N. Giurgiteanu – författare

The present monograph analyses the FitzHugh-Nagumo (F-N) model Le. , the Cauchy problem for some generalized Van der Pol equation depending on three real parameters a, band c. This model, given in (1. 1. 17), governs the initiation of the cardiac impulse. The presence of the three parameters leads to a large variety of dy­ namics, each of them responsible for a specific functioning of the heart. For physiologists it is highly desirable to have aglobai view of all possible qualitatively distinct responses of the F-N model for all values of the pa­ rameters. This reduces to the knowledge of the global bifurcation diagram. So far, only a few partial results appeared and they were spread through­ out the literature. Our work provides a more or less complete theoretical and numerical investigation of the complex phase dynamics and bifurca­ tions associated with the F-N dynamical system. This study includes the static and dynamic bifurcations generated by the variation of a, band c and the corresponding oscillations, of special interest for applications. It enables one to predict all possible types of initiations of heart beats and the mechanism of transformation of some types of oscillations into others by following the dynamics along transient phase space trajectories. Of course, all these results hold for the F-N model. The global phase space picture enables one to determine the domain of validity of this model.

This application-oriented monograph presents a comprehensive theoretical and numerical investigation of all types of oscillators and bifurcations (such as Hopf, Bogdanov-Takens, Bautin, and homoclinic) generated by the FitzHugh-Nagumo model. The wide diversity of the oscillators as used in electrophysiology, biology, and engineering is emphasised. Various asymptotic behaviours are revealed. The dramatic changes in oscillations connected with the emergence or disappearance of concave limit cycles are investigated. Codimension of bifurcations is minutely analysed. New types of codimension one and two bifurcations of planar systems were found. A detailed global bifurcation diagram concludes the work. Audience: This volume will be of interest to researchers and graduate students whose work involves the mathematics of biology, ordinary differential equations, approximations and expansions, cardiac electrophysiology, biological transport, and cell membranes.