Detection of dynamical states of neuronal networks with persistent homology tools

The Brunel network is a neuronal network model composed of excitatory and inhibitory leaky integrate-and-fire spiking neurons. This network is known to present four distinguished activity states, which can be mainly described by the synchronicity and regularity of the firing of its neurons. Topological data analysis (TDA) is a field from algebraic topology and computational geometry developed during the 2000s, designed to study the topological structure of datasets. The main tool of TDA is persistent homology. This algebraic method allows to study the topological structure of a network by monitoring how its cavities or holes at different dimensions evolve. The aim of this project is to show that persistent homology can be useful to identify the dynamics of neuronal networks. In order to do so, we apply persistent homology to identify the four activity states produced by the Brunel network. Moreover, we use persistent homology to identify the dynamics induced by the presence of synaptic plasticity (specifically short-time depression) in the Brunel network