The successor representation (SR) is a candidate principle for generalization in reinforcement learning, computational accounts of memory, and the structure of neural representations in the hippocampus. Given a sequence of states, the SR learns a predictive representation for every given state that encodes how often, on average, each upcoming state is expected to be visited, even if it is multiple steps ahead. A discount or scale parameter determines how many steps into the future SR's generalizations reach, enabling rapid value computation, subgoal discovery, and flexible decision-making in large trees. However, SR with a single scale could discard information for predicting both the sequential order of and the distance between states, which are common problems in navigation for animals and artificial agents. Here we propose a solution: an ensemble of SRs with multiple scales. We show that the derivative of multi-scale SR can reconstruct both the sequence of expected future states and estimate distance to goal. This derivative can be computed linearly: we show that a multi-scale SR ensemble is the Laplace transform of future states, and the inverse of this Laplace transform is a biologically plausible linear estimation of the derivative. Multi-scale SR and its derivative could lead to a common principle for how the medial temporal lobe supports both map-based and vector-based navigation.
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