Brain Modes, Components, and Gradients: Theory and Applications

The dynamics of many physical systems are often shaped by the constraints imposed by their underlying structure, and the brain is no exception. While many studies have explored relationships between brain anatomy and function, a unified framework for understanding how brain dynamics emerge from its relatively stable neuroanatomical scaffold has been lacking. Moreover, the specific anatomical properties that fundamentally constrain neuronal dynamics remain unclear. In our recent work, we have highlighted the previously underappreciated role of brain geometry in constraining brain dynamics. Specifically, we showed that geometric eigenmodes—eigenmodes derived from the brain’s cortical and subcortical geometry—can effectively capture diverse experimental human fMRI data from spontaneous and task-evoked recordings.

In this talk, I will provide an overview of the potential of geometric eigenmodes in understanding brain structure and dynamics. Key topics will include: (1) An accessible description of the theoretical foundation of geometric eigenmodes and their connection to neural field theory, a popular class of macroscopic mathematical model of the brain; (2) A practical demonstration of how geometric eigenmodes of the cortex and subcortex can be extracted from T1-weighted MRI data using our open-access code at https://github.com/NSBLab/BrainEigenmodes; (3) A description of how geometric eigenmodes can be applied to reconstruct diverse neuroimaging data; (4) A discussion of the advantages of using geometric eigenmodes, as well as common implantation challenges that one need to; (5) A demonstration that geometric eigenmodes are tied to a generative model of wave dynamics, which can reproduce numerous canonical features of functional brain organization; (6) A discussion of the generalizability of geometric eigenmodes across individuals and species; and (7) An overview of some examples of diverse applications of geometric eigenmodes in studying brain structure, function, and organization. 

The talk will overview conceptual and methodological advances to study gradients in functional brain organization. It will focus on diffusion map embedding, which is a widely used non-linear technique to reduce the dimensionality of large-scale brain mapping datasets. It works by computing eigenvectors and eigenvalues of a diffusion operator applied to the data. Diffusion map embedding thereby translates short- and long-range connectivity into distance relations within a lower dimensional embedding space, where regions with similar connectivity profiles and strong inter-connectivity are close together, while regions with little inter-connectivity are dissimilar connectivity profiles are further apart.

Diffusion map embedding has several strengths: (i) it can capture non-linear relationships between data points; (ii) it maintains the global structure of the underlying data and produces easily interpretable results (e.g., connectivity gradients); and (iii) compared to other methods, diffusion map embedding is relatively robust to noise and computationally inexpensive. Limitations of diffusion maps include ease of interpretability, stability, and simplicity compared to linear methods such as PCA, which have been found to often detect similar underlying patterns in large-scale brain mapping datasets.

This talk will cover the following specific aspects of the theory and implementation of diffusion map embedding, including a (very brief and accessible) overview into its mathematics, a differentiation from other dimensionality reductions techniques, the choice of related affinity kernels and alignment techniques, and its increasing adoption to study lower dimensional spatial patterns of functional as well as structural brain organization in both humans and non-human animals.

I will furthermore provide practical guidance on how to derive, align, and visualize brain gradients via the open access BrainSpace toolbox (http://brainspace.readthedocs.io). Here, we will provide a run through to demonstrate how a few lines of python (or matlab) code can suffice to identify lower-dimensional gradients of brain organization, based on open access neuroimaging datasets.  

This talk will provide an overview of the practical applications of ICA to investigate individual differences in functional organization in health and disease. The talk will begin with a description of different outputs from the ICA-dual regression pipeline: subject-specific mode spatial maps, timeseries, and amplitudes. I will summarize the rich literature investigating the relationships of these ICA outputs to behavior and symptomatology in large-scale datasets including the Human Connectome Project and UK Biobank. Specific examples related to arousal, mental health, genetics, and the general positive-negative behavioral axis will be discussed.

The software run-through will provide the audience with intuition for the role of dimensionality in ICA decomposition by demonstrating how relatively larger networks get split into multiple separate subcomponents as dimensionality increases. In the strengths/limitations section, ICA will be contrasted with other statistical network decomposition approaches, such as seed-based analysis and probabilistic functional modes. Seed-based analysis involves selecting a seed region of interest and computing the whole-brain connectivity map with the seed region. Benefits of seed-based analysis include simplicity and interpretability, while disadvantages include seed sensitivity and lack of multivariate network modelling. Probabilistic functional modes offer a Bayesian alternative to ICA with benefits from improved sensitivity to individual differences and ability to model network overlap. By comparison, ICA allows multivariate network modelling while being limited by the lack of sensitivity to network overlap due to the independence constraint. The talk will close with a brief discussion of the challenges posed by analytical flexibility in brain representations. For example, it can be difficult to take two papers on the same topic using different mode-based analyses and determine whether the results are consistent or not. As such, the field needs new tools to facilitate cross-study comparison and interpretation.  

