Robust stabilization of delayed neural fields with partial measurement and actuation

Neural fields are integro-differential equations describing spatiotemporal activity of neuronal populations. When considering finite propagation speed of action potentials, neural fields are affected by space-dependent delays. In this paper, we provide conditions under which such dynamics can be robustly stabilized by a proportional feedback acting only on a portion of the neuronal population and by relying on measurements of this subpopulation only. To that aim, in line with recent works, we extend the concept of input-to-state stability (ISS) to generic nonlinear delayed spatiotemporal dynamics and provide a small-gain result relying on Lyapunov-Krasovskii functionals. Exploiting the robustness properties induced by ISS, we provide conditions under which a uniform control signal can be used for the whole controlled subpopulation and we analyze the robustness of the proposed strategy to measurement and actuation delays. These theoretical findings are compared to simulation results in a model of pathological oscillations generation in Parkinson's disease.