Optimal control of nonlinear models of neural population dynamics

This dissertation investigates optimal control strategies for nonlinear models of neural population dynamics to deepen our understanding of the mechanisms that govern coordinated neural activity. Nonlinear dynamical systems theory provides a framework for modeling neural activity across various scales and enables to study brain functions theoretically. The exploration of optimal control strategies for neural systems is guided by two principal objectives: Elucidating principles of interaction within neural systems, and designing interventions that guide neural states toward desired outcomes. As natural selection likely promotes energy efficiency in neural signaling, optimal control theory may offer a theoretical foundation for studying naturalistic interaction and communication strategies that regulate brain function. As experimental and clinical stimulation techniques become increasingly precise, a rigorous mathematical approach to optimal control may enhance the efficiency with which stimulation goals are achieved. A cost functional formalizes the concept of stimulation efficiency and balances proximity to a target activity with the strength of the control input. The optimization of stimulation efficiency is framed as a problem of minimizing the cost functional under the constraints imposed by the system dynamics. The derivative of the cost functional with respect to the control signal is computed using the adjoint method, which combines the calculus of variations with Lagrange multipliers. We implement an open-source software tool for the computation of optimal control signals through iterative gradient descent that is available for various models of neural population dynamics. Two representatives of a common motif of recurrently coupled populations of excitatory and inhibitory neurons are studied. Both exhibit a similar state space in isolation: We identify stable states of constant high, constant low, or oscillatory activity, as well as bistable states where stable high- and low-activity states coexist. These states persist in networks, complemented by a variety of multistable states and oscillatory patterns. Upon examination of the same control task across models, we observe that optimal control signals share qualitative features, but exhibit quantitative differences. This dissertation investigates two categories of control tasks: The study of transitions in multistability and the enforcement of specific oscillatory behavior or phase relations. The former category is inspired by the hypothesis that the brain is a multistable system, exhibiting different stable states that correspond to distinct operational regimes. The latter category is inspired by the pivotal function of phase relations and synchrony in neural information processing. Within categories, we find principles that elucidate how the dynamics shape optimal control strategies. Optimal transitions in bistability are achieved by steering the system minimally into the basin of attraction of the target state, thereby exploiting its attractiveness in an optimal manner. The underlying control signals can be discerned from the bifurcation diagram. Network nodes that are crucial for whole-brain desynchronization can be predicted directly from the network graph, while advanced methods from linear network control theory offer no additional benefit. Both the optimal phase adjustment of single-node systems and the optimal synchronization and desynchronization of whole-brain networks depend critically on the relative timing of state and control. The phase response of the uncoupled oscillator precisely determines optimal phase shifts in a single neural population; however, it fails to predict optimal control strategies in a whole-brain network. This underscores the need for a more nuanced understanding of the differences between single-node and network control.