Spike-adding in parabolic bursters: The role of folded-saddle canards

Abstract

The present work develops a new approach to studying parabolic bursting, and also proposes a novel four-dimensional canonical and polynomial-based parabolic burster. In addition to this new polynomial system, we also consider the conductance-based model of the Aplysia R15 neuron known as the Plant model, and a reduction of this prototypical biophysical parabolic burster to three variables, including one phase variable, namely the Baer-Rinzel-Carillo (BRC) phase model. Revisiting these models from the perspective of slow-fast dynamics reveals that the number of spikes per burst may vary upon parameter changes, however the spike-adding process occurs in an explosive fashion that involves special solutions called canards. This spike-adding canard explosion phenomenon is analysed by using tools from geometric singular perturbation theory in tandem with numerical bifurcation techniques. We find that the bifurcation structure persists across all considered systems, that is, spikes within the burst are incremented via the crossing of an excitability threshold given by a particular type of canard orbit, namely the true canard of a folded-saddle singularity. However there can be a difference in the spike-adding transitions in parameter space from one case to another, according to whether the process is continuous or discontinuous, which depends upon the geometry of the folded-saddle canard. Using these findings, we construct a new polynomial approximation of the Plant model, which retains all the key elements for parabolic bursting, including the spike-adding transitions mediated by folded-saddle canards. Finally, we briefly investigate the presence of spike-adding via canards in planar phase models of parabolic bursting, namely the theta model by Ermentrout and Kopell.

Parabolic bursting, spike-adding, dips and slices in a minimal model

“…This type of dynamics also arises in three-time-scale systems as recently shown in [14]. An important object in this spike-adding scenario, unveiled in [13] and justified through qualitative arguments and numerical bifurcation analysis, is a so-called folded homoclinic bifurcation shown as blue and red curves in Figures 2 and 3; see also [14]. This object refers to a homoclinic bifurcation in the DRS which then, due to the time rescaling, corresponds in the RS to a limiting curve starting at the folded saddle and returning back to it with a finite passage time, hence connecting the faux canard and the true singular canards of the folded saddle.…”

“…We analyse this minimal system by putting the emphasis on the spike-adding scenario which distinguishes the first transition where spike addition is based on folded-saddle canards, whereas the subsequent ones are added through jump-on points (still near a folded saddle). We also describe the folded homoclinic bifurcation that takes places in the slow subsystem and which was already identified in [13] as a key ingredient for spike adding in parabolic bursters. We then show a progressive reduction of the initial model to a one-dimensional ODE with periodic forcing allowing comparison with the forced van der Pol system.…”

“…This is best understood by analysing the integrable structure of the DRS; see below and also Figures 2 and 3. As showcased in [13], the presence of a folded saddle on the critical manifold of a parabolic burster organises the transition from the subthreshold (oscillatory) regime to the spiking and then the bursting regime, with the canard associated with the folded saddle organising spike-adding transitions. This type of dynamics also arises in three-time-scale systems as recently shown in [14].…”

“…the sodium maximal conductance) and, from slow spikeless cycles in some parameter regime, the solutions can gain more and more spikes as a parameter is varied. These spike-adding transitions were recently analysed in the framework of geometric singular perturbation theory in [13] and it was shown that they are organised by canards of folded-saddle type; see also [14].…”

“…Minimal phase models of parabolic bursters were first introduced by Ermentrout and Kopell with the so-called theta model [18], sometimes also referred to as Atoll model [28]; see also [45]. Spike-adding via canards in the Atoll model was related to jump-on points in [13]. More generally, canards in slow-fast planar systems defined on the two-torus were studied by Guckenheimer and Ilyashenko in [24] and then by Schurov and collaborators in [41][42][43] however without any link made to parabolic bursting or spike-adding.…”