Neural circuits get more than pairwise edges: directed simplicial complexes record what joint neuron suppression does

The object used to explain a network's internal representation moves from a graph of pairwise edges to a directed simplicial complex that can hold multi-neuron joint effects; vertices are internal neurons, and edges and higher-dimensional simplices record causal links between neuron activations for a given input set. Earlier neural circuit-discovery methods represent neuron interactions as pairwise edges in a graph and cannot express the joint effect on another neuron when a group of neurons is suppressed together. On two handcrafted multilayer perceptrons whose internal computations are known, the complexes for one-hot inputs recover the designed feature pathways: each one-hot complex of the monosemantic network contains ten edges and four 2-simplices, and the four complexes coincide up to relabelling of neurons.

The construction treats the network as a structural causal model and uses coherent counterfactual ablations that stay on the neural manifold to progressively suppress a set of predictor neurons, then tests whether the target neuron's activation change exceeds a target-scaled relevance margin. Setting those neurons to zero directly would push the activation vector off the neural manifold and distort the delicate coordination within a feature representation, so the method instead follows counterfactual trajectories that remain on the manifold. On the two small multilayer perceptrons for a synthetic two-dimensional four-quadrant classification task, the one-hot complexes contain ten edges plus four 2-simplices for the monosemantic network and four to twelve edges plus two to four 2-simplices for the polysemantic network, matching each network's structure.

The 1-skeleton of the complex agrees with established circuit-discovery methods on the edges both can express, namely those where the predictor feeds the target directly. Established methods return a subgraph of the network and can contain only directly connected edges; the complex also yields within-layer and backward edges and simplices of dimension two and above. Such representable edges make up 40% of the monosemantic complexes and 33% to 50% of the polysemantic complexes; in most comparisons the method keeps all of them, and the methods' scores rank these edges with a minimum AUC of 0.65.

The demonstrated scope is a synthetic two-dimensional four-quadrant classification task on two small ReLU multilayer perceptrons, with reference computations supplied by handcrafted networks; the comparison with ACDC, EAP and EAP-IG covers only 1-skeleton edges where the predictor feeds the target directly. The author's stated next step is to apply the construction to larger networks and natural data.

Whether the complexes identify polysemantic neurons remains open: in the polysemantic network the neuron belonging to the most maximal simplices is the one representing the fourth quadrant, not the six neurons that mix both coordinates. Whether higher-dimensional simplices show higher-order interactions is also open, because the relevance test checks only that the joint ablation has an effect without comparing it with the effects of individual predictors, and a simplex is tested only if its faces are admitted. The construction is somewhat sensitive to the choice of input distribution, the multiple-testing correction is applied separately per dimension so the error rate across dimensions, input sets and networks is not controlled, and the coherent counterfactual ablation trajectories are computed from the local Jacobian, so at boundaries between ReLU regions they can fail to continue.