chaos-synchrony-ei-networks - SKILL.md Agent Skill

name: chaos-synchrony-ei-networks description: "Extended Sompolinsky-Crisanti-Sommers (SCS) theory for two-population Excitatory-Inhibitory networks with target-specific inhibition. DMFT derivation of phase diagrams showing quiescence, asynchronous chaos, persistent activity, structured chaos, and coherent oscillations. Shows target-specific inhibition determines which collective instability dominates. Activation: SCS E/I theory, DMFT neural networks, chaos-synchrony transition, E/I balance, neural phase diagram, 兴奋抑制网络混沌."

Chaos to Synchrony in E/I Networks with Target-Specific Inhibition

Extended SCS framework for two-population firing-rate networks with segregated excitatory/inhibitory neurons and target-specific inhibitory couplings that break E/I balance. Uses Dynamic Mean-Field Theory (DMFT) to derive unified phase diagrams linking inhibitory architecture to large-scale dynamical regimes.

Paper Reference

Title: From Chaos to Synchrony in Recurrent Excitatory-Inhibitory Networks with Target-Specific Inhibition
Authors: Carles Martorell, Rubén Calvo, Alessia Annibale, Miguel A. Muñoz
arXiv: 2605.14916v1 (cond-mat.dis-nn)
Date: 2026-05-14
PDF: https://arxiv.org/pdf/2605.14916.pdf

Background: SCS Theory

The seminal Sompolinsky-Crisanti-Sommers (SCS) theory showed random recurrent networks undergo a transition from quiescence to asynchronous chaos as connectivity strength increases:

Random connectivity → dynamical instability → internally generated fluctuations
Dynamic Mean-Field Theory (DMFT) became the standard framework for phase diagrams

Extended Framework: Two-Population E/I with Target-Specific Inhibition

Network Architecture

Two-population firing-rate network:
  - Excitatory population (E): N/2 neurons
  - Inhibitory population (I): N/2 neurons
  - Target-specific inhibitory couplings (β, δ):
    β: E→I inhibitory strength
    δ: I→I inhibitory strength

The connectivity matrix has i.i.d. entries with mean J₀/N and variance J²/N. Parameters β, δ > 0 control relative inhibitory coupling strengths — a minimal extension that breaks E/I balance.

Key Question

How does target-specific inhibition reorganize the SCS phase diagram? Specifically: Can persistent activity coexist with chaos? What determines whether the system enters structured chaos vs. collective oscillations?

DMFT Equations

The mean-field theory yields self-consistent equations for:

Mean activities Mₓ(t), Mᵧ(t) — macroscopic order parameters
Autocorrelation functions Cₓ(τ), Cᵧ(τ) — fluctuation statistics
Cross-correlations between E and I populations

These are solved self-consistently to determine phase boundaries.

Three Qualitatively Distinct Phase Diagram Organizations

The nature of the phase diagram is classified by the dominant eigenvalue λₘ of an effective matrix M_{β,δ} determined by (β, δ):

Type 1: Inhibition-Dominated (λₘ real and positive)

Phases: Quiescent (Q) → Asynchronous Chaos (AC) → Persistent Activity (PA)

Classical SCS phenomenology preserved
E/I network behaves like extended single-population model

Type 2: Strictly Balanced (λₘ complex with positive real part)

Phases: Quiescent (Q) → Asynchronous Chaos (AC) → Coherent Oscillatory Activity (COA)

New regime: collective oscillations emerge
Chaos-to-oscillation transition via competition mechanism

Type 3: Excitation-Dominated (λₘ with negative real part)

Phases: Quiescent (Q) → Asynchronous Chaos (AC) only

Simple phase diagram — only disorder-driven instability

Emergent Phases Beyond Asynchronous Chaos

Persistent Activity (PA)

Non-zero mean activity fixed point
Stable when λₘ is real and positive
Analogous to working memory states

Structured/Synchronous Chaos (SC)

Chaos with non-vanishing mean activity
Extends synchronous chaos from single-population to E/I system
Chaotic fluctuations around structured (non-zero) mean trajectory

Coherent Oscillatory Activity (COA)

