Stéphanie Jacquemont

François Jacquenet

Marc Sebban

Discovering Patterns in Flows: a Privacy Preserving Approach with the ACSM Prototype ⋆

In this demonstration, we aim to present the ACSM prototype that deals with the discovery of frequent patterns in the context of flow management problems. One important issue while working on such problems is to ensure the preservation of private data collected from the users. The approach presented here is based on the representation of flows in the form of probabilistic automata. Resorting to efficient algebraic techniques, the ACSM prototype is able to discover from those automata sequential patterns under constraints. Contrary to standard sequential pattern techniques that may be applied in such contexts, our prototype makes no use of individuals data.

Introduction

Sequential pattern mining has been a very active domain of research since the mid 90's (see [START_REF] Dong | Sequence Data Mining[END_REF] for an overview of the domain) and it has provided various powerful tools used in many applications. Nevertheless, those researches have mainly focused on increasing the efficiency of algorithms in terms of speed and space consumption. Parallel to this line of research, there has been a great interest in privacy preserving data mining techniques [START_REF]Privacy-Preserving Data Mining: Models and Algorithms[END_REF], but not directly dedicated to the specific domain of sequence mining, except the work of Zhan et al. [START_REF] Zhan | Privacy-preserving collaborative sequential pattern mining[END_REF], and more recently that of Kapoor et al. [4] or Kim et al. [START_REF] Kim | Privacy preserving data mining of sequential patterns for network traffic data[END_REF]. However, in those approaches, it is supposed that we have a set of sequences on which we can apply some specific techniques in order to do some kind of anonymization, directly or indirectly on the data to be processed.

We can note that there are many situations where the database of sequences results from the study of flows. For example, we may study the flow of cars in a town and get some set of paths used by car drivers on which the sequence mining task could be achieved. We may study the flow of visits on a particular web site and get some history files of sequences of pages visited by some users. We may study the flow of IP packets in some networks and get some set of routes used by IP packets sent and received by users. We may also study the flow of customers in a store and get some video recordings of customer behaviors in the store, etc. In such situations, anonymization techniques may be used on the data with more or less difficulties. However, this would result in a huge amount of pre-processing and moreover, and would require to collect non anonymous datasets that might be used by malicious agents before the end of the anonymization process. The approach we propose with the ACSM (Automata-based Constrained Sequence Mining) prototype is completely different. We assume that the underlying problem can be modeled in the form of a probabilistic automaton. This hypothesis is particularly verified in flow management problems such as those previously mentioned. In such a context, ACSM achieves a sequence mining task directly from the automaton using algebraic methods. Hence, it does not need to get the sequences of data making the flow but rather the structure of the flow itself (an automaton) and the values assigned to each path of the flow.

The ACSM prototype

Due to space limitation we cannot present the theoretical aspects behind ACSM. For that purpose, the reader may refer to [START_REF] Jacquemont | Mining probabilistic automata: A statistical view of sequential pattern mining[END_REF]. From an operational point of view, ACSM takes as input an XML file describing a flow graph. This file is loaded by ACSM that generates the corresponding probabilistic automaton while satisfying some statistical constraints.

ACSM can then be used to extract frequent sequential patterns given a support threshold. Our software has been implemented to satisfy statistical constraints in order to reduce the risk to extract false frequent patterns or to overlook true frequent ones. In this probabilistic framework, the user can then parameterize ACSM according to a priori fixed type I and type II errors. We dealt with the control of the false positive and false negative rates in [START_REF] Jacquemont | A lower bound on the sample size needed to perform a significant frequent pattern mining task[END_REF] by providing a lower bound of the sample size on which the sequence mining task has to be performed. This means that we can formally decide when we can stop observing the flow, pick up the XML file and begin to mine the data.

The probabilities on the edges and vertices of the automaton are used to calculate the frequency of each discovered sequence and ACSM returns the paths (sequences), made up of not necessarily consecutive edges, whose frequency is greater than the fixed support threshold. Efficient algebraic calculus are used to optimize the extraction process, based on the LU factorization.

Overview of the demonstration

To put forward the interest of ACSM, we show an application (called Traf-ficMiner) in the domain of car flow management that has been designed on top of it. Using ACSM to discover car flow patterns in a town, that is sequences of non necessarily consecutive streets frequently used by car drivers of that town, is a way to ensure privacy preservation of drivers' behavior. Indeed, until now, if we wanted to discover frequent routes in a town we had to install cameras in each street and then record the routes traversed by each driver. This solution is of course unrealistic because of its cost, but moreover it would be a great breach to drivers privacy and this would not be acceptable. The solution we propose consists in just counting, in each street of the part of the town we want to study, the number of cars using this street. The map of the town and the resulting counters provide a probabilistic automaton that can be processed by ACSM.

Figure 1 shows a screenshot of TrafficMiner discovering frequent routes in the city of Arlington (Virginia, USA). The demonstration shows the way we can design the automaton associated with the map using a specific interface and the way we can simulate some traffic on this map. Then, we look at the XML file describing the flows of cars focusing on each of its markup. Then, we run ACSM to get the frequent routes used by the virtual car drivers.At the demo, it will also be possible for users to provide specific flows that may have some interests for them. The input data will have to be provided in an XML form explained on site. Then running ACSM will provide patterns for the users.

Contribution of ACSM for the ML and DM communities

What makes our piece of software unique and special? To our knowledge, ACSM is the first prototype able to deal with flow management problems and to extract sequential patterns in datasets completely preserving the privacy of users involved in the flows.

What are the innovative aspects or in what way/area does it represent the state of the art? The innovative aspects of our system concerns privacy preservation issues. Contrary to standard sequential pattern mining algorithms, our prototype does not need any user dependent sequential data. No anonymization step is required because the sequential data are no more needed.

For whom is it most interesting/useful? Our prototype, and the ideas behind, may be very useful for the industrial practitioners. To give an example, we may consider webmasters who might want to mine visits of users without using classical history files that lead to many known problems due to proxies, caches, etc. Instead of a tedious pre-processing step on history files, they may easily proceed by putting counters on each page and link of their site and then use ACSM on the automaton built using the structure of the website and the probabilities assigned thanks to the counters.

