Traveling waves and breathers in an excitatory-inhibitory neural field

摘要

We study existence and stability of traveling activity bump solutions in an excitatory-inhibitory (E-I) neuralfield with Heaviside firing rate functions by deriving existence conditions for traveling bumps and an Evansfunction to analyze their spectral stability. Subsequently, we show that these existence and stability resultsreduce, in the limit of wave speed c → 0, to the equivalent conditions developed for the stationary bump case.Using the results for the stationary bump case, we show that drift bifurcations of stationary bumps serve as amechanism for generating traveling bump solutions in the E-I neural field as parameters are varied. Furthermore,we explore the interrelations between stationary and traveling types of bumps and breathers (time-periodicoscillatory bumps) by bridging together analytical and simulation results for stationary and traveling bumps andtheir bifurcations in a region of parameter space. Interestingly, we find evidence for a codimension-2 drift-Hopfbifurcation occurring as two parameters, inhibitory time constant τ and I-to-I synaptic connection strength w_(ii) ,are varied and show that the codimension-2 point serves as an organizing center for the dynamics of these fourtypes of spatially localized solutions. Additionally, we describe a case involving subcritical bifurcations that leadto traveling waves and breathers as τ is varied.