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.
Significance Statement
Oscillatory extracellular fields modulate neural activity at multiple spatiotemporal scales and hence play major roles in physiology and cognition. Traditionally, these fields’ organization is described via harmonic decompositions into θ, γ and other “brain waves”. Here we argue that these constructs are only approximations to the physical oscillatory motifs–the oscillons, which represent the actual temporal architecture of synchronized neural dynamics. Focusing on the low-frequency θ-oscillons, we demonstrate correspondences with the traditional θ-waves for averaged, lento-changing characteristics, and discuss several new properties and dynamics that heretofore remained unexplored. Specifically, speed-coupled frequency modulations support oscillatory models of brain wave dynamics, suggesting a novel, “FM” perspective on the information exchange in hippocampo-cortical network and linking electrophysiological data to theoretical models of neuronal synchronization.
Introduction
Synchronized neural activity induces rhythmically oscillating electrical fields that modulate circuit dynamics at multiple spatiotemporal scales (Buzsáki, 2011; Buzsáki et al., 2012; Colgin, 2016). The corresponding local field potentials (LFPs) are easily recorded and widely used for describing varieties of neurophysiological phenomena (Kopell et al., 2010; Thut et al., 2012). However, our perspective on the LFP structure and properties, as well as our interpretation of its functions depend inherently on the techniques used in data analyses. Currently, most studies are based on Fourier paradigm, in which the oscillating LFP are viewed as a superposition of harmonics with constant frequencies (van Vugt et al., 2007; Brigham, 1988). The familiar θ, γ, and other “brain waves” are combinations of such harmonics, occupying a particular frequency band (Buzsáki, 2011; Colgin, 2016) (Fig. 1A).
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
Figure 1-1
Comparative waveforms of Fourier θ-wave (4 − 12 Hz) and θ-oscillon (top panel) and slow-γ wave, Fourier-filtered between 20 and 40 Hz, compared to slow-γ oscillon (bottom panel). Despite similar oscillation rates, the wave shapes are different. Download Figure 1-1, TIF file.
Despite their widespread use, Fourier decompositions are known to offer limited detail in capturing LFP dynamics, particularly for transient, irregular, or noisy data, and may therefore obscure the underlying brain wave structure. A recently proposed alternative, the Discrete Padé Transform (Throughout the text, terminological definitions are given in italics). (DPT), allows replacing the packets of Fourier harmonics with solitary waves that adapt their frequencies to the fields’ ongoing structure (Bessis, 1996; Bessis and Perotti, 2009; Perotti et al., 2013) (Fig. 1B). As it turns out, there are two kinds of such “flexible” frequencies: those that change over time in a regular manner, leaving distinct, contiguous traces on the spectrogram, and those that assume sporadic values from moment to moment. Mathematically, the “irregular” harmonics represent noise, ξ(t), while the “regular” frequencies define the oscillatory component of the signal (Perotti et al., 2019; Zobaer et al., 2022) (Fig. 1C).
At high temporal resolutions, the “timelines” of regular frequencies produce undulant patterns—the spectral waves (Fig. 1C). These waves vacillate around their respective lento-changing means, and span over the frequency domains attributed to the traditional, Fourier-defined rhythms. For example, the lowest-frequency spectral wave, prominent during active behavior, occupies the range that roughly corresponds to the traditional θ-domain (Jung and Kornmüller, 1938; Buzsáki, 2002, 2005; Burgess and O’Keefe, 2005; Colgin, 2013). The following spectral wave occupies the slow-γ domain (Colgin, 2015), and so forth (Fig. 1C). The gap between the mean frequencies is typically larger than the undulations’ magnitudes, which allows keeping the standard nomenclature, e.g., using the notation
Importantly, all oscillon parameters are obtained empirically from the recorded data: the amplitudes, the shapes of the spectral waves and the noise exhibit robust, tractable characteristics that reflect physical organization of the synchronized neuronal activity (Restrepo et al., 2005; Arenas et al., 2008; Liao et al., 2011; Burton et al., 2012; Mi et al., 2013). In other words, the elements of the decomposition 2 can be interpreted not only as structural motifs, but also as functional units within the LFP waves. This raises many questions: what are the physiological properties of the oscillons? Which of their features were previously captured through Fourier analyses, and how accurately? Do the oscillons reveal any new operational aspects of the brain waves?
