Driver assistance systems are designed to increase comfort and safety by automating parts of the driving task. At the same time, modern in-vehicle information systems with large touchscreens provide the driver with numerous options for entertainment, information, or communication, and are a potential source of distraction. However, little is known about how driving automation affects how drivers interact with the center stack touchscreen, i.e., how drivers self-regulate their behavior in response to different levels of driving automation. To investigate this, we apply multilevel models to a real-world driving dataset consisting of 31,378 sequences. Our results show significant differences in drivers' interaction and glance behavior in response to different levels of driving automation, vehicle speed, and road curvature. During automated driving, drivers perform more interactions per touchscreen sequence and increase the time spent looking at the center stack touchscreen. Specifically, at higher levels of driving automation (level 2), the mean glance duration toward the center stack touchscreen increases by 36% and the mean number of interactions per sequence increases by 17% compared to manual driving. Furthermore, partially automated driving has a strong impact on the use of more complex UI elements (e.g., maps) and touch gestures (e.g., multitouch). We also show that the effect of driving automation on drivers' self-regulation is greater than that of vehicle speed and road curvature. The derived knowledge can inform the design and evaluation of touch-based infotainment systems and the development of context-aware driver monitoring systems.