Corentin Vall Ée

Pierre Maurel

Isabelle Corouge

Chrisitan Barillot

Acquisition duration in resting-state arterial spin labeling. How long is enough?

Keywords: Functional Magnetic Resonance Imaging, Arterial Spin Labeling, Resting-state fMRI, Acquisition duration, Modeling

Resting-state Arterial Spin Labeling (rs-ASL) is a rather confidential method compared to restingstate BOLD but drives great prospects with respect to potential clinical applications. By enabling the study of CBF maps, rs-ASL can lead to significant clinical subject-scaled applications as CBF is a biomarker in neuropathology. An important parameter to consider in functional imaging is the acquisition duration. Despite directly impacting practicability and functional networks representation, there is no standard for rs-ASL. Our work here focuses on strengthening the confidence in ASL as a rs-fMRI method and on studying the influence of the acquisition duration.

To this end, we acquired a long rs-ASL sequence and assessed the quality of typical functional brain networks quality over time compared to gold-standard networks. Our results show that after 14 min of duration acquisition, functional networks representation can be considered as stable.

INTRODUCTION

Functional MR imaging (fMRI) builds the links between location and function in the brain. The two main sub-domains in fMRI are task-based fMRI and resting-state fMRI. In task-based fMRI, a functional location is considered to be where the acquired signal matches with the task guidelines given to the subject. In resting-state fMRI, as no task is given, the focus is on fluctuations in voxels time-series induced by spontaneous neural activations. Similarities in these time-series in different areas have shown to be not random, but matching function of the brain [START_REF] Biswal | Functional connectivity in the motor cortex of resting human brain using echo-planar MRI[END_REF]. These similarities define the functional connectivity of the brain and show the underlying cerebral architecture organized into functional specialized units communicating with each other. Resting-state functional imaging aims to identify functional areas of the brain and depict how they interact outside any structural connectivity consideration [START_REF] Van Den Heuvel | Exploring the brain network: A review on resting-state fMRI functional connectivity[END_REF] . Healthy and diseased subjects differ in functional networks cartography and in the intensity of functional connectivity for major disorders such as Parkinson's disease (3), Alzheimer's disease (4), severe depression [START_REF] Craddock | Disease state prediction from resting state functional connectivity[END_REF] or schizophrenia [START_REF] Lynall | Functional Connectivity and Brain Networks in Schizophrenia[END_REF].

Another subdivision in fMRI concerns the way the signal is obtained. The two major techniques are Blood Oxygen Level Dependent (BOLD) fMRI and functional Arterial Spin Labeling (fASL). Based on neurovascular coupling effects, BOLD techniques rely on the local signal variation induced by the neuron consumption of blood oxygen. Arterial Spin Labeling (ASL) is an MRI perfusion technique which uses magnetically labeled arterial water protons as an endogenous tracer. An inversion pulse labels the inflowing blood and after a delay called post-labeling delay, a labeled image of the volume of interest is acquired. The subtraction of the labeled image from a control image, i.e., non labeled, reflects the quantity of spins that have perfused the imaged volume, producing what is commonly called a perfusion-weighted (PW) image.

The PW map can be used to quantify the cerebral blood flow (CBF) under some assumptions [START_REF] Buxton | A general kinetic model for quantitative perfusion imaging with arterial spin labeling[END_REF][START_REF] Borogovac | Arterial spin labeling (ASL) fMRI: Advantages, theoretical constrains and experimental challenges in neurosciences[END_REF]. The quantification of CBF is the main advantage of ASL over BOLD. Indeed, the latter provides an indirect and non-quantitative measurement of neural activity, as it results from a combination of variations in CBF, cerebral blood volume and cerebral metabolic rate of oxygen. While the pathologies mentioned above were studied with BOLD, ASL allows to study a new set of pathologies with fMRI, such as acute stroke [START_REF] Wang | The value of arterial spin-labeled perfusion imaging in acute ischemic stroke: Comparison with dynamic susceptibility contrast-enhanced MRI[END_REF] or chronic fatigue syndrome [START_REF] Boissoneault | Abnormal resting state functional connectivity in patients with chronic fatigue syndrome: An arterial spin-labeling fMRI study[END_REF] since CBF abnormalities can characterize pathologies.

The main drawback of ASL is its lower signal-to-noise ratio compared to BOLD fMRI. The repetition time (TR) is also twice to three times higher in fASL compared to BOLD fMRI, which impacts its temporal resolution. Furthermore, ASL can be implemented through numerous MRI sequences and meta-analyses can be difficult to set up, for ASL shows a high sequence parameter dependency [START_REF] Grade | A neuroradiologist's guide to arterial spin labeling MRI in clinical practice[END_REF][START_REF] Mutsaerts | Multi-vendor reliability of arterial spin labeling perfusion MRI using a near-identical sequence: Implications for multi-center studies[END_REF]. Nevertheless consensus seems to overcome with years [START_REF] Alsop | Recommended implementation of arterial spin-labeled Perfusion mri for clinical applications: A consensus of the ISMRM Perfusion Study group and the European consortium for ASL in dementia[END_REF]. Predominant in clinical usage and in academic research, BOLD is still considered as the gold standard in fMRI. However, as it provides quantification of CBF, ASL can be a serious contender to BOLD when it comes to pathologies evaluation, especially for Alzheimer's disease [START_REF] Alsop | Arterial Spin Labeling Blood Flow MRI: Its Role in the Early Characterization of Alzheimer's Disease[END_REF][START_REF] Wolk | Arterial spin labeling MRI[END_REF][START_REF] Zhang | Cerebral blood flow measured by arterial spin labeling MRI at resting state in normal aging and Alzheimer's disease[END_REF]. The absence of contrast agent injection makes ASL well suited for longitudinal studies, particularly for pediatric population or for population with poor venous access or contrast agent contraindication.

The acquisition duration is an important parameter in an rs-fMRI study with strong practical consequences.

