Simplicial cascades are orchestrated by the multidimensional geometry of neuronal complexes

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 $k$-dimensional ``geometrical 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.