Conclusion

We think this demonstration of the ACSM system proves the interest of processing automata instead of sequences in various situations. This is the case for example if privacy concerns are crucial issues of the applications. Interoperability with ACSM is very easy as it only requires an XML file as an input, which gives ACSM the ability to be easily encapsulated in some specific applications.

Fig. 1 .

 1 Fig. 1. TrafficMiner running on a map of Arlington.

⋆ This work has been supported in part by the french national research agency under the Bingo2 project


Michel Thiebaut
email: michel.thiebaut@gmail.com

Yaniv Schotten

Heidi Assaf

Johansen-Berg

Yaniv Assaf

Heidi Johansen-Berg

Schotten

The role of diffusion MRI in neuroscience

published or not. The documents may come   L'archive ouverte pluridisciplinaire

"We admire the contrivance of the fibre of every muscle, and ought still more to admire their disposition in the Brain, where an infinite number of them contained in a very small space, each execute their particular offices without confusion or disorder" [START_REF] Steno | Discour de Monsieur Stenon sur l'anatomie du cerveau[END_REF] .

As noted by Steno, post mortem dissections of the brain reveal an astonishing level of complexity, particularly within the white matter fibre pathways, composed of trillions of axons. As Isaac Newton intuitively suggested a few years later, these axons propagate electricity "along the solid filament of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles" [START_REF] Newton | Philosophiae Naturalis Principia Mathematica[END_REF] . Anatomical exploration of the white matter connections of the human brain then becomes a new challenge, mixing medical knowledge with advanced practical skills of dissection and illustration (Figure 1). These anatomical descriptions were a stepping-stone for the elaboration of new theories of brain function. This can be seen for example toward the end of the XIXth century, when Meynert extended Newton's vision even further, by introducing the possibility of reasoning occurring through the association of specialised areas of the brain via their white matter connections. The concept of associationism thus was born [START_REF] Meynert | A Clinical Treatise on Diseases of the Fore-brain Based Upon a Study of Its Structure, Functions, and Nutrition[END_REF] .

Empirical validation of associationism came soon after with the study of patients with brain lesions and well-defined symptoms. Such studies required patience, as individual brain anatomy could only be revealed after death. So research projects might well last a lifetime for some or remain unachieved for others. Nevertheless, studies of single cases or case series revealed the first white matter biomarkers for neurological syndromes such as conduction aphasia [START_REF] Wernicke | Der Aphasische Symptomencomplex. Ein psychologische Studie auf anatomischer Basis[END_REF] , transcortical motor aphasia, subcortical motor aphasia, transcortical sensory aphasia, subcortical sensory aphasia [START_REF] Lichtheim | On aphasia[END_REF] , alexia without agraphia [START_REF] Dejerine | Contribution a l'étude anatomo-pathologique et clinique des differentes variétés de cécité-verbale[END_REF] , bilateral and unilateral apraxia [START_REF] Liepmann | Lehrbuch der Nervenkrakheiten[END_REF] . These historical contributions demonstrated that the proper functioning of the brain can be disrupted even by distant lesions through disconnection and diaschisis mechanisms [START_REF] Monakow | [END_REF][START_REF] Monakow | Die Lokalisation im Grosshirn und der Abbau der Funktion durch kortikale Herde[END_REF] .

Fundamental advances in our understanding of brain function came from this work.

Nevertheless this static vision of the brain did not account for classical conditioning effects [START_REF] Pavlov | Conditioned Reflexes: An Investigation of the Physiological Activity of the Cerebral Cortex[END_REF] or memory [START_REF] Lashley | Studies of Cerebral Function in Learning. II. The Effects of Longcontinued Practice upon Localization[END_REF] , or diaschisis [START_REF] Price | Dynamic diaschisis: anatomically remote and context-sensitive human brain lesions[END_REF][START_REF] Finger | The Monakow concept of diaschisis: origins and perspectives[END_REF][START_REF] Carrera | Diaschisis: past, present, future[END_REF] phenomena that required notions of plasticity. Donald Hebb introduced this concept as follows "Let us assume that the persistence or repetition of a reverberatory activity (or "trace") tends to induce lasting cellular changes that add to its stability.… When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A's efficiency, as one of the cells firing B, is increased." With these words, Hebbian theory acknowledged the possibility for changes to occur in the brain, and provided a framework for conceptualising learning through the mechanisms of plasticity ). DWI measures the diffusion of water molecules along different directions. Given that axons are impermeable they constrain water diffusion to their main direction. Hence, DWI indirectly assesses white matter microstructure. Further by piecing together local estimates of water diffusion orientation, main bundles of axons can be reconstructed. This technique clearly boosted research because of its non-invasiveness. Anatomical studies performed in living human brains were able to refine the description of large fiber bundles [START_REF] Catani | Virtual in vivo interactive dissection of white matter fasciculi in the human brain[END_REF][START_REF] Catani | Perisylvian language networks of the human brain[END_REF] , allowing for replication in large numbers of participants [START_REF] Catani | A diffusion tensor imaging tractography atlas for virtual in vivo dissections[END_REF][START_REF] De Schotten | Visualization of disconnection syndromes in humans[END_REF] and enabling the discovery of inter-individual variability [START_REF] Ciccarelli | Diffusion-based tractography in neurological disorders: concepts, applications, and future developments[END_REF][START_REF] De Schotten | Atlasing location, asymmetry and inter-subject variability of white matter tracts in the human brain with MR diffusion tractography[END_REF] . Functional models could now benefit from whole brain connectomes for the first time [START_REF] Bullmore | Complex brain networks: graph theoretical analysis of structural and functional systems[END_REF] , identifying new disconnection syndromes [START_REF] De Schotten | Direct evidence for a parietalfrontal pathway subserving spatial awareness in humans[END_REF] and allowing for new divisions of the cortex based on axonal inputs and outputs or connectivity based parcellation [START_REF] Johansen-Berg | Changes in connectivity profiles define functionally distinct regions in human medial frontal cortex[END_REF][START_REF] Behrens | Non-invasive mapping of connections between human thalamus and cortex using diffusion imaging[END_REF] . Finally, the structural connectome of an individual could be assessed different time points allowing for the investigation of white matter plasticity for the first time [START_REF] Sagi | Learning in the fast lane: new insights into neuroplasticity[END_REF] .

