matrix composites, scale transition models, local stress

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

Introduction:

Composites materials based on high glass transition temperature matrices are likely to be used as structural parts for next generation aircraft and, thus, will have to face oxidative atmosphere at moderate or high temperatures (typically 100-300°C) during several thousands of hours [1]. The lifetime scheduled for the structural elements of supersonic aircraft approximately corresponds to 20,000 cycles in a range of temperatures varying between -55°C and 120°C, with a total length of the plateau at 120°C of 80,000 h (10 years) [START_REF] Bellenger | Ageing of a carbon epoxy composite for aeronautic applications[END_REF]. Such high temperatures over long periods cause a thermal ageing of the organic matrix constituting the composite parts leading to thermal oxidation of the material external plies. The problem occured when such organic matrix composites have been considered for applications under severe environmental conditions (temperatures of about 250°C, air pressure higher than 1 atmosphere) [START_REF] Hergenrother | Development of composites, adhesives and sealants for highspeed commercial airplanes[END_REF][START_REF] Brunner | Development of the California State University, Long Beach composite durability laboratory[END_REF].

The laminated composites exposed to high temperatures are subject to degradation caused mainly by thermal oxidation of their organic matrix. It induces the growth of oxidized layer of reduced thickness in the subsurface of the composite parts. This superficial phenomenon leads to weight loss and alters the mechanical behavior of the material [START_REF] Bowles | Specimen Geometry Effects on Graphite/PMR-15 Composites During Thermo-oxidative Aging[END_REF][START_REF] Madhukar | Thermo-oxidative stability and fiber surface modification effects on the Inplane shear properties of graphite/PMR-15 Composites[END_REF][START_REF] Olivier | developement of experimental, theorical and numerical tools for studying thermo-oxydation of CFRP composites[END_REF], inducing a fragility of its subsurface, the properties of which are different to those of the bulk, generally remaining unaffected by oxidation. The purpose of this work is the multiscale analysis of the consequences of the thermal-oxidation of epoxy matrices on the intrinsic mechanical properties of the external composite ply, on the one hand, and on the internal mechanical states of composite structure according to the considered mechanical loading, on the other hand. This work includes five sections following this introduction.

The second section is dedicated to the influence of thermal oxidation on the mechanical properties of composite ply. The third section presents Eshelby-Kröner self-consistent model: its constitutive laws are given for the estimation of effective thermal elastic properties of composite ply (stiffness tensor and coefficients of thermal expansion). The fourth section is devoted to the coupling of the classical lamination theory and analytical scale-transition relations for local stress determination in both constituents of composite plies subjected to a thermo-elastic loading. The fifth section describes the conclusions of the present study.

Effective properties of a thermo-oxidized composite ply

Oxidation model and results

Various models enabling to achieve the numerical computation of oxidation process in pure epoxies or epoxy matrix composites have been proposed in the literature [START_REF] Colin | Kinetic modelling and simulation of gravimetric curves: application to the oxidation of bismaleimide and epoxy resins[END_REF][START_REF] Tandon | Modeling of oxidative development in PMR-15 resin[END_REF]. Simulations of isothermal aging at 150°C of an epoxy resin demonstrated that the oxidized layer thickness was not growing beyond 200 m whereas the oxidation products concentration were continuously increasing in the affected layer [START_REF] Olivier | developement of experimental, theorical and numerical tools for studying thermo-oxydation of CFRP composites[END_REF]. Ultra-micro indentation tests on epoxy resin, after 100h, 600h and 1000h of isothermal aging at 150°C

show that a relation between elastic indentation modulus and oxidation products concentration occurs (Eq. 1) [START_REF] Olivier | developement of experimental, theorical and numerical tools for studying thermo-oxydation of CFRP composites[END_REF]:

) Q . ( Exp Y m 79 0 1460 5479 - - = (1)
where Y m is the elastic modulus of the epoxy matrix, expressed in GPa, while Q is the oxidation products concentration (mol/L). The corresponding profile for Y m is displayed on figure 1. The matrix is considered as an isotropic material, which Poisson's ratio 0.35 ν m = is assumed, due to the lack of available information on its evolution, as independent of the state of thermal oxidation process.

