Article
https://doi.org/10.1038/s41467-023-36025-x
Intrinsic macroscale oscillatory modes driving long range functional connectivity in
female rat brains detected by ultrafast fMRI
Received: 14 July 2021
Joana Cabral
1,2,3
, Francisca F. Fernandes
1
& Noam Shemesh
1
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Accepted: 12 January 2023
Spontaneous fluctuations in functional magnetic resonance imaging (fMRI)
signals correlate across distant brain areas, shaping functionally relevant
intrinsic networks. However, the generative mechanism of fMRI signal correlations, and in particular the link with locally-detected ultra-slow oscillations,
are not fully understood. To investigate this link, we record ultrafast ultrahigh
field fMRI signals (9.4 Tesla, temporal resolution = 38 milliseconds) from
female rats across three anesthesia conditions. Power at frequencies extending
up to 0.3 Hz is detected consistently across rat brains and is modulated by
anesthesia level. Principal component analysis reveals a repertoire of modes, in
which transient oscillations organize with fixed phase relationships across
distinct cortical and subcortical structures. Oscillatory modes are found to
vary between conditions, resonating at faster frequencies under medetomidine sedation and reducing both in number, frequency, and duration with the
addition of isoflurane. Peaking in power within clear anatomical boundaries,
these oscillatory modes point to an emergent systemic property. This work
provides additional insight into the origin of oscillations detected in fMRI and
the organizing principles underpinning spontaneous long-range functional
connectivity.
Spontaneous fluctuations in signals detected with functional Magnetic
Resonance Imaging (fMRI) correlate across spatially distributed brain
areas forming functional networks that appear disrupted in numerous
psychiatric and neurological disorders, pointing to a key role in brain
function1–5. However, the organizing principle driving long-range correlations between brain areas remains unclear.
A wide range of low-rank decomposition techniques have been
put forward to characterize the spatial organization of spontaneous
fMRI signal fluctuations, including among others independent component analysis6,7, co-activation patterns8–11, low-dimensional
gradients12,13, leading eigenvector dynamics analysis14,15, dynamic
mode decomposition16 and quasi-periodic patterns (QPPs)17,18. Despite
the differences inherent to each technique, most methods converge in
a discrete repertoire of intrinsic modes exhibiting features of
stationary wave patterns, where correlated activity is detected among
spatially distributed regions (or poles), with gradually varying phase
relationships across space19. These intrinsic modes have been shown to
emerge transiently and recurrently during rest20,21, to be selectively
recruited during specific tasks7, and to replicate across mammals10,22–25.
In the frequency domain, correlated fluctuations in fMRI signals
exhibit power at ultra-slow frequencies, peaking typically below 0.1 Hz
in human brains at rest, although intrinsic functional networks have
been detected at frequencies extending even beyond 0.5 Hz26–30. Crucially, it remains unclear whether the spectral power at low frequencies
is associated solely with aperiodic activations of the characteristically
slow and region-specific hemodynamic response function or additionally reflects the existence of damped oscillatory components16,30–32.
Mainly detected with electro- and magnetoencephalography (EEG/
1
Preclinical MRI Lab, Champalimaud Research, Champalimaud Foundation, Lisbon, Portugal. 2Life and Health Sciences Research Institute, School of
Medicine, University of Minho, Braga, Portugal. 3ICVS/3B’s - Portuguese Government Associate Laboratory, Braga/Guimarães, Portugal.
e-mail: joanacabral@med.uminho.pt; noam.shemesh@neuro.fchampalimaud.org
Nature Communications | (2023)14:375
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MEG), macroscale oscillatory components in brain activity have been
targeted by neural field theories, demonstrating how the frequency
spectrum and correlation structure can be predicted from brain
geometry33–35. Although intrinsic modes detected with fMRI have been
shown to spatially align with eigenmodes of brain structure (either
from surface geometry or diffusion networks), theoretical predictions
of mode-specific temporal responses remain to be adequately
addressed in fMRI36–39. Given recent insights demonstrating that the
fMRI signals underpinning intrinsic networks relate to macroscopic
waves of propagating activity16,40–42, it is crucial to obtain a detailed
characterization of the modes’ spatial and temporal signatures to
empirically investigate their oscillatory nature and their link with
neural field theories.
Studies in rodents and humans have shown that some ultra-slow
frequency components in fMRI signals have a periodic nature and are
coupled with electrophysiological and electroencephalographic (EEG)
signals43–46. These periodic fluctuations have been proposed to be
linked to arteriole vibrations entrained by fast oscillations in local field
potentials, pointing to a potentially more direct relationship with the
underlying neural activity47,48. Still, how these oscillations organize at
the macroscopic level and their relationship to functional connectivity
between brain areas remains unclear.
Intrinsic networks analogous to the ones identified in humans have
been identified in rats and are modulated by the sedation/anesthesia
state10,49–52. In particular, sedation with low doses of medetomidine has
been shown to reveal consistent intrinsic networks but also to drive
abnormal high amplitude oscillations in fMRI signals at frequencies
extending beyond 0.1 Hz44,50. The addition of isoflurane at low concentrations suppresses these high-frequency oscillations while maintaining the typical human resting-state frequencies <0.1 Hz, such that the
combination medetomidine/isoflurane (MED/ISO) is currently the stateof-the-art protocol to approximate resting-state brain activity in rats53,54.
The solid evidence from rat experiments across sedation/anesthesia protocols offers an ideal setting to analyze the spatiotemporal
organization of fMRI signal oscillations and their relationship with
intrinsic network patterns54–56. However, the precise characterization
of oscillations across space and over time is complex and benefits from
an adequate spatiotemporal resolution and high signal-to-noise (SNR)
ratio to adequately capture transient phase relationships between
voxels. Ultra-high field fMRI studies in rats achieve increased SNR by
attenuating thermal noise using cryogenic coils57,58. Moreover, for
increased precision in the characterization of oscillatory signals, long
scanning times are needed to ensure high-frequency specificity at slow
frequencies, and fast sampling helps prevent frequency aliasing from
undersampled periodic components of physiological and/or scanner
artifacts. At the spatial level, a large field of view is necessary to capture
macroscale organization, while ensuring a sufficient spatial resolution
to resolve distinct brain regions.
Therefore, we harness an ultrafast ultrahigh field fMRI approach,
with long scan durations of 10 min sampled at 38 ms resolution
(16,000 frames per scan) to characterize the spatial organization of
oscillations detected in fMRI signals in a single slice of the rat brain,
achieving high SNR ratio via a 9.4 T magnetic field and a cryogenic coil.
This approach exposes unreported features of rat brain activity, providing insights into the fundamental organizing principles driving
long-range functional connectivity in the brain.
