Simplicial cascades are orchestrated by the multidimensional geometry of neuronal complexes

Abstract: Cascades arise in many contexts (e.g., neuronal avalanches, social contagions, and system failures). Despite evidence that propagations often involve higher-order dependencies, cascade theory has largely focused on models with pairwise/dyadic interactions. Here, we develop a simplicial threshold model (STM) for nonlinear cascades over simplicial complexes that encode dyadic, triadic and higher-order interactions. We study STM cascades over small-world'' models that contain both short- and long-range $k$-simplices, exploring how spatio-temporal patterns manifest as a frustration between local and nonlocal propagations. We show that higher-order coupling and nonlinear thresholding can coordinate to robustly guide cascades along a simplicial-generalization of paths that we call $k$-dimensionalgeometrical channels''. We also find this coordination to enhance the diversity and efficiency of cascades over a ``neuronal complex'', i.e., a simplicial-complex-based model for a neuronal network. We support these findings with bifurcation theory and a data-driven approach based on latent geometry. Our findings and mathematical techniques provide fruitful directions for uncovering the multiscale, multidimensional mechanisms that orchestrate the spatio-temporal patterns of nonlinear cascades.