(A) Task design. A maze (left) that has a single central stem ( M) which can be reached from three possible starting points (segments A-C) through four possible trajectories (brown, magenta, green, blue). Similarly, there are three possible goal locations (segments H-J) with four possible trajectories. Analyses focus on the M segment (t=0) so that points A-E are in the past, and F-J in the future. (B) On this task, a hypothetical splitter cell at M could distinguish the recent past (D vs E, t = –1) while being independent of the far past (same activity as for D vs E, even though A vs C at t = –2 are now also different, bottom left panel). Alternatively, its activity could depend on both the recent and the far past (different activity compared to D vs E, bottom right panel). (C) Different timescales of encoding traces of the past result in different representational similarity matrices comparing ensemble neural activity at M across different trajectories. Δ–1 indicates comparisons with a difference at t-1, Δ–2 indicates comparisons with a difference at t-2. If only the recent past is encoded in ensemble hippocampal activity (left matrix), trajectories that share the recent segment (no difference at t = –1, tiles with no Δ–1 in the matrix) are correlated (red matrix elements), while trajectories that differentiate the recent segment (tiles with Δ–1) are uncorrelated. In contrast, an ensemble that differentiates both recent and far past in a graded manner (right matrix) will show some positive correlation when the far past is shared (no Δ–2, yellow matrix elements). (D) Similarly, different possible hypotheses about what is encoded by the splitter cell signal (full trajectory, goal, task state) result in different RSA matrices. For the full trajectory RSA matrix (left panel), correlations are driven by the number of overlapping trajectory segments (high correlation if only 2 segments are different, 2Δ; low correlation if all four segments of the trajectory are different, 4Δ). If ensemble similarity is based on goal (center panel), correlations are driven only by how close in space the goal is (highest for the same goal, I, I; intermediate for adjacent goals such as H, I and J, I; low for far goals). Finally, if neural activity encodes task state (i.e. what trajectory needs to be executed in order to get reward; right panel), every trajectory is equally uncorrelated with every other trajectory, because no spatial information is required to distinguish states. Underlying these RSA predictions is the idea that computations based on state have different information content compared to those based on trajectory: state-based computations, minimally, only require the states to be different. In contrast, trajectories are spatial, and have similarity structure based on factors like distance and amount of overlap.