Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding

TL;DR

This work develops Semi-Simplicial Neural Networks (SSNs) that operate on directed semi-simplicial sets to capture higher-order directional interactions, addressing a key limitation of traditional graph and topological deep learning models. By introducing Routing-SSNs (R-SSNs), the authors achieve scalable learning over face-map–induced relations, with provable expressivity improvements over standard GNNs and existing TDL approaches. They build a principled brain-dynamics framework using Dynamical Activity Complexes (DACs), showing SSNs on DACs can recover a broad suite of neurotopological invariants that characterize brain activity, surpassing prior methods. Empirically, SSNs deliver state-of-the-art brain-stimulus classification on a biologically realistic neocortical microcircuit and demonstrate strong performance on auxiliary node/edge tasks, underscoring the value of principled topological models for structured brain data. The work also provides practical guidance on model routing, theoretical guarantees, and avenues for future extensions and open problems in neurotopology-informed learning.

Abstract

Graph Neural Networks (GNNs) excel at learning from pairwise interactions but often overlook multi-way and hierarchical relationships. Topological Deep Learning (TDL) addresses this limitation by leveraging combinatorial topological spaces. However, existing TDL models are restricted to undirected settings and fail to capture the higher-order directed patterns prevalent in many complex systems, e.g., brain networks, where such interactions are both abundant and functionally significant. To fill this gap, we introduce Semi-Simplicial Neural Networks (SSNs), a principled class of TDL models that operate on semi-simplicial sets -- combinatorial structures that encode directed higher-order motifs and their directional relationships. To enhance scalability, we propose Routing-SSNs, which dynamically select the most informative relations in a learnable manner. We prove that SSNs are strictly more expressive than standard graph and TDL models. We then introduce a new principled framework for brain dynamics representation learning, grounded in the ability of SSNs to provably recover topological descriptors shown to successfully characterize brain activity. Empirically, SSNs achieve state-of-the-art performance on brain dynamics classification tasks, outperforming the second-best model by up to 27%, and message passing GNNs by up to 50% in accuracy. Our results highlight the potential of principled topological models for learning from structured brain data, establishing a unique real-world case study for TDL. We also test SSNs on standard node classification and edge regression tasks, showing competitive performance. We will make the code and data publicly available.

Figures (14)

• Figure 1: Overview of the Semi-Simplicial Neural Networks framework for Brain Dynamics Classification. Given a structured brain sample and its corresponding response to external stimulation, we jointly model them as an attributed semi-simplicial set $\mathcal{K}$ that captures higher-order co-activation patterns $X^l$. A set of relations $\mathcal{R}$ induced by the complex’s topology is then selected, and the structure, data, and relations are processed by Semi-Simplicial Neural Networks (SSNs)—a novel class of architectures capable of leveraging higher-order directional information. Our experiments demonstrate that this approach is essential for accurately predicting the originating stimulus.
• Figure 1: Topological invariant computation across architectures.
• Figure 2: (Top) Digraph; (Bottom) Directed flag complex.
• Figure 2: Counts of directed simplices in the Microcircuit complex $\mathcal{K}_\mathcal{G}$.
• Figure 3: (Top) Dynamic binary graph; (Bottom) associated DAC. Binary vectors indicate simplex activation over 4 time steps.

Theorems & Definitions (47)

• Proposition 1
• Theorem 1
• Theorem 2
• Theorem 3: Informal
• Theorem 4
• Proposition 2
• proof
• Proposition 3
• proof
• Lemma 1