We employed Poincaré return mappings for a parameter interval to an exemplary elliptic bursting model, the FitzHugh-Nagumo-Rinzel model. Using the interval mappings, we were able to examine in detail the bifurcations that underlie the complex activity transitions between: tonic spiking and bursting, bursting and mixed-mode oscillations, and finally, mixed-mode oscillations and quiescence in the...

The Fitzhugh-Nagumo (fn) mathematical model characterizes the action potential of the membrane. The dynamics of the Fitzhugh-Nagumo model have been extensively studied both with a view to their biological implications and as a test bed for numerical methods, which can be applied to more complex models. This paper deals with the dynamics in the (FH) model. Here, the dynamics are analyzed, qualit...

Numerical studies indicate that the FitzHugh–Nagumo system exhibits stable traveling pulses with oscillatory tails. In this paper, the existence of such pulses is proved analytically in the singular perturbation limit near parameter values where the FitzHugh–Nagumo system exhibits folds. In addition, the stability of these pulses is investigated numerically, and a mechanism is proposed that exp...

We study the well-posedness of a coupled system of PDEs and ODEs arising in the numerical simulation of electrocardiograms. It consists of a system of degenerate reaction-diffusion equations, the so-called bidomain equations, governing the electrical activity of the heart, and a diffusion equation governing the potential in the surrounding tissues. Global existence of weak solutions is proved f...

In this paper, the transport phenomena of synaptic electric impulses are considered. The FitzHugh--Nagumo and FitzHugh--Rinzel models appear mathematically appropriate for evaluating these scientific issues. Moreover, applications such arise in several biophysical different fields as, instance, biology, medicine electronics, where, by means nanoscale memristor networks, scientists seek to repro...

We introduce a technique based on algebraic topology for quantifying spatio-temporally chaotic dynamics. The technique is illustrated using the Gray-Scott and the FitzHugh-Nagumo models.

A space clamped neural axon along which flows a constant current exhibits current dependent oscillations of the action potential. With increasing current a characteristic Hopf bifurcation scenario can be observed: approximately harmonic small amplitude oscillations arise and bifurcate subsequently into relaxation oscillations of relatively large amplitude. Amplitude and period of the oscillatio...

The electrophysiology of cardiac myocytes is tightly coupled to mechanics. One aspect of this is apparent in the effects of mechanical strain on cardiac action potential dynamics. These can be considered through adjustments to classic electrophysiological models. The standard FitzHugh Nagumo model is a reduced two-variable model from Hodgkin-Huxley that can reproduce basic mechanisms of action ...