Theta Oscillons in Behaving Rats

Abstract

Recently discovered constituents of the brain waves-the oscillons-provide a high-resolution representation of the extracellular field dynamics. Here, we study the most robust, highest-amplitude oscillons recorded in actively behaving male rats, which underlie the traditional θ-waves. The resemblances between θ-oscillons and the conventional θ-waves are manifested primarily at the ballpark level-mean frequencies, mean amplitudes, and bandwidths. In addition, both hippocampal and cortical oscillons exhibit a number of intricate, behavior-attuned, transient properties that suggest a new vantage point for understanding the θ-rhythms' structure, origins and functions. In particular, we demonstrate that oscillons are frequency-modulated waves, with speed-controlled parameters, embedded into a weak noise background. We also use a basic model of neuronal synchronization to contextualize and to interpret the oscillons. The results suggest that the synchronicity levels in physiological networks are fairly low and are modulated by the animal's physiological state.

Keywords: hippocampo-cortical circuit; neural synchronization; oscillons; theta rhythm.

Figure 1.

Oscillons and spectral waves in cortical LFP. A, Fourier spectrogram: the high-power stripe between about 4 and 12 Hz marks the conventional θ-band, the slow-γ domain lays between 20 and 45 Hz. B, The corresponding full Padé spectrogram, same time resolution, shows a pattern of “flexible” frequencies, both stable and unstable. The vertical position of each dot marks a specific frequency value, the horizontal position marks the corresponding time, and the color indicates the instantaneous amplitude (colorbar on the right). C, Most frequencies, (typically over $80\mathrm{\%}$, dark segment on the pie diagram), are unstable or “noise-carrying”. Removing them reveals the denoised Padé spectrogram, on which the stable frequencies trace out regular timelines–spectral waves. Color change along each spectral wave encodes the corresponding time-dependent amplitude (Eq. 1). D, Combining a particular spectral wave, ν_q(t), with its amplitude, A_q(t), yields an individual oscillon, as indicated by Equation 1. Shown is a one-second-long segment of the cortical θ-oscillon (red trace) and the slow-γ oscillon (blue trace). Notice that summing just these two oscillons (first two terms in Eq. 2 approximates the full LFP profile (gray line) quite closely. For more θ and slow-γ waveforms see Extended Data Figure 1-1. E, Numerical reliability: dotted cyan line shows a simulated spectral wave with the mean frequency ν₀ = 8 Hz, a single modulating frequency, Ω₁/2π ≈ 0.6 Hz, and the modulation depth ν₁ ≈ 2 Hz, $\nu (t)={\nu }_0+{\nu }_1\cos (\mathrm{\Omega }t)$. Dark red dots mark the stable frequencies reconstructed via the DPT procedure directly from the corresponding oscillon, $\ell (t)=e^{2\pi i\int \nu (t)dt}$, just as is done with the data. The reconstructed and the input spectral waves match. F, The superposition of all oscillatory inputs (magenta) nearly matches the original LFP signal (gray trace). The difference is due to noise (ξ(t), dotted black), carried by the unstable frequencies, which typically accounts for less than 5–7% of the signal’s net power during active behaviors (pie diagram), and about 10–15% during quiescence. Data sampled in a 6 months old, wake male rat during active behavior.

Figure 2.

Waves and oscillons. A, Data recorded during a fast move over the long straight segment of a linear track. The food wells shown by blue dots. B, A Fourier-defined θ-wave (gray), its amplitude (dotted line on top) and its instantaneous frequency produced by Hilbert transform (black line, scale on the right), compared to the θ-oscillon (pink) and its amplitude (dotted brown line). See also Extended Data Figure 2-1. A segment of the spectral wave is shown on Figure 1C. C, W-spectrogram: black line shows Welch-power profile computed for a particular 600 ms long segment a hippocampal spectral wave, centered at 2.4 s. Arranging such profiles next to each other in natural temporal order yields a 3D landscape that illustrates frequency dynamics. D, The instantaneous Fourier frequency yields a defeatured W-spectrogram that does not resolve rapid frequency modulations. E, DTW comparison of two profiles (red and black). Paired points shown by gray lines. One-to-many connections mark the stretchings that compensate shape mismatches. F, Two concurrent segments of the hippocampal (blue), and the cortical (orange), spectral waves, containing ∼300 data points each (time-wise–about 40 ms), normalized by their respective means and shifted vertically into the [0, 1] range. After the alignments, the number of points increases by $50\mathrm{\%}$ (note the stretched-out x-axis). The net separation between the aligned curves, measured in Euclidean metric and normalized to the original curve lengths quantifies the shape difference between the waveforms, in this case $\sim 7\mathrm{\%}$.

Figure 3.

