Control of Nonlinear Wave Solutions to Neural Field Equations

Abstract: Neural field equations offer a continuous description of the dynamics of large populations of synaptically coupled neurons. This makes them a convenient tool to describe various neural processes, such as working memory, motion perception, and visual hallucinations, to name a few. Due to the important applications, the question arises how to effectively control solutions in such systems. In this work, we investigate the problem of position control of traveling wave solutions to scalar neural field equations on the basis of singular perturbation analysis. Thereby, we consider different means of control such as spatio-temporal modulations of the neural firing threshold, asymmetric synaptic coupling kernels, and additive inputs. Treating these controls as perturbations to the neural field system, one obtains an equation of motion for traveling wave solutions in response to the applied controls. Subsequently, we pose the inverse question of how to design controls that lead to a propagation of the solution following a desired velocity protocol. In particular, we make use of a specific excitation of the solution's translational modes which enables an explicit calculation of a necessary control signal for a given velocity protocol without evoking shape deformations. Moreover, we derive an equivalent control method relying on a modulation of the neurons' synaptic footprint.