Inferring the relationship between different brain maps is a topic of substantial interest. Identifying true associations requires knowledge about the distribution of correlations that arise by chance in the presence of smoothness (or spatial autocorrelation; SA) in these maps. This null distribution can be generated from an ensemble of surrogate brain maps that preserve the intrinsic SA but break the correlations between maps. This educational session introduces the use of “eigenstrapping”, a novel method involving the spectral decomposition of cortical and subcortical surfaces in terms of their geometric eigenmodes, and then random rotations of these modes to produce SA-preserving surrogate brain maps. The null distributions generated by this method properly reflect the null hypothesis of no association between brain maps in the presence of SA, offering a powerful tool for brain map inference.

This session will provide an accessible introduction to geometric eigenmodes, emphasising their role in brain mapping and their mathematical foundations. Participants will learn how the method works both theoretically and practically, and how it can be a powerful tool for improving the reliability of brain map analysis. A practical demonstration will involve participants in the step-by-step implementation of the method using Jupyter notebooks, with several applications to real-world brain-imaging data. Additionally, we will present a case study illustrating how this method enhances the statistical robustness of findings in neuroscience.

Designed for researchers, students, and clinicians in neuroscience and computational fields at all levels, this session bridges theoretical concepts with practical application. Participants will be engaged with quizzes and short problem-solving tasks and will leave with a deeper understanding of the potential of geometric eigenmodes in improving brain map analysis and statistical inference. The session will conclude with a guided discussion, inviting participants to explore how they can integrate this method into their own research. 

Brain activity unfolds over an intricately connected network of white matter fibers – the structural connectome. By generalising the well-known Fourier transform to the network structure of the connectome, the mathematics of structural eigenmode decomposition allows us to predict expected patterns of brain activity. Structural eigenmodes are distributed patterns of brain activity, each associated with a specific spatial frequency, from coarse- to fine-grained. Although this approach tells us how the spatial organisation of brain activity arises from the connectome, a, a puzzle remains: How does this static network give rise to the rich dynamics that characterise the living brain? What governs the relative prevalence of structural eigenmodes at different points in time? The answer comes from neuromodulation. Neurons express a wide array of neurotransmitter receptors, which control cellular and ultimately regional excitability and receptivity to incoming inputs, according to macroscopic gradients.

This educational session introduces the combined use of structural eigenmodes and pharmacological-fMRI as a means of interrogating the relationship between brain structure and dynamics. Through specific examples with real data, attendees will learn how to extract and interpret structural eigenmodes’ contribution to brain dynamics, and their relative change under different pharmacological perturbations. We will place emphasis on using structural eigenmodes to disentangle how different perturbations can induce convergent effects on dynamics, which are obscured by traditional approaches to neuroimaging data analysis. We will also show how using the same decomposition (eigenmodes of the structural connectome) and imaging modality (functional MRI) provides a ‘common currency’ to compare brain dynamics and their perturbations in different species. Finally, we will guide attendees on how to combine structural eigenmodes with publicly available PET maps of receptor density and meta-analytic maps of task-related activity in the human brain, to model transitions between different activity patterns of interest. Throughout, we will emphasise how practitioners can identify and use appropriate null models to test neurobiological hypotheses using structural eigenmodes. An interactive session will demonstrate this principle on concrete hypotheses provided by session participants, to ensure relevance for their own work.

The goal of this session is to enable participants with the conceptual tools and practical know-how to integrate the method of structural eigenmodes into their own work. We intend to ensure that the field can converge towards best practices that are statistically rigorous and biologically meaningful. To lower barriers to adoption and ensure broad accessibility across the diverse backgrounds of OHBM attendees, the session will be self-contained, without requiring formal training in mathematics. Rather, emphasis will be placed on balancing rigour and intuitive understanding with practical relevance. We will conclude by addressing conceptual pitfalls, practical caveats and open challenges of this approach in an open discussion. 

