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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 offunctional 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. 

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. 

In this paper we derive exact and approximate conditions for the (local) stability of the cluster synchronization manifold for sparsely interconnected oscillators with heterogeneous and weighted Kuramoto dynamics. Cluster synchronization, which emerges when the oscillators can be partitioned in a way that their phases remain identical over time within each group, is critically important for normal and abnormal behaviors in technological and biological systems ranging from the power grid to the human brain. Yet, despite its importance, cluster synchronization has received limited attention, so that the fundamental mechanisms regulating cluster synchronization in important classes of oscillatory networks are still unknown. In this paper we provide the first conditions for the stability of the cluster synchronization manifold for general weighted networks of heterogeneous oscillators with Kuramoto dynamics. In particular, we discuss how existing results are inapplicable or insufficient to characterize the stability of cluster synchronization for oscillators with Kuramoto dynamics, provide rigorous quantitative conditions that reveal how the network weights and oscillators' natural frequencies regulate cluster synchronization, and offer examples to quantify the tightness of our conditions. Further, we develop approximate conditions that, despite their heuristic nature, are numerically shown to tightly capture the transition to stability of the cluster synchronization manifold.