Persistent homology broadens the controllable subspace in human structural connectomes

Local Interaction Rules Drive Global Organization of the Human Connectome.

INTRODUCTION The human connectome exhibits nontrivial large-scale organization despite emerging from decentralized local biological interactions. Most existing generative models reproduce connectomic features through global optimization principles, predefined wiring targets, or developmental templates, leaving unresolved which properties arise from locality alone and which require additional nonlocal mechanisms. METHODS We implemented a simulation framework showing that global coherence can emerge from local compatibility constraints. Networks were generated exclusively through bounded spatial interactions, probabilistic local edge formation, and suppression of incompatible configurations, without global objectives, target topologies, or long-range coordination. Simulated ensembles were analyzed using graph-theoretical metrics, scaling relationships, and rule-based structural classification relative to published reference values of human connectome descriptors. RESULTS Simulations consistently generated mesoscopic organization characterized by high clustering, modular structure, motif enrichment, and strong short-range connectivity bias. Degree distributions were broad and right-skewed, while edge-length distributions showed pronounced spatial localization. In contrast, several higher-order integrative properties were not reproduced, including empirical connectivity scale, rich-club organization, and long-range hub-to-hub connectivity. Although global metrics displayed substantial quantitative divergence from reported empirical values, several structural regimes and scaling relationships were preserved across parameter ranges. DISCUSSION Our results distinguish connectome properties structurally compatible with local compatibility constraints from those underdetermined under locality alone. We provide a diagnostic framework designed to isolate the explanatory contribution of local interaction rules to connectome organization through simulations that identify which structural properties emerge directly from locality and which require additional mechanisms beyond local constraints.

ENCORE: Fast geometric framework for aligning brain structural connectivity on cortical manifolds.

Brain networks are typically represented by adjacency matrices, where each node corresponds to a brain region. In traditional brain network analysis, nodes are assumed to be matched across individuals, but the methods used for node matching often overlook the underlying connectivity information. This oversight can result in inaccurate node alignment, leading to inflated edge variability. To overcome this challenge, we propose a novel framework for registering high-resolution continuous connectivity (ConCon), defined as a continuous function on a product manifold space - specifically, the cortical surface - capturing structural connectivity between all pairs of cortical points. Leveraging ConCon, we formulate an optimal diffeomorphism problem to align both connectivity profiles and cortical surfaces simultaneously. We introduce an efficient algorithm to solve this problem and validate our approach using data from the Human Connectome Project (HCP). Results show that ENCORE consistently improves inter-subject correspondence of fine-grained connectivity features and yields higher accuracy in structural pathway localization compared with existing surface-based registration methods.

Resilience Beyond Pairwise Networks

We derive a one-dimensional reduction for nonlinear dynamics on simplicial complexes containing both pairwise and triangular (higher-order) interactions. The effective state is defined using a mixed weight determined by the pairwise and triangular degrees of each node. The resulting reduced equation retains two structural coefficients, associated separately with the pairwise and higher-order coupling channels. A fluctuation expansion identifies the closure assumptions underlying the reduction and shows how deviations of individual node states from the effective state contribute to the approximation error. We numerically validate the proposed framework on Gene-regulatory dynamics, the double-well system, and SIS spreading. The states of the reduced model are compared with full-network simulations through coupling-parameter sweeps, steady-state branch calculations, and progressive node-removal experiments on synthetic and real-world networks. The reduced model successfully reproduces the principal transitions and steady-state branches in all three dynamical systems considered. Agreement is strongest for relatively homogeneous networks and deteriorates when structural heterogeneity produces a broader distribution of node states. The closure diagnostics account for this loss of accuracy and indicate when a single effective state is no longer sufficient. The reduction therefore provides a tractable description of resilience in systems with coexisting pairwise and higher-order interactions.

Network Curvature as a Geometric Biomarker for Identifying Disease-Related Subnetworks

Nonlinear model reduction of complex networks via spectral submanifolds

Complex networked systems are prevalent in biology, engineering, and the social sciences, yet their high-dimensional, nonlinear dynamics pose major challenges for analysis and prediction. A mathematically rigorous route to simplification is to represent system behavior on a low-dimensional, smooth invariant manifold known as a spectral submanifold (SSM). Here we present a comprehensive SSM reduction framework and its globalized extension (gSSM) for dimensionality reduction in large-scale nonlinear networks. Our approach yields accurate global and node-level predictions across synthetic and real networks, including highly heterogeneous topologies and systems with higher-order interactions. Crucially, SSM is a robust tipping-point predictor: even at low truncation order (e.g., $O(2)$) it reliably identifies the onset of sustained activity, while higher orders and gSSM capture post-onset amplitudes and saturation. Consistently, the reduction collapses the full network dynamics to a one-dimensional system, offering clarity and efficiency. Across all the realizations, SSM/gSSM consistently outperform classical spectral and mean-field methods in modeling critical transitions at both microscopic and macroscopic scales, establishing SSM-based reduction as a robust, interpretable tool for nonlinear networked systems with broad applicability to epidemiology, ecology, and engineered networks.

Emergent topological structure in spontaneous brain-organoid activity

Neural activity is widely held to organize on low-dimensional structure embedded in a high-dimensional state space. Persistent homology reads such structure directly from the pattern of pairwise correlations, without assuming in advance which variables are relevant. We apply persistent homology to microelectrode-array (MEA) recordings of spontaneous activity from human (Lancaster) and mouse (Paşca) cortical organoids, spanning 26–234 simultaneously sorted units, and ask whether topological data analysis resolves structure at the node counts that neural recordings actually deliver. Building weighted networks in correlation space and characterizing them by Vietoris–Rips filtration, we find that the first homology (H1, loops) rises significantly above a rate- and population-preserving null in 14 of 18 datasets. This loop structure occupies a non-redundant core: it is robust to random removal of units yet disrupted by targeted removal of the units that carry it. Topological richness grows with network size, and second homology (H2) emerges significantly above the null only in the larger networks. These results show that persistent homology resolves structured topology in neural recordings at the scale experiments actually deliver.