Amplitude-modulated spiking as a novel route to bursting: Coupling-induced mixed-mode oscillations by symmetry breaking

Abstract

Mixed-mode oscillations consisting of alternating small- and large-amplitude oscillations are increasingly well understood and are often caused by folded singularities, canard orbits, or singular Hopf bifurcations. We show that coupling between identical nonlinear oscillators can cause mixed-mode oscillations because of symmetry breaking. This behavior is illustrated for diffusively coupled FitzHugh–Nagumo oscillators with negative coupling constant, and we show that it is caused by a singular Hopf bifurcation related to a folded saddle-node (FSN) singularity. Inspired by earlier work on models of pancreatic beta-cells [Sherman, Bull. Math. Biol. 56, 811 (1994)], we then identify a new type of bursting dynamics due to diffusive coupling of cells firing action potentials when isolated. In the presence of coupling, small-amplitude oscillations in the action potential height precede transitions to square-wave bursting. Confirming the hypothesis from the earlier work that this behavior is related to a pitchfork-of-limit-cycles bifurcation in the fast subsystem, we find that it is caused by symmetry breaking. Moreover, we show that it is organized by a FSN in the averaged system, which causes a singular Hopf bifurcation. Such behavior is related to the recently studied dynamics caused by the so-called torus canards.

Mixed-mode oscillations in coupled FitzHugh-Nagumo oscillators: blowup analysis of cusped singularities

“…Here, we continue our study of two identical FHN units that are nonoscillatory when uncoupled and investigate the mechanism underlying the result that repulsive coupling can lead to mixed-mode oscillations [32]. We show that the coupled system possess a degenerate singularity in the singular limit → 0, and demonstrate -through a center manifold computation -that this singularity corresponds to a cusp.…”

“…In our recent study of coupled bursting oscillators [32], we revisited the finding by Sherman [37] who showed that coupling of spiking cells can lead to bursting via slow desynchronization so that each burst is preceded by a large number of full action potentials (spikes). Before the transition to bursting, the averaged membrane potentials show SAOs reflecting amplitudemodulated spiking [32].…”

“…In our recent study of coupled bursting oscillators [32], we revisited the finding by Sherman [37] who showed that coupling of spiking cells can lead to bursting via slow desynchronization so that each burst is preceded by a large number of full action potentials (spikes). Before the transition to bursting, the averaged membrane potentials show SAOs reflecting amplitudemodulated spiking [32]. We showed that the dynamical structure of the system obtained by averaging was captured by a system of two coupled FitzHugh-Nagumo (FHN) units [12,28] with repulsive coupling, and that this simpler system exhibits MMOs organized by a singular Hopf bifurcation related to a folded saddle-node (FSN) singularity (type II) [14].…”

Spatial distribution of heterogeneity as a modulator of collective dynamics in pancreatic beta-cell networks and beyond

“…Here, we explore transitions between quiescent states and collective oscillations in coupled networks of heterogeneous, excitable nodes. For the node dynamics, we treat two types of system, the first being the FitzHugh–Nagumo (FHN) model, which is a prototypical model of electrical excitability in biological cells ( Fitzhugh, 1961 ; Nagumo, 1962 ), is often used to investigate collective network dynamics, and has been used as a model for beta cells ( Scialla et al, 2021 ; Pedersen et al, 2022 ). For the second, we consider a conductance-based model of pancreatic beta cells designed to more closely reproduce the signalling dynamics in these cells ( Sherman et al, 1988 ).…”

Intermittent bursting oscillations and the bifurcation analysis in an excited Rayleigh-Duffing oscillator

“…Bursting oscillation is a complex dynamical behavior consisting of comparatively big-amplitude waves and approximately simple harmonic micro-amplitude vibrations, and is universally studied with the help of the slow/fast decomposition analysis that is proposed by in Rinzel [17] in 1985. For example, Pedersen et al [18] identified a novel bursting form related to the pitchfork bifurcation of cycle attractors, which was caused by symmetry breaking. Patsios et al [19] introduced a complex mixed-mode oscillation triggered by the jump mechanism rather than the standard canard phenomenon.…”