Spike-adding structure in fold/hom bursters

Square-wave or fold/hom bursting is typical of many excitable dynamical systems, such as pancreatic or other endocrine cells. Besides, it is also found in a great variety of fast-slow systems coming from other neural models, chemical reactions, laser dynamics, and so on. We focus on the spike-adding process and its connection with the homoclinic structure of the system. The creation of new fast spikes on a bursting neuron is an important phenomenon as it increases the duty cycle of the neuron. Here we mainly work with the Hindmarsh-Rose neuron model, a prototype of fold/hom bursting, but also with the pancreatic β-cell model, where, as already known from the literature, homoclinic bifurcations play an important role in bursting dynamics. Based on several numerical simulations, we present a theoretical scheme that provides a complete scenario of bifurcations involved in the spike-adding process and their connection with the homoclinic bifurcations on the parametric space. The global scheme explains the different phenomena of the spike-adding processes presented in literature (continuous and chaotic processes after Terman analysis) and moreover, it also indicates where each kind of spike-adding process occurs. Different elements are involved in the theoretical scheme, such as homoclinic isolas, canard orbits, inclination and orbit flip codimension-two bifurcation points and several pencils of period doubling and fold bifurcations, all of them illustrated with different numerical techniques. Some of these bifurcations needed in the process may be not visible on some numerical simulations because the organizing points are in different parametric planes due to the high dimension of the whole parameter space, but their effects are present. Therefore, we introduce a mechanism of the spike-adding process in fold/hom bursters in the whole space of parameters, even if apparently no role is played by the “far-away” homoclinic bifurcations. This fact is illustrated showing how the theoretical scheme provides a theoretical explanation to the different interspike-interval bifurcation diagrams (IBD) that have appeared in the literature for different models.RB and SS have been supported by the Spanish Research projects MTM2015-64095-P, PGC2018-096026-B-I00, the Universidad de Zaragoza-CUD project UZCUD2019-CIE-04 and European Regional Development Fund and Diputación General de Aragón (E24-17R and LMP124-18). SI and LP have been supported by Spanish Research projects MTM2014-56953-P and MTM2017-87697-P. LP has been partially supported by the Gobierno de Asturias project PA-18-PF-BP17-072

Hopf-Zero singularities truly unfold chaos

We provide conditions to guarantee the occurrence of Shilnikov bifurcations in analytic unfoldings of some Hopf-Zero singularities through a beyond all order phenomenon: the exponentially small breakdown of invariant manifolds which coincide at any order of the normal form procedure. The conditions are computable and satisfied by generic singularities and generic unfoldings. The existence of Shilnikov bifurcations in the case was already argued by Guckenheimer in the 80’s. About the same time, endowing the space of unfoldings with a convenient topology, persistence and density of the Shilnikov phenomenon was proved by Broer and Vegter in 1984. However, since the proof involves the use of flat perturbations, this approach is not valid in the analytic context. What is more, none of the mentioned approaches provides a computable criteria to decide whether a given unfolding exhibits Shilnikov bifurcations or not. Many people appeals to the appearance of Hopf-Zero singularities to explain the emergence of chaos in a huge number of applications. However, no one can refer to a specific theorem establishing the conditions that a given unfolding should satisfy to ensure that chaotic dynamics are exhibited. We fill this gap by providing an ultimate result about the appearance of Shilnikov bifurcations in analytic unfoldings of a certain class of Hopf-Zero singularities. These conditions are computable and satisfied by generic families. One of these conditions depends on the full jet of the singularity and comes from a beyond all order phenomenon. It can be related with Stokes constants. The other conditions only depend on the 2-jet of the family.Peer Reviewe