Most current studies work with a duration from 8 min to 13 min and a TR from 3 s to 4 s (i.e. 120 to 260 images). Intuitively, one would assume the longer the duration, the better the sampling of the signal correlation across the brain and thus the better the acquisition. But this requires to define what "better" actually means and does not consider the practical questions of clinical implementation and subject restingstate upholding. To the best of our knowledge, some papers already studied the influence of duration in rs-BOLD [START_REF] Van Dijk | Intrinsic Functional Connectivity As a Tool For Human Connectomics: Theory, Properties, and Optimization[END_REF][START_REF] Bouix | Evaluating acquisition time of rfmri in the human connectome project for early psychosis. How much is enough?[END_REF][START_REF] Birn | The effect of scan length on the reliability of resting-state fMRI connectivity estimates[END_REF][START_REF] Anderson | Reproducibility of single-subject functional connectivity measurements[END_REF][START_REF] Laumann | Functional System and Areal Organization of a Highly Sampled Individual Human Brain[END_REF], whereas in rs-ASL, it has not been explored yet. In this work, we first focus on the feasibility of detecting functional connected regions of the brain from rs-ASL. We remain as close as what a typical investigator of rs-ASL would experience by implementing usual sequence, processing and functional networks detection methods. We then assess a trend over the duration influence on rs-ASL detected networks quality: we do not directly assess whether an acquisition is good at a given time, but rather how it evolves with longer durations. After describing the scores used and the modeling approach, an in-depth analysis for the Default-Mode Network (DMN) will be presented in order to illustrate scores evolution on the most typical resting-state network. Finally, we will show results for all the functional networks under consideration and discuss an optimal sequence duration in rs-ASL.

MATERIAL

Subjects

Seven healthy male right-handed subjects aged from 21 to 28 years (23.5 yo±2.5) were involved in this study. All subjects gave informed written consent before participating in the study. We have maintained the homogeneity of the population in order to limit the influence of factors such as gender or age.

MR Acquisitions

The subjects were scanned on a 3.0T whole body Siemens MR scanner (Magnetom Verio, Siemens Healthcare, Erlangen, Germany) with a 32-channel head coil. A 3D anatomical T1-weighted MP2RAGE image was acquired for each subject. The resting-state ASL imaging was performed using a 2D EPI pseudocontinuous (pCASL) sequence. Subjects were asked to keep their eyes closed, to relax (mind-wandering) without falling asleep. We used the most common parameters reported in the literature: TR = 3500 ms, FoV = 224 x 224 mm 2 , TE = 12 ms, LD = 1500 ms and a 1250 ms post-labeling delay (PLD). Volumes were made of 24 slices of 64 × 64 voxels with 5 mm slice thickness with 20% gap for a total resolution of 3.5×3.5×6 mm 3 . The number of volumes was 420 for a total duration of 24 min 30 s. For the 1250 ms PLD, we chose a balance between a longer duration, about 1800 ms, recommended to optimize the quality of the CBF estimate [START_REF] Alsop | Recommended implementation of arterial spin-labeled Perfusion mri for clinical applications: A consensus of the ISMRM Perfusion Study group and the European consortium for ASL in dementia[END_REF][START_REF] Chen | Characterizing Resting-State Brain Function Using Arterial Spin Labeling[END_REF], and a shorter duration, 600 ms, which seems to give a better functional representation [START_REF] Liang | A k-space sharing 3D GRASE pseudocontinuous ASL method for whole-brain resting-state functional connectivity[END_REF][START_REF] Liang | Graph analysis of resting-state ASL perfusion MRI data: Nonlinear correlations among CBF and network metrics[END_REF]. We kept the PLD quite long as the main advantage of ASL is ultimately to compute CBF although we will focus on functional areas representation in this paper.

Data preprocessing

For each subject, the raw pCASL series is divided into 46 sub-series. The duration of these sub-series ranges from nearly 2 min (34 volumes) to 24 min 30,s (420 volumes) with a time step of 30 s. For the sake of simplicity, we will only mention rounded durations hereafter. All these subdivisions are made before any preprocessing: the preprocessing is done independently on each sub-series. For the preprocessing steps and their parameters we chose the most common ones found in bibliography. All steps are shown in Figure 1. For the preprocessing steps we used Matlab CONN toolbox (www.nitrc.org/projects/conn, RRID:SCR 009550) [START_REF] Whitfield-Gabrieli | Conn : A Functional Connectivity Toolbox for Correlated and Anticorrelated Brain Networks[END_REF].

METHODS

Detecting networks with Seed-Based Analysis

To obtain the mapping of invidivual functional networks, we rely on seed-based analysis (SBA) [START_REF] Van Den Heuvel | Exploring the brain network: A review on resting-state fMRI functional connectivity[END_REF]. The principle of this method, the first proposed to define functional connectivity [START_REF] Biswal | Functional connectivity in the motor cortex of resting human brain using echo-planar MRI[END_REF], is quite straightforward.

Considering a similarity measure (usually linear correlation, but many others exist (26)), SBA builds functional areas by gathering voxels which exhibit a matching signal, in the sense of the chosen measure, to that of a ROI, called the seed in this context. Even if SBA is a very polymorphic modeling method, we use its most common form in our work. Hence we consider linear correlation as the similarity measure and use a set of 20 single voxels as seeds. Seeds are spread in the expected location of six usual functional networks: DMN, Sensori-motor, Language, Salience, Visual and Cerebellum. The exact positions of the seeds in the MNI152 space are provided in the appendix section and were suggested by the CONN toolbox.

To build a functional map for each seed, we statistically test whether the signal between the seed and a candidate voxel is positively correlated with a risk of 1% FWER-corrected. This is a tough conservative testing compared to most of rs-ASL (even fMRI in general) studies, but we agree with the recommendation of ( 27) on false positive underestimation in fMRI litterature.

Evaluation scores

The resting-state BOLD litterature suggests different acquisition durations: 6 min (17), 10 min (18), 12 min [START_REF] Birn | The effect of scan length on the reliability of resting-state fMRI connectivity estimates[END_REF], 25 min [START_REF] Anderson | Reproducibility of single-subject functional connectivity measurements[END_REF], and even 100 min [START_REF] Laumann | Functional System and Areal Organization of a Highly Sampled Individual Human Brain[END_REF]. The main reason of their apparent discrepancy is the modeling. Indeed, there is many ways to properly define a model to assess to role of acquisition duration (a fortiori how much duration is enough), even if they lead to different conclusions. As a pioneer work on rs-ASL, we want our modeling to reflect an investigator experience with the impact of acquisition duration on functional network estimation. Figure 2 illustrates the investigation of acquisition duration we will model. The DMN estimation is validated after 14 min on Figure 2 because it matches with how the functional network is expected to look like. In modeling terms, it is basically assessing the overlap of the estimated network with a reference network. In order to investigate a trend afterwards and decide which acquisition duration is enough, the individual functional maps will be compared to a reference, like process described in Figure 2. For that purpose, we rely on the Multi-Subjects Dictionary Learning atlas (MSDL) by [START_REF] Varoquaux | Multi-subject dictionary learning to segment an atlas of brain spontaneous activity[END_REF]. MSDL is an atlas of 17 resting-state functional networks containing our 6 networks of interest, and from which our seeds are independent. The key idea is to have functional maps close to what an expert would expect to observe when looking for the typical functional areas investigated here.

