With this procedure, it is not possible to obtain ( β − S ) -pairs below a certain value of S. The reason is that the system (with mean-field control) is undergoing a saddle-node bifurcation as visualized in the right of Fig. 3.
This will be important if we want to understand the fate of any solution, discontinuous or not, under mean field feedback control.
A linear mean field feedback control moves this saddle-node bifurcation toward distinct localized wave segments with a characteristic form (shape, size) and behind this bifurcation these waves become transient objects; see Fig. 1, Fig. 3, and Video 4.
Localized structures have also been discussed in the context of cortical spreading depression (SD) in migraine before, in particular a model with narrowly tuned parameters that shows transient waves [13, 52, 53] and a model with mean field feedback control that allows for localized waves [30].
In fact, the effect of mean field inhibitory feedback control can best be understood, if we compare these patterns in models with and without this control.
This leads to the key idea of our model, namely to introduce mean field inhibitory feedback control.
Before we further consider the effect of mean field inhibitory feedback control, we have to describe the behavior of continuous waves (closed wave fronts without open ends) when excitability is decreased, e.g., by increasing β, without mean field inhibitory feedback control.
In the design of our model, we make use of the fact that in a model without mean field inhibitory feedback control spiral waves do not curl-in anymore, but become half plane waves at a low critical excitability, called the rotor boundary ∂ R ∞ [33, 34].