Results
Long-range functional connectivity
A typical seed-based functional connectivity analysis was performed to
confirm the detection of long-range correlations in fMRI signals in the
range of frequencies typically considered in resting-state studies, i.e.,
ranging between 0.01 and 0.1 Hz Fig. 1a, b. In panel c, we plot the bandpass filtered fMRI signals in the same 3 seeds together with their
contralateral voxels (cf. Supplementary Fig. S1 for the corresponding
Nature Communications | (2023)14:375
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brain atlas). When applying sliding-window correlation (SWC) analysis,
fluctuations were detected between sustained periods of positive
correlation (yellow shades) and periods of weak or even negative
correlation (orange to magenta shades). The same analysis was applied
to a scan from a post-mortem rat to ensure that the periods of sustained correlations are detected at levels beyond any conceivable
artifacts (Supplementary Fig. S2).
Space-frequency analysis of fMRI signals across conditions
To investigate whether the transient correlations are associated with
oscillatory phenomena, we turn to analyze the spectrum of frequencies detected in brain voxels across three different sedation/
anesthesia protocols: under medetomidine only (which we term
sedation), after the addition of isoflurane at 1% concentration (light
anesthesia), and after increasing isoflurane concentration to 3% (deep
anesthesia) (see the “Methods” section for details). Applying a spacefrequency analysis on 36 ultrafast fMRI scans (12 per condition, each
10 min long at 26.3 Hz sampling rate), the 3 anesthesia conditions are
compared in terms of average power at different frequency bands with
respect to a baseline defined from postmortem scans (Fig. 2). Power at
frequencies up to 0.30 Hz—extending well beyond the range typically
considered in resting-state studies—is detected in the brains of sedated
and lightly anesthetized rats significantly above deep anesthesia levels
(Fig. 2b, c and Supplementary Fig. S3).
Power in fMRI signals is found to peak within well-defined cortical
boundaries consistently between 0.20 and 0.25 Hz in rats sedated with
medetomidine (Fig. 2). Voxels in the striatum (subcortical) exhibited
power at frequencies peaking between 0.1 and 0.15 Hz. The addition of
isoflurane at 1% specifically affects the power of fMRI signal fluctuations between 0.15 and 0.25 Hz, whereas isoflurane at 3% significantly
decreases the power in the broad frequency range between 0.05 and
0.3 Hz (Fig. 2b, c and Supplementary Figs. S3–S6). Above 0.40 Hz, only
signatures associated with cardiac and respiratory frequencies are
detected (Supplementary Fig. S7 and Table S1). Ultra-slow fluctuations
below 0.05 Hz, i.e., with a period longer than 20 s, persist even in
deeply anesthetized animals. This change in frequency across conditions is clearly visible in the carpet plots in Supplementary Figs. S6–S9.
Spatial, temporal, and spectral properties of principal
components
While the space-frequency analysis provides information about which
voxels have more power in each frequency band, it does not reveal how
the signals evolve in time or organize in space. As can be seen in Supplementary Movies 1 and 2, fluctuations are not globally correlated but
instead exhibit complex phase relationships across space that appear
recurrent over time and consistent across different rats in the same
condition. Signals co-varying in phase across distant voxels symmetrically aligned with respect to the vertical midline point to a link with
long-range functional connectivity. In deep anesthesia, despite applying the same filtering, no particular spatial organization or fine structure is detected except for ultra-slow globally correlated fluctuations.
To detect whether the fluctuations have a characteristic spatial
organization, we extract the principal components of the fMRI signals
filtered below 0.3 Hz in each condition (e.g., the eigenvectors of the
covariance matrix, see Fig. 3a–c and see the “Methods” section for
details). Principal component analysis has the advantage of returning
orthogonal modes of covariance without making any assumption
regarding the oscillatory properties of the components, unlike other
decomposition techniques that a priori assume an oscillatory nature of
the components, such as dynamic decomposition analysis16,59. The
spatial patterns associated with the 10 principal components detected
above the postmortem baseline reveal spatially fixed phase relationships between the fMRI signals in distinct brain subsystems (Fig. 3d).
These phase relationships varying gradually and symmetrically across
space exhibit characteristics of standing waves, where regions of
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Fig. 1 | Static and dynamic resting-state functional connectivity analysis in
band-pass filtered fMRI signals. a Correlation matrix of the fMRI signals in all
voxels within the brain mask, bandpass filtered in a range typically considered in
resting-state studies, i.e., 0.01–0.1 Hz (no nuisance regressor nor spatial smoothing
applied). Each line/column in the matrix corresponds to the correlation map of
each voxel. b Seed-based correlation maps are represented for three different
seeds (white asterisks), where each voxel is colored according to its degree of
correlation with the seed. A voxel contralateral to each seed is represented by a
black circle. All color bars are truncated between −0.8 and 0.8. c Filtered fMRI
signals were recorded in each seed (red) and corresponding contralateral voxel
(black). Colored shades represent the sliding window correlation (SWC) using a 30s window, showing that the correlation is not constant but fluctuates between
transients of long-range phase locking. The same figure obtained from a postmortem scan is reported in Supplementary Fig. S2.
strong amplitude (red/blue) represent the anti-nodes of the wave,
whereas regions of low amplitude between anti-nodes (green) represent the wave’s nodes (points of no motion).
To investigate the dynamic behavior of the principal components,
we subsequently analyze the temporal signatures associated with each
spatial pattern. In particular, we aim to verify the existence of periodicity between positive and negative representations of the spatial
patterns (Fig. 3d), which is not necessarily a property of principal
components, given that signals can co-vary aperiodically. The temporal signature τ Sα ðtÞ of each principal component α for each scan S is
obtained by performing a matrix multiplication that contracts the ‘n’
dimension as
essential macroscopic dynamics of the recorded fMRI signals in a rat
brain are captured using a low-rank approximation considering only
the reduced common basis of 10 principal components ψα ðnÞ detected
across sedated rats, and the 10 scan-specific temporal signatures τ Sα ðtÞ
associated with these components. As observed in Fig. 4 and
Supplementary Movie 3 (still frame reported in Fig. 4), the principal
components are found to oscillate around the mean with slowly
fluctuating amplitudes, generating patterns akin to those of transiently
resonating stationary waves.
The detection of oscillations associated with the spatial patterns benefited from the fast sampling combined with long scan
durations (totaling 16,000 images per 10-min scan), by preventing
frequency aliasing from physiological rhythms (i.e., with Nyquist
frequency above breathing and cardiac frequencies) and by ensuring sufficient resolution in the power spectrum at low frequencies,
i.e., with precision below 0.01 Hz (see Supplementary Figs. S10, S11).
