Gaël Varoquaux

Bertrand Thirion

Gael Varoquaux
email: gael.varoquaux@inria.fr

How machine learning is shaping cognitive neuroimaging

Keywords: machine learning, neuroimaging, cognition, fMRI, decoding, encoding

Functional brain images are rich and noisy data that can capture indirect signatures of neural activity underlying cognition in a given experimental setting. Can data mining leverage them to build models of cognition? Only if it is applied to well-posed questions, crafted to reveal cognitive mechanisms. Here we review how predictive models have been used on neuroimaging data to ask new questions, i.e., to uncover new aspects of cognitive organization. We also give a statistical learning perspective on these progresses and on the remaining gaping holes.

Introduction

Functional neuroimaging has opened the door to quantitative yet non invasive experiments on brain function. These experiments contribute to bridging the gap between cognitive sciences and neuroscience: the former analyse thought and mind while the latter probes the nervous system at various spatial and temporal scales. To study high-level aspects of human cognition, the two modalities of choice are functional Magnetic Resonance Imaging (fMRI) and electro-and magneto-encephalography (EEG/MEG), both can be used to observe brain activity with good spatial resolution for fMRI and temporal temporal resolution for EEG/MEG. The concurrent progress of scanners and experimental paradigms has made it possible to accumulate very rich imaging data that quantify specific correlates of brain function in an uncountable variety of cognitive tasks and processes. In parallel, the advent of machine learning has brought huge progress to data processing of large datasets. But these techniques are geared towards well-posed predictive tasks. The key question is then; how can they be leveraged to push forward understanding of the brain, beyond merely predicting a numerical signal? This paper presents a subjective view on the work that has been done combining machine learning with functional neuroimaging to advance the understanding of brain function. It dwells mostly on modeling considerations: how and what do the predictive models teach us about the brain? But it also touches upon machine learning and statistical issues. This review focuses on fMRI in humans, that represents most of the accumulated functional neuroimaging data; however, most of the concepts carry to other imaging modalities. FMRI provides images of the brain at the mm scale, however it is only sensitive to the metabolic counterpart of neural activity and suffers from a poor temporal resolution. The first two sections of this paper discuss supervised learning, used first to model brain activity from the stimuli, then to predict the task performed from the evoked activity. The last section reviews the use of unsupervised learning to extract relevant structures in functional images: the interaction structure that underlies brain function, or their natural spatial organization.

Encoding: richer models of evoked activity

The keystone to the use of fMRI in cognitive neuroscience is the standard massunivariate analysis framework. It consists of modeling the brain response evoked via an experimental paradigm as the linear combination of different experimental conditions [START_REF] Penny | Statistical Parametric Mapping: The Analysis of Functional Brain Images[END_REF][START_REF] Poldrack | Handbook of Functional MRI Data Analysis[END_REF]. A statistical test is performed at each voxel to delineate regions recruited differently by the various conditions. The art of fMRI experiment design and analysis then consists in crafting the succession of conditions so that, when properly contrasted, they reveal the neural support corresponding to the cognitive function of interest. With regards to brain function, this statistical analysis answers naturally a "where" question, but to a lesser extent a "how" question. Indeed the tests for differences between experimental conditions are statistically well-posed, but not very expressive to refine cognitive models.

In contrast, the study of neural coding, lead historically via intra-cellular recordings of neural activity, has opened the door to breaking down many cognitive functions into atomic steps implemented by ensembles of neurons. The seminal work of Hubel and Wiesel [START_REF] Hubel | Receptive fields, binocular interaction and functional architecture in the cat's visual cortex[END_REF] showed that neurons in the primary visual cortex have receptive fields tuned to a variety of image features, from simple cells sensitive to local orientation in an image, to more complex cells capturing in addition, motion and length of local image features. Progress on uncovering the link between stimuli and neural response revealed neurons tuned to richer and higher-level descriptions of the stimulus, such as receptive fields specific to complex shapes [START_REF] Logothetis | Shape representation in the inferior temporal cortex of monkeys[END_REF], but also a richer description of neural responses, in particular coding distributed across a population of neurons [START_REF] Georgopoulos | Neuronal population coding of movement direction[END_REF]. Beyond individual neurons, at the spatial scales probed in fMRI [1] , and high-level cognition arises from functional integration of multiple specialized brain regions [START_REF] Tononi | Complexity and coherency: integrating information in the brain[END_REF].

The stepping stones of this line of work are to find the right features of the stimuli and neuronal population that can be matched closely. How well the former explains the latter gives a natural figure of merit of these models, in a setting known as encoding [START_REF] Naselaris | Encoding and decoding in fMRI[END_REF]. Given models that explain neural responses at the spatial scales captured by fMRI [START_REF] Edelman | Toward direct visualization of the internal shape representation space by fMRI[END_REF][START_REF] O'doherty | Model-based fMRI and its application to reward learning and decision making[END_REF] rather than at the neural level, encoding research can be lead with fMRI data, which benefits from full-brain coverage. Technically, designing an encoding model is not different from specifying the design matrix in a standard fMRI analysis and can be seen as model-based fMRI [START_REF] O'doherty | Model-based fMRI and its application to reward learning and decision making[END_REF]. However relinquishing the methodology of contrasts for more diverse, albeit indirect, statistical tests opens the door to richer modeling. In particular, it is possible to tackle more complex stimuli, such as natural stimuli [START_REF] Kay | Identifying natural images from human brain activity[END_REF], very high-level and diverse descriptions of the stimuli [START_REF] Mitchell | Predicting human brain activity associated with the meanings of nouns[END_REF], or a cognitive model of the observed behavior [START_REF] O'doherty | Model-based fMRI and its application to reward learning and decision making[END_REF]. This increase in model complexity is the driving force behind the use of machine learning in encoding. First it entails fitting many parameters on limited data, and thus conventional in-sample statistical testing is thorny. For this reason, goodness of fit of the encoding model is best assessed via its cross-validated ability to predict brain signals [START_REF] Nishimoto | Reconstructing visual experiences from brain activity evoked by natural movies[END_REF]. Similarly, the predictive engine that links stimuli features to brain signal is best chosen amongst machine learning tools, that balance modeling flexibility and regularization, such as a naive Bayes predictor [START_REF] Mitchell | Predicting human brain activity associated with the meanings of nouns[END_REF], sparse [START_REF] Nishimoto | Reconstructing visual experiences from brain activity evoked by natural movies[END_REF] or ridge [START_REF] Huth | A continuous semantic space describes the representation of thousands of object and action categories across the human brain[END_REF] regression. Finally, the computational models that derive encoding features from the stimuli often draw from the domain-specific feature extraction techniques developed in applied machine learning research. These provide simple quantitative proxies for the cognitive features of interest. For instance, to map semantic concepts [START_REF] Mitchell | Predicting human brain activity associated with the meanings of nouns[END_REF] and [START_REF] Huth | A continuous semantic space describes the representation of thousands of object and action categories across the human brain[END_REF] used natural language processing techniques: word co-occurrence or an ontology on words. The ties between brain science and machine learning are strikingly close in the study of vision: computer vision, i.e., the use of computers and machine learning to analyze and interpret images, has built upon, but also fostered our understanding of the brain visual system. David Marr's seminal work [START_REF] Marr | Vision: A Computational Investigation Into the Human Representation and Processing of Visual Information[END_REF] formalized the idea of hierarchical levels of representation that tie together the receptive fields observed in visual cortex, but is also reflected in modern stateof-the-art computer vision architecture based on convolutional networks [START_REF] Lecun | Convolutional networks and applications in vision[END_REF]. Very recently, Yamins et al. [START_REF] Yamins | Performance-optimized hierarchical models predict neural responses in higher visual cortex[END_REF] have shown a striking correspondence between 296 neural recordings in the infero-temporal cortex of the monkey and intermediate layers of computer-vision convolutional networks. This work is a quintessential example of machine learning in encoding models: a predictive engine performs the same task as the brain system under study; machine learning is used to fit its parameters on a set of stimuli and the final architecture matches neural data.

Transferring such results to fMRI would open doors to studying the full complete brain of healthy human subjects rather than 296 neurons in implanted monkeys. However, it poses significant challenges. Indeed, fMRI is an indirect and noisy measurement of brain activity, that captures the average effect of many spikes and does not resolve cortical columns, let alone individual neurons. The concept of population receptive field [START_REF] Dumoulin | Population receptive field estimates in human visual cortex[END_REF] is sometimes used to refer to the aggregate properties of neurons in one voxel. Thus, encoding models need to be adapted to the resulting structured noise and signal convolutions. Model evaluation and selection is in itself often a major roadblock.

Decoding: towards principled reverse inference

In the study of neural recordings, decoding models reconstruct stimuli or behavior from the neural code [START_REF] Abbott | Decoding neuronal firing and modelling neural networks[END_REF][START_REF] Georgopoulos | Neuronal population coding of movement direction[END_REF]. More generally, the decoding approach can be seen as solving the inverse problem to the encoding model, even when applied on fMRI signals that do not capture individual neural firing [START_REF] Dehaene | Inferring behavior from functional brain images[END_REF][START_REF] Thirion | Dealing with the shortcomings of spatial normalization: Multi-subject parcellation of fMRI datasets[END_REF].

