ei-network-chaos-synchrony-theory

# E/I Network Chaos-Synchrony Theory Extended SCS theory for recurrent Excitatory-Inhibitory networks with target-specific inhibition. Provides theoretical framework for understanding dynamical regime transitions in biological neural circuits. ## Background: Classic SCS Theory The seminal Sompolinsky-Crisanti-Sommers (SCS) theory showed that random recurrent networks undergo a transition from quiescence to asynchronous chaos as connectivity strength increases. This established the link between: - Random connectivity → dynamical instability → internally generated fluctuations ## Extended Framework: Target-Specific Inhibition ### Network Architecture ``` Two-population firing-rate network: ├── Excitatory neurons (E) → self-excitation + cross-excitation └── Inhibitory neurons (I) → target-specific inhibition (breaks E-I balance) ``` ### Key Parameters - **g_E**: Excitatory coupling strength - **g_I**: Inhibitory coupling strength - **η**: Target-specificity of inhibition - **E-I balance ratio**: Determines dynamical regime ## Three Dynamical Regimes ### Regime 1: Inhibition-Dominated / Strictly Balanced - **Behavior**: Quiescent activity OR asynchronous chaos - **Characteristics**: - Mean activity → 0 (quiescent) - Fluctuation-driven chaos with vanishing mean - Similar to classic SCS but with E/I structure ### Regime 2: Excitation-Dominated (Synchronous Chaos) - **Behavior**: Persistent activity + synchronous chaos - **Characteristics**: - Non-vanishing mean activity - Chaotic fluctuations around the mean - Stability matrix eigenvalues determine transition ### Regime 3: Excitation-Dominated (Coherent Oscillations) - **Behavior**: Persistent activity + coherent oscillations - **Characteristics**: - Periodic mean trajectory - **Key finding**: Chaotic fluctuations are SUPPRESSED - Onset of oscillations eliminates chaos (input-induced suppression) ## Critical Discovery: Chaos Suppression by Oscillations Coherent oscillations do NOT coexist with chaotic fluctuations. Instead, oscillation onset SUPPRESSES the chaotic component. This is reminiscent of input-induced suppression of chaos: - Periodic drive → entrainment → chaos suppression - Similar to Poincaré-Lindqvist behavior in forced oscillators ## Mean-Field Theory Derivation ### Self-Consistent Equations ``` Mean activities: m_E(t), m_I(t) Autocorrelations: C_E(τ), C_I(τ) ``` The DMF equations: 1. `m_E(t) = ∫ Dz φ(√(C_E(0))z + m_E(t))` 2. `m_I(t) = ∫ Dz φ(√(C_I(0))z + m_I(t))` 3. `C_α(τ) = g_α² ∫ Dz' Dz φ(...)φ(...)` (α ∈ {E, I}) ### Stability Analysis - **Mean-driven instability**: Eigenvalues of stability matrix cross real axis - **Fluctuation-driven instability**: Largest Lyapunov exponent becomes positive - **Phase boundaries**: Determined by coupling strengths and target-specificity ## Phase Diagram ``` g_I (inhibition) ↑ │ Quiescent │ Asynchronous ────────────────┤ Chaos │ ────────────────┤────────────────── Coherent │ Synchronous Oscillations │ Chaos │ └──────────────────→ g_E (excitation) ``` ## Computational Applications ### 1. Brain State Transitions - Sleep-wake transitions (quiescence → chaos) - Seizure dynamics (chaos → oscillations) - Cognitive switching (regime transitions) ### 2. Neuromodulation Control - Neuromodulators as parameters shifting the system between regimes - Dopamine: shifts E/I balance - Acetylcholine: modulates target-specificity ### 3. Network Design - Designing reservoirs for specific computational regimes - Optimal regime for different tasks (memory vs. computation) ## Implementation Patterns ### Stability Matrix Construction ```python import numpy as np from scipy.linalg import eigvals def stability_matrix(g_E, g_I, eta, m_E, m_I, phi_prime): """Construct stability matrix for E/I network.""" J_EE = g_E * (1 - eta) * phi_prime(m_E) J_EI = -g_I * eta * phi_prime(m_I) J_IE = g_E * eta * phi_prime(m_E) J_II = -g_I * (1 - eta) * phi_prime(m_I) J = np.array([[J_EE, J_EI], [J_IE, J_II]]) return J def check_stability(J): """Check stability via eigenvalue analysis.""" eigenvalues = eigvals(J) max_real = max(np.real(eigenvalues)) return max_real < 0 # Stable if all eigenvalues have negative real part ``` ### Phase Diagram Computation ```python def compute_phase_diagram(g_E_range, g_I_range, eta=0.5): """Compute phase diagram over parameter space.""" phase_map = np.zeros((len(g_E_range), len(g_I_range))) for i, g_E in enumerate(g_E_range): for j, g_I in enumerate(g_I_range): m_E, m_I = solve_mean_field(g_E, g_I, eta) J = stability_matrix(g_E, g_I, eta, m_E, m_I) lyap = largest_lyapunov(g_E, g_I, m_E, m_I) if max(np.real(eigvals(J))) < 0: if lyap < 0: phase_map[i, j] = 0 # Quiescent else: phase_map[i, j] = 1 # Asynchronous chaos else: if lyap < 0: phase_map[i, j] = 2 # Coherent oscillations else: phase_map[i, j] = 3 # Synchronous chaos return phase_map ``` ## Key Insights 1. **Target-specific inhibition** is a key control parameter for large-scale dynamics 2. **E-I structure** fundamentally changes the SCS phase diagram 3. **Coherent oscillations** emerge as a distinct regime, not just a variant of chaos 4. **Chaos suppression** by oscillations is a robust phenomenon 5. **Biological plausibility**: E/I segregation is ubiquitous in cortex ## Related Skills - **chaos-synchrony-ei-networks**: Dynamical mean-field theory for E/I networks - **neural-population-dynamics**: Methods for analyzing neural population dynamics - **rhythm-switching-adaptive-time-constants-rnn**: RNN rhythm switching mechanisms - **hermes-brain-connectivity**: Brain connectivity analysis tools ## References - Martorell et al., "From Chaos to Synchrony in Recurrent Excitatory-Inhibitory Networks with Target-Specific Inhibition", arXiv:2605.14916 (2026) - Sompolinsky, Crisanti, Sommers, "Chaos in Random Neural Networks", PRL (1988) - Rajan, Abbott, Sompolinsky, "Stimulus-dependent suppression of chaos in recurrent neural networks", PRE (2010)