Effects of curvature on speed and stability of traveling spots with synaptic depression

Determine how surface curvature affects the propagation speed and stability of traveling spot solutions in the synaptic-depression neural field model given by ∂t u = −u + ∫_D w(x,y) q(y) f[u(y)] dy and τ ∂t q = 1 − q − β q f[u], when posed on curved surfaces such as the bumpy sphere, including whether curvature increases or decreases speed and whether it stabilizes or destabilizes the traveling spot.

Background

The authors simulate traveling spots in a neural field model with synaptic depression on a bumpy spherical surface and observe that curvature deflects trajectories relative to great-circle paths. While qualitative deflection is evident, the quantitative impact of curvature on dynamical properties such as speed and stability is not established.

Prior work on planar domains suggests that spatial inhomogeneities can pin or disrupt waves. Extending such analyses to curved surfaces and specifically to traveling spots with adaptation would clarify how geometric features modulate propagation characteristics.