Non-equilibrium geometric forces steer spiral waves on folded surfaces

TL;DR

This work shows that surface geometry actively controls spiral-wave dynamics by modulating diffusion through the Laplace-Beltrami operator, turning curvature into a drive for defect motion. Using the complex Ginzburg-Landau equation on curved surfaces and isothermal coordinates, the authors show that gradients in an effective diffusion coefficient $D_{ ext{eff}}$ produce geometric forces, yielding radial, tangential, and orbital defect motions, with curvature gradients stabilizing spirals on realistic brain cortical geometries. The results include near-field analytical treatments on cones, validation with simulations, and a unifying diffusion-gradient framework that generalizes to arbitrary curved geometries, including Gaussian bumps and cortical folding. The findings imply that brain geometry can actively shape neural dynamics and that curvature-mediated diffusion control offers a universal mechanism for pattern formation across oscillatory, chemical, and active-matter systems, with potential for curvature-based control and design of dynamical media.

Abstract

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.

Figures (6)

• Figure 1: Curvature determines spiral wave defect dynamics. Simulations of spiral waves on simple surfaces, such as disk and spherical caps, exhibit equilibrium steady state defect configurations (left). On surfaces with complex curvature landscapes like a sinusoidal geometry, defects undergo non equilibrium dynamics (center). Experimental observations on the cortical surface reveal motile defects with trajectories spanning cortical regions (right). Experimental data obtained from Ref. doi:10.1126/sciadv.abf2709.
• Figure 2: Validation of near field theory on conical geometry. (a) Unrolled flat wedge coordinates $(\tilde{r}, \tilde{\theta})$, obtained by cutting and flattening the cone into the plane. The azimuthal angle spans $\tilde{\phi} \in [0, 2\pi(1-\chi))$, reflecting the angular deficit at the apex. The conical singularity is indicated by the star at the origin. A physical topological defect is shown as a black dot in the principle wedge, while the white circles in the dashed wedges represent the $n-1$ image defects; the example shown corresponds to $\chi = 2/3$ and $n = 3$. (b) Isothermal coordinates $(r,\theta)$ obtained by conformally mapping the cone to the plane such that the full azimuthal angle $\phi \in [0,2\pi)$ is restored. The relation between the two coordinate systems is indicated schematically, with the radial rescaling $r = [(1-\chi)\tilde{r}]^{1/(1-\chi)}$ and angular mapping $\phi = \tilde{\phi}/(1-\chi)$. With $\alpha = 0.5$ and $\chi = \pm 0.25$, analytical and simulated trajectories are compared for positive (c) but not negative (d) Gaussian curvature. Simulated defect trajectory for $\alpha=0.3$ on a cone corresponding to $\chi=0.1$ (e). At low parameters ($\alpha = 0.1$, $\chi = \pm 0.1$), analytical and simulated trajectories match remarkably well for both positive (f) and negative (g) Gaussian curvature.
• Figure 3: Defect trajectories near a bump exhibit rotational motion due to nonlocal curvature coupling. Two trajectories, near (blue) and further away (red) both converge and begin to orbit (a). In time, there is not an exact specific stable radius but small oscillators around $r^* \approx 9$ (b) in what appears to be a limit cycle (c).
• Figure 4: Analytical predictions match simulations for simple $D_{\text{eff}}$ gradients. (a) Trajectory of spiral defects in a linear $D_{\text{eff}}$ gradient. (b) Trajectory in a quadratic $D_{\text{eff}}$ gradient. (c) Comparison of simulated and analytical (dashed lines) velocity components for the linear gradient case, showing excellent agreement between theory and simulation when $\alpha=0$. (d) Velocity comparison for the quadratic gradient case match for $\alpha=0.1$.
• Figure 5: Spatial gradients in the diffusion coefficient drive defect motion on flat surfaces. (a) Defect trajectories (lines) under a circular step diffusion profile. Panels (b) and (c) show the deflection of the spiral wave front (see Eq. \ref{['eq:bending']}): when the inner diffusion coefficient is less than the outer value, the wavefront bend inwards which propels defects anticlockwise (b); on the other hand, reversing the $D_{in}$ and $D_{out}$ ratio causes opposite behavior (c). In (b) and (c), $\kappa$ is normalized by the deflection for a spiral wave located at the same position but with no change in $D_\text{eff}$. The total deflection (d), and the distance traveled (e) follow the same curve against the ratio of $D_{\text{eff}}$ and show a proportionality (f). Initializing the defect further from the reduced D region causes it to travel less (g) as does reducing the size of the region (h), but it eventually comes and settles into the same orbital radius (i).