The rostrocaudal patterning of the neural tube is a key event in early brain development. This process is mainly driven by a gradient of WNT, which defines the fate of the present neural progenitor cells in a dose dependent matter and leads to a subdivision of the tube into forebrain, midbrain and hindbrain. Although this process is extensively studied experimentally both in vivo and in vitro, an integrated view of the responsible genetic circuitry is currently lacking. In this work, we present a minimal gene regulatory model for rostrocaudal neural tube patterning. The model's nodes and architecture are determined in a data driven way, leading to a tristable configuration of mutually repressing brain regions. Analysis of the parameter sensitivity and simulations of knockdown and overexpression cases show that repression of hindbrain fate is a promising strategy for the improvement of current protocols for the generation of dopaminergic neurons in vitro. Furthermore, we combine the model with an existing model for dorsoventral neural tube patterning, to test its capabilities in an in vivo setting, by predicting the steady state pattern of a realistic three-dimensional neural tube. This reveals that the rostrocaudal pattern stacks dorsoventrally in the caudal half of the neural tube. Finally, we simulate morphogen secretion overexpression, which highlights the sensitivity of neural tube patterning to the morphogen levels.