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  1. Transcranial electrical stimulation (tES) technology and neuroimaging are increasingly coupled in basic and applied science. This synergy has enabled individualized tES therapy and facilitated causal inferences in functional neuroimaging. However, traditional tES paradigms have been stymied by relatively small changes in neural activity and high inter-subject variability in cognitive effects. In this perspective, we propose a tES framework to treat these issues which is grounded in dynamical systems and control theory. The proposed paradigm involves a tight coupling of tES and neuroimaging in which M/EEG is used to parameterize generative brain models as well as control tES delivery in a hybrid closed-loop fashion. We also present a novel quantitative framework for cognitive enhancement driven by a new computational objective: shaping how the brain reacts to potential “inputs” (e.g., task contexts) rather than enforcing a fixed pattern of brain activity. 
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  2. System identification poses a significant bottleneck to characterizing and controlling complex systems. This challenge is greatest when both the system states and parameters are not directly accessible, leading to a dual-estimation problem. Current approaches to such problems are limited in their ability to scale with many-parameter systems, as often occurs in networks. In the current work, we present a new, computationally efficient approach to treat large dual-estimation problems. In this work, we derive analytic back-propagated gradients for the Prediction Error Method which enables efficient and accurate identification of large systems. The PEM approach consists of directly integrating state estimation into a dual-optimization objective, leaving a differentiable cost/error function only in terms of the unknown system parameters, which we solve using numerical gradient/Hessian methods. Intuitively, this approach consists of solving for the parameters that generate the most accurate state estimator (Extended/Cubature Kalman Filter). We demonstrate that this approach is at least as accurate in state and parameter estimation as joint Kalman Filters (Extended/Unscented/Cubature) and Expectation-Maximization, despite lower complexity. We demonstrate the utility of our approach by inverting anatomically-detailed individualized brain models from human magnetoencephalography (MEG) data. 
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  3. Equilibria, or fixed points, play an important role in dynamical systems across various domains, yet finding them can be computationally challenging. Here, we show how to efficiently compute all equilibrium points of discrete-valued, discrete-time systems on sparse networks. Using graph partitioning, we recursively decompose the original problem into a set of smaller, simpler problems that are easy to compute, and whose solutions combine to yield the full equilibrium set. This makes it possible to find the fixed points of systems on arbitrarily large networks meeting certain criteria. This approach can also be used without computing the full equilibrium set, which may grow very large in some cases. For example, one can use this method to check the existence and total number of equilibria, or to find equilibria that are optimal with respect to a given cost function. We demonstrate the potential capabilities of this approach with examples in two scientific domains: computing the number of fixed points in brain networks and finding the minimal energy conformations of lattice-based protein folding models. 
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