Shaded areas indicate the Parietal Window (PWb), the transformation circuit, head direction modulation, and boundary vector cells (BVCs). Example cells are represented as stylized firing rate maps in a simple square environment. Firing related to the North, East, South and West walls is depicted in four colors (blue, yellow, purple, red, respectively). Only 4 PWb cells, 16 transformation circuit neurons, 4 BVCs and four head-direction modulations (North, East, South, West) are shown for simplicity. PWb neurons have egocentric receptive fields (RFs, dashed ovals, shown left of and within each square) that are attached to the agent (black triangle). The RFs respond to boundaries at a specific distance and egocentric direction (ahead, left, right, behind). As the agent moves around the environment, any boundary can fall into the egocentric RF, depending on the agent’s orientation (four example positions and orientations shown), resulting in a firing rate map with firing related to all four boundaries (i.e. the blue, yellow, purple, and red bands, each conditional on the agent facing in a different direction). Considering the PWb cell with the RF ahead of the agent (top left, green star): due to the HD modulation a different RSC cell is receptive to input from that PWb neuron depending on the agent‘s current orientation. For example when the agent is facing East the 2nd row of the RSC transformation circuit is receptive to inputs from the PWb, and the first PWb neuron projects to the second cell in that row (green arrow). That RSC cell in turn projects to a BVC with a RF to the East (downward light grey arrow). The Eastward BVC also gets inputs from the other 3 PWb cells when the agent faces in the other three directions, via the other RSC neurons in the second column (connectivity not shown, but indicated by matching symbols: hexagon, triangle, square). Thus, this BVC can fire whenever the agent is near to the East wall, irrespective of the agent’s orientation. In top-down mode (imagery), PWb cells are driven by different BVCs depending on the facing direction of the animal. The PWb cell with the RF ahead of the agent (top left, green star) recieves connections from all transformation circuit neurons shown with star symbols: conveying input from the Eastward BVC when facing East, the Northward BVC when facing North etc. In this way, it is driven to fire whenever there is a boundary ahead of the agent. All connections between PWb and BVC cells and transformation circuit neurons are bidirectional, to enable both bottom-up and top-down operation.