Nagumo equation

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.

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... 

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 ... 

We consider next a front propagating into a metastable state for the Nagumo equation, F p) = 6 p - e)/o(l - p) p - a). The front velocity in the absence of a cutoff is vq = l/ /2 - a /l. Equation (4.86) yields v Exact analytical results for the front solutions can be obtained if the reaction term F(p) with a cutoff is replaced by a piecewise linear approximation. The dependence of the velocity shift on the cutoff displays good agreement with the results by Brunet and Derrida and Kessler et al. [495]. 

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. ... 

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... 

While historically things were described a little differently, all in all, Hodgkin and Huxley had one equation for the membrane potential and three more for describing conductance. They had a total of four differential equations. Four differential equations is a manageable number and one that might benefit from reduction because two is very close to four. Indeed, two prevents chaos and allows two dimensional qualitative analysis of the differential equation. The reduction to two dimensions is precisely what Nagumo et al. (1962) and Fitzhugh (1961) have done. They have done a... 

Determine the shift in the front velocity for the RD equation with a Nagumo reaction term, when a cutoff is imposed at the state p = 0. Use the variational principle to show that... 

May, R. M. (1976) Simple mathematical models with very complicated dynamics. Nature 261, 459 McKean, H. P. (1970) Nagumo s equation. Adv. Math. 4, 209... 

A second example of a reaction-diffusion equation is the FitzHugh-Nagumo (FITN) model for two chemical concentrations u and v ... 

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