Aug 2026· Chaos· Vol 36 8· 0 citations· 45 references
Medicine
TL;DR
A novel large-scale delayed neural network with dual-star architecture, incorporating both second-order and third-order interactions, and the accurate function fitting achieved by the model provides new insights for the design and control of neural networks.
Abstract
Understanding large-scale neural network dynamics represents a fundamental challenge in both biological and artificial neural systems. Traditional pairwise interaction models inadequately capture the complex connectivity patterns observed in real neural networks, where groups of three or more neurons interact simultaneously. This paper presents a novel large-scale delayed neural network with dual-star architecture, incorporating both second-order and third-order interactions. Two analytical methods are employed to derive the characteristic polynomial for the proposed network: the Coates flow graph formula and the Schur complement method. Both approaches circumvent the computational intractability of conventional determinant-based methods for high-dimensional systems. Using the holistic element concept, rigorous conditions for local stability and Hopf bifurcation are established. Comprehensive numerical simulations validate theoretical findings and reveal that higher-order interactions exhibit non-monotonic effects on bifurcation thresholds, self-feedback mechanisms provide robust stabilization, and network size significantly influences bifurcation characteristics. Finally, the accurate function fitting achieved by our model provides new insights for the design and control of neural 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.
Amit Tiwari, C. Hens, Prosenjit Kundu· 0 citations
The Kuramoto model is a canonical paradigm for characterizing the synchronization dynamics of coupled systems. Recently, Kuramoto systems with higher-order coupling have emerged as a growing research hotspot, yet most studies have largely focused on dynamical behavior analysis, with little attention on the control of global network behaviors. Building on the network stochastic resonance theory proposed by Wang et al (2026 Proc. R. Soc. A: Math. Phys. Eng. Sci. 482 20250945), this study extends the control channel from three-body coupling modulation to pairwise coupling modulation, and proposes an open-loop dynamical regulation strategy for global collective behaviors. This strategy leverages network stochastic resonance to achieve effective control without the need for real-time monitoring of the network. The findings of this research provide useful insights for the regulation of collective dynamics in higher-order coupled Kuramoto networks and the open-loop dynamic control of complex oscillator networks.
Wenchang Qi, Zheng Wang, Jinjie Zhu et al.· Journal of Physics: Complexi...· 0 citations
Higher-order networks exhibit rich critical phenomena that cannot be captured by traditional pairwise models. Here, we develop an analytical dimension-reduction framework that maps higher-order networked dynamics onto an effective low-dimensional system, allowing accurate prediction of tipping boundaries, bistability regions, and the nature of phase transitions. We demonstrate the power of this framework across a range of dynamical processes, revealing distinct effects of higher-order interactions on transition continuity and hysteresis. Furthermore, we find that system resilience exhibits a profound dependence on the alignment between pairwise and higher-order connectivity, with assortative mixing enhancing tipping toward active states. Our findings establish a general theory for understanding the critical transitions in higher-order networks, offering new insights for anticipating and managing systemic risk in complex systems.
A one-dimensional reduction for dynamical systems on networks with purely higher-order interactions is developed, formulated through an effective higher-order interaction strength, associated with the triangular interactions of the underlying network and the dynamical system's effective state.
Can asymmetry between two interacting networks fundamentally change how they synchronize? We identify double explosive transitions in the forward direction, backward direction, or a combination thereof, with single or double hysteresis loops in an adaptive bilayer multiplex network of Kuramoto oscillators with pairwise and three-body interactions and asymmetric phase lags. Using the Ott-Antonsen reduction, we derive a low-dimensional system and perform a stability analysis. The reduced model accurately captures the full microscopic dynamics and enables analytical expressions for the bifurcation points. Systematic mapping across multiple parameter planes reveals eight distinct synchronization regimes. The relative ordering of saddle-node and pitchfork bifurcation points---controlled by phase-lag asymmetry, cross-layer adaptation, and the higher-order interaction strength---creates two distinct coherent branches (weak and strong), giving rise to double explosive transitions. Crucially, phase-lag asymmetry acts as a robust control knob: while symmetric phase lags suppress explosive transitions, layer-specific differences promote multistability and double explosive transitions. The higher-order interaction strength $K_2$ and adaptation strengths $q$, $p$, and $h$ further modulate these transitions in a complex, parameter-dependent manner. Excellent agreement between analytical predictions and numerical simulations confirms the robustness of our reduced description.
Explosive death - a discontinuous, first-order transition from oscillatory dynamics to a steady state - is investigated in a framework of interacting networks under mixed coupling. Unlike previous studies focused on monolayer topologies, this work explores the interplay between local diffusive coupling among peripheral nodes and their respective hubs, and conjugate (dissimilar variable) interactions between the hubs of different layers. We demonstrate that by tuning the intralayer and interlayer coupling strengths, the system exhibits a sudden collapse of oscillations into a steady state. The generality of this explosive transition is verified across a diverse range of dynamical systems, including the Stuart-Landau limit cycle oscillator, and the Hindmarsh-Rose bursting neuron model with star and scale-free networks. Our results suggest that the combination of star-type or scale-free network connectivity and mismatch in coupling variables provides a robust mechanism for inducing abrupt transitions to quenching in multilayered nonlinear systems within the explored parameter space.
Ranjib Banerjee, Dibakar Ghosh, Amit Sharma· Europhysics letters· 0 citations
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