How Long Is the Coast of a Neuron? Resolution-Dependent Interaction Order and Computational Capacity
Nima Dehghani
q-bio.NC, cond-mat.dis-nn, cond-mat.stat-mech, cs.NE, physics.bio-ph
2026-10-14
Projecting a six-contact, third-order neuron onto the soma lifts the direct fraction from 55% to 100% and cuts available information by 85%.
A point neuron is a coarse ruler. The cell is reduced to a handful of state variables and one input-output rule, and theories of recurrent dynamics and population computation sit on that reduction. The statistical literature looks compatible with it. In small, sparsely firing populations, a pairwise maximum-entropy model, matching only first- and second-order moments, already accounts for a good deal of collective activity. A recent maximum-entropy analysis takes the same claim down to single cells: equal-time direct dependencies on relatively few recorded neurons explain more than 90% of output entropy in mouse recordings, and 62% in C. elegans.
Those percentages are the relation left after projection. The weights are functional dependencies, not a synaptic inventory. Dendrites hold spatially separate nonlinear subunits, so a population model and a dendritic model are two rulers on one object, joined by a many-to-one map. The paper does not treat arbors as fractals.
The direct family allows an intercept plus a separate function of each coordinate. Binary inputs reduce it to logistic regression. If a coordinate has several states, each state gets its own value, so the labels 0, 1, and 2 are not forced into a fake order. Projecting the true conditional onto this family yields three entropies in descending order: output entropy, the direct model's cross-entropy, and the true conditional entropy. Available information, the mutual information between the visible inputs and the output, splits into a direct piece and a conditional KL residual. The residual is non-additivity relative to that family, not a named synergy score.
R divides the direct piece by output entropy. The mouse and worm analyses report R. F divides it by the information still available on that ruler. Whenever any information remains, R cannot exceed F. Coarse-graining cannot increase available information. F can still rise. When the residual falls faster than the available information, F moves toward 100% while total predictive information shrinks. Reporting that rise without the denominator is the denominator trap. If the direct family is closed under the projection, the absolute residual cannot grow. Without closure, it has no universal direction.
Interaction order moves with the coordinates. A distribution that contains only a three-body coupling can have perfectly uniform pairwise margins. Compress two spins into a new variable and the third-order dependence reappears as ordinary mutual information. How much survives depends on the compression. Order belongs to the ruler.
One firing rule is then read at three resolutions. Six binary inputs are split across two branches. Firing probability has a baseline of 0.06, rises to 0.46 when one branch triplet is fully on, and reaches 0.86 when both are. The law stays fixed. What changes is which input states an observer can still separate. At branch resolution each branch is a saturating sum capped at 2, so two active contacts and three become one reading and the coincidence is no longer visible. A symmetric somatic average also erases which branch fired. A 0.6/0.4 mix changes only that symmetry. The three steps schematize NMDA-like coincidence detection, driving-force saturation, and electrotonic attenuation. Three is not a physiological threshold, and none of this is a fitted membrane equation.
The reverse experiment asks what a finer ruler makes available to a restricted readout. The target has three pairwise products and one third-order term. Inputs are drawn symmetrically from -1 to 1, so a linear readout cannot pick the products up from a shared positive mean. An aligned model puts 18 contacts into three nonlinear subunits matched to those pairs, and does not insert the third-order term by hand. A shuffled model keeps the same contact count and subunit count and randomizes the groups. A nested sigma-pi net, products inside each subunit followed by a local nonlinearity, reveals nine resolutions from 1 subunit to 24, with the original six inputs always present. Every new feature is a function of those inputs, so mutual information about the target does not increase. The linear readout's hypothesis class does.
At coupling K = 1.8, exact enumeration gives 0.823 bits of connected third-order information. The pairwise maximum-entropy description has entropy of exactly 3 bits. An AND projection leaves 0.261 bits of pairwise mutual information. An equality projection leaves 0.823 bits, the whole third-order quantity, now as a pairwise dependence. At K = 0 the projected variables are independent.
