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Open Access

Peer-reviewed

Research Article

Noise-like fluctuations drive shifts in neuronal timescales and 1/f power spectra across brain states

  • Axel Hutt,

    Roles Conceptualization, Formal analysis, Investigation, Methodology, Project administration, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review & editing

    Affiliation MLMS, NECTARINE, Université de Strasbourg, CNRS, Inria, ICube, Strasbourg, France

    ⨯
  • Matteus McCulloch,

    Roles Conceptualization, Investigation, Software, Visualization, Writing – original draft

    Affiliation Department of Cellular and Molecular Medicine, University of Ottawa, Ottawa, Ontario, Canada

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  • Anthony G. Hudetz,

    Roles Data curation, Investigation

    Affiliation Department of Anesthesiology, Center for Consciousness Science, University of Michigan, Ann Arbor, Michigan, United States of America

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  • Aref Pariz,

    Roles Investigation, Project administration, Resources, Software, Visualization, Writing – review & editing

    Affiliations Institute of Mental Health Research at The Royal, Ottawa, Ontario, Canada, Department of Biology, University of Ottawa, Ottawa, Ontario, Canada, Krembil Brain Institute, University Health Network, Toronto, Ontario, Canada

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  • Jérémie Lefebvre

    Roles Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing

    jeremie.lefebvre@uottawa.ca

    Affiliations Department of Cellular and Molecular Medicine, University of Ottawa, Ottawa, Ontario, Canada, Department of Biology, University of Ottawa, Ottawa, Ontario, Canada, Krembil Brain Institute, University Health Network, Toronto, Ontario, Canada

    ⨯

Abstract

The aperiodic, broadband components of many aggregated electrophysiological recordings such as LFP, EEG and ECoG exhibit characteristic features, notably power-law scaling and a spectral knee. These features of the power spectral density (PSD) vary with behavioral state, arousal, pharmacological interventions, and are linked to transitions between distinct brain states. We investigate the origins of this variability using large-scale recurrent neural networks with sparse, balanced, and random connectivity, driven by state-dependent fluctuations. By integrating recent advances in random matrix theory, we develop an analytical framework that characterizes how such fluctuations shape key features of the PSD, accounting for both nonlinear and stochastic contributions. Our results show that the variability of broadband spectral features can arise as a generic property of nonlinear recurrent networks driven by noise-like fluctuations. In particular, the emergence and modulation of the spectral knee reflect shifts in effective neuronal timescales, linking noise-like fluctuations to state-dependent temporal organization in large-scale networks. Together, these findings provide a mechanistic account of how broadband spectral features can emerge from intrinsic network dynamics, and suggest that changes in spectral features may reflect noise-driven, nonlinear transitions in recurrent neural systems, rather than the action of a single underlying biophysical mechanism.

Author summary

The electrical activity of the brain contains both rhythmic oscillations and a broadband background signal that follows a characteristic 1/f pattern. Although this background activity is routinely measured with techniques such as EEG and local field potentials, its biological origin and functional significance remain debated. In particular, its shape changes systematically across sleep, wakefulness, anesthesia, and neurological disorders, but the mechanisms responsible for these changes are still unclear. Here, we use mathematical analyses and large-scale computer simulations of neural networks to show that many of these spectral changes can emerge naturally from the interaction between recurrent network dynamics and ongoing, noise-like fluctuations. Rather than requiring a specific cellular or molecular process, changes in broadband spectral features arise because fluctuations alter the effective timescales over which neural populations evolve. This shifts the characteristic “knee” of the power spectrum and changes its overall shape. Our findings provide a simple mechanistic explanation for how brain activity can reorganize across different functional states and suggest that commonly used spectral biomarkers are in fact dynamic and represent a generic feature of recurrent neural networks.

Citation: Hutt A, McCulloch M, Hudetz AG, Pariz A, Lefebvre J (2026) Noise-like fluctuations drive shifts in neuronal timescales and 1/f power spectra across brain states. PLOS Complex Syst 3(9): e0000133. https://doi.org/10.1371/journal.pcsy.0000133

Editor: Andre Fujita, University of Sao Paulo, BRAZIL

Received: January 6, 2026; Accepted: August 25, 2026; Published: September 25, 2026

Copyright: © 2026 Hutt et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: The algorithms and numerical methods that support the findings of this study are available from GitHub (https://github.com/Jeremie-Lefebvre/Hutt-et-al-PLOS-complex-systems-2026).

