Nonlinear Analysis of Breathing Pulses in a Synaptically Coupled Neural Network

We analyze the weakly nonlinear stability of a stationary pulse undergoing a Hopf bifurcation in a neural field model with an excitatory or Mexican hat synaptic weight function and Heaviside firing rate nonlinearity. The presence of a spatially localized input inhomogeneity $I(x)$ precludes the 0 eigenvalue related to translation invariance of the pulse. Consequently, in the spectral analysis of the linearization about the stationary pulse U(x), there are two spatial modes, either of which can undergo a Hopf bifurcation in the Mexican hat network to produce a periodic orbit that either expands/contracts (breather) or moves side-to-side (slosher). We derive the normal form for each mode becoming critical in the Hopf bifurcation by (i) the method of amplitude equations and (ii) center manifold reduction, which are shown to agree. Importantly, the critical third order coefficient of the normal form is found to be in strong agreement with numerical simulations of the full model, particularly when the bifurcation switches from super- to subcritical. The motivation of this work is to establish the framework for a perturbative analysis of the breather in a neighborhood of the Hopf bifurcation point to study weakly interacting breathers analytically.