Fitzhugh-Nagumo model

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.

The nullclines of that model are shown in Fig. 3.6 for several parameter values. We see the remarkable similarity with the reduced Oregonator or the FitzHugh-Nagumo models of Sect. 3.1.4 Also, the dynamics of phytoplankton is faster than zooplankton (r b, c),... 

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

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. 

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. 

Fig. 1.6. Noise-induced excitations in the stochastic FitzHugh-Nagumo model under inhibitory white noise driving. Parameters for all plots 6o = 1-05, t = 0.01, 7 = 1.4, Q = 1/3. Left top o = 0.001. Left center (T = 0.009. Left bottom

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

What we believe to be particularly important in the result [Eq. (52)] is that the impulse speed depends strongly on the sodium current activation rate. Thus by measuring the impulse speed we obtain information not only about passive electric characteristics of the nerve fiber but also about the dynamics of the molecular structures responsible for the fiber s activity. A more comprehensive comparison of the above theory with experiment, in particular with the computer-aided treatment of the H-H model carried out in Reference (24), is given elsewhere, in which theory modifications that are more adequate to the H-H model are also analyzed. It should be noted, besides, that qualitatively similar results were obtained by Rinzel and Keller who studied impulse propagation in a FitzHugh-Nagumo model (which takes into account the inertial nature of the variable in the same manner as it does potassium conductance). 

Fisher, C, 9-9-9-11 Fitts, P.M., 77-10,78-5 FitzHugh, R., 22-17 FitzHugh-Nagumo model, 22-17 Flash, T., 77-6 Fleishman and quainlance s... 

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. 

Fig. 6. Twisted scroll wave in the FitzHugh-Nagumo model with stepwise refractoriness (from [62] reprinted with permission from A. M. Pertsov).

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