In the following, we focus on the hippocampal and the cortical θ-oscillons recorded in male rats, shuttling between two food wells on a linear track. The experimental specifics are briefly outlined in Section 2, while computational details are provided in Bessis (1996), Bessis and Perotti (2009), Perotti et al. (2013, 2019), and Zobaer et al. (2022). In Section 3, we demonstrate that some qualities of the θ-oscillons parallel those of the traditional θ-waves; yet many of their characteristics have not been addressed by conventional analyses. In Section 4, we utilize a basic model of neuronal synchronization to illustrate and contextualize our empirical observations. Finally, in Section 5, we highlight a new perspective on information exchange within the hippocampo-cortical network, as suggested by the newfound architecture of θ-rhythmicity.
Methods
Experimental procedures
Data were recorded from the CA1 pyramidal cell layer of male Long-Evans rats’ hippocampus and from anterior cingulate cortex (Domenico et al., 2021). The animals were trained in a familiar room to run back and forth on a rectangular, 3.5-meters-long, track for food reward (Fig. 2A). The daily recording procedure consisted of two 30 min running sessions. A typical run between the food wells takes about 30 s. Uninterrupted fast moves typically last less than 5 s, followed by slowdowns at the corners and track ends. Two diodes were mounted over the animal’s head to track its positions, at 33 Hz rate, with a spatial resolution of about 2 mm. Further details on surgery, tetrode recordings and other procedures can be found in Domenico et al. (2021). The data analyzed here are not included in Domenico et al. (2021), but experimental details are identical.
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
Figure 2-1
Hilbert transform. A. Fourier-defined θ-wave (gray waveform in the background) shown with its amplitude (dotted line on top) and the corresponding instantaneous frequency (black line, placed according to the right scale) produced by Hilbert transform. Pink waveform in the foreground shows θ-oscillon and its amplitude computed from the net contribution of stable poles contributing to spectral wave (dotted brown line). B. Fourier-defined slow-γ wave (20 − 40 Hz), compared to the slow-γ oscillon. Amplitudes and frequencies are as above. The amplitude of the latter is lower because it does not include the noise component. Download Figure 2-1, TIF file.
Signal processing
The original LFP data were sampled at Sr = 8 kHz rate, and interpolated to
Since the poles are computed independently at each time step, based on a finite number of data points, the patterns of reconstructed frequencies contain gaps and irregularities. To capture the underlying continuous physical processes, we interpolated the “raw” spectral traces over uniformly spaced time points and used Welch transform to extract the embedded frequencies.
Hilbert transform is used to evaluate the imaginary part of a signal, and with it the instantaneous phases, frequencies and amplitudes (Fig. 2B). The resulting instantaneous frequency corresponds to DPT’s mean frequency, but its dynamic is much less detailed, and the embedded (modulating) frequencies are not resolved at all. In DPT, the oscillons’ net amplitude is obtained as the net sum of amplitudes carried by stable frequencies. The oscillons’ mean frequency is obtained as their spectral waves’ moving mean, over periods exceeding the characteristic undulation span.
Welch transform was designed for estimating power spectra in transient signals (Welch, 1967). The signal is first split into a large number of highly overlapping shorter segments, and then the power spectrum of each segment is evaluated independently. The power profile of each segment thus marks the most prominent frequencies appearing over the corresponding time interval. Arranging such power profiles next to each other in natural order, one gets three-dimensional (3D) W-spectrograms illustrated on Figure 2C.
By construction, the lateral sections of W-spectrograms are the instantaneous power profiles, whereas the longitudinal sections show the peaks’ dynamics, used to trace the evolution of the embedded frequencies. For an illustration, a W-spectrogram built for the instantaneous Fourier frequency, captures major frequency disturbances, but does not resolve rapid frequency modulations (Fig. 2D).
Dynamic time warping (DTW) is used to quantify similarity between two series of features, e.g., for recognizing similar speech patterns. The method is based on applying series of stretches that maximize alignment between two profiles at minimal cost, without omitting elements or scrambling their order (Sakoe and Chiba, 1978; Berndt and Clifford, 1994; Salvador and Chan, 2007). DTW is particularly suitable for comparing dynamic waveforms, whose parts may alternately lag and outpace one another, such as spectral waves (Berndt and Clifford, 1994; Salvador and Chan, 2007; Neamtu et al., 2018). Analyses performed using MATLAB, including its dtw and alignsignals functions (Fig. 2E,F).
