The Langevin approach Phase portraits under fluctuations

The Langevin approach Phase portraits under fluctuations 

Low noise intensity Excitability case A noise intensity 

When the parameter that controls the excitation threshold of an excitable element fluctuates, then we end up with a system of coupled equations of Langevin type. In the case of the FitzHugh-Nagumo system this situation is modeled by the following Eqs.  

Due to fluctuations the stable fixed point can be destabilized and the system is by chance brought out of the rest state. Here (t) is an arbitrary zero mean stochastic process that describes fluctuations in the excitability parameter b — b t) = 6o - - (i) around a mean value bo. In Fig. 1.4 we show different realizations for the FitzHugh-Nagumo Eqs. 1.31, that permit us to describe its essential properties. 

l) of Fig. 1.4. In panel A.l) the system simply relaxes to the instantaneous fixed points [xo,yo)b t) In panel C.l), due to a high excitability, small stable limit cycles are induced by noise . If the intensity of the fluctuations is increased the system can occasionally escape the vicinity of the fixed point and performs excitation loops, compare panel A.l) with A.2) or C.l) with C.2). 

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