Back when I was at MIT and starting to think about the predictive capabilities of reservoirs, I wanted to pull out the dynamical systems textbook and answer all my questions about prediction.
The first dynamical systems textbook I pulled out was Strogatz. I realized that I could hit the problem of prediction with that textbook in the limit of weak input, when the basinattractor portrait failed to change, and that resulted in this paper. The second book I looked up was one on "Random Dynamical Systems". I got one chapter in when I realized something this book wasn't answering any of my questions on prediction. I realized that a conceptual shift was needed. These dynamical systems were not "random". They were filters of input, and the input was the signal, and it was incorrect for all the problems I was working on to treat the input as noise. I barely care about what the state of the system is; I only care about how the state of the system relates to the past of the input, something that may be harder to keep track of. The fix to this, I think, is to look at the joint state space of predictive features of the input and the state of the system and find dynamics on that joint state space. I did this in this paper. You have to know something about the input. The math gets a bit complicated, but I'm hoping that slight fixes to the Strogatz textbook can be imported in the more general case! Altogether, I think a new textbook on dynamical systems with input is in order, one that includes more recent work on reservoir computing. These inputdependent dynamical systems actually do a computation, and so many fields from biophysics to theoretical neuroscience care about quantifying exactly how well that computation is done. Considering the input as noise is the opposite of solving the problems in these fields. I think this is a classic example of how bioinspired math could spark an entirely new textbook.
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November 2022
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