Recurrent neural networks (RNN) have long been studied to explain how fixed-point attractors may emerge from noisy, high-dimensional dynamics. Recently, computational neuroscientists have devoted sustained efforts to understand how RNN could embed attractor manifolds of finite dimension, in particular in the context of the representation of space by mammals. A natural issue is the existence of...

Dependent type theory imposes a type system on Zemelo-Fraenkel set theory (ZFC). From a mathematics and physics perspective dependent type theory naturally generalizes the Bourbaki notion of structure and provides a universal notion of isomorphism and symmetry. This comes with a universal substitution theorem --- isomorphic objects are inter-substitutable in well-typed contexts (Hofmann and...

Naturalistic animal behavior exhibits a striking amount of variability in the temporal domain along at least three independent axes: hierarchical, contextual, and stochastic. First, a vast hierarchy of timescales links movements into behavioral sequences and long-term activities, from milliseconds to minutes. Second, action timing can be modulated by changes in context, of either internal...