FitzHugh-Nagumo system model

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) ... 

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. ... 

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 ... 

This system is closely related to the Fitzhugh-Nagumo model of neural activity see Murray (1989) or Edelstein-Keshet (1988) for an introduction. 

Figure 3.4 Nullclines of the FitzHugh-Nagumo model (3.56)-(3.57). The straight line is G(x, y) = 0, and the curve is F(x, y) = 0. Fixed points are at their intersections. In all panels a = 0.25. a) 7 = 3, I = 0.2. There is only one fixed point, which is stable, and the dynamics is qualitatively similar to the one in the first row of Fig. 3.3. b) 7 = 3, / = 0.1. There is a single unstable fixed point. For small e, say e < 0.04, the behavior is oscillatory and qualitatively similar to the one displayed in the second row of Fig. 3.3. c) 7 = 3, / = 0. There is a single stable fixed point, which is close to the lower turning point of the F(x, y) = 0 nullcline. For small e the dynamics is excitable, as in the third row of Fig. 3.3. d) 7 = 9, / = 0. There are three fixed points. The central one is unstable and the other two are linearly stable. Thus the system presents bistability.

A simple model for the local temporal dynamics of such systems is the FitzHugh-Nagumo model (3.59)-(3.60), written here as ... 

To characterize the level of coherence of noise-induced excitations we analyze the time evolution of the activator concentration x in the FitzHugh-Nagumo model, see Fig. 1.6. In this representation the excitation loops shown previously in Fig. 1.4 become spikes spaced out by intervals during which the system performs noisy relaxation oscillations aroimd its stable state. The phenomenon of coherence resonance manifests itself in the three realizations of x t) for different noise intensities given in Fig. 1.6. For very low noise intensity (upper panel) an excitation is a rare event which happens at random times. In the panel at the bottom, for high noise intensity, the systems fires more easily but still rather randomly. In the panel in the center instead, at an optimal noise intensity, the system fires almost periodically. 

R. Toral, C. Mirasso, and J. D. Gunton. System size coherence resonance in coupled FitzHugh-Nagumo models. Europhys. Lett., 61 162, 2003. 

Such a system can be described by a mathematical model known as the FitzHugh - Nagumo (M IN) scheme, originally applied to the propagation of electric excitation pulses in nerve tissues. In the chemical context, these (mathematical) waves may correspond to time- and space-dependent concentration of some particular substance. 

The model which my colleagues and I have used over the past several years in our studies of spiral waves is simple, two-variable reaction-diffusion model of the Fitzhugh-Nagumo type [ 13,19]. It is a mathematical caricature of what is thought to take place in many real excitable systems. The model has the virtue of providing particularly fast time-dependent numerical simulations of... 

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