The oxidized matrix is stiffer than the unaffected matrix in a ratio varying between 1 and 1.38. The matrix properties of a thermally oxidized ply can be very different from those of the corresponding unaffected ply, so that three main consequences due to the thermal oxidation of composite structure can be expected: a variation of i-the effective macroscopic properties of composite plies, ii-the profiles of macroscopic stress in the structure and iii-the localization of the macroscopic mechanical states (stress and strain) in both constituents of each ply (fiber and matrix). The three above listed expected consequences of thermo-oxidation are multi-scale phenomena, either explicitly, in the case of items i-and iii-, (i-being an homogenization problem, whereas iii-is related to localization problem), or implicitly, in the case of item ii-, which implies the knowledge of the macroscopic effective properties of the composite ply (resulting from the estimates achieved in i-). In order to be able to treat each of these aspects, a scale-transition model is required. This model is described below in Section 3.

Multi-scale Analysis

Introduction

Mean-field scale transition models involve a multi-scale representation of the studied materials. In the case of composite materials, the following two-scale model is often sufficient:

-The macroscopic scale corresponds to that of the composite ply. It is denoted by the superscript I , whereas, 5/23 -the so-called "pseudo-macroscopic" [START_REF] Sprauel | First European Powder Diffraction International Conference on X-Ray stress analysis[END_REF] properties and mechanical states of the constituents (either the resin or its reinforcements,) are respectively indicated by the superscripts m and f . The composite structure is actually constituted by an assembly of the above described composite plies, each of them possibly having the principal axis of their reinforcements differently oriented from one to another. This approach enables to treat the case of multidirectional laminates, as shown, for example, in [START_REF] Fréour | On an analytical self-consistent model for internal stress prediction in fiber-reinforced composites submitted to hygroelastic load[END_REF]. In the present work, the local properties of the i-superscripted constituents are considered to be known (i.e. the pseudomacroscopic stiffnesses, L i and coefficients of thermal expansion M i ), whereas the corresponding effective macroscopic properties of the composite structure (respectively, L I and M I ) are a priori unknown and results from computations.

Among the numerous, available in the literature scale transition models, able to handle such a problem, most involve rough-and-ready theoretical frameworks, such as: Voigt [START_REF] Voigt | Lehrbuch der Kristallphysik[END_REF], Reuss [START_REF] Reuss | Berechnung der Fliessgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle[END_REF], Neerfeld-Hill [14-15] Tsai-Hahn [START_REF] Tsai | Introduction to Composite Materials[END_REF] and Mori-Tanaka [START_REF] Tanaka | The Hardening of Crystals by Non-deforming Particules and Fibers[END_REF][START_REF] Mori | Average Stress in Matrix and Average Elastic Energy of Materials with Misfitting Inclusions[END_REF] approximates In spite of the lack of the physical realism of the mathematical framework involved in the aforementioned models, some of them do nevertheless provide a numerically satisfying estimation of the effective properties of a composite ply, by comparison with the experimental values or others, more rigorous models. Both Tsai-Hahn and Mori-Tanaka models fulfill this interesting condition [START_REF] Jacquemin | A hygro-elastic self-consistent model for fiberreinforced composites[END_REF][START_REF] Fréour | Extension of Mori-Tanaka approach to hygroelastic loading of fiber-reinforced composites -Comparison with Eshelby-Kröner self-consistent model[END_REF]. Nevertheless, in the field of scale transition modelling, the best candidate remains Kröner-Eshelby selfconsistent model [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF][START_REF] Kröner | Berechnung der elastischen Konstanten des Vielkristalls aus des Konstanten des Einkristalls[END_REF], because this model takes into account a rigorous treatment of the thermo-hygro-elastic interactions between the homogeneous macroscopic medium and its heterogeneous constituents, as well as this model enables handling the microstructure (i.e. the particular morphology of the constituents, especially that of the reinforcements).

