mean-field-low-rank-adaptation-oscillations

name	mean-field-low-rank-adaptation-oscillations
version	1.0.0
description	Dynamical mean-field theory for low-rank recurrent networks with firing-rate adaptation. Identifies four oscillatory regimes and bifurcation mechanisms linking chaos, Hopf bifurcation, and noise-sustained oscillations to biological rhythms (Up-Down states, waxing-and-waning episodes).
trigger_phrases	["mean-field theory","low-rank recurrent network","firing-rate adaptation","oscillatory dynamics","chaos onset","Hopf bifurcation","Up-Down states","coherent state","noise-sustained oscillations"]
activation_keywords	["neural oscillations","working memory","sleep rhythms","anesthesia","criticality","dynamical systems","reduced-order model","bifurcation analysis"]
tags	["q-bio.NC","computational neuroscience","dynamical systems","mean-field theory","oscillations","adaptation"]
created	2026-06-30T00:00:00.000Z
arxiv_id	2606.30366
arxiv_url	https://arxiv.org/abs/2606.30366
authors	["Bowen W. Zheng","Earl K. Miller","Ila R. Fiete"]
affiliations	["MIT (Earl K. Miller lab)","Stanford (Ila R. Fiete lab)"]
publication_date	2026-06-29T00:00:00.000Z

Mean-Field Theory of Rich Oscillatory Dynamics in Low-Rank Recurrent Networks with Activity-Dependent Adaptation

Abstract

Dynamical mean-field theory for random recurrent networks with low-rank structure and firing-rate-driven adaptation. When random connectivity is strong enough to generate chaos, increasing adaptation strength drives the network through four regimes: static coherent state → noise-sustained oscillations (regular → irregular) → stochastic switching between symmetric wells → global limit cycle. The theory identifies two instability mechanisms (chaos onset from random connectivity + Hopf bifurcation of coherent mode) and shows how adaptation shapes both through frequency-dependent single-neuron transfer function. A reduced 3D model captures bifurcation structure. Above chaos threshold, coherent population-level oscillations coexist with heterogeneous firing rates and network-generated stochasticity at single-neuron level. Produces waxing-and-waning rhythmic episodes, persistent state switching, and slow Up-Down alternations — dynamics observed during wakefulness, sleep, and anesthesia.

Core Insights

1. Four Dynamical Regimes

Adaptation strength creates a progression through qualitatively distinct regimes:

• Regime I (Static Coherent State): Weak adaptation, stable fixed point
• Regime II (Noise-Sustained Oscillations): Regular → irregular oscillations via noise amplification near bifurcation
• Regime III (Stochastic Switching): Bistable regime with noise-driven transitions between symmetric attractor wells
• Regime IV (Global Limit Cycle): Strong adaptation drives deterministic oscillations

2. Two Instability Mechanisms

• Chaos onset: Driven by random connectivity strength (classical mean-field instability)
• Hopf bifurcation of coherent mode: Adaptation creates oscillatory instability even in non-chaotic regime

These mechanisms interact: adaptation reshapes the frequency-dependent single-neuron transfer function, modifying both chaos threshold and oscillation frequency.

3. Reduced 3D Model

Full network dynamics captured by 3D ODE system tracking:

• Mean population activity (m)
• Adaptation variable (a)
• Variance of activity (Δ)

This reduction preserves bifurcation structure and enables analytical tractability.

4. Coexistence of Scales

Above chaos threshold:

• Population level: Coherent oscillations (macroscopic order)
• Single-neuron level: Heterogeneous firing rates + network-generated stochasticity (microscopic variability)