| name | ei-network-chaos-synchrony-theory |
| description | Extended Sompolinsky-Crisanti-Sommers (SCS) chaos theory for Excitatory-Inhibitory recurrent networks with target-specific inhibition. Derives mean-field theory for E/I networks, identifies three dynamical regimes (quiescence, asynchronous chaos, coherent oscillations), and shows coherent oscillations suppress chaos. Activation: chaos-synchrony, SCS theory, E/I balance, target-specific inhibition, dynamical mean-field theory, neural phase diagram, recurrent network dynamics, excitation-inhibition balance, neural chaos theory, 兴奋抑制平衡, 混沌同步
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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:
m_E(t) = ∫ Dz φ(√(C_E(0))z + m_E(t))m_I(t) = ∫ Dz φ(√(C_I(0))z + m_I(t))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
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
Phase Diagram Computation
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
else:
phase_map[i, j] = 1
else:
if lyap < 0:
phase_map[i, j] = 2
else:
phase_map[i, j] = 3
return phase_map
Key Insights
- Target-specific inhibition is a key control parameter for large-scale dynamics
- E-I structure fundamentally changes the SCS phase diagram
- Coherent oscillations emerge as a distinct regime, not just a variant of chaos
- Chaos suppression by oscillations is a robust phenomenon
- 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)