6/23

Determination of the effective macroscopic thermo-elastic properties of composite plies

Within Eshelby-Kröner self-consistent model, the homogenization law relating the macroscopic elastic stiffness of a composite ply to the properties of the constituents is:

( ) [ ] m f, i 1 : : = - + - = I L L E L L I i I i I (2)
In equation ( 2), I stands for the fourth-order identity tensor, whereas Hill tensor, or

Morris tensor E I (depending on which work from [START_REF] Hill | Continuum micro-mechanics of elastoplastic polycrystals[END_REF][START_REF] Morris | Elastic constants of polycrystals[END_REF] is considered) expresses the dependence of the localization tensor on the morphology assumed for the matrix and its reinforcements [START_REF] Hill | Continuum micro-mechanics of elastoplastic polycrystals[END_REF]. Its components are functions of the macroscopic stiffness L I (some examples are given in [START_REF] Kocks | Texture and Anisotropy[END_REF][START_REF] Mura | Micromechanics of Defects in Solids[END_REF]).

The homogenization relation satisfied by the macroscopic coefficients of thermal expansion of a composite ply writes as follows:

( ) [ ] m f, i 1 1 : : : : = - - + - = i i I i I I I M L I L L E L M (3)
Only a few study studies were dedicated to the oxidation effects on the mechanical behavior of organic matrix composites. It was nevertheless shown in [START_REF] Bowles | The effect of fiber surface modification and thermal aging on composite toughness and its measurement[END_REF] that the properties of carbon fibers are stable below 300°C. As a consequence, the variation of mechanical properties of thermo-oxidized laminates only depends on the evolution of the matrix properties. Table 1 displays the local properties of carbon fiber considered in order to achieve the homogenization calculations.

Evolutions of the effective longitudinal and transverse macroscopic elastic moduli, as a function of the oxidation products concentration are shown on Figures 2 and3, for a composite containing a carbon fiber volume fraction equal to 60 %. The effective mechanical properties of the oxidized composite ply were determined, during the oxidation process, through Eshelby-Kröner self-consistent homogenization procedure (Eq. 2) (curve "KESC"). The results were compared to the corresponding properties estimated according to the finite element method (FEM "curve").

Figure 4 shows the evolutions obtained for the effective macroscopic coefficients of thermal expansion of a composite ply, according to equation (3), during the thermal oxidation process experienced by the epoxy. According to figures 2 and 4, the longitudinal effective properties (i.e. in the direction parallel to axis of the reinforcing fibers), are, in first approximation, independent from the state of thermal oxidation process of the polymer matrix constituting the considered composite ply. Actually, a relative deviation weaker than 1 % is observed at the end of the oxidation process for the Young's modulus, while the longitudinal macroscopic coefficient of thermal expansion remains almost constant (and tends towards zero). This result is explained by the fact that in the fiber direction, a unidirectional ply presents thermo-mechanical properties which are controlled by the properties of a single constituent: the fiber. Experimental results on composite plies in similar circumstances, i.e. in cases when the carbon fibers exhibit constant properties, while those of the matrix significantly vary, corroborates the results of our simulations: longitudinal effective macroscopic properties of composite plies do not vary as a function of the properties polymer matrix (at least in the case that the composite ply contains a high volume fraction of fibers) [START_REF] Patel | Hygrothermal effects on the structural behaviour of thick composite laminates using higher-order theory[END_REF]. On the contrary, the effective macroscopic properties of a composite ply, in the direction transverse to the reinforcements axis, do significantly depend on the state of thermal oxidation process experienced by the polymer matrix constituting the ply. In fact, the relative variation of transverse Young modulus of oxidized ply compared to the reference value (unaffected ply) can reach 12 % at the term of oxidation process (and spreads so that the variation for transversal coefficient of thermal expansion is 3.5 %). This effect strongly depends on the volume fraction of reinforcements in the composite ply and may reach 38% in the pure matrix at the end of the oxidation process. The matrix was modeled as undergoing two time-dependent processes, and the fibers were assumed to be stable (i.e. impervious to any oxidation effects on their properties). The first process is a matrix mechanism that is partly an extension of the cure process, in which the matrix changes by chemical and physical aging.

