that an accurate HRF model may significantly improve **detection** performance. To capture this variability, robust HRF **estimation** is necessary which can be achieved only **in** voxels or regions that elicit an evoked response to a given stimulus [2]. So far, many works have addressed this issue either by considering linear or nonlinear HRF models [3–5], parametric, semi-parametric or non-parametric (i.e. FIR models) descriptions [6–8], and by performing univariate (voxelwise) [4, 7], multivariate (regionwise) [9, 10] or even multiscale, i.e. spatially adaptive inference [11]. However, to the best of our knowledge, all these existing works assume the spatial support of the HRFs, either defined at the voxel or region-**level**, to be pre-specified. The proposed methodology takes place **in** the **Joint** **Detection**-**Estimation** (JDE) framework introduced **in** [9] and extended **in** [10, 12, 13] to account for spatial correlation between voxels. Standard JDE-based inference requires a pre-specified decomposition of the brain into functionally homogeneous parcels (groups of connected voxels) but with no guarantee of their optimality. These parcels should be small enough to guarantee the invariance of the HRF within each parcel, but large enough to contain reliable information for its inference [14]. Several attempts have been conducted to provide a robust **parcellation** such as **in** [15–19]. However, these approaches do not fully account for hemodynamics variability. Here, we introduce the concept of hemodynamic territory as a set of parcels which share a common HRF pattern. To determine such sets, we incorporate an additional layer **in** the JDE hierarchy, namely an adaptive parcel identification step based upon local hemodynamic properties. **In** this novel **Joint** **Parcellation**-**Detection**-**Estimation** (JPDE) model (Section II), for all the parcels of a given territory, HRFs are voxelwise but defined as local stochastic perturbations of the same HRF pattern. Then, hemodynamics **estimation** reduces to the identification of a limited number (say K) of such HRF patterns and parcel identification reformulates as a clustering problem where each voxel is assigned an HRF group among K. The HRF group assignment variables are governed by a hidden Markov model to enforce spatial correlation, i.e. favor group assignments that are spatially homogeneous. Finally, the overall scheme iteratively identifies hemodynamic territories as pairs of one HRF pattern and a set of parcels assigned to the corresponding HRF group.

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I. I NTRODUCTION
Functional Magnetic Resonance Imaging (**fMRI**) is a powerful tool to non-invasively study the relation- ship between a sensory or cognitive task and the ensuing evoked neural activity through the neurovascular coupling measured by the BOLD signal [1]. Since the 90’s, this neuroimaging modality has become widely used **in** brain mapping as well as **in** functional connectivity study **in** order to probe the specialization and integration processes **in** sensory, motor and cognitive brain regions [2–4]. **In** this work, we focus on the recovery of localization and dynamics of local evoked activity, thus on specialized cerebral processes. **In** this setting, the key issue is the modeling of the link between stimulation events and the induced BOLD effect throughout the brain. Physiological non-linear models [5–8] are the most specific approaches to properly describe this link but their computational cost and their identifiability issues limit their use to a restricted number of specific regions and to a few experimental conditions. **In** contrast, the common approach, being the focus of this paper, rather relies on linear systems which appear more robust and tractable [2, 9]. Here, the link between stimulation and BOLD effect is modelled through a convolutive system where each stimulus event induces a BOLD response, via the convolution of the binary stimulus sequence with the Hemodynamic Response Function (HRF 1 ). There are two goals for such BOLD analysis: the **detection** of where cerebral activity occurs and the **estimation** of its dynamics through the HRF identification. Commonly, the **estimation** part is ignored and the HRF is fixed to a canonical shape which has been derived from human primary visual area BOLD response [10, 11]. The **detection** task is performed by a General Linear Model (GLM), where stimulus-induced components are assumed to be known and only their relative weighting are to be recovered **in** the form of effect maps [2]. However, spatial intra-**subject** and between-**subject** variability of the HRF has been highlighted [12–14], **in** addition to potential timing fluctuations induced by the paradigm (e.g. variations **in** delay [15]). To take this variability into account, more flexibility can be injected **in** the GLM framework by adding more regressors. **In** a parametric setting, this amounts to adding a function basis, such as canonical HRF derivatives, a set of gamma or logistic functions [15, 16]. **In** a non-parametric setting, all HRF coefficients are explicitly encoded as a Finite Impulse Response (FIR) [17]. The major drawback of these GLM extensions is the multiplicity of regressors for a given condition, so that the **detection** task becomes more difficult to perform and that statistical power is decreased. Moreover, the more coefficients to recover, the more ill-posed the problem becomes. The alternative approaches that aim at keeping a single regressor per condition and add also a temporal regularization constraint to fix the ill-posedness

