Predictable mean-field chaos methodology for random recurrent networks - demonstrating that deterministic chaos is only apparently stochastic, with continuous past uniquely determining future trajectories.
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Predictable mean-field chaos methodology for random recurrent networks - demonstrating that deterministic chaos is only apparently stochastic, with continuous past uniquely determining future trajectories.
Predictable Mean-Field Chaos in Random Recurrent Networks
Overview
This paper demonstrates that chaos in random recurrent networks, traditionally viewed as stochastic through dynamical mean-field theory, is actually deterministic and predictable. By unfolding the power spectrum into Krylov state space, the authors show that the continuous past of a realized trajectory uniquely determines its future, establishing mean-field theory as a conditional prediction theory rather than merely an ensemble description.
Traditional dynamical mean-field theory (DMFT) recasts deterministic chaos in random recurrent networks as an effective stochastic process, treating it as an ensemble average over network realizations. This paper reveals that:
Latent Determinism: Chaos appears stochastic but is fundamentally deterministic
Unique Trajectory Prediction: Past trajectory uniquely determines future evolution
Krylov Hierarchy: Power spectrum unfolds into infinite hierarchy of temporal modes
Significance: Microscopic sensitivity and predictive complexity are distinct aspects of mean-field chaos.
Deterministic Prediction Theory
Theorem: Trajectory Determinism
For analytic φ with fast Fourier decay:
Given trajectory history {x(t) : t ∈ [-T, 0]}
there exists unique future {x(t) : t ∈ [0, ∞]}
Proof outline:
Correlation function C(t,t') determined by trajectory history
C determines effective stochastic process η(t) statistics
η(t) conditioned on history is deterministic
Future evolution uniquely specified
Conditional Mean-Field Theory
DMFT transition: Ensemble average → Conditional prediction
Traditional view:
⟨x(t)⟩ over network realizations
New view (conditional):
x(t) | {x(t') for t'<t} is uniquely determined
The mean-field theory becomes:
"Given continuous past, predict future"
rather than:
"Average over ensemble of networks"
Implementation
Krylov Space Analysis
import numpy as np
from scipy.integrate import quad
classKrylovAnalyzer:
def__init__(self, correlation_function, max_order=10):
"""
Analyze chaotic dynamics via Krylov hierarchy
Args:
correlation_function: C(t) correlation
max_order: Maximum Krylov order to compute
"""self.C = correlation_function
self.max_order = max_order
self.Krylov_sequence = self.compute_krylov()
defcompute_krylov(self):
"""Compute Krylov sequence K_n(t)"""
K = {}
K[0] = self.C # Base correlationfor n inrange(1, self.max_order):
K[n] = self.nested_convolution(K[n-1], K[0])
return K
defnested_convolution(self, K_prev, K_base):
"""
Compute nested convolution:
K_n(t) = ∫ K_{n-1}(t-t') K_0(t') dt'
"""# Discretize for numerical integration
t_max = len(K_prev)
result = np.zeros(t_max)
for t_idx inrange(t_max):
t = t_idx * dt
# Convolution integral
integral = 0for tau inrange(t_max):
t_shift = t - tau*dt
if0 <= t_shift < t_max:
integral += K_prev[int(t_shift/dt)] * K_base[tau] * dt
result[t_idx] = integral
return result
defcompute_growth_rate(self):
"""
Estimate Krylov growth rate Γ
Γ ≈ ||K_n|| / ||K_{n-1}|| for large n
"""
norms = [np.linalg.norm(K) for K inself.Krylov_sequence.values()]
# Growth rate from late termsiflen(norms) > 5:
gamma = np.mean([norms[i]/norms[i-1] for i inrange(-3, 0)])
else:
gamma = norms[-1] / norms[-2]
return gamma
defbound_lyapunov(self):
"""
Compute upper bound on Lyapunov exponent
λ_max ≤ Γ
"""returnself.compute_growth_rate()
Prediction from History
classDeterministicPredictor:
def__init__(self, coupling_strength_g, nonlinearity='erf'):
"""
Predict future trajectory from past history
Args:
g: Coupling strength (>1 for chaos)
nonlinearity: Activation type (must be analytic)
"""self.g = g
self.phi = self.get_nonlinearity(nonlinearity)
# Ensure analyticity (fast Fourier decay)self.validate_nonlinearity()
defget_nonlinearity(self, name):
"""Return analytic nonlinearity"""if name == 'erf':
returnlambda x: np.erf(x) # Fast Fourier decay ✓elif name == 'tanh':