This mechanism, which depends on temperature and time, is associated with small amounts of matrix shrinkage and mass loss (less than 1 percent) and with density and material property changes. In thermoset materials, this mechanism is dominated by changes in chemical cross-linking, so the mechanism will be referred to here as the cross-linking mechanism. The second process is a degradation mechanism due to oxidation and/or another mechanism that breaks down the polymer structure of the matrix. Temperature and oxygen availability govern this mechanism. In thick parts of the matrix, this mechanism appears to be diffusion limited and, thus, occurs mostly on the surface, causing material property changes (mostly degradations) and mass loss. Mass can be lost by erosion of the surface (making the part smaller), formation of voids, reduction in density of the matrix, or shrinkage of the matrix. In practice, density changes are 9/23 not observed, whereas some void formation and major shrinkage are observed [START_REF] Gates | Acceleration of Aging in Graphite/Bismaleimide and Graphite/Thermoplastic Composites[END_REF]. This second mechanism will be referred to as the mass-loss mechanism. When matrix shrinkage occurs (i.e. in oxidative environment), the high strain gradients present in matrix areas, close to fibres which have a very high stiffness, lead to high local stresses that can be favorable to fibre/matrix debonding or to crack initiation [START_REF] Lafarie-Frenot | Influence of oxidative environments on damage in c/epoxy laminates subjected to thermal cycling[END_REF]. The total strain in matrix m T ε ε ε ε is:

m S m E m T ε ε ε ε ε ε ε ε ε ε ε ε + = (4)
Where m E ε is the elastic strain of matrix, whereas m S ε which denotes the strain due to the shrinkage of matrix, depends on its oxidation state.

Q δ 0 0 0 δ 0 0 0 δ m 11 m 11 m 11             = m S ε ε ε ε (5)
where

max max S m 11 Q ) Q ( ε = δ , and % 5 , 2 ) Q ( max S = ε
is the maximal strain due to the matrix shrinkage [START_REF] Decelle | Oxidation induced shrinkage for thermally aged epoxy networks[END_REF].

The matrix shrinkage associated to degradation causes deformations at the ply level. The fibers are assumed to be dimensionally stable, and hence restrain the shrinking of the matrix. This behavior is analogous to the thermal expansion of a ply consisting of a fiber and a matrix with different thermal expansion coefficients. It results in a macroscopic strain that is intermediate between the strain that would be experienced by the unrestrained matrix material, from the one hand and the bare fiber, one the other hand.

Determination of mechanical states within the constituents of a composite ply subjected to thermo-elastic loading:

Analytical relations satisfying the fundamental assumptions of Eshelby-Kröner selfconsistent model, enabling to proceed to the localization of the macroscopic strains within the matrix were established in [START_REF] Fréour | On an analytical self-consistent model for internal stress prediction in fiber-reinforced composites submitted to hygroelastic load[END_REF]. Due to the length of these equations, they will not be provided in the present work, so that the interested reader should refer to equations [START_REF] Mori | Average Stress in Matrix and Average Elastic Energy of Materials with Misfitting Inclusions[END_REF][START_REF] Jacquemin | A hygro-elastic self-consistent model for fiberreinforced composites[END_REF] of [START_REF] Fréour | On an analytical self-consistent model for internal stress prediction in fiber-reinforced composites submitted to hygroelastic load[END_REF]. The variation of the internal macroscopic stresses ii) the epoxy matrix experiences an increasing stress state during the thermal oxidation process, which can be attributed to the increased stiffness of this constituent induced by the thermal-oxidation. Actually, composites are generally designed so that the reinforcements concentrate the stresses. Moreover, in practice, damage and failure initially often takes place in the subsurface of a mechanical part. Thus, an increased stress experienced by the resin of the external ply is, a priori not favorable to the durability of the composite structure. These considerations can help to explain the worst durability of thermo-oxidized composite structures, observed in practice, by comparison with the reference behaviour of the unaffected by oxidation structure.

