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