In this study we justify rigorously the approximation of the steep firing rate functions with a unit step function in a two-population neural firing rate model with steep firing rate functions. We do this justification by exploiting the theory of switching dynamical systems. It has been demonstrated that switching dynamics offer a possibility of simplifying the dynamical system and getting approximations of the solution of the system for any specific choice of parameters. In this approach the phase space of the system is divided into regular and singular domains, where the limit dynamics can be written down explicitly. The advantages of this method are illustrated by a number of numerical examples for different cases of the singular domains (i.e. for black, white and transparent walls) and for specific choices of the parameters involved. General conditions have been formulated on these parameters to give black, white and transparent walls. Further, the existence and stability of regular and singular stationary points have been investigated. It has been shown that the regular stationary points (i.e. stationary points inside the regular domains) are always stable and this property is preserved for smooth and sufficiently steep activation functions. In the most technical part of the paper we have provided conditions on the existence and stability of singular stationary points (i.e. those belonging to the singular domains). For the existence results, the implicit function theorem has been used, whereas the stability of singular stationary points is addressed by applying singular perturbation analysis and the Tikhonov theorem.

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. (C) 2012 Elsevier Ltd. All rights reserved.

Figure 1 A. Sketch of the two population neural field model with excitatory (h e