  We study numerically and analytically first- and second-order phase
transitions in neuronal networks stimulated by shot noise (a flow of random
spikes bombarding neurons). Using an exactly solvable cortical model of
neuronal networks on classical random networks, we find critical phenomena
accompanying the transitions and their dependence on the shot noise intensity.
We show that a pattern of spontaneous neuronal activity near a critical point
of a phase transition is a characteristic property that can be used to identify
the bifurcation mechanism of the transition. We demonstrate that bursts and
avalanches are precursors of a first-order phase transition, paroxysmal-like
spikes of activity precede a second-order phase transition caused by a
saddle-node bifurcation, while irregular spindle oscillations represent
spontaneous activity near a second-order phase transition caused by a
supercritical Hopf bifurcation. Our most interesting result is the observation
of the paroxysmal-like spikes. We show that a paroxysmal-like spike is a single
nonlinear event that appears instantly from a low background activity with a
rapid onset, reaches a large amplitude, and ends up with an abrupt return to
lower activity. These spikes are similar to single paroxysmal spikes and sharp
waves observed in EEG measurements. Our analysis shows that above the
saddle-node bifurcation, sustained network oscillations appear with a large
amplitude but a small frequency in contrast to network oscillations near the
Hopf bifurcation that have a small amplitude but a large frequency. We discuss
an amazing similarity between excitability of the cortical model stimulated by
shot noise and excitability of the Morris-Lecar neuron stimulated by an applied
current.
