Notebook 02 · 2026-09-19 · interactive · unit 1 weights from phase5_eight_channel.py

The E/I Unit on a Pattern Pair

One excitatory/inhibitory unit acting on the two dominant patterns of an 8-channel EEG state. Two knobs, gain asymmetry and coupling, and what each does to the purity of the state.

The setup

From the 8-channel purification run (S001, eyes closed, seconds 10–14), the two largest weights are pattern 0, the whole-head in-phase mode, and pattern 1, the front-versus-occiput mode. In their own basis the pair is diagonal:

Λ1 = 0.660000.231

The unit's generator has two parameters. Gain asymmetry g is how much faster pattern 0 grows than pattern 1. Coupling κ is how strongly each pattern feeds the other.

K1 = g−κκ−g

Running the unit for time t moves the pair to eK₁t Λ₁ eK₁ᵀt, and the whole 8-dim state is renormalised to trace one. The other three units are switched off, so the other six weights sit still and only shrink or grow in proportion. Everything below is that map, computed exactly.

Turn the knobs

0.30
0.10
2.0
regime
—
purity, full state
—
start 0.493 · pure = 1
entropy
—
nats · start 0.998
pair weights now
—
were 0.660 / 0.231
The pair as an ellipse in the (pattern 0, pattern 1) plane. Axis lengths are the square roots of the two weights; the dashed outline is where it started. A rotation turns the ellipse without reshaping it — that is why coupling alone cannot change purity.
Purity of the full 8-channel state against time for the current g and κ. The marker is the current t. Grey curves are the two single-knob cases for comparison.

The whole landscape

Purity after a fixed time, over every combination of gain and coupling. The diagonal g = κ is the exceptional point of the generator: above it the gain wins and the state purifies, below it the coupling wins and the state oscillates. The surface has a ridge along that line.

3.0
Purity of the full state at time t, as a function of gain asymmetry (−1 to 1) and coupling (0 to 1). Drag to rotate.

The second surface fixes the coupling and lets gain and time vary. Below the coupling line the surface ripples and never climbs; above it, it climbs and stays up. The transition is not gradual — it is the exceptional point.

0.40
Purity against gain asymmetry (0 to 1) and time (0 to 8) at fixed coupling κ. The crease at g = κ separates the oscillating region from the purifying one.

What the pictures say

Gain alone changes purity by multiplication: pattern 0 by e2gt, pattern 1 by e−2gt. Positive gain purifies. Negative gain favours the weak pattern, so purity dips through its minimum where the two weights cross and then climbs again on the other side.

Coupling alone is a rotation, and a rotation cannot change the eigenvalues of anything. Purity is flat, exactly, for all time. An E/I unit with balanced gain is invisible to this observable.

With both, the sign of g² − κ² decides everything. Coupling-dominated: purity oscillates with period π / √(κ² − g²) and has no trend. Gain-dominated: purity climbs to 1 along a direction tilted from pattern 0 by an angle set by κ / g.

So the field can purify robustly, and it can only mix transiently — through a negative gain that hasn't yet crossed over, or an oscillation caught on the way down. That is the prediction to carry to the within-seizure, time-resolved entropy on CHB-MIT: purifiers should be monotonic, mixers should not.