  This Thesis explores how tools from Statistical Physics and Information
Theory can help us describe and understand complex systems. In the first part,
we study the interplay between internal interactions, environmental changes,
and effective representations of complex stochastic systems. We model the
environment as an unobserved stochastic process and investigate how the mutual
information between internal degrees of freedom encodes internal and
environmental processes, and how it can help us disentangle them. Then, we
attempt to build effective representations via information-preserving
projections that preserve the dependencies of a complex system. In the
paradigmatic case of underdamped systems, we find that optimal effective
representations may be unexpectedly singular, revealing fundamental limits of
approximating complex models. In the second part of this Thesis, we apply these
approaches and ideas from criticality to neural systems at different scales,
showing that unobserved neural activity may lead to power-law neuronal
avalanches in the absence of criticality. Remarkably, the properties of mutual
information suggest that, whereas avalanches may emerge from stochastic
modulation, interactions between neural populations are at the source of
scale-free correlations. We further explore the role of the structure of
interactions by showing how interaction networks change the nature of the phase
transition of a well-known cellular automaton. Finally, we test a
phenomenological coarse-graining procedure for neural timeseries in different
settings, and in particular in the presence of stochastic environments. Our
results shed light on the role of the interaction network, its topological
features, and unobserved modulation in the emergence of collective patterns of
brain activity and general complex systems.
