Pulse Stationary Solutions To A Neural Field Equation

Analysis of the dynamical mechanisms underlying spatially structured activity statesin neural tissue is crucially important for understanding a wide range of neurobiologicalphenomena, both naturally occurring and pathological. Neural feld models can exhibit arich repertoire of spatiotemporal dynamics, including solitary traveling fronts and pulses,stationary pulses and spatially localized oscillations(breaters), spiral waves and Turing-like patterns. In recent years, neural felds have been used to model orientation tuningin primary visual cortex, short term working memory, control of head direction, motionperception and so on.We mainly studied the standing pulse solution of neural feld equation with multi-parameter oscillatory connection functions and nonsaturating piecewise-linear gain func-tion.First of all, we transformed the neural feld equation into higher order ordinarydiferential equation(ODE) with Fourier transform and its inverse transform, while theODE had diferent forms in diferent section, but satisfed some boundary conditions.Then we worked with the boundary value problem(BVP) when the gain function wasHeaviside, thus we obtained the existence of standing pulse solution through its defni-tion.Secondly, we restricted the parameter of the connection function to fnd a sufcientcondition of the existence of single-pulse standing solution, and proposed a necessary andsufcient condition of its existence through numerical simulation. Then we linearized theperturbed equation to get some compact linear operator. Through the analysis of theeigenvalue of the linear operator, we came to the conclusion of linear stability of thesingle-pulse standing solution. We relaxed the parameter of the connection functionto extend the results about the existence and stability of single-pulse standing solutionbased on the recently references.Finally, we prove the existence of traveling wave solutions to a neural feld equationwith multi-parameter oscillatory connection functions, and give some discussions on the wave speed.