Networks in nature do not act in isolation, but instead exchange information and depend on one another to function properly [1][2][3] . Theory has shown that connecting random networks may very easily result in abrupt failures [3][4][5][6] . This finding reveals an intriguing paradox 7,8 : if natural systems organize in interconnected networks, how can they be so stable? Here we provide a solution to this conundrum, showing that the stability of a system of networks relies on the relation between the internal structure of a network and its pattern of connections to other networks. Specifically, we demonstrate that if interconnections are provided by network hubs, and the connections between networks are moderately convergent, the system of networks is stable and robust to failure. We test this theoretical prediction on two independent experiments of functional brain networks (in task and resting states), which show that brain networks are connected with a topology that maximizes stability according to the theory.The theory of networks of networks relies largely on unstructured patterns of connectivity between networks 3,4,6 . When two stable networks are fully interconnected with one-to-one random connections, such that every node in a network depends on a randomly chosen node in the other network, small perturbations in one network are amplified by the interaction between networks 3,6 . This process leads to cascading failures, which are thought to underpin catastrophic outcomes in man-made infrastructures, such as blackouts in power grids 3,4 . By contrast, many stable living systems, including the brain 9 and cellular networks 10 , are organized in interconnected networks. Random networks are very efficient mathematical constructs to develop theory, but the majority of networks observed in nature are correlated 11,12 . Correlations, in turn, provide structure and are known to influence the dynamical and structural properties of interconnected networks, as has been recently shown 13 . Most natural networks form hubs, increasing the relevance of certain nodes. This adds a degree of freedom to the system, in determining whether hubs broadcast information to other networks or, conversely, whether cross-network communication is governed by nodes with less influence in their own network.We develop a full theory for systems of structured networks, identifying a structural communication protocol that ensures the system of networks is stable (less susceptible to catastrophic failure) and optimized for fast communication across the entire system. The theory establishes concrete predictions of a regime of correlated connectivity between the networks composing the system. We test these predictions with two different systems of brain connectivity based on functional magnetic resonance imaging (fMRI) data. The brain organizes in a series of interacting networks 9,14 , presenting a paradigmatic case study for a theory of connected correlated networks. We show that for two independent experiments of functional networks in ta...

Monitoring is a complex multidimensional neurocognitive phenomenon. Patients with fronto-insular stroke (FIS), behavioural variant frontotemporal dementia (bvFTD) and Alzheimer’s disease (AD) show a lack of self-awareness, insight, and self-monitoring, which translate into anosognosia and daily behavioural impairments. Notably, they also present damage in key monitoring areas. While neuroscientific research on this domain has accrued in recent years, no previous study has compared monitoring performance across these brain diseases and none has applied a multiple lesion model approach combined with neuroimaging analysis. Here, we evaluated explicit and implicit monitoring in patients with focal stoke (FIS) and two types of dementia (bvFTD and AD) presenting damage in key monitoring areas. Participants performed a visual perception task and provided two types of report: confidence (explicit judgment of trust about their performance) and wagering (implicit reports which consisted in betting on their accuracy in the perceptual task). Then, damaged areas were analyzed via structural MRI to identify associations with potential behavioral deficits. In AD, inadequate confidence judgments were accompanied by poor wagering performance, demonstrating explicit and implicit monitoring impairments. By contrast, disorders of implicit monitoring in FIS and bvFTD patients occurred in the context of accurate confidence reports, suggesting a reduced ability to turn self-knowledge into appropriate wagering conducts. MRI analysis showed that ventromedial compromise was related to overconfidence, whereas fronto-temporo-insular damage was associated with excessive wagering. Therefore, joint assessment of explicit and implicit monitoring could favor a better differentiation of neurological profiles (frontal damage vs AD) and eventually contribute to delineating clinical interventions.

During the COVID-19 pandemic, the scientific community developed predictive models to evaluate potential governmental interventions. However, the analysis of the effects these interventions had is less advanced. Here, we propose a data-driven framework to assess these effects retrospectively. We use a regularized regression to find a parsimonious model that fits the data with the least changes in the $$R_t$$ R t parameter. Then, we postulate each jump in $$R_t$$ R t as the effect of an intervention. Following the do-operator prescriptions, we simulate the counterfactual case by forcing $$R_t$$ R t to stay at the pre-jump value. We then attribute a value to the intervention from the difference between true evolution and simulated counterfactual. We show that the recommendation to use facemasks for all activities would reduce the number of cases by 200,000 ($$95\%$$ 95 % CI 190,000–210,000) in Connecticut, Massachusetts, and New York State. The framework presented here might be used in any case where cause and effects are sparse in time.

Connectionist and dynamic field models consist of a set of coupled first-order differential equations describing the evolution in time of different units. We compare three numerical methods for the integration of these equations: the Euler method, and two methods we have developed and present here: a modified version of the fourth-order Runge Kutta method, and one semi-analytical method. We apply them to solve a well-known nonlinear connectionist model of retrieval in single-digit multiplication, and show that, in many regimes, the semi-analytical and modified Runge Kutta methods outperform the Euler method, in some regimes by more than three orders of magnitude. Given the outstanding difference in execution time of the methods, and that the EM is widely used, we conclude that the researchers in the field can greatly benefit from our analysis and developed methods.