A simple model - the FitzHugh-Nagumo system

Here e, a, and b are real, positive parameters, e is chosen small in order to guarantee a clear timescale separation between the the fast x-variable (activator) and the slow y-variable (inhibitor). The variables a and b determine the position of the so-called nullclines, the two functions y x) that are determined by setting time derivatives dx/dt = 0 and dy/dt = 0. Depending on the parameters the FHN system has different dynamical regimes. Fig. 1.2 shows phase space portraits together with the nullclines and timeseries for three qualitatively different cases. 

In the upper row we see the excitable regime. The solid lines represent the nullclines of the system, the dashed line a typical trajectory. Each dash represents a fixed time interval, i.e. where the system moves faster through phase space the dashes become longer. The system possesses one fixed point (intersection of the nullclines) which is stable. Small perturbations decay. A super-threshold perturbation leads to a large response (spike) after which the system returns to the fixed point. After that a new perturbation is possible if the system from outside is brought again over the threshold. 

In the middle row the oscillatory regime is illustrated. The systems exhibits continuous oscillations. Perturbations have at this stage little influence on the dynamics. 

In the lower row we find the bistable parameter regime. The system possesses two stable fixed points. A first overcritical external perturbation 

Once the system is in the vicinity of this second fixed point a new external perturbation leads to a new excursion which ends at the initial point. 

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