How Long Is the Coast of a Neuron? Resolution-Dependent Interaction Order and Computational Capacity
Authors: Nima Dehghani
Organizations: McGovern Institute for Brain Research, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA · The NSF AI Institute for Artificial Intelligence and Fundamental Interactions (IAIFI), Cambridge, Massachusetts, USA
Mandlebrot's coastline paradox reframes measurement as a joint property of an object and its ruler. Here we ask how observational resolution shapes the apparent structure of neural computation. We introduce a scale-indexed decomposition that separates the predictive information captured by direct dependencies from the residual outside that model family. Exact constructions show that projection can convert higher-order dependence into pairwise structure. Projection can increase the direct fraction of surviving information even as total predictive information decreases. A schematic branch-to-soma cascade grounds these effects in local coincidence detection, saturating integration and somatic mixing. Resolving organized nonlinear subunits improves approximation and raises effective representational rank. In a nested hierarchy, these gains occur without increasing information about the target. This distinction separates information availability from access by a restricted readout. A point-neuron or direct population model therefore establishes an effective description at a specified scale. It does not uniquely identify the finer computation that generated the observed relation. Which resolution is adequate depends on the observable and perturbation of interest.
Figures & tables
Figure 1: A higher-order dependency becomes pairwise after projection. (a) Exact enumeration at K=1.8 gives 0.823 bits of connected third-order information at the fine scale and 0.261 bits of pairwise mutual information after the disjoint coincidence projection x=b1 , z=b2∧b3 . The projected variables are independent at K=0 . (b) The information that survives as pairwise depends on the map: an XNOR/parity projection retains all of I(3) , whereas the coincidence map retains only part.
Figure 2: Direct dependencies become more complete along an exact cascade. (a) Blue bars show F for the coordinate-invariant categorical-additive family; pink shows the residual share. Purple diamonds show the more restrictive linear-moment family, which yields 55.0%, 81.3% and 98.9%. (b) Coarse-graining reduces the available information Iavail , the captured direct information Idir and, more sharply, the conditional-KL residual Ihigher . Values are exact sums over all 64 input states.
Figure 3: Organized multi-contact features improve held-out approximation. (a) Mean held-out MSE for a linear point neuron, a three-subunit model with shuffled contacts and a three-subunit model with contacts aligned to the target’s pair structure. Error bars are 95% normal intervals across 100 independent train–test trials (6,000 samples in each split). (b) Mean held-out MSE of a generic fixed sigmoid random-feature network. The first tested width whose mean reaches the aligned model is 40. Shading shows 95% normal intervals across 40 trials.
Figure 4: Approximation and effective dimension grow in a nested hierarchy. (a) Mean held-out MSE as progressively more nonlinear sigma–pi subunits are exposed from the same architecture; the dashed line is the direct-input point baseline. Error bars and baseline shading are 95% normal intervals across 60 independent architectures and train–test sets (5,000 samples per split). (b) Normalized inverse error and normalized effective rank with log-space power-law fits. Exponent intervals are 95% percentile-bootstrap intervals over trials (2,000 resamples).
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Figure S1: The final direct fraction depends on mixing and measurement resolution. (a) Linear-moment F for the exact cascade after weighted somatic mixing and quantization. (b) Available information retained by the same projection. At Δ=0.1 , the white cross marks symmetric 0.5/0.5 mixing and the cyan diamond marks asymmetric 0.6/0.4 mixing. Heat-map values are exact; discontinuous bands arise when quantization merges different branch-state pairs.