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How Many Bits? The Information Theory of a Brain Link

Treat the BBI as a cascade of noisy channels and the ceiling becomes explicit. The honest verdict: a few bits per minute, orders of magnitude below ordinary speech.

The cascade and the data-processing inequality

The message has to survive two noisy stages in series: decode brain A, then encode into brain B. Information theory gives a hard ceiling. By the data-processing inequality, whatever brain B ends up sharing with brain A cannot exceed what the message m already captured, nor what the write stage can deliver.

I(x_A; x_B) \le \min\big(\, I(x_A; m),\; I(m; x_B) \,\big)

The data-processing inequality: end-to-end mutual information is capped by the weaker of the two stages — and the weaker stage is almost always the write.

This is the information-theoretic restatement of the read/write asymmetry. No amount of clever decoding on the sender fixes a channel whose ceiling is set by the phosphene you can barely write.

Cascaded errors add up

p = p_1 + p_2 - 2\,p_1 p_2

Two binary symmetric channels in series (read error p1, write error p2) compose to a single error p. Errors accumulate; if either half is bad, so is the whole.

With realistically coarse decode and write, both p_1 and p_2 sit well away from zero, and the composed error climbs toward the useless limit of $0.5$ quickly. The classic fix is a codebook with repetition: send each bit several times and vote, trading rate for reliability. That is exactly what the binary-word human demonstrations did — which is also why they were so slow.

Interactive: turn up the noise and watch the composed error rise and the effective bit-rate collapse. This is the cascade equation made tangible.

Bit-rate in the real demonstrations

The honest numbers are humbling, and we should give ranges rather than false precision. The binary-word human demonstrations ran on the order of a few bits per minute; BrainNet moved a handful of rotate/don't-rotate decisions over minutes of play. These are bits per minute, not bits per second.

B = \log_2 N + P\log_2 P + (1-P)\log_2\!\left(\frac{1-P}{N-1}\right)

The Wolpaw information-transfer rate in bits per selection, for N possible symbols at accuracy P. Multiply by selections per minute to compare systems on a common footing.

Now the sobering comparison. Ordinary speech conveys information at roughly tens of bits per second across languages. A BBI as demonstrated is therefore something like three to four orders of magnitude slower than talking. That gap — between the information rate of a brain-to-brain channel and the information bandwidth we take for granted in conversation — is the true distance between demonstration and the thought-sharing aspiration.

Buying reliability with more brains

If one channel is too noisy, do not build a better one — build several and vote. Independent decoders combined by majority vote drive the error down, the familiar wisdom-of-crowds effect. This is the honest engine behind multi-brain systems, and it points straight to consensus decoding in the next guide.

P_{\mathrm{err}}(N) = \sum_{k > N/2} \binom{N}{k}\, p^{k}(1-p)^{N-k} \;\xrightarrow[\,p < 1/2\,]{N \to \infty}\; 0

Majority-vote error over N independent decoders, each with error p below one-half, shrinks toward zero as N grows — reliability bought with redundancy.