Mathematical methods for manifold learning, pose representation and path integration in biological neural networks

In this thesis, I study how the brain's internal map of space can be read out from neural population activity, and how such a map could be built and maintained by recurrent dynamics. I begin by introducing the cognitive map hypothesis and the two conceptual tools used throughout. I start by discussing how to frame population activity as a low-dimensional manifold, and stable activity patterns as neural attractors. I then analyse hippocampal CA1 recordings from an experiment in which visual landmarks were manipulated to decouple a rat's internal sense of position from its true physical location, a dissociation quantified by the hippocampal gain. I use manifold learning to decode the resulting hippocampal gain continuously in time. I show that this works on both spike-sorted and raw tetrode data, without behavioural labels and at a finer temporal resolution than the existing spectral method. Next, I develop a continuous attractor network, LieCAN, defined on the special Euclidean group SE(2). By applying the path-integration offset as a group operation rather than as coordinate addition, the egocentric to allocentric transformation is embedded in the recurrent connectivity itself, allowing the network to integrate egocentric velocity directly. I derive the model's core theoretical properties, compare it against an allocentric network differing in how the offset is applied, and the class of connectivity kernel, and show that its stationary state reproduces the curved positional uncertainty induced by heading noise, which a Euclidean kernel cannot represent. Finally, I propose extending the framework to SE(3) and discuss the evidence for three-dimensional spatial coding in bats.