To study the quality of the detected networks as a function of the acquisition duration, we evaluate the overlap between the SBA estimated functional maps and the MSDL references (simply called "reference" hereafter) through two measures: the Jaccard's index and the area under curve (AUC).

Let E be a set, let (v i ) i k∈N be observations in E and (M 1 ; M 2 ) ∈ {0; 1} E × {0; 1} E binary categorical variables. Let A,B,C,D be four sets with respective cardinals a,b,c,d defined by:

       A := {v i | M 1 (v i ) = 1, M 2 (v i ) = 1} B := {v i | M 1 (v i ) = 1, M 2 (v i ) = 0} C := {v i | M 1 (v i ) = 0, M 2 (v i ) = 1} D := {v i | M 1 (v i ) = 0, M 2 (v i ) = 0} (1)
Almost all common similarity measures (Sokal measure family, Sørensen-Dice, correlation etc.) can be defined with a,b,c,d. If one of the binary categorical variable can be considered as the truth, let's say the functional maps to be compared (binary here, but the definition can easily be extended to probability maps).

Jaccard's index

When comparing two spatially distributed data, the most obvious measure is the Jaccard's index: the ratio between the size of their intersection and their union. It is defined by J = a / (a + b + c) in our notation system. It provides intuitive and visual information about the overlap between one tested correlation map and one reference. It is also test-dependent: changing the risk or the multiple comparisons correction at the detection step will also change the shape and extent of the functional area, generally modifying Jaccard's index. This may be considered as a drawback but in fact, a statistical test is usually used at some point when investigating functional data.

Receiver operating characteristic analysis

In this section, we assume that the binary categorical variables are parameterized by at least one parameter.

For example, in our case, it could be the risk for the statistical test of correlation α or a threshold on correlation r. Let r be our parameter, a r , b r , c r , and d r the previously defined cardinals in (1) now parametrized by r, and let define a set x (r) , y (r

) , r ∈ [-1, 1] ⊂ [0, 1] 2 by:      x(r) = 1 - d r d r + b r y(r) = a r a r + c r (2)
The implicitly defined function f : x → y is called the Receiver operating characteristic curve (ROCcurve) and its integral 1 0 f (x)dx is simply called the Area Under Curve (AUC). In the case where M 2 is considered to be the truth, f is just informally f : 1 -Specificity → Sensitivity. The AUC is not test-dependent as it covers all possible values of the threshold parameter (i.e. risk/correlation). It illustrates how a functional map can be close to the reference by considering all values of the considered parameter, while the Jaccard's index reflects how it is close to the reference by considering one value of the given parameter. Hence AUC is a better way to assess the trend of interest from a theoretical point of view. However, it is further away from the practical proximity of the Jaccard's index modeling offers, so we will eventually consider both scores.

Modeling trend with respect to the duration

Both Jaccard's index and AUC are computed for each subject, each seed, each duration and each functional network reference from MSDL. The next step is to model the trend of these two scores evolution according to the acquisition duration for all subjects and for each combination between one seed and one reference. Assuming rs-ASL sequence lasts long enough to cover all usage, extrapolation for a duration longer than 24 min 30 s seems superfluous. There is no theoretical model, even in BOLD, on the dependence between acquisition duration and quality of functional networks detection: we are not interested in an explicit formula. Moreover, even processed independently, neighboring within-subject time-points have a strong dependency as they come from the same acquisition. Under these conditions, a local non-parametric regression is very well-suited. We chose to use the Loess method. Loess can be understood as a local polynomial regression on a subset of the whole dataset, defined by a weighted K-nearest neighbors algorithm. For a more comprehensive description, see [START_REF] Cleveland | Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting[END_REF]. We used second degree polynomial functions with a 0.8 span.

RESULTS

Effect of the acquisition duration for the Default Mode Network

In this section, we present an in-depth analysis of the DMN. In the set of 20 seeds we used, many should not be inspected when used in combination with MSDL DMN. The main reason is that most of the combinations has no objective basis for detecting the DMN. Otherwise, the seed may have failed to detect precisely the networks it was meant to detect, which is expected with very short acquisition duration. A good way to get an idea of the quality of the overlap between the functional maps associated with a seed and a reference for all durations is to check the boxplots of the Jaccard's index as in Figure 3. Boxplots give an overview of the results for rs-ASL: for the DMN reference, Jaccard's indices have higher values for the seeds placed in order to detect it. Prefrontal and posterior seeds seem to work well while lateral DMN seeds provide lower scores but still higher than any other seeds. Figure 4 shows the evolution of the estimated DMN with the prefrontal seed and corresponding scores for one subject. The depiction made by the scores of the overlap between the estimation of the DMN and the MSDL reference match with the four stages identified in Figure 2. Figure 5 shows the Jaccard's index, AUC, Sensitivity and Predicted Positive Value (PPV), for each subject and at each acquisition duration. Loess on Jaccard's index, as well as on AUC, models quite well what can be observed by looking directly at the functional map. Jaccard's index seems to stabilize after 12-13 min and AUC at an earlier acquisition duration around 9-10 min. We could have expected sensitivity and PPV to follow the same trend. Actually, sensitivity just grows over time, but more slowly for longer durations. Interestingly, PPV reaches a peak in the second stage mentioned above. The seven subjects show different level of response but good correlations (except for the subject 2 with AUC), i.e. the trend is the same among subjects, rather than an average effect induced by the Loess.

Moreover results observed for the DMN, can be generalized for almost every combination of seeds and references as we will see.

Effect of the acquisition duration for all functional networks

Visual inspection of the acquisition data and of the estimated functional networks estimation is always a good practice (30) However, with more than 6000 functional maps generated (20 seeds, 7 subjects, 46 acquisition durations), a visual checking of all the maps is not practicable. As seen for DMN, many combinations between seeds and reference should not be investigated since they are not functionnally meaningful and will yield to very low overlapping scores (e.g. prefrontal seed with visual cortex). For Jaccard's index, we selected combinations for which at least 50% of observations have J 0.1. For AUC, the median is also considered, with a threshold of 0.7. These thresholds on the median values may seem rather low, but let us remember that all the acquisition durations are taken into account, even the shortest ones. Figure 6 shows the median values for all the combinations between seeds and references. The two thresholds lead to an almost identical choice for the selection of combinations. All seeds have their best scores with the expected reference, and each of the six functional networks are considered to be sufficiently well detected with SBA for Jaccard's index in accordance with our selection rules. The AUC suggests as good enough one more seed for cerebellum but consider that salience is not detected well enough with our set of seeds.

