The human brain is a complex system with high metabolic demands and extensive connectivity that requires control to balance energy consumption and functional efficiency over time. How this control is manifested on a whole-brain scale is largely unexplored, particularly what the associated costs are. Using the network control theory, here, we introduce a novel concept, time-averaged control energy (TCE), to quantify the cost of controlling human brain dynamics at rest, as measured from functional and diffusion MRI. Importantly, TCE spatially correlates with oxygen metabolism measures from the positron emission tomography, providing insight into the bioenergetic footing of resting-state control. Examining the temporal dimension of control costs, we find that brain state transitions along a hierarchical axis from sensory to association areas are more efficient in terms of control costs and more frequent within hierarchical groups than between. This inverse correlation between temporal control costs and state visits suggests a mechanism for maintaining functional diversity while minimizing energy expenditure. By unpacking the temporal dimension of control costs, we contribute to the neuroscientific understanding of how the brain governs its functionality while managing energy expenses. Author Summary Understanding how the brain balances functional efficiency with energy conservation is a central question in neuroscience. The network control theory (NCT) views this question from a network perspective where the brain manages signal propagations along its structural connections to transition across desired activity states. Our study thus presents a novel framework based on the NCT to analyze the costs associated with transitioning across resting states, revealing that regions with high control costs on average are also metabolically demanding in terms of oxygen use. Our findings further show that transitions between sensory and association states are infrequent due to high control costs, while transitions within these states are more common. This suggests that the brain employs a mechanism to preserve functional diversity while minimizing energy costs.

Capturing and tracking large-scale brain activity dynamics holds the potential to deepen our understanding of cognition. Previously, tools from topological data analysis, especially Mapper, have been successfully used to mine brain activity dynamics at the highest spatiotemporal resolutions. Even though it is a relatively established tool within the field of topological data analysis, Mapper results are highly impacted by parameter selection. Given that noninvasive human neuroimaging data (e.g., from fMRI) is typically fraught with artifacts and no gold standards exist regarding “true” state transitions, we argue for a thorough examination of Mapper parameter choices to better reveal their impact. Using synthetic data (with known transition structure) and real fMRI data, we explore a variety of parameter choices for each Mapper step, thereby providing guidance and heuristics for the field. We also release our parameter exploration toolbox as a software package to make it easier for scientists to investigate and apply Mapper to any dataset.

Characterizing large-scale dynamic organization of the brain relies on both data-driven and mechanistic modeling, which demands a low versus high level of prior knowledge and assumptions about how constituents of the brain interact. However, the conceptual translation between the two is not straightforward. The present work aims to provide a bridge between data-driven and mechanistic modeling. We conceptualize brain dynamics as a complex landscape that is continuously modulated by internal and external changes. The modulation can induce transitions between one stable brain state (attractor) to another. Here, we provide a novel method—Temporal Mapper—built upon established tools from the field of topological data analysis to retrieve the network of attractor transitions from time series data alone. For theoretical validation, we use a biophysical network model to induce transitions in a controlled manner, which provides simulated time series equipped with a ground-truth attractor transition network. Our approach reconstructs the ground-truth transition network from simulated time series data better than existing time-varying approaches. For empirical relevance, we apply our approach to fMRI data gathered during a continuous multitask experiment. We found that occupancy of the high-degree nodes and cycles of the transition network was significantly associated with subjects’ behavioral performance. Taken together, we provide an important first step toward integrating data-driven and mechanistic modeling of brain dynamics. Author Summary Brain dynamics are often described by data-driven models or mechanistic dynamical systems models to understand how specific brain states persist or change (transition). However, there lacks a computational framework that explicitly connects states/transitions discovered by data-driven methods to those of mechanistic models, leading to a disconnection between data analysis and theoretical modeling. To begin bridging this gap, we develop a data-driven method, the Temporal Mapper, to extract dynamical systems features from time series and represent them as attractor transition networks. The Temporal Mapper can reconstruct ground-truth transition networks of mechanistic models. When applied to human fMRI data, the method helps predict behavioral performance from the topology of transition networks. Potential applications include characterizing brain dynamic organization in health and diseases and designing brain stimulation protocols.