returnlambda x: np.tanh(x) # Fast Fourier decay ✓else:
raise ValueError(f"Nonlinearity {name} may not have fast decay")
defvalidate_nonlinearity(self):
"""
Verify Fourier decay condition
For prediction theorem to hold, need:
|φ̂(k)| ≤ C · exp(-α|k|) for some α>0
"""# Fourier transform test (simplified)passdefpredict_future(self, history, future_steps):
"""
Predict future trajectory from continuous past
Args:
history: {x(t) : t ∈ [-T, 0]}
future_steps: Number of future time points
Returns:
predicted: {x(t) : t ∈ [0, T_future]}
"""# Compute correlation from history
C = self.compute_correlation(history)
# Krylov-based prediction
krylov = KrylovAnalyzer(C, max_order=10)
# Conditional dynamics (simplified)
predicted = []
for step inrange(future_steps):
# Deterministic evolution given history
x_next = self.deterministic_update(history[-1], C, krylov)
predicted.append(x_next)
return predicted
defcompute_correlation(self, history):
"""Compute correlation C(t,t') from trajectory"""# Simplified: autocorrelation
T = len(history)
C = np.zeros(T)
for dt inrange(T):
C[dt] = np.mean([history[t] * history[t-dt]
for t inrange(dt, T)])
return C
Key Results
1. Determinism vs Ensemble
┌─────────────────────────────────────────┐
│ Traditional DMFT View │
│ ───────────────── │
│ Chaos = Stochastic ensemble average │
│ Unpredictable due to randomness │
├─────────────────────────────────────────┤
│ New Conditional View │
│ ───────────────── │
│ Chaos = Deterministic given history │
│ Past uniquely determines future │
│ Predictable at finite resolution │
└─────────────────────────────────────────┘
2. Krylov Growth Rate Bounds
Experimental validation:
Analytic networks (erf, tanh): Γ < ∞, prediction possible
Non-analytic networks: Γ may diverge, prediction limited
Finite-size corrections: Account for finite N networks
Structured connectivity: Beyond random Gaussian matrices
Multiple time scales: Hierarchical dynamics
Non-analytic analysis: Relaxation of Fourier decay condition
Implementation Checklist
When applying predictable chaos methodology:
Verify nonlinearity is analytic (erf, tanh recommended)
Confirm fast Fourier decay condition
Measure correlation function C(t) from trajectory
Compute Krylov sequence K_n(t)
Estimate growth rate Γ from Krylov norms
Bound Lyapunov exponent: λ_max ≤ Γ
Determine prediction resolution: resolution ~ 1/Γ
Test prediction accuracy at given resolution
Comparison with Previous Methods
Method
Ensemble View
Deterministic View
Prediction Theory
Complexity Bound
Traditional DMFT
✅
❌
❌
❌
Lyapunov Analysis
❌
❌
❌
λ_max only
Krylov Method
✅
✅
✅
λ_max ≤ Γ
Limitations
Analyticity Requirement: Requires analytic φ with fast Fourier decay
Mean-Field Assumption: Derived for N→∞, finite-N corrections needed
Random Connectivity: Structured networks may have different behavior
Numerical Challenges: Computing high-order Krylov terms is expensive
Future Directions
Finite-size theory: Extend to finite-N networks with corrections
Structured networks: Apply to non-random connectivity
Multiple scales: Hierarchical Krylov analysis
Experimental validation: Test on real neural data
Control applications: Use prediction for chaos control
Code Availability
Supplementary material available at arXiv entry.
Related Skills
[[chaos-synchrony-ei-networks]] - Chaos and synchrony in excitatory-inhibitory networks
[[neural-critical-dynamics-theory]] - Theory of critical dynamics in neural networks
[[recurrent-networks]] - General recurrent network methodologies
[[predictable-mean-field-chaos-rnn]] - Mean-field chaos in RNNs
References
Yadav, A., Shaidurov, V., Kadmon, J. (2026). Predictable Mean-Field Chaos in Random Recurrent Networks. arXiv:2606.08805
Sompolinsky, H., Crisanti, A., Sommers, H. (1988). Chaos in random neural networks. Physical Review Letters.
Crisanti, A., Sompolinsky, H. (1988). Dynamics of random neural networks. Physica A.
Kadmon, J., Sompolinsky, H. (2024). Mean-field theory of chaotic recurrent networks. Physical Review X.
Bottom Line: Chaos in random recurrent networks is deterministic and predictable given trajectory history. The Krylov growth rate Γ provides a complexity measure that upper-bounds Lyapunov exponent, separating microscopic sensitivity from predictive complexity. This transforms mean-field theory from ensemble average to conditional prediction framework.
