Nagumo kinetics

A. Malevanets and R. Kapral, Phys. Rev. E, 55, 5657 (1997). A Microscopic Model for FitzHugh-Nagumo Kinetics. 

To be specific, we consider logistic kinetics and Nagumo kinetics [274], F(p) = rp(l —p) p—b), as examples for cases A and B, respectively. For the logistic case, a linear stability analysis of the stationary states (0,0) and (1,0) provides their eigen-... 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

We have seen that the Belousov-Zhabotinsky reaction, even in the restricted parameter range for which some elementary analysis can be done, has a large variety of behaviors, which makes it the ideal model system to illustrate nonlinear dynamics of chemical systems. We briefly mention here a kinetic system of a rather different origin, the FitzHugh-Nagumo (FN) model (Murray, 1993 Meron, 1992) ... 

Fig. 1.3. Trajectories of the FitzHugh-Nagumo model for sub- and super-threshold perturbations (red and blue curves respectively). Bach point plotted at constant time intervals ti = ij-i At witli At = 0.005. Kinetic parameters 6 = 1.4, a = 1/3 and 7 = 2. Left high time scale separation for = O.Ol. Right low time scale separation for = 0.1.

When e, /, and q are kept fixed, (j) controls the kinetics in the same way as b does in the FitzHugh-Nagumo model For small 4> the kinetics is oscillatory and for (f) > 4>hb = 4.43 10 , it becomes excitable via a super-critical Hopf bifurcation. 

For numerical calculations, three specific models have been used extensively. These are the Oregonator model of the BZ reaction [12, 14, 24], the FitzHugh-Nagumo model [25, 26], and the (piecewise linear) Pushchino kinetics (also called Kinetics B ) [24, 27, 28]. These are all two-variable models with a diagonal diffusion matrix. 

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