Effect of localized input on bump solutions in a two-population neural-field model

We investigate a two-population Wilson–Cowan model extended with stationary and spatially dependent localized external inputs and study the existence and stability of localized stationary (bump) solutions. The generic situation for this model in the absence of external inputs corresponds to two bump pairs, one narrow and one broad pair. For spatially wide localized external inputs we find this generic picture to be unchanged. However, for strongly localized external inputs we find that three and even four bump pairs, all with symmetric activity profiles around the center of the localized external input, may coexist. We next investigate the stability of these bump pairs using two different techniques: a simplified phase–space reduction (Amari) technique and full stability analysis. Examples of models, i.e., choices of model parameters, exhibiting up to three stable bump pairs are found. The Amari technique is further found to be a poor predictor of stability in the case of strongly localized external inputs. The bump-pair states are also probed numerically using a fourth order Runge–Kutta method, and an excellent agreement is found between numerical simulations and the analytical predictions from full stability analysis.