Since a decoding model often predicts quantities that are directly observable, it can provide a very useful tool to validate an encoding model. Indeed, decoding performance is an omnibus test of goodness of fit: it tests the overall significance of the model, but does not test which variables have a significant contribution to the model. As an omnibus test, decoding is used with explicit sophisticated encodings [START_REF] Kay | Identifying natural images from human brain activity[END_REF][START_REF] Nishimoto | Reconstructing visual experiences from brain activity evoked by natural movies[END_REF][START_REF] Huth | A continuous semantic space describes the representation of thousands of object and action categories across the human brain[END_REF][START_REF] Mitchell | Predicting human brain activity associated with the meanings of nouns[END_REF][START_REF] Naselaris | Encoding and decoding in fMRI[END_REF], but also with simple fMRI analysis to perform an omnibus test at the region level [START_REF] Haxby | Distributed and overlapping representations of faces and objects in ventral temporal cortex[END_REF] or on a wide family of regions as in searchlight analysis [START_REF] Kriegeskorte | Information-based functional brain mapping[END_REF]. Interestingly, an early fMRI study [START_REF] Edelman | Toward direct visualization of the internal shape representation space by fMRI[END_REF] on neural representation hypothesized that "objects are represented by a relatively widely distributed activity of functional modules", but considered this statement to be insufficiently quantitative to allow a statistical test. Nowadays this study would probably be formulated in an encoding/decoding framework [START_REF] Naselaris | Encoding and decoding in fMRI[END_REF], using a multivariate predictor to provide evidence for the author's hypothesis, as in [START_REF] Haxby | Distributed and overlapping representations of faces and objects in ventral temporal cortex[END_REF]. It is often considered that multi-voxel analysis, as used in decoding, provides an increase in sensitivity compared to standard mass-univariate analysis [START_REF] Norman | Beyond mind-reading: multi-voxel pattern analysis of fmri data[END_REF]; however, we stress that it does not correspond to an increase in statistical power, but rather to a different test performed: decoding performs a global (omnibus) test of the model, while voxel-level tests are useful to delineate regions, but are subject to corrections for multiple comparisons.

As noted in [START_REF] Peelen | Using multi-voxel pattern analysis of fMRI data to interpret overlapping functional activations[END_REF], decoding analysis provides a good framework to interpret overlapping activation patterns. Brain maps in encoding and decoding settings carry actually a different meaning. An inference with an encoding model, or in the fMRI standard analysis framework, is a statement on whether or not the signal in a brain region is well explained by the model that we have of the task: we can conclude that th task implies this brain activation, and we say that the region is recruited by the task. A decoding analysis tells us that if we observe a certain brain activity, we can deduce properties of the task or the stimulus. Such a conclusion is the converse implication of the encoding settings, sometimes dubbed reverse inference [START_REF] Poldrack | Can cognitive processes be inferred from neuroimaging data[END_REF]. Reverse inference, i.e., drawing conclusions on behavior and mental processes from the brain activations, answers natural questions in cognitive neuroimaging, e.g.,: what is the function of neural sub-system? But reverse inferences drawn from maps, estimated using encoding models, are a logical fallacy [START_REF] Poldrack | Can cognitive processes be inferred from neuroimaging data[END_REF]. On the other hand, decoding models provide a path to principled reverse inferences [START_REF] Poldrack | Inferring mental states from neuroimaging data: from reverse inference to large-scale decoding[END_REF]. However, it is important to keep in mind that, in general, a decoding experiment does not tell us anything about tasks and cognitive processes that it did not probe. For example, an experiment studying brain regions discriminating images of faces from images of houses [START_REF] Haxby | Distributed and overlapping representations of faces and objects in ventral temporal cortex[END_REF] does not inform us on how these regions are related to recognizing letters. The appealing idea of inferring brain processes from brain activation only carries meaning if the decoding model has captured a large variety of brain processes. Beyond interpretation of brain images, the basic neuroscience questions at stakes here are that of functional specificity. For instance, while many brain regions are more activated under physical pain, a decoding analysis including many different aspects of pain showed that a network comprising parts of the thalamus, the insulae, and the somatosensory cortex was specific of physical pain [START_REF] Wager | An fMRI-based neurologic signature of physical pain[END_REF]. At the spatial scale probed by fMRI, the multiplicity of regions needed to come to precise conclusions on the cognitive function recruited is consistent with the modern view that high-level cognitive processes arise from distributed networks. This calls for multivariate decoding engines.

Going beyond a specific cognitive domain, such a vision or pain, and studying functional specialization in a broad sense require probing more functions than can be addressed in one experimental paradigm. For this reason, investigators have turned to accumulating data across experiments. Using 8 studies, covering 22 different cognitive concepts, Poldrack et al. [START_REF] Poldrack | Decoding the large-scale structure of brain function by classifying mental states across individuals[END_REF] were able to predict the concepts involved from activation images in unseen subjects. The use of a variety of studies, with different experimental protocols, can overcome the idiosyncrasies of each protocol that are not relevant to cognitive concepts of interest; for instance, to study high-level decision mechanisms independently of the modality used to present stimuli -visual or auditory. However, in [START_REF] Poldrack | Decoding the large-scale structure of brain function by classifying mental states across individuals[END_REF], the train set contained images from the same protocols as the test set; thus, the hypothesis that the decoder was actually detecting protocols rather than cognitive concepts cannot be ruled out. To generalize to unseen protocols, the challenge is to describe them in terms that are common enough to be shared across many protocols, but also sufficiently rich to capture their cognitive content. Schwartz et al. [START_REF] Schwartz | Mapping cognitive ontologies to and from the brain[END_REF] used an ontology of experimental paradigms and multi-label classification: labeling 83 different conditions, from 19 studies, with a set of different terms from the ontology. The resulting predicting engine can not only describe the content of an unseen experiment from the corresponding brain activation, but also give brain maps associated with each term in a reverse inference. Covering more cognitive concepts requires accumulating many brain images. Sharing data across institutions is a practical means to this end, for instance relying on the OpenfMRI project [START_REF] Poldrack | Toward open sharing of task-based fMRI data: the OpenfMRI project[END_REF] that hosts to this day 24 different fMRI studies. Another interesting alley is to collect from the literature the coordinates, in standard brain space, of observed activation foci, as in the Neurosynth project [START_REF] Yarkoni | Large-scale automated synthesis of human functional neuroimaging data[END_REF].

Although decoding gives a principled methodological framework for reverse inference, there are some tough statistical challenges. Indeed, the discriminant brain maps extracted may be the most relevant information captured by the model from a neuroscience perspective. However, decoders solve a high-dimensional multivariate statistical estimation problem that is very ill-posed [START_REF] Haufe | On the interpretation of weight vectors of linear models in multivariate neuroimaging[END_REF] given the typical small sample size. Many different brain maps will give rise to similar predictive performance. Worst yet, minimizing a prediction risk does not lead to any control on the brain maps. For instance, if two neighboring voxels carry the same information but one is less noisy than the other, a decoder might favor selecting only that one. For related reasons, sparse models can only capture a subset of relevant voxels [START_REF] Varoquaux | Small-sample brain mapping: sparse recovery on spatially correlated designs with randomization and clustering[END_REF]. Injecting priors -or regularization-in the estimation makes it well-posed and shapes the brain maps extracted. Capturing large-scale brain systems calls for spatial regularization such as sparsity and spatial smoothness [START_REF] Grosenick | Interpretable whole-brain prediction analysis with graphnet[END_REF] or total-variation (TV) for piecewise smooth maps [START_REF] Michel | Total variation regularization for fMRI-based prediction of behavior[END_REF]. In particular TV-ℓ 1 regularization, combining sparsity and total-variation, selects well the predictive regions [START_REF] Gramfort | Identifying predictive regions from fMRI with TV-L1 prior[END_REF]. Unlike widespread belief, multivariate tools used commonly, such as support vector machines or searchlight, seem to do a worse job at selecting predictive regions than univariate tools [START_REF] Gramfort | Identifying predictive regions from fMRI with TV-L1 prior[END_REF].

Encoding and decoding models explore the two directions linking brain activation to stimuli and cognitive processes [START_REF] Naselaris | Encoding and decoding in fMRI[END_REF]. Both of these methodologies do not form credible models of how the brain creates cognition. They are rather experimental devices to test hypotheses and retrieve brain maps, where the critical modeling work goes in the formal description of the cognitive concepts associated with the brain signals under study. This description is most often a non-trivial transformation of the stimuli, non-linear [START_REF] Yamins | Performance-optimized hierarchical models predict neural responses in higher visual cortex[END_REF] or calling for concept ontologies [START_REF] Huth | A continuous semantic space describes the representation of thousands of object and action categories across the human brain[END_REF][START_REF] Poldrack | Decoding the large-scale structure of brain function by classifying mental states across individuals[END_REF][START_REF] Schwartz | Mapping cognitive ontologies to and from the brain[END_REF]. Following the concepts of neural coding and Marr's vision that good representations give rise to powerful computational processing [START_REF] Marr | Vision: A Computational Investigation Into the Human Representation and Processing of Visual Information[END_REF], encoding and decoding models are often understood as revealing a representational space, distributed representations in the cortex that reflect fundamental concepts [START_REF] Edelman | Toward direct visualization of the internal shape representation space by fMRI[END_REF][START_REF] Haxby | Decoding neural representational spaces using multivariate pattern analysis[END_REF]. However, the combination of the lack of temporal dynamics in fMRI and the linear models that we rely upon naturally create such an understanding of the data in terms of representations, while for some functions studied, the actual neural implementation may be closer to processes [START_REF] Davis | Measuring neural representations with fMRI: practices and pitfalls[END_REF] dynamically sustained information, as in theories of conscious processing [START_REF] Dehaene | Towards a cognitive neuroscience of consciousness: basic evidence and a workspace framework[END_REF]. In this light, the use of linear models for decoding may be criticized as too simple to capture non-linear interactions. However, from the neuroscience point-of-view they lead to probing well-posed questions [START_REF] Naselaris | Encoding and decoding in fMRI[END_REF] and from the statistical learning point of view, they can be relatively well-behaved even in very high dimensional settings with the typical small sample sizes faced by fMRI [START_REF] Varoquaux | Small-sample brain mapping: sparse recovery on spatially correlated designs with randomization and clustering[END_REF].

Finding hidden structure: parcellations and connectomes

In machine learning applications, it is often easier to accumulate unlabeled data than labeled data. This is also the case in neuroimaging, as controlling the cognitive state of the subject is very challenging and calls for careful experimental design and analysis. Data collection speed is limited by the timescale of psychological experiments. On the opposite, accumulating so-called resting-state acquisitions, in which the mental state of the subject is not controlled, is much more tractable [START_REF] Biswal | Toward discovery science of human brain function[END_REF], and is applicable to diminished populations [START_REF] Greicius | Resting-state functional connectivity in neuropsychiatric disorders[END_REF]. The interest of resting-state data for cognitive neuroimaging is not immediate, as it does not carry explicit cognitive information; however, they reflect on-going activity, which is an important part of brain function. Indeed, on-going activity shares a common substrate with activity explicitly evoked by controlled cognition, priming cognition but also shaped by task [START_REF] Sadaghiani | The relation of ongoing brain activity, evoked neural responses, and cognition[END_REF]. Unsupervised learning on resting-state scans holds the promise of extracting intrinsic brain structures [START_REF] Biswal | Toward discovery science of human brain function[END_REF].