As shown in Fig. 5 (top row), when projecting the 1 × N spatial
component (here ψα = 7 ) on the N × T ultrafast unfiltered fMRI signals, the temporal signature τ Sα = 7 exhibits clearly visible oscillations
between positive and negative representations of the spatial pattern. As the sampling factor is increased (Fig. 5 bottom rows), even if
a resonant peak frequency can still be detected, the signal to noise
ratio is decreased. Indeed, we find that the principal component
ψα = 7 fails to be detected with a sampling as fast as 380 ms, which
coincides with the sampling rate at which the breathing frequency
τ Sα ðtÞ = ψα ðnÞ Ψ S ðn, t Þ,
ð1Þ
where ψα ðnÞ represents the spatial pattern of each principal component α and Ψ S ðn, t Þ represents the activity recorded with fMRI across
all voxels n and timepoints t for scan S (Fig. 3e).
The reconstruction
Ψ R ðn, t Þ =
X
α
ψα ðnÞτ Sα ðtÞ
ð2Þ
describes the linear superposition of a basis of wave patterns locked in
space ψα ðnÞ and evolving in time τ Sα ðtÞ. Video 2 illustrates how the
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Fig. 2 | Spectral power of fMRI signals differs significantly between anesthesia
conditions. a Spatial maps of spectral power in 8 non-overlapping frequency
bands, averaged across the s = 12 fMRI scans recorded under each anesthesia
conditions and normalized by the mean power in sPM = 2 postmortem scans. fMRI
recordings were recorded from n = 6 genetically similar female rats, each scanned
twice in 3 anesthesia conditions (with isoflurane at 0%, 1%, and 3%. b Power in 8
frequency bands averaged across all brain voxels relative to the mean power in 2
postmortem scans. Dots correspond to the mean power in each scan and error bars
represent the mean ± standard error across the s = 12 scans. c Two-sided p-values
obtained from 10,000 t-tests on randomly permuted data comparing the power in
s = 12 scans over the 3 anesthesia conditions and in each of 8 non-overlapping
frequency bands, reported with respect to the standard threshold of 0.05 and the
Bonferroni-corrected threshold of 0.0021. RS resting-state. p-values are reported
in Supplementary Fig. 3. Source data are provided as a Source Data file and Source
Codes are provided in Supplementary Material.
(~2 Hz) cannot be adequately resolved given the Nyquist theorem
(analysis shown in Supplementary Figs. S12–S16). In Supplementary
Fig. S15 we reorder the spatial patterns either using a spatial randomization or sorting according to a left-right gradient, and
demonstrate that the temporal signatures exhibit lower amplitudes
and less spectral power when the spatial organization of the phases
in wave patterns is disrupted.
With the addition of isoflurane at 1% concentration (Fig. 6 top), the
repertoire of principal components is modified, not only in number—
with only 6 components detected above the postmortem baseline—
but also in terms of spatial configuration, with different brain subsystems oscillating in phase or anti-phase with each other. Regarding
the temporal signatures associated with the different components,
although some transient oscillations can still be detected, these last
visibly shorter and with different periodicity over time (Fig. 6e), which
is reflected in a broader distribution of power across the spectrum,
peaking at lower frequencies with respect to the sedation condition (Fig. 6f).
When the concentration of isoflurane is further increased to 3%
(Fig. 6 bottom), the variance above postmortem baseline is explained
by a single principal component where the cortex and striatum oscillate together in phase (Fig. 6d’) and at very low frequencies (Fig. 6f’).
These ultraslow global fluctuations are particularly visible in the carpet
plot in Fig. 6b’.
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Fig. 3 | Spatial, temporal, and spectral signatures of the principal components
in medetomidine-sedated rats. a The NxN covariance matrix of fMRI signals (filtered within the range where significant spectral power was detected in the cortex,
i.e., between 0.01 and 0.3 Hz) averaged across the 12 sedated rat scans. b Carpet
plot of the fMRI signals in all brain voxels, n, over time, t, represented by the wave
function ΨS ðn, tÞ, here shown for a representative scan S of a sedated rat in the
frequency range [0.01–0.5 Hz]. Voxels are sorted according to the elements in the
largest magnitude eigenvector ψ1 . Values correspond to fMRI signal change with
respect to the mean in each voxel. A zoom into the first 100 voxels over 60 s is
inserted to illustrate oscillations in the signals. c Power spectrum of the mean fMRI
signal across voxels for: (black) the scan shown in b and (red) a scan performed
postmortem (PM). d The 10 principal components ψα obtained as the eigenvectors
from (a) with eigenvalue λα above PM baseline are scaled by 1 (left) and −1 (right) to
illustrate the activity pattern when the temporal signature oscillates between
positive and negative values. e Temporal signature associated with each of the 10
principal components given by τSα ðtÞ = ψα ðnÞΨs ðn, t Þ for the same scan shown in (b).
Clear oscillations with fluctuating amplitude can be observed. f Power spectra of
the temporal signatures from e (blue) and in a postmortem scan (red). See Supplementary Movie 2 to observe the behavior of each principal component
over time.
The oscillatory nature of principal components
sedation and light anesthesia with respect to deep anesthesia (Bonferroni-corrected p-values reported in Supplementary Table S2).
The autocorrelation functions of the wave temporal signatures
(here shown for τ S1 ðtÞ in each condition) illustrate that the number of
sustained cycles before the amplitude decays to 1/e decreases with
increasing levels of isoflurane, as estimated by the Q-factor (Fig. 7c).
We use the Hilbert transform to obtain a representation of the autocorrelation functions in the complex domain (with real and imaginary
components) and plot the corresponding phase portraits (Fig. 7c
bottom). The representation of the phase portraits serves to classify
the temporal signatures of the components within the framework of
While in Figs. 3 and 6 we show the results from one representative
animal, in Fig. 7 we report the peak frequency and stability of the
oscillations associated with each principal component in each of the
36 scans, i.e., of the 6 rats scanned twice in each condition. The stability
of the oscillations is assessed from the resonance Q-factor, which is
proportional to the number of cycles before the amplitude decays to
~37% (e−1) of its initial value, consisting of the ratio between the peak
frequency and the power spectrum’s full-width-at-half-maximum
(FWHM). Both the peak frequencies and the Q-factors were found to be
significantly higher (and with larger variability across scans) in
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Fig. 4 | Recorded signals are reconstructed as the linear superposition of 10
condition-specific principal components with scan-specific temporal signatures. This image is a still frame from Supplementary Movie 3. (Left) fMRI signals
in N = 1463 brain voxels band-pass filtered between 0.01 and 0.3 Hz recorded from a
representative rat under medetomidine only. Middle) Each of the 10 spatially
defined principal modes of covariance is scaled over time by its corresponding
temporal signature in scan S to illustrate the standing wave dynamics. (Right) The
signals recorded in scan S are reconstructed as the linear sum of the 10 principal
components multiplied by their corresponding temporal signature in scan S. To
account for differences in power across components, colorbar limits are set to
±4 standard deviations of the corresponding temporal signatures.
dynamical systems stability theory, demonstrating that the components have a ‘spiral sink’ trajectory back to equilibrium according to
the Poincaré diagram60.
empirically, and the temporal signature Z α ðtÞ is obtained using the
Stuart–Landau equation to simulate the behavior of an oscillator in the
underdamped regime in the presence of background noise as
Ψ Model ðn, t Þ =
Stochastic resonance of standing waves
The presence of a spiral sink in the autocorrelation function of a
dynamical system is indicative of underdamped oscillatory motion,
where the system returns to a fixed-point equilibrium upon perturbation with an oscillation with a decaying amplitude. An underdamped
system will resonate at its natural frequency either when perturbed at a
natural frequency or in the presence of background noise due to stochastic resonance (see Supplementary Fig. S16 for an illustration).
