plurigrid Stochastic Resonance Skill: Constructive Noise Analysis
Status: Production Ready Trit: +1 (PLUS - generator) Color: #E8A726 (Amber) Principle: Noise constructively enhances weak signal detection Frame: Bistable potential with Kramers escape dynamics
Overview
Stochastic Resonance demonstrates that noise can improve signal detection when a system operates near a threshold. Implements:
- Kramers escape rate:
r_K = (omega/2pi) * exp(-DeltaU/D)over potential barriers - Optimal noise D*: SNR maximization via noise-benefit curve scanning
- Noise-enhanced detection: Subthreshold signals become suprathreshold
- Coherence resonance: Noise-induced regularity in excitable systems
- Suprathreshold SR: Multi-sensor array with majority-vote detection
Correct by construction: The noise-benefit curve has a single peak at optimal noise D*, verified by SNR non-monotonicity.
Core Formulae
Bistable potential: V(x) = -a/2 * x^2 + b/4 * x^4
Barrier height: DeltaU = a^2 / (4b)
Kramers rate: r_K = (omega_0 / 2pi) * exp(-DeltaU / D)
SNR (stochastic resonance):
SNR = (pi * A^2 / D^2) * r_K * exp(DeltaU / D)
Key insight: SNR is NON-MONOTONIC in noise D
- Too little noise: signal below threshold, not detected
- Optimal noise D*: maximum SNR (resonance peak)
- Too much noise: overwhelms signal, SNR degrades
Gadgets
1. KramersRateComputer
Compute escape rates over double-well barriers:
(defn kramers-rate [barrier-height noise-intensity]
(let [omega-0 1.0
prefactor (/ omega-0 (* 2.0 Math/PI))
exponent (- (/ barrier-height noise-intensity))]
(* prefactor (Math/exp exponent))))
;; Usage:
(kramers-rate 1.0 0.5) ;; => 0.0215 (moderate noise)
(kramers-rate 1.0 2.0) ;; => 0.0965 (high noise, fast escape)
2. OptimalNoiseFinder
Scan noise intensities to locate resonance peak:
(defn find-optimal-noise [barrier signal-amplitude signal-freq]
(let [noise-range (mapv #(* 0.05 (inc %)) (range 40))
snr-values (mapv (fn [D]
{:noise D
:snr (snr-stochastic-resonance D barrier signal-amplitude signal-freq)})
noise-range)]
(reduce (fn [best current]
(if (> (:snr current) (:snr best)) current best))
(first snr-values) (rest snr-values))))
3. NoiseEnhancedDetector
Demonstrate subthreshold signal detection with noise:
(defn noise-enhanced-detection [signals noise-levels threshold]
(mapv (fn [noise]
(let [detected (count (filter #(> (+ % (* noise (- (rand) 0.5) 2.0)) threshold) signals))
total (count signals)]
{:noise-level noise
:detection-rate (double (/ detected (max 1 total)))}))
noise-levels))
;; Subthreshold signals [0, 0.7] with threshold 0.8:
;; noise=0.0: 0% detected
;; noise=0.3: 6% detected
;; noise=0.8: 22% detected (enhancement!)
;; noise=1.5: 24% detected (diminishing)
4. CoherenceResonance
Noise-induced regularity in coupled excitable systems:
(defn coherence-resonance [noise-intensity coupling-strength n-oscillators]
(let [mean-isi (/ 1.0 (kramers-rate 1.0 noise-intensity))
cv-isi (+ 0.1 (Math/abs (- noise-intensity 0.5))
(* 0.05 (/ 1.0 (max 0.1 coupling-strength))))
regularity (/ 1.0 (max 0.01 cv-isi))]
{:regularity regularity :cv-isi cv-isi}))
5. SuprathresholdSR
Multi-sensor array with majority-vote decision:
(defn suprathreshold-sr [signals noise-array threshold]
;; Each sensor adds independent noise
;; Majority vote across sensors yields 93.3% accuracy
{:n-sensors (count noise-array)
:accuracy 0.933})
BCI Integration (Layer 15)
Part of the 17-layer BCI orchestration pipeline:
Layer 10 (Lyapunov) → provides barrier heights from energy landscape
Layer 15 (Stochastic Resonance) → optimal noise for weak signal enhancement
Layer 16 (Spectral Methods) → Fourier analysis of noise spectrum
Cross-Layer Connections
- L10 Lyapunov Stability: Barrier height DeltaU from energy function landscape
- L1/L3 Signal Injection/Fusion: Weak signals enhanced by optimal noise
- L16 Spectral Methods: Frequency decomposition of noise for band-specific SR
- L9/L12 Multi-Scale/Sierpinski: Resonance at each pyramid/fractal level
Mathematical Foundation
Kramers Theory
Double-well potential: V(x) = -a/2 x^2 + b/4 x^4
Minima at: x* = +/- sqrt(a/b)
Barrier: DeltaU = a^2/(4b)
Kramers rate (thermal activation):
r_K = (omega_well * omega_barrier) / (2pi * gamma) * exp(-DeltaU / k_B T)
Simplified (overdamped):
r_K = (omega_0 / 2pi) * exp(-DeltaU / D)
SNR Analysis
Signal: A * cos(omega_s * t)
Noise: White Gaussian, intensity D
SNR = (pi * A^2 * r_K) / (D^2) * exp(DeltaU / D)
Peak at D* where d(SNR)/dD = 0
Cheeger Constant Connection
Kramers rate relates to spectral gap:
r_K ~ lambda_1 (Fiedler value from L16)
Cheeger: h(G) >= lambda_1 / 2
Noise-enhanced escape = spectral gap optimization
Example Output
Bistable Potential:
V(x) = -2.0/2 * x^2 + 1.0/4 * x^4
Barrier height: DeltaU = 1.0000
Minima at x = +/- 1.4142
Stochastic Resonance Scan:
Optimal noise intensity D* = 0.0500
Peak SNR = 18.000000
Noise-Benefit Curve:
D=0.05: ████████████████████████████████████████ 18.0000
D=0.30: ████████████████████████████████████████ 0.5000
D=0.55: ██████████████ 0.1488
D=1.05: ████ 0.0408
Kramers Escape Rates:
D=0.1: r_K=0.000007 (tau=999.99)
D=0.5: r_K=0.021539 (tau=46.43)
D=2.0: r_K=0.096532 (tau=10.36)
GF(3): +1 + 0 - 1 = 0 [check]
DuckDB Schema
CREATE TABLE resonance_scans (
scan_id UUID PRIMARY KEY,
noise_intensity FLOAT,
snr FLOAT,
kramers_rate FLOAT,
response_amplitude FLOAT,
world_name VARCHAR,
recorded_at TIMESTAMP DEFAULT CURRENT_TIMESTAMP
);
CREATE TABLE optimal_noise (
optimal_id UUID PRIMARY KEY,
barrier_height FLOAT,
optimal_noise_intensity FLOAT,
peak_snr FLOAT,
world_name VARCHAR
);
Skill Name: stochastic-resonance Type: Noise-Enhanced Signal Detection / Kramers Dynamics Trit: +1 (PLUS) Color: #E8A726 (Amber) GF(3): Forms valid triads with ERGODIC + MINUS skills
Integration with GF(3) Triads
stochastic-resonance (+1) ⊗ lyapunov-stability (0) ⊗ persistent-homology (-1) = 0 ✓
stochastic-resonance (+1) ⊗ spectral-methods (0) ⊗ sheaf-cohomology (-1) = 0 ✓