Source
Tononi, Sporns & Edelman (1994), PNAS 91:5033–5037, LIT-998, read in full in NOTE-767: Theory (Eqs. 1–6), Results (Figs. 1–3) and Discussion.
What was actually shown
The measure. For a system X of n components with stationary activity, integration is I(X) = Σᵢ H(xᵢ) − H(X). Its average ⟨I(X_j^k)⟩ over all subsets of size k rises monotonically from 0 at k = 1 to I(X) at k = n (from Eq. 3). Neural complexity is
C_N(X) = Σ_{k=1}^{n} [(k/n) I(X) − ⟨I(X_j^k)⟩] = Σ_{k=1}^{n/2} ⟨MI(X_j^k; X − X_j
the shortfall below the linear rise, equivalently the average mutual information between a subset and the rest, summed over sizes (Eqs. 4, 6). C_N ≥ 0. For Gaussian activity, I(X) = −ln|CORR(X)|/2.
The two extremes (Fig. 1). Toeplitz covariance matrices, n = 64, with correlation length varied from near-independence to all coefficients near one. C_N is very low at both ends and maximal when the coefficients span the whole range from 0 to 1, though I keeps rising. This could have come out otherwise: C_N could have tracked I.
Connectivity (Fig. 2). In a simulated primary visual area (512 neuronal groups, intrinsic Gaussian noise, no input), the patchy pattern modelled on cortex gave C_N = 2244.7. The same number and strength of intergroup connections spread uniformly gave a higher I(X), a globally synchronised state, and C_N = 1240.2. Halving the density of the patchy pattern gave C_N = 542.35. In small linear systems (n = 8, fixed total weight), concentrating connections on fewer partners raised C_N/I (Fig. 3A, 1,000 networks per point). At constant I, more reciprocal pairs tended to go with higher C_N (Fig. 3B, 10,000 networks).
What this does not say
- It does not say the brain has high C_N. Nothing was recorded. The authors’ claim that neuroimaging covariance shows high C_N rests on unpublished data, and the prediction about waking and slow-wave sleep is a prediction.
- It does not prove the two-extremes property. It is shown for one family of matrices. At independence C_N is zero by definition; “low” at full dependence is shown, not derived.
- It does not find units. C_N is one number for a system whose components are fixed in advance. It names no boundary, block or scale, and it has no threshold or penalty for estimating structure from finite data. So it does not compete with a gap-and-Occam block test (CLAIM-154, CLAIM-157) as a way to pick out a unit. It only says that grouped, patchy structure raises the score.
- It does not say high complexity is consciousness. The Discussion links C_N to “conscious perception and behavior” only as a hypothesis for later work, and conditionally on extending the measure to nonstationary activity.
- The connectivity results are single realisations (Fig. 2) or tendencies (Fig. 3B), and two effects (grouping, reciprocity at lower w) are “data not shown”.
Connections
- QUESTION-042 asks whether blocks that clear an Occam-priced test
fall at anatomical boundaries in a recorded brain. This account does not
answer it, and no
answers:is declared. It supplies the comparison the question was given LIT-998 for. The paper treats a system as one only if “no bipartition yields two statistically independent subsets”, which is a oneness test with a zero threshold. CLAIM-157’s gap-beats-Occam criterion is the priced version of the same idea. - THEORY-097 puts long-range blocks at about 70 ms. A stationary measure like C_N cannot see a block that short.
- THEORY-063 cites Tononi and Edelman’s later dynamic core (“maximal causal density”, “distinct functional borders”). That is the descendant that adds a border. Nothing there rests on this paper.