[Paper Review] Noise dynamically suppresses chaos in random neural networks

Sven Goedeke, Jannis Schuecker+1 more|arXiv (Cornell University)|Mar 6, 2016

Neural Networks and Applications3 citations

TL;DR

This paper demonstrates that additive white noise dynamically suppresses chaos in randomly coupled neural networks by shifting the onset of chaos to significantly higher coupling strengths than predicted by traditional stability analysis. Using dynamical mean-field theory, the authors derive an exact condition for the transition to chaos and show that noise enables a coexistence of expansive dynamics and stable long-term behavior, with temporal correlations peaking slightly above the critical coupling.

ABSTRACT

Noise is ubiquitous in neural systems due to intrinsic stochasticity or external drive. For deterministic dynamics, randomly coupled neural networks display a transition to chaos at a critical coupling strength. Here, we investigate the effect of additive white noise on the onset of chaos. We develop the dynamical mean-field theory yielding the statistics of the activity and the maximum Lyapunov exponent. An exact condition determines the transition from stable to chaotic dynamics. Noise suppresses chaos by a dynamic mechanism, shifting the transition to significantly larger coupling strengths than predicted by local stability analysis. A regime emerges, where expansive dynamics and stable long-term behavior coexist. Furthermore, the time scale of the temporal correlations does not diverge at the transition, but peaks slightly above the critical coupling strength.

Motivation & Objective

• To understand how additive white noise affects the onset of chaos in randomly coupled neural networks.
• To develop a dynamical mean-field theory that captures the statistics of neural activity and the maximum Lyapunov exponent under noise.
• To determine the exact condition for the transition from stable to chaotic dynamics in the presence of noise.
• To investigate whether noise can stabilize systems that would otherwise be chaotic, even when local stability analysis predicts instability.
• To examine the behavior of temporal correlation time scales near the transition point under noisy conditions.

Proposed method

• The authors employ dynamical mean-field theory to analytically describe the statistical properties of neural activity in large random networks under additive white noise.
• They derive the maximum Lyapunov exponent as a function of coupling strength and noise intensity to assess stability and chaos.
• An exact analytical condition for the transition from stable to chaotic dynamics is derived using the theory, valid beyond local linear stability analysis.
• The method accounts for the nonlinearity of neural dynamics and the stochastic nature of noise, enabling a global characterization of the system's behavior.
• The theory predicts the peak of temporal correlation time scales, which occurs slightly above the critical coupling strength rather than diverging at the transition.

Experimental results

Research questions

• RQ1 How does additive white noise alter the critical coupling strength at which chaos emerges in random neural networks?
• RQ2 Can noise induce a regime where expansive dynamics coexist with stable long-term behavior?
• RQ3 Does the temporal correlation time scale diverge at the transition to chaos, or does it exhibit a peak near the critical point?
• RQ4 How does the dynamical mean-field theory capture the interplay between noise, coupling strength, and chaos in finite-size but large-scale neural networks?
• RQ5 What is the exact condition for the transition to chaos when noise is present, and how does it differ from predictions based on local stability analysis?

Key findings

• Noise dynamically suppresses chaos by shifting the onset of chaos to significantly higher coupling strengths than predicted by local stability analysis.
• A stable regime emerges where expansive dynamics and long-term stability coexist, enabled by the presence of noise.
• The maximum Lyapunov exponent remains negative for coupling strengths beyond the deterministic critical point, indicating sustained stability under noise.
• The temporal correlation time scale does not diverge at the transition but instead peaks slightly above the critical coupling strength.
• The exact condition for the transition to chaos is derived analytically and depends on both noise intensity and network coupling strength.
• The dynamical mean-field theory successfully captures the non-trivial interplay between noise, nonlinearity, and network structure in suppressing chaotic behavior.