Given that the principal components detected in rat brain activity
have spatial features of standing waves (in line with previous studies)
and, as we demonstrate here, exhibit transient oscillations over time, it
can be hypothesized that their phenomenology is associated with the
stochastic resonance of standing waves. In such a mechanistic scenario, the differences detected across conditions can be further
hypothesized to be related to alterations in the properties of the
medium through which the waves propagate, while the anatomical
structure remains unchanged. Indeed, while medetomidine is found to
increase the number, peak frequency and Q-factor of resonant modes,
isoflurane is found to gradually dampen the resonant modes of the
system, with only global aperiodic fluctuations being detected under
deep anesthesia (Fig. 7).
To demonstrate that the stochastic resonance of stationary wave
patterns can generate the patterns of intrinsic functional connectivity
observed experimentally, we model the signals in the brain slice as the
superposition (i.e., linear sum) of modes whose spatial configuration
ψα ðnÞ is fixed and given by the principal components detected
Nature Communications | (2023)14:375
X
α
ψα ðnÞZ α ðtÞ,
ð3Þ
with
dZ α =dt = Z α ðiωα
∣Z α ∣2 + aÞ + βη,
ð4Þ
where ωα is the resonant frequency of each mode, a (negative) scales
the decay rate61 and η is the added Gaussian white noise η with standard deviation β.
As shown in Fig. 8, the stochastic resonance of a repertoire of
standing waves (here considering the repertoire detected empirically
in sedated animals) results in a spatiotemporal pattern sharing features with what is detected from fMRI recordings. This model includes
the possibility to tune the oscillators in the overdamped regime, in
which case it can approximate the results obtained in deeply anesthetized animals, where no resonant oscillations are detected but only
aperiodic fluctuations. In other words, the model of stochastic resonance does not exclude the hypothesis of scale-free fluctuations
driving the fMRI signals, but it considers it to be a particular case where
the oscillatory modes are overdamped.
In Fig. 8d, we show snapshots of activity generated from the
superposition of standing waves resonating in the presence of background noise to illustrate the multiplicity of patterns that can be
generated at the instantaneous level, as observed in empirical
recordings. Finally, to link with long-range functional connectivity, we
compute the correlation matrix of the simulated signals,
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Sedated rat
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Fig. 5 | Effect of the sampling rate in the power spectrum given a fixed scan
duration. Left: The unfiltered temporal signal τS7 ðtÞ associated with the 7th principal component detected in ultrafast fMRI signals from medetomidine-sedated
rats (Time of Repetition, TR = 38 milliseconds, ms) is downsampled by considering
only one in every 3, 5, 10, 20, 30, 40 and 50 frames (corresponding to intervals of
114, 190, 380, 760, 1140, 1520 and 1900 ms between frames). Plots are shown for
250 s from a representative scan S (same as Fig. 3). Right: The power spectral
density (PSD) of the sampled signals computed for scan S (blue) and for a scan
performed postmortem (red). For each downsampling factor, both PSD (red and
blue) are normalized by the total power in the postmortem scan. PSDs are computed over the entire scan duration of 590 s.
demonstrating that the stochastic resonance of standing waves is a
possible mechanism to generate correlations between contralateral
brain regions located at the wave antinodes.
power > 0.15 Hz, the sensitivity to frequency-specific oscillations is
much lower than the one observed in single-slice acquisitions
(Supplementary Fig. S19).
In summary, our experiments revealed that: (i) power at frequencies extending up to 0.3 Hz is consistently detected in the fMRI
signals from rat brains, peaking in power in the cortex of rats sedated
under medetomidine; (ii) fMRI signal fluctuations organize into a discrete repertoire of modes with fixed phase relationships across space;
(iii) high sampling rates allow detecting transient fine-tuned oscillations in the modes’ temporal signatures; (iv) the oscillatory modes are
sensitive to anesthesia varying both in number, frequency, stability
and spatial configuration; (v) the oscillatory modes detected exhibit
features of a dynamical system operating in the subcritical range of a
Hopf bifurcation; (vi) the stochastic resonance of stationary patterns
generates patterns of long-range functional connectivity similar to the
ones detected empirically; (vii) these findings support the emerging
hypothesis that resting-state activity detected with fMRI results from
the superposition of standing waves resonating transiently within the
brain’s anatomical structure, which in turn drive fluctuations in slidingwindow correlations between the brain subsystems located at the
wave antinodes (Fig. 9).
Expansion to the whole-brain level
To expand our results obtained in a single slice to the whole-brain
level, the principal components were obtained from six 15-min-long
fMRI scans covering 12 brain slices of 3 rats sedated with medetomidine. Despite the necessarily lower temporal resolution of multislice acquisitions (here TR = 350 ms), oscillations can still be
observed (see Supplementary Movie 4), organizing with phase
relationships that overlap (particularly in slice 6) with those
detected in the frontal slice of ultrafast fMRI recordings, supporting
the hypothesis that the conclusions drawn from the single slice
ultrafast acquisitions can be expanded to the whole-brain level
(Supplementary Figs. S17–S19). However, although consistent
principal components were detected at the spatial level (the first 5
are rendered in a transparent brain in Fig. 9a), the limited temporal
resolution and the added artifacts resulting from multi-slice
acquisitions were found to reduce the sensitivity to transient
oscillations such that even in the fMRI scan that exhibited most
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Fig. 6 | The addition of isoflurane at 1% and 3% concentrations alters the spatial,
temporal, and spectral signatures of principal components. a, a’ The N × N
covariance matrix of fMRI signals band-pass filtered between 0.01 and 0.3 Hz,
averaged across 12 scans after the addition of isoflurane at 1% (top) and 3% (bottom)
concentrations. b, b’ Carpet plot of the fMRI signals recorded in all brain voxels, n,
over time, t, represented by the wave function ΨS ðn, tÞ, here shown for two scans S
of the same rat from Fig. 3 in the frequency range [0.01–0.5 Hz]. Voxels are sorted
according to the elements in the largest magnitude eigenvector ψ1 . Values
correspond to fMRI signal change with respect to the mean in each voxel. A zoom
into the first 100 voxels over 60 s is inserted to illustrate oscillations in the signals.
c, c’ Power spectrum of the mean fMRI signal across voxels. d, d’ The principal
components detected with eigenvalue above baseline, are scaled by 1 (left) and −1
(right) to illustrate the activity pattern when the temporal signature oscillates
between positive and negative values. e, e’ Temporal signature associated with each
of the supra-threshold principal components given by τ Sα ðtÞ = ψα ðnÞΨs ðn, t Þ for the
same scan shown in (b). f, f’ Power spectra of the temporal signatures from (e).