Funding: We thank the the New Frontiers in Research Fund (Grant No. NFRFE-2023-00354 to J.L.) National Sciences and Engineering Research Council of Canada (Grant No. NSERC RGPIN-2026-04461 to J.L.) for support of this research. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

1. Introduction

The spectral characteristics of the so-called broadband brain activity – specifically the aperiodic component of mesoscopic electrophysiological recordings such as LFP, EEG and ECoG [1–4] – have been repetitively shown to conform to a power-law scaling, where parameterizes the decay of signal amplitude at higher frequencies [1,2,5–7]. For instance, signals with a high value of are dominated by large-amplitude, slow-wave synchronous fluctuations, and are commonly observed during sleep and anesthesia [8–11]. In contrast, a decrease in , reflecting a less steep power spectrum, is associated with less synchronized activity, as seen during arousal and wakefulness [8,10–12].

Features of the aperiodic power spectrum show substantial variability with behavior, arousal, and pharmacological interventions (see [2] and references therein). For instance, shifts in the exponent have emerged as a robust index of mesoscopic brain state [13], across variations in arousal and anesthesia [10], sleep stages [8,9], cognitive load [14], task demands [15], and aging [16]. Beyond the exponent , the aperiodic spectrum is also structured by a spectral knee, a characteristic bend that delineates distinct frequency regimes and reflects an underlying neuronal timescale [6,7,10,17,18]. The presence of a spectral knee implies that the PSD does not follow a single law, but instead multiple scaling regimes across frequencies, each associated with its own effective exponent. Variability in the spectral knee suggests that neuronal timescales are not fixed, but instead vary in a state-dependent manner. The spectral knee affects estimates of the spectral slope by determining the frequency range over which a linear approximation of the log–log spectrum is valid, namely at frequencies above the knee. Changes in the resulting critical exponent have been linked to fluctuations in the collective, distributed, and heterogeneous activity of large neuronal assemblies characterizing EEG and ECoG signals [1,6,7,17], reflecting complex cortical dynamics shaped by neural interactions, correlations, filtering, and excitation–inhibition balance [1,2,19–21].

While its origin remains highly debated [3,11,22–26], power-law scaling (also referred to as scale-free dynamics) is closely linked to dynamical stability in complex systems. In particular, has been associated with proximity to bifurcation points and a general propensity for pattern formation and self-organized criticality [11,17,22,25,26] (but see [3,23,24]). Such scaling has been reported across a range of physical [11,22] and neural phenomena [19], including long-range temporal correlations [1,2,5], neuronal timescales [6,7], neuronal avalanche statistics [27,28] and random ion diffusion [29]. Yet, the origin and functional significance of these scaling laws in the nervous system remain unresolved.

The presence and variability of the spectral knee and scaling are consistent with nonlinear systems undergoing state transitions, whereby brain dynamics shift between distinct regimes accompanied by changes in coherence and temporal structure [30–32]. Such transitions have been linked to changes in endogenous fluctuations in neural activity, which are particularly prominent in awake, aroused states [30,33–44]. These fluctuations reflect elevated neural activity arising from sensory inputs, brainstem modulation (e.g., the Ascending Reticular Arousal System), and inter-area communication [44], and are known to contribute to the function and stability of brain networks [31,33,37,40,41,43–48].

To better understand the variability of the spectral knee and power-law scaling, we consider a generic large-scale nonlinear neural network with sparse, balanced excitatory and inhibitory connectivity, driven by state-dependent fluctuations that emulate changes in brain activation, such as those occurring during transitions between sleep and wakefulness or anesthesia [44,49,50]. Such models have been widely used to study interacting neural populations across multiple spatial and temporal scales [47,48,51–57], and have successfully reproduced key spectral features of mesoscopic electrophysiological signals [17,44]. They provide a versatile framework to capture the variability, scale, and structural complexity of neural systems exhibiting power-law-like dynamics (see [11,27,30] and references therein). Building on this framework, and leveraging recent advances in random neural networks [58], we examine how state-dependent fluctuations—modeled here as noise of changing variance—shape both the scaling properties and the emergence of the spectral knee in the power spectral density (PSD). Using random matrix theory, we derive analytical expressions for the mean PSD that account for both nonlinear and stochastic contributions. Our results show that variability in scaling and spectral knee can arise as a generic property of nonlinear recurrent networks driven by noise-like fluctuations, whereby changes in network activity induce shifts in effective scaling regimes and associated neuronal timescales. Together, these findings indicate that broadband spectral features may reflect noise-driven nonlinear transitions in recurrent networks, rather than the action of a single underlying biophysical mechanism.

