Extreme value theory of evolving phenomena in complex dynamical systems: Firing cascades in a model of a neural network

We extend the scope of the dynamical theory of extreme values to include phenomena that do not happen instantaneously but evolve over a finite, albeit unknown at the onset, time interval. We consider complex dynamical systems composed of many individual subsystems linked by a network of interactions. As a specific example of the general theory, a model of a neural network, previously introduced by other authors to describe the electrical activity of the cerebral cortex, is analyzed in detail. On the basis of this analysis, we propose a novel definition of a neuronal cascade, a physiological phenomenon of primary importance. We derive extreme value laws for the statistics of these cascades, both from the point of view of exceedances (that satisfy critical scaling theory in a certain regime) and of block maxima.