Can turning up the input make the brain model use its on-off switching?

E48 — Rectifying at zero makes the network scale-symmetric, and it measures as nearly affine

The whole brain reduced to a 7x8 matrix (E47). Sweeping the operating point shows why, and the demonstration is exact rather than statistical. Driving the same compositions through the same brain at two input scales:

A hundred times the input gives exactly a hundred times the output, the same firing pattern, and the same affine fidelity to four decimals. relu(a v) = a relu(v) for positive a, so a rectifying rate network integrated from rest with no per-neuron threshold is positively homogeneous of degree one, and its response to a I is exactly a times its response to I. Turning the drive up therefore cannot engage the nonlinearity - which is why sweeping the input scale was wasted effort.

This does not say the network cannot compute nonlinear functions, and an earlier draft of this entry wrongly said it did. Positive homogeneity is a symmetry under scaling, not linearity: max(a - b, 0) is positively homogeneous and nonlinear, because changing inputs relative to each other crosses a rectification boundary even though scaling them together never does. The evidence that this network is close to affine is the measured affine fidelity below, not the scaling argument. The log-concentration encoding is itself nonlinear in composition.

A uniform tonic current is worse, not better: it lifts 99% of cells above zero so nothing rectifies at all and switching falls to 0.1%.

The threshold has to be per neuron, and it has to be solved rather than set. Giving each cell a threshold at the median of the membrane state it reaches breaks the homogeneity. Set in one pass it fails - imposing the threshold changes the rates, which changes the recurrent drive, which moves the very states the threshold was read from, and the population falls to 1.4% switching. Iterated to a fixed point, reached in 7 rounds with 92.2% of neurons alive:

This is the same fixed point the mushroom body's gain needed (2.2, 2.6) and the same class of fault as every earlier one in this project: the circuit was never wrong, the operating point was. The firing ceiling as parameterised clips almost everything and is dropped rather than tuned.

At 94.6% switching the affine map still reproduces the heading at cosine 0.823, so about 57% of the heading magnitude is not affine. Whether that residue is worth 163,972 neurons is not a question the geometry answers, and the pre-registered condition asks for the search, not the cosine.