2. Results

To better understand the mechanisms underlying the variability of the spectral knee and power-law scaling, we considered a generic large-scale nonlinear neural network with sparse balanced excitatory and inhibitory connections. This network was exposed to fluctuations whose intensity was manipulated to emulate different brain states, which we loosely define here as dynamical regimes characterized by different levels of activation (e.g., stages of anesthesia, degrees of arousal, etc). In our model, these state-dependent fluctuations were modeled as additive noise (see Methods) whose variance reflects the importance of endogenous broadband signals resulting from the recruitment cells and/or neural populations [44,59] This framework offers a flexible approach for capturing the diversity, intrinsic variability, and multiscale structural complexity of neural systems exhibiting critical or power-law–like behavior, and have further been shown to accurately capture the power distribution of aggregated electrophysiological recordings such as LFP, EEG and ECoG [17,44,50]. Specifically, we considered a network of N local populations and the dynamics of their mesoscopic synaptic potentials obeying

(1)

where denotes a linear relaxation operator, and the vector is a sigmoidal activation function and I is a constant input. The variable represents the potential in population i, reflecting a coarse-grained average of microscopic neuronal activity [60]. The matrix denotes the connectivity of the network, which is sparse and balanced with connection probability c: excitatory and inhibitory weights are drawn from normal distributions while their sum at each node is zero [53,56,57]. To emulate changes in brain state and different levels of neural activation, stochastic forces drive the network. Specifically, each node in Eq. (1) is exposed to state-dependent fluctuations, here modeled as additive, independent Gaussian white noise processes of variance D, i.e., with (see Methods, Section 4.1 for additional details). Note that we have also examined numerically the case of additive colored noise with increasing correlation times (see S1 Text).

A representative example of the resulting dynamics can be seen in Fig 1. The variance D of the state-dependent fluctuations—modeled here as additive noise—was allowed to fluctuate slowly and randomly in time, representing smooth variation in brain activation, or brain state. Note that later in the analysis, the variance D will be considered constant. These fluctuations emulate changes in endogenous neural activity, steering brain dynamics towards regimes of low-activation (i.e., noise variance D is small; Fig 1a) and high-activation (i.e., noise variance D is large; Fig 1b) (e.g., gradual transitions between anesthesia and wakefulness). These distinct regimes are characterized by distinct power spectral density (PSD) profiles: as shown in Fig 1c–d, increasing the variance of the fluctuations suppresses low-frequency contributions and enhances higher-frequency power. Importantly, the broadband component of these PSDs differs by a visible shift in the position of a characteristic bend—the spectral knee—which separates distinct scaling regimes. The spectral knee reflects a characteristic neuronal timescale, corresponding to the inverse of the frequency beyond which temporal autocorrelations in the population activity begin to decay rapidly. Accordingly, state-dependent fluctuations primarily induce shifts in the position of the spectral knee, thereby altering the frequency range over which power-law scaling can be meaningfully assessed. In contrast, at sufficiently high frequencies—beyond the knee—the scaling exponent approaches an asymptotic value that remains largely invariant across noise levels, consistent with theoretical predictions (see Methods). As can be seen in Fig 1e, these spectral changes echo transitions in the network dynamics: low-noise epochs are characterized by high-amplitude, low-frequency, synchronous activity associated with slower integration, whereas higher noise levels give rise to faster, irregular, and asynchronous activity. These transitions suggest state-dependent changes in network stability, whereby fluctuations in neural activity reshape the network’s intrinsic dynamics and facilitate qualitatively distinct dynamical regimes [17,46,48,56,57].

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Fig 1. State-dependent variability in neural timescales and dynamics.