Capturing brain interactions

The brain is a heavily interacting system. Mapping its connections in the form of a connectome [START_REF] Sporns | The human connectome: a structural description of the human brain[END_REF] can help to understand the flow of information in the brain. Fluctuations in brain activity, for example, observed at rest, reveal functional interactions and thus can be used to estimate a functional connectome [START_REF] Varoquaux | Learning and comparing functional connectomes across subjects[END_REF]. From a statistical learning perspective, the estimation of a connectome can be formalized as extracting the conditional independence structure from observed correlations [START_REF] Varoquaux | Markov models for fMRI correlation structure: is brain functional connectivity small world, or decomposable into networks?[END_REF]. The challenge here is that of the paucity of data, and can be tackled with graph estimators that have good small-sample structure recovery properties, such as sparse covariance models [START_REF] Varoquaux | Brain covariance selection: better individual functional connectivity models using population prior[END_REF][START_REF] Smith | Network modelling methods for fMRI[END_REF].

Learning functional units

Interest in resting-state data arose originally from the observation that voxel-based signals observed at rest could be used to segment spatial structures known from task studies [START_REF] Biswal | Functional connectivity in the motor cortex of resting human brain using echo-planar MRI[END_REF]. Subsequently, researchers realized that these could exhibit some additional functional structures [START_REF] Raichle | A default mode of brain function[END_REF]. What spatial distributed brain networks are modulated during rest? This question can be formulated as that of blind source separation, and independent component analysis (ICA) provides a good algorithm to recover these networks [START_REF] Kiviniemi | Independent component analysis of nondeterministic fmri signal sources[END_REF][START_REF] Beckmann | Investigations into resting-state connectivity using independent component analysis[END_REF]. Datasets of increasing size and quality lead to extracting more networks, that break up in a set of smaller regions, paving the brain in a parcellation [START_REF] Kiviniemi | Functional segmentation of the brain cortex using high model order group PICA[END_REF]. Breaking down the brain into homogeneous units is a long quest in neuroscience that can be traced back to Brodmann areas. Such parcellations have been historically driven by anatomical features. Resting-state fMRI provides valuable data to learn a functional parcellation, as it gives a spatially-resolved window into intrinsic brain function. Indeed, functionally-homogeneous regions can be extracted by clustering voxels with similar fMRI time-series [START_REF] Yeo | The organization of the human cerebral cortex estimated by intrinsic functional connectivity[END_REF][START_REF] Craddock | A whole brain fMRI atlas generated via spatially constrained spectral clustering[END_REF]. The unmixing model underlying ICA can be adapted to extracting regions by formulating it in the more general framework of dictionary learning [START_REF] Varoquaux | Multi-subject dictionary learning to segment an atlas of brain spontaneous activity[END_REF] and adding sparsity-inducing penalty that also favor clustered spatial components, thus yielding region segmentations [START_REF] Abraham | Extracting brain regions from rest fMRI with total-variation constrained dictionary learning[END_REF]. While identifying intrinsic functional brain modules is crucial from a basic neuroscience point of view, brain parcellation can also provide useful data reduction even if they don't capture true functional units [START_REF] Thirion | Dealing with the shortcomings of spatial normalization: Multi-subject parcellation of fMRI datasets[END_REF][START_REF] Varoquaux | Small-sample brain mapping: sparse recovery on spatially correlated designs with randomization and clustering[END_REF]. These different purposes give rise to different methodological trade-offs [START_REF] Thirion | Which fMRI clustering gives good brain parcellations? Name[END_REF]. Beyond resting-state data, applying similar methods to databases of evoked activity exploring a large variety of cognitive concepts can have the additional benefit of appending cognitive labels to the spatial units extracted [START_REF] Smith | Correspondence of the brain's functional architecture during activation and rest[END_REF][START_REF] Varoquaux | Cohort-level brain mapping: learning cognitive atoms to single out specialized regions[END_REF].

However, care must be exercised when applying the brain-parcellation techniques. By construction, such methods will return a parcellation, even if there is little to no structure in the data. They do not build upon well-posed statistical hypothesis testing. The methods can often be unstable, with a small modification of the input data leading to large changes in the results. This unstability can be explained by, on one hand the lack of explicit noise model, and on the other hand the fact that unsupervised learning is an intrinsically hard problem from the statistical standpoint. Validation of the functional units is very challenging beyond a simple confirmation bias that boils down to checking for known structures, the variability of which is unknown and uncontrolled. Some researchers have explored quantifying variability of the patterns [START_REF] Varoquaux | A group model for stable multi-subject ICA on fMRI datasets[END_REF][START_REF] Craddock | A whole brain fMRI atlas generated via spatially constrained spectral clustering[END_REF][START_REF] Thirion | Which fMRI clustering gives good brain parcellations? Name[END_REF][START_REF] Abraham | Extracting brain regions from rest fMRI with total-variation constrained dictionary learning[END_REF] or controlling how well they explain the data [START_REF] Abraham | Extracting brain regions from rest fMRI with total-variation constrained dictionary learning[END_REF][START_REF] Thirion | Which fMRI clustering gives good brain parcellations? Name[END_REF] but these are weak proxys of the neuroscientific questions on brain organization.

Practical considerations: methods and implementations matter

The focus of this review is not on methodological details, but on general concepts and approaches that further our understanding of brain function. However, it is important to stress that many of the roadblocks to the use of machine-learningrelated techniques in cognitive neuroimaging lie in the methods. From a theoretical point of view, the statistical control is seldom warranted by the models used [START_REF] Varoquaux | Small-sample brain mapping: sparse recovery on spatially correlated designs with randomization and clustering[END_REF][START_REF] Gramfort | Identifying predictive regions from fMRI with TV-L1 prior[END_REF]. On the empirical side of things, best practices are not established. The highdimensionality of the statistical models and the plurality of methods considered mean that, at the level of the literature, machine-learning techniques probably give rise to more variability, although they do come with more expressiveness and power.

A final critical aspect, all too often overlooked, is that of software. The standard GUI-based fMRI data processing environments, such as SPM, FSL [START_REF] Smith | Advances in functional and structural mr image analysis and implementation as fsl[END_REF] or AFNI [START_REF] Cox | AFNI: software for analysis and visualization of functional magnetic resonance neuroimages[END_REF], do not implement most of the modeling approaches described in this review. FSL and AFNI do provide some methods tailored to fMRI uses (respectively ICA [START_REF] Beckmann | Investigations into resting-state connectivity using independent component analysis[END_REF] and basic decoding [START_REF] Laconte | Decoding fMRI brain states in real-time[END_REF]). There is progress on dedicated tools such as PyMVPA [START_REF] Hanke | PyMVPA: A python toolbox for multivariate pattern analysis of fmri data[END_REF], but these require the practitioners to learn new skills, in particular some understanding of machine learning and basic programming. The challenges of a good environment for machine-learning on fMRI data is that it should be simple enough to be within reach of the practitioner, yet leverage a powerful machine-learning toolkit, such as the popular scikit-learn package in Python [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF], and offer flexibility to assemble new models, encoding, decoding, or unsupervised [START_REF] Abraham | Machine learning for neuroimaging with scikit-learn[END_REF].

Conclusions

The goals of cognitive neurosciences are to link cognition with its neural basis. FMRI gives a noisy and incomplete window on neural mechanisms. Nevertheless, to map effects at a large scale, it is priceless, as it can be applied massively on healthy human subjects, and thus enables the systematic study of high-level cognition. Machine learning tools are instrumental in making the most of this data, as they do not require a precise mechanistic understanding of the signal, but rather to frame a prediction problem that captures some relevant aspects of brain function, as in encoding or decoding. However, for progress in neuroscience, black-box prediction engines do not suffice as the key to understanding brain function lies in the properties of the signal used for prediction. For these reasons, the statistics aspects in statistical learning cannot be neglected: different methods give rise to different results and the figure of merit does not simply boil down to predictive power.

Figures

Figure 1 Schematics of the distinction between encoding and decoding in brain imaging.

[START_REF] Penny | Statistical Parametric Mapping: The Analysis of Functional Brain Images[END_REF] It is unlikely that standard fMRI acquisitions, even after analysis with powerful multivariate methods, capture information at the level of the cortical column[START_REF] Freeman | Orientation decoding depends on maps, not columns[END_REF].

Acknowledgements

The authors acknowledge fruitful discussions with Catherine Wacongne and thank the two reviewers, Krzysztof Gorgolewski and Tal Yarkoni, for their review which improved the manuscript.

Abbreviations

fMRI: functional Magnetic Resonnance Imaging EEG:

Electro EncephaloGraphy MEG:

Magneto EncephaloGraphy TV:

Total-Variation ICA:

Independent Component Analysis GUI:

Graphical User Interface


Marc-Antoine Coppo
email: coppo@unice.frcandel@unice.fr

Bernard Candelpergher

Inverse binomial series and values of Arakawa-Kaneko zeta functions

Keywords: Mathematical Subject Classification (2010): 11M06, 11M41, 11Y60, 33B30 Harmonic sums, Binomial transformations, Bell polynomials, Polylogarithms, Log-sine integrals, Mellin transforms, Zeta values

In this article, we present a variety of evaluations of series of polylogarithmic nature. More precisely, we express the special values at positive integers of two classes of zeta functions of Arakawa-Kaneko-type by means of certain inverse binomial series involving harmonic sums which appeared fifteen years ago in physics in relation with the Feynman diagrams. In some cases, these series may be explicitly evaluated in terms of zeta values and other related numbers. Incidentally, this connection allows us to deduce new identities for the

) considered by S. Ramanujan in his notebooks.

Introduction

The function β defined for (s) > 0 by the Dirichlet series

β(s) = ∞ n=1
(-1) n-1 (2n -1) s has the integral representation

β(s) = 1 Γ(s) +∞ 0 e -t 1 + e -2t t s-1 dt = 1 Γ(s) +∞ 0 e -t 1 -e -2t Li 0 1 -e -2t 2 t s-1 dt
where Li k denotes the classical polylogarithm Li k (z) = ∞ n=1 z n n k . One may also observe that 1 Γ(s) +∞ 0 e -t 1 -e -2t Li 1 (1 -e -2t ) t s-1 dt = (2 -2 -s )s ζ(s + 1) .

