Front bifurcations in an excitatory neural network

We show how a one-dimensional excitatory neural network can exhibit a symmetry breaking front bifurcation analogous to that found in reaction diffusion systems. This occurs in a homogeneous network when a stationary front undergoes a pitchfork bifurcation leading to bidirectional wave propagation. We analyze the dynamics in a neighborhood of the front bifurcation using perturbation methods, and we establish that a weak input inhomogeneity can induce a Hopf instability of the stationary front, leading to the formation of an oscillatory front or breather. We then carry out a stability analysis of stationary fronts in an exactly solvable model and use this to derive conditions for oscillatory fronts beyond the weak input regime. In particular, we show how wave propagation failure occurs in the presence of a large stationary input due to the pinning of a stationary front; a subsequent reduction in the strength of the input then generates a breather via a Hopf instability of the front. Finally, we derive conditions for the locking of a traveling front to a moving input, and we show how locking depends on both the amplitude and velocity of the input.