Excitable dynamics and threshold sets in nonlinear systems

Abstract

Following our previous work [J. Zagora et al., Faraday Discuss. 120, 313 (2001)], we present a quantitative definition of a threshold that separates large-amplitude excitatory responses and small-amplitude nonexcitatory responses to a perturbation of an excitable system with a single globally attracting steady state. For systems with two variables, finding the threshold set is formulated as a boundary value problem supplemented by a condition of a maximum separation rate. For this highly nonlinear problem we formulate a numerical method based on the use of multiple shooting and continuation methods. The threshold phenomena are examined by using an example dynamical system with chemical reaction--the bromate-sulfite-ferrocyanide system. In a model of this reaction we find the threshold set, construct a bifurcation diagram and discuss how excitability can vanish. These results are compared with recent experiments. We also discuss relevance of other definitions of the excitability threshold including the concept of nullclines.

Fronts and pulses in an enzymatic reaction catalyzed by glucose oxidase

“…Many models possessing pH bistability and oscillations also exhibit excitability (29)(30)(31). The region of excitability is often located near the Hopf bifurcation that leads to oscillations or near the saddle-node bifurcation that results in bistability (32,33). Our first task was thus to locate the boundary between the bistable region and the high pH SS (SSI), where excitability is possible.…”