Homoclinic bifurcations of both equilibria and periodic orbits are argued to be critical for understanding the dynamics of the Hindmarsh–Rose model in particular, as well as of some squarewave bursting models of neurons of the Hodgkin–Huxley type. They explain very well various transitions between the tonic spiking and bursting oscillations in the model. We present the approach that allows for ...

Pancreatic islets exhibit bursting oscillations that give rise to oscillatory Ca (2+) entry and insulin secretion from β-cells. These oscillations are driven by a slowly activating K (+) current, Kslow, which is composed of two components: an ATP-sensitive K (+) current and a Ca (2+) -activated K (+) current through SK4 channels. Using a mathematical model of pancreatic β-cells, we analyze how ...

Hundreds of thousands of hashtags are generated every day on Twitter. Only a few become bursting topics. Among the few, only some can be predicted in real-time. In this paper, we take the initiative to conduct a systematic study of a series of challenging real-time prediction problems of bursting hashtags. Which hashtags will become bursting? If they do, when will the burst happen? How long wil...

We employed Poincaré return mappings for a parameter interval to an exemplary elliptic bursting model, the FitzHugh-Nagumo-Rinzel model. Using the interval mappings, we were able to examine in detail the bifurcations that underlie the complex activity transitions between: tonic spiking and bursting, bursting and mixed-mode oscillations, and finally, mixed-mode oscillations and quiescence in the...

When a system’s activity alternates between a resting state (e.g. a stable equilibrium) and an active state (e.g. a stable periodic orbit), the system is said to exhibit bursting behavior. We use bifurcation theory to identify three distinct topological types of bursting in one-dimensional mappings and 20 topological types in two-dimensional mappings having one fast and one slow variable. We sh...

The upcoming IEEE 802.11e is an extension of the 802.11 Wireless Local Area Network (WLAN) standard. We use this new standard with its priority provisioning mechanism to address fairness issues that occur when geographically co-located WLANs share the radio channel in the so-called Overlapping Basic Service Set environment. By allowing a new mechanism that is part of the 802.11e, called EDCF-TX...

Tonically spiking as well as bursting neurons are frequently observed in electrophysiological experiments. The theory of slow–fast dynamical systems can describe basic scenarios of how these regimes of activity can be generated and transitions between them can be made. Here, we suggest a biophysically plausible mechanism based on homoclinic bifurcations of a saddle periodic orbit which explains...

We describe a simple conductance-based model neuron that includes intra- and extracellular ion concentration dynamics and show that this model exhibits periodic bursting. The bursting arises as the fast-spiking behavior of the neuron is modulated by the slow oscillatory behavior in the ion concentration variables and vice versa. By separating these time scales and studying the bifurcation struc...

We compare multistability in central pattern generator (CPG) motifs comprised either of biologically plausible Hodgkin-Huxley models of bursting leech heart interneurons or of phenomenologically reduced phase models. We propose a novel computational tool for detailed examination of polyrhythmic bursting in biophysical CPG models with coupling asymmetries and arbitrary coupling strength. We carr...