Synchronization is crucial for the correct functionality of many natural and man-made complex systems. In this work we characterize the formation of synchronization patterns in networks of Kuramoto oscillators. Specifically, we reveal conditions on the network weights and structure and on the oscillators' natural frequencies that allow the phases of a group of oscillators to evolve cohesively, yet independently from the phases of oscillators in different clusters. Our conditions are applicable to general directed and weighted networks of heterogeneous oscillators. Surprisingly, although the oscillators exhibit nonlinear dynamics, our approach relies entirely on tools from linear algebra and graph theory. Further, we develop a control mechanism to determine the smallest (as measured by the Frobenius norm) network perturbation to ensure the formation of a desired synchronization pattern. Our procedure allows us to constrain the set of edges that can be modified, thus enforcing the sparsity structure of the network perturbation. The results are validated through a set of numerical examples.

The mechanisms controlling dynamical patterns in spontaneous brain activity are poorly understood. Here, we provide evidence that cortical dynamics in the ultra-slow frequency range (<0.01–0.1 Hz) requires intact cortical-subcortical communication. Using functional magnetic resonance imaging (fMRI) at rest, we identify Dynamic Functional States (DFSs), transient but recurrent clusters of cortical and subcortical regions synchronizing at ultra-slow frequencies. We observe that shifts in cortical clusters are temporally coincident with shifts in subcortical clusters, with cortical regions flexibly synchronizing with either limbic regions (hippocampus/amygdala), or subcortical nuclei (thalamus/basal ganglia). Focal lesions induced by stroke, especially those damaging white matter connections between basal ganglia/thalamus and cortex, provoke anomalies in the fraction times, dwell times, and transitions between DFSs, causing a bias toward abnormal network integration. Dynamical anomalies observed 2 weeks after stroke recover in time and contribute to explaining neurological impairment and long-term outcome.

Neurological deficits following stroke are traditionally described as syndromes related to damage of a specific area or vascular territory. Recent studies indicate that, at the population level, post-stroke neurological impairments cluster in three sets of correlated deficits across different behavioral domains. To examine the reproducibility and specificity of this structure, we prospectively studied first-time stroke patients (n = 237) using a bedside, clinically applicable, neuropsychological assessment and compared the behavioral and anatomical results with those obtained from a different prospective cohort studied with an extensive neuropsychological battery. The behavioral assessment at one-week post stroke included the Oxford Cognitive Screen (OCS) and the National Institutes of Health Stroke Scale (NIHSS). A principal component analysis was used to reduce variables and describe behavioral variance across patients. Lesions were manually segmented on structural scans. The relationship between anatomy and behavior was analyzed using multivariate regression models. Three principal components (PC) explained ≈50% of the behavioral variance across subjects. PC1 loaded on language, calculation, praxis, right side neglect, and memory deficits; PC2 loaded on left motor, visual, and spatial neglect deficits; PC3 loaded on right motor deficits. These components matched those obtained with a more extensive battery. The underlying lesion anatomy was also similar. Neurological deficits following stroke are correlated in a low dimensional structure of impairment, related neither to the damage of a specific area or vascular territory. Rather they reflect widespread network impairment caused by focal lesions. These factors showed consistency across different populations, neurobehavioral batteries and, most importantly, can be described using a combination of clinically applicable batteries (NIHSS and OCS). They represent robust behavioral biomarkers for future stroke population studies.

Mathematical theories and empirical evidence suggest that several complex natural and man-made systems are fragile: as their size increases, arbitrarily small and localized alterations of the system parameters may trigger system-wide failures. Examples are abundant, from perturbation of the population densities leading to extinction of species in ecological networks [1], to structural changes in metabolic networks preventing reactions [2], cascading failures in power networks [3], and the onset of epileptic seizures following alterations of structural connectivity among populations of neurons [4]. While fragility of these systems has long been recognized [5], convincing theories of why natural evolution or technological advance has failed, or avoided, to enhance robustness in complex systems are still lacking. In this paper we propose a mechanistic explanation of this phenomenon. We show that a fundamental tradeoff exists between fragility of a complex network and its controllability degree, that is, the control energy needed to drive the network state to a desirable state. We provide analytical and numerical evidence that easily controllable networks are fragile, suggesting that natural and man-made systems can either be resilient to parameters perturbation or efficient to adapt their state in response to external excitations and controls.

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