Controls
0.20
0.50
0.05
1.00
1.20
1.00
2.00
Metric:
Balanced accuracy = ½·(hit rate + specificity).
Hit rate H:
H = 1 − Φ((θ − g·s)/(g·σ)), where σ = √(σ²int + (α·σext)²)
False alarm F:
F = 1 − Φ(θ/(g·σ)). Specificity = 1 − F.
Why inverted‑U?
Small noise helps subthreshold signals cross the effective boundary; too much noise raises false alarms.
More notes
Gain g stands in for dopaminergic modulation (steeper input–output). Lower g ⇒ you need more external noise to reach the sweet spot (right‑shift).
α is how strongly external broadband noise (e.g., white noise) couples into effective internal noise. Continuous, non‑salient noise acts like “dither.”
This is a deliberately minimal, static model (no dynamics, no attention). It’s meant to illustrate the Moderate Brain Arousal intuition, not prove it.
Group A (gA) — peak at –
Group B (gB) — peak at –