Localized activity patterns in two-population neuronal networks

We investigate a two-population neuronal network model of the Wilson–Cowan type with respect to existence of localized stationary solutions (“bumps”) and focus on the situation where two separate bump solutions (one narrow pair and one broad pair) exist. The stability of the bumps is investigated by means of two different approaches: The first generalizes the Amari approach, while the second is based on a direct linearization procedure. A classification scheme for the stability problem is formulated, and it is shown that the two approaches yield the same predictions, except for one notable exception. The narrow pair is generically unstable, while the broad pair is stable for small and moderate values of the relative inhibition time. At a critical relative inhibition time the broad pair is typically converted to stable breathers through a Hopf bifurcation. In our numerical example the broad pulse pair remains stable even when the inhibition time constant is three times longer than the excitation time constant. Thus, our model results do not support the claim that slow excitation mediated by, e.g., NMDA-receptors is needed to allow stable bumps.