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The brain can be conceptualized as a control system facilitating transitions between states, such as from rest to motor activity. Applying network control theory to measurements of brain signals enables characterization of brain dynamics through control properties, including controllability. However, most prior studies that have applied network control theory have evaluated brain dynamics under unperturbed conditions, neglecting the critical role of external perturbations in accurate system identification, which is a fundamental principle in control theory. The incorporation of perturbation inputs is therefore essential for precise characterization of brain dynamics. In this study, we combine a perturbation input paradigm with a network control theory framework and propose a novel method for estimating the controllability Gramian matrix in a simple, theoretically grounded manner. This method provides insights into brain dynamics, including overall controllability (quantified by the Gramian's eigenvalues) and specific controllable directions (represented by its eigenvectors). As a proof of concept, we applied our method to transcranial magnetic stimulation (TMS)-induced electroencephalographic (EEG) responses across four motor-related states and two resting states. We found that states such as open-eye rest, closed-eye rest, and motor-related states were more effectively differentiated using controllable directions than overall controllability. However, certain states, like motor execution and motor imagery, remained indistinguishable using these measures. These findings indicate that some brain states differ in their intrinsic control properties as dynamical systems, while others share similarities that make them less distinguishable. This study underscores the value of control theory-based analyses in quantitatively how intrinsic brain states shape the brain's responses to stimulation, providing deeper insights into the dynamic properties of these states. This methodology holds promise for diverse applications, including characterizing individual response variability and identifying conditions for optimal stimulation efficacy.
Competing Interest Statement
Hartwig R. Siebner has received honoraria as a speaker and consultant from Lundbeck AS, Denmark, and as an editor (Neuroimage Clinical) from Elsevier Publishers, Amsterdam, The Netherlands. He has received royalties as a book editor from Springer Publishers, Stuttgart, Germany; Oxford University Press, Oxford, UK; and from Gyldendal Publishers, Copenhagen, Denmark.