General properties. A, The lowest spectral wave occupies the domain that is generally attributed to the θ-frequency band. The color of each dot represents the amplitude, as on Figure 1. The spectral peaks and troughs range from about 17 to about 2 Hz (gray boxes). B, The hippocampal θ-oscillon’s spectral wave, made visible through three “frequency slits” that represent three most commonly used θ-bands, 4–12 Hz, 5–15 Hz, and 6–11 Hz (gray stripes). The frequencies that fit into a slit produce the corresponding Fourier wave, shown as red-shaded traces on the bottom right panel. Note that the spectral waves are crosscut by all θ-bands. The Fourier waves are close to each other (DTW distances $D(f_{{\theta }_1}^h,f_{{\theta }_2}^h)\approx 6\pm 2\mathrm{\%}$, $D(f_{{\theta }_2}^h,f_{{\theta }_3}^h)\approx 5\pm 1.9\mathrm{\%}$, $D(f_{{\theta }_3}^h,f_{{\theta }_1}^h)\approx 8\pm 2.1\mathrm{\%}$) than from the θ-oscillon, (e.g., $D({\vartheta }_{\theta }^h,f_{{\theta }_2}^h)\approx 6.1\pm 2\mathrm{\%}$), which, in turn, is closer to the original LFP, $D({\ell }^h,{\vartheta }_{\theta }^h)=4\pm 1.4\mathrm{\%}$. All p < 10⁻⁹, CIs $95\mathrm{\%}$. C, A longer segment of a hippocampal-θ spectral wave. The nearly constant solid black trace in the middle shows the instantaneous Fourier θ-frequency. The dashed purple line shows the spectral wave’s moving average, which provides a lucid description of the θ-rhythm’s trend. The corresponding waveform, shown at the bottom is regular when the mean is steady (pink box), and corrugates when the mean is perturbed (blue boxes). D, The moving mean of the hippocampal and cortical θ-oscillons’ spectral waves (blue and red curves respectively) follow the speed’s time profile, s(t) (dashed gray). The latter is scaled vertically for illustrative purposes (see also Extended Data Fig. 3-1). E, The amplitudes of hippocampal θ-oscillon also co-vary with the speed—an effect captured previously via Fourier analyses (Young et al., 2021; Kennedy et al., 2022).

Figure 4.

Spectral waves and embedded frequencies. A, The spectral patterns produced via shifting-window evaluation of instantaneous frequencies are intermittent (Fig. 3A). To recapture the underlying continuous spectral dynamics, we interpolated the raw datapoints over a uniform time series, thus recovering the hippocampal (left) and the cortical (right) spectral waves with uninterrupted shapes. B, The contiguous data series allow constructing 3D W-spectrogram on which each peak along the frequency axis highlights the dynamics of a particular embedded frequency. Altitudinal shadowing emphasizes higher peaks (colorbar along the vertical axis). Note that most peaks in both hippocampal (left) and the cortical (right) W-spectrograms are localized not only in frequency but also in time, indicating short-lived spectral perturbations. For more examples, see Extended Data Figure 4-1. The dynamics of these frequencies is coupled with the speed—higher speeds drive up the magnitudes of the embedded frequencies. The speed profile is scaled vertically and shifted horizontally to best match the frequency magnitudes (orange and black trace, respectively). While the response of the hippocampal frequency to speed is nearly immediate (about τ^h = 90 ± 24 ms delay, p < 10⁻⁷), the cortical response is delayed by about two θ-periods (τ^c = 250 ± 50 ms, p < 10⁻⁷). C, Examples of the individual cortical peaks’ sampled magnitudes (heights of the dots on the panels B) and the corresponding speeds (heights of the crosses) exhibit clear quasi-linear dependencies. D, The net magnitude of the spectral wave co-varies with the speed in the both hippocampus (delay in this case τ^h = 92 ms, left) and in the cortex (delay τ^c = 289 ms, right).

Figure 5.

Spectral wave, noise and speed. A, Hippocampal spectral wave grows magnitude with speed (dashed black curve), which reflects the increasing level of synchronization (see below). Shaded area highlights a period of slow motion, during which the noise escalates. The original LFP amplitude is shown in the background (gray trace), for reference. B, The dynamics of the regular part of the LFP (red trace) and the noise component (dotted black trace), obtained for a 12-second lap. The original LFP is in the background (gray). C, The hippocampal (left panel) and cortical (right panel) noise levels follow speed, but more loosely than the oscillon’s amplitude.

Figure 6.

Kuramoto model. 1,000 oscillators (phasors) with base frequencies normally distributed around 8 Hz with the variance 1 Hz, coupled via Equation 7, produce a mean field characterized by a single spectral wave—a solitary Kuramoto oscillon (gray trace in the background, scaled up on the top panel 10 times relative to the other panels, for visibility). On all panels, the instantaneous amplitude is defined by the color scale, as on Figure 4A,B. A, At small couplings, K-oscillon has low amplitude and its spectral wave often reshapes and disrupts (blue boxes). The W-spectrogram (right panel) shows that the embedded frequencies restructure at ∼100 ms timescale. B, as the coupling between phasors grows, the synchronized amplitude builds up and the K-oscillon’s shape regularizes. Note that when the spectral wave flattens out, the oscillon is nearly sinusoidal (strong synchronization, red boxes), and the dynamics of the embedded frequencies during these periods are suppressed (right panel). C, At large couplings, synchronization dominates: the spectral wave narrows, the embedded frequencies die out and the oscillon reduces to a nearly sinusoidal harmonic. D, A hippocampal θ-oscillon’s spectral wave regularizes and the amplitude grows when the rat’s speed is steady (gray dashed line, shifted by ∼80 ms); desynchronization occurs when the speed is low or transient. E, The K-oscillon’s amplitude (orange curve), the magnitude of its spectral wave (purple), and the noise level, ξ (gray), for different coupling strengths. As the system synchronizes ( $1.7⪅\lambda ⪅3$), the amplitude grows, while the spectral undulations and the noise subside. At higher couplings noise is nearly fully suppressed.