Abstract
Although hippocampal grid cells are thought to be crucial for spatial navigation, their computational purpose remains disputed. Recently, they were proposed to represent spatial transitions and to convey this knowledge downstream to place cells. However, a single scale of transitions is insufficient to plan long goal-directed sequences in behaviorally acceptable time.
Here, a scale-space data structure is suggested to optimally accelerate retrievals from transition systems, called Transition Scale-Space (TSS). Remaining exclusively on an algorithmic level, the scale increment is proved to be ideally 
for biologically plausible receptive fields. It is then argued that temporal buffering is necessary to learn the scale-space online. Next, two modes for retrieval of sequences from the TSS are presented, namely top-down and bottom-up. The two modes are evaluated in symbolic simulations, i.e., without biologically plausible spiking neurons. Additionally, a TSS is used for short-cut discovery in a simulated Morris water maze. Finally, the presented results are discussed in depth with respect to biological plausibility, and several testable predictions derived. Moreover, relations to other grid cell models, multi-resolution path planning, and scale-space theory are highlighted. Summarized, reward-free transition encoding is shown here, in a theoretical model, to be compatible with the observed discretization along the dorso-ventral axis of medial Entorhinal Cortex (MEC).
Because the theoretical model generalizes beyond navigation, the TSS is suggested to be a general-purpose cortical data structure for fast retrieval of sequences and relational knowledge.