a. State of low activation, characterized by weak fluctuations, modeled as additive noise of small variance (i.e., D is small). b. Whenever state-dependent fluctuations are increased, the network is instead found in a state of high activation (i.e., D is large). c. Power spectral density of the mean network activity over an epoch of low activation (PSD measured over a time interval of sec in panel e), exhibiting strong low frequency power (gray shaded area) and steep decay at high frequencies. The bent delimited by the light green line segment corresponds to the spectral knee. Insert shows the dynamics of the mean network activity (i.e., ), computed by performing an ensemble average over N network nodes. d. Power spectral density of the mean network activity over an epoch of high activation (PSD measured over a time interval of sec in panel e, where low frequency power has been suppressed (gray shaded area). The spectral knee (light red line segment) is shifted towards higher frequencies. The slope of the PSD beyond the knee is less steep, indicating a shift in exponent () caused by additive state-dependent fluctuations. e. Response of the network in Eq. (1) to state-dependent fluctuations whose variance varies in time. The system exhibits intermittent shifts in dynamics (network average ; black line; middle), driven by slow changes in noise variance (D; blue line; top). These are marked by high-amplitude, oscillatory, and synchronous activity in the low activation state (green box), alternating with epochs of high-frequency, irregular, and asynchronous patterns in the high activation state (red box), visible from the time-frequency spectrum (bottom). The parameter D varies randomly according to a Ornstein-Uhlenbeck process with long correlation time 1000, and was positive rectified. Parameters are N = 250, c = 0.05, r = 0.8, , , , and . Numerical integration was performed over a trial of duration T = 50s with a time step of ms. The time-frequency spectrum was estimated by a Short Time Fourier Transform and the power spectra were estimated using the Welch method with sampling rate 1000Hz, sliding window duration of 2048ms and overlap of 1024ms.

https://doi.org/10.1371/journal.pcsy.0000133.g001

Characterizing PSD features in high-dimensional, nonlinear systems such as Eq. (1) is a notoriously challenging task. It is thus common practice to perform a linearization around a set of deterministic equilibria [58,61] to compute the PSD and its associated features. However, such approaches offer limited insight: they are difficult to perform by matrix inversion in large networks [58,61], and further fail at capturing nonlinear effects. In fact, the calculations of the mean PSD (i.e., the average power spectrum taken across network nodes) has recently been shown to simplify greatly, adopting the Lorentzian distribution , where D refers to the variance of the state-dependent fluctuations in our model, and where corresponds to the spectral radius [58] (cf. Section 4.2). Indeed, by construction, our network model is subject to the circular law of random matrix theory [62], in which the eigenvalues are constrained with high probability in a disk of radius in the complex plane [63]. The location of that disk in the complex plane, and size of are determining factors of the stability of Eq. (1), reflecting the joint influence of network size, connectivity and heterogeneity on the dynamics. For a thorough introduction and discussion of spectral analysis, the circular law, and their relevance to neuroscience, we direct the reader to Ref [64].

However, there is an important caveat: approaches relying on deterministic premises preclude any state-dependent change in the position of the spectral knee and associated spectral slope. Indeed, such approaches predict a PSD profile and power-law scaling laws that does not vary with noise: the resulting Lorentzian spectral profile (i.e., ; see Eq. (8)) exhibits fixed spectral knee and spectral slope (or critical exponent when considering the large frequency limit), independent of the variance of the fluctuations (i.e., D). In such cases, changes in D merely rescales the overall amplitude of the spectrum, and hence cannot produce any shifts in the position of the spectral knee or alter the frequency ranges over which scaling is observed. This contrasts with both experimental findings and the extensive simulations performed on our network model (i.e., see Fig 1), which display marked, state-dependent variability in spectral knee position and resulting spectral slope and/or critical exponent . Taken together, these results indicate that nonlinear effects are required to account for the observed variability in broadband spectral structure.

To bypass these limitations and obtain further insight about the mechanism supporting this variability, we first leveraged the theory of random matrices and adapted it to the presence of additive noise. Specifically, we derived a variational equation inspired by Floquet theory and the master stability function. to determine stability, not by linearizing about deterministic equilibria as is commonly done, but instead about the stochastic system evolution assuming the stationary probability density function of of Gaussian shape. This assumption is motivated by the uniqueness of the (noiseless) fixed point [54,56,57] and further confirmed by extensive numerical simulations supporting this approximation (see Fig 2a-b).

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Fig 2. State-dependent fluctuations drive shifts in stability.

a. Probability density function (PDF) of the network activity, computed numerically for different noise variances: (bright red line), (red line) and (dark red line). These adopt a Gaussian profile. b. The variance of the PDF computed numerically (dark red line) scales almost linearly with noise variance (D; dashed black line). c. Spectral disk of the corresponding Jacobian , cf. Eq. (11), delimiting the eigenvalues (gray circles) in the complex plane. Increasing the variance of state-dependent fluctuations (i.e., D) decreases the spectral radius (see Eq. (14)) and thus contracts the spectral disk (D = 0 (dashed black circle); (bright red circle); (red circle) and (dark red circle)). This reinforces linear behavior and hence stability, pushing the network eigenspectra away from the instability threshold (; vertical dashed black line). Parameters are N = 250, c = 0.05, r = 0.8, , , , and . Numerical integration was performed over a trial of duration T = 5s using ms.

