The role of external and endogenous noise in neural network dynamics and statistics

Noise is ubiquitous, stemming from the surrounding environment or arising from the inherent stochasticity of the system under consideration. Its presence may qualitatively change the behavior of a physical system, possibly leading to surprising and unexpected phenomena, and, as such, it should be accommodated for in realistic models. In this work, I present several models, that bear interest in neuroscience, in which noise plays a role of paramount importance. Throughout my thesis, investigations are conducted by means of both analytical and computational methods. First, I introduce, and further develop key analytical tools for tackling analytically the dynamics of a stochastic system. More specifically, I develop a perturbative technique which allows for computing the statistics of such systems even if they do not obey a gradient dynamics. Second, I focus on purely stochastic oscillators. I show that a collection of such oscillators, occupying the nodes of a generic network, can organize at the macroscopic level yielding noise-sustained spatiotemporal pattern with long-range correlations. Then, the same oscillators are organized in a directed unidirectional lattice with adjacent connections. The endogenous component of noise, coupled to a certain topology of the embedding space, seeds a coherent amplification of the signal across the lattice. Almost periodic oscillations emerge that I thoroughly investigate. Finally, I demonstrate that the coherent amplification of an imposed noisy perturbation destabilizes the synchronous state of an ensemble made of deterministic oscillators also when a conventional linear stability analysis would deem the system resilient to small external disturbances.