Classification of bursting patterns: A tale of two ducks.

Serafim Rodrigues,John Rinzel,Mathieu Desroches

doi:10.1371/journal.pcbi.1009752

Abstract

Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product, it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram and colleagues, and then by Golubitsky and colleagues, which, together with the Rinzel-Izhikevich proposals, provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least 2 slow variables is then added, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the 2 main families of folded-node bursters, depending upon the phase (active/spiking or silent/nonspiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast subsystem approach.

Highlights

The fascination of experimentalists, physicists, and mathematicians toward spontaneous and complex oscillations dates back to the early 20th century, through observations of electrochemical systems [1]
The folded-node bursting dynamics is further showcased in panel (d) of Figs 8, 9 and 10, where we show it in the (β, z, x) 3D projection of the 4D phase space together with the 2D critical manifold S0 of the full system, its 2 fold curves F and the folded-node point lying on the lower fold curve F À, labeled fn and indicated by a black dot
The mathematical classification of bursting patterns was initiated with seminal papers published in the mid-1980s with 3 proposed classes of bursting oscillations [6,19,41]

Summary

OPEN ACCESS

We review a recent bursting example that falls outside the previous classification systems Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. We describe the 2 main families of folded-node bursters, depending upon the phase (active/spiking or silent/nonspiking) of the bursting cycle during which folded-node dynamics occurs We classify both families and give examples of minimal systems displaying these novel bursting patterns. We provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast subsystem approach.

Author summary

Introduction

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Conclusions