Nagumo reaction

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... 

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 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... 

A formal synthesis of pamamycin-607 was reported by Nagumo, who constructed the tetrahydrofuran ring by making use of novel ether-ring transformation via a phenonium ion. The reaction mechanism is depicted below <02TL5333>. 

Yamamoto, T., Eki, T., Nagumo, S., Suemune, H., and Sakai, K., Drastic ring transformation reactions of fused bicyclic rings to bridged bicyclic rings. Tetrahedron, 48, 4517, 1992. 

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) ... 

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... 

In 1998, Kawahara and Nagumo reported the first total synthesis of a member of the TAN1251 series [63] and five years later both authors revisited the TAN1251A alkaloid by means of a new enantioselective synthesis (see Section 5.6). The retro synthetic analysis of TAN 1251A is outlined in Scheme 37. The target compound could be obtained by aldol reaction of tricyclic lactam 119, whose disconnection at the amide bond led to the bicyclic amino acid 120, which could be prepared from azaspirocyclic compound 121 by means of alkylation of the secondary amine and Mitsunobu-type chemistry. Azabicycle 121 may be prepared by an intramolecular alkylation of 122, which in turn could be available from allyl derivative 123. The latter can be prepared from carboxylic acid 124 by alkylation and subsequent Curtius rearrangement. 

Since Wardrop had reported the conversion of (+)-119 into (-)-TAN 1251A using the procedure already described by Kawahara and Nagumo in the racemic series, the enantiocontrolled synthesis of (+)-119 achieved by these authors constitutes a new formal synthesis of 2. This approach is noteworthy for its use of a proline derivative as the starting material and the elaboration of the carbocyclic ring by an aldol reaction. 

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]. 

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

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. 

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... 

Oishi T, Nagumo Y, Hirama M. Convergent synthesis of the trans-fused 6-n-6-6 (n = 7-10) tetracyclic ether system based on a ring-closing metathesis reaction. Ghent. Gont-mm. 1998 1041-1042. 

---