These preliminary observations lead us to introduce two families of functions α k and β k defined by the Mellin transforms

α k (s) = 1 Γ(s) +∞ 0 e -t 1 -e -2t Li k 1 -e -2t t s-1 dt for (s) > 0 and k ≥ 1 , β k (s) = 1 Γ(s) +∞ 0 e -t 1 -e -2t Li k 1 -e -2t
2 t s-1 dt for (s) > 0 and k ≥ 0 , so that α 1 (s) = 2 -s (2 s+1 -1)sζ(s + 1), and β 0 (s) = β(s) .

We point out that the function α k (s) introduced here is (apart from a factor 2 -s ) nothing less than the special value at x = 1/2 of the (generalized) Arakawa-Kaneko zeta function ξ k (s, x) previously defined in [START_REF] Coppo | The Arakawa-Kaneko Zeta function[END_REF], whereas the function β k (s) is a new function of the same type. Let us remind that the original Arakawa-Kaneko zeta function ξ k (s) = ξ k (s, 1) was introduced by Arakawa and Kaneko in 1999 (cf. [START_REF] Arakawa | Multiple zeta values, Poly-Bernoulli numbers and related zeta functions[END_REF]) and formed the subject of recent works and further generalizations (cf. [START_REF] Candelpergher | A new class of identities involving Cauchy numbers, harmonic numbers and zeta values[END_REF], [START_REF] Coppo | The Arakawa-Kaneko Zeta function[END_REF], [START_REF] Sasaki | On generalized poly-Bernoulli numbers and related L-functions[END_REF], [START_REF] Young | Symmetries of Bernoulli polynomial series and Arakawa-Kaneko zeta functions[END_REF]).

In the case where s is a positive integer, the special values α k (s) and β k (s) can be expressed by means of certain inverse binomial series studied by Kalmykov and Davydychev in relation with the Feynman diagrams (cf. [START_REF] Davydychev | Massive Feynman diagrams and inverse binomial sums[END_REF]). More precisely, we obtain the following identities: (1) n , . . . , O (j) n , . . . , O (m) n ) (for k ≥ 1) ,

α k (m + 1) = ∞ n=1 2 2n-1 2n n n k+1 P m (O
β k (m + 1) = ∞ n=1 2 n-1 2n n n k+1
P m (O (1) n , . . . , O (j) n , . . . , O (m) n ) (for k ≥ 0) , where P m is the modified Bell polynomial of order m and O (j) n = n k=1 1 (2k-1) j is the "odd" harmonic number of order j. For small values of k ans s, these series may be explicitly evaluated in terms of zeta values and other related constants which are real periods in the sense of Kontsevich and Zagier (cf. [START_REF] Kontsevich | Periods[END_REF]). For instance, we have the following evaluations:

α 1 (2) = ∞ n=1 2 2n-1 2n n O n n 2 = 7 2 ζ(3) , β 1 (2) = ∞ n=1 2 n-1 2n n O n n 2 = 7 4 ζ(3) - π 2 G , α 2 (2) = ∞ n=1 2 2n-1 2n n O n n 3 = 7ζ(3) ln 2 - π 4 32 -8G(1) , α 3 (1) = ∞ n=1 2 2n-1 2n n 1 n 4 = π 2 2 (ln 2) 2 - 7 2 ζ(3) ln 2 + π 4 96 + 4G(1) ,
where we use the following notations:

O n := O (1) n = n j=1 1 2j -1 , G := β(2) is the Catalan constant, G(1) := ∞ n=1
O n (2n) 3 is the Ramanujan constant (cf. [START_REF] Berndt | Ramanujan's Notebooks Part I[END_REF] p. 257, [START_REF] Sitaramachandrarao | A formula of S[END_REF]).

The evaluation of α 1 (2) above was given previously in [START_REF] Coppo | Nouvelles expressions des formules de Hasse et de Hermite pour la fonction Zêta d'Hurwitz[END_REF], whereas the following ones are new. In particular, the two last relations provide new interesting formulae for the Ramanujan's constant G(1) as explained in detail in section 6.

2 Bell polynomials and "odd" harmonic numbers Definition 1. The modified Bell polynomials are the polynomials

P m ∈ Q[x 1 , x 2 , ..., x m ]
defined for all natural numbers m by P 0 = 1 and the generating function

exp ∞ k=1 x k z k k = ∞ m=0 P m (x 1 , • • • , x m ) z m ,
The general explicit expression for P m is

P m (x 1 , ..., x m ) = k 1 +2k 2 +•••+mkm=m 1 k 1 !k 2 ! . . . k m ! x 1 1 k 1 x 2 2 k 2 . . . x m m km . Example 1.
For the first values of m, one has

P 0 = 1 , P 1 = x 1 , P 2 = 1 2 x 2 1 + 1 2 x 2 , P 3 = 1 6 x 3 1 + 1 2 x 1 x 2 + 1 3 x 3 , P 4 = 1 24 x 4 1 + 1 4 x 2 1 x 2 + 1 8 x 2 2 + 1 3 x 1 x 3 + 1 4 x 4 .

Notation.

For s ∈ C with (s) ≥ 1 and an integer n ≥ 1, let O (s) n be the "odd" harmonic sum:

O (s) n = n k=1 1 (2k -1) s and O n := O (1)
n .

In the notation of [START_REF] Coppo | The Arakawa-Kaneko Zeta function[END_REF], one has

O (s) n = 2 -s h (s) n-1 (1/2) with h (s) n (x) = n j=0 1 (j + x) s .
Proposition 1. For all integers m ≥ 0 and n ≥ 1,

P m (O n , . . . , O (m) n ) = n 2n n 2 2n-1 +∞ 0 e -t (1 -e -2t ) n-1 t m m! dt . ( 1
)
Proof. Since

P m (O n , . . . , O (m) n ) = 2 -m P m (h (1)
n-1 (1/2), . . . , h (m) n-1 (1/2)) , formula (1)
follows from Lemma 1 of [START_REF] Coppo | The Arakawa-Kaneko Zeta function[END_REF] in the special case x = 1/2. To be self-contained, we give a direct proof below. We show that

∞ m=0 P m (O n , • • • , O (m) n ) z m = n j=1 2j -1 2j -1 -z = n 2n n 2 2n-1 +∞ 0 e tz (1 -e -2t
) n-1 e -t dt , and then we shall obtain formula (1) by identification of the coefficients of z m . On one side, one has

n j=1 2j -1 2j -1 -z = n j=1 (1 - z 2j -1 ) -1 = exp(- n j=1 log(1 - z 2j -1 )) = exp( n j=1 +∞ k=1 z k k(2j -1) k ) = exp( +∞ k=1 z k k n j=1 1 (2j -1) k ) , thus n j=1 2j -1 2j -1 -z = exp( ∞ k=1 O (k) n z k k ) = ∞ m=0 P m (O n , • • • , O (m) n ) z m .
On the other side, one has

n j=1 2j -1 2j -1 -z = 1 √ π Γ(n + 1/2)Γ(-z/2 + 1/2) Γ(n -z/2 + 1/2) = 1 √ π Γ(n + 1/2) Γ(n) Γ(n)Γ(-z/2 + 1/2) Γ(n -z/2 + 1/2) = n 2 2n 2n n B(n, -z/2 + 1/2) ,
where B is the Euler Beta function. Thus, for 0 < |z| < 1, one has

n j=1 2j -1 2j -1 -z = n 2n n 2 2n 1 0 u n-1 (1 -u) -z/2-1/2 du ,
and making the change of variable u = 1 -e -2t , one then obtains:

n j=1 2j -1 2j -1 -z = n 2n n 2 2n-1 +∞ 0 e tz (1 -e -2t ) n-1 e -t dt ,
and finally

∞ m=0 P m (O n , • • • , O (m) n ) z m = n 2n n 2 2n-1 +∞ 0 e tz (1 -e -2t
) n-1 e -t dt .

D(a)(x) = +∞ 0 e -t 1 -e -t (1 -e -t ) x a(t) dt , S(a)(x) = +∞ 0 e -t 1 -e -t (1 -e -xt ) a(t) dt . Proposition 2. For all integers n ≥ 1, one has S(a)(n) = n k=1 a(k) ,
and for all integer n ≥ 0,

D(a)(n + 1) = n k=0 (-1) k n k a(k + 1) .
Proof. The first relation follows from

S(a)(n) = +∞ 0 e -t -e -(n+1)t 1 -e -t a(t)dt = +∞ 0 ( n k=1 e -kt ) a(t)dt = n k=1 +∞ 0 e -kt a(t)dt .
The second relation results from the binomial expansion of (1 -e -t ) n since

D(a)(n + 1) = +∞ 0 e -t (1 -e -t ) n a(t)dt = +∞ 0 e -t n k=0 (-1) k n k e -kt a(t)dt = n k=0 (-1) k n k +∞ 0 e -t e -kt a(t)dt = n k=0 (-1) k n k a(k + 1) .
Example 2. For s with (s) ≥ 1 and x ∈ P , let

a(x) = 1 (2x -1) s . One has a(x) = +∞ 0 e -(2x-1)t t s-1 Γ(s) dt = +∞ 0 e -xt e t 2 ( t 2 ) s-1 2Γ(s) dt .
Thus, for all integer n ≥ 1,

D(a)(n) = +∞ 0 e -t 2 (1 -e -t ) n-1 ( t 2 ) s-1 2Γ(s) dt = +∞ 0 e -t (1 -e -2t ) n-1 t s-1 Γ(s) dt . (2)
By ( 1), one has

+∞ 0 e -t (1 -e -2t ) n-1 t m m! dt = 2 2n-1 n 2n n P m (O n , . . . , O (m) n ) .
Thus, if s is an integer, s = m + 1 with m ≥ 0, then we get for all integers n ≥ 1 the following formula

D( 1 (2x -1) m+1 )(n) = 2 2n-1 n 2n n P m (O n , . . . , O (m) n ) . ( 3
)
Lemma 1. The operators D and S are linked by the following relation:

D 1 x S(a) = 1 x D(a) for all x ∈ P .
Proof of the lemma. By definition of S(a),

1 x S(a)(x) = +∞ 0 1 -e -xt x [ e -t 1 -e -t a(t)] dt ,
integrating by parts, we get

1 x S(a)(x) = +∞ 0 e -xt ∞ t e -u 1 -e -u a(u) du dt , this gives 1 x S(a)(t) = ∞ t e -u 1 -e -u a(u) du . Thus D( 1 x S(a))(x) = +∞ 0 e -t (1 -e -t ) x-1 ∞ t e -u 1 -e -u a(u) du dt ,
and integrating again by parts, we get

D( 1 x S(a))(x) = +∞ 0 1 x (1 -e -t ) x e -t 1 -e -t a(t) dt = 1 x D(a)(x) .
Remark 1. The relation between D and S given above is a reformulation of a result that we called the "harmonic property" in an earlier paper: cf. [START_REF] Candelpergher | A new class of identities involving Cauchy numbers, harmonic numbers and zeta values[END_REF] Theorem 6 (where the operator S is denoted by A). Proposition 3. For all complex numbers z such that |z| < 1 2 , one has

+∞ n=1 D(a)(n) z n = - +∞ n=1 a(n)( z z -1 ) n , +∞ n=1 D(a)(n) z n n = - +∞ n=1 1 n S(a)(n)( z z -1 ) n . ( 4
)
Proof. For the first relation we write

+∞ n=0 D(a)(n + 1)z n+1 = +∞ 0 e -t z 1 -(1 -e -t )z a(t)dt = - +∞ 0 ( z z -1 ) e -t 1 -e -t z z-1 a(t)dt .
The expansion

( z z -1 ) e -t 1 -e -t z z-1 a(t) = +∞ n=1 e -nt ( z z -1 ) n a(t) gives +∞ n=0 D(a)(n + 1)z n+1 = - +∞ 0 +∞ n=1 e -nt ( z z -1 ) n a(t)dt = - +∞ n=1 ( z z -1 ) n +∞ 0 e -nt a(t)dt ,
the order of +∞ 0 and ∞ n=1 may be interchanged because

+∞ 0 +∞ n=1 e -nt ( |z| 1 -|z| ) n | a(t)|dt = ( |z| 1 -|z| ) +∞ 0 e -t 1 -e -t |z| 1-|z| | a(t)| < +∞ .
The second relation ( 4) is an immediate consequence of the first one by Lemma 1 above.

Proposition 4. For all integers

p ≥ 1, one has ∞ n=1 D(a)(n) p n n k = +∞ 0 e -t 1 -e -t Li k ( 1 -e -t p ) a(t) dt . ( 5
)
Proof. Let p be a positive integer, then

∞ n=1 D(a)(n) p n n k = ∞ n=1 1 p n n k +∞ 0 e -t 1 -e -t (1 -e -t ) n a(t) dt = +∞ 0 e -t 1 -e -t ∞ n=1 (1 -e -t ) n p n n k a(t) dt = +∞ 0 e -t 1 -e -t Li k ( 1 -e -t p ) a(t) dt ,
the order of +∞ 0 and ∞ n=1 may be interchanged since, by the hypothesis on a,

| a(t)| ≤ Ce αt for all t ∈ ]0, +∞[ , which gives +∞ 0 e -t 1 -e -t ∞ n=1 (1 -e -t ) n p n n k | a(t)| dt ≤ C +∞ 0 e -t 1 -e -t Li k ( 1 -e -t
p )e αt dt < +∞ .

α k (s) = 1 Γ(s) +∞ 0 e -t 1 -e -2t Li k 1 -e -2t t s-1 dt (for k ≥ 1) , β k (s) = 1 Γ(s) +∞ 0 e -t 1 -e -2t Li k 1 -e -2t 2 t s-1 dt (for k ≥ 0) . Example 3. α 1 (s) = 2 Γ(s) +∞ 0 e -t 1 -e -2t t s dt = 2 -s (2 s+1 -1)sζ(s + 1) , β 1 (s) = 1 Γ(s) +∞ 0 e -t 1 -e -2t ln 2 -ln(1 + e -2t ) t s-1 dt .
Remark 2 (link with the Arakawa-Kaneko zeta function). In [START_REF] Coppo | The Arakawa-Kaneko Zeta function[END_REF], we introduced the function (s, x) → ξ k (s, x) defined for (s) > 0 and x > 0 by :

ξ k (s, x) = 1 Γ(s) +∞ 0 e -xt Li k (1 -e -t )
1 -e -t t s-1 dt which is a very natural extension of the zeta function of Arakawa and Kaneko in the same way as the Hurwitz zeta function ζ(s, x) generalizes the Riemann zeta function. In the simplest case k = 1, ξ 1 (s, x) is nothing else than sζ(s + 1, x), and moreover one deduces immediately from the previous definition that

α k (s) = 2 -s ξ k (s, 1 2
) .

Proposition 5. If s is such that (s) ≥ 1, then

α k (s) = ∞ n=1 1 n k D( 1 (2x -1) s )(n) (for k ≥ 1) , β k (s) = ∞ n=1 1 2 n n k D( 1 (2x -1) s )(n) (for k ≥ 0) .
Proof. This is an immediate consequence of formula (5) applied to the function

a(x) = 1 (2x -1) s (for p = 1, 2) since a(t) = e t 2 ( t 2 ) s-1 2Γ(s)
as already seen in Example 2.

Corollary 1. For all integers m ≥ 0, then

α k (m + 1) = ∞ n=1 2 2n-1 2n n n k+1 P m (O (1) n , . . . , O (m) n ) (for k ≥ 1) , ( 6
)
β k (m + 1) = ∞ n=1 2 n-1 2n n n k+1 P m (O (1) n , . . . , O (m) n ) (for k ≥ 0) . ( 7
)
Proof. This is an immediate consequence of Proposition 5 by formula (3).

Example 4. Since α 1 (s) = 2 -s (2 s+1 -1)sζ(s + 1), then for all integers m ≥ 1,

(2 -2 -m )mζ(m + 1) = ∞ n=1 2 2n-1 2n n n 2 P m-1 (O (1) n , . . . , O (m-1) n ) . ( 8
)
In particular, for m = 2, a nice formula for Apéry's constant (cf. [START_REF] Coppo | Nouvelles expressions des formules de Hasse et de Hermite pour la fonction Zêta d'Hurwitz[END_REF], p. 81) is regained:

∞ n=1 2 2n 2n n O n n 2 = 7ζ(3) . ( 9
)
Example 5. Since β 0 (s) = β(s), one has for all integers m ≥ 1,

β(2m) = ∞ n=1 2 n-1 2n n n P 2m-1 (O (1) n , . . . , O (2m-1) n ) . ( 10
)
In particular, for m = 1, a nice formula for Catalan's constant (cf. [START_REF] Berndt | Ramanujan's Notebooks Part I[END_REF] p. 293, Entry 34) is regained:

∞ n=1 2 n 2n n O n n = 2G, (11)
and for m = 2, formula ( 10) is translated into

∞ n=1 2 n 2n n (O n ) 3 n + 3 ∞ n=1 2 n 2n n O n O (2) n n + 2 ∞ n=1 2 n 2n n O (3) n n = 12β(4) . ( 12
)
Remark 3. In a similar way (cf. [3] § 5.5), one can prove for ξ k (s) := ξ k (s, 1) the following identity:

ξ k (s) = ∞ n=1 1 n k D( 1 x s )(n) (for (s) ≥ 1 and k ≥ 1) ,
and, furthermore, one has (cf. [START_REF] Candelpergher | A new class of identities involving Cauchy numbers, harmonic numbers and zeta values[END_REF] § 3)

D( 1 x m+1 )(n) = P m (H (1) n , . . . , H (m) n ) n with H (j) n = n k=1 1 k j (j = 1, 2, • • • , m) ,
which gives for instance Euler's famous identity:

ξ 1 (2) = ∞ n=1 H n n 2 = 2ζ(3) .

The function β 1

The Euler series transformation (Proposition 3 above) provides an alternative expression for β 1 .

Proposition 6. For all s ∈ C with (s) ≥ 1, one has

β 1 (s) = ∞ n=1 (-1) n-1 O (s) n n , hence, for each integer m ≥ 1, ∞ n=1 (-1) n-1 O (m) n n = ∞ n=1 2 n-1 2n n n 2 P m-1 (O (1) n , . . . , O (m-1) n ) . ( 13
)
Proof. By ( 4), one has for all |z| < 1 2 ,

+∞ n=1 D(a)(n) n z n = - +∞ n=1 1 n S(a)(n)( z z -1 ) n . If the series +∞ n=1 D(a)(n) n 1
2 n is convergent, then, by the classical Abel lemma, we get

+∞ n=1 D(a)(n) n 1 2 n = +∞ n=1 1 n S(a)(n)(-1) n-1 .
By formula (5), the series

∞ n=1 1 2 n n D( 1 (2x-1) s )(n) is convergent and β 1 (s) = ∞ n=1 1 2 n n D( 1 (2x -1) s )(n) = +∞ n=1 (-1) n-1 n S( 1 (2x -1) s )(n) .
Then, using formula (1) for D( 1 (2x -1) m )(n), one obtains [START_REF] Young | Symmetries of Bernoulli polynomial series and Arakawa-Kaneko zeta functions[END_REF].

Example 6. ∞ n=1 (-1) n-1 O n n = ∞ n=1 2 n-1 2n n 1 n 2 = π 2 16 , ∞ n=1 (-1) n-1 O (2) n n = ∞ n=1 2 n-1 2n n O n n 2 = 7 4 ζ(3) - π 2 G ( 14
)
( [START_REF] Davydychev | Massive Feynman diagrams and inverse binomial sums[END_REF] (2.36) and (2.37) with u = 2 and Remark 4. In complete analogy with [START_REF] Young | Symmetries of Bernoulli polynomial series and Arakawa-Kaneko zeta functions[END_REF], one also has the following relation

θ = π 2 ) , ∞ n=1 (-1) n-1 O (3) n n = ∞ n=1 2 n 2n n (O n ) 2 (2n) 2 + ∞ n=1 2 n 2n n O (2) n (2n) 2 = π 4 64 -G 2 (15)
∞ n=1 (-1) n-1 H (m) n n = ∞ n=1 1 2 n n 2 P m-1 (H (1) n , . . . , H (m-1) n ) (16)
which is equivalent to that given by Choi and Srivastava ([4] p. 66, formula (4.29)).