Hence, in this review of the role of DWI in neuroscience we will first describe the anatomical advances provided by this method and will then survey its importance in the establishment of new brain models. Finally, we will review studies that investigated white matter plasticity. Hopefully, the ideas expressed in this review will provide a comprehensive understanding of the scientific question investigated with DWI and encourage new generations to pursue exploit this powerful technology to gain novel insights into human brain wiring.

Part 1/ Anatomy

Diffusion MRI provides different types of information about human brain anatomy. Local (voxel-wise) measures of diffusion properties can give insight into local white matter microstructure, whereas exploiting these measures to perform in-vivo tractography can provide information on the organisation of white matter at a systems level.

Local measures of white matter microstructure

Voxel-wise measures of diffusion properties, such as fractional anisotropy or mean diffusivity, are modulated by local tissue microstructure [START_REF] Beaulieu | The biological basis of diffusion anisotropy[END_REF] . For example, studies in model systems show that fractional anisotropy increases with increasing packing density or decreasing axon diameter [START_REF] Takahashi | Magnetic resonance microimaging of intraaxonal water diffusion in live excised lamprey spinal cord[END_REF] , and decreases with reductions in myelin [START_REF] Gulani | Apparent diffusion tensor measurements in myelin-deficient rat spinal cords[END_REF] . However, although empirical or simulation studies can clearly demonstrate that varying these tissue properties modulates diffusion parameters, we are still faced with an inverse problem when trying to interpret an observed difference in diffusion parameters (e.g. reduced fractional anisotropy), as there is not a one to one relationship between a given diffusion parameter and the underlying tissue structure [START_REF] Zatorre | Plasticity in gray and white: neuroimaging changes in brain structure during learning[END_REF][START_REF] Basser | Historical perspectives and future outlook of diffusion MRI[END_REF][START_REF] Alexander | Imaging brain microstructure with diffusion MRI: Practicality and applications[END_REF][START_REF] Caspers | Decoding the microstructural correlate of diffusion MRI[END_REF][START_REF] Jeurissen | Diffusion MRI Fiber tractography of the brain[END_REF] . Nevertheless, diffusion parameters provide useful noninvasive measures of local tissue microstructure, with sensitivity to features including membrane integrity, myelin thickness, axon diameter and packing density. These physical characteristics of the white matter fibre bundle will have consequences for the physiological functioning of that bundle, affecting properties such as conduction time, refractory time, probability of transmission or even synchronisation of signals across a distributed corticocortical network. Variations in these physiological properties may in turn be expected to give rise to variation in behavioural outputs [START_REF] Zatorre | Plasticity in gray and white: neuroimaging changes in brain structure during learning[END_REF][START_REF] Kanai | The structural basis of inter-individual differences in human behaviour and cognition[END_REF][START_REF] Johansen-Berg | Behavioural relevance of variation in white matter microstructure[END_REF] .

For instance, left and right hemispheres show variation in function as well as in anatomy.

Diffusion parameters' comparison between the two hemispheres indicated increased fractional anisotropy in the left hemisphere in the external capsule [START_REF] Kubicki | Uncinate fasciculus findings in schizophrenia: a magnetic resonance diffusion tensor imaging study[END_REF] , cingulum bundle [START_REF] Kubicki | Cingulate fasciculus integrity disruption in schizophrenia: a magnetic resonance diffusion tensor imaging study[END_REF] and perisylvian white matter [START_REF] Park | White matter hemisphere asymmetries in healthy subjects and in schizophrenia: a diffusion tensor MRI study[END_REF] thought to support left hemispheric dominance for language functions [START_REF] Piervincenzi | Multimodal assessment of hemispheric lateralization for language and its relevance for behavior[END_REF][START_REF] Budisavljevic | Age-Related Differences and Heritability of the Perisylvian Language Networks[END_REF] . Right asymmetries were also identified in the dorsal fronto-parietal white matter [START_REF] Park | White matter hemisphere asymmetries in healthy subjects and in schizophrenia: a diffusion tensor MRI study[END_REF] that could be related to the right hemispheric dominance for spatial processing of information. However, the high variability between subject and the lack of behavioural measures in these preliminary studies did not allow a solid relationship to be drawn between voxelwise measure of interhemispheric differences and functional dominance. Particularly, the relationship between language lateralisation and structural connectivity measures in the language system remains intricate [START_REF] Piervincenzi | Multimodal assessment of hemispheric lateralization for language and its relevance for behavior[END_REF][START_REF] Lopez-Barroso | Word learning is mediated by the left arcuate fasciculus[END_REF] .

Further studies have shown that individual differences in measures of local white matter microstructure correlate with variations in physiological properties of the fiber pathways [START_REF] Wahl | Human motor corpus callosum: topography, somatotopy, and link between microstructure and function[END_REF][START_REF] Kloppel | The cortical motor threshold reflects microstructural properties of cerebral white matter[END_REF][START_REF] Whitford | Predicting inter-hemispheric transfer time from the diffusion properties of the corpus callosum in healthy individuals and schizophrenia patients: a combined ERP and DTI study[END_REF][START_REF] Stufflebeam | A non-invasive method to relate the timing of neural activity to white matter microstructural integrity[END_REF][START_REF] Roberts | Maturational differences in thalamocortical white matter microstructure and auditory evoked response latencies in autism spectrum disorders[END_REF][START_REF] Vandermosten | White matter lateralization and interhemispheric coherence to auditory modulations in normal reading and dyslexic adults[END_REF][START_REF] O'donnell | Advances in computational and statistical diffusion MRI[END_REF] .

For example, paired-pulse transcranial magnetic stimulation can be used to probe the functional connectivity of a cortico-cortical connection. A conditioning pulse applied to dorsal premotor cortex of one hemisphere will modulate the excitability of the primary motor cortex in the other hemisphere; the degree of modulation can be used to estimate functional connectivity between the two cortical areas. Individual differences in this measure were found to correlate with variation in white matter fractional anisotropy (FA), such that individuals with stronger functional connectivity had higher FA in callosal or subcortical white matter pathways between premotor and motor cortex [START_REF] Boorman | Individual differences in white-matter microstructure reflect variation in functional connectivity during choice[END_REF] . This supports the idea that variation in white matter microstructure is associated with variation in the physiological properties of those fiber bundles.

Advanced diffusion weighting imaging analyses provide estimates of the axon diameter distribution or fibre composition along with other physical properties, such as the intra-axonal resistance, membrane resistance and capacitance, etc, help determine many important functional properties of nerves, such as their conduction velocity or rate of information transfer [START_REF] Hursh | The properties of growing nerve fibers[END_REF][START_REF] Ritchie | On the relation between fibre diameter and conduction velocity in myelinated nerve fibres[END_REF][START_REF] Waxman | Determinants of conduction velocity in myelinated nerve fibers[END_REF][START_REF] Waxman | The axon : structure, function, and pathophysiology[END_REF][START_REF] Tasaki | On the Relation Between the Conduction-Rate, the Fiber-Diameter and the Internodal Distance of the Medullated Nerve Fiber[END_REF] . Neural pathways that are characterized by fast reaction times (e.g. within the motor system) will also exhibit higher percentage of large-diameter axons. On the other hand, pathways that show slow conduction velocity will also exhibit a larger population of smaller axons or even non-myelinated axons. Therefore, it is only logical to assume that the properties of axon diameter distribution will have critical role on healthy central nervous system functioning and will be dramatically affected in abnormal conditions and disease.

Despite the fundamental new insights it provides, diffusion weighted imaging has several important limitations. The key limitation is implicit in the diffusion tensor model (DTI, [START_REF] Basser | Inferring microstructural features and the physiological state of tissues from diffusion-weighted images[END_REF] ). This assumption oversimplifies the true motion of water within brain tissue. The practical consequence of this over-simplicity of the diffusion tensor model is that the common DTI indices, mean diffusivity and FA, are non-specific to any particular tissue compartment.

Features of microstructure, such as cell size, density, permeability and orientation distribution, all affect DTI indices and changes in the indices are impossible to associate with more specific changes in microstructural features. Recent trends aim to use more sophisticated models of diffusion in order to measure microstructural features directly [START_REF] Assaf | Composite hindered and restricted model of diffusion (CHARMED) MR imaging of the human brain[END_REF][START_REF] Cohen | High b-value q-space analyzed diffusion-weighted MRS and MRI in neuronal tissues -a technical review[END_REF] .

One such approach for modelling the diffusion signal (instead of DTI) is to devise a mathematical formalism that is guided by tissue geometry. Yet, anyone who has looked down a microscope at a brain section would have been impressed by the complexity of the white matter geometry [START_REF] Caspers | Decoding the microstructural correlate of diffusion MRI[END_REF][START_REF] Roebroeck | Ex vivo diffusion MRI of the brain: Technical challenges and recent advances[END_REF] . Cells of different shapes and sizes with processes spreading at different scales and orientations depict a mess of geometries at the micron-scale. Yet even at a lower-level of magnification, where cell layers and fibre bundles are visible, complexity arises where fibres cross, disperse and fan within hundreds of microns.

Pioneering work by Stanisz and collaborators attempted to reduce the complexity of neural tissue modelling by introducing a two-compartment geometrical framework that includes diffusion within spheres and cylinders [START_REF] Stanisz | An analytical model of restricted diffusion in bovine optic nerve[END_REF] . Further works have suggested that diffusion within the confined boundaries of cells and axons can be regarded as restricted diffusion. Under this approach it is hypothesized that the geometry of the tissue affects the diffusivity of water molecules. For example, it is reasonable to assume that water diffusion within the axon will be restricted by the myelin membrane while elsewhere it will be only hindered or free. This is the basis for the composite hindered and restricted model of diffusion (CHARMED) [START_REF] Assaf | Composite hindered and restricted model of diffusion (CHARMED) MR imaging of the human brain[END_REF][START_REF] Assaf | New modeling and experimental framework to characterize hindered and restricted water diffusion in brain white matter[END_REF] . In CHARMED, restricted diffusion in the intra-axonal space is modelled as diffusion within impermeable cylinders [START_REF] Assaf | New modeling and experimental framework to characterize hindered and restricted water diffusion in brain white matter[END_REF][START_REF] Neuman | SPIN-ECHO OF SPINS DIFFUSING IN A BOUNDED MEDIUM[END_REF][START_REF] Codd | Spin Echo Analysis of Restricted Diffusion under Generalized Gradient Waveforms: Planar, Cylindrical, and Spherical Pores with Wall Relaxivity[END_REF] . As a consequence, the CHARMED model allows enhanced characterization of the axonal water compartment. One of the advantages of CHARMED is that it is possible to reconstruct the 3D displacement distribution function for each of the components (hindered and restricted). Thus, it is possible to extract physically meaningful parameters for each of the diffusing components; these parameters include the diffusivity of the extra-axonal matrix (diffusivity of the hindered part), the axonal density (the volume fraction of the restricted part), and the fibres directions (the orientation density function of the restricted part). While the CHARMED model provides a conceptual framework to separate different modes of diffusion and relate them to tissue compartments, it allows modelling of 2-3 different fibre orientations with accuracy limited by the number of measured directions [START_REF] Assaf | Composite hindered and restricted model of diffusion (CHARMED) MR imaging of the human brain[END_REF] . Subsequent works tried to expand the CHARMED model to include fibre dispersion and fanning within a voxel and increase the accuracy and orientation estimation of the model [START_REF] Jespersen | Neurite density from magnetic resonance diffusion measurements at ultrahigh field: comparison with light microscopy and electron microscopy[END_REF][START_REF] Jespersen | Modeling dendrite density from magnetic resonance diffusion measurements[END_REF][START_REF] Zhang | NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain[END_REF][START_REF] Santis | Improved precision in CHARMED assessment of white matter through sampling scheme optimization and model parsimony testing[END_REF][START_REF] Panagiotaki | Compartment models of the diffusion MR signal in brain white matter: a taxonomy and comparison[END_REF] . Noteworthy is the NODDI framework which expands CHARMED to model also dispersion of fibre orientation but with optimized acquisition and analysis pipelines [START_REF] Zhang | NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain[END_REF] .

Neural tissue is complex, and it should be realized that there will not be a single model that can capture all the details and complexity of different neural compartments [START_REF] Roebroeck | Ex vivo diffusion MRI of the brain: Technical challenges and recent advances[END_REF][START_REF] Novikov | Quantifying brain microstructure with diffusion MRI: Theory and parameter estimation[END_REF] . The different abovementioned models (and others) try to separate different features of interest of the tissue. Obviously including too many free parameters in the model will lead to computational problems such as over fitting and other optimization and data fitting issues.

Thus, the optimal model for diffusion imaging depends on the research question. For example, an important feature of white matter that is not modelled in CHARMED or NODDI is axon diameter [START_REF] Sepehrband | Parametric Probability Distribution Functions for Axon Diameters of Corpus Callosum[END_REF][START_REF] Benjamini | White matter microstructure from nonparametric axon diameter distribution mapping[END_REF][START_REF] Drobnjak | OGSE, and sensitivity to axon diameter in diffusion MRI: Insight from a simulation study[END_REF] . Being an important feature of brain connectivity with implications for conduction velocity and information transfer efficiency within the brain [START_REF] Liewald | Distribution of axon diameters in cortical white matter: an electron-microscopic study on three human brains and a macaque[END_REF][START_REF] De Schotten | A lateralized brain network for visuospatial attention[END_REF] , the in-vivo measurement of axonal diameter became another modelling challenge. In an extension of the CHARMED framework, called AxCaliber, the estimation of the axon diameter distribution became also feasible [START_REF] Assaf | AxCaliber -A method to measure the axon diameter distribution and density in neuronal tissues[END_REF][START_REF] Assaf | AxCaliber: a method for measuring axon diameter distribution from diffusion MRI[END_REF] . The idea behind AxCaliber is that each axon size will experience restricted diffusion at a different diffusion time. For example, an axon with a diameter of 1um will experience restricted diffusion already at very short diffusion times while a larger axon will experience restricted diffusion only when the diffusion time is increased. By acquiring a multi-diffusion time CHARMED data set it is possible to accurately estimate the axon diameter distribution function with the AxCaliber framework. In AxCaliber, unlike CHARMED, water diffusion is measured exactly perpendicular to the long axis of the fibers (this is a prerequisite of the model). It was found that by using a gamma function (with only 2 free parameters) it is possible to adequately estimate the axon diameter distribution function by AxCaliber [START_REF] Assaf | AxCaliber: a method for measuring axon diameter distribution from diffusion MRI[END_REF] . The AxCaliber framework was verified on excised samples of optic and sciatic nerves.

These two nerve samples have very different axon diameter distribution functions (Figure 2).

The AxCaliber diameter distribution functions were in good agreement with the histological ones (Figure 2) [START_REF] Assaf | AxCaliber: a method for measuring axon diameter distribution from diffusion MRI[END_REF] . In addition, AxCaliber was implemented to study the morphology of the corpus callosum (in-vivo). AxCaliber analysis was performed on a voxel-by-voxel basis providing for each the axon diameter distribution function for each voxel. These distributions were used as an input to a clustering algorithm in order to visualize regions with significantly different axon diameter distribution (ADD). The clustering based on the ADD was able to segment the corpus callosum into several regions that fit known morphological zones of those samples 80 (Figure 3). Recently the axon diameter distribution was demonstrated in the human brain and also was shown to correlate with conduction velocity measures of interhemispheric transmission time [START_REF] Horowitz | In vivo correlation between axon diameter and conduction velocity in the human brain[END_REF] . Hence correlations between in vivo measures of axonal diameters and reaction time measures during cognitive paradigms will be of particular interest in near future.

It should be noted that the original AxCaliber framework suffers from several inherent limitations. The first is the need to measure the diffusion properties exactly perpendicular to the fibre orientation. This limited the application of the method to the corpus callosum. In a further development of the CHARMED and NODDI frameworks it was suggested that the mean axon diameter could be directly measured for any orientated fibre system [START_REF] Zhang | Axon diameter mapping in the presence of orientation dispersion with diffusion MRI[END_REF] . The ActiveAx acquisition and analysis pipeline, although not measuring the entire axonal distribution probability function, provides an approach to estimate the mean axon diameter for the entire brain. Another limitation to measuring axonal diameter properties with diffusion imaging is the need to have high diffusion gradient amplitude in order to have comparable accuracy over a wide spread of axon diameters. It was shown, theoretically, that small axons (<1micron) will be inaccurately estimated with current gradient technology and will bias the overall measured axon diameter [START_REF] Dyrby | Contrast and stability of the axon diameter index from microstructure imaging with diffusion MRI[END_REF][START_REF] Dyrby | Dependence of axon diameter index on maximum gradient strength[END_REF] . This measurement artefact is significantly minimized with new gradient technology providing high amplitudes of diffusion gradients (300mT/m) [START_REF] Huang | The impact of gradient strength on in vivo diffusion MRI estimates of axon diameter[END_REF] .

Yet, other works suggested that the modelling algorithm can be further developed to cope with the abovementioned limitations by introducing additional features such as restricted diffusion within the extracellular space [START_REF] Fieremans | In vivo observation and biophysical interpretation of time-dependent diffusion in human white matter[END_REF][START_REF] Novikov | Revealing mesoscopic structural universality with diffusion[END_REF][START_REF] Santis | Including diffusion time dependence in the extra-axonal space improves in vivo estimates of axonal diameter and density in human white matter[END_REF] . Yet, despite this limitation, the measurement of axonal properties provides an additional micro-structural feature that can better characterize brain connections and connectivity.