The human cortex is intricately organized along multiple overlapping spatial axes, which can be revealed using advanced dimensionality reduction techniques like PCA and diffusion map embedding. These methods enable us to uncover spatial patterns from in vivo structural and functional connectomes and explore how they relate to other brain features, such as microstructure and functional mapping. While these approaches provide a compelling framework to understand the brain's spatial organization, significant questions remain, particularly regarding individual differences. For instance: How do these gradients vary from person to person? What is the relationship between structural and functional gradients? And what practical insights can we gain by conceptualizing the brain through its gradient organization? Answering these questions requires integrating theory-driven research with diverse datasets and innovative methodologies. By investigating the unique ways these spatial axes appear in individuals, we can better understand their roles in health and disease.

This educational session will introduce participants to cutting-edge approaches for studying individual variability in brain axes, drawing on examples ranging from biophysical modeling to the influence of the menstrual cycle. Through real-world data demonstrations, attendees will learn how to identify, interpret, and apply gradients in the context of individual differences. The session emphasizes a multimodal perspective to uncover variability while highlighting critical caveats in interpreting findings. Key theoretical frameworks, such as the dual origin theory and structural model, will be explored to provide a deeper understanding of brain gradients.

Participants will also have the opportunity to engage in hands-on learning through a guided tutorial, where they will create and interpret gradients using their own data. In addition, the session will provide access to an online resource, including a dedicated website containing tutorials, best practices, and additional information on gradient analysis. These resources aim to support participants in applying what they’ve learned long after the session ends.

The session will underscore best practices for statistical evaluation across individuals and modalities, offering practical workflows to investigate relationships and test hypotheses. Interactive components will enable participants to apply these methods directly to their research questions, ensuring relevance and actionable insights.

The goal is to equip attendees with both the conceptual framework and practical expertise to integrate gradient analysis into their work. This session is designed to make these advanced methods accessible to a wide audience, regardless of formal training in mathematics or neuroscience. Emphasis will be placed on balancing theoretical depth with practical application, making the content approachable for OHBM attendees from diverse backgrounds.

We will conclude with an open discussion, addressing common pitfalls, practical challenges, and open questions in the field. By fostering a collaborative environment and providing long-term resources like tutorials and the website, we aim to advance best practices that are both statistically sound and biologically meaningful, ultimately enabling a broader adoption of gradient-based approaches in neuroscience research.

Summary of learning goals:
- Gain a clear understanding of the spatial axes (gradients) underlying cortical organization and their relevance to structural and functional brain connectivity.
- Learn how gradients vary across individuals and how these differences relate to factors such as brain microstructure, functional mappings, and external variables (e.g., menstrual cycle).
- Acquire practical experience in creating and interpreting gradients using real data, including guidance on statistical evaluation and visualization.
- Explore key theories such as the dual origin theory and structural model to deepen understanding of gradient-based brain organization.
- Engage with peers in interactive discussions to address challenges, brainstorm applications, and refine understanding of the method.
 

The relation between functional activity and the underlying structure in the brain has revealed to be complex and has gained increasing attention. In this context, graph signal processing (GSP) represents a novel framework allowing to link dynamic functional activity signals, e.g., from functional magnetic resonance imaging (fMRI), but also other modalities such as electro- and magneto-encephalography (EEG, MEG), with the brain structural architecture in a non-trivial way. Structural brain modes are extracted from a structural connectivity graph using Laplacian eigendecomposition, and functional activity patterns are represented as graph signals, defined on top of the graph. The Graph Fourier Transform allows then to decompose functional signals into structural bases and graph spectral filtering can be used to distinguish portions of functional activity that are more or less smooth on the graph; i.e., coupled or decoupled from brain structure, respectively. The application of this framework to healthy and pathological brain data has revealed key features of brain structure-function coupling, such as: (i) a cortical distribution along a gradient opposing lower to higher level cognitive regions; (ii) its individual specificity; (iii) its changes in resting-state and different tasks; (iv) a dynamic behavior; (v) the capability of highlighting abnormal brain features in diverse clinical contexts.
In this talk, I will give a detailed outline of the methodological framework for brain GSP, guiding the audience through the main conceptual bases on which GSP is founded; i.e.: (i) Graph and Graph signal definition, in the context of brain imaging; (ii) Graph Laplacian eigendecomposition to obtain structural brain modes or harmonics, constituting the graph spectral domain; (iii) projection of the functional signals into the graph spectral domain through the application of the Graph Fourier Transform; (iv) filtering of the functional signals in the spectral domain, e.g., with high/low pass filters allowing to reconstruct portions of the signals including only a partial number of harmonics; (v) extrapolation of node-wise and edge-wise structure-function coupling metrics; i.e., structural decoupling index and coupled / decoupled functional connectivity.
A thorough review of the existent literature on the topic will be included, where limitations and future challenges of the GSP framework will be highlighted and discussed.
A practical demonstration of the brain GSP pipeline by use of NiGSP, open source Python-based toolbox that we developed for this purpose. A real dataset of functional and diffusion MRI will be used to demonstrate applied examples within the toolbox. 