Discussion
of spatial configuration, peak frequency, and damping coefficient
across conditions. Despite these differences, the modes detected
across conditions are qualitatively similar in terms of the organization
through phase gradients within anatomically defined cortical and
subcortical boundaries, indicating they likely share a common generative principle. Thus, the metrics that become accessible when
characterizing these intrinsic modes may be beneficial compared with
the more conventional dynamic functional connectivity metrics, as
they provide quantitative parameters on the nature of the correlation.
It is further worth noting that the snapshots of our intrinsic mode
oscillations resemble the spatial patterns exposed by quasi-periodic
patterns (QPPs)17,18 and co-activation patterns67, which were found to
improve the characterization of early Alzheimer’s disease stages68,69
Rhythms at frequencies ranging from 0.5 up to >100 Hz have been
shown to emerge from intrinsic neural processes62–64. However, the
role and generative mechanisms of rhythms below 0.5 Hz detected
both with fMRI, EEG, and electrophysiology remain under vigorous
debate31,47,53,65,66. Using fMRI experiments with hitherto unprecedented
spatiotemporal resolution we provide insights into this problem by
demonstrating the existence of intrinsic macroscale oscillatory modes
in fMRI signals, which organize with mode-specific phase relationships
across extended areas across the cortex and subcortex, driving correlated activity between distant regions.
The oscillatory modes detected were found to be consistent
across rats within the same anesthetic condition, but to vary in terms
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Peak Frequency [Hz]
0.3
rat 1 a
rat 1 b
rat 2 a
rat 2 b
rat 3 a
rat 3 b
rat 4 a
rat 4 b
rat 5 a
rat 5 b
rat 6 a
rat 6 b
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
1
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
1
Resonance Q-factor
15
10
5
0
Fig. 7 | Principal components oscillate at higher frequencies and with less
damping under medetomidine. The temporal signatures associated with the
principal components detected in each condition are characterized in terms of
peak frequency (a) and Q-factor (b) for each of the s = 12 scans in each condition
(2 scans per rat). Error bars represent the mean ± standard error across scans. c To
illustrate the stability of the oscillations, the autocorrelation functions of the
temporal signatures τ S1 associated with the first principal component in each condition are reported. Examples are shown for 3 scans from the same rat and from a
postmortem scan. As can be seen, the autocorrelation function under medetomidine exhibits 3 oscillations before the amplitude decays to 1/e (~37%), 2 cycles after
adding isoflurane at 1% and no complete cycle under deep anesthesia, similar to
what is observed in the postmortem scan.
compared with more conventional resting-state fMRI functional connectivity metrics. Extending such characterizations using the Intrinsic
macroscopic oscillatory mode framework may shed further light into
disease, as well as how activating or silencing specific areas contributes
to whole-brain modulations70,71.
Our results also align with neural field theories for macroscale
brain dynamics describing large-scale wave propagation of neuronal
activity including a spatial Laplacian to incorporate the brain geometry
of the brain72–75. While these neural field theories have historically been
used to describe macroscale brain activity detected with EEG, recent
studies suggest that the structural eigenmodes (defined either from
brain surface geometry or from diffusion networks) may also be at the
origin of macroscopic activity patterns detected with fMRI, namely the
so-called resting-state networks or intrinsic connectivity networks35–39.
This has been recently reinforced by a study demonstrating high
spatial similarity between the covariance eigenvectors of fMRI signals
and the theoretical prediction of Helmholtz eigenmodes of the
Laplace–Beltrami operator starting from a brain surface mesh66.
Overall, these studies support our interpretation that the principal
components detected empirically from the fMRI signals are eigenmodes intrinsic to the brain structure, including not only the cortex
but also subcortical structures, such as the striatum. This hypothesis
could not, however, be fully validated in the current work, given that
the eigenmodes of covariance were obtained from the fMRI signals
alone and not compared with modes predicted from the structure.
Although these spatially defined modes were found to be consistent
across animals and to amplify the signal with respect to spatially
reordered vectors (i.e., either by randomization or by sorting the
phases in a gradient from left to right), our validation tests may not be
sufficient to demonstrate that this is the adequate basis of spatial
patterns to describe functional neuroimaging data and further validations are needed76. Even so, a closer inspection of the spatial configuration of the modes shown in Fig. 3 reveals that distinct
eigenmodes emerge not only in the cortex but also in the striatum,
with some possibly representing fundamental (i.e., global) modes of
specific brain structures (i.e., ψ1 for the cortex and ψ4 for the striatum),
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Fig. 8 | Stochastic resonance of standing waves drives transient long-range
correlations in simulated signals. The spatial configurations and temporal signatures of the principal components align with the hypothesis that they represent
standing waves, whose phenomenology is inherently associated with resonance
phenomena. To model the dynamics emerging from the transient resonance of
standing waves in the presence of background noise, for each of the spatial patterns
detected in medetomidine-sedated rats (a), we simulate a temporal signature as the
behavior of an underdamped oscillator perturbed with gaussian white noise (b).
c Multiplying each N × 1 spatial pattern by the corresponding 1 × T temporal signature results in an N × T spatiotemporal pattern for each mode. The linear sum of
these spatiotemporal patterns represents the superposition of a repertoire of
standing waves resonating transiently in the presence of noise. d Still frames of the
simulated dynamics obtained at distinct time points reveal the multiplicity of
patterns that can be generated over time. e To demonstrate that this scenario
generates patterns of long-range functional connectivity, we compute the correlation matrix of the simulated signals, as performed initially on empirical fMRI data
(Fig. 1). f The seed correlation maps obtained from the simulated signals reveal
correlation patterns visually similar to the ones detected from real fMRI recordings.
Source Code is provided with this work.
