Optimal control of deterministic and stochastic neural models in finite and infinite dimension. Application to the control of neural dynamics by Optogenetics.

Summary The aim of this thesis is to propose different mathematical models of neurons for Optogenetics and to study their optimal control. We first define a controlled version of finite dimensional deterministic models, called conductance models. We study a minimal time problem for an affine single-input system whose singularities we study. We apply a direct numerical method to observe optimal trajectories and controls. Optogenetic control appears as a new way to judge the ability of conductance models to reproduce the experimentally observed characteristics of the membrane potential dynamics. We then define a stochastic model in infinite dimension to take into account the randomness of ion channel mechanisms and the propagation of action potentials. It is a controlled piecewise deterministic Markov process (PDMP) with values in a Hilbert space. We define a large class of controlled PDMPs in infinite dimension and prove the strongly Markovian character of these processes. We treat an optimal control problem with finite time horizon. We study the Markovian decision process (MDP) included in the PDMP and show the equivalence of the two problems. We give sufficient conditions for the existence of optimal controls for the MDP, and hence the PDMP. We discuss variants for the stochastic Optogenetic model in infinite dimension. Finally, we study the extension of the model to a reflexive Banach space, and then, in a special case, to a non-reflexive Banach space.