### The oscillons

We implemented a “Short Time Padé Transform” (STPT), in which a short segment of the time series (that fits into a window of a width *T*_{W}) is analyzed at a time. This allows us to follow the signal’s spectral composition on moment-to-moment basis and to illustrate its spectral dynamics using Padé spectrograms (the analogues of to the standard Fourier spectrograms^{18,19}).

Applying these analyses to the hippocampal LFPs recorded in awake rodents during habituation stage^{20}, we observed that there exist two types of time-modulated frequencies (Fig. 1). First, there is a set of frequencies that change across time in a regular manner, leaving distinct, continuous traces—the *spectral waves*. As shown on Fig. 1A, the most robust, continuous spectral waves with high amplitudes (typically three or four of them) are confined to the low frequency domain and roughly correspond to the traditional *θ*– and *γ*-waves^{13,16}. The higher frequency (over 100 Hz) spectral waves are scarce and short, representing time-localized oscillatory phenomena that correspond, in the standard Fourier approach, to fast *γ* events^{21}, sharp wave ripples (SWRs)^{22} or spindles^{23}. Second, there exists a large set of “irregular” frequencies that assume sporadic values from one moment to another, without producing contiguous patterns and that correspond to instantaneous waves with very low amplitudes.

From the mathematical perspective, the existence of these two types of instantaneous frequencies can be explained based on several subtle theorems of Complex Analysis, which point out that the “irregular” harmonics represent the signal’s noise component, whereas the “regular,” stable harmonics define its oscillatory part (see^{24,25,26,27} and the Mathematical Supplement). Thus, in addition to revealing subtle dynamics the frequency spectrum, the DPT method allows a context-free, impartial identification of noise, which makes it particularly important for the biological applications^{28,29}.

As it turns out, the unstable, or “noisy,” frequencies typically constitute over 95% of the total number of harmonics (Fig. 1A). However, the superposition of the harmonics that correspond to the remaining, *stable* frequencies captures the shape of the signal remarkably well (Fig. 1B). In other words, although only a small portion of frequencies are regular, they contribute over 99% of the signal’s amplitude: typically, the original LFP signal differs from the superposition of the stable harmonics by less than 1%. If the contribution of the “irregular” harmonics (i.e., the noise component *ξ*(*t*)) is included, the difference is less than 10^{−4}–10^{−6} of the signal’s amplitude.

These results suggest that the familiar Fourier decomposition of the LFP signals into a superposition of plane waves with *constant* frequencies,

$$r(t)={{rm{Sigma }}}_{p=1}^{N},{a}_{p}{e}^{i{omega }_{p}t},$$

(1)

should be replaced by a combination of a few phase-modulated waves embedded into a weak noise background *ξ*(*t*),

$$s(t)={{rm{Sigma }}}_{q=1}^{M},{A}_{q}{e}^{i{varphi }_{q}(t)}+xi (t),$$

(2)

which we call *oscillons*. We emphasize that the number (Mll N) of the oscillons in the decomposition (2), their amplitudes *A*_{q}, their phases *ϕ*_{q} and the time-dependent frequencies *ω*_{q}(*t*) = ∂_{t}*ϕ*_{q}(*t*) (i.e., the spectral waves shown on Fig. 1A) are reconstructed on moment-by-moment basis from the local segments of the LFP signal in a hands-off manner: we do not presume *a priori* how many frequencies will be qualified as “stable,” when these stable frequencies will appear or disappear, or how their values will evolve in time, or what the corresponding amplitudes will be. Thus, the structure of the decomposition (2) is obtained *empirically*, which suggests that the oscillons may reflect the actual, physical structure of the LFP rhythms.

### The spectral waves

We studied the structure the two lowest spectral waves using high temporal resolution spectrograms (Fig. 2A). Notice that these spectral waves have a clear oscillatory structure,

$${omega }_{q}(t)={omega }_{q,0}+{omega }_{q,1},sin ({{rm{Omega }}}_{q,1}t+{phi }_{q,1})+{omega }_{q,2},sin ({{rm{Omega }}}_{q,2}t+{phi }_{q,2})+ldots ,,q=1,2,$$

(3)

characterized by a mean frequency *ω*_{q,0}, as well as by the amplitudes, *ω*_{q,i}, the frequencies, Ω_{θ,i}, and the phases, *φ*_{θ,i}, of the modulating harmonics. The lowest wave has the mean frequency of about 8 Hz and lies in the domain 2 ≤ *ω*/2*π* ≤ 17 Hz, which corresponds to the *θ*-frequency range^{13}. The second wave has the mean frequency of about 35 Hz and lies in the low-*γ* domain 25 ≤ *ω*/2*π* ≤ 45 Hz^{16}. Importantly, the spectral waves are well separated from one another: the difference between their mean frequencies is larger than their amplitudes, which allows indexing them using the standard brain wave notations, as *ω*_{θ}(*t*) and ({omega }_{{gamma }_{l}}(t)) respectively, e.g.,

$${omega }_{theta }(t)={omega }_{theta ,0}+{omega }_{theta ,1},sin ({{rm{Omega }}}_{theta ,1}t+{phi }_{theta ,1})+{omega }_{theta ,2},sin ({{rm{Omega }}}_{theta ,2}t+{phi }_{theta ,2})+ldots ,$$

