sequential-chaotic-oscillations-ei-networks

name: sequential-chaotic-oscillations-ei-networks description: "Sequential chaotic oscillations (SCOs) in excitatory-inhibitory threshold-linear networks - dynamical mechanism for sequential metastability in brain dynamics. Activation: sequential metastability, chaotic itinerancy, E-I oscillation, SCO, threshold-linear network, brain dynamics, metastable states."

Sequential Chaotic Oscillations in E-I Threshold-Linear Networks

arXiv:2606.00373 - Submitted May 29, 2026

Authors: Jie Zang, Carina Curto

Core Innovation: Proposes Sequential Chaotic Oscillations (SCOs) as a candidate dynamical mechanism for sequential metastability observed in healthy brain function, providing a dynamical-systems framework for the balance between integration and segregation.

Key Concepts

Sequential Chaotic Oscillations (SCOs)

• Definition: A simple form of chaotic itinerancy occurring in excitatory-inhibitory threshold-linear networks (E-I TLNs) under constant input
• Characteristics:
  • Sequence of metastable states with predictable transition order determined by underlying graph structure
  • Requires unstable singleton fixed points and sufficiently strong inhibition
  • Captures sequential metastability phenomenon observed in brain dynamics

Graph Rules for E-I TLNs

• Developed new graph rules to characterize fixed point structure
• Applied to paths and cycles networks
• Identified parameter regime for SCO emergence:
  1. Unstable singleton fixed points required
  2. Strong inhibition necessary
  3. Network topology determines transition sequence

E-I Oscillation Modes

• Z-mode: Captures excitatory differences between neurons
• Mean mode: Represents overall network activity
• Decomposition enables distinction of attractors associated with full-support fixed points
• E-I oscillations need NOT be synchronized - novel finding

Methodology

Threshold-Linear Network (TLN) Framework

Mathematical formulation for E-I TLN dynamics:

ẋ_i = -x_i + [∑_j W_ij x_j + b_i]_+  (excitatory neurons)
ẋ_i = -x_i + [-∑_j W_ij x_j + b_i]_+  (inhibitory neurons)

Where [·]_+ denotes threshold-linear function (ReLU-like).

Fixed Point Analysis

• Singleton fixed points: Single active neuron state
• Full-support fixed points: All neurons active
• Instability of singleton fixed points is necessary condition for SCOs

Graph-Theoretic Characterization

• Transition sequence predictable from graph topology
• Paths → directed transition sequences
• Cycles → periodic-like behavior with chaotic modulation

Theoretical Significance

Sequential Metastability Mechanism

• Bridges gap between empirical brain observations and dynamical systems theory
• Provides mechanistic explanation for metastable state transitions
• Predictable transition order → structured chaos

Integration-Segregation Balance

• Metastable states reflect balance between:
  • Integration: Network-wide coordination (mean mode)
  • Segregation: Local specialization (z-mode)
• SCOs provide formal framework for this balance

Non-Synchronized E-I Oscillations

• Counterintuitive finding: E-I populations can oscillate independently
• Traditional assumption: E and I populations synchronized
• Novel insight: Different modes capture distinct aspects of network dynamics

Applications

Brain Dynamics Modeling

• Framework for understanding metastable dynamics in:
  • Resting state networks
  • Cognitive state transitions
  • Task-related activity sequences

Neural Network Architecture

• Insights for designing E-I balanced networks with:
  • Predictable state transition dynamics
  • Controlled chaotic behavior
  • Structured metastability

Computational Neuroscience

• Graph rules enable:
  • Predicting network dynamics from connectivity
  • Characterizing fixed point landscapes
  • Designing networks with specific oscillation patterns

Key Findings

1. SCO Emergence Conditions:

   • Unstable singleton fixed points (necessary)
   • Strong inhibition (sufficient)
   • Specific network topology (predicts transitions)

2. Graph Rules Validation:

   • Characterization works for paths and cycles
   • Transition sequence predictable from graph
   • Fixed point structure determines dynamics

3. Mode Decomposition:

   • Z-mode + Mean mode = complete dynamics description
   • Enables attractor classification
   • Separates local and global activity patterns

Connection to Existing Research

• Metastability: Links to empirical findings in fMRI, EEG studies
• Chaotic Itinerancy: Provides simplified realization of complex phenomenon
• E-I Balance: Extends classical E-I oscillation framework
• Network Dynamics: Complements attractor network theories

Mathematical Framework

State Space

• Metastable states as saddle points or transient attractors
• Transition dynamics governed by:
  • Inhibition strength
  • Graph topology
  • Initial conditions

Predictability

• Transition sequence deterministic given graph
• Timing chaotic (unpredictable)
• Order predictable → "sequential" character

Limitations & Future Directions

• Current work focuses on paths and cycles
• Extension to general graphs needed
• Biological validation pending
• Comparison with empirical brain data required

Practical Implications

• Design principles for neuromorphic circuits with structured chaos
• Framework for analyzing metastability in neural recordings
• Graph-based network design for specific dynamic regimes

References

• Carina Curto's work on fixed points in neural networks
• Chaotic itinerancy literature
• Brain metastability empirical studies

Activation Keywords

sequential metastability, chaotic itinerancy, E-I oscillation, threshold-linear network, SCO, metastable states, brain dynamics, excitatory-inhibitory network, graph rules, fixed point analysis, neural oscillation modes, integration-segregation balance