Noise-driven bifurcations in a nonlinear Fokker-Planck system describing stochastic neural fields

Abstract

The existence and characterisation of noise-driven bifurcations from the spatially homogeneous stationary states of a nonlinear, non-local Fokker-Planck type partial differential equation describing stochastic neural fields is established. The resulting theory is extended to a system of partial differential equations modelling noisy grid cells. It is shown that as the noise level decreases, multiple bifurcations from the homogeneous steady state occur. Furthermore, the shape of the branch at a bifurcation point is characterised locally. The theory is supported by a set of numerical illustrations of the condition leading to bifurcations and of the corresponding local patterns that emerge around the bifurcation point.

On the fluctuations of an SDE system modelling grid cells

“…We also remark that the Fokker-Planck PDE (1.4), in the case of diffusion term σ " σ 0 constant, has been further analyzed in [12,13]. Under particular assumptions on the drift term, it is shown that for every noise strength σ 0 ą 0 there exists a unique stationary space homogeneous solution (from the modelling point of view this corresponds to the animal completely loosing track of its position in space).…”

“…In particular, as the brain is inherently noisy, understanding how the grid cell network is affected by noise is one of the currently open challenges in the field. This question has recently been addressed from several directions [7,9,10,23,26,31,12,13].…”

“…In [12] the authors studied a system of Fokker-Planck PDEs derived by adding noise to the attractor network models in [8,14] and formally taking the mean field limit. The PDE has been further analyzed in [13]. The passage to the mean field limit has been rigorously proved in [11], where the authors derive the limit for a generalized model, covering diverse concrete models commonly proposed (cf.…”