Proposition 7.

Let

H n = n k=1 (-1) k-1 k . If s is such that (s) > 1, then ∞ n=1 H n (2n -1) s = (1 -2 -s )ζ(s) ln 2 + ∞ n=1 (-1) n-1 n(2n -1) s -β 1 (s) . ( 17
)
Thus, for each integer m > 1,

∞ n=1 H n (2n -1) m = (1-2 -m )ζ(m) ln 2+(-1) m ln 2+2 m k=1 (-1) m-k β(k)-β 1 (m) . (18)
Proof. The first relation is a direct consequence of the following elementary result:

If the series ∞ n=1 a n , ∞ n=1 b n , ∞ n=1 a n b n and ∞ n=1 b n n k=1 a k are convergent, then the series n≥1 a n n k=1 b k is convergent and we have ∞ n=1 a n n k=1 b k = ∞ n=1 a n ∞ n=1 b n + ∞ n=1 a n b n - ∞ n=1 b n n k=1 a k .
(-1) k (2n -1) k = (-1) m (2n -1) -m -1 2n , hence (-1) n-1 n(2n -1) m = (-1) m (-1) n-1 n + 2 m k=1 (-1) m-k (-1) n-1 (2n -1) k
which gives formula (18).

Example 7. Formula (18) gives respectively for m = 2 and m = 3 the following identities:

∞ n=1 H n (2n -1) 2 = π 2 8 ln 2 + ln 2 - π 2 + 2G - 7 4 ζ(3) + πG 2 , ( 19
) ∞ n=1 H n (2n -1) 3 = 7 8 ζ(3) ln 2 -ln 2 + π 2 -2G + π 3 16 + G 2 - π 4 64 . ( 20
)
5 The values α k (1) and β k [START_REF] Arakawa | Multiple zeta values, Poly-Bernoulli numbers and related zeta functions[END_REF] The special values of α k and β k at s = 1 are evaluated in terms of (generalized) log-sine functions (cf. [START_REF] Davydychev | Massive Feynman diagrams and inverse binomial sums[END_REF], [START_REF] Lewin | Polylogarithms and associated functions[END_REF]).

Proposition 8. For each integer k ≥ 1 and α ∈ R such that 0 ≤ α ≤ 1, let L k (α) be the log-sine-type integral1 :

L k (α) = απ 0 u ln k-1 2 sin u 2 du = π 2 α 0 x ln k-1 2 sin πx 2 dx .
Then, one has

2α k (1) = ∞ n=1 2 2n
2n n

1 n k+1 = 2 k-1 k i=1 (-1) i-1 (ln 2) k-i (i -1)!(k -i)! L i (1) , ( 21
)
2β k (1) = ∞ n=1 2 n 2n n 1 n k+1 = k i=1 (-1) i-1 2 i-1 (ln 2) k-i (i -1)!(k -i)! L i ( 1 2
) .

(

) 22
Proof. The proof is similar to that given in [14] § 4. Let

J k (x) = 1 2 k ∞ n=1 (2x) 2n 2n n n k+1 .
Then, one has for k ≥ 1,

J k (x) = x 0 J k-1 (u) u du .
By a classical identity due to Euler (cf. [START_REF] Zhang | Values of the riemann zeta function and integrals involving log(2 sin θ 2 )[END_REF], [START_REF] Zucker | On the series 2k k -1 k -n and related sums[END_REF]), one also has

J 1 (x) = (arcsin x) 2 , hence J 0 (x) = 2x arcsin x √ 1 -x 2 . It is easily verified that 2α k (1) = 2 k J k (1) = 2 k 1 0 J k-1 (x)
x dx , and

2β k (1) = 2 k J k ( √ 2 2 ) = 2 k √ 2 2 0 J k-1 (x) x dx .
By (k -1) integrations by parts and the change of variable x = sin u 2 , we get

J k (1) = (-1) k-1 (k -1)! 1 0 ln k-1 (x) J 0 (x) x dx = (-1) k-1 2(k -1)! π 0 u ln k-1 (sin u 2 )du ,
and

J k ( √ 2 2 ) = (-1) k-1 (k -1)! √ 2 2 0 ln k-1 ( √ 2x) J 0 (x) x dx = (-1) k-1 2(k -1)! 1 2 π 0 u ln k-1 ( √ 2 sin u 2 )du.
It remains to use the binomial expansions of

ln k-1 (sin u 2 ) = ln( 1 2 ) + ln(2 sin u 2 ) k-1
, and

ln k-1 ( √ 2 sin u 2 ) = ln( 1 √ 2 ) + ln(2 sin u 2 ) k-1
to obtain formulas (21) and ( 22).

Proposition 9.

For all α such that 0 ≤ α ≤ 1, we have

a) L 1 (α) = π 2 2 α 2 , b) L 2 (α) = ζ(3) - ∞ n=1 cos(πnα) n 3 -απ ∞ n=1 sin(πnα) n 2 , c) L 3 (α) = π 4 16 (α 4 - 8 3 α 3 + 2α 2 ) + 2 ∞ n=1 H n (n + 1) 3 cos(π(n + 1)α) + 2πα ∞ n=1 H n (n + 1) 2 sin(π(n + 1)α) - 1 2 ζ(4) . ( 23
)
Proof. The assertion a) is trivially verified and b) is a classical identity (cf. [START_REF] Lewin | Polylogarithms and associated functions[END_REF], formula (7.53)). It remains to prove c). We use the following expansion :

Log 2 (1 -z) = 2 ∞ n=1 H n z n+1 n + 1 to get Log 2 1 -e -iπx = 2 ∞ n=1
H n e -iπx(n+1) n + 1 .

Since

Log Integrating, this gives

α 0 x ln 2 2 sin πx 2 dx = π 2 4 α 0 x(x -1) 2 dx + 2 ∞ n=1 H n n + 1 α 0 x cos(π(n + 1)x) dx . ( 24
)
The permutation of and in ( 24) is justified by the following Lemma 2 and the dominated convergence theorem. The integrals in the right-hand side of (24) are easily computed by

α 0 x(x -1) 2 dx = α 4 4 - 2α 3 3 + α 2 2 ,
and α 0

x cos(π(n + 1)x)dx = cos(π(n + 1)α)

π 2 (n + 1) 2 + α π(n + 1) sin(π(n + 1)α) - 1 π 2 (n + 1) 2 .
Thus, we deduce from (24) the following expression for L 3 (α):

L 3 (α) = π 4 4 ( α 4 4 - 2α 3 3 + α 2 2 ) + 2 ∞ n=1 H n (n + 1) 3 cos(π(n + 1)α) + 2πα ∞ n=1 H n (n + 1) 2 sin(π(n + 1)α) -2 ∞ n=1 H n (n + 1) 3 .

Moreover, one has

H n n + 1 x cos(π(n + 1)x) = k n=1 S n (x)( H n n + 1 - H n+1 n + 2 ) + H k k + 1 S k (x) ,
and one has

|S n (x)| = x n j=1
cos(π(j + 1)x) ≤ 2x sin(πx/2) .

It follows that, for all x ∈]0, 1[,

k n=1 xH n cos(π(n + 1)x) n + 1 ≤ 2x sin(πx/2) ( H 1 2 - H k+1 k + 2 + H k k + 1 ) ≤ Cx sin(πx/2) ≤ C .

New formulae for Ramanujan's constant G(1)

In Chapter 9 of his notebooks (cf. [START_REF] Berndt | Ramanujan's Notebooks Part I[END_REF] p. 255, Entry 11), Ramanujan introduced two generating functions2 :

F (x) := ∞ n=1 O n x 2n (2n) 2 and G(x) := ∞ n=1 O n x 2n (2n) 3 ,
then, he writes the following functional relation :

G(x) + G 1 -x 1 + x = F (x) log(x) + F 1 -x 1 + x log 1 -x 1 + x - 1 16 log 2 (x) log 2 1 -x 1 + x + C , (ii) with C = π 4 ∞ n=0 (-1) n (4n + 1) 3 - π 3 √ 3 ∞ n=0 1 (2n + 1) 3 .
Unfortunately, this beautiful formula for C given by Ramanujan turns out to be erroneous since, letting x tend to 1 in (ii), one sees easily that the constant C must be equal to

G(1) = ∞ n=1
O n (2n) 3 (cf. [START_REF] Berndt | Ramanujan's Notebooks Part I[END_REF] p. 257, or [START_REF] Sitaramachandrarao | A formula of S[END_REF] for more details).

However, the calculation of α 3 (1) and α 2 (2) provides two interesting formulae for the constant G(1).

Proposition 10. Let G(1) = ∞ n=1 On (2n) 3 be the Ramanujan constant. One has G(1) = 7 8 ζ(3) ln 2 - π 4 384 - 1 8 π 2 (ln 2) 2 + 2 ∞ n=1 2 2n 2n n 1 (2n) 4 . ( 25
)
Proof. One has

G(1) = ∞ n=1 O n (2n) 3 = ∞ n=1 H 2n -1 2 H n (2n) 3 = ∞ n=1 H 2n (2n) 3 - ∞ n=1 1 2 H n (2n) 3 = ∞ n=1 1 + (-1) n 2 H n n 3 - ∞ n=1 1 2 4 H n n 3 = 1 2 ∞ n=1 H n n 3 + 1 2 ∞ n=1 (-1) n H n n 3 - 1 2 4 ∞ n=1 H n n 3 = 35 64 ζ(4) - 1 2 ∞ n=1 (-1) n-1 H n n 3 .
Moreover, by (23) with α = 1, one also has Thus

L 3 (1) = 25 8 ζ(4) -2 ∞ n=1 (-1) n-1 H n n 3 = π 4 96 + 4G(1) . ( 26
F (1) = ∞ n=1 O n (2n) 2 = 1 4 ∞ n=1 2 2n 2n n O n (2n) 2 . ( 28
)
The calculation of α 2 (2) provides a nice expression of the Ramanujan constant G(1) similar to (28). O n (2n) 3 .