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Index Terms— Brain activity, hemodynamics, JDE, **fMRI**, Bayesian inference, multisession
1. INTRODUCTION
**In** the context of **fMRI** data analyses, the present paper is a contribu- tion to encoding methods. **In** such studies, two main concerns arise at the **subject**-**level** analysis: (i) a precise localization of evoked brain activity elicited by sensorimotor or cognitive tasks, and (ii) a robust **estimation** of the underlying hemodynamic response associated with these activations. Since these two steps are inherently linked, the **Joint** **Detection**-**Estimation** (JDE) approach [1, 2], has been proposed to face these issues **in** a coordinated formalism. This approach per- forms a multivariate inference for both **detection** and **estimation**. It makes use of a regional bilinear generative model of the BOLD re- sponse and constrains parameter **estimation** by physiological priors using temporal and spatial information **in** a Markovian model. The efficiency and usefulness of this approach has been validated at the group **level** **in** [3] considering single-session datasets.

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Besides, results on HRF estimates are reported for the two JDE versions and compared to the canonical HRF, as well as maps of regularization factor estimates.
Fig. 12 shows results for the VA contrast. High positive values are bilaterally recovered **in** the occipital region and the overall cluster localizations are consistent for both MCMC and VEM algorithms. The only difference lies **in** the temporal auditory regions, especially on the right side, where VEM yields rather more negative values than MCMC. Thus VEM seems more sensitive than MCMC. The results obtained by the classical GLM (see Fig. 12 [right]) are comparable to those of JDE **in** the occipital region with roughly the same **level** of re- covered activations. However, **in** the central region, we observe activations **in** the white matter that can be interpreted as false positives and that were not exhibited using the JDE formalism. The bottom part of Fig. 12 compares the estimated values of the regularization factors between VEM and MCMC algorithms for two experimental conditions involved **in** the VA contrast. Since these estimates are only relevant **in** parcels which are activated by at least one condition, a mask was applied to hide non-activated parcels. We used the following criterion to classify a parcel as activated: (and nonactivated otherwise). These maps of estimates show that VEM yields more contrasted values between the visual and auditory conditions. Table III provides the estimated values **in** the highlighted parcels of interest. The auditory condition does not elicit evoked activity and yields lower values **in** both parcels whereas the visual condition is associated with higher values. The latter comment holds for both algorithms but VEM provides much lower values than MCMC for the non-activated condition. For the activated condition, the situation is comparable, with and . This illustrates a noteworthy difference between VEM and MCMC. Probably due to the mean Þeld and variational approximations, the hidden Þeld may not have the same behaviour (different regularization effect) between the two algorithms. Still, this discrepancy is not visible on the NRL maps.

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This paper is structured as follows. For the sake of self- consistency, the classical **fMRI** analysis framework is sum- marized **in** Section 2. The JDE approach is presented **in** Sec- tion 3. It relies on a prior **parcellation** of **fMRI** data, which derives from a clustering procedure that preserves connectiv- ity and functional homogeneity. Then, at the parcel **level** the JDE framework allows us to specify and estimate a specific BOLD model. Section 4 is devoted to group studies **in** **fMRI** where the principles of random effect analysis are reminded. **In** Section 5, results obtained at the group **level** using different **subject**-**level** inferences are compared on two salient contrasts of interest of a quick **fMRI** mapping experiment. A special at- tention is paid to the HRF variability **in** the motor and parietal regions. Conclusions are drawn **in** Section 6.