Results
Shape of the θ-oscillons. Similarly to the traditional hippocampal and cortical θ-waves, denoted below as
To put these observations into a perspective, note that there exist several consensual θ-frequency bands, e.g., from 4 to 12 Hz, from 5 to 10 Hz, from 6 to 10 Hz, from 6 to 12 Hz, from 5 to 15 Hz, etc. Brandon et al. (2013), Huxter et al. (2008), Mizuseki et al. (2009), Harris et al. (2002), Jezek et al. (2011), Kropff et al. (2021), Richard et al. (2013), Young et al. (2021), Chen et al. (2011), Ahmed and Mehta (2012), Zheng et al. (2015), and Kennedy et al. (2022). The shapes of the corresponding Fourier θ-waves differ by about
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
Figure 3-1
Speed vs. mean θ-frequency coupling. A. Additional examples demonstrating covariance between the moving mean of the hippocampal (blue) and the cortical (red) θ-frequency with the rat’s speed (dashed brown curve). The latter is scaled vertically and shifted as on Fig. 4, to match the frequency ranges. The instantaneous frequency of the traditional, Fourier-defined θ-waves is shown by solid black curve, as on Fig. 3. A five-fold (top panel) and ten-fold (bottom panel) vertical stretch of the Fourier-frequency produces the dotted black curve, whose similarity to the spectral waves’ means explains the general correspondence between our results and conventional evaluations of speed-frequency couplings. B. Mean frequency profile of a hippocampal θ-oscillon, left-shifted by 300 ms (right scale), vs. the mean frequency of a hippocampal slow-γ oscillon, dark lilac, right-shifted by 500 ms (left scale), shown with the rat’s speed profile. Download Figure 3-1, TIF file.
Overall, θ-oscillons are similar to their Fourier counterparts: the DTW-difference between their waveforms during active moves are small,
Coupling to locomotion. The θ-oscillons’ amplitudes and their mean frequencies are coupled to speed, as one would expect based on a host of the previous studies of θ-rhythmicity (Fig. 3C, Richard et al., 2013; Kropff et al., 2021; Young et al., 2021; Kennedy et al., 2022). Nevertheless, this observation is informative, since oscillons are qualitatively different constructions. While the traditional brain waves’ frequency is evaluated by tracking their Fourier-envelope, the oscillons’ mean frequency is obtained as their spectral waves’ moving mean, over periods comparable with largest undulation span (∼200 ms). Yet, the two outcomes are consistent, although the oscillon provides a more nuanced description of the trend dynamics: the mean θ-oscillon frequencies in both the hippocampus and cortex,
Spectral undulations are the oscillons’ distinguishing feature that captures subtle details of the θ-rhythm’s cadence (Fig. 3C). The semi-periodic appearance of the spectral waves suggests that their ups and downs should be decomposable into a harmonics series,
To test these possibilities, we interpolated the “raw” frequency patterns over uniformly spaced time points and obtained contiguous spectral waves over about 5.5 s–the maximal time that the rat can run uninterruptedly over the straight segment of the track (Fig. 4A). This wave was then split into 600 ms long, strongly overlapping (by
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 τh = 90 ± 24 ms delay, p < 10−7), the cortical response is delayed by about two θ-periods (τc = 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 τh = 92 ms, left) and in the cortex (delay τc = 289 ms, right).
Figure 4-1
Additional examples of hippocampal (left column) and cortical (right column) W-spectrograms, illustrating the embedded frequency dynamics for the θ-oscillons. Dark red arrows point at the appearances of isolated peaks and the black arrows point at the “seedbeds” of peaks recurring at the same frequency. Download Figure 4-1, TIF file.
The results demonstrate that both hippocampal and cortical spectral waves are highly dynamic and complex. First, most peaks are localized not only in frequency but also in time: a typical peak grows and wanes off in about 200–300 ms, i.e., the embedded frequencies and the depth of modulation are highly transitive, changing faster than the mean frequencies by an order of magnitude (Zobaer et al., 2022). In other words, the representation 3 holds over relatively short periods (typically 1 s or less Perotti et al., 2019), and then requires corrections in order to account for rapidly accumulating changes. Thus, θ-oscillons may be viewed as steady oscillatory processes with a cycling frequency that drifts on the behavioral timescale around an average of νθ,0 ≈ 8 Hz and is modulated by a series of swift, transient vibrations.
Curiously, certain peaks in the W-spectrograms appear and disappear repeatedly near the same location along the Ω-axis, i.e., fast moves can consistently incite θ-vibrations at the same embedded frequencies, indicating restorative network dynamics (Goutagny et al., 2009). Other peaks appear sporadically, possibly reflecting spontaneously generated oscillations, resonances or brief external contributions (Fig. 4B, Zobaer et al., 2022). Furthermore, these events are coupled with the rat’s ongoing behavior: as the speed increases, the power flows into higher embedded frequencies (Ωis over 5 Hz) and then recedes as the speed drops, i.e., fast moves appear to drive spectral undulations both in the hippocampus and in the cortex.