https://doi.org/10.1371/journal.pcsy.0000133.g002

This enabled us to determine an effective spectral radius , which takes into account the interaction between the nonlinearity and the additive noise present in Eq. (1) and their mutual impact on stability. This approach revealed that the size of the spectral disk - constraining the eigenvalues associated with the corresponding Jacobian in the complex plane - is directly controlled by the variance of state-dependent fluctuations. Indeed, as illustrated in Fig 2c, increasing D contracts the spectral radius , thereby steering the system away from instability and amplifying linear behavior. We note that these results are fully consistent with the predictions made about the stabilizing role of noise in networks with similar architectures [40,43].

To confront these theoretical findings, we revisited the expression for the mean PSD, but now using the effective spectral radius (i.e., , see Section 4.3). The resulting effective mean PSD, , possesses an explicit, nonlinear dependence on the variance of the state-dependent fluctuations (i.e., D). As can be seen in Fig 3, this approach accurately captured the variability of the PSD induced by state-dependent fluctuations. Notably, our analysis predicts a state-dependent shift in neuronal timescale, associated with a change in spectral knee position. Indeed, the characteristic knee frequency (i.e., ) was found to increase with noise variance, indicating a reduction in the effective neuronal timescale and a corresponding shift in the organization of scaling across frequencies in the PSD. This prediction was further supported by numerical simulations, averaged across multiple, independent realizations of initial conditions and well preserved over different levels of connectivity heterogeneity (i.e., , see Eq. (3)), and across noise variances over multiple orders of magnitude (Fig 3a–c). This alignment provides strong support for the theoretical analysis pursued here and its underlying assumptions (see Section 4.3). Importantly, explicitly incorporating noise (i.e., fluctuations) into our framework reveals that the system is far more stable than predicted by standard linearization techniques: low frequency power is suppressed. Significant deviations emerge between the actual mean PSD and that predicted by linearization (i.e., 3a–c, black lines) around the deterministic trivial fixed point. These deviations are accurately captured by our stochastic framework. This effect is particularly pronounced near the instability threshold (due to higher connectivity variance with ) and for larger noise variances, where deviations between the noise-corrected PSD and deterministic predictions are largest (see Fig 3c). As the noise variance is decreased, these nonlinear contributions to the PSD vanish (Fig 3a–b). To generalize these observations, we have also confirmed numerically that this phenomenon occurs assuming Dale’s law in the network topology, (see S1 Fig, S1 Text) and in presence of colored noise over a wide range of correlation timescales (see S2 Fig, S2 Text).

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Fig 3. Stochastic analysis captures the variability of the power spectrum.

The effective mean power spectral density (see Eq. (17); normalized) is shown for increasing noise variance (i.e., , , , and , from pale to dark). The power-law scaling is accurately predicted by the analytical derivations relying on noise-induced, nonlinear effects (i.e., Eq. (17)), especially at larger noise variances. Scatter points correspond to numerical simulations, averaged over independent network realizations (i.e., Eq. (1)), whereas dashed lines represent the theoretical prediction taking into account nonlinear, stochastic effects (i.e., Eq. (17)). The black bold line in each panel represent the prediction ignoring noise contributions (i.e., Eq. (8)). The vertical dotted lines correspond to the position of the spectral knee (i.e., Eq. (18), whose characteristic frequency (i.e., timescale) increases with noise variance. Across each panel, the connectivity weight variance is increased, i.e., a. (blue); b. (orange); c. (green). Due to normalization and the absence of non-linearity in panel a, theoretical predictions and data align and superpose: the slope of the PSD and spectral knee remain the same, irrespective of noise variance. However, deviations in panels b and c reveal noise-induced effects, and a displacement of the spectral knee whose frequency scales with D. Data points are plotted SED, and denote the error range corresponding to the standard deviation taken over independent trials. Other parameters are N = 300, c = 0.05, r = 0.8, and . Results were averaged over 10 independent trials of duration T = 30 s with a time step of ms.