Tractography to estimate long-range connectivity

As mentioned above, local estimates of dominant diffusion directions can be followed to reconstruct estimates of fiber pathways. There are many different methodological approaches to performing diffusion tractography, as reviewed elsewhere in this special issue [START_REF] Jeurissen | Diffusion MRI Fiber tractography of the brain[END_REF][START_REF] Dell'acqua | Reconstructing fiber orientations with diffusion MRI[END_REF] . From the neuroanatomical perspective, tractography allows us to estimate the organisation of major fibre pathways in the human brain (Figure 4) and to explore variation in the strength and organisation of these pathways between individuals or over time. Early applications of diffusion tractography were able to provide the first 'in vivo dissections' and mapping of major white matter fiber bundles in the human brain with an increasing precision [START_REF] Catani | Virtual in vivo interactive dissection of white matter fasciculi in the human brain[END_REF][START_REF] Catani | A diffusion tensor imaging tractography atlas for virtual in vivo dissections[END_REF][START_REF] Mori | In vivo visualization of human neural pathways by magnetic resonance imaging[END_REF][START_REF] Conturo | Tracking neuronal fiber pathways in the living human brain[END_REF][START_REF] Jones | Non-invasive assessment of axonal fiber connectivity in the human brain via diffusion tensor MRI[END_REF][START_REF] Mori | Imaging cortical association tracts in the human brain using diffusion-tensor-based axonal tracking[END_REF][START_REF] Lawes | Atlas-based segmentation of white matter tracts of the human brain using diffusion tensor tractography and comparison with classical dissection[END_REF][START_REF] Oishi | Human brain white matter atlas: identification and assignment of common anatomical structures in superficial white matter[END_REF][START_REF] Mori | Stereotaxic white matter atlas based on diffusion tensor imaging in an ICBM template[END_REF][START_REF] Catani | Short frontal lobe connections of the human brain[END_REF][START_REF] Takemura | A Major Human White Matter Pathway Between Dorsal and Ventral Visual Cortex[END_REF][100] and led to the publication of several atlases [101][102][103] .

The use of tractography for anatomical studies has mostly depicted associative tracts (i.e. cortico-cortical connection) as they were defined in the early XIXth century (Figure 5a).

Handling of tractography was indeed much easier than standard post mortem white matter dissections and led to the discovery of new associative tracts which were consequently replicated using standard post-mortem Klingler dissections 104,105 . In that sense tractography has boosted our anatomical knowledge of white matter pathways by providing an easier access to white matter connections 106,107 . For instance, much evidences from tractography, cross validated with post-mortem dissections, now converges toward a model of the arcuate fasciculus splitted into three branches (temporo-parietal, fronto-parietal and posterior parietotemporal, Figure 5b). However, caution is required to be taken as many limitations in tractography may mislead neuroanatomists by piecing together different tracts segments into large single bundles [START_REF] Jeurissen | Diffusion MRI Fiber tractography of the brain[END_REF][START_REF] Dell'acqua | Reconstructing fiber orientations with diffusion MRI[END_REF] . This is particularly the case for projection (i.e. subcortico-cortical) and commisural pathways (i.e. interhemispheric), which have been more challenging to dissect due to severe limitations in the tractography of complex white matter configurations (i.e crossing, fanning and kissing configuration) (Figure 5c). Although crossing has been the focus of interest of much work for the past 5 years, with very promising results particularly evident with the use of spherical deconvolution tractography [108][109][110][111][112][113] (Figure 5d), fanning and kissing configurations still remain a clear challenge in this field of research. The fanning limitation is characterised by an underestimation of the fibre count reaching an area and the kissing limitation by the misidentification of the correct group of fibres to track. Together, these two problems hamper progress in the depiction of the projection and commissural pathways 114- 116 .

Tractography to divide grey matter into subregions

At a cortical level, SMA and pre SMA, two areas working hierarchically in the control of action 118 have a different pattern of connection with the rest of the brain. Exploiting this difference, connectivity information gained from diffusion tractography can be used to define the boundary between these two regions based purely on connectivity, driven by the fact that SMA shows a closer relationship with primary motor cortex whereas pre-SMA is more connected to the prefrontal cortex [START_REF] Johansen-Berg | Changes in connectivity profiles define functionally distinct regions in human medial frontal cortex[END_REF] . Indeed, connectivity-based parcellation has been used to define subdivisions across a range of different human cortical areas [119][120][121][122][123][124][125][126] , as well as to demonstrate homologies between human and monkey cortex [127][128][129][130][131][132][133] . Recently, diffusion tractography was used along with other multimodal structural and functional MRI data to generate a novel whole brain cortical parcellation using data from the Human Connectome Project (Figure 7a, 126 ).

Alongside enthusiasm for the newfound ability to visualise white matter bundles in living human brains, have been notes of caution to keep in mind that tractography is not the same as invasive tract tracing. 'Gold standard' invasive tract tracing typically involves injecting a tracer into an area of interest, allowing it to be taken up by cells, and then observing it in interconnected regions after it has been transported there via anterograde and/or retrograde transport mechanisms. As such, invasive tract tracing involves tracing axonal pathways. By contrast, in diffusion tractography we are typically following estimates of the path of least resistance to diffusion. In many cases, this corresponds roughly to the path of the dominant fibre orientation, but in other cases it does not. There is therefore awareness that, as with any method in biology, diffusion tractography is susceptible to false negatives (inability to track pathways that we know exist) and false positives (tracking spurious pathways). Much valuable work has been done to try to validate diffusion tractography, giving us a clearer idea of the limitations and strengths of the method [134][135][136][137][138][139][140][141] . This is challenging work, requiring development of novel phantoms with realistic properties 115,142 , or meticulous reconstruction of fibre pathways from brain sections taken from tract tracing experiments 143,144 . Such work is encouraging but there is more to be done to allow neuroscientists to be able to interpret diffusion tractography results with confidence 145 .

Part 2/ Functional models

Functional models of the brain originally emerged from the study of brain damaged patients 146 . The localisation of the lesion in the brain was indeed interpreted as the core origin of the functional impairment 147 . However, the localisation of the lesion appeared insufficient to explain the existence of patients with different lesions and similar symptoms.

The idea emerged that the communication between these regions was impaired through a mechanism of white matter disconnection. Traditional model of disconnection syndromes includes conduction aphasia [START_REF] Wernicke | Der Aphasische Symptomencomplex. Ein psychologische Studie auf anatomischer Basis[END_REF] , transcortical motor aphasia, subcortical motor aphasia, transcortical sensory aphasia, subcortical sensory aphasia [START_REF] Lichtheim | On aphasia[END_REF] , alexia without agraphia 8 , bilateral and unilateral apraxia [START_REF] Liepmann | Lehrbuch der Nervenkrakheiten[END_REF] . With the advent brain imaging the neuroimaging community focused on voxels independently [148][149][150] without any attempts to capture correlation across them 151,152 , effectively masking disconnection syndromes 153,154 . The access to visualisation and mapping of white matter connections instead allowed for the investigation of white matter connections in patients and led to the discovery of new disconnection syndromes. For instance, visuospatial neglect, a severe neurological condition characterised by the loss of half of the visual field's awareness, is associated with the disruption of the fibres connecting the frontal to the parietal lobe [START_REF] De Schotten | Direct evidence for a parietalfrontal pathway subserving spatial awareness in humans[END_REF]153,155 . Tractography also contributed to the establishment or confirmation of a new disconnection hypothesis for Gerstmann syndrome 156 , and neurodevelopmental hypoconnectivity hypotheses for psychopathy 157 , dyslexia 158 as well as congenital prosopagnosia 159 . Conversely, aberrant/increased connections (i.e. hyperconnection) hypotheses have recently been suggested as a potential mechanisms for inattention in attention deficit hyperactive disorders 160 , auditory [161][162][163] and visual hallucinations 164 in schizophrenia. All these results demonstrate the utmost importance of brain connection to the proper functioning of the brain. Indeed brain areas deprived of their inputs (i.e. afferent connections) or outputs (i.e. efferent connections) will no longer be able to contribute anymore to the elaboration of the cognition and the behaviour.

With regards to effects on behaviour, studies have shown that individual differences in behavioural performance for a given task correlate with variation in white matter microstructure of task-relevant pathways, even in young healthy populations 39,165 38 . For example, as summarised in Figure 7, variation in bimanual co-ordination skill correlates with variation in fractional anisotropy in the body of the corpus callosum, a white matter area that contains transcallosal pathways between primary and supplementary motor areas 166 .

Similar effects have been found in a wide range of sensory, motor and cognitive domains including for example vision [167][168][169] , audition 170 , motor skills [171][172][173][174][175] and language 176,177 , literacy [178][179][180] , emotion 181 and motivation 182 , visuospatial 183 , memory 184 and executive function 175,185,186 and individual characteristics such as creativity 187 , musical skills 188 and personality 189,190 . Such evidences suggest that the function of a brain area can be defined by its afferent connections from and efferent connection to other areas [192][193][194] . As reported in the previous section, brain regions can be characterised by their structural connectivity with the rest of the brain. Therefore, boundaries based on structural connectivity might well give a functional parcellation of the brain. Preliminary evidence starts to abound in this direction.

For example, delineating subregions of human thalamus is important to be able to study the function of these regions in vivo and to be able to target specific nuclei for surgical interventions such as deep brain stimulation for movement disorders. However, divisions between thalamic nuclei are not visible on standard structural MRI. Using diffusion tractography, thalamic subregions can be reliably identified based on cortical connectivity patterns [START_REF] Behrens | Non-invasive mapping of connections between human thalamus and cortex using diffusion imaging[END_REF] . These connectivity-based divisions have been shown to have functional relevance 195,196 , and could be useful for individualised neurosurgical planning 197 . Recently, tractography revealed a division of the frontal lobes in 12 areas characterised by their connection with the rest of the brain and a clear-cut functional specificity (Figure 5b) 126 .

Hence model of structural connectivity assessed with tractography seems to capture the functional organisation of the brain.

However, the past two decades have been marked by the discovery of large interindividual variability in brain structure and function (i.e. different phenotype), creating additional challenges in furthering the understanding of the brain and a lack of models to explain its functioning [198][199][200][201] . Studies have begun to explore and describe these pathways across large numbers of individuals [START_REF] De Schotten | Atlasing location, asymmetry and inter-subject variability of white matter tracts in the human brain with MR diffusion tractography[END_REF]116,202 . Just as studies have found that individual differences in behaviour correlate with variation in local white matter microstructure, so too have tractography studies found that variation in the 'strength' or organization of white matter fibres relate to variations in behaviour. For instance, language areas and their interconnections are predominantly represented in the left hemisphere for most but not all healthy participants 203,204 . However, individuals vary in the degree of lateralisation and people with more symmetric patterns of connection have been shown to perform better at episodic memory tasks 203 . These differences are important consideration for the field of clinical neuroscience. Forkel et al. 205 demonstrated that the phenotype of the structural network supporting language (i.e. arcuate fasciculus) may interact with recovery after a stroke in the left hemisphere. Additionally, Lunven et al. 206 reported that the strength of interhemispheric communication is important for the recovery of visuospatial neglect after a stroke in the right hemisphere (see 207,208 for extensive reviews on the role of diffusion MRI in clinical neurosciences). Hence the study of inter-individual variability is a new line of research directly accessible in large scale projects such as the Human Connectome Project (https://www.humanconnectome.org), where high quality diffusion MRI data have been acquired in 1200 subjects and openly shared with the community 209 . Using diffusion weighted imaging tractography to characterise the functional specialisation of brain areas at the individual level may therefore benefit the domain of inter-individual variability, providing a more tailored brain model when doing group comparisons eventually making methods like spatial registration and smoothing obsolete.

Part 3/ Measures of Plasticity

Until the mid '90s, research on white matter was limited to histological investigations of its anatomy, for two reasons: 1) There was no in-vivo and non-invasive imaging method that could provide quantitative measures of the white matter; 2) it was thought that in the fully developed and healthy brain, the micro-structure of white matter should be roughly stable and fixed. Nevertheless, recent studies suggest that the white matter can undergo changes following training or cognitive experience. Such studies coined the term white matter plasticity.

Neuroplasticity, the functional and structural re-organization capacity of the brain may occur at several levels (from molecular to regional changes) and time-scales (seconds 210,211 to years 212 ). Obviously, the definition of neuroplasticity and its investigation are centred around neurons and less on white matter. Nevertheless, the scope of neuroplasticity is wide, ranging from short-term changes at the synapse through long-term potentiation (LTP) and depression (LTD) mechanisms through long-lasting new neuronal connections (dendritic trees) [213][214][215][216][217][218] . Pioneering studies revealed that new white matter projections can be formed following extensive learning procedures 219,220 . These studies, performed on post-mortem monkey brains using tracer methods revealed that the brain can rewire itself concomitantly to cortical neuroplasticity 221 . Recent works utilized diffusion tensor imaging to reveal white matter changes following long term training (weeks to months) 180,188,222 . These studies suggested that diffusion imaging can be sensitive to white matter plasticity. Recent evidences showed that following only a 5-day water maze task, in addition to significant cortical plasticity, white matter plasticity in the corpus callosum could be detected with DTI 223 . Histology showed that this plasticity, characterized by increase in FA, is also manifested by elevated levels of myelin (Figure 6). In addition, a recent molecular imaging study suggests that already following action potentials, signalling mechanisms control oligodendrocyte myelin formation 224 . This experiment was conducted on cultured neural tissue. Indeed, a recent study showed that 2 hours of spatial navigation task are sufficient to cause diffusion changes in the fornix.

All these studies on plasticity are not exempt of limitation 225 and future research is needed in order to strengthen that field.

These experiments, although performed on different species and samples (human, rodents and tissue culture) suggest that the white matter is a dynamic tissue. Yet, what exactly happens in the white matter is not fully understood and the temporal evolution of this phenomenon is uncharted. Specifically, a significant impact on neuroscience will be the demonstration of white matter plasticity in the human brain following short episodes (minutes/hours) of training. We anticipate that short-term white matter plasticity will be stimulate different mechanism from long term plasticity, however, DTI might be not sensitive and specific enough to demonstrate that difference. More specific methods such as CHARMED and AxCaliber could provide the enhanced sensitivity and specificity needed for comprehensive characterization of this phenomenon. Nevertheless, once white matter plasticity mechanisms can be resolved with MRI, these kinds of experiments could shed a new light on brain physiology and provide a whole new definition to the concept of brain connectivity.

Conclusion

Over the past 20 years, diffusion weighted imaging has proven to be an indispensable approach for the study of white matter. It has provided invaluable insight into neuroanatomy, which has led to the discovery of new pathways, and refinement of what we already knew.

Notably, diffusion weighted imaging has demonstrated the existence of a clear organisation and a true inter-individual variability in the way that the brain is connected, and that this variability directly impacts the functioning of the brain. Finally, diffusion imaging offered the unique opportunity to directly quantify brain plasticity. However, while reducing the gap between direct biological and indirect DWI quantification of brain tissues continues to be a challenge, it is a necessity for future generations who may wish to exploit this powerful technology to gain novel insights.

Figure 1 :

 1 Figure 1: First drawings of white matter connection in the human brain. Left, brain dissection from Vieussens 3 and right, brain dissection from Johann Christian Reil 4 .

17 .

 17 However, measures of plasticity were quite challenging, as they required the anatomical investigation of living samples with very invasive methods. Decades later, magnetic resonance imaging (MRI) brought a new window on human brain anatomy. In 1985, a breakthrough came with the emergence diffusion-weighted MRI (DWI, 18

Figure 2 :

 2 Figure 2: AxCaliber of porcine optic and sciatic nerves: (A) and (B) are electron microscope section of optic nerve (A) and sciatic nerve (B). (C) AxCaliber axon diameter distribution -(optic -blue, sciatic -red). (D) Axon diameter distribution derived from electron microscopy

Figure 3 :

 3 Figure 3: Cluster analysis of Ax-Caliber's ADD along the corpus callosum: (A) A midsagittal T2-weighted MRI with the AxCaliber clusters super-imposed, enlarged at (B). (C) The AxCaliber averaged ADDs for the different clusters given in (A) and (B).

Figure 4 :

 4 Figure 4: Virtual dissection of the major fibre pathways in the human brain 90 .

Figure 5 :

 5 Figure 5: White matter anatomy from early tractography to advanced models. a) Standard anatomical description of the arcuate fasciculus (Left panel, 104 ) replicated with diffusion weighted imaging tractography (Right panel, 19 ) b) New anatomical model of the arcuate fasciculus (Left panel, 20 ) validated with Klingler post-mortem dissection (Right panel, 95 ) c) Model of white matter configurations that challenge classical tractography (Left panel), example of the tractography of the corpus callosum splenium showing the same limitation (Right panel) . d) First branch of the superior longitudinal fasciculus crossing with the corona radiata discovered in humans with advanced spherical deconvolution

Figure 6 :

 6 Figure 6: Connectivity-based parcellation of the frontal lobes. a) 3D lateral view of the brain and the 12 regions defined by the structural connections with the rest of the brain. b) functional specificity of the 12 connectivity based regions.

Figure 7 :

 7 Figure 7: Correlations between individual differences in behavioural performance for a given task and variation in white matter microstructure. Axial slices are displayed in neurological convention. Numbers indicate MNI152 axial coordinates 191 . L, Left ; R, Right.

Figure 8 :

 8 Figure 8: White matter plasticity following a 5-day water maze. (a) Statistical parametric maps of interaction in FA values between scan time (pre vs. post water maze) and group (learning vs. control). An enlargement of the corpus callosum region is shown on the right. (b) Immunohistochemical staining (x10 magnification) of the CC for MBP. Note the increase in MBP immuno-reactivity in the CC after the water maze task. (c) Quantification of the immunoreactivity (staining intensity) in the CC (P< 0.05).

Acknowledgements

We wish to thank Lauren Sakuma for useful discussion. This work was supported by the 'Agence Nationale de la Recherche' [grants number ANR-10-IAIHU-06 and number ANR-13-JSV4-0001-01].