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

## TL;DR

This paper extends extreme value theory to evolving phenomena in complex systems, exemplified by neural network cascades, providing new statistical laws for analyzing such events.

## Contribution

It introduces a novel framework for applying extreme value theory to non-instantaneous, evolving phenomena in complex dynamical systems, with a focus on neural cascades.

## Key findings

- Derived extreme value laws for neural cascades.
- Proposed a new definition of neuronal cascade.
- Analyzed cascade statistics using exceedances and block maxima.

## Abstract

We extend the scope of the dynamical theory of extreme values to cover 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 neural network, introduced 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 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) and of block maxima.