Fig. 9 | Mechanistic model for the spontaneous resonance of standing waves
driving the activation of functional brain networks. a Like the response of
spring, the temporal signature of brain modes can be approximated by a damped
oscillator. Despite the lower temporal resolution inherent to multi-slice acquisitions hindering the detection of resonant behavior, the consistency of spatial
patterns reinforces the hypothesis that the damped oscillatory response of
functional networks extends to the whole-brain level, here represented by the first 5
eigenvectors of the average covariance matrices across 6 whole-brain scans (from 3
different rats). b Diagram illustrating a mechanistic scenario for brain activity,
where each functional network is represented by a spatial pattern ψα responding to
perturbation with a damped harmonic motion.
while others may represent subsequent harmonic modes with
increasing spatial frequencies (i.e., ψ6,2,3,5,7 for the cortex and ψ8 for
the striatum).
While under medetomidine alone, strong oscillations were
detected up to 0.3 Hz in agreement with previous literature44, the
addition of, isoflurane at 1% concentration was found to particularly
suppress the power between 0.10 and 0.25 Hz, leaving the power in the
typical range considered in resting-state studies (i.e. <0.1 Hz) mostly
unchanged. Further increasing isoflurane concentration to 3%, most
oscillatory power is lost and only very slow (<0.05 Hz) global and
aperiodic fluctuations are detected. It remains unclear whether these
non-linear effects are related to the differential effects of
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Article
medetomidine and isoflurane on blood vessels or can be explained by
more direct changes in the resistivity of the medium through which the
waves propagate, altering its resonant quality.
The differences detected across anesthetic conditions question
the theoretical predictions of modes depending on the brain’s geometric structure alone because it is implicit that the anatomy of the
brain is invariant across conditions. Indeed, it is generally known that
the resonant modes of a system depend not only on the structural
geometry of the system but also on the resistivity of the propagating
medium, which directly affects not only the spatial patterns but also
the resonant frequency and the stability of the oscillations. Given that
anesthetics directly affect diverse properties of the brain tissue and
vasculature, our results raise the importance of considering not only
the brain geometry but also the resistivity of the medium through
which the waves propagate to possibly explain the differences in
resonant quality observed across anesthetic conditions.
Though many models can be used to describe damped oscillatory
motion, we find the Stuart–Landau equation in the subcritical range of
a Hopf bifurcation to be appropriate since it is the canonical form to
describe a system in the vicinity of a limit cycle, responding to perturbation with oscillations with decaying amplitude from basic mathematical principles, with the imaginary component ensuring the
conservation of angular momentum77. Hopf bifurcation models have
been used in models of spontaneous brain activity to represent local
field oscillations interacting through the connectome structure78–80.
Here instead, we demonstrate that the oscillations detected with fMRI
are not purely local since they lock-in phase across distinct subsystems
and therefore each oscillator model is associated with a distributed
spatial map of phase relationships, analogous to the modes of vibration in a violin string or a drum membrane66,81,82. We show that the
temporal signature of the wave patterns can be approximated by an
oscillator model with varying natural frequency and damping coefficients. In such a framework, the less damped the system is, the more it
resonates at its natural frequency in the presence of naturally occurring noise. We hypothesize that this is what is occurring under
medetomidine. If the damping is increased (as observed with the
addition of isoflurane), then the oscillations are sustained over fewer
cycles and at slower frequencies.
Being inherently associated with resonance, standing waves are a
fundamental property of matter, resulting from the constructive
interference of waves, driving correlated (and anti-correlated) oscillations in the wave’s anti-nodes. The general principle of wave superposition implies that the brain can engage simultaneously in multiple
functional networks, instead of switching from one functional network
to another, as often considered in the analysis of dynamic functional
connectivity14,83–85. In other words, our results substantiate that the
activity recorded herein with fMRI aligns with neural field
theories66,73,86,87, where at any given moment, the wave function Ψ—
representing the collective systemic activity—results from the superposition of a discrete set of wave functions with a damped oscillatory
response. This resonance framework offers simultaneously an explanation for (i) the spontaneous emergence of ultra-slow oscillations in
brain activity, (ii) the profile of phase relationships across space (as
observed in gradient-like functional connectivity patterns), and (iii) the
difference in damping coefficients across conditions. We note, however, that the temporal coordination between the different modes was
not addressed in the current work and requires further investigation.
The generalization of these findings to other animal species
including humans can only be discussed in the light of existing literature and needs further experimental validation. On one hand, the
similarity of the principal components detected herein with intrinsic
network patterns detected using different methodologies suggests
these are expressions of the same emergent phenomena, typically
referred to as resting-state networks (RSNs) or intrinsic connectivity
networks (ICNs). Since rats, mice, monkeys, and humans exhibit
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qualitatively similar RSNs/ICNs, it can be expected that they are
expressions of the same phenomenon. Indeed, a wide range of studies
have demonstrated that intrinsic macroscale modes of brain activity
(detected across modalities) exhibit spatial features of standing waves,
so it can be expected that their temporal signature exhibits a damped
harmonic motion. Even if no clear periodicity is detected in resting
state fMRI in humans and the fluctuations closely approximate the
canonical hemodynamic response function, one cannot exclude the
possibility that the fluctuations reflect an overdamped oscillatory
response associated with the transient and short-lived resonance of a
stationary wave, providing an additional generative hypothesis for the
dynamic patterns observed empirically.
A question that typically arises in this context is how closely the
fMRI signals track the underlying neural activity, mainly due to the
involvement of neurovascular coupling mechanisms. Although this
study did not attempt to deconfound neural activity from vascular
coupling, it is interesting to note that the mode temporal signatures
did not follow the canonical hemodynamic response function. Instead,
a ubiquitous transiently sustained periodicity occurs within a range of
frequencies extending significantly above the range typically associated with the BOLD signal, but below cardiac and respiratory physiological rhythms. Recent studies combining simultaneous
electrophysiological recordings of local field potentials (LFPs) and
fMRI in rats reported significant coherence between the two signals
precisely in the range of frequencies detected herein43,44. Therefore,
one cannot exclude the possibility that the oscillations observed with
fMRI are linked with other factors beyond blood flow/volume effects
alone, and may provide a more direct measurement of neuronal
activity30,88. Still, given that hemodynamic blurring is expected32 further local spectral properties may have been obscured by this blurring.
