Stochastic neural field theory of wandering bumps on a sphere

Abstract

We use a combination of group theoretic and perturbation methods to analyze the stochastic wandering of bump solutions in a neural field model on the sphere S2. We first construct an explicit bump solution in the absence of external inputs and noise, by taking the synaptic weight distribution to be the sum of first-order spherical harmonics. The corresponding neural field equation is equivariant under the action of the special orthogonal group SO(3), which implies that the bump is marginally stable with respect to rotations of the sphere. We then carry out an amplitude–phase decomposition of the solution in the presence of a weakly biased external input and weak noise, and use this to derive a pair of stochastic differential equations for the wandering of the bump, expressed in terms of angular coordinates on the sphere. The stochastic dynamics is a non-trivial generalization of the corresponding phase dynamics describing the wandering of a bump on a ring network with SO(2) symmetry, since SO(3) is non-abelian and S2 is a curved manifold.