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It is typically assumed that large networks of neurons exhibit a large repertoire of nonlinear behaviours. Here we challenge this assumption by leveraging mathematical models derived from measurements of local field potentials via intracranial electroencephalography and of whole-brain blood-oxygen-level-dependent brain activity via functional magnetic resonance imaging. We used state-of-the-art linear and nonlinear families of models to describe spontaneous resting-state activity of 700 participants in the Human Connectome Project and 122 participants in the Restoring Active Memory project. We found that linear autoregressive models provide the best fit across both data types and three performance metrics: predictive power, computational complexity and the extent of the residual dynamics unexplained by the model. To explain this observation, we show that microscopic nonlinear dynamics can be counteracted or masked by four factors associated with macroscopic dynamics: averaging over space and over time, which are inherent to aggregated macroscopic brain activity, and observation noise and limited data samples, which stem from technological limitations. We therefore argue that easier-to-interpret linear models can faithfully describe macroscopic brain dynamics during resting-state conditions.

 

The wide availability of data coupled with the computational advances in artificial intelligence and machine learning promise to enable many future technologies such as autonomous driving. While there has been a variety of successful demonstrations of these technologies, critical system failures have repeatedly been reported. Even if rare, such system failures pose a serious barrier to adoption without a rigorous risk assessment. This article presents a framework for the systematic and rigorous risk verification of systems. We consider a wide range of system specifications formulated in signal temporal logic (STL) and model the system as a stochastic process, permitting discrete-time and continuous-time stochastic processes. We then define the STL robustness risk as the risk of lacking robustness against failure . This definition is motivated as system failures are often caused by missing robustness to modeling errors, system disturbances, and distribution shifts in the underlying data generating process. Within the definition, we permit general classes of risk measures and focus on tail risk measures such as the value-at-risk and the conditional value-at-risk. While the STL robustness risk is in general hard to compute, we propose the approximate STL robustness risk as a more tractable notion that upper bounds the STL robustness risk. We show how the approximate STL robustness risk can accurately be estimated from system trajectory data. For discrete-time stochastic processes, we show under which conditions the approximate STL robustness risk can even be computed exactly. We illustrate our verification algorithm in the autonomous driving simulator CARLA and show how a least risky controller can be selected among four neural network lane-keeping controllers for five meaningful system specifications.