Coarse-grained dynamics of an activity bump in a neural field model

We study a stochastic nonlocal partial differential equation, arising in the context of modelling spatially distributed neural activity, which is capable of sustaining stationary and moving spatially localized 'activity bumps'. This system is known to undergo a pitchfork bifurcation in bump speed as a parameter (the strength of adaptation) is changed; yet increasing the noise intensity effectively slowed the motion of the bump. Here we study the system from the point of view of describing the high-dimensional stochastic dynamics in terms of the effective dynamics of a single scalar 'coarse' variable, i.e. reducing the dimensionality of the system. We show that such a reduced description in the form of an effective Langevin equation characterized by a double-well potential is quantitatively successful. The effective potential can be extracted using short, appropriately initialized bursts of direct simulation, and the effects of changing parameters on this potential can easily be studied. We demonstrate this approach in terms of (a) an experience-based 'intelligent' choice of the coarse variable and (b) a variable obtained through data-mining direct simulation results, using a diffusion map approach.