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J. Idier, Ph.D., IRCCyN (CNRS), 1 rue de la Noë, BP 92101 44321 Nantes cedex 3, France jerome.idier@irccyn.ec-nantes.fr
(activating or non-activating). The parameter controlling the strength of the spatial correlation is set by hand, as the smoothing **level** used when spatially filtering the data. The combination of these prior distributions with the likelihood allows us to derive the target posterior distribution using Bayes’ rule. We then resort to Gibbs sampling to draw realizations from this posterior law. The posterior mean (PM) estimates of the HRF, the Neural Response Levels (NRLs) and the corresponding labels are directly computed from the generated samples **in** a Markov Chain Monte Carlo (MCMC) procedure. **In** [2], this SMM approach was shown to give a significant gain **in** terms of sensitivity and specificity on artificial **fMRI** data, compared to the IMM counterpart. Since our primary interest was the comparison of SMM with IMM, our simulations followed the generative model and did not truly reflect a “realistic” BOLD data structure. Here, we assess the robustness against putative misspecifications from the assumed model and present first results on real **fMRI** data obtained during an event-related paradigm.

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1 Introduction
**In** **fMRI**, one usually resorts to spatial filtering to enhance the signal-to-noise ratio at the expense of a loss of spatial resolution. A more challenging approach works on the unsmoothed data by introducing some prior knowledge on the sought spatial structures through for instance local interaction models such as Markov Random Fields (MRFs). Discrete MRFs, which have been used **in** seg- mentation and clustering, typically involve a set of hyper-parameters: the smaller this number the less complex the patterns modelled by the corresponding MRF. For instance, a single temperature **level** controls the amount of spatial correlation **in** symmetric Ising fields. **In** the considered **fMRI** application [1], such Ising fields are hidden since the activation **detection** process is modelled a priori through a two-class Spatial Mixture Model (SMM). Moreover, their definition varies within a brain **parcellation** that segregates the 4D data into Γ functionally homogeneous

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The **detection** of the evoked activity and the **estimation** of the dynamics have been mainly addressed as two separate tasks while each of them depends on the other. A precise localization of activations depends on a reliable HRF estimate, while a robust HRF shape is only achievable **in** brain regions eliciting task-related activity [11,12]. **In** this context, the **joint** **detection** **estimation** (JDE) model per- forms both tasks simultaneously [13–15]. **In** the JDE model, a single HRF shape is considered for a specific parcel (group of voxels). Although the JDE model jointly detects the evoked activity within the brain and estimates the HRF, it still requires a prior **parcellation** of the brain into functionally homogeneous regions. This challenge motivated the development of the **joint** **parcellation** de- tection **estimation** (JPDE) model [16,17] that performs online **parcellation** along with the **detection** and **estimation** tasks by setting voxels that share the same HRF pattern **in** the same HRF group (parcel). The JPDE model can be inferred using the VEM algorithm. However, this model still requires manual settings of the number of parcels. To overcome this issue, a model selection procedure was proposed **in** [18] to select the optimum number of parcels. This procedure depends mainly on free energy calculations where the model that maximizes the free energy is the best fit for the data. The limitation of this procedure arises from the fact that it needs to be run for each candidate model which can be time consuming especially if no prior information exists about the number of parcels. The standard JPDE model has been adopted **in** a Bayesian non-parametric ap- proach [19] by making use of the Dirichlet process mixtures model combined with a hidden Markov random field to automatically infer the number of parcels and their shapes simultaneously with the **estimation** and **detection** tasks. **In** this paper, a new approach is proposed to estimate the number of parcels from the **fMRI** BOLD signal. More precisely, we propose to embed the adaptive mean shift algorithm (which is a common clustering algorithm) within the variational inference framework associated with the JPDE model to estimate the parcels and their corresponding HRF profiles.

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Index Terms— **joint** **detection**-**estimation**, hemodynam- ics, Gaussian mixtures, **parcellation**, brain
1. INTRODUCTION
Functional MRI (**fMRI**) is an imaging technique that indi- rectly measures neural activity through the Blood-oxygen- **level**-dependent (BOLD) signal [1], which captures the vari- ation **in** blood oxygenation arising from an external stimula- tion. This variation also allows the **estimation** of the under- lying dynamics, namely the characterization of the so-called hemodynamic response function (HRF). The hemodynamic characteristics are likely to spatially vary, but can be consid- ered constant up to a certain spatial extent. Hence, it makes sense to estimate a single HRF shape for any given area of the brain. To this end, parcel-based approaches that segment **fMRI** data into functionally homogeneous regions and per- form parcelwise **fMRI** data analysis provide an appealing framework [2].

