System Identification and Optimal Control of Complex Brain Networks

The communicability between topologically associated brain regions may provide a globally connected architecture to support communication through coherence. In chapter 1 of the dissertation, connectivity statistics are discovered by inverting a generative model of dynamic functional connectivity (FC) based on the dynamic communicability between brain regions. Dynamic FC and communicability are related by a set of affine transformations that preserve the group structure of Hermitian positive-definite (HPD) matrices. Infinitesimal transformations are integrated over time on a manifold of HPD matrices, describing parametric geodesic curves that represent dynamic changes in communicability. A hierarchical geodesic regression model is fit to a dataset involving 30 participants who performed a modified Eriksen Flanker task, to learn within- and between- subject connectivity statistics. Differences in dynamic FC are highly predictive of trial characteristics, as a linear classifier can predict whether participants respond with left/right hand and whether the response is correct or incorrect. The remaining chapters of the dissertation present new methods that enable causal investigations of functional networks using non-invasive brain stimulation. Chapter 2 simulates a transcranial electrical stimulation (TES) experiment, using a closed-loop feedback controller to drive the system into a priori target states (i.e., activating or deactivating a set of target ROIs) by identifying an optimal stimulation policy that balances difference performance criteria, such as stimulation dose and state error. The optimal control algorithm minimizes an expected future cost function to reduce stimulation dose for participants while achieving the desired effect - driving the system into different functional networks such as the Default Mode Network (DMN) and Cognitive Control Network (CCN). A projected gradient method is used to model variations in controllability due to individual differences (e.g., skull thickness, cortical morphometry, electrode placement and orientation), and an input sensitivity matrix is systematically varied to monotonically increase the controllability of the system. The effect of controllability is assessed under different levels of noise (measurement noise + input noise + system disturbances) for each transition, demonstrating the feasibility of the work under realistic conditions. In chapter 3, a data-driven systems identification is employed to learn dynamic causal models for TES based on simulations of white noise stimulation on a realistic head model. The algorithms learn low-rank realizations of the system that capture the dominant dynamical features of the data with a small number of modal parameters, using only the input-output timeseries data. An interpretable model is derived by constructing a linear mapping between the low-rank realization and the source level model, describing fluctuations in current source density (CSD). This approach relies on a three-step process: 1. Learn the impulse response function consisting of a Markov parameter sequence using the input-output data timeseries; 2. Estimate a reduced-order model from the singular value decomposition of a circulant matrix of repeated Markov parameter sequences; 3. Project the reduced-order model onto source space using the linear map constructed from a realistic finite-element head model. This approach simultaneously removes stimulation artifacts, source localizes scalp EEG, and predicts the dynamics of the system in response to TES