Non-equilibrium geometric forces steer spiral waves on folded surfaces

Spiral waves are ubiquitous signatures of non equilibrium dynamics, appearing across chemical, biological, and active systems. Yet, in many living systems these waves unfold on curved and folded surfaces whose geometry has rarely been treated as a dynamical factor. Here we show that surface curvature fundamentally shapes spiral wave behavior and can contribute to the organization of neural activity in the brain. Via analytical theory and simulations of the complex Ginzburg Landau equation (CGLE) on curved surfaces, we demonstrate that curvature enters through the Laplace Beltrami operator as a spatial modulation of effective diffusion. Gradients of this effective diffusion generate a geometric force on spiral defects, and the complex nature of the CGLE produces a complex mobility that leads to non central and non reciprocal responses. Applied to realistic cortical surfaces of the human brain, the model predicts that the pattern of cortical folding stabilizes and localizes spiral waves, while progressive smoothing of the surface erases these non equilibrium structures. This reveals that brain geometry is not a passive scaffold but an active physical constraint that shapes neural dynamics. More broadly, the same geometric mechanism provides a universal route by which curvature and topology control pattern formation across oscillatory, chemical, and active matter systems.