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From a methodological point of view, we have shown that our **joint** **detection**- **estimation** technique is able to identify deactivations **in** the brain. This is owing to the introduction of a third class **in** the prior mixture model associ- ated to the NRLs. Nonetheless, we did not exhibit real deactivations on the analysed datasets. **In** the future, we should therefore validate the 3-class exten- sion on specific datasets. A good candidate could be a dataset acquired during an event-related auditory paradigm **in** which silence events are presented ran- domly to compare activations to a baseline derived from such events. As al- ready shown **in** (Ciuciu et al., 2003), silence events may generate deactivations **in** the temporal lobe if they are presented to the **subject** when the gradients of the scanner are switched off. This will be the **subject** of further work. Smoothing the data spatially provides a reliable manner for recovering clusters of activation instead of isolated spots, at the expense of a loss of resolution. To avoid this preprocessing, the proposed method could be extended by **in**- troducing spatial correlation **in** the prior model. This could be done either on the NRLs ( a) or on the underlying states (labels q). We argue **in** favour of the second solution for simplicity reasons. As already derived for Gaussian mixtures **in** (Vincent et al., 2007b,a), it is quite simple to sample from an Ising (2-class model) or Potts (3-class model) Markov random field (MRF) that enforce neighbouring voxels to be classified **in** the same state (e.g., ac- tivating). This approach actually seems more reasonable **in** terms of compu- tational load than considering edge-preserving MRF based on non-quadratic potentials (Green, 1990; Geman and McClure, 1987). Also, for computational reasons this extension has been developed **in** a supervised framework meaning that the hyper-parameter encoding spatial regularity of the hidden MRF is set by hand. Future work will be focused on an spatially adaptive extension **in** which this parameter is estimated as well, as already done **in** (Woolrich et al., 2005; Woolrich and Behrens, 2006).

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This paper is structured as follows. For the sake of self- consistency, the classical **fMRI** analysis framework is sum- marized **in** Section 2. The JDE approach is presented **in** Sec- tion ??. It relies on a prior **parcellation** of **fMRI** data, which derives from a clustering procedure that preserves connectiv- ity and functional homogeneity. Then, at the parcel **level** the JDE framework allows us to specify and estimate a specific BOLD model. Section ?? is devoted to group studies **in** **fMRI** where the principles of random effect analysis are reminded. **In** Section ??, results obtained at the group **level** using dif- ferent **subject**-**level** inferences are compared on two salient contrasts of interest of a quick **fMRI** mapping experiment. A special attention is paid to the HRF variability **in** the motor and parietal regions. Conclusions are drawn **in** Section ??.

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BM NeuroSpin center, Bˆ at. 145, F-91191 Gif-sur-Yvette, France
Abstract. Although the study of cerebral vasoreactivity using **fMRI** is mainly conducted through the BOLD **fMRI** modality, owing to its relatively high signal-to-noise ratio (SNR), ASL **fMRI** provides a more interpretable measure of cerebral vasoreactivity than BOLD **fMRI**. Still, ASL suffers from a low SNR and is hampered by a large amount of physiological noise. The current contribution aims at improving the re- covery of the vasoreactive component from the ASL signal. To this end, a Bayesian hierarchical model is proposed, enabling the recovery of per- fusion levels as well as fitting their dynamics. On a single-**subject** ASL real data set involving perfusion changes induced by hypercapnia, the approach is compared with a classical GLM-based analysis. A better goodness-of-fit is achieved, especially **in** the transitions between baseline and hypercapnia periods. Also, perfusion levels are recovered with higher sensitivity and show a better contrast between gray- and white matter.

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Titre: Un cadre de détection-**estimation** conjointe pour analyser les données individuelles d’IRMf
Philippe Ciuciu 1 , Thomas Vincent 1 , Laurent Risser 2 and Sophie Donnet 3
Abstract: **In** this paper, we review classical and advanced methodologies for analysing within-**subject** functional Magnetic Resonance Imaging (**fMRI**) data. Such data are usually acquired during sensory or cognitive experiments that aims at stimulating the **subject** **in** the scanner and eliciting evoked brain activity. From such four-dimensional datasets (three **in** space, one **in** time), the goal is twofold: first, detecting brain regions involved **in** the sensory or cognitives processes that the experimental design manipulates; second, estimating the underlying activation dynamics. The first issue is usually addressed **in** the context of the General Linear Model (GLM), which a priori assumes a functional form for the impulse response of the hemodynamic filter. The second question aims at estimating this shape which makes sense **in** activating regions only. **In** the last five years, a novel **Joint** **Detection**-**Estimation** (JDE) framework addressing these two questions simultaneously has been proposed **in** [59, 60, 102]. We show to which extent this methodology outperforms the GLM approach **in** terms of statistical sensitivity and specificity, which additional questions it allows us to investigate theoretically and how it provides us with a well-adapted framework to treat spatially unsmoothed real **fMRI** data both **in** the 3D acquisition volume and on the cortical surface.

