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Finding an interpretable and compact representation of complex neuroimaging data is extremely useful for understanding brain behavioral mapping and hence for explaining the biological underpinnings of mental disorders. However, hand-crafted representations, as well as linear transformations, may inadequately capture the considerable variability across individuals. Here, we implemented a data-driven approach using a three-dimensional autoencoder on two large-scale datasets. This approach provides a latent representation of high-dimensional task-fMRI data which can account for demographic characteristics whilst also being readily interpretable both in the latent space learned by the autoencoder and in the original voxel space. This was achieved by addressing a joint optimization problem that simultaneously reconstructs the data and predicts clinical or demographic variables. We then applied normative modeling to the latent variables to define summary statistics ('latent indices') and establish a multivariate mapping to non-imaging measures. Our model, trained with multi-task fMRI data from the Human Connectome Project (HCP) and UK biobank task-fMRI data, demonstrated high performance in age and sex predictions and successfully captured complex behavioral characteristics while preserving individual variability through a latent representation. Our model also performed competitively with respect to various baseline models including several variants of principal components analysis, independent components analysis and classical regions of interest, both in terms of reconstruction accuracy and strength of association with behavioral variables., Competing Interests: The authors have declared that no competing interests exist., (Copyright: © 2024 Zabihi et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

Clinical neuroimaging data availability has grown substantially in the last decade, providing the potential for studying heterogeneity in clinical cohorts on a previously unprecedented scale. Normative modeling is an emerging statistical tool for dissecting heterogeneity in complex brain disorders. However, its application remains technically challenging due to medical data privacy issues and difficulties in dealing with nuisance variation, such as the variability in the image acquisition process. Here, we approach the problem of estimating a reference normative model across a massive population using a massive multi-center neuroimaging dataset. To this end, we introduce a federated probabilistic framework using hierarchical Bayesian regression (HBR) to complete the life-cycle of normative modeling. The proposed model provides the possibilities to learn, update, and adapt the model parameters on decentralized neuroimaging data. Our experimental results confirm the superiority of HBR in deriving more accurate normative ranges on large multi-site neuroimaging datasets compared to the current standard methods. In addition, our approach provides the possibility to recalibrate and reuse the learned model on local datasets and even on datasets with very small sample sizes. The proposed method will facilitate applications of normative modeling as a medical tool for screening the biological deviations in individuals affected by complex illnesses such as mental disorders., Competing Interests: I have read the journal’s policy and the authors of this manuscript have the following competing interests: Ole A. Andreassen is a consultant to HealthLytix and received speaker’s honorarium from Lundbeck. Christian F. Beckmann is shareholder and director of SBG Neuro. This does not alter our adherence to PLOS ONE policies on sharing data and materials., (Copyright: © 2022 Kia et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)