Topological and phenomenological classification of bursting oscillations

Abstract

We describe a classification scheme for bursting oscillations which encompasses many of those found in the literature on bursting in excitable media. This is an extension of the scheme of Rinzel (in Mathematical Topics in Population Biology, Springer, Berlin, 1987), put in the context of a sequence of horizontal cuts through a two-parameter bifurcation diagram. We use this to describe the phenomenological character of different types of bursting, addressing the issue of how well the bursting can be characterized given the limited amount of information often available in experimental settings.

A unified model for two modes of bursting in GnRH neurons

“…These numerical results show that the mechanism underlying this bursting behavior is similar to that of the canonical “Type II” parabolic burster, where the fast subsystem possesses a saddle-node on invariant circle (SNIC) bifurcation that defines a threshold for burst initiation and termination (Bertram et al 1995). In the case of this bursting model, however, the fast subsystem is bistable rather than monostable in the parameter regime corresponding to the active phase of the burst.…”

“…This cycle is then repeated for each subsequent parabolic burst. The phenomenon of moving back and forth relative to the SN/HC is also observed in the Chay-Cook model of parabolic bursting (Bertram et al 1995), where the turning point for the full model trajectory depends on a single slow variable representing intracellular Ca 2+ concentration. Although for our analysis, there is an additional slow variable h 2.HVA , it does not add any complexity to the underlying dynamics of the burst.…”

“…The dynamic equations of these three variables constitute the “slow” subsystem, whereas the “fast” subsystem is obtained by treating the three slow variables as quasi-static. The fast subsystem has a three-dimensional critical or “slow” manifold (Bertram et al 1995) defined in terms of the fast variable V by the surface V = V ∞ ( m s , h 2,HVA , Ca ) of steady state solutions to Eq. (1).…”

“…The protocol for conducting the slow-fast bifurcation analysis is adopted from previous studies (Rinzel and Lee 1987; Bertram et al 1995; Izhikevich 2000) and is summarized as follows: (1) a set of slow variables are identified in the model through inspection of parameters; (2) a “fast” subsystem is formulated by treating the slow variables in the full system as quasistatic, i.e., slow variables are model parameters in the fast subsystem; (3) a single burst is simulated by the full model, and the values of the fast and slow variables are recorded; (4) at discrete time points along the burst, we compute a one-parameter bifurcation diagram for the fast subsystem with respect to one of the slow variables, and the other slow variables are set to their values obtained in step (3); and (5) for each bifurcation diagram, the phase plane solution of the full model is superimposed and the relationship between the phase trajectory and the bifurcation structure of the fast subsystem is examined.…”

Electrophysiological Features of Neurons in the Mesencephalic Trigeminal Nuclei

“…Coherent network oscillations are correlated with different behavioral states in the brain and are determined by intrinsic resonance properties [13]. As parts of the intrinsic neuronal properties, resonance and membrane oscillation have been found both in the CNS [15,16,17,18,19,20,21,22] and in peripheral sensory neurons including Mes V, TG and DRG neurons [23,24]. …”