The human brain exhibits cryptic spatiotemporal dynamics that can be investigated through the lens of coupled dynamical systems and spectral decomposition methods. In this talk, I will present how concepts from harmonic analysis, Markov processes, and oscillatory dynamics provide a rigorous mathematical framework for deciphering the hidden rules governing large-scale brain organisation.

More precisely, I will show how brain patterns switch as a Markov process; how eigenmodes exhibit damped oscillatory motion; and how collective rhythms emerge from metastable synchronisation. This work illustrates a bidirectional bridge: mathematical formalisms illuminate neuroscience phenomena, while brain data offers concrete applications for advancing theoretical frameworks—ultimately enabling us to understand the dynamical principles underlying cognitive function, consciousness and mental health.