Source
Sánchez-Cañizares (2026), LIT-1992, §4 (PDF pp. 7–12) and §5 (pp. 13–18); read in NOTE-1506. The potential is Haken, Kelso and Bunz 1985 (LIT-588; THEORY-1227).
What was actually shown
The landscape (p. 8). φ̇ = −a sin φ − 2b sin 2φ. With b = 1 there are minima at 0 and π for |a| < 4 and one minimum otherwise. Checked: V″ = a cos φ + 4b cos 2φ is positive at π iff a < 4b, which is THEORY-1227’s anti-phase boundary.
The schedule (eq. 3, p. 8) and the statistics (eqs. 4–7, p. 11). Means and standard deviations of φ and cos φ over φ(0) uniform on (−π, π], with noise √q ξ and q = 1. Fig. 3 shows the standard deviations bump in 1 ≤ t < 2 and decrease afterwards.
Re-simulated here (Euler–Maruyama, dt = 10⁻³, 20,000 paths). The SD of cos φ:
| t | 0.99 | 1.5 | 1.99 | 5 | 10 | 15 |
|---|---|---|---|---|---|---|
| q = 0 | 0.71 | 0.93 | 0.98 | 0.82 | 0.12 | 0.01 |
| q = 1 | 0.71 | 0.87 | 0.89 | 0.48 | 0.40 | 0.41 |
At q = 1 the late values are the stationary ones of p(φ) ∝ exp(2a cos φ/q) with a = 1: mean of cos φ 0.698, SD 0.405. So the rise and fall is robust, and the fall “to 0” (p. 12) needs noise much weaker than q = 1 Gaussian white noise. The paper’s noise is underspecified (bounded ξ ∈ [−1, 1] and delta-correlated, pp. 8–9), which may explain the difference.
The comparison (§5). Visual, against reproduced figures from Sergent et al. 2021 (SD of the ERP peaking around 400–500 ms at the critical SNR), Wolff et al. 2019 (trial-to-trial variability peaking near 100 ms and declining after 200 ms) and, by re-mapping the initial regime, Hua et al. 2025. Two mismatches are stated: the ERP returns to baseline faster than the model, and at low SNR the experimental SD is smaller (p. 15, fn. 17).
What this does not say
- That HKB fits the data: no parameter was fitted, and the author says the agreement is “only qualitative” (p. 18).
- That the trial-to-trial variability in the experiments arises from spread of initial conditions, as the model’s statistics do.
- Anything about the mind. That is THEORY-1302.