Conclusions

In this work, the consequences due to the shrinkage induced by a thermal oxidation process of the epoxy resin constituting a composite structure were investigated, for the first time, through a scale transition approach. The obtained results were compared to finite elements calculations, with a rather good agreement. The scale transition approach demonstrates that the effective properties and mechanical states of the external ply of a composite structure having experienced a thermal oxidation process do not strongly differ from the corresponding reference values of the ply unaffected by the thermal oxidation. On 12/23 the opposite, the local properties and mechanical states of the matrix constituting the ply affected by thermal oxidation and the matrix shrinkage do strongly vary as a function of the thermal oxidation process state. It was demonstrated that both the evolutions of the mechanical properties and mechanical states in the epoxy after the thermal oxidation were not compatible with an enhanced durability of the structure, especially in the case that a transverse external load had to be undergone. Nevertheless, the present approach does not underline that the service life of a composite plate submitted to longitudinal or thermal load would be affected by the thermal oxidation of the epoxy. In further works, others features related to the thermal ageing of organic matrix composites structures, such as high temperature creep, will be investigated. Tables Table 1. Properties of T300 carbon fiber [START_REF] Soden | Lamina Properties Lay-up Configurations and Loading Conditions for a Range of Fiber-reinforced Composite Laminates[END_REF].

Y 1 (GPa) Y 2 , Y 3 (GPa)

4 4 . 1

 41 Study of the multi-scale mechanical states experienced by the composite structure Matrix shrinkage effect on the mechanical states of the thermo-oxidized composite

  figure 5-6 respectively. On the same figures, the stress states predicted for the thermo-

Figure 1 :

 1 Figure 1: Evolution of the elastic modulus as a function of the oxidation products concentration. Comparison with the behavior of unaffected matrix.

Figure 2 .

 2 Figure 2. Longitudinal macroscopic effective elastic modulus of a composite ply, as a function of the oxidation products concentration.

Figure 3 .

 3 Figure 3. Transverse macroscopic effective elastic modulus of a composite ply, as a function of the oxidation products concentration.

Figure 4 .

 4 Figure 4. Evolution of the longitudinal (M 11 ) and transverse (M 22 ) macroscopic coefficients of thermal expansion, as a function of the oxidation products concentration.

Figure 5 :

 5 Figure 5: Longitudinal (σ 11 ) and transverse (σ 22 ) stresses concentrations in the constituents of the oxidized external ply, the composite structure being subjected to a transverse stress σ 22 =50MPa. ST: scale-transition, comp: composite, mat: matrix, FEM: finite element method. Q stands for the oxidation products concentration.

Figure 6 .

 6 Figure 6. Effects of shrinkage on the longitudinal (σ 11 ) and transverse (σ 22 ) stresses concentrations in the constituents of the oxidized external ply, the composite structure being subjected to a transverse stress σ 22 =50 MPa. ST: scale-transition, comp: composite, mat: matrix, FEM: finite element method. Q stands for the oxidation products concentration.

Figure 1 :Figure 2 .Figure 3 .Figure 4 .

 1234 Figure 1: Evolution of the elastic modulus as a function of the oxidation products concentration. Comparison with the behavior of unaffected matrix.

Figure 5 :Figure 6 .

 56 Figure 5: Longitudinal (σ 11 ) and transverse (σ 22 ) stresses concentrations in the constituents of the oxidized external ply, the composite structure being subjected to a transverse stress σ 22 =50MPa. ST: scale-transition, comp: composite, mat: matrix, FEM: finite element method. Q stands for the oxidation products concentration.