mean-field-oscillatory-low-rank-rnn

name	mean-field-oscillatory-low-rank-rnn
description	Mean-field theory for rich oscillatory dynamics in low-rank recurrent networks with activity-dependent adaptation. Analyzes how low-rank structure and adaptation interact to produce complex oscillatory and chaotic behavior in recurrent neural networks.
trigger_words	["mean field oscillatory","low rank recurrent","activity dependent adaptation","oscillatory dynamics RNN","rich oscillatory","RNN mean field theory"]
category	neuroscience

Overview

This paper (arXiv:2606.30366) develops mean-field theory for recurrent neural networks with both low-rank structure and activity-dependent adaptation. Shows how these two mechanisms interact to produce rich dynamical regimes including oscillations, chaos, and multistability.

Core Theory

Model Structure

dx/dt = -x + W_lowrank * φ(x) + W_full * φ(x) - g_adapt * a * φ(x)
da/dt = (-a + φ(x)) / τ_adapt

• W_lowrank: Low-rank structured connectivity (rank r << N)
• W_full: Random full-rank connectivity (chaotic driver)
• g_adapt: Adaptation strength controlling oscillation onset
• τ_adapt: Adaptation timescale

Mean-Field Reduction

1. Project dynamics onto low-rank subspace
2. Use dynamical mean-field theory for random component
3. Derive closed equations for order parameters
4. Analyze fixed points, limit cycles, and chaotic regimes

Key Regimes

1. Fixed point regime: Low adaptation, dominated by low-rank structure
2. Oscillatory regime: Moderate adaptation creates limit cycles
3. Chaotic regime: Strong random connectivity drives chaos
4. Mixed regime: Adaptation + chaos produces complex metastable dynamics

Implementation

Mean-Field Equations

m_dot = f(m, q, a)    # Order parameter dynamics
q_dot = g(m, q, a)    # Variance dynamics  
a_dot = (-a + h(m)) / τ_adapt  # Adaptation dynamics

Bifurcation Analysis

• Vary g_adapt and σ_W (random connectivity strength)
• Track fixed point stability via Jacobian eigenvalues
• Identify Hopf bifurcations leading to oscillations
• Map chaos onset via largest Lyapunov exponent

Pitfalls

• Low-rank assumption: Results depend on r << N; for higher rank, mean-field breaks down
• Timescale separation: Analysis assumes τ_adapt >> 1; fast adaptation needs different treatment
• Activation function: Results derived for specific φ(x); different nonlinearities change dynamics

Applications

• Working memory: Oscillatory regimes support persistent activity
• Decision making: Multistability enables choice between alternatives
• Motor control: Oscillations drive rhythmic motor patterns

Verification

1. Simulate full RNN and compare with mean-field predictions
2. Verify bifurcation diagram matches numerical continuation
3. Test different activation functions (ReLU, tanh, sigmoid)
4. Validate low-rank assumption by varying rank r

Activation

mean field theory, oscillatory dynamics, low rank RNN, activity dependent adaptation, bifurcation analysis, RNN dynamics, working memory, neural oscillations