Devil's staircase inside shrimp-shaped regions reveals periodicity of plateau spikes and bursts

TL;DR

This paper investigates a discrete-time model of cardiac cell activity, revealing complex periodic structures called devil's staircases that relate to arrhythmias and abnormal heart rhythms.

Contribution

It uncovers the intricate internal structure of shrimp-shaped parameter regions, showing infinite transitions between periodicities in cardiac spike models.

Findings

EADs are mainly periodic attractors.

DADs often exhibit chaotic behavior.

Shrimp-shaped regions contain infinite periodic transitions.

Abstract

Slow-fast dynamics are intrinsically related to complex phenomena and are responsible for many of the homeostatic dynamics that keep biological systems healthy functioning. We study a discrete-time membrane potential model that can generate a diverse set of spiking behavior depending on the choice of slow-fast time scales, from fast spiking to bursting, or plateau action potentials -- also known as cardiac spikes since they are characteristic in heart myocytes. The plateau of cardiac spikes can lose stability, generating early or delayed afterdepolarizations (EADs and DADs, respectively), both of which are related to cardiac arrhythmia. We show the periodicity changes along the transition from the healthy action potentials to these impaired oscillations. We show that while EADs are mainly periodic attractors, DADs usually come with chaos. EADs are found inside shrimp-shaped regions of the parameter space. However, in our system, multiple periodic attractors live within a shrimp-shaped region, giving it an internal structure made of infinite transitions between periodicities forming a complete devil's staircase. Understanding the periodicity of plateau attractors in slow-fast systems could be useful in unveiling the characteristics of heart myocyte behaviors that are linked to cardiac arrhythmias.