Figure 7 shows the range of durations where scores are not significantly different from their maximum values (5% risk) for each selected reference/seed combination. Colors on heatmap are scaled between minimum and maximum values of the corresponding score and matches with the stages already described for DMN in the previous section. Indeed, for every combination between seeds and references, both scores rapidly increase, and start to stabilize after a certain duration. However, for both measures, the 95% confidence interval around the maximum suggests a later start in the stabilization than suggested directly by the Loess curve values. While some combinations scores look already stabilized at 12 min, almost all of them are close to their maximum value at 16 min. Figure 8 shows a collection of functional areas obtained at a duration of 14 min. While language seed struggles to detect spatial components far from the seeds, all the other ones provide good detection of expected functional networks. The two bottom rows show the same subjects and the same reference with different seeds.

DISCUSSION

On methods

In order to estimate functional networks, the two most common methods are SBA and Independent Component Analysis (ICA). We did not work with ICA because the association between independent components and functional areas of the brain is intrinsically tedious [START_REF] Mckeown | Independent component analysis of functional MRI: What is signal and what is noise?[END_REF] [START_REF] Cole | Advances and pitfalls in the analysis and interpretation of resting-state FMRI data[END_REF], especially when it comes to comparison between subjects. Moreover, in order to investigate the relationship between quality and duration for all subjects, we must estimate the functional areas in the same way for each subject. Indeed, keeping the same seeds and the same test for SBA is trivial, while keeping the same number of independent components is equivocal, since it should be decided by a goodness of fit criterion.

We chose to report the positive predicted value rather than specificity for two main reasons. On the one hand, true negatives can have multiple definitions in fMRI, since it depends on the voxels considered: the whole volume, only the brain or any smaller ROI like grey matter. Although it is logical to consider only brain voxels for functional activity, this implies an extremely high number of true negatives, since the volume of a functional network is ten to a hundred times smaller than the one of the whole brain.

Therefore, the specificity reaches values too high to provide relevant information on similarity between functional areas. On the other hand, like specificity, PPV plays a similar role with respect to sensitivity: specificity gives a complementary information to sensitivity in the totality of voxels whereas PPV gives a complementary information in the union of the reference and the estimated functional area.

On results

Our two objectives were to confirm the feasibility of resting state ASL and to evaluate the influence of the acquisition duration on the estimation of functional areas. Figure 4 and Figure 8 with corresponding scores in Figure 6 and in Figure 7 confirm that, even with the basic preprocessing and straightforward methodology we used, ASL is fully viable as a resting-state method. Regarding the impact of acquisition duration, the most important result is the stabilization of the functional areas representation after a certain duration for both measures, Jaccard's index and AUC, with a strong inter-subjects correlation (i.e. not a mean-effect induced by the Loess modeling). Since the acquisition should have the shortest duration possible for clinical implementation, the recommended duration eventually corresponds to the start of the stabilization stage. Strict definitions of the stabilization stage lead to longer duration since they would rely heavily on the Loess maximum by considering as stable just a narrow interval around of the maximum. However, since after 12 min to 14 min the score variations are low, a slight change in preprocessing or in the population could also lead to unstable maximum, without changing the trend. Relaxed definitions would keep optimal duration stability, but they may consider a functional area as good enough when an human investigator would not. Actually, early stages of acquisition are associated with poor representation of functional areas disconnected from the seed. Based on our different results, 14 min seems to be an interesting compromise.

Note that the DMN, the sensori-motor cortex, and the cerebellum have an almost consensual spatial definition among the authors, unlike language, visual and salience, which show a greater spatial variability (see for example http://neurosynth.org/). As we provide an evaluation only with one set of references (from the MSDL), one could have expected this to be a limitation of our work. However the spatial variability of the areas of interest in atlases is low enough to change only the scores but not the trend observed in this paper. Preprocessing influence should also be considered as positive: since we use typical and basic preprocessing, more advanced techniques should provide the same or an earlier stabilization, still keeping our suggestion as a sufficient duration.

The same is true for ASL readout approach. Using a 3D readout is probably an improvement in our sequence, as it tends to outperform 2D EPI, but not every investigator has access to 3D sequence (13) 1 . Not to mention the resting-state and readout approach, ASL is very sensitive to changes in its parameters. Since we are studying the influence of acquisition duration for a given set of parameters, the optimal duration of 14 min could be strongly influenced by the sequence parameters. Although the influence of each of them is to be kept in mind, most of them should not disturb the investigators, as most of them have a specific bibliography that goes well beyond the issue of the optimal acquisition duration. However, two of them may have a deep impact on our results: post-labeling delay and repetition time (TR). As mentioned in section 2.2, we already have some clues on how the PLD can influence functional networks representation.

The critical parameter in our opinion is repetition time. It defines the sample frequency of the resting-state 1 It is anticipated that single-shot 3D readout may be the preferred option in the future, but these methods are not yet sufficiently well tested to recommend for general use at this time. Multislice singleshot 2D echo-planar imaging (EPI) or spiral readout should be considered a viable alternative to segmented 3D sequences, because they are available on all systems and are insensitive to image artifacts from motion. signal and turning our 14 min suggestion into a 240 volumes since we only work numerically on the signal. Its variation may shift the stabilization step toward a higher/lower number of volumes and hence a longest/shortest duration, without changing the stabilization of the functional networks representation after a certain number of volumes (i.e. same signal but different sampling frequency). Moreover in rs-ASL, TR values are typically between 3 s and 5 s, which is too wide to assume the locally linear dependence between TR and optimal duration. As a preliminary work on optimal duration in rs-ASL, we focus more on the modeling rather than investigating the influence of the TR. However, a specific study on the relationship between repetition time, volume number and quality of acquisition would be, in our opinion, highly beneficial to better define the optimal duration and also would be useful when an investigator sets up a sequence.

Last, we work on a homogeneous sample of subjects. While it greatly limits the influence of variables related to the population description and not included in our modeling, it also narrows the population represented by our sample of subjects. Hence, a natural perspective is to include new subjects in order to extend the population represented and checks if the influence of variables like gender, age, laterality could be excluded from the population variability. It should be noted that geriatric, pediatric and pathologic populations should be treated separately rather than included. Indeed, these populations can show extreme differences in average (or local) perfusion, which possibly hinders the functional networks estimation, and hence, when the stabilization starts.

