In our research, we develop theories and models of the dynamic interactions between cognitive processes of perception, attention, and motor function with special emphasis on the control of eye movements. Experimentally, we use eye movement measurements, where, in many cases, gaze-position dependent displays are utilized. Our mathematical models are motivated by neurocognitive principles and are implemented as stochastic dynamical systems that are intended to replace traditional research programs in the form of state-dependent and rule-based models. These mechanistic models are validated and further developed via data assimilation (Bayesian inference). For all new projects, we emphasize open science and publish experimental data and source code freely accessible.