Key discovery: Collective oscillations suppress chaotic fluctuations
Transition from AC → COA reflects competition between:
- Disorder-driven chaotic state (high-dimensional)
- Low-dimensional collective oscillatory mode
Deeper in oscillatory phase: collective mode fully suppresses chaos
Mechanism reminiscent of stimulus-induced suppression of chaos, but here the oscillatory drive is generated endogenously by structured E/I feedback

Chaos Quantification

Largest Lyapunov Exponent (LLE)

Positive LLE → chaotic regime
Zero/negative LLE → non-chaotic (fixed point or periodic)
AC-SC boundary identified by LLE analysis

Kuramoto Order Parameter

Measures phase synchronization across neurons
COA regime: high Kuramoto parameter (synchronized)
AC regime: low Kuramoto parameter (desynchronized)

Stability Analysis

Two distinct routes out of quiescence:

Type I: Mean-Driven Instability

Governed by eigenvalue λₘ of M_{β,δ}
Determines PA or COA transition
Depends on target-specific inhibition parameters (β, δ)

Type II: Fluctuation-Driven Instability

Governed by autocorrelation stability condition
Determines AC transition
Analogous to classical SCS instability

Phase Diagram Rescaling

After axis redefinition: (J₀/J), (1/gJ) — quiescent-state stability boundaries collapse onto a common curve across different (β, δ). However, transitions between non-quiescent states depend explicitly on β and δ.

Key Findings

Target-specific inhibition reorganizes the phase diagram — it selects which collective instability (chaotic vs. oscillatory) becomes dominant
Oscillations suppress chaos — COA transition eliminates chaotic fluctuations, not a phase of chaos around oscillatory mean
Endogenous chaos suppression — unlike stimulus-induced suppression, here the oscillatory drive is internally generated
Three robust classes of phase diagrams based on dominant eigenvalue properties of the inhibitory structure
Unified DMFT framework linking inhibitory architecture to large-scale dynamical regime organization

Implications for Neuroscience

Criticality Hypothesis

Cortical networks may operate near phase transitions where activity displays scale invariance and favorable computational properties. The E/I structure determines which transitions are accessible.

Computational Regimes

Asynchronous chaos: High-dimensional computation, rich internal dynamics
Persistent activity: Working memory, sustained representations
Coherent oscillations: Rhythmic coordination, synchronized processing
Structured chaos: Complex computations with partial order

Biological Relevance

E/I balance breaking is ubiquitous in biological circuits
Target-specific inhibition (different β, δ for different targets) is biologically realistic
Phase diagram organization predicts which dynamical regimes are accessible given circuit architecture

Mathematical Framework

Model Equations

τ dxᵢ/dt = -xᵢ + Σⱼ Wᵢⱼ φ(xⱼ)
τ dyᵢ/dt = -yᵢ + Σⱼ W'ᵢⱼ φ(yⱼ)

Where W, W' are random connectivity matrices with E/I structure.

DMFT Self-Consistency

Mₓ(t) = ∫ D[z] φ(√qₓ z + Mₓ(t))
Cₓ(τ) = ∫∫ Dz Dz' φ(·) φ(·)

With appropriate self-consistency conditions on qₓ, qᵧ.

Applications

Cortical circuit modeling: Understanding E/I balance in cortical dynamics
Neural computation theory: Linking circuit architecture to computational regimes
Brain state transitions: Modeling transitions between dynamical regimes
Neuromorphic design: Architecture-guided dynamical regime selection
E/I balance disorders: Modeling pathological dynamics in schizophrenia, epilepsy

Activation Keywords

E/I network SCS theory
chaos-synchrony transition
DMFT neural networks
target-specific inhibition
E/I balance breaking
neural phase diagram
asynchronous chaos
coherent oscillations
structured chaos
兴奋抑制网络混沌理论

Related Skills

ei-network-chaos-synchrony-theory — Existing SCS E/I theory skill
neural-population-dynamics — Neural population analysis methods
neural-critical-dynamics-theory — Critical dynamics theory

Limitations

Firing-rate model (not spiking neurons)
Random connectivity (not structured biological connectivity)
Two-population simplification
Mean-field approximation (thermodynamic limit N→∞)
No external input dynamics considered

Future Directions

Spiking neuron version of the framework
Structured (non-random) connectivity patterns
Multi-population extensions (>2 populations)
External input / sensory-driven dynamics
Learning rules that adapt (β, δ) dynamically