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Mindful attention is characterized by acknowledging the present experience as a transientmental event. Early stages of mindfulness practicemay require greater neural effort for later efficiency. Early effort may self-regulate behavior and focalize the present, but this understanding lacks a computational explanation. Here we used network control theory as a model of how external control inputs—operationalizing effort—distribute changes in neural activity evoked during mindful attention across the white matter network.We hypothesized that individuals with greater network controllability, thereby efficiently distributing control inputs, effectively self-regulate behavior. We further hypothesized that brain regions that utilize greater control input exhibit shorter intrinsic timescales of neural activity. Shorter timescales characterize quickly discontinuing past processing to focalize the present.We tested these hypotheses in a randomized controlled study that primed participants to either mindfully respond or naturally react to alcohol cues during fMRI and administered text reminders and measurements of alcohol consumption during 4 wk postscan. We found that participants with greater network controllability moderated alcohol consumption. Mindful regulation of alcohol cues, compared to one’s own natural reactions, reduced craving, but craving did not differ from the baseline group. Mindful regulation of alcohol cues, compared to the natural reactions of the baseline group, involved more-effortful control of neural dynamics across cognitive control and attention subnetworks. This effort persisted in the natural reactions of the mindful group compared to the baseline group. More-effortful neural states had shorter timescales than less effortful states, offering an explanation for how mindful attention promotes being present.

•Relationship between emergent dynamics, functional hierarchy, and controllability.•DOC patients exhibit less hierarchical and emergent neural dynamics.•DOC patients’ structural connectomes exhibit compromised modal controllability.•Whole-brain model based on patient connectomes recapitulates functional alterations. High-level brain functions are widely believed to emerge from the orchestrated activity of multiple neural systems. However, lacking a formal definition and practical quantification of emergence for experimental data, neuroscientists have been unable to empirically test this long-standing conjecture. Here we investigate this fundamental question by leveraging a recently proposed framework known as “Integrated Information Decomposition,” which establishes a principled information-theoretic approach to operationalise and quantify emergence in dynamical systems — including the human brain. By analysing functional MRI data, our results show that the emergent and hierarchical character of neural dynamics is significantly diminished in chronically unresponsive patients suffering from severe brain injury. At a functional level, we demonstrate that emergence capacity is positively correlated with the extent of hierarchical organisation in brain activity. Furthermore, by combining computational approaches from network control theory and whole-brain biophysical modelling, we show that the reduced capacity for emergent and hierarchical dynamics in severely brain-injured patients can be mechanistically explained by disruptions in the patients’ structural connectome. Overall, our results suggest that chronic unresponsiveness resulting from severe brain injury may be related to structural impairment of the fundamental neural infrastructures required for brain dynamics to support emergence.

1. An important dimension of a species' role is its ability to alter the state and maintain the diversity of its community. Centrality metrics have often been used to identify these species, which are sometimes referred as \"keystone\" species. However, the relationship between centrality and keystoneness is largely phenomenological and based mostly on our intuition regarding what constitutes an important species. While centrality is useful when predicting which species' extinctions could cause the largest change in a community, it says little about how these species could be used to attain or preserve a particular community state. 2. Here we introduce structural controllability, an approach that allows us to quantify the extent to which network topology can be harnessed to achieve a desired state. It also allows us to quantify a species' control capacity—its relative importance—and identify the set of species that are critical in this context because they have the largest possible control capacity. We illustrate the application of structural controllability with ten pairs of uninvaded and invaded plant-pollinator communities. 3. We found that the controllability of a community is not dependent on its invasion status, but on the asymmetric nature of its mutual dependences. While central species were also likely to have a large control capacity, centrality fails to identify species that, despite being less connected, were critical in their communities. Interestingly, this set of critical species was mostly composed of plants and included every invasive species in our dataset. We also found that species with high control capacity, and in particular critical species, contribute the most to the stable coexistence of their community. This result was true, even when controlling for the species' degree, abundance/interaction strength, and the relative dependence of their partners. 4. Synthesis. Structural controllability is strongly related to the stability of a network and measures the difficulty of managing an ecological community. It also identifies species that are critical to sustain biodiversity and to change or maintain the state of their community and are therefore likely to be very relevant for management and conservation.

