  It is well known that bursting activity plays an important role in the
processes of transmission of neural signals. In terms of population dynamics,
macroscopic bursting can be described using a mean-field approach. Mean field
theory provides a useful tool for analysis of collective behavior of a large
populations of interacting units, allowing to reduce the description of
corresponding dynamics to just a few equations. Recently a new phenomenological
model was proposed that describes bursting population activity of a big group
of excitatory neurons, taking into account short-term synaptic plasticity and
the astrocytic modulation of the synaptic dynamics [1]. The purpose of the
present study is to investigate various bifurcation scenarios of the appearance
of bursting activity in the phenomenological model. We show that the birth of
bursting population pattern can be connected both with the cascade of period
doubling bifurcations and further development of chaos according to the
Shilnikov scenario, which leads to the appearance of a homoclinic attractor
containing a homoclinic loop of a saddle-focus equilibrium with the
two-dimensional unstable invariant manifold. We also show that the homoclinic
spiral attractors observed in the system under study generate several types of
bursting activity with different properties.
