The theory of structural controllability allows us to assess controllability of a network as a function of its interconnection graph and independently of the edge weights. Yet, existing structural ...controllability results require the weights to be selected arbitrarily and independently from one another and provide no guarantees when these conditions are not satisfied. In this note, we develop a new theory for structural controllability of networks with symmetric, thus constrained, weights. First, we show that network controllability remains a generic property even when the weights are symmetric. Then, we characterize necessary and sufficient graph-theoretic conditions for structural controllability of networks with symmetric weights: a symmetric network is structurally controllable if and only if it is structurally controllable without weight constraints. Finally, we use our results to assess structural controllability from one region of a class of empirically-reconstructed brain networks.
Abstract
Oscillatory activity is ubiquitous in natural and engineered network systems. The interaction scheme underlying interdependent oscillatory components governs the emergence of network-wide ...patterns of synchrony that regulate and enable complex functions. Yet, understanding, and ultimately harnessing, the structure-function relationship in oscillator networks remains an outstanding challenge of modern science. Here, we address this challenge by presenting a principled method to prescribe exact and robust functional configurations from local network interactions through optimal tuning of the oscillators’ parameters. To quantify the behavioral synchrony between coupled oscillators, we introduce the notion of
functional pattern
, which encodes the pairwise relationships between the oscillators’ phases. Our procedure is computationally efficient and provably correct, accounts for constrained interaction types, and allows to concurrently assign multiple desired functional patterns. Further, we derive algebraic and graph-theoretic conditions to guarantee the feasibility and stability of target functional patterns. These conditions provide an interpretable mapping between the structural constraints and their functional implications in oscillator networks. As a proof of concept, we apply the proposed method to replicate empirically recorded functional relationships from cortical oscillations in a human brain, and to redistribute the active power flow in different models of electrical grids.
Abstract
Dynamical brain state transitions are critical for flexible working memory but the network mechanisms are incompletely understood. Here, we show that working memory performance entails ...brain-wide switching between activity states using a combination of functional magnetic resonance imaging in healthy controls and individuals with schizophrenia, pharmacological fMRI, genetic analyses and network control theory. The stability of states relates to dopamine D1 receptor gene expression while state transitions are influenced by D2 receptor expression and pharmacological modulation. Individuals with schizophrenia show altered network control properties, including a more diverse energy landscape and decreased stability of working memory representations. Our results demonstrate the relevance of dopamine signaling for the steering of whole-brain network dynamics during working memory and link these processes to schizophrenia pathophysiology.
In this paper, we study cluster synchronization in networks of oscillators with heterogenous Kuramoto dynamics, where multiple groups of oscillators with identical phases coexist in a connected ...network. Cluster synchronization is at the basis of several biological and technological processes; yet, the underlying mechanisms to enable the cluster synchronization of Kuramoto oscillators have remained elusive. In this paper, we derive quantitative conditions on the network weights, cluster configuration, and oscillators' natural frequency that ensure the asymptotic stability of the cluster synchronization manifold; that is, the ability to recover the desired cluster synchronization configuration following a perturbation of the oscillators' states. Qualitatively, our results show that cluster synchronization is stable when the intracluster coupling is sufficiently stronger than the intercluster coupling, the natural frequencies of the oscillators in distinct clusters are sufficiently different, or, in the case of two clusters, when the intracluster dynamics is homogeneous. We validate the effectiveness of our theoretical results via numerical studies.
Optimizing direct electrical stimulation for the treatment of neurological disease remains difficult due to an incomplete understanding of its physical propagation through brain tissue. Here, we use ...network control theory to predict how stimulation spreads through white matter to influence spatially distributed dynamics. We test the theory’s predictions using a unique dataset comprising diffusion weighted imaging and electrocorticography in epilepsy patients undergoing grid stimulation. We find statistically significant shared variance between the predicted activity state transitions and the observed activity state transitions. We then use an optimal control framework to posit testable hypotheses regarding which brain states and structural properties will efficiently improve memory encoding when stimulated. Our work quantifies the role that white matter architecture plays in guiding the dynamics of direct electrical stimulation and offers empirical support for the utility of network control theory in explaining the brain’s response to stimulation.
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•Tools from network control theory are extended to model empirical brain stimulation•A model of stimulation spreading along axon bundles predicts changes in activity•The model posits how brain activity and connectivity facilitate targeted stimulation
Stiso et al. report evidence that network control theory can explain the propagation of electrical stimulation through the human brain and quantify how white matter connectivity is crucial for driving spatially distributed changes in activity. Furthermore, they use network control theory to predict stimulation outcome in specific cognitive contexts.