CONCLUSION

We model and process data in order to get results as close as an investigator would do. All the considered functional areas were well detected by rs-ASL. Our results show a quality stabilization after a certain volume number/duration for both scores in all but one combination between seeds and references: very long sequences should not hence be considered. The main objective was to answer "How long is enough?".

While we can suggest, for our set of sequence parameters, the optimal volume number of 240 / optimal duration of 14 min, any methods that improves the detection of functional networks are likely to provide an earlier stabilization start, i.e. optimal volume number/duration. Since we use a basic and typical sequence, preprocessing and estimation, 240 volumes / 14 min even if not optimal should be enough for most rs-ASL usage. Last, the exploration of the impact of the TR and PLD on the optimal acquisition duration was beyond the scope of this article but would be highly beneficial for sequence implementation, since we forecast they are the two parameters that may shift the stabilization start toward higher number of volumes.

M 2 ,

 2 therefore a becomes the number of True Positives, b of False Positives, c of False Negatives and d of True Negatives. We also trivially have the Sensitivity: a/ (a + c), Specificity: d/ (d + b), and the Positive Predicted Value (PPV): a/ (a + b). In fMRI, the v i are the voxels and the variables M 1 , M 2 are

Figure 1 .

 1 Figure 1. Preprocessing steps of the rs-ASL images. After a surround substraction of labels and controls, preprocessing starts with the realignment of the perfusion weighted-map. All the functional volumes are registered with the first one. Second step is the indirect normalization of the functional volumes. It starts with registration of the functional data on the anatomical 3D T1 (a). Then comes the registration of the anatomical image on the MNI 152 template (b). Finally, transformations of (a) and (b) are composed to register functional data on the MNI template (c). We then used Gaussian smoothing with typical 6 mm kernel and pass-band filtering with a range from 0.005 Hz to 0.1 Hz. Final denoising was made with COMPCOR (33).

Figure 2 .

 2 Figure 2. Seed-based estimation of the DMN with prefrontal seed at different acquisition durations. Four stages can be identified. At 2 min (1), the map shows only false positive noise detection. Between 4 min and 8 min (2), the false positive noise has disappeared while the frontal component of the DMN starts growing and the posterior starts being detected. Between 10 min and 12 min (3), the frontal component is well detected, the posterior grows and the lateral components are barely being detected. At 14 min and after (4), DMN detection is good and interestingly, stable.

Figure 3 .

 3 Figure 3. Each boxplot corresponds to one seed and shows the distribution of Jaccard's index between the estimated functional area corresponding to the considered seed on the one hand, and the MSDL DMN reference on the other hand, for all subjects and all durations. The seeds are grouped by color, each corresponding to one of the six functional areas considered. As expected, the seeds located in the expected DMN location (in pink) give the best results.

Figure 4 .

 4 Figure 4. Subject 4 DMN detection (in blue) with prefrontal seed and MDSL DMN reference (in red) over a 2 min to 24 min duration with 2 min steps. Maps are shown in MNI152 space.

Figure 5 .

 5 Figure 5. Jaccard's indices (A), AUC (B), Positive Predicted Value (C) and Sensibility (D) evolution with time with their associated Loess for the seed associated with prefrontal DMN. On all subjects, Jaccard's index increases with duration before 10-12 min, then stabilizes. The AUC shows the same trend with an earlier stabilization, around 9-10 min. PPV grows rapidly, reaches a peak at 8 min, then decreases slowly. Finally, Sensitivity increases with time, but more slowly for longer durations. Subjects show different level of response but good correlations.

Figure 6 .

 6 Figure 6. Median values of Jaccard's indices and AUC for all combinations of seeds/references. Green circles show where seeds/reference combinations are selected with respect to our thresholding rules (0.1 for the Jaccard median and 0.7 for the AUC median).

Figure 7 .

 7 Figure 7. Color maps of Jaccard's Index and AUC Loess value with respect to the acquisition duration for all selected reference/seed combination. Every combination has a fast increasing score followed by a stabilization stage. The PPV peak shows on all combinations around 9 min and appears always just before score start stabilizing. Maximum and its 95% confidence interval may be unstable since score variations are often low after PPV peak.

Figure 8 .

 8 Figure 8. Collection of functional areas at 14 min with the corresponding scores: Jaccard's Index, Area Under the Curve, Sensitivity and PPV. The two bottom rows show the same subjects and the same reference but with different seeds. The third row shows the estimated (blue) and reference (red) visual network for the same subject (subject 7) but different seeds ("dorsal left" seed on the left and "dorsal right" seed on the right). Similarly, the bottom row shows the same subject (subject 1) with different seeds ("Prefrontal" and "Posterior").

This is a provisional file, not the final typeset article

Frontiers

ACKNOWLEDGMENTS AND FUNDINGS

MRI data acquisition was supported by the Neurinfo MRI research facility from the University of Rennes I.

Neurinfo is granted by the the European Union (FEDER), the French State, the Brittany Council, Rennes Metropole, Inria, Inserm and the University Hospital of Rennes. This manuscript has been released as a pre-print at: https://www.hal.inserm.fr/inserm-02129799v1

DATA AVAILABILITY STATEMENT

In accordance with the consent form signed by the subjects, authors are not allowed to share MRI acquisitions. However, requests to see the raw data can be sent to the corresponding author. For the preprocessing steps we used Matlab CONN toolbox (www.nitrc.org/projects/conn, RRID:SCR 009550) [START_REF] Whitfield-Gabrieli | Conn : A Functional Connectivity Toolbox for Correlated and Anticorrelated Brain Networks[END_REF]. The rest of the code used for evaluation scores and Loess is available upon request from the corresponding author.

CONFLICT OF INTEREST STATEMENT

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

AUTHOR CONTRIBUTIONS

FIGURES APPENDIX -SEED LOCATION IN MNI152


Guillaume Agnesina

Redha Taiar

Nicolas Houel

Kevin Guelton

Philippe Hellard

Yuli Toshev

BRG.LifeMOD™ modeling and simulation of swimmers impulse during a grab start

Keywords: 3D modeling, swimming, grab start, kinematics, joint moments, 0 6000

INTRODUCTION

Whatever the swimming disciplines the study of swimmers' performance involves the identification of three technical phases: start, turn and strokes phases. Analysis of the temporal distribution showed that the start phase accounts for 15 % and 7.7 % of total time, respectively for 50 m and 100 m freestyle events. In short distance races (50m and 100m) the start represents a particularly important factor. For example, at the Athens Olympic Games (2004), the time separating the eight finalists in the men's 50m freestyle final was 0.44 s, which represents 2% of the winner's total race time (21.93 s). The difference in performance among the finalists' may find an explanation in the time lost during the start phase.

