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Control Knobs: K, alpha, zeta, Psi

Four knobs parameterise the UPDE. The supervisor adjusts them to steer coherence.

The Equation

dtheta_i/dt = omega_i
            + sum_j K_ij sin(theta_j - theta_i - alpha_ij)
            + zeta sin(Psi - theta_i)

K -- Coupling Strength

K_ij is the (i,j) entry of the Knm matrix. Controls how strongly oscillator j pulls oscillator i.

  • Increasing K raises synchrony (R increases).
  • Decreasing K allows oscillators to decouple.
  • Matrix is symmetric, non-negative, zero diagonal.
  • Built from base_strength * exp(-decay_alpha * |i-j|) by default.

Supervisor use: Boost global K when R_good is low (DEGRADED regime). Reduce K on specific layers to suppress R_bad.

alpha -- Phase Lag

alpha_ij shifts the coupling function. At alpha=0, coupling is purely attractive. At alpha=pi/2, coupling becomes repulsive (Sakaguchi-Kuramoto model).

  • Models transport delays, propagation latencies.
  • Non-zero alpha creates phase-locked states with constant offset.
  • The imprint model can shift alpha based on exposure history.

Supervisor use: Increase alpha on bad layers to desynchronise pathological lock-in (retry storms, seizure-like cascades).

zeta -- Driver Strength

Scalar strength of the external periodic drive zeta sin(Psi - theta_i). Pulls all oscillators toward the reference phase.

  • zeta=0: free-running (no external drive).
  • zeta>0: entrainment toward Psi. Higher zeta = stronger pull.
  • Used for pacing, reference tracking, or forced entrainment.

Supervisor use: Increase zeta during CRITICAL regime to damp oscillators toward a known stable phase.

Psi -- Reference Phase

The target phase for the external drive. Only effective when zeta > 0.

  • In biological systems, Psi can represent a circadian reference.
  • In engineering, Psi can represent a clock signal or coordination target.

Supervisor use: Adjusted during RECOVERY regime to guide re-entrainment.

Scope

Each ControlAction specifies a scope:

  • "global" -- applies to all oscillators.
  • "layer_{n}" -- applies to oscillators in hierarchy layer n.

The ActuationMapper resolves scope to specific actuator commands. The ActionProjector clips values to bounds and enforces rate limits.

Constraints

Knob Typical range Rate limit
K [0, 5.0] 0.1 per step
alpha [-pi, pi] 0.05 per step
zeta [0, 2.0] 0.2 per step
Psi [0, 2*pi) no limit

Ranges and rate limits are domain-specific, configured in the binding spec actuators section.