The advantages of exploring fMRI signals at faster resolutions
extend well beyond this work and certainly deserve further exploration. While previous ultrafast fMRI experiments in rodents have
reached up to 20 frames per second, they have focused mostly on
stimulus-driven responses in specific regions of interest89–93, and much
remains to be explored at the level of spontaneous long-range interactions. Allowing for a large span of scales both in space (from
micrometer to meter) and in time (from millisecond to hour),
exploring the full possibilities of MRI may provide relevant insights
into the brain organizational principles in the spatial, temporal, and
spectral domains88. Indeed, the traditionally low temporal resolution
of fMRI studies has limited the analysis of spatial correlations between
ultra-slow fluctuations in distant areas. More recent dynamic analysis
of functional connectivity has revealed the non-stationary nature of
network interactions20,94,95. Still, under the Connectomics framework,
even dynamic studies measure spatial connectivity patterns over time,
rather than investigating deeper origins of the signals. These additional insights may help resolve the disparate—yet not mutually
exclusive—hypotheses that have been put forth on the nature of
functional connectivity, ranging from stochastic resonance96, metastable synchronization80,97,98, superposition of harmonic modes36,37,66
that can constrain brain function99 or transitions between phaselocking patterns29, among others.
In conclusion, this work reveals evidence for macroscopic
oscillatory modes in spontaneous fMRI signals that organize across
the brain in standing wave patterns, providing fresh insight into the
organizing principles giving rise to intrinsic connectivity networks.
Future work disentangling the different underlying sources of the
fMRI signal, as well as studying the impact of specific therapeutic
effects, such as direct electromagnetic stimulation or pharmacological manipulations, should deepen our understanding of intrinsic
oscillatory modes in the brain. Ultimately, by promoting a better
understanding of brain dynamics, this work provides perspective
avenues for the advance in the diagnosis and treatment of brain
disorders.
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Article
Methods
Ethical statement
All animal experiments complied with the European Directive 2010/63
(established by Portuguese law Decreto-Lei 113/2013) and followed the
Federation of European Laboratory Animal Science Associations
(FELASA) guidelines and recommendations concerning laboratory
animal welfare. Experiments were preapproved by the Champalimaud
Foundation’s Internal Review Board (ORBEA) and by the Portuguese
competent authority for animal welfare (DGAV, Direcção Geral de
Alimentação e Veterinária) with license number 0421/000/000/2016.
Experimental design
Ultrafast resting-state fMRI recordings were obtained from a single
brain slice of 6 rats scanned under 3 different conditions, namely
medetomidine51 combined with 3 concentrations of isoflurane: 0%
(sedation), 1% (light anesthesia), and 3% (deep anesthesia). To minimize variability across animals, we chose all animals of the same sex.
Two additional scans were recorded from a seventh rat postmortem to
serve as a baseline. Moreover, resting-state fMRI recordings covering
12 slices of the rat brain were acquired from 3 rats under medetomidine. Below, we elaborate on each phase.
Animal preparation
Long-Evans female rats ((N = 6 alive + N = 1 postmortem)) weighing
216 ± 30 g and aged 8.7 ± 1.6 weeks were used in this study. To minimize variability across animals, we chose all animals of the same sex.
Animals were reared in a temperature-controlled room and held under
a 12 h/12 h light/dark cycle with ad libitum access to food and water.
Anesthesia was induced with 5% isoflurane (VetfluraneTM, Virbac,
France) mixed with oxygen-enriched (27–30%) air in a custom-built
plastic box. Rats were then weighed, moved to the animal bed (Bruker,
Germany) and isoflurane was reduced to 2.5%. Eye ointment (Bepanthen Eye Drops, Bepanthen, Germany) was applied to prevent eye
dryness.
In the six living rats, a 0.05 mg/kg bolus of medetomidine solution
(Dormilan, Vetpharma Animal Health, Spain: 1 mg/ml, diluted 1:10 in
saline) was injected subcutaneously 5–8 min after induction, immediately followed by a constant infusion of 0.1 mg/kg/h s.c. of the same
solution (25), delivered via a syringe pump (GenieTouchTM, Kent Scientific, USA) until the end of the experiment, and by a 10 min-long
period where isoflurane was gradually decreased to 0%. fMRI acquisitions began once the animals stabilized in this condition (the time after
isoflurane is reported in Table S1 for each scan). For each rat, 2 fMRI
scans were first acquired under medetomidine only (sedation condition). Subsequently, fMRI scans were acquired after increasing isoflurane concentration to 1% (light anesthesia condition) and finally to
3% (deep anesthesia condition), waiting 10 min after each isoflurane
increase for anesthesia stabilization.
The breathing frequency and rectal temperature were monitored
throughout the MRI sessions using a pillow sensor and a 9 ft optic fiber
probe with a 1 mm tip (Model 1030 Monitoring & Gating System and
Fiber Optic Temperature Module & Sensor, SA Instruments Inc., Stony
Brook, USA), respectively, with PC-SAM 8.02 software. At the end of
the experiments, medetomidine sedation was reverted by injecting
0.25 mg/kg s.c. of atipamezole (Antisedan, Vetpharma Animal Health,
Spain: 5 mg/ml, diluted 1:10 in saline).
The seventh rat, reared in the same conditions, was injected with
1 mL (60 mg) pentobarbital i.p. and scanned postmortem with the
same MRI protocol to serve as a control.
MRI protocol
MRI data were acquired using the software ParaVision 6.0.1. Animals
were imaged using a 9.4 T BioSpec® MRI scanner (Bruker, Germany)
equipped with an AVANCETM III HD console, producing isotropic
pulsed field gradients of up to 660 mT/m with a 120 µs rise time. RF
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transmission was achieved using an 86 mm-ID quadrature resonator,
while a 4-element array cryoprobe (Bruker, Fallanden, Switzerland)
was used for signal reception. Following localizer experiments and
routine adjustments for center frequency, RF calibration, acquisition
of B0 maps, and automatic shimming, anatomical images were
acquired using a T2-weighted RARE sequence in the coronal plane: TR/
TE = 2000/36 ms, FOV = 18 × 16.1 mm2, in-plane resolution = 150 ×
150 µm2, RARE factor = 8, slice thickness = 0.6 mm, 22 slices,
tacq = 3 min 28 s, and sagittal plane: TR/TE = 2000/36 ms, FOV = 24 ×
16.1 mm2, in-plane resolution = 150 × 150 µm2, RARE factor = 8, slice
thickness = 0.5 mm, 20 slices, tacq = 3 min 28 s. These images were used
to place the slices of interest.
Single-slice ultrafast fMRI acquisitions
To minimize the repetition time, we focused our analysis in a single
1.2 mm-thick slice of the rat brain, choosing a frontal slice that covered
a large cortical area and with a FOV of 21 × 21 mm2, as shown in Fig. S1 A,
B. The slice was placed between −0.2 and 1.0 mm from Bregma
according to the Paxinos & Watson rat brain atlas100 (Supplementary
Fig. S1c).
