Daily experience suggests that we perceive distances near us linearly. However, the actual geometry of spatial representation in the brain is unknown. Here we report that neurons in the CA1 region of rat hippocampus that mediate spatial perception represent space according to a non-linear hyperbolic geometry. This geometry uses an exponential scale and yields greater positional information than a linear scale. We found that the size of the representation matches the optimal predictions for the number of CA1 neurons. The representations also dynamically expanded proportional to the logarithm of time that the animal spent exploring the environment, in correspondence with the maximal mutual information that can be received. The dynamic changes tracked even small variations due to changes in the running speed of the animal. These results demonstrate how neural circuits achieve efficient representations using dynamic hyperbolic geometry.

The interaction between hippocampus and cortex is key to memory formation and representation learning. Based on anatomical wiring and transmission delays, we proposed a self-supervised recurrent neural network (PredRAE) with a predictive reconstruction loss to account for the cognitive functions of hippocampus. This framework extends predictive coding in the time axis and incorporates predictive features in Bayes filters for temporal prediction. In simulations, we were able to reproduce characteristic place cell features such as one-shot plasticity, localized spatial representation and replay, which marks the trace of memory formation. The simulated place cells also exhibited precise spike timing, evidenced by phase precession. Trained on MNIST sequences, PredRAE learned the underlying temporal dependencies and a spontaneous representation of digit labels and rotation dynamics in the linear transformation of its hidden unit activities. Such learning is robust against 16 different image corruptions. Inspired by the brain circuit, the simple and concise framework has great potential to approximate human performance with its capacity to robustly disentangle representations and generalize temporal dynamics.

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