To evaluate this effect, we identified the lateral sections of the W-spectrogram, w(t), that best aligned with the speed profile, s(t)—that is, those with the minimal DTW-difference, D(w, s)—and then determined the time shift, τ, required for optimal alignment. The results indicate that cortical responses tend to delay by about
Another surprising observation is that the peak magnitudes associated with the individual embedded spectral frequencies, Ωi (the coefficients of the oscillatory terms in the Eq. 2), tend to grow roughly proportionally to speed,
On the one hand, the coupling 4 between the embedded frequency magnitudes and speed is foreseeable: linear modulation of extracellular fields’ base frequency by speed,
However, there are a few empirical distinctions between the physical speed-controlled oscillators and the idealized models (O’Keefe and Recce, 1993; Burgess et al., 2007; Hasselmo et al., 2007; Burgess, 2008; Burgess and O’Keefe, 2011). First, the actual coupling may deviate from strict linearity. Second, the frequency expansion may contain several oscillatory, speed-modulated terms similar to 3 or be overtly nonlinear, whereas the model expansion 5 contains only one linear term. Third, this dependence is highly dynamic, not steady. Four, the modulation parameters are particularized, i.e., peak- and time-specific, albeit they may be distributed around well-defined means. Also note that the W-spectrograms of the instantaneous Fourier frequencies do not capture these structures–they average the spectral dynamics out (Fig. 2D).
Noise. As mentioned in the Introduction, the qualitative difference between the regular and the irregular frequencies allows delineating the LFP’s noise component. While in most empirical studies “noise” is identified ad hoc, as a cumulation of irregular fluctuations or unpredictable interferences within the signal (Stein et al., 2005; Rowe et al., 2007; Ermentrout et al., 2008; Faisal et al., 2008; McDonnell and Ward, 2011), here noise is defined conceptually, based on intrinsic properties of Padé approximants to the signal’s z-transform that are mathematically tied to stochasticity (Bessis, 1996; Bessis and Perotti, 2009; Perotti et al., 2013). Being that noise is qualitatively distinct from oscillatory dynamics, its properties provide an independent, complementary description of the network’s state.
As indicated on Figure 1C, in a typical LFP, only a few frequencies exhibit regular behavior, and yet their combined contribution is dominant: the stochastic component, ξ(t), usually accounts for less than
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.
Kuramoto Oscillon
What is the genesis of the oscillons? It is commonly believed that the rhythmic LFP oscillations emerge from spontaneous synchronization of neuronal activity, although the specifics of these processes remain unknown (Restrepo et al., 2005; Arenas et al., 2008; Liao et al., 2011; Burton et al., 2012; Mi et al., 2013). A comprehensive analysis of the recorded data or modeling at the physiological level of detail is prohibitively complex and technically out of reach. Nevertheless, the essence of synchronization can be illustrated with simple computational models, which helps clarifying the observed phenomena. Specifically, the celebrated Kuramoto model allows tracing the onset of synchronization in a population of oscillating units, using a single parameter that describes the coupling strength (Kuramoto, 1975; Strogatz, 2000). Under fairly generic assumptions, these oscillators, or phasors, can qualitatively represent (map to) the recurring activity of individual neurons (Izhikevich, 1999a, 1999b; Hoppensteadt and Izhikevich, 1997, 1998, 1999), so that their net population dynamics captures the ebb and flow of the mean extracellular field.
The behavior of each unit is described by a time-dependent phase, ϕm(t), that changes between 0 and 2π, inducing an oscillatory output,
From perspective of this discussion, this phenomenon is of interest because it yields a solitary “synthetic” oscillon, that helps illustrating properties of the physiological oscillons. As shown on Figure 6A, if the phasors’ proper frequencies are distributed closely (within about ±3 Hz) to the mean frequency of νθ,0 ≈ 8 Hz, then the net field’s dynamics is characterized by a single spectral wave that changes its properties according to the synchronization level. For weak couplings (small λs), synchronization is fragmentary: segments with steady frequency extend over a few oscillations, outside of which the spectral wave has a large magnitude (note that the oscillon’s amplitude, AK, remains low) and is carried by many embedded frequencies, as indicated by abundance of transient peaks on W-spectrograms (Fig. 6A, right panel). As λ grows, the net amplitude, AK, increases and the segments of synchronicity lengthen, while the embedded oscillations subdue (Fig. 6B). As λ grows further, the embedded frequencies reduce in number and loose magnitudes, notably over the periods of increased synchronicity. As λ gets even higher, the oscillon turns into a simple harmonic, and its spectral wave degenerates into a line (Fig. 6C). Ultimately, spectral undulations get suppressed, as synchronization becomes fully dominant.