(29)

Proof. Applying (2.44) and (2.45) of [START_REF] Davydychev | Massive Feynman diagrams and inverse binomial sums[END_REF] with u = 4 and θ = π, one obtains, after calculations, an expression of α 2 (2) involving L 3 (1) which may be simplified using formula (26). Finally, we get the following relation: O n (2n) 3 (31) which gets closer to the erroneous formula given by Ramanujan for C.

Conclusion

With the aim of defining a natural framework for the study of the special values of zeta functions of Arakawa-Kaneko-type, we were led to consider polylogarithmic series in the generic form: (-1) j n j a(j + 1) .

We studied the important cases a(x) = x -s and a(x) = (2x -1) -s for z = 1 and z = 1/2. Though limited in practice to small values of k and s, our approach provided plenty of nice formulae with interest both in number theory and in physics.

4 The functions α k and β k Definition 3 .

 3 Let k be a positive integer. The functions α k and β k are respectively defined for all s ∈ C with (s) > 0 by

([ 7 ]

 7 (2.38), (2.39), (2.40) and (C.4) with u = 2 and θ = π 2) .

Applied to a n = 1 (

 1 2n-1) s and b n = (-1) n-1 n , this relation gives (17). The second relation is a consequence of the first one by the following observation: If s is an integer, s = m, then we have m k=1

  for x ∈]0, 1[. Proof of the lemma. Let S n (x) = x n j=1 cos(π(j + 1)x). A sommation by parts gives k n=1

Example 8 .

 8 Formulae (21) and (22) give for k = 2 the following identities: which were known of Ramanujan (cf.[START_REF] Berndt | Ramanujan's Notebooks Part I[END_REF], p. 269).

3 ( 1 Remark 5 . 1 ( 2

 31512 )Then, applying formula (21) with k = 3, it results from (26) that 2α We have seen before (cf. Example 4, formula[START_REF] Lewin | Polylogarithms and associated functions[END_REF]) that α

Proposition 11 .

 11 Let G(1) = ∞ n=1 On (2n) 3 be the Ramanujan constant. One has the following formula:

α 2 ( 2

 22

F

  k (a, z) = ∞ n=1 D(a)(n) z n n k with k ∈ N, z ∈ C,andD(a)(n + 1) = n j=0

3 The operators D and S, and the Euler series transformation Definition 2. Let

	a be an analytic function in P = {x | (x) ≥ 1} defined by
	a(x) =	+∞	e -xt a(t)dt for all x ∈ P ,
	0
	where a ∈ C 1 ([0, +∞[) is such that there exists α < 1, and C > 0 with
	| a(t)| ≤ Ce αt for all t ∈ ]0, +∞[ .

For x ∈ P , we define the functions x → D(a)(x) and x → S(a)(x) by

  2 1 -e -iπx = Log 2 e -iπx/2 (e iπx/2 -e -iπx/2 )

		= (-iπx/2 + iπ/2 + ln(2 sin	1 2	xπ)) 2
		= -	π 2 4	(x -1) 2 + ln 2 (2 sin	πx 2	x) + i Log 2 1 -e -iπx ,
	one has	ln 2 (2 sin	πx 2	) =	π 2 4	(x -1) 2 +		Log 2 1 -e -iπx ,
	hence	ln 2 2 sin	πx 2		=	π 2 4	(x -1) 2 + 2	∞ n=1	H n	cos(π(n + 1)x) n + 1	.

With the notations of[START_REF] Lewin | Polylogarithms and associated functions[END_REF], L k (α) is -Ls (1) k+1 (απ).

These functions are respectively denoted by φ and ψ in the original manuscript : cf.[START_REF] Ramanujan | [END_REF] p. 108.

Acknowledgements. The authors gratefully thank the referee for his relevant comments.


Olivier Cardin
email: olivier.cardin@univ-nantes.fr

Damien Trentesaux
email: damien.trentesaux@univ-valenciennes.fr

André Thomas
email: andre.thomas@univ-lorraine.frhind.bril@univ-lorraine.fr

Pierre Castagna
email: pierre.castagna@univ-nantes.fr

Thierry Berger
email: thierry.berger@univ-valenciennes.fr

Hind Bril

Coupling predictive scheduling and reactive control in manufacturing: state of the art and future challenges

Keywords: holonic control, predictive/reactive, performance indicator, discrete-event observer, flexible manufacturing system

come   L'archive ouverte pluridisciplinaire

Introduction

Classical (historical) predictive approaches consist in using a centralized predictive scheduling system loosely coupled with a reactive control system that implements it. The scheduling models are based on a mathematical representation of the production system from which an optimization or heuristic algorithm is designed and computed in a centralized way. This approach leads to determine or approximate the optimal sequence of tasks to be executed in the system in order to maximize one or several criterion(s) somehow related to productivity, customer satisfaction, etc. The result of the calculation is then used by the Manufacturing Execution System (MES) for the Production Activity Control of the production system [START_REF] Trentesaux | Benchmarking flexible job-shop scheduling and control systems[END_REF]. This approach is considered optimal as long as the modeling of the production system is realistic but also deterministic. In such an approach, parameters are simplified in order to fasten up the calculations. If stochastic changes of parameters are significant (e.g. duration of manual operations, breakdowns or failures), the execution of the schedule in the production system gives results that are generally far from optimal or even inapplicable [START_REF] Cardin | A study of the robustness of the group scheduling method using an emulation of a complex FMS[END_REF]. Traditionally, the production system is halted at the time when a disruption is detected during the execution of the scheduler then waits for a new schedule to be generated. If the rescheduling phase is long or if disruptions happen frequently, the duration of the rescheduling phase may lead to a drastic reduction of the overall performance. As a consequence, this approach, despite the fact that it has been widely used for several years in a number of industries, cannot be considered as sufficiently efficient nowadays since reactivity issues grow more and more important. Since few years, a research field, dealing with proactive scheduling has emerge. The main idea is to increase the robustness of the predictive schedule, and as a consequence, to limit the "nervousness" of the scheduling/rescheduling iterations, see for example [START_REF] Chaari | A genetic algorithm for robust hybrid flow shop scheduling[END_REF]. These technics typically use redundancy (temporal or resource-oriented), probabilistic methods, contingent methods, or objective functions that integrate robustness criteria evaluating the risk to not respect a candidate scheduling given possible perturbations. A growing activity from operation research has emerged in the last few years in that field [START_REF] Ghezail | Analysis of robustness in proactive scheduling: A graphical approach[END_REF].

Reactive approaches consider every event in real time, with no anticipation. Several approaches can be identified depending on the fact that they are centralized or distributed. When centralized, priority rules (e.g. heuristics-based) are defined and used on the fly, that is, whenever a decision must be taken. The choice of the rule to apply can also be decided dynamically. When distributed, control decisions are distributed among a set of cooperative control entities, being agents or holons, with no hierarchical relationships among them. Distributed approaches have been studied by researchers massively in the 90's, see for example [START_REF] Prabhu | Modelling and Analysis of Heterarchical Manufacturing Systems Using Discontinuous Differential Equations[END_REF], one of the historical reference in this field. These approaches are known to generate applicable solution since decisions are taker according to the real state of the production system. Despite this, they are also known to have their performances rapidly decreasing with time compared to pure predictive ones if no perturbation occurs.

Due to the limitations of these two historical approaches facing the current industrial needs, researchers are more and more considering a last kind of approach by trying to propose scheduling and control architectures and models that couple local reactive mechanisms implemented into agents/holons with global predictive mechanisms, being robust or not. In such predictive/reactive or proactive/reactive approaches (denoted hybrid approaches in the remaining of this paper), some of these control holons/agents are typically interfaced with the predictive scheduling system that provides them with an optimal or approximated scheduling and at the same time, interfaced with components like physical products or resources of the physical production system to control them [6] [7].

In such hybrid architectures, the fundamental decision facing perturbation is whether to still follow the predictive/proactive schedule (predictive mode) or not. If not, they may switch to a reactive mode where events and decision are handled in real time with the intention to switch back to a predictive mode as soon as possible. The main issue for researchers is then to provide accurate mechanisms to define the best switching dates (and/or the best switching decision-making levels) for control holons/agents so that they behave in a sense that the whole behavior of the hybrid architecture stay globally optimized despite disturbances. This predictive/proactivereactive coupling issue is not easy to solve: for example, if a broken machine can be repaired quickly, then it may not be necessary for its control holon/agent to switch in reactive mode if the pre-determined schedule will be still accurate because of some slack in the original schedule. Another issue is related to the possible nervousness of the architecture that may often switch from one mode to another [START_REF] Barbosa | Nervousness in Dynamic Self-organized Holonic Multi-agent Systems[END_REF].

From our point of view, this global issue can be broken down into the following scientific challenges. First, it is necessary to provide tools that enable the estimation of future performances, including disturbance detection, diagnosis and prognostic mechanisms (i.e. evaluation of the impact of a disturbance on the global performances). Second, based on these estimators, it is necessary to design efficient synchronization mechanisms, leading typically to the design of a proper indicator to determine if necessary when it is pertinent to switch back to predictive mode. Third, efficient switching strategies based on these synchronization mechanisms must be designed. These strategies must lead to a fair use of reactive modes (sufficiently to absorb uncertainties, but used as less as possible to avoid decreasing the performance). These strategies must be integrated into a control system considering balanced articulation between hierarchical mechanisms and heterarchical ones, while avoiding nervousness. The following sections propose a literature review structured following these three underlined challenges in the context of hybrid architectures.

2

First challenge: estimation of future performances

One fundamental reason explaining the lack of studies devoted to the predictive/proactive-reactive coupling issue in hybrid architectures, is related to the difficulty for researchers to design models enabling them to estimate future performances because of the difficulty to accurately observe the real state of the production system (eg., locate products and their state), and extrapolate possible evolution scenarios in the near future. This feature is mandatory in order to detect at which moment the control should switch from a predictive to a reactive mode. This detection might only be based on prediction models, split into two classes: analytic models, rapidly limited by the size of the considered systems because of their algorithmic complexity, and discrete-event simulation models, able to handle large systems but extremely timeconsuming. This last characteristic often limits their use in the context of real-time decision making.