•The ALECM considers the complex interactions along the white matter network.•The ALECM reveals that SZ and BD require higher energy for Hetero-state transition.•The ALECM successfully induced Hetero-state transition in the patients' brains. The brain, as a complex system, achieves state transitions through interactions among its regions and also performs various functions. An in-depth exploration of brain state transitions is crucial for revealing functional changes in both health and pathological states and realizing precise brain function intervention. Network control theory offers a novel framework for investigating the dynamic characteristics of brain state transitions. Existing studies have primarily focused on analyzing the energy required for brain state transitions, which are driven either by the single brain region or by all brain regions. However, they often neglect the critical question of how the whole brain responds to external control inputs that are driven by control energy from multiple brain regions, which limits their application value in guiding clinical neurostimulation. In this paper, we proposed the Adaptive Local Energy Control Model (ALECM) to explore brain state transitions, which considers the complex interactions of the whole brain along the white matter network when external control inputs are applied to multiple regions. It not only quantifies the energy required for state transitions but also predicts their outcomes based on local control. Our results indicated that patients with Schizophrenia (SZ) and Bipolar Disorder (BD) required more energy to drive the brain state transitions from the pathological state to the healthy baseline state, which is defined as Hetero-state transition. Importantly, we successfully induced Hetero-state transition in the patients' brains by using the ALECM, with subnetworks or specific brain regions serving as local control sets. Eventually, the network similarity between patients and healthy subjects reached baseline levels. These offer evidence that the ALECM can effectively quantify the cost characteristics of brain state transitions, providing a theoretical foundation for accurately predicting the efficacy of electromagnetic perturbation therapies in the future.

The energetic and entropic organization of the brain's functional activity in mild cognitive impairment (MCI) has yet to be fully characterized. Network Control Theory (NCT) is a multi‐modal approach that captures alterations in the brain's energetic landscape by combining the brain's functional activity and the structural connectome. Entropy is another complementary metric that can quantify the complexity and predictability in a neural time series, offering insights into the brain's dynamic functional activity. Our study aims to explore the differences in the brain's energetic and entropic landscape between people with MCI and healthy controls (HC). Four hundred ninety‐nine HC and 55 MCI patients were included. First, k‐means clustering was applied to functional MRI (fMRI) time series to identify commonly recurring brain activity states. Second, NCT was used to calculate the minimum energy required to transition between these brain activity states, otherwise known as transition energy (TE). The entropy of the fMRI time series as well as PET‐derived amyloid beta (Aβ) and tau deposition were measured for each brain region. The TE and entropy were compared between MCI and HC at the network, regional, and global levels using linear models where age, sex, and intracranial volume were added as covariates. The association of TE and entropy with Aβ and tau deposition was investigated in MCI patients using linear models where age, sex, and intracranial volume were controlled. Commonly recurring brain activity states included those with high (+) and low (‐) amplitude activity in visual (+/‐), default mode (+/‐), and dorsal attention (+/‐) networks. Compared to HC, MCI patients required lower transition energy in the limbic network (adjusted p = 0.028). Decreased global entropy was observed in MCI patients compared to HC (p = 7.29e‐7). There was a positive association between TE and entropy in the frontoparietal network (p = 7.03e‐3). Increased global Aβ was associated with higher global entropy in MCI patients (ρ = 0.632, p = 0.041). Lower TE in the limbic network in MCI patients may indicate either neurodegeneration‐related neural loss and atrophy or a potential functional upregulation mechanism in this early stage of cognitive impairment. Future studies that include people with Alzheimer's Disease (AD) are needed to better characterize the changes in the energetic landscape in the later stages of cognitive impairment. We examined the shifts in the brain energetics computed with network control theory and entropy in people with mild cognitive impairment (MCI) compared to healthy controls (HC). MCI patients exhibit lower transition energy in the limbic network, reduced global entropy, and a positive Aβ‐entropy association compared to HC, suggesting a disrupted energetic and entropic landscape as potential neuroimaging biomarkers of MCI.

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. 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.

Background Trigeminal neuralgia (TN) involves complex neural network alterations beyond the trigeminal system. Network Control Theory (NCT) offers a novel framework to quantify how brain network architecture constrains neural dynamics. This study investigated structural network controllability in TN to elucidate disease-specific alterations in brain network control properties. Methods Eighty-two TN patients and 42 healthy controls (HCs) underwent diffusion tensor imaging. Structural connectomes were constructed using deterministic tractography and parcellated with the Brainnetome atlas. Average controllability (AC), reflecting the ease of driving networks toward accessible states, and modal controllability (MC), indicating the capacity for difficult state transitions, were calculated at whole-brain, network, and regional levels. Age-related effects on controllability were examined. Results TN patients demonstrated significantly reduced whole-brain AC ( P  = 0.009) and increased MC ( P  = 0.009) compared to HCs. Network-level analyses revealed decreased AC and increased MC in the dorsal attention network ( P  = 0.018) and default mode network ( P  = 0.009), with reduced AC in subcortical regions ( P  = 0.041). No regional differences survived False Discovery Rate correction. Notably, controllability metrics correlated significantly with age in TN patients across multiple networks, whereas HCs showed no age-related correlations. Neither pain laterality nor neurovascular compression influenced controllability patterns. Conclusions TN is characterized by aberrant network controllability, manifesting as reduced efficiency in routine state transitions and increased energy requirements for network control. The unique age-controllability relationship in TN suggests disease-specific alterations in network dynamics distinct from normal aging. These findings establish NCT as a valuable framework for understanding TN pathophysiology and highlight the disorder’s network-level rather than focal nature. Graphical Abstract

