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Information processing in echo state networks at the edge of chaos

Abstract

We investigate information processing in randomly connected recurrent neural networks. It has been shown previously that the computational capabilities of these networks are maximized when the recurrent layer is close to the border between a stable and an unstable dynamics regime, the so called edge of chaos. The reasons, however, for this maximized performance are not completely understood. We adopt an information-theoretical framework and are for the first time able to quantify the computational capabilities between elements of these networks directly as they undergo the phase transition to chaos. Specifically, we present evidence that both information transfer and storage in the recurrent layer are maximized close to this phase transition, providing an explanation for why guiding the recurrent layer toward the edge of chaos is computationally useful. As a consequence, our study suggests self-organized ways of improving performance in recurrent neural networks, driven by input data. Moreover, the networks we study share important features with biological systems such as feedback connections and online computation on input streams. A key example is the cerebral cortex, which was shown to also operate close to the edge of chaos. Consequently, the behavior of model systems as studied here is likely to shed light on reasons why biological systems are tuned into this specific regime.

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Notes

  1. This initial separation has to be chosen carefully. It should be as small as possible, but still large enough so that its influence will be measurable with limited numerical precision on a computer. We found 10−12 to be a robust value in our simulations, which is also recommended by Sprott (2004) for the precision used in this study.

  2. The TE can be formulated using the l previous states of the source. However, where only the previous state is a causal information contributor (as for ESNs), it is sensible to set l = 1 to measure direct transfer only at step. n

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Acknowledgments

We thank the Australian Commonwealth Scientific and Research Organization’s (CSIRO) Advanced Scientific Computing group for access to high performance computing resources used for simulation and analysis. Joschka Boedecker acknowledges travel support from the CSIRO Complex Systems Science network. Michael Mayer thanks the National Science Council of Taiwan for their support (grant number 98-2218-E-194-003-MY2). In addition, we thank the anonymous reviewers for their helpful comments on the manuscript.

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Correspondence to Joschka Boedecker.

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Boedecker, J., Obst, O., Lizier, J.T. et al. Information processing in echo state networks at the edge of chaos. Theory Biosci. 131, 205–213 (2012). https://doi.org/10.1007/s12064-011-0146-8

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  • DOI: https://doi.org/10.1007/s12064-011-0146-8

Keywords

  • Recurrent neural networks
  • Reservoir computing
  • Information transfer
  • Active information storage
  • Phase transition