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Updated: Jan 10, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
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Slow Transition to Low-Dimensional Chaos in Heavy-Tailed Recurrent Neural Networks.

Yi Xie1,2, Stefan Mihalas1, Lukasz Kusmierz1

  • 1Allen Institute, Seattle, WA, USA.

Biorxiv : the Preprint Server for Biology
|November 24, 2025
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Summary

Recurrent neural networks (RNNs) with heavy-tailed synaptic weights show a robust transition to chaos, unlike Gaussian networks. This biological realism offers a tradeoff between dynamic stability and neural activity richness.

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Area of Science:

  • Computational neuroscience
  • Theoretical neuroscience
  • Machine learning

Background:

  • Synaptic weights in the brain often follow heavy-tailed distributions.
  • Most theoretical analyses of recurrent neural networks (RNNs) assume Gaussian connectivity, which may not be biologically accurate.

Purpose of the Study:

  • To investigate the dynamics of RNNs with biologically plausible Lévy alpha-stable distributed weights.
  • To analyze the transition between quiescent and chaotic states in finite-size heavy-tailed RNNs.
  • To understand the tradeoff between robustness and dimensionality in neural activity.

Main Methods:

  • Systematic study of RNNs with random weights from Lévy alpha-stable distributions.
  • Finite-size analysis to identify transitions between quiescent and chaotic dynamics.
  • Theoretical prediction and simulation-based validation of the transition gain.
  • Analysis of Lyapunov dimension to assess attractor dimensionality.

Main Results:

  • Finite heavy-tailed RNNs exhibit a sharp transition between quiescent and chaotic dynamics, contrasting with mean-field predictions.
  • A broader gain regime near the edge of chaos is observed in heavy-tailed RNNs, indicating a slower transition to chaos.
  • Heavier tails lead to a reduced Lyapunov dimension, implying lower effective dimensionality of the attractor.
  • A biologically aligned tradeoff exists between the robustness of dynamics and the richness of neural activity.

Conclusions:

  • Finite-size effects are crucial for understanding dynamics in heavy-tailed RNNs, where mean-field theory breaks down.
  • Heavy-tailed connectivity provides a more realistic model for neural circuits, offering insights into brain function.
  • The study provides a tractable framework for analyzing dynamics in realistically sized, heavy-tailed neural circuits.