RESUMEN

We study wave propagation in the buffered bistable equation, i.e., the bistable equation where the diffusing species reacts with immobile buffers that restrict its diffusion. Such a model describes wave front propagation in excitable systems where the diffusing species is buffered; in particular, the study of the propagation of waves of increased calcium concentration in a variety of cell types depends directly upon the analysis of such buffered excitability. However, despite the biological importance of these types of equations, there have been almost no analytical studies of their properties.Here, we study the question of whether or not the inclusion of multiple buffers can eliminate propagated waves. First, we prove that a unique (up to translation) traveling wave front exists. Moreover, the wave speed is also unique. Then we prove that this traveling wave front is stable, i.e., that any initial condition which vaguely resembles a traveling wave front (in a way we make precise) evolves to the unique wave front.We thus prove that multiple stationary buffers cannot prevent the existence of a traveling wave front in the buffered bistable equation and may not eliminate stable wave fronts. This suggests (although we do not prove) that the same result is true for more complex and realistic models of calcium wave propagation, a result of direct physiological relevante

RESUMEN

This paper explores analytically and numerically, in the context of the FitzHugh-Nagumo model of nerve membrane excitability, an interesting phenomenon that has been described as a delay or memory effect. It can occur when a parameter passes slowly through a Hopf bifurcation point and the system's response changes from a slowly varying steady state to slowly varying oscillations. On quantitative observation it is found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects; the dynamic aspect of the parameter variation. This delay and its dependence on the speed of the parameter variation are described. The model involves several parameters and particular singular limits are investigated. One in particular is the slow passage through a low frequency Hopf bifurcation where the system's response changes from a slowly varying steady state to slowly varying relaxation oscillations. We find in this case the onset of oscillations exhibits an advance rather than a delay. This paper shows that in general delays in. the onset of oscillations may be expected but that small amplitude noise and periodic environmental perturbations of near resonant frequency may decrease the delay and destroy the memory effect. This paper suggests that both deterministic and stochastic approaches will be important for comparing theoretical and experimental results in systems where slow passage through a Hopf bifurcation is the underlying mechanism for the onset of 'oscillations.