  We use tools of bifurcation theory to characterize dynamics of
astrocytic~IP$_3$ and~Ca$^{2+}$ for different~IP$_3$ regimes from a
mathematical point of view. We do so following a bottom-up approach, starting
from a compact, well-stirred astrocyte model to first identify
characteristic~IP$_3$ pathways whereby~Ca$^{2+}$ (and~IP$_3$) dynamics
"bifurcate", namely change from stable (constant) concentration levels, to
oscillatory dynamics. Then we extend our analysis to the elemental case of two
astrocytes, coupled by~IP$_3$ diffusion mediated by gap junction channels,
putting emphasis on the mechanisms of emergence of chaotic oscillations.
Finally, we complete our analysis discussing spatiotemporal~Ca$^{2+}$ dynamics
in a spatially-extended astrocyte model, gaining insights on the possible
physical mechanisms whereby random Ca$^{2+}$~generation could be orchestrated
into robust, spatially-confined intracellular~Ca$^{2+}$ oscillations.
