Abstract. 
Bursting is a manifestation of the complex, multiple time scale dynamics
observed in diverse neuronal models. A description list of the nonlocal
bifurcations leading to its onset is far from being complete and
presents a dire need for cross-disciplinary neuroscience and the
dynamical systems theory. There has been a recent breakthrough in this
direction that explains a few novel mechanisms of transitions between
tonic spiking and bursting activity, as well as their co-existence in
models of leech interneurons through homoclinic saddle-node bifurcations
of periodic orbits including a blue sky catastrophe. We will discuss the
bifurcation theory that underlies theses transitions, as well as one on
a spike adding route: as a parameter shifting the membrane potential of
half-inactivation slow potassium current is monotonically changed, a
sequence of bifurcations occurs causing incremental change of the number
of spikes in a burst. Of our special interest is the origin of the
sequence, where each transition is accompanied by chaos. To figure out
the transition dynamics we construct a one-parameter family of the onto
Poincare return mappings on the central manifolds of slow motions. We
show that the transitions in question are due to the bifurcations of
homoclinics of a repelling point of the map setting a threshold between
tonic spiking and hyperpolarized states of the neuron model.

We show that the regulation of bursting activity in cells forming
networks with mixed, inhibitory and excitatory fast chemical and
electrical, synapses is a crucial skill for controlling rhythmic
movements in motifs, which are the building blocks of CPGs governing
various motor behaviors. We have found that the order parameter of such
networks is the ratio of the burst durations of the cells, so that the
designated pace makers, which are identified by either intrinsic
properties of the cell nearby  the tonic spiking threshold, or by the
architecture of the network under consideration,  are able to
synchronize other strongly uncorrelated or desynchronized neurons in the
network, thereby determining the network's paces and rhythms. We analyze
different topologies and synaptic configurations of motifs to determine
the mechanisms for universality and synergy of bursting patterns
observed in dissimilar networks. We  discuss also multistability of
rhythms of networks and causes for intertransitions between various rhythms.