Optimal data fitting on lie groups: A coset approach

This work considers the problem of fitting data on a Lie group by a coset of a compact subgroup. This problem can be seen as an extension of the problem of fitting affine subspaces in n to data which can be solved using principal component analysis. We show how the fitting problem can be reduced for biinvariant distances to a generalized mean calculation on an homogeneous space. For biinvariant Riemannian distances we provide an algorithm based on the Karcher mean gradient algorithm. We illustrate our approach by some examples on SO(n). © 2010 Springer -Verlag Berlin Heidelberg.

Optimal recall from bounded metaplastic synapses: predicting functional adaptations in hippocampal area CA3.

A venerable history of classical work on autoassociative memory has significantly shaped our understanding of several features of the hippocampus, and most prominently of its CA3 area, in relation to memory storage and retrieval. However, existing theories of hippocampal memory processing ignore a key biological constraint affecting memory storage in neural circuits: the bounded dynamical range of synapses. Recent treatments based on the notion of metaplasticity provide a powerful model for individual bounded synapses; however, their implications for the ability of the hippocampus to retrieve memories well and the dynamics of neurons associated with that retrieval are both unknown. Here, we develop a theoretical framework for memory storage and recall with bounded synapses. We formulate the recall of a previously stored pattern from a noisy recall cue and limited-capacity (and therefore lossy) synapses as a probabilistic inference problem, and derive neural dynamics that implement approximate inference algorithms to solve this problem efficiently. In particular, for binary synapses with metaplastic states, we demonstrate for the first time that memories can be efficiently read out with biologically plausible network dynamics that are completely constrained by the synaptic plasticity rule, and the statistics of the stored patterns and of the recall cue. Our theory organises into a coherent framework a wide range of existing data about the regulation of excitability, feedback inhibition, and network oscillations in area CA3, and makes novel and directly testable predictions that can guide future experiments.