Quantifying brain state transition cost is a fundamental problem in systems neuroscience. Previous studies utilized network control theory to measure the cost by considering a neural system as a deterministic dynamical system. However, this approach does not capture the stochasticity of neural systems, which is important for accurately quantifying brain state transition cost. Here, we propose a novel framework based on optimal control in stochastic systems. In our framework, we quantify the transition cost as the Kullback-Leibler divergence from an uncontrolled transition path to the optimally controlled path, which is known as Schrödinger Bridge. To test its utility, we applied this framework to functional magnetic resonance imaging data from the Human Connectome Project and computed the brain state transition cost in cognitive tasks. We demonstrate correspondence between brain state transition cost and the difficulty of tasks. The results suggest that our framework provides a general theoretical tool for investigating cognitive functions from the viewpoint of transition cost. In our daily lives, we perform numerous tasks with different kinds and levels of cognitive demand. To successfully perform these tasks, the brain needs to modulate its spontaneous activity to reach an appropriate state for each task. Previous studies utilized optimal control in deterministic systems to measure the cost for the brain state transition. However, no unified framework for quantifying brain state transition cost that takes account of the stochasticity of neural activities has been proposed. Here, we describe a novel framework for measuring brain state transition cost, utilizing the idea of optimal control in stochastic systems. We assessed the utility of our framework for quantifying the cost of transitioning between various cognitive tasks. Our framework can be applied to very diverse settings because of its generality.

The control of networked dynamical systems opens the possibility for new discoveries and therapies in systems biology and neuroscience. Recent theoretical advances provide candidate mechanisms by which a system can be driven from one pre-specified state to another, and computational approaches provide tools to test those mechanisms in real-world systems. Despite already having been applied to study network systems in biology and neuroscience, the practical performance of these tools and associated measures on simple networks with pre-specified structure has yet to be assessed. Here, we study the behavior of four control metrics (global, average, modal, and boundary controllability) on eight canonical graphs (including Erdős–Rényi, regular, small-world, random geometric, Barábasi–Albert preferential attachment, and several modular networks) with different edge weighting schemes (Gaussian, power-law, and two nonparametric distributions from brain networks, as examples of real-world systems). We observe that differences in global controllability across graph models are more salient when edge weight distributions are heavy-tailed as opposed to normal. In contrast, differences in average, modal, and boundary controllability across graph models (as well as across nodes in the graph) are more salient when edge weight distributions are less heavy-tailed. Across graph models and edge weighting schemes, average and modal controllability are negatively correlated with one another across nodes; yet, across graph instances, the relation between average and modal controllability can be positive, negative, or nonsignificant. Collectively, these findings demonstrate that controllability statistics (and their relations) differ across graphs with different topologies and that these differences can be muted or accentuated by differences in the edge weight distributions. More generally, our numerical studies motivate future analytical efforts to better understand the mathematical underpinnings of the relationship between graph topology and control, as well as efforts to design networks with specific control profiles.

Quantifying the relationship between the brain’s functional activity patterns and its structural backbone is crucial when relating the severity of brain pathology to disability in multiple sclerosis (MS). Network control theory (NCT) characterizes the brain’s energetic landscape using the structural connectome and patterns of brain activity over time. We applied NCT to investigate brain-state dynamics and energy landscapes in controls and people with MS (pwMS). We also computed entropy of brain activity and investigated its association with the dynamic landscape’s transition energy and lesion volume. Brain states were identified by clustering regional brain activity vectors, and NCT was applied to compute the energy required to transition between these brain states. We found that entropy was negatively correlated with lesion volume and transition energy, and that larger transition energies were associated with pwMS with disability. This work supports the notion that shifts in the pattern of brain activity in pwMS without disability results in decreased transition energies compared to controls, but, as this shift evolves over the disease, transition energies increase beyond controls and disability occurs. Our results provide the first evidence in pwMS that larger lesion volumes result in greater transition energy between brain states and decreased entropy of brain activity. We investigated the brain-state dynamic and energy landscapes in healthy individuals and people with multiple sclerosis (pwMS). We also investigated the entropy of brain activity and its association with transition energy between brain states and lesion volume. We clustered regional brain activity time series to identify the brain states. Then, we applied network control theory using structural connectivity network to identify the minimum required energy to transition between brain states. We observed the pwMS without disability showed decreased transition energy, while pwMS with evidence of disability showed increased transition energy compared to healthy individuals. Lower entropy of brain activity was associated with greater lesion load and larger transition energy. This study provides a possible mechanism of how MS-related damage to the brain’s structural backbone can impact brain dynamics, entropy, and energetics.