Linear Dynamics & Control of Brain Networks

The brain is an intricately structured organ responsible for the rich emergent dynamics that support the complex cognitive functions we enjoy as humans. With around $10^{11}$ neurons and $10^{15}$ synapses, understanding how the human brain works has proven to be a daunting endeavor, requiring concerted collaboration across traditional disciplinary boundaries. In some cases, that collaboration has occurred between experimentalists and technicians, who offer new physical tools to measure and manipulate neural function. In other contexts, that collaboration has occurred between experimentalists and theorists, who offer new conceptual tools to explain existing data and inform new directions for empirical research. In this chapter, we offer an example of the latter. Specifically, we focus on the simple but powerful framework of linear systems theory as a useful tool both for capturing biophysically relevant parameters of neural activity and connectivity, and for analytical and numerical study. We begin with a brief overview of state-space representations and linearization of neural models for non-linear dynamical systems. We then derive core concepts in the theory of linear systems such as the impulse and controlled responses to external stimuli, achieving desired state transitions, controllability, and minimum energy control. Afterwards, we discuss recent advances in the application of linear systems theory to structural and functional brain data across multiple spatial and temporal scales, along with methodological considerations and limitations. We close with a brief discussion of open frontiers and our vision for the future.