1063-6706 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TFUZZ.2019.2936807, IEEE Transactions on Fuzzy Systems 1 Fuzzy General Linear Modeling for Functional Magnetic Resonance Imaging Analysis Alejandro Veloz, Claudio Moraga, Alejandro Weinstein, Luis Hern´ andez- Garc´ ıa, Steren Chabert, Rodrigo Salas, Rodrigo Riveros, Carlos Bennett, and H´ ector Allende. Abstract—Functional Magnetic Resonance Imaging (fMRI) is a key neuroimaging technique. The classic fMRI analysis pipeline is based on the assumption that the haemodynamic response (HR) is the same across brain regions, time, and subjects. Although convenient, there is ample evidence that this assumption does not hold, and that these differences result in inaccuracies in brain activity detection. This work presents a new fMRI processing pipeline that captures the intrinsic intra- and inter-subject variability of the HR. At the core of this new pipeline is the definition of a fuzzy haemodynamic response function (HRF). The proposed pipeline includes a new fuzzy General Linear Model (GLM) able to handle the fuzzy HRF, including a practical realization based on the LR representation of fuzzy numbers. The work also describes how to obtain activation maps from the fuzzy GLM, and a methodology to compute the statistical power of the analysis. The method is evaluated in synthetic and real fMRI data and compared with other state-of-the-art techniques. The experiments based on synthetic data show that the fuzzy GLM approach is more robust under uncertainty regarding the true specific shape of the HR. The experiments based on real data show an increased volume of the activated brain areas, suggesting that the proposed method is able to prevent false negative errors in the boundaries of target brain regions in which HR should be negligible. Index Terms—Fuzzy numbers, fuzzy haemodynamic response function, fuzzy general linear model, functional magnetic reso- nance imaging. I. I NTRODUCTION F UNCTIONAL Magnetic Resonance Imaging (fMRI) is a technique widely used in the fields of neuroscience and experimental psychology. It is also used clinically for planning brain surgeries or for radiation therapy. This neuroimaging technique allows to detect activation of a small group of neurons thanks to the series of biophysical changes known in the literature as the BOLD (Blood Oxygenation Level Alejandro Veloz (corresponding author) is with the Department of Infor- matics, Universidad T´ ecnica Federico Santa Mar´ ıa, the School of Biomedical Engineering, and Centro de Investigaci ´ on y Desarrollo en Ingenier´ ıa en Salud, Universidad de Valpara´ ıso, Chile, e-mail: alejandro.veloz@uv.cl. Claudio Moraga is with the TU Dortmund University, Germany, e-mail: claudio.moraga@tu-dortmund.de. Luis Hern´ andez-Garc´ ıa is with the Department of Biomedical Engineering, University of Michigan, Ann Arbor, USA, e-mail: hernan@umich.edu. Alejandro Weinstein, Steren Chabert, and Rodrigo Salas are with the School of Biomedical Engineering, and Centro de Investigaci ´ on y Desarrollo en Ingenier´ ıa en Salud, Universidad de Valpara´ ıso, Chile, e-mails: alejan- dro.weinstein@uv.cl, steren.chabert@uv.cl, rodrigo.salas@uv.cl. Carlos Bennett and Rodrigo Riveros are with the Hospital Carlos van Buren, and the School of Medicine, Universidad de Valpara´ ıso, Chile, e-mails: carlos.bennett@uv.cl, rodrigo.riveros@uv.cl. H´ ector Allende is with the Department of Informatics, Universidad T´ ecnica Federico Santa Mar´ ıa, Chile, e-mail: hallende@inf.utfsm.cl. Dependent) effect [1]. This phenomenon is mainly related to the increase of oxygen consumption required by these neurons to achieve metabolic equilibrium. Functional brain mapping using Magnetic Resonance Imaging (MRI) is based on measuring the response caused by the changes induced by the BOLD effect, which is referred to as BOLD response or haemodynamic response (HR). A key element of the data processing pipeline used in fMRI is the characterization of the haemodynamic response (HR) of the subject [2]–[4]. The HR exhibits large variability in its characteristic param- eters (time-to-peak, response delay, dispersion and amplitude) between brain areas [5]–[7], across the life span [8], [9], and due to structural or functional changes following brain damage [10]–[15]. However, the standard hypothesis-driven method for modeling the haemodynamic response is based on a single fixed function that is used as a “one size fits all” approach that ignores the intrinsic intra- and inter-subject variability. This function, called the canonical Haemodynamic Response Function (HRF), cannot accommodate individual HR response variations, resulting in inaccuracies in the detection of brain activity [6], [7], [16]. There are two main categories for fMRI experiments: resting- state and task-related. In resting-state experiments there are no external stimuli presented to the subject. This paradigm is used to evaluate the spontaneous interactions of brain areas which are rendered as brain connectivity maps. The HR variability also impacts directly on brain connectivity analyses performed to discover brain networks, by missing connections or inferring spurious connections, rendering the results inaccurate or uninterpretable [17]. In task-related experiments a series of known external stimuli are presented to the subject. This paradigm is used to find the areas of the brain activated by these stimuli. Task-related fMRI analyses are used to map the functional anatomy of the brain, and for the localization of eloquent areas prior to surgical procedures. However, the increase in he type II error observed in empirical studies due to the HR variability has recently caused criticism within the clinical community [10]–[15], [18], [19]. In clinical applications, a type II error can lead to the removal of vital brain tissue during a surgery. This work focuses on task-related fMRI experiments. Many attempts have been made to model the variability of the HR response. However, they lack the expressiveness to model the wide range of variations observed empirically or they do not incorporate knowledge related to the biophysics of the HR. The details of such methods are described in the next section.