emma

I love linear dynamical systems and control theory When I got into neuroscience, I had high hopes that all my Strogatz-level nonlinear dynamical systems knowledge could be put to use. The brain is highly nonlinear, and dynamic. It'd be great to explain the brain solely in terms of limit cycles, bifurcations, and chaos. But you drive a NDS with time-varying inputs, the transient properties of your vector field dominate the asymptotic structure. One way to solve this is geometric control theory, which make use of lots of Lie stuff but appears to be a niche and mostly beyond my comprehension. The alternative is standard control theory (which is mostly, or at least locally, linear). You pay a huge price by giving up nonlinearity, but as a reward you get an insanely expressive (complete?) theory for how inputs and internal dynamics interact. So many intuitions become beautifully clear in linear dynamical systems world. The exact correspondence between autoregressive, state-space, and convolution representations of dynamical systems. Or, how truncated BPTT (at one time step) can learn arbitrarily long range dependencies and this is a property of observability and persistence of excitation not truncation. Or, how LQR and KF are dual -- the same problem but with arrows flipped. Or, how cleanly the translation between discrete and continuous time systems become (no Euler steps) etc