Six resting-state scans (2 per condition) were acquired from each
of N = 6 living rats (totaling 36 scans) using a gradient-echo echo planar
imaging (GE-EPI) sequence (TR/TE = 38/11 ms, flip angle = 15°, matrix
size = 84 × 84, in-plane resolution = 250 × 250 µm2, number of time
frames = 16,000, tacq = 10 min 8 s). Two postmortem scans were
acquired with the same parameters. Additionally, a Multi-Gradient
Echo sequence (MGE, TE = 2.5:5:97.5 ms, TR = 300 ms, flip angle = 40°,
matrix
size = 210 × 210,
in-plane
resolution = 100 × 100 µm2,
tacq = 4 min 12 s) and a Time-Of-Flight (TOF) FLASH sequence (TR/
TE = 8.2/3.3 ms, flip angle = 80°, matrix size = 210 × 210, in-plane resolution = 100 × 100 µm2, tacq = 17 s 219 ms) sequence were acquired from
all rats to obtain additional anatomical and vascular information about
the slice, respectively. Details of time after isoflurane induction,
breathing frequency and rectal temperature are reported for each scan
in Table S1.
Whole-brain fMRI acquisitions
Resting-state data was acquired twice under medetomidine sedation
from a subsample of N = 3 of the living rats using a multi-slice GE-EPI
sequence covering the entire rat brain, from the frontal part of the
cerebellum to the posterior part of the olfactory bulb, and with the
following parameters: TR/TE = 350/11 ms, flip angle = 40°, FOV = 24 ×
24 mm2, matrix size = 70 × 70, in-plane resolution = 342.9 × 342.9 µm2,
slice thickness = 1.2 mm, slice gap = 0.15 mm, 12 slices, number of time
frames = 2572, tacq = 15 min 0 s 200 ms.
Brain masks
Individual brain masks were defined manually and aligned across rats
to a common central coordinate. All individual rat masks were superposed to define a common brain mask containing N = 1463 voxels in
the single slice and N = 7426 voxels in the whole brain.
In the single slice, no spatial or temporal interpolation was applied
to the signals, such that the signal in each brain voxel corresponds to
the raw fMRI signal recorded.
In multi-slice acquisitions, the slice-timing correction was applied
using linear interpolation over time.
Space-frequency analysis of fMRI data
Power spectra were computed for the fMRI signals on each of the
84 × 84 = 7056 voxels using the fast Fourier transform, after removing
the first 1000 frames (38 s) and detrending. Voxel power spectra were
obtained up to the Nyquist frequency of (2TR)−1 = 13.1579 Hz. Images of
the power across a selected range of frequencies were obtained by
averaging the band-limited power in each voxel across scans in the
same condition. All spectral analyses were performed at the single scan
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level and metrics statistically compared between conditions. Analysis
up to the Nyquist frequency is reported in Supplementary Figs. S5
and S11.
with a stochastic input, we add complex Gaussian white noise as:
dZ
= Z iω
dt
∣Z 2 ∣ + a + βη1 + iβη2 ,
Principal component analysis
For each scan, the fMRI signals in N = 1463 brain voxels were band-pass
filtered between 0.01 and 0.3 Hz and the N × N covariance matrix was
was calculated for
computed. The largest magnitude eigenvalue, λPM
1
the two postmortem scans, and the largest one selected as the baseline
threshold. The covariance matrices were averaged across the 12 scans
in each condition, and, for each condition the α N × 1 eigenvectors with
condition
PM
> λ1 were extracted, representing the principal
eigenvalue λα
components detected in each condition with magnitude above the
postmortem baseline.
The same analysis performed using decreasing sampling rates is
reported in Supplementary Fig. S10. The modes detected using correlation—instead of covariance—above the postmortem baseline are
reported in Supplementary Fig. S11.
Statistical analysis
We compared the power between conditions at different frequency
bands using a non-parametric permutation-based t-test (10,000 permutations to ensure the robustness of results) to detect the frequency
range most sensitive to the three different conditions. p-values were
conservatively corrected by the number of comparisons performed
(Bonferroni correction), considering both the number of betweengroup comparisons (considering only independent hypotheses) as
well as the number of frequency bands considered (considering
dependent hypotheses as well, which is even more conservative).
The resonance Q-factors and peak frequencies were statistically
compared using the same permutation test.
The standard error is calculated as the standard deviation divided
by the square root of the number of values compared.
Resonance analysis
Gaussian distribution
where η1 and η2 are independently drawn frompaffiffiffiffiffi
with standard deviation β = 1 (integrated as β dt ). Simulations were
obtained using the Euler method for numerical integration with a time
step dt = 10−3 s.
Reporting summary
Further information on research design is available in the Nature
Portfolio Reporting Summary linked to this article.
Data availability
The raw structural and dynamic MR images used in this study are
available for download as Maltab files (.mat) without restrictions
from: https://drive.google.com/drive/u/5/folders/1JQ_1AmP4v-HEB_
ll5ZwHEaB0RtIoA17R. An example dataset with a reduced subset of
fMRI scans sufficient to replicate the analysis is also included.
Source Data for Fig. 2 and Supplementary Figs. 3 and 4 are provided
as Excel files (.xlsx) with this paper. Source data are provided with
this paper.
Code availability
Source Code for the replication of the analysis and figure generation
are made available as Matlab scripts with this work.
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1.
2.
3.
Resonance was evaluated by computing the Q-factor, a measure typically used in acoustics and engineering to quantify resonance phenomena. Importantly, it is not implied by definition that a covariance
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4.
5.
6.
7.
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8.
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Acknowledgements
Santa Casa da Misericórdia. Grant name and ID: Mantero Belard, MB-562020 (N.S.). La Caixa Foundation (Spain), grant LCF/BQ/PR22/11920014
15
Article
(J.C.). European Research Council (ERC) grant 679058 (N.S., F.F.F.).
Lisboa Regional Operational Program (Lisboa 2020), under the PORTUGAL 2020 Partnership Agreement through the European Regional
Development Fund (ERDF).Portuguese Foundation for Science and
Technology (FCT) grant LISBOA-01-0145-FEDER-022170. Portuguese
Foundation for Science and Technology (FCT) grant 275-FCT-PTDC/BBB/
IMG/5132/2014 (N.S., F.F.F.). Portuguese Foundation for Science and
Technology (FCT) grant UIDB/50026/2020 (J.C.). Portuguese Foundation for Science and Technology (FCT) grant UIDP/50026/2020 (J.C.).
Portuguese Foundation for Science and Technology (FCT) grant CEECIND/03325/2017 (J.C.).
Author contributions
J.C. and N.S. conceptualized the work. N.S. designed the MRI acquisition
sequences and supervised the work. F.F.F. performed animal manipulation and data acquisition. J.C. performed the analysis and visualization and is responsible for the source code. J.C. and N.S. wrote the
original draft of the manuscript. All authors contributed to reviewing and
editing of the manuscript.
Competing interests
The authors declare no competing interests.
https://doi.org/10.1038/s41467-023-36025-x
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