Fast–Slow Bursters in the Unfolding of a High Codimension Singularity and the Ultra-slow Transitions of Classes

Abstract

Bursting is a phenomenon found in a variety of physical and biological systems. For example, in neuroscience, bursting is believed to play a key role in the way information is transferred in the nervous system. In this work, we propose a model that, appropriately tuned, can display several types of bursting behaviors. The model contains two subsystems acting at different time scales. For the fast subsystem we use the planar unfolding of a high codimension singularity. In its bifurcation diagram, we locate paths that underlie the right sequence of bifurcations necessary for bursting. The slow subsystem steers the fast one back and forth along these paths leading to bursting behavior. The model is able to produce almost all the classes of bursting predicted for systems with a planar fast subsystem. Transitions between classes can be obtained through an ultra-slow modulation of the model’s parameters. A detailed exploration of the parameter space allows predicting possible transitions. This provides a single framework to understand the coexistence of diverse bursting patterns in physical and biological systems or in models.

Natural extension of fast-slow decomposition for dynamical systems

“…The full system dynamics can be approximated by assuming that the fast variables drift along a sequence of attractors, punctuated by occasional rapid transitions between different attractors at certain bifurcation events, while the slow variables trace the path imposed by the slow dynamics. In particular when a single slow variable is used, the classical modeling approach includes an ordinary differential equation (ODE) for that variable and considers the locus of zero speed, or nullcline of the slow variable with respect to the fast variables, to specify its direction of motion [7][8][9][10][11]; another approach is to view the slow dynamics as a non-automomous external force [12][13][14].…”

Evidence for spreading seizure as a cause of theta-alpha activity electrographic pattern in stereo-EEG seizure recordings

“…In epilepsy, there is large evidence that the majority of seizure onset and offset transitions can be understood as bifurcations (Touboul et al, 2011;Jirsa et al, 2014;Bernard et al, 2014), although also other forms of transitions have been recognized, for instance in absence seizures so-called false bifurcations, in which significant changes occur rather smoothly than discretely (Marten et al, 2009). Bifurcation analysis permits the construction of a seizure taxonomy based on purely dynamic features (Jirsa et al, 2014;Saggio et al, 2017), which guides further research and may, for instance, lead to the discovery of novel attractors (Houssaini et al, 2015). From the perspective of bifurcations, the TAA patterns would be classified as a dynamical system undergoing a supercritical Hopf bifurcation, whose distinguishing feature is the gradually increasing amplitude of the oscillations after crossing the bifurcation (Jirsa et al, 2014;Saggio et al, 2017).…”

“…Bifurcation analysis permits the construction of a seizure taxonomy based on purely dynamic features (Jirsa et al, 2014;Saggio et al, 2017), which guides further research and may, for instance, lead to the discovery of novel attractors (Houssaini et al, 2015). From the perspective of bifurcations, the TAA patterns would be classified as a dynamical system undergoing a supercritical Hopf bifurcation, whose distinguishing feature is the gradually increasing amplitude of the oscillations after crossing the bifurcation (Jirsa et al, 2014;Saggio et al, 2017). As we have demonstrated here, the TAA pattern can also be produced by spreading seizures with fast traveling waves.…”

The Hidden Repertoire of Brain Dynamics and Dysfunction

“…We can thus describe the evolution of a seizure with a few collective variables acting on different timescales: fast variables that, depending on the value of their parameter, can produce either resting or oscillatory activity with bifurcations separating the different regimes; slow variables describing the processes that brings the fast variables across the onset and offset bifurcations (Figure 2). Across multiple patients (Saggio, Spiegler, Bernard, & Jirsa, 2017), most had seizures characterized by different bifurcations in different moments, which implies that different classes of seizure types coexist and can be described with the same model, so that ultra-slow changes in the parameters of the fast variables can bring the patient closer to one or the other seizure type. From the perspective of dynamical system modeling, this states that there must exist some slow variable dynamics (under the assumption of autonomous systems).…”