(4)

for the *θ* spectral wave an

$${omega }_{{gamma }_{l}}(t)={omega }_{{gamma }_{l},0}+{omega }_{{gamma }_{l},1},sin ({{rm{Omega }}}_{{gamma }_{l},1}t+{phi }_{{gamma }_{l},1})+{omega }_{{gamma }_{l},2},sin ({{rm{Omega }}}_{{gamma }_{l},2}t+{phi }_{{gamma }_{l},2})+ldots $$

(5)

for the low-*γ* spectral wave, etc.

We verified that these structures are stable with respect to the variations of the STPT parameters, e.g., to changing the sliding window size, *T*_{W}. The size of the sliding window, and hence the number of points *N* that fall within this window can be changed by over 400%, without affecting the overall shape of the spectral waves (Fig. 2B). The smallest window size (a few milliseconds) is restricted by the requirement that the number of data points captured within *T*_{W} should be bigger than the physical number of the spectral waves. On the other hand, the maximal value of *T*_{W} is limited by the temporal resolution of STPT: if the size of the window becomes comparable to the characteristic period of a physical spectral wave, then the reconstructed wave looses its undulating shape and may instead produce a set of sidebands surrounding the mean frequency^{3}. This effect limits the magnitude of the *T*_{W} to abut 50 milliseconds—for larger values of *T*_{W}, the undulating structure begins to straighten out, as shown on Fig. 1A for *T*_{W} = 80 msec.

In contrast with this behavior, the values of the irregular frequencies are highly sensitive to the sliding window size and other DPT parameters, as one would expect from a noise-representing component. The corresponding “noisy” harmonics can therefore be easily detected and removed using simple numerical procedures (see Mathematical Supplement). Moreover, we verified that the structure of the Padé Spectrogram, i.e., the parameters the oscillons remain stable even if the amount of numerically injected noise exceeds the signal’s natural noise level by an order of magnitude (about 10^{−4} of the signal’s mean amplitude), which indicates that the oscillatory part of the signal is robustly identified.

### Parameters of the low frequency oscillons

To obtain a more stable description of the underlying patterns, we interpolated the spectral waves over the uniformly spaced time points (Fig. 3A) and then studied the resulting “smoothened” spectral waves using the standard DFT tools. In particular, we found that, for studied LFP signals, the mean frequency of the *θ*-oscillon is about *ω*_{θ,0}/2*π* = 7.5 ± 0.5 Hz and the mean frequency of the low *γ*-oscillon is ({omega }_{{gamma }_{l,0}}/2pi =34pm 2) Hz, which correspond to the traditional (Fourier defined) average frequencies of the *θ* and the low *γ* rhythms.

The amplitudes of the *θ* and the low *γ* spectral waves—7.0 ± 1.5 Hz and 10.1 ± 1.7 Hz respectively—define the frequency domains (spectral widths) of the *θ* and the low *γ* rhythms (Fig. 3B). The amplitudes of the corresponding oscillons constitute approximately *A*_{θ}/*A* ≈ 62% and ({A}_{{gamma }_{l}}/Aapprox mathrm{17 % }) of the net signals’ amplitude *A*, i.e., the *θ* and the low *γ* oscillons carry about 80% of the signals’ magnitude.

The oscillatory parts of the spectral waves are also characterized by a stable set of frequencies and amplitudes: for the first two modulating harmonics we found *ω*_{θ,1}/2*π* ≈ 4.3 Hz, *ω*_{θ,2}/2*π* ≈ 3.2 Hz for the *θ* spectral wave (4) and ({omega }_{{gamma }_{lmathrm{,1}}}/2pi approx 6.1) Hz, ({omega }_{{gamma }_{lmathrm{,2}}}/2pi approx 4.3) Hz for the *γ* spectral wave (5). The corresponding modulating frequencies for the *θ*-oscillon are Ω_{θ,1} = 4.3 ± 0.45 Hz, Ω_{θ,2} = 7.3 ± 0.48 Hz, …, (Fig. 3C). The lowest modulating frequencies for the *γ*-oscillon are slightly higher: ({{rm{Omega }}}_{{gamma }_{l},1}=5.3pm 0.41) Hz, ({{rm{Omega }}}_{{gamma }_{l},2}=8.3pm 0.51) Hz, …. In general, the modulating frequencies tend to increase with the mean frequency.

Importantly, the reconstructed frequencies sometimes exhibit approximate resonance relationships (Fig. 3C), implying that some of the higher order frequencies may be overtones of a smaller set of prime frequencies that define the dynamics of neuronal synchronization^{30,31,32}.