  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.