The brain connectome is multidimensional, characterized by multiple axes that reflect the spatial configurations of macroscale organizational principles. These axes can be uncovered using dimensionality reduction algorithms, such as Principal Component Analysis (PCA) and diffusion map embedding, which simplify complex connectivity patterns into interpretable gradients. The resulting low-dimensional representations of brain organization often reveal systematic spatial patterns, such as the hierarchical axis from unimodal to transmodal regions, but they also exhibit substantial variability across individuals. This variability raises critical questions: How can we align these low-dimensional representations to enable meaningful comparisons between individuals, capturing shared features while preserving unique, individual-specific characteristics? Furthermore, how can these alignment methods be generalized to cross-species comparison to uncover conserved and species-specific principles of brain organization?
Building on these questions, this session will introduce linear and nonlinear dimensionality reduction methods, focusing on principal component analysis and diffusion embedding algorithms to capture gradients of the functional connectome. We will highlight the individual variability observed in these gradients and the necessity of aligning them across individuals for meaningful comparisons. We will start with the traditional approach, Procrustes matching, which uses rigid transformations (e.g., rotation, scaling, translation) to align the direction, order, and scale of gradients across individuals. In addition, we will introduce the joint-embedding technique, which embeds multiple connectomes simultaneously to extract shared and aligned components, enabling direct comparisons across individuals and species. Furthermore, we will discuss scalable frameworks for joint-embedding methods, designed to facilitate alignment for large datasets across developmental stages.
Attendees will gain a comprehensive understanding of dimensionality reduction algorithms, the concept of low-dimensional axes, their variability across individuals, and the factors contributing to this variation. Through practical examples and shared resources, participants will learn how to calculate, align, and evaluate gradients for high-dimensional data across diverse populations. The session will also include discussions on best practices for calculating, aligning, and evaluating these gradients in human and nonhuman primate data.
 

Independent component analysis (ICA) is one of the most widely used approaches for estimating intrinsic networks from fMRI data and has been in use for over 25 years now. Over that time, numerous extensions and variations of ICA, a method that moves beyond second order methods and focuses on higher order statistics under the broader umbrella of (joint) blind and semi-blind source separation, have been developed including group ICA, spatial and temporal ICA, spatially constrained ICA, hierarchical ICA, nonlinear ICA, dynamic ICA, multimodal ICA, and much more. ICA has contributed in no small part to a transformation of our study of functional neuroimaging data. In this talk I will attempt to summarize the methods developed over the past 25 years, as well as provide an overview of how ICA has been used to decompose neuroimaging data, estimate intrinsic networks, and visualize transitions and relationships within and between networks. I will attempt to provide comprehensive classification of the many ICA approaches into categories, highlighting methodological aspects and their relationship to one another and other methods such as seed-based approaches, gradient approaches, and others. I will also provide examples of the scientific contributions these approaches have made to our understanding of the healthy and disordered human brain. In addition to this, my talk will summarize the various tools and approaches that are available to the community for implementing ICA on one’s own data. A few hands-on examples using the GIFT/FIT tools will be provided, including interoperable versions that work with the brain imaging data structure (BIDS) and which provides easy to use, containerized tools for use on any computer environment. The talk will be designed to be didactic and interactive, allowing for a robust engagement with the attendees with a goal of facilitating not just facts but also an intuitive understanding of the approaches and providing a pathway for individuals to use the tools for themselves. I will also place ICA within the larger context of brain modes and gradients, which can help those that use these approaches understand the correspondence, as well as strengths and limitations of the many published studies to date in addition to setting the stage for future innovative developments and finding going forward. In sum, the goal of this talk is to showcase the exciting ways ICA has been used with a eye towards enabling attendees to leverage existing approaches to generate new findings and extensions going forward.