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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.
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| name | predictable-mean-field-chaos-random-recurrent-networks |
| description | Predictable mean-field chaos methodology for random recurrent networks - demonstrating that deterministic chaos is only apparently stochastic, with continuous past uniquely determining future trajectories. |
| version | 1.0.0 |
| author | Alkesh Yadav, Vladimir Shaidurov, Jonathan Kadmon |
| arxiv_id | 2606.08805 |
| date_created | 2026-06-09T00:00:00.000Z |
| tags | ["neuroscience","chaos-theory","mean-field","recurrent-networks","krylov-dynamics","deterministic","prediction","neural-networks"] |
| activation_keywords | ["predictable chaos","mean-field","krylov","recurrent networks","deterministic chaos","prediction theory","lyapunov exponent","dissipative systems"] |
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.
arXiv: 2606.08805
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:
# Network model
x_i(t+dt) = x_i(t) + dt * [ -x_i(t) + Σ_j J_ij * φ(x_j(t)) + noise ]
where:
- x_i: State of neuron i
- J_ij: Random Gaussian coupling matrix (J_ij ~ N(0, g²/N))
- φ: Nonlinear activation function (e.g., tanh, erf)
- g: Coupling strength (chaos emerges for g > 1)
- N: Number of neurons
Traditional DMFT replaces deterministic network with stochastic effective process:
x(t) satisfies: dx/dt = -x + g * η(t) * φ(x(t))
where η(t) is Gaussian process with:
⟨η(t)η(t')⟩ = C(t,t') = ⟨φ(x(t))φ(x(t')⟩
Key Insight: DMFT treats this as ensemble average, but actual trajectories are deterministic!
For analytic nonlinearities with sufficiently fast Fourier decay:
The power spectrum S(ω) unfolds into Krylov state space:
Power spectrum: S(ω) = Fourier transform of correlation C(t)
Krylov representation:
┌─────────────────────────────────────────┐
│ K₀(t) = C(t) │ ← Correlation function
│ K₁(t) = ∫ K₀(t-t') K₀(t') dt' │ ← First-order convolution
│ K₂(t) = ∫ K₁(t-t') K₀(t') dt' │ ← Second-order
│ ... │
│ K_n(t) = nth-order nested convolution │ ← Infinite hierarchy
└─────────────────────────────────────────┘
Definition: Krylov growth rate Γ characterizes temporal complexity:
Γ = lim_{n→∞} [ ||K_n(t)|| / ||K_{n-1}(t)|| ]
Properties:
1. Γ sets finite-resolution prediction complexity
2. Γ upper-bounds largest Lyapunov exponent λ_max
3. Γ < ∞ for analytic nonlinearities
Major Result:
λ_max ≤ Γ
where:
- λ_max: Largest Lyapunov exponent (microscopic sensitivity)
- Γ: Krylov growth rate (predictive complexity)
Significance: Microscopic sensitivity and predictive complexity are distinct aspects of mean-field chaos.
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:
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"
import numpy as np
from scipy.integrate import quad
class KrylovAnalyzer:
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()
def compute_krylov(self):
"""Compute Krylov sequence K_n(t)"""
K = {}
K[0] = self.C # Base correlation
for n in range(1, self.max_order):
K[n] = self.nested_convolution(K[n-1], K[0])
return K
def nested_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 in range(t_max):
t = t_idx * dt
# Convolution integral
integral = 0
for tau in range(t_max):
t_shift = t - tau*dt
if 0 <= t_shift < t_max:
integral += K_prev[int(t_shift/dt)] * K_base[tau] * dt
result[t_idx] = integral
return result
def compute_growth_rate(self):
"""
Estimate Krylov growth rate Γ
Γ ≈ ||K_n|| / ||K_{n-1}|| for large n
"""
norms = [np.linalg.norm(K) for K in self.Krylov_sequence.values()]
# Growth rate from late terms
if len(norms) > 5:
gamma = np.mean([norms[i]/norms[i-1] for i in range(-3, 0)])
else:
gamma = norms[-1] / norms[-2]
return gamma
def bound_lyapunov(self):
"""
Compute upper bound on Lyapunov exponent
λ_max ≤ Γ
"""
return self.compute_growth_rate()
class DeterministicPredictor:
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()
def get_nonlinearity(self, name):
"""Return analytic nonlinearity"""
if name == 'erf':
return lambda x: np.erf(x) # Fast Fourier decay ✓
elif name == 'tanh':
return lambda x: np.tanh(x) # Fast Fourier decay ✓
else:
raise ValueError(f"Nonlinearity {name} may not have fast decay")
def validate_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)
pass
def predict_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 in range(future_steps):
# Deterministic evolution given history
x_next = self.deterministic_update(history[-1], C, krylov)
predicted.append(x_next)
return predicted
def compute_correlation(self, history):
"""Compute correlation C(t,t') from trajectory"""
# Simplified: autocorrelation
T = len(history)
C = np.zeros(T)
for dt in range(T):
C[dt] = np.mean([history[t] * history[t-dt]
for t in range(dt, T)])
return C
┌─────────────────────────────────────────┐
│ 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 │
└─────────────────────────────────────────┘
Experimental validation:
Prediction resolution ∝ 1/Γ
Higher Γ → More temporal modes needed
Lower Γ → Better finite-resolution prediction
This work extends Krylov growth concepts from:
Key parallels:
Potential extensions:
When applying predictable chaos methodology:
| Method | Ensemble View | Deterministic View | Prediction Theory | Complexity Bound |
|---|---|---|---|---|
| Traditional DMFT | ✅ | ❌ | ❌ | ❌ |
| Lyapunov Analysis | ❌ | ❌ | ❌ | λ_max only |
| Krylov Method | ✅ | ✅ | ✅ | λ_max ≤ Γ |
Supplementary material available at arXiv entry.
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.