Introduction to Focus Issue: Mixed Mode Oscillations: Experiment, Computation, and Analysis

Abstract

Mixed mode oscillations ͑MMOs͒ occur when a dynamical system switches between fast and slow motion and small and large amplitude. MMOs appear in a variety of systems in nature, and may be simple or complex. This focus issue presents a series of articles on theoretical, numerical, and experimental aspects of MMOs. The applications cover physical, chemical, and biological systems. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2903177͔An oscillator is a dynamical system which goes through the same-or almost the same-states again and again. Oscillators occur everywhere in nature as rhythms and vibrations. Simple oscillators are based on a single mechanism, but in more complex systems different mechanisms are active during different phases of the oscillation. This can give rise to oscillations which shift between slow and fast motion and small and large amplitude. Such mixed mode oscillations (MMOs) are the subject of a substantial current research effort, and include experimental, computational, and theoretical approaches which shed light on important issues in physics, chemistry, and biology.Oscillatory behavior occurs everywhere in nature. The harmonic oscillator is a fundamental mathematical structure that any science student meets. Real world oscillators do, however, rarely possess the uniformity of the harmonic oscillator but typically switch between slow and fast motion and small and large amplitudes, and, hence, display MMOs. Here we make a slightly narrower definition of MMOs. We primarily refer to MMOs as complex patterns that arise in dynamical systems, in which oscillations with different amplitudes are interspersed. These amplitude regimes differ roughly by an order of magnitude. In each regime, oscillations are created by a different mechanism and their amplitudes may have small variations. Additional mechanisms govern the transition among regimes. MMOs of this type are ubiquitous in nature, and have first been observed in chemical reactions more than 100 years ago, 1 with the BelouzovZhabotinsky ͑BZ͒ reaction discovered in the 1970s being the most thoroughly studied example. MMOs may also occur through the canard phenomenon, first discovered in the van der Pol equation. 30,31 Here a limit cycle born in a Hopf bifurcation experiences the transition from a small, almost harmonic cycle to a large relaxation oscillation in a narrow parameter interval. The intermediate limit cycles existing during the transition are the canard cycles which are characterized by following a slow manifold for a substantial time and distance on its unstable part. The canard phenomenon occurs in the parameter range of the system where it is a singular perturbation problem, and it has subsequently been identified in a number of other 2D singularly perturbed oscillators. 5,12 If such a system is modified either by adding further variables or by noise MMOs may occur in larger regions as the dynamics switches between small amplitude oscillations ͑SAOs͒ and large amplitude oscillations ͑LAOs͒. Dynamics related to canards is a...

Biophysical model for gamma rhythms in the olfactory bulb via subthreshold oscillations

“…A closer approximation for the MCs might be ''resonate and fire'' neurons (26), for which there is not yet a theory of synchronization. Such a theory is beyond the scope of this article, but might be facilitated by the work that has been done on mixed-mode oscillations (20,(27)(28)(29).…”

Canard Theory and Excitability

“…Recently, canard theory combined with an appropriate global return mechanism was used based on the multiple time-scale structure of the underlying models to explain these complicated dynamics [2,3,6,22,43,57,61,63]. This is now one widely accepted explanation for MMOs; see, e.g., [5,8,14,16,27,38,54,56,60] and the current review [10].…”

“…Canard induced mixed mode oscillations (MMOs) are a prominent example of a complex rhythm that can be traced to folded node singularities. We refer the interested reader to, e.g., [5,6,10,43,61].…”

On Chaotic Subthreshold Oscillations in a Simple Neuronal Model

“…Mixed-mode oscillations are believed to stand behind a variety of phenomena in neuronal dynamics and in physical chemistry (see e.g. the Focus Issue [3]). In Fig.4 projections of phase portraits for different stages of the evolution of the spiking state are shown.…”