Cross-frequency phase-amplitude coupling (PAC) describes the phenomenon where the power of a high-frequency oscillation evolves with the phase of a low-frequency one. It has been widely observed in ...the brain and linked to various brain functions. In this paper, we show that Stuart-Landau oscillators coupled in a nonlinear fashion can give rise to PAC in two commonly accepted architectures, namely, (1) a high-frequency neural oscillation driven by an external low-frequency input and (2) two interacting local oscillations with distinct, locally generated frequencies. We characterize the parameters that affect PAC behavior, thus providing insight on this phenomenon observed in neuronal networks. Inspired by some empirical studies, we further present an interconnection structure for brain regions wherein cross-region interactions are established only by low-frequency oscillations. We then demonstrate that low-frequency phase synchrony can integrate high-frequency activities regulated by local PAC and control the direction of information flow between distant regions.
In this paper, we study representation learning for multi-task decision-making in non-stationary environments. We consider the framework of sequential linear bandits, where the agent performs a ...series of tasks drawn from different environments. The embeddings of tasks in each environment share a low-dimensional feature extractor called representation , and representations are different across environments. We propose an online algorithm that facilitates efficient decision-making by learning and transferring non-stationary representations in an adaptive fashion. We prove that our algorithm significantly outperforms the existing ones that treat tasks independently. We also conduct experiments using both synthetic and real data to validate our theoretical insights and demonstrate the efficacy of our algorithm.
In this paper, we propose a framework to control brain-wide functional connectivity by selectively acting on the brain's structure and parameters. Functional connectivity, which measures the degree ...of correlation between neural activities in different brain regions, can be used to distinguish between healthy and certain diseased brain dynamics and, possibly, as a control parameter to restore healthy functions. In this work, we use a collection of interconnected Kuramoto oscillators to model oscillatory neural activity, and show that functional connectivity is essentially regulated by the degree of synchronization between different clusters of oscillators. Then, we propose a minimally invasive method to correct the oscillators' interconnections and frequencies to enforce arbitrary and stable synchronization patterns among the oscillators and, consequently, a desired pattern of functional connectivity. Additionally, we show that our synchronization-based framework is robust to parameter mismatches and numerical inaccuracies, and validate it using a realistic neurovascular model to simulate neural activity and functional connectivity in the human brain.
Reverse engineering the brain holds the promise to overhaul the quality of life of human beings and vastly benefit mankind. Further advances towards this goal will lead to the reversal of cognitive ...decline, the creation of pioneering neural prostheses, the establishments of novel treatments for neurological disorders, and the development of human augmentation methods. This dissertation presents a cross-disciplinary approach to the study of the structure-function relationship in neural systems. Specifically, we study the brain as a dynamical system that obeys network-wide principles, and address three foundational challenges by using mathematically grounded methods that lie at the intersection of control theory, network science, and neuroscience. First, through the application of control and graph-theoretic paradigms, we investigate how the spatial organization of anatomical brain network components governs and constrains its complex dynamical behaviors. The first chapters are dedicated to the study of structural brain networks – that is, empirically reconstructed large-scale networks that describe the interconnection scheme between different brain regions. We rigorously reveal that brain state transitions can be controlled by a single region, and that structural brain networks possess distinct controllability profiles with respect to random networks of the same size. Second, we address the modeling and analysis of neural activity synchronization across different brain areas – which can be described by functional brain networks. To do so, we juxtapose a bottom-up approach and a top-down approach. In the former, we utilize data-driven dynamical models to reveal that the synchronization of brain network dynamics is resilient to data heterogeneity, thus supporting the utilization of large heterogeneous repositories of brain recordings. In the latter, we abstract rhythmic activity of a neural system as the output of a network of diffusively coupled oscillators, and derive prescriptive conditions for the emergence of cluster synchronization. Such a phenomenon emerges when different groups of synchronized components coexist in a network, and regulates the functional interactions among network components. Third and final, we build upon our previous findings and take aim at the tantalizing idea of controlling the synchronization of brain dynamics through minimally invasive local interventions. We derive a method to optimally intervene on the structural network parameters to achieve desired cluster-synchronized trajectories and, thus, prescribed functional interactions. Additionally, we show that our synchronization-based framework is robust to mismatches in network parameters, and validate it using a realistic neurovascular model to simulate neural activity and functional connectivity in the human brain.
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