To solve this issue, an observer able to detect abnormal behavior (difference between theoretical expected behavior and observed behavior -state reconstructor abili-ties) and to evaluate the impact of this difference on the global behavior of the system (diagnoser abilities) must be designed [START_REF] Zaytoon | Overview of fault diagnosis methods for Discrete Event Systems[END_REF]. For example, if it is obviously necessary to detect the delay in execution of a task from the predictive schedule, some of these delays might not be critical for the behavior of the system, either because they are very short, or thanks to the available free margin. Many modelling formalisms are classically used to build diagnosers, including automata [START_REF] Sampath | Diagnosability of discrete-event systems[END_REF] and their timed and probabilistic extensions, Petri nets [START_REF] Cabasino | Fault detection for discrete event systems using Petri nets with unobservable transitions[END_REF] [START_REF] Dotoli | Identification of the unobservable behaviour of industrial automation systems by Petri nets[END_REF], statecharts and hierarchical state machines [START_REF] Paoli | Diagnosability Analysis of a Class of Hierarchical State Machines[END_REF]. The most promising perspective here would be to implement the diagnoser using online simulation, which is an efficient but hard to implement forecasting tool. These programs are usually dedicated to the dimensioning phase (offline), but are increasingly used as actual systems control tools, included in the control loop (online) [START_REF] Cardin | Using online simulation in Holonic manufacturing systems[END_REF].

Second challenge: designing efficient synchronization mechanisms

The "switch down" mechanism consists in switching from a predictive mode where a predictive scheduling is to be executed by holons to a reactive mode where real-time holon decision may override these predictive scheduling decisions. This kind of switching is widely addressed in the literature (event-driven or thresholddriven switch),see for example [START_REF] Pach | ORCA-FMS: a dynamic architecture for the optimized and reactive control of flexible manufacturing scheduling[END_REF]. But a first issue appears: researchers do not really pay attention to the real need to "switch down" (e.g. if a machine breakdown is shortly repaired, then slack may be used to avoid real-time overriding decisions).

In addition, the "switch back" mechanism that concerns the way the predictive mode is reused after and instead of the reactive mode is rarely addressed or even mentioned. All these decisions must be taken according to global performances objectives targeted by the production manager. To correctly address this challenge, two questions relevant to synchronization have first to be answered to:

•

What are the most pertinent criteria to switch down or back? • How to reinsert these concerned in-progress products in the remaining of the material flow (switch down case), or how to synchronize the new re-optimized schedule with the state of the manufacturing system after this optimized schedule is obtained (switch back case)?

The first question is relative to performance indicators system leading to be able to estimate when it is pertinent to switch down or back according to the circumstances (the physical context: flexible manufacturing system, shop floor, constraints, management rules, etc…). It is obvious that objectives and performance indicators must be determined according to the industrial context and it seems difficult to design generic indicators useful (and applicable) to a particular system. They have probably to be design according to the physical context or at least according to an industrial system class.

Indicator system according to the class of physical context and built on a learning system might be pertinent. Multicriteria optimization based on Choquet integrals could, according to the estimators obtained thanks to the first challenge, lead to establish switching points according to measured drifts and situations. This approach is close to the one proposed by [START_REF] Chan | The development of intelligent decision support tools to aid the design of flexible manufacturing systems[END_REF]. The authors proposed an integrated approach for the automatic design of flexible manufacturing systems using simulation and multicriteria decision-making techniques. In this work, the selection of the most suitable design, based on a multi-criteria decision-making technique, the Analytic Hierarchy Process AHP, is employed to analyze the output from the flexible manufacturing system simulation models. Intelligent tools such as expert systems, fuzzy systems and neural networks, were developed to support the design process of the flexible manufacturing system.

E. Muhl proposed for the automotive industry a way to optimize, in a centralized way, the schedule of the car assembly line according to a unique performance indicator and the determination of the pertinent parameters which were periodically recalculated to assure the best synchronization between the real shop-floor state and the new schedule [START_REF] Muhl | Optimization of physical flows in an automotive manufacturing plant: some experiments and issues[END_REF]. Another way to design this indicator system could be found thanks to learning mechanisms as neural networks, fuzzy approaches or Choquet integrals usage [START_REF] Thomas | Multilayer perceptron for simulation models reduction: Application to a sawmill workshop[END_REF]. During his PhD work, C. Herrera first merged the two centralized and distributed approaches applied to a similar industrial case [START_REF] Thomas | Architecture de systèmes contrôlés par le produit pour un environnement de juste à temps[END_REF], [START_REF] Herrera | Cadre générique de planification logistique dans un contexte de décisions centralisées et distribuées[END_REF]. He proposed a multilevel parametric model to solve this re-scheduling problem. But the performance indicator leading to the switch decision has been taken as a hypothesis and the distributed decisions were limited to the first choice with a simple splitting decision to reinsert the remaining parts in the existing predictive schedule. Another research work focusing on the synchronization problem was done by H. El Haouzi [START_REF] Haouzi | Design and validation of a productdriven control system based on a six sigma methodology and discrete event simulation[END_REF]. She proposed an original architecture to control manufacturing flows on two assembly lines. In case of disturbances, products can arrive early or late at the synchronization point between the main assembly line and its feeders. The architecture was composed of an ERP and a distributed decision system. The on-line information was provided by Auto-ID technologies.

4

Third challenge: designing efficient switching strategies integrated into a hybrid control architecture Several European projects addressed the designing of distributed/hybrid control architectures into the so-called "smart factories". PABADIS and PABADIS PROMISE are amongst the firsts EU projects in that direction. More recently, let's mention GRACE, SMARTPRODUCT and ARUM projects 1 . The GRACE project is in line with the current need to build modular, intelligent and distributed manufacturing control systems and studied more precisely the impact of manufacturing operation on quality. The distributed control architecture is interfaced with a Manufacturing Execution System (MES). The SMARTPRODUCT project focused the work on the embedding of "proactive knowledge" into smart products. "Proactive" Smart products "talk", "guide", and "assist" designers, workers and consumers dealing with them. Some proactive knowledge will be co-constructed with the product, while other parts are gathered during the product lifecycle using embedded sensing and communication. Neither GRACE nor SMARTPRODUCT addressed the optimization of the control architecture, being hybrid or not. More recently, an interesting initiative, the ARUM project, aimed at designing a holonic multi-agent system combined with a service architecture designed to improve performance and scalability beyond the state of the art. The proposed solution integrates multiple layers of sensors, legacy systems and agent-based tools for beneficial services like learning, quality, and risk and cost management, including ecological footprints aspects.

In the scientific literature, there also exist different hybrid scheduling and control architectures. [START_REF] Pach | ORCA-FMS: a dynamic architecture for the optimized and reactive control of flexible manufacturing scheduling[END_REF] pointed out that the main idea is to take advantages of two basic structuration mechanisms: hierarchical (vertical relationships, toward centralization of information and decisions) and heterarchical (horizontal relationships, towards distribution of information and decisions) mechanisms. By doing this, it is expected to avoid their respective drawback (typically: lack of reactivity for hierarchies and myopia for heterarchies). Thus, usually, the hierarchical part of the architecture is responsible for the predictive and global optimization, while the heterarchical part allows reactivity and local optimization. Famous flagship hybrid architectures are PROSA [START_REF] Van Brussel | Reference architecture for holonic manufacturing systems: PROSA[END_REF], ADACOR [START_REF] Leitao | ADACOR: A Collaborative Production Automation and Control Architecture[END_REF] or D-MAS [START_REF] Verstraete | Engineering Manufacturing Control Systems Using PROSA and Delegate MAS[END_REF]. Such hybrid architectures are composed of cooperative decisional control entities, typically modeled as holons or agents. In [START_REF] Pach | ORCA-FMS: a dynamic architecture for the optimized and reactive control of flexible manufacturing scheduling[END_REF] was proposed an original literature review of several hybrid architectures. Indeed, there are numerous ways to combine the introduced structuration mechanisms. The authors identified two lines of study. The first "structure dynamics" concerns the possibility of the control structure to evolve with time (dynamic structure, e.g. full change from pure hierarchical to a pure heterarchical architecture) or not (static structure). The second "control homogeneity" deals with the way the control is applied: in the same way for every decisional agent/holon (homogeneous control) or not (heterogeneous control). In terms of structure dynamics and homogeneity control, hybrid architectures have thus been positioned according four sub-classes.

Non static hybrid architectures are very promising since they provide (self-) adaptation mechanisms needed to improve the agility of the control system [START_REF] Barbosa | Nervousness in Dynamic Self-organized Holonic Multi-agent Systems[END_REF]. In such architectures, an important mechanism of switching is responsible to switch holons/agents from/to predictive to/from reactive modes (for a holon/agent or a group of holons/agents). This mechanism adapts dynamically the structure of the control architecture to the production uncertainties in ensuring the performance. Of course, more generally, there may be different intermediary levels and mode between a fully predictive and a fully reactive mode. As a consequence, attention must be paid when designing and optimizing hybrid architectures. Some first ideas have been proposed [START_REF] Pach | ORCA-FMS: a dynamic architecture for the optimized and reactive control of flexible manufacturing scheduling[END_REF] , but they were provided aside the main topic of the referenced paper. Thus, this initial work was clearly insufficient and not really formalized in a generic and effective way. For example, the production order set was assumed to be provided as a whole, in a static manner, with no "on the fly" orders. Under that condition, the switching down was made only once, and with few attention paid to the real need to switch down. Moreover, the switch back was made only at the end of the production of the order set.

This challenge is complex to address and despite the growing number hybrid architectures proposed in the literature, the way prediction and reaction are coupled is neither optimized nor even clearly justified. This contributes clearly to a lack of applications of such contributions in real situations in industries despite the fact that they respond to a real industrial need. As an illustration, to the best of our knowledge, only P2000+ [START_REF] Bussmann | An agent-based approach to the control of flexible production systems[END_REF] was applied in Daimler but it failed because of issues related to the proposed research topic (and others issues, such as global cost).

Conclusion and perspectives

This paper depicted a state of the art of predictive-reactive control architectures of manufacturing systems. Even though these hybrid architectures show promising performances on academic examples, three main challenges are still to be investigated from the authors' perspective. Several leads are given to orient future research activities in this field, with the objective of making these concepts applicable on industrial shop floors in the next few years.

http://grace-project.org/, http://www.smartproducts-project.eu/ and http://arum-project.eu/