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t07 — Population information and the noise-correlation ceiling

notes.md

# t07 — Population information and the noise-correlation ceiling

**Design.** No external data. Exact linear algebra for N Gaussian-tuned neurons with
covariance sigma^2[(1-rho)I + rho J], plus seeded grid maximum-likelihood decoding
(400 trials per cell).

**Result.** For identical tuning, I(s) = g^2 N/(sigma^2 (1+(N-1)rho)), checked by matrix
inversion to N = 200 (error 7e-15). The ceiling in units of
g^2/sigma^2 is 1/rho: 100, 20, 5 and 2 for rho = 0.01, 0.05, 0.2 and 0.5, or 4.18,
3.03, 2.06 and 1.46 bits under a uniform prior. Saturation is fast:
N90 = 9(1-rho)/rho falls from 891 neurons at rho = 0.01 to 9 at rho = 0.5, and a
100-neuron population reaches 50 per cent of its ceiling at rho = 0.01, 99 per cent
at 0.5. The 100th neuron adds under 1 per cent once rho exceeds 1.0e-4, but
1.0101 per cent at rho = 0, so that bar mostly restates the 1/N return.

Simulation agrees: at N = 500, rho = 0.5 the mean squared error is 1.01 [0.88, 1.16]
times the Cramér–Rao bound and flat in N (exponent 0.03 [-0.02, 0.09]).

**Where it fails.** The ceiling needs redundant tuning. With tuning spread over the
circle, sum f' is about zero and information grows linearly at every rho < 1: 58
times the claimed ceiling at N = 500, rho = 0.5, exponent 0.98 [0.92, 1.04]. That is
the note's refutation condition. Heterogeneity does not rescue the claim; it confines
it to redundant tuning.

**Verdict.** Partially supported.

Included code and results

• tests/t07/code.py
• tests/t07/results.json
• tests/t07/series.csv