Regardless of underwater factors, the start phase depends primarily on the quality of the swimmer's impulse on the starting platform [START_REF] Vilas-Boas | Swimming starts and turns: Determining factors of swimming performance[END_REF]. Although recent studies have been undertaken, using both dynamics and kinematics approaches, they do not yield additional information concerning the relationship between the swimmers' movements and their actual performance. Few studies have addressed the modeling (dynamic and/or kinematics) of the parameters that determine the performance according to swimmers' movement during start phases (Holtes and [START_REF] Holthe | Kinematic comparison of grab and track starts in swimming[END_REF]. It is to note that modeling methods used for movements study in others sports, for example skiing [START_REF] Houel | Interests and limits of the kinematic analysis by imagery. Contribution to the realization of models kinematics and dynamic of the movement[END_REF]), seem to be the most effective for detailed understanding of movements and for performance prediction.

The main aim of the study is to develop an approach for 3D modeling and simulation of swimmers impulse during a grab start in order to optimize the performance.

II. METHODS

Four national level swimmers were instructed to perform a grab start. Subjects' average height and mass were respectively 183.5 cm (± 3.41) and 75.77 kg (± 3.89). Swimmers were equipped with 16 passive markers (8 on the right and 8 on the left side) attached at the level of each important articulation: foot, ankle, knee, hip, shoulder, elbow, wrist, finger (Fig. 1).

For each start, 3 high speed video cameras (125 Hz) were used. The most important data were obtained by the two cameras in the two sagittal (lateral) planes (right and left). Simultaneously, the ground reaction forces were measured using a 3D force-plate AMTI OR6-7-2000 mounted on the start block (Fig. 1) with sampling frequency of 1000 Hz.

Swimmer's segments centres of mass velocity were calculated by integration of acceleration values. Before each start, the video cameras and the force-plate were calibrated and clock synchronized (0.008 s accuracy). Two-dimensional video analysis was carried out for the impulse phase, i.e. during the period when the athletes were in contact with the force-plate mounted on the start block. The video analysis allowed determining the angles between the subjects' segments axes and the horizontal plane. The data were fitted using polynomials' method [START_REF] Tavernier | Optimisation de l'ajustement de trajectoires par lissage avec des polynômes d'ordres multiples élevés: comparaison avec le filtre de Butterworth du second ordre[END_REF][START_REF] Winter | Biomechanics and motor control of human movement[END_REF]. The swimmer's knee and elbow joint angles (right and left knee elbow) were obtained using high sped video cameras and passive markers and are shown on Fig. 2 and Fig. 3. After that they were used as input data for the 3D modeling. The morphological parameters of the subjects were defined using the measured values for the height and the mass and the anthropometric tables from [START_REF] Dempster | The anthropometry manual work space for the seated subjects[END_REF]. The sum of segments energies was obtained using the equations for the sum of segments energies as defined in [START_REF] Winter | Biomechanics and motor control of human movement[END_REF].

During the impulse phase, subjects were represented using an open tree structure composed of eight straight segments connected via frictionless joints (Fig. 1). Input data for the model consisted of the fitting angles calculated at each joint (Fig. 2 and Fig. 3) and the subjects' morphological parameters. Based on analysis of swimmers' forces and joint moments exerted during the impulse, the model predicts the total body power during the impulse and the position, the initial angle and the velocity of the segments centres of mass at the instant of takeoff.

III. RESULTS

For every swimmer, 3D model was generated using BRG.LifeMOD TM Biomechanics Modeler on the base of individual morphologic parameters and kinematic data obtained by high speed video camera and passive markers attached at the level of each important articulation. 3D swimmer's body models for four characteristic moments during the grab start are shown: for the initial moment of the impulse phase, t = 0 s (Fig. 4); for the maximal efforts moment, t = 0.6013 s (Fig. 5); for the final start moment, t = 0.8781 s (Fig. 6).

For each joint, the dynamic torque, force and power were determined using the inverse dynamic equations [START_REF] Winter | Biomechanics and motor control of human movement[END_REF][START_REF] Houel | Interests and limits of the kinematic analysis by imagery. Contribution to the realization of models kinematics and dynamic of the movement[END_REF]. The knee joint torques/time functions (right and left) for the time period 0.00 s ÷ 0.92 s are shown on Fig. 7. The velocities of the segments centres of mass were also calculated. The velocity/time functions of the pelvis centre of mass in the sagittal plane for the time period 0.00 s ÷ 0.92 s are shown on Fig. 8 (along the x-axis and the y-axis). The model was validated by comparing the predicted ground reactions forces during the start time period and the measured by AMTI force-plate ground reactions (Fig. 9).

IV. CONCLUSION

The proposed approach allows predicting swimmer's joint moments for each important articulation during the impulse phase of the grab start and to analyze the segmental coordination for each studied swimmer in order to optimize the performance. The model was successfully validated by comparing the predicted ground reactions during take-off with the experimental ground reactions data collected in situ.

Figure 1 .

 1 Figure 1. Passive markers (8 on the right and 8 on the left side) were attached at the level of each important articulation. AMTI force-plate was mounted on the block.

Figure 2 .Figure 3 .

 23 Figure 2. The swimmer's knee joint angles (right and left knee joint) obtained using high sped video cameras and passive maerkers and after that used as input data for the 3D swimmer body model. Time period: 0.00s ÷ 0.92s

Figure 4 .Figure 5 .

 45 Figure 4. 3D suimmer's body model in the initial moment of the impulse phase. Time = 0 s;

Figure 6 .

 6 Figure 6. 3D suimmer's body model i n the final grab start moment. Time = 0.784 s.

Figure 7 .

 7 Figure 7. The knee joint torques/time functions (right and left). Time period = 0.00 s ÷ 0.92 s.

Figure 8 .Figure 9 .

 89 Figure 8. The velocity/time functions of the pelvis centre of mass in the sagittal plane for the time period 0.00 s ÷ 0.92 s


Eitan Altman
email: eitan.altman@inria.fr

Mandar Matar

Gerard Burnside
email: gerard.burnside@nokia-bell-labs.com

Corinne Touati
email: corinne.touati@inria.fr

Normalized equilibrium in Tullock rent seeking game

Games with Common Coupled Constraints represent many real life situations. In these games, if one player fails to satisfy its constraints common to other players, then the other players are also penalised. Therefore these games can be viewed as being cooperative in goals related to meeting the common constraints, and non cooperative in terms of the utilities. We study in this paper the Tullock rent seeking game with additional common coupled constraints. We have succeded in showing that the utilities satisfy the property of diagonal strict concavity (DSC), which can be viewed as an extention of concavity to a game setting. It not only guarantees the uniqueness of the Nash equilibrium but also of the normalized equilibrium.