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Index Terms— Variational EM, MRF, Biomedical signal **detection**,
Magnetic resonance imaging.
1. INTRODUCTION
Functional Magnetic Resonance Imaging (**fMRI**) is a powerful tool to non-invasively study the relation between cognitive task and cere- bral activity through the analysis of the hemodynamic BOLD sig- nal [4]. Within-**subject** analysis **in** event-related **fMRI** first relies on (i) a **detection** step to localize which parts of the brain are acti- vated by a given stimulus type, and second on (ii) an **estimation** step to recover the temporal dynamics of the brain response. Most ap- proaches to detect neural activity rely on a single a priori model for the temporal dynamics of activated voxels also known as the hemodynamic response function (HRF) [5]. A canonical HRF is usually assumed for the whole brain although there has been evi- dence that this response can vary **in** space or region, across subjects and groups [6]. **In** addition, a robust and accurate **estimation** of the HRF is possible only **in** regions that elicit an evoked response to an experimental stimulus [7]. Both issues of properly detecting evoked activity and estimating the HRF then play a central role **in** **fMRI** data analysis. They are usually dealt with independently with no possi- ble feedback although both issues are strongly connected one to an- other. To introduce more flexibility regarding the assumptions on the HRF model, a novel approach referred to as the **Joint** **Detection** Es- timation (JDE) framework has been introduced **in** [1] and extended **in** [2] to account for spatial correlation between neighboring voxels **in** the brain volume (regular lattice **in** 3D). **In** this latter approach, the HRF can be estimated while simultaneously detecting activity, **in** a region-based analysis, that is on a set of pre-specified regions also named parcels partitioning the whole brain data set. This approach is mainly based on: (i) the non-parametric modelling of the HRF at a regional spatial scale (parcel-**level**) that provides a fair compromise

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3.4 Conclusion
**In** this chapter, we introduced the JPDE model which is an extension of the parcel-based JDE model. This model assumes that a single unknown HRF shape is driving hemodynamic responses **in** a given parcel. Activated voxels within the parcel are then localized by inferring a spatially regularized bi- linear model. One major limitation of the JDE model that it requires the **parcellation** to be fixed a priori by, e.g., using clustering algorithms. The JPDE model solves this problem by avoiding the pre-defined **parcellation**. This model allows the grouping of the regions that share a similar HRF pat- tern and relaxing the hard constraint of a single HRF profile over a given parcel to cope with possible **parcellation** errors. These concerns were ad- dressed by introducing HRF patterns represented by Gaussian distributions and assigned to representative voxels using latent variables. These latent variables are governed by a hidden Markov random field, namely a Potts model that enforces spatial correlation between neighbouring voxels. How- ever, the number of the hemodynamic territories (parcels) has to be specified a priori for the JPDE model. This number has a huge influence of the detec- tion and **estimation** tasks and its adjustment is generally a non-trivial task. **In** this context, we proposed a variational model selection procedure based on the free energy calculation. This procedure was added as an extension to the JPDE model where the free energy was calculated for different candidate models after convergence. Each of these models is characterized by a given number of parcels and the model maximizing the free energy is the best fit for the **fMRI** data. **In** other words, if we have Ω different models then we have to run the JPDE model with the model selection procedure Ω times and compute the value of the free energy each time. The proposed extension was validated using synthetic and real data experiments. For synthetic data, the proposed procedure managed to estimate the correct number of parcels (when compared to the ground truth) for all the experiments. As regards real data, the region of interest was the temporal lobes and the model with two parcels was selected as the best data fit. These results are coherent with those obtained by the JPDE model **in** ( Chaari et al. , 2012 ) where two similar HRF profiles were estimated **in** the left component and one HRF profile was estimated **in** the right component.

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