The brain, composed of billions of interconnected neurons, forms a complex network that exhibits an incredibly wide array of behaviors. Treating this structure as a dynamical system provides a multitude of tools to use to model and understand the relationship between structure and function in the brain. As subnetworks exist at all levels in the brain, ranging from networks of individual neurons to networks of entire regions, a diverse set of models, each with different properties have been considered to study different dynamic brain behaviors. One such class of models are firing rate models, which monitor the average spike rate of populations of neurons. A particular firing rate model is the linear-threshold network model, which exhibits a wide range of rich behaviors based on the underlying network structure and inputs. This ability to exhibit a variety of behaviors motivates the use of this model to study a variety of dynamical behaviors observed in the brain.

This dissertation considers three problems within the realm of modeling dynamical brain behaviors with the linear-threshold model. First, motivated by the appearance of oscillatory behavior when observing brain activity, we study the existence of oscillations in the linear-threshold dynamics. In order to provide sufficient conditions for oscillations in specific network topologies we also provide conditions for the stability of equilibrium points that maintain a specific support. Second, we discuss the dynamical brain behavior of selective inhibition and recruitment. We consider thalamocortical networks with both hierarchical and star-connected topologies, and focus on how the inclusion of the thalamus can improve the stabilizability properties of the linear-threshold dynamics relevant to the application. We finish by investigating the problem of reference tracking for the linear-threshold dynamics, which can be used to frame many brain behaviors. We approach this both analytically and with a data-driven approach to better match observations on how the brain processes information.