INTRODUCTION

Games with constraints have long been used for modeling and studying non-cooperative behavior in various areas. This includes road traffic [START_REF] El-Azouzi | Constrained traffic equilibrium in routing[END_REF][START_REF] Larsson | Side constrained traffic equilibrium models -traffic management through link tolls[END_REF] and telecommunications [START_REF] Hsiao | Optimal decentralized flow control of Markovian queueing networks with multiple controllers[END_REF]. Various types of constraints may appear in every day game situations; the simplest consisting of orthogonal constraints, where the strategies of the players are restricted independently of each other [START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF]. A second type of constraints are called Common Coupled Constraints (CCC) [START_REF] Altman | Constrained Games: the impact of the attitude to adversary's constraints[END_REF][START_REF] Pavel | An extension of duality to a game-theoretic framework[END_REF][START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF] in which all players have a common convex nonorthogonal multi-strategy space. This model can be viewed as constraints that are common to all users. A unilateral deviation of a player from some feasible multi-strategy (one that satisfies the constraints) to another strategy that is feasible for that player, does not result, therefore, in the violation of constraints of other users. CCC have often been used in telecommunications networking problems as well as in power transfer over a smart grid, where capacity constraints of links are naturally common. Games with this type of constraints are a special case of General Constrained Games (GCG) [START_REF] Debreu | A social equilibrium existence theorem[END_REF], see also [START_REF] Altman | Constrained Games: the impact of the attitude to adversary's constraints[END_REF][START_REF] Altman | Constrained Markov Games: Nash Equilibria[END_REF][START_REF] Altman | Constrained Cost-Coupled Stochastic Games with Independent State Processes[END_REF][START_REF] Gomez-Ramırez | Saddle-point cal-culation for constrained finite Markov chains[END_REF][START_REF] Kallenberg | Linear Programming and Finite Markovian Control Problems[END_REF][START_REF] Shimkin | Stochastic games with average cost constraints[END_REF].

In this paper we study the well known Tullock rent seeking game with Common Coupled Constraints. This game describes contest over resources. Each player bids an amount that she is ready to pay. She then pays an amount proportional to her bid and receives in turn a payoff that is proportional to her bid divided by the sum of bids of all players.

Gamenets 2019 Paris, France

Copyright is held by author/owner(s).

The presence of a capacity constraint results in infinitely many equilibria and we are faced with a question of equilibrium selection. Using Kuhn Tucker conditions to the best response, we can solve a relaxed game instead of the original constrained game, which has however the same equilibria as the original game. The Lagrange multipliers can be interpreted as a shadow cost that a manager sets in order to guarantee that the equilibrium achieved satisfies the constraints. This approach may however be completey unscalable since KKT Theorem does not guarantee that the price per resource unit is the same for all players. In fact, since the Lagrange multipliers are obtained for the best response function, they could depend not only on the player but also on the policy of all other players, rending the approach even less scalable. We are interested in finding such shadow cost which is fixed per resource unit. Such an equilibrium along with a fixed shadow price is called a normalized equilibrium.

The Tullock rent-seeking game has been used recently to model and study several game phenomena in networking. It was used to model contests over timelines in social netorks for maximizing visibility [START_REF] Reiffers-Masson | Game theory approach for modeling competition over visibility on social networks[END_REF]. Each player i controls the rate λiai of a Poisson process of posts that player i sends into a common timeline of length K. This rate is given by a basic popularity rate λi times the acceleration effort (e.g. through advertisement) given by ai. Using basic queueing theory, the authors show that the stationary expected number of posts in the timeline originating from player i is given by

K λiai N j=1 λjaj
This visibility measure is the payoff in Tullock's model, while the cost for acceleration at a rate ai is proportional to ai as in Tullick's model. Another application of the Tullock rent seeking game is in the study of contests between miners in blockchain [START_REF] Altman | Mining competition in a multi-cryptocurrency ecosystem at the network edge: A congestion game approach[END_REF].

A few words on rent seeking. According to Wikipedia, "In public choice theory and in economics, rent-seeking involves seeking to increase one's share of existing wealth without creating new wealth. Rent-seeking results in reduced economic efficiency through poor allocation of resources, reduced actual wealth-creation, lost government revenue, increased income inequality, and (potentially) national decline."

Wikipedia further describes the origin of the idea: "The idea of rent-seeking was developed by Gordon Tullock in 1967, while the expression rent-seeking itself was coined in 1974 by Anne Krueger [START_REF] Krueger | The Political Economy of the Rent-Seeking Society[END_REF]. The word "rent" does not refer specifically to payment on a lease but rather to Adam Smith's division of incomes into profit, wage, and rent. The origin of the term refers to gaining control of land or other natural resources."

Our first contribution is to show that the utilities satisfy a property that extends concavity to games, and is called Diagonally Strict Concavity. This is shown to imply the existence and uniqueness of a normalized equilibrium. We shall then show that this property further extends to the case of contests over several resources.

A SINGLE RESOURCE

Consider an N players game. Player m bids a quantity x m . We have minimum constraints x m ≥ for all m.

The payoff from this contest to player m is

P m = x m M j=1 x j
.

This comes at a cost of x m γ to player m where γ is a constant. The utility for player m is thus

U m (x) = x m M j=1 x j -x m γ. Theorem 1. (i)
The utility of player m is concave in its action and is continuous in the actions of other players. (ii) For any strictly positive value of γ, the above game has a unique Nash equilibrium in pure policies.

Proof. Direct calculation leads to (i). The existence then directly follows from [START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF]. Uniqueness is established in [START_REF] Altman | Generalising diagonal strict concavity property for uniqueness of Nash equilibrium[END_REF], see also [START_REF] Kaae | Existence, Uniqueness and Comparative Statics in Contests[END_REF]. Other related uniqueness results in the asymmetric case can be foudn in [START_REF] Reiffers-Masson | Game theory approach for modeling competition over visibility on social networks[END_REF][START_REF] Szidarovszky | On the Existence and Uniqueness of Pure Nash Equilibrium in Rent-Seeking Games[END_REF].

NORMALIZED EQUILIBRIUM

The games we have seen so far involved orthogonal constraints. By that we mean that the actions that a player can use do not depend on the actions of other players. We next introduce capacity constraint. We require that the following holds for some contant V :

M j=1 x j ≤ V (1)
Capacity constraints may represent physical bounds on resources, such as bounded power, or resources that are bounded by regulation. For example, legislation may impose bounds on the power used or on the emission of CO2 by cars. With the additional capacity constraint, the Nash equilibrium is no more unique and there may in fact be an infinite number of equilibria. We call this the game with capacity constraint.

