ABSTRACT Neurons in the brain are submerged into oscillating extracellular Local Field Potential (LFP) created by the synchronized synaptic currents. The dynamics of these oscillations is one of the principal characteristics of the brain activity at all levels: from the synchronized spiking of the individual neurons and neuronal ensembles to the high-level cognitive processes. A physiological interpretation of the LFP data depends on the mathematical and computational approaches used for its analysis. Traditionally, the oscillatory nature of LFP motivates using Fourier methods, which have indeed dominated LFP research for the last several decades and currently constitute the only systematic framework for understanding brain oscillations. Yet these methods are not well suited for handling two fundamental attributes of biological signals: noise and nonstationarity, and may therefore obscure the actual physiological structure of the brain rhythms. To address this problem, we developed an approach based on the Padé Approximation techniques—a powerful novel technique that allows a much more nuanced analysis of the LFP oscillations. Previously, our method was successfully applied to studying various physical signals, e.g., to detecting gravitational waves in gravitational antennas. Applying this method in biological realm also lead us immediately to new observations. Specifically, we discovered that the hippocampal and the cortical LFPs recorded in rats consist of a small set of frequency-modulated waves, which we call oscillons. We hypothesize that oscillons represent the actual, physical structure of the brain waves (such as, e.g., θ- wave or γ-waves) that was previously obscured by the traditional, less powerful techniques. Another key feature of our method is that it possesses an impartial marker of the noise component, which allows us to identify and remove the “noise shell” from the signal and then to investigate not only the noise itself, but also the interplay between the noise and the oscillatory dynamics. The goal of the proposed research is to carry and extensive scope of detailed studies of this new level of the brain rhythms’ structure through this newly discovered computational lens. We anticipate that our work will lead us a fundamentally better understanding of the brain wave structure in wakefulness and in sleep, and produce new insights into the underlying neurophysiological and cognitive phenomena.