Author Summary Modern neuroimaging promises to transform how we understand human brain function, as well as how we diagnose and treat mental disorders. However, this promise hinges on the development of computational tools for distilling complex, high-dimensional neuroimaging data into simple representations that can be explored in research or clinical settings. The Mapper approach from topological data analysis (TDA) can be used to generate such representations. Here, we introduce several improvements to the underlying algorithm to aid scalability and parameter selection for high-dimensional neuroimaging data. We also provide new analytical tools for annotating and extracting neurobiological and behavioral insights from the generated representations. We hope this new framework will help facilitate translational applications of precision neuroimaging in clinical settings. Abstract For better translational outcomes, researchers and clinicians alike demand novel tools to distill complex neuroimaging data into simple yet behaviorally relevant representations at the single-participant level. Recently, the Mapper approach from topological data analysis (TDA) has been successfully applied on noninvasive human neuroimaging data to characterize the entire dynamical landscape of whole-brain configurations at the individual level without requiring any spatiotemporal averaging at the outset. Despite promising results, initial applications of Mapper to neuroimaging data were constrained by (1) the need for dimensionality reduction and (2) lack of a biologically grounded heuristic for efficiently exploring the vast parameter space. Here, we present a novel computational framework for Mapper—designed specifically for neuroimaging data—that removes limitations and reduces computational costs associated with dimensionality reduction and parameter exploration. We also introduce new meta-analytic approaches to better anchor Mapper-generated representations to neuroanatomy and behavior. Our new NeuMapper framework was developed and validated using multiple fMRI datasets where participants engaged in continuous multitask experiments that mimic “ongoing” cognition. Looking forward, we hope our framework will help researchers push the boundaries of psychiatric neuroimaging toward generating insights at the single-participant level across consortium-size datasets.

While brain imaging tools like functional magnetic resonance imaging (fMRI) afford measurements of whole-brain activity, it remains unclear how best to interpret patterns found amid the data’s apparent self-organization. To clarify how patterns of brain activity support brain function, one might identify metric spaces that optimally distinguish brain states across experimentally defined conditions. Therefore, the present study considers the relative capacities of several metric spaces to disambiguate experimentally defined brain states. One fundamental metric space interprets fMRI data topographically, that is, as the vector of amplitudes of a multivariate signal, changing with time. Another perspective compares the brain’s functional connectivity, that is, the similarity matrix computed between signals from different brain regions. More recently, metric spaces that consider the data’s topology have become available. Such methods treat data as a sample drawn from an abstract geometric object. To recover the structure of that object, topological data analysis detects features that are invariant under continuous deformations (such as coordinate rotation and nodal misalignment). Moreover, the methods explicitly consider features that persist across multiple geometric scales. While, certainly, there are strengths and weaknesses of each brain dynamics metric space, wefind that those that track topological features optimally distinguish experimentally defined brain states. Author Summary Time-varying functional connectivity interprets brain function as time-varying patterns of coordinated brain activity. While many questions remain regarding how brain function emerges from multiregional interactions, advances in the mathematics of topological data analysis (TDA) may provide new insights. One tool from TDA, “persistent homology,” observes the occurrence and persistence of n -dimensional holes in a sequence of simplicial complexes extracted from a weighted graph. In the present study, we compare the use of persistent homology versus more traditional metrics at the task of segmenting brain states that differ across experimental conditions. We find that the structures identified by persistent homology more accurately segment the stimuli, more accurately segment high versus low performance levels under common stimuli, and generalize better across volunteers.

In this article, we present an open source neuroinformatics platform for exploring, analyzing, and validating distilled graphical representations of high-dimensional neuroimaging data extracted using topological data analysis (TDA). TDA techniques like Mapper have been recently applied to examine the brain’s dynamical organization during ongoing cognition without averaging data in space, in time, or across participants at the outset. Such TDA-based approaches mark an important deviation from standard neuroimaging analyses by distilling complex high-dimensional neuroimaging data into simple—yet neurophysiologically valid and behaviorally relevant—representations that can be interactively explored at the single-participant level. To facilitate wider use of such techniques within neuroimaging and general neuroscience communities, our work provides several tools for visualizing, interacting with, and grounding TDA-generated graphical representations in neurophysiology. Through Python-based Jupyter notebooks and open datasets, we provide a platform to assess and visualize different intermittent stages of Mapper and examine the influence of Mapper parameters on the generated representations. We hope this platform could enable researchers and clinicians alike to explore topological representations of neuroimaging data and generate biological insights underlying complex mental disorders. Author Summary To develop biologically grounded psychiatric diagnosis, researchers and clinicians need tools for distilling complex high-dimensional neuroimaging data into simple yet interactive and clinically relevant representations. Further, for translational outcomes, these representations should be conceivable at the single-participant level. Topological data analysis techniques such as Mapper allow generation of these representations. Here, we introduce a set of tools that can facilitate wider acceptance of Mapper within the neuroscience community and provide a series of easy-to-follow steps for visualizing Mapper-generated graphical representations. We provide detailed examples to reveal the “under-the-hood” workings of Mapper.