Let y be an equilibrium in the above game and let y [-m] denote the action vectors of all players other than m. By KKT Theorem, since for each m, U m is concave in x m , there is a Lagrange multiplier λ m (y [-m] such that y m maximizes the Lagrangian

L m (x m ) = U m (x, y [-m] ) -λ m (y [-m] ) V k - M j=1 x j and λ m (y [-m] ) V - M j=1
x j = 0 (complementarity property). We call the game with the Lagrangians L m replacing the utilities U m the relaxed game.

The Lagrange multipliers can be interpreted as shadow prices: if a price is set on player m such that when other players are at equilibrium, the player pays x m λ m (y [-m] ) for its use of the capacity, then y is an equillibrium in the game with capacity constraints. Yet this pricing is not scalable since for the same use of the resources it may vary from user to user and it further depends on the the chosen equilibrium. For billing purposes one would prefer λ m not to depend on y nor on m, but to be a constant.

Does there exist a constant Lagrange multiplier λ independent of strategies of the payers and of the idendity m of the player, along with an associated equilibrium y for the corresponding relaxed game? If the answer is positive then y is called a normalized equilibrium [START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF].

Our goal is to establish the existence and uniquenesss of the normalized equilibrium.

DIAGONAL STRICT CONCAVITY

For a vector of real nonnegative numbers r, define

σ(x, r) = N m=1 rmU m (x) g(x, r) =       r1 ∂ ∂x 1 U 1 (x1, x-1) r2 ∂ ∂x 2 U 2 (x2, x-2) . . . rN ∂ ∂x N U N (xN , x-N )       (2)
σ is called diagonally strict concave (DSC) for a given r if for every distinct x 0 and x 1 , (x 1 -x 0 ) g(x 0 , r) + (x 0 -x 1 ) g(x 1 , r) > 0 Let G(x,r) be the Jacobian of g(x,r) with respect to x and let Gi,j be i th row and j th column of G(x,r). Then a sufficient condition for σ to be diagonally strict concave for some r is that for all x, [G(x, r) + G (x, r)] is negative definite.

Our interest in diagonally strict concave utility functions is due to the following properties of games possesing such utilities.

Theorem 2. (Theorem 4 from [START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF]) Let σ be diagonally strict concave for some r. Then there exists a unique normalized equilibrium.

PROOF OF DSC

In this section we establish that the Tullock game with capacity constraint has a DSC structure and thus has a unique normalized equilibrium.

In our case we have

g(x, r) =      r 1 x -1 X r 2 x -2 X . . . r N x -N X      (3)
where

X = N i=1 xi and x-m = N i=1,i =m xi Gi,j = ∂ ∂xj ∂ ∂xi rixi X (4) ri ∂ ∂xj x-i X 2 = ri -2x i X 3 if i = j ri x i -x -i X 3 if i = j (5)
For

[G + G ] consider Gi,j + Gj,i = -4r i x -i X 3 if i = j r i( x i -x -i) +r j (xj-x-j) X 3 if i = j (6) [G + G ] is negative definite if A [G + G ] A < 0, ∀A, A = 0
where A is the column vector

A =    a1 . . . aN    A G + G A = (7) N i=1   N j=1,j =i aiaj ri (xi -x-i) + rj (xj -x-j) X 3   -a 2 i 4rix-i X 3
We choose ri = 1 for all i. Then (7) equals -Z/X 3 where Z is given by

N i=1 a 2 i 4x-i +   N j=1,j =i aiaj ((x-i -xi) + (x-j -xj))   (8) = N i=1 a 2 i 4 (X -xi) + N j>i 4aiaj (X -xi -xj) (9) =4 N i=1 a 2 i (X -xi) + N j>i aiaj (X -xi -xj) (10) =4 N i=1   a 2 i N j=1,j =i xi + N j>i aiaj N k=1,k =j,k =i x k   (11) = N i=1 4xi   N j=1,j =i a 2 j + aj N k>j,k =i a k   (12)
Now ( 12) is positive for any positive value of x and hence [G'+G] matrix is negative definite.

SEVERAL RESOURCES

We consider next the following extension to the case of K resources. Each player m of the M players has a budget B(m) that he can invest by bidding x m k of resource k. The following (orthogonal) constraint should hold:

K k=1 x m k ≤ B(m).
The payoff for player m is the sum of payoffs in all K contests, i.e.

P m (x) = K k=1 P m k (x k )
where x k is the vector x 1 k , ..., x M k and where

P m k (x k ) = x m k M j=1 x j k .
and the cost of a contest k to player m is γ(k)x m k . Player m' s utility is thus

U m (x) = K k=1 x m k M j=1 x j k -γ k x m k
For the study of such games, see [START_REF] Reiffers-Masson | Game theory approach for modeling competition over visibility on social networks[END_REF].

We next define capacity constraint on each of the K resources. Let V be the column vector with the kth entry being a constant V k . We then require for each k that

N m=1 x m k ≤ V k
Note that when applying KKT conditions to the best response at equilibrium, we shall have K Lagrange multiplyers. We wish to find a vector of K Lagrange multiplyers that do not depend on the player nor on the policy of other players, such that the Nash equilibrium for the relaxed game will be an equilibrium for the original constrained game and in particular the constraints would be met and would satisfy the complentarity conditions. This is the vector version of a normalized equilibrium.

According to Theorem 4 of Rosen [START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF], we have to show that the set of utilities is diagonally strict concave in order to have existence and uniqueness of the normalized equilibrium. The latter follows from the fact that DSC holds for each resource separately and then apply the proof of Corollary 2 in [START_REF] Orda | Competitive routing in multi-user communication networks[END_REF].

CONCLUSIONS AND FUTURE WORK

We have shown that the utilities in the Tullock game are strict diagonal concave. This allows to conclude using Rosen's result that in absence of common correlated constraints, the Nash equilibrium exists and is unique, while in presence of such constraints, the normalized equilibrium exists in pure strategies and is unique. Note that while the statements on the Nash equilibrium have already been available in [START_REF] Altman | Generalising diagonal strict concavity property for uniqueness of Nash equilibrium[END_REF] which proposed an extension to the DSC property, that reference does provide tools to handle the normalized equilibrium.

Another advantage from the derivation of the DSC structure is that one can use dynamic distributed algorithms to converge to the normalized equilibrium and convergence is guaranteed under DSC, see [START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF].