FitzHugh-Nagumo

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 ... 

Figure C3.6.7 Cubic (jir = 0) and linear (r = 0) nullclines for tire FitzHugh-Nagumo equation, (a) The excitable domain showing trajectories resulting from sub- and super-tlireshold excitations, (b) The oscillatory domain showing limit cycle orbits small inner limit cycle close to Hopf point large outer limit cycle far from Hopf point.

Figure C3.6.8 (a) A growing ring of excitation in an excitable FitzHugh-Nagumo medium, (b) A spiral wave in tlie same system.

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

The FitzHugh-Nagumo (FHN) model [82] is a simplification of the HH model and involves a tunnel diode (Fig. 23). 

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

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

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) ... 

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 similar, but somewhat more complex system in the family of autocatalytic-type processes is that of an excitable reaction dynamics. This requires multiple reactions with significantly different characteristic timescales. We consider excitable dynamics occurring in the same two flows as in the previous cases (Neufeld et al., 2002c), and as a specific reaction example we focus on the FitzHugh-Nagumo... 

Figure 7.11 Total concentration (C ) in the stationary state vs Da for the FitzHugh-Nagumo dynamics in the open blinking vortex-sink flow showing two discontinuous transitions.

Figure 7.12 Spatial distribution of C for the FitzHugh-Nagumo dynamics under the closed sine-flow (2.66). Note the double line structure of the excited filaments. (From Neufeld et al. (2002c))...

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. 

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. 

In the previous section the stochastic FitzHugh-Nagumo system has been treated using the Langevin eqs. 1.31. Alternatively it can be described by the Fokker-Planck equation (FPE) (cf. subsec. 1.3.3). In the case of the... 

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

Pattern formation in dichotomously driven, locally coupled FitzHugh-Nagumo systems... 

In this subsection we study a spatially extended version of the FitzHugh-Nagumo system ... 

M. Kostur X. Sailer and L. Schimansky-Geier. Stationary probability distributions for FitzHugh-Nagumo systems. Fluctuation and Noise Letters, 3 155, 2003. 

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

PitzHugh-Nagumo The FitzHugh-Nagumo equations are also called the Bonhoeffer-Van der Pol equations and have been used as a generic system that shows excitability and oscillatory activity. FitzHugh [1969] showed that much of the behavior of the Hodgkin-Huxley equations can be reproduced by a system of two differential equations ... 

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