Rigorous analytical treatment of curvature-driven effects in neural fields on surfaces

Develop a rigorous analytical framework that characterizes curvature-driven effects on spatiotemporal solutions of neural field equations posed on smooth, closed surfaces (such as deformed spheres and realistic cortical meshes). In particular, establish how surface geometry influences the existence, stability, and evolution of observed patterns including labyrinthine corridors and localized activity, under laterally inhibitory weight kernels and standard firing-rate nonlinearities.

Background

The paper presents numerical simulations of neural field equations on curved geometries, including deformed spheres and a realistic human cortical surface, using high-order RBF-based quadrature and interpolation. These simulations reveal that surface curvature substantially shapes spatiotemporal activity patterns, such as labyrinthine corridors and localized spots.

While the numerical framework reliably captures geometry-dependent phenomena, the authors explicitly note the absence of a rigorous analytical treatment that explains and predicts these curvature-induced effects on solutions of neural field models posed on non-Euclidean domains. Formal analysis would clarify mechanisms, parameter dependencies, and stability properties of these observed dynamics.