On the cascade, the categorical direct fraction rises while absolute information falls.
| Ruler | Direct fraction F | Available information | Higher-order residual | Share of output entropy R |
| 6 contacts | 55.0% | 0.156742 bits | 0.0706 bits | 13.6% |
| 2 saturated branches | 96.1% | 0.028607 bits | 0.0011 bits | 4.3% |
| Somatic average | 100% | 0.023526 bits | 0 | 3.7% |
A stricter linear-moment family scores 55.0%, 81.3%, and 98.9%. The paper states that the denominator contracts by 85%. The somatic 100% is definitional: one coordinate is left, and any function of a single variable lies in the direct family. Switching the mix to 0.6/0.4 drops linear-family F from 98.9% to 78.4%. The generative law is unchanged.
Held-out error separates organization from contact count. Over 100 splits with 6,000 samples each, mean MSE is 0.0437 for the point readout, 0.0393 for shuffled contacts, and 0.0309 for three aligned subunits. Among fixed sigmoid random features, the first tested width whose mean error matches the aligned model is 40. That is not a lower bound on network width. In the nested hierarchy, 60 architectures and 5,000 samples per split move mean MSE from 0.0421 at one exposed subunit to 0.0152 at 24. The direct-input baseline, with no nonlinear subunit, is about 0.0436. Normalized inverse error scales about as the subunit count to the 0.31, interval 0.30 to 0.32, log-space R^2 0.8613. Effective rank, the participation ratio, rises from about 6.86 to 14.03, exponent about 0.24, interval 0.23 to 0.24, R^2 0.9907.
A high R is an effective description at one scale. In mouse hippocampus and visual cortex, R above 0.9 means any unobserved variable can add at most a tenth of the output entropy to the same binarized output under the same distribution of population states. The bound caps predictive gain. It does not distinguish linear summation from a nonlinear combination. A monosynaptic edge fails the same test: if its contribution changes with local dendritic state, the direct family keeps a positive residual.
Sampling geometry does the companion damage. In the hippocampal analysis reported in the paper, holding each conditional response fixed and changing only the input-state weights drops the median share attributed to first-order terms from 0.75 under empirical weights to 0.68 under product-marginal weights and 0.46 under balanced weights. Higher-order structure leaks into low-order parameters.
For architecture search the comparison is narrow. On this target, three aligned subunits reach an error that takes 40 unstructured sigmoid features, and shuffling the groups gives much of that gain back. Mutual information about the target stays flat while the hypothesis class grows. Copying cell counts or cell-level adjacency does not decide which contacts share a subunit. A point neuron is the weak-input regime of the membrane equation. Voltage-dependent NMDA conductance takes the weighted sum apart.
The cascade and the three-spin system are exhaustive constructions. The paper says the ratios are not a fit to the mouse or worm recordings. Both scaling exponents belong to this sigma-pi ensemble. The 0.31 relation has log-space R^2 of 0.8613, a trend inside one model family, not a law of dendritic capacity.
The somatic 100% is the least informative number, because a single coordinate cannot leave a residual among coordinates. The branch ruler is the sharper result: F is already 96.1% and the residual is 0.0011 bits, while the third-order products are still in the generative law. The 0.6/0.4 mix sends linear F back to 78.4%. Directness follows the projection. Remove the closure assumption and the residual has no monotone theorem.
There is no new in vivo measurement. XOR and ten-bit parity come from cited pyramidal-cell simulations, which the paper marks as not evidence of in vivo computation. A sufficiency test would hold the output fixed, measure how much predictive information finer variables add, and then perturb branch inhibition, spatial clustering, or the temporal regime. That experiment is specified and not run. The approximation trials use 5,000 to 6,000 samples in six dimensions, so they also miss the scarce-data regime in which the paper notes that a coarser model can generalize better.