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 (
These observations suggest that the spectral wave’s width (frequency deviation from the mean, Δθ (t) = νθ(t) − νθ,0(t)–modulation depth), may serve as an indicator of the ongoing synchronization level. From this perspective, the fact that both hippocampal and cortical θ-spectral waves are generally wide (±4 Hz, see Figs. 3A,C, 4A), implies that the physiological synchronization level is fairly low.
Also note that Kuramoto oscillons produced by weakly coupled ensembles often exhibit brief periods of regularity, with reduced spectral undulations, usually accompanied by higher amplitudes (Fig. 6A,B). Similar effects are observed in the empirical, physiological oscillons, where higher-synchronicity episodes, lasting about 50–70 ms in cortex and 100–200 ms in the hippocampus, p < 10−4, CI
Surprisingly, the Kuramoto model, being fully deterministic, also produces a noise component. At weak couplings, about
Discussion
The intricate structure of the synchronized extracellular fields can be anatomized using different decomposition techniques. The constituents brought forth by a particular decomposition provide a specific semantics for reasoning about the LFP functions. Being that these semantics may differ substantially, one may inquire which approach better reflects the physical structure of the brain rhythms. The oscillatory nature of LFPs suggests partitioning the signal into Fourier harmonics—an approach that dominates the field since the discovery of the brain waves (Buzsáki, 2011; Buzsáki et al., 2012; Colgin, 2016). However, its is also known that the Fourier techniques obscure the structure of noisy and nonstationary data—precisely the kind of signals that are relevant in biology (Grünbaum, 2003). In particular, these considerations apply to the LFPs, since their constituents—the oscillons—have transient structures enveloped by noise, i.e., are by nature noisy and nonstationary (Perotti et al., 2019; Zobaer et al., 2022).
Since the oscillons are constructed empirically, using a high time-frequency resolution technique, and exhibit stable, reproducible features that dovetail with theoretical models of synchronization, they likely capture the physical architecture of the extracellular fields, whereas the traditional, Fourier-defined brain waves provide approximative descriptions. Thus far, oscillons were observed in rodents’ hippocampal and the cortical LFPs, but similar structure should be expected in generic brain rhythms. Their systematic analyses should help linking electrophysiological data to the synchronization mechanisms and reveal the dynamics of the noise component.
Lastly, the oscillons suggest a fresh vantage point for understanding the principles of information transfer in the hippocampo-cortical network. Traditionally, the coupling of LFP rhythms to neuronal activity is traced through modulations of the brain waves’ amplitudes and phases (Chen et al., 2011; Ahmed and Mehta, 2012; Richard et al., 2013; Zheng et al., 2015; Kropff et al., 2021; Young et al., 2021; Kennedy et al., 2022). In contrast, frequency-modulated (FM) oscillons imply a complementary format, in which a slow-changing mean frequency defines the “channel of communication,” over which the information is carried by the rapid phase and frequency alterations, reflecting fast endogenous dynamics and abrupt external inputs (Ziemer and Tranter, 2010; Zobaer et al., 2022). In other words, Fourier analyses emphasize amplitude modulation (AM), while the DPT decomposition highlights the FM principles of information transfer, carried over several discrete channels (Izhikevich, 1999a, 1999b; Hoppensteadt and Izhikevich, 1997, 1998, 1999). In the specific case of θ-rhythms, the AM-format is important for slower, larger-scale phenomena, e.g., for the couplings between the speed with the steadily changing mean frequencies, amplitudes, narrowing and widening of the θ-band, etc., whereas information about rapid activities may transmit across the hippocampo-cortical network via alterations of the embedded θ-frequencies. The nature of the information carried by frequency-modulated oscillons and the mechanisms by which they couple with neuronal activity remain poorly understood; however, tracking these dynamics could represent a significant direction for future research.
Footnotes
The work was supported by NIH grants R01NS110806 (MSZ and YD), R01AG074226 (NL and YD), R01DA054977 (CMD and DJ) and NSF grant 1422438 (CH and YD).
The authors declare no competing financial interests.
- Correspondence should be addressed to Yuri Dabaghian at yuri.a.dabaghian{at}uth.tmc.edu.
