Learning Continuous Chaotic Attractors with a Reservoir Computer

TL;DR

This work trains a 1000-neuron RNN-a reservoir computer (RC)-to abstract a continuous dynamical attractor memory from isolated examples of dynamical attractsor memories, and proposes a theoretical mechanism of this abstraction by combining ideas from differentiable generalized synchronization and feedback dynamics.

Abstract

Neural systems are well known for their ability to learn and store information as memories. Even more impressive is their ability to abstract these memories to create complex internal representations, enabling advanced functions such as the spatial manipulation of mental representations. While recurrent neural networks (RNNs) are capable of representing complex information, the exact mechanisms of how dynamical neural systems perform abstraction are still not well-understood, thereby hindering the development of more advanced functions. Here, we train a 1000-neuron RNN -- a reservoir computer (RC) -- to abstract a continuous dynamical attractor memory from isolated examples of dynamical attractor memories. Further, we explain the abstraction mechanism with new theory. By training the RC on isolated and shifted examples of either stable limit cycles or chaotic Lorenz attractors, the RC learns a continuum of attractors, as quantified by an extra Lyapunov exponent equal to zero. We propose a theoretical mechanism of this abstraction by combining ideas from differentiable generalized synchronization and feedback dynamics. Our results quantify abstraction in simple neural systems, enabling us to design artificial RNNs for abstraction, and leading us towards a neural basis of abstraction.

Figures (5)

• Figure 1: Schematic of a reservoir computer learning a limit cycle memory. (a) Time series of a limit cycle that drives the RNN reservoir to the state of the limit cycle. Weighted sums of the reservoir states are trained to reproduce the original time series (b) by creating the $W$ matrix. (c) The reservoir uses the weighted sums in $W$ to evolve, closing the feedback loop in the RC. (d) The RC now evolves autonomously along a trajectory that closely follows the expected dynamics of the original limit cycle. Here, color represents time.
• Figure 2: Successful abstraction in learning a continuous limit cycle memory. (a) 5 shifted limit cycles are learned by the reservoir as 5 isolated examples. (b) 2D plot of the predicted output of the autonomous reservoir whose initial state has been prepared between the 5 training examples. The shift magnitude, or the distance of each translation, of the initial state is colored from green to black. (c) To visualize the abstraction that occurred in an additional dimension along the direction of the translation, we show a 3D plot of the predicted reservoir time series projected onto the $\Delta \bm{r}$ vector in Eq. \ref{['eq:deltar']}, and the first two principal components after removing the $\Delta r$ projection. The cutout highlights the "height" of the the continuous attractor, formed from the abstraction along the $\Delta \bm{r}$ axis.
• Figure 3: Obtaining a Lyapunov spectrum from a Lorenz attractor. (a) A 3D plot of the Lyapunov perturbation orbits (colored), which check the stability of a trajectory, about the Lorenz attractor (black, gray), obtained by evolving the orbits about the Jacobian of the Lorenz system evaluated at each point, followed by an orthonormalization. (b) A plot of the three Lyapunov exponents over $\approx 30$ seconds for the Lorenz system, whose average over $T=100$ seconds yields the estimated Lyapunov exponents.
• Figure 4: LEs of an RC after learning a continuous limit cycle. Heat maps of the first 4 LEs of the RC with different spectral radii ($\rho$) on the $x$-axis and time constants ($\gamma$) on the $y$-axis.
• Figure 5: LEs of a RC after learning a continuous Lorenz attractor. Heat maps of the first 5 LEs of the RC with different spectral radii ($\rho$) on the $x$-axis and time constants ($\gamma$) on the $y$-axis.