  Analytical expressions for scaling of brain wave spectra derived from the
general nonlinear wave Hamiltonian form show excellent agreement with
experimental "neuronal avalanche" data. The theory of the weakly evanescent
nonlinear brain wave dynamics reveals the underlying collective processes
hidden behind the phenomenological statistical description of the neuronal
avalanches and connects together the whole range of brain activity states, from
oscillatory wave-like modes, to neuronal avalanches, to incoherent spiking,
showing that the neuronal avalanches are just the manifestation of the
different nonlinear side of wave processes abundant in cortical tissue. In a
more broad way these results show that a system of wave modes interacting
through all possible combinations of the third order nonlinear terms described
by a general wave Hamiltonian necessarily produces anharmonic wave modes with
temporal and spatial scaling properties that follow scale free power laws. To
the best of our knowledge this was never reported in the physical literature
and may be applicable to many physical systems that involve wave processes and
not just to neuronal avalanches.
