Abstract
Bursting phenomena are found in a wide variety of fast-slow systems. In this article, we consider the Hindmarsh-Rose neuron model, where, as it is known in the literature, there are homoclinic bifurcations involved in the bursting dynamics. However, the global homoclinic structure is far from being fully understood. Working in a three-parameter space, the results of our numerical analysis show a complex atlas of bifurcations, which extends from the singular limit to regions where a fast-slow perspective no longer applies. Based on this information, we propose a global theoretical description. Surfaces of codimension-one homoclinic bifurcations are exponentially close to each other in the fast-slow regime. Remarkably, explained by the specific properties of these surfaces, we show how the Hindmarsh-Rose model exhibits isolas of homoclinic bifurcations when appropriate two-dimensional slices are considered in the three-parameter space. On the other hand, these homoclinic bifurcation surfaces contain curves corresponding to parameter values where additional degeneracies are exhibited. These codimension-two bifurcation curves organize the bifurcations associated with the spike-adding process and they behave like the "spines-of-a-book," gathering "pages" of bifurcations of periodic orbits. Depending on how the parameter space is explored, homoclinic phenomena may be absent or far away, but their organizing role in the bursting dynamics is beyond doubt, since the involved bifurcations are generated in them. This is shown in the global analysis and in the proposed theoretical scheme.
Highlights
Fast–slow dynamics is a quite common phenomenon in theoretical and practical models in many disciplines where different time scales are present
We consider the Hindmarsh–Rose neuron model, where, as it is known in the literature, there are homoclinic bifurcations involved in the bursting dynamics
We recall some basic aspects about homoclinic bifurcations and fast–slow dynamics, including a description of foldhom bursters, one of the mechanisms exhibited by the HR model for the creation of bursting orbits
Summary
Fast–slow dynamics is a quite common phenomenon in theoretical and practical models in many disciplines where different time scales are present. In Ref. 20, a theoretical scheme giving a complete scenario of bifurcations involved in the spike-adding processes in fold/hom bursters was introduced This theoretical scheme provides a complete description of the connections of the different codimension-two points and the organization of the homoclinic curves for ε fixed. 4, 5, and 7 in Ref. 27), they use two different notations for a unique curve of homoclinic bifurcation, changing the label from hom(n) to hom(n+1 a) after a sharp fold of the curve is crossed, pointing out that the number of spikes changes from n to n + 1. We find “spines-ofa-book,” bearing in mind the codimension-two bifurcation curves, holding pencils of bifurcations of periodic orbits Both structures give rise to the theoretical model proposed in this article.
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