chaos-synchrony-ei-networks

# 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: 1. **Mean activities** Mₓ(t), Mᵧ(t) — macroscopic order parameters 2. **Autocorrelation functions** Cₓ(τ), Cᵧ(τ) — fluctuation statistics 3. **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 1. **Target-specific inhibition reorganizes the phase diagram** — it selects which collective instability (chaotic vs. oscillatory) becomes dominant 2. **Oscillations suppress chaos** — COA transition eliminates chaotic fluctuations, not a phase of chaos around oscillatory mean 3. **Endogenous chaos suppression** — unlike stimulus-induced suppression, here the oscillatory drive is internally generated 4. **Three robust classes** of phase diagrams based on dominant eigenvalue properties of the inhibitory structure 5. **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 1. **Cortical circuit modeling:** Understanding E/I balance in cortical dynamics 2. **Neural computation theory:** Linking circuit architecture to computational regimes 3. **Brain state transitions:** Modeling transitions between dynamical regimes 4. **Neuromorphic design:** Architecture-guided dynamical regime selection 5. **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 1. Spiking neuron version of the framework 2. Structured (non-random) connectivity patterns 3. Multi-population extensions (>2 populations) 4. External input / sensory-driven dynamics 5. Learning rules that adapt (β, δ) dynamically