https://doi.org/10.1371/journal.pcsy.0000133.g003

Collectively, these findings suggest that, within the constraints of our model, state-dependent fluctuations—modeled here as additive noise—tune the effective neuronal timescale of the network. This effect is directly reflected in the position of the spectral knee and, consequently, in estimates of the spectral slope. To quantify this, we compared the characteristic knee frequency () and exponent obtained from theory (see Section 4.5) to numerical simulations, using linear regression of the mean PSD over varying frequency ranges and multiple independent trials. As shown in Fig 4a, both theory and simulations predict that increasing noise variance shifts toward higher frequencies, indicating a reduction in the effective neuronal timescale. This shift reorganizes the frequency ranges over which power-law scaling is observed. Accordingly, as illustrated in Fig 4b, displacement of the spectral knee leads to a decrease in the estimated value of within fixed frequency bands, reflecting a flattening of the spectrum. This effect is not captured by standard approaches that neglect nonlinear and noise-induced contributions. We note that our analytical derivations slightly underestimate the decrease in observed numerically across the explored range of noise variances, due to approximations introduced in the theoretical treatment (see Methods). Taken together, these results indicate that variability in PSD features can arise from state-dependent changes in neuronal timescales, reflecting a nonlinear and generic property of recurrent networks driven by stochastic fluctuations.

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Fig 4. State-dependent fluctuations shift spectral knee position and reorganize scaling regimes.

a. The position of the spectral knee shifts toward higher frequencies as the noise variance (D) increases. Results are shown for increasing connectivity variance ( (blue), (orange), and (green)). Theoretical predictions based on noise-induced nonlinear effects are shown as dashed lines, whereas horizontal lines represent predictions neglecting noise contributions. The shaded area denotes the standard deviation across independent trials. b. Apparent changes in the exponent () of the mean power spectral density with increasing noise variance (D). Theoretical predictions based on noise-induced nonlinear effects are shown as dashed lines, whereas horizontal lines correspond to predictions ignoring noise contributions, evaluated at a reference frequency Hz. The dots denote the exponent estimated via linear regression over the Hz range and averaged across independent network realizations. The shaded area denotes the standard deviation across trials. The color code is the same as in a. Notably, this frequency range lies near the spectral knee, such that variations in primarily reflect shifts in the knee position rather than changes in asymptotic scaling. The exponent is estimated over the interval Hz as is swept from 5 to 50 Hz; the knee frequency is defined as the value of at which the exponent crosses 1 in each trial. Results were averaged over 10 (for b) and 20 (for a) independent trials of duration T = 30 s with time step ms. Other parameters are N = 300, c = 0.05, r = 0.8, and .

https://doi.org/10.1371/journal.pcsy.0000133.g004

Previous theoretical and experimental studies on the origin of PSD of neural population activity have indicated that neural ion diffusion contributes primarily to -scaling in LFP data [24] and this ion diffusion is random in nature [29]. In light of these studies, we aim to investigate whether the hypothesized role of random fluctuations in the present work is consistent with experimental LFP data. Fig 5a shows the PSD of LFP recordings from two animals under both light and deep anesthesia. The spectral knee is visible in one LFP trial (lower panel) and appears at lower frequency in deeper anesthesia level, whereas the spectral knee is less visible in the other LFP trial. Moreover, power-law fits () over a predefined frequency range reveal steeper spectra under deep anesthesia compared to light anesthesia. This trend is further supported at the group level in Fig 5(b), which displays the distribution of estimated exponents across seven paired trials for each anesthesia condition. The results indicate a statistically significant increase in with increasing anesthesia depth (p < 0.05). This is consistent with previous findings in intracranial EEG (cf. Fig 5 in [10,65]). Collectively, in addition these findings are consistent with the hypothesis proposed in [44], which suggests that deeper anesthesia reduces intrinsic endogenous fluctuations, potentially via suppression of the ascending reticular arousal system. In this framework, deeper anesthesia corresponds to a lower intrinsic noise level. Taken together, Figs 4 and 5 qualitatively support the prediction that decreasing the variance of endogenous fluctuations leads to a slower spectral knee frequency and an increase in the spectral exponent .

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Fig 5. Estimation of exponent from experimental data.

a. Power Spectral Density (PSD) of two Local Field Potential trials (open dots) under light (black) and deep (red) anaesthesia levels and corresponding linear fits in the frequency range [5Hz;50Hz] (solid line). The inset plots illustrate the difference of spectral power between light and deep anaesthesia , where are the corresponding functions fit with respect to frequency f. These functions increase monotonously and cross the horizontal axis with increasing frequency, which reflects a lower slope for light anaesthesia than for deep anaesthesia. b Estimated exponents for a set of seven experimental Local Field Potential trials. The dashed lines denote the trial pairs. A paired bootstrap inference test found statistically significant different exponents for low and high anaesthesia levels (p = 0.018).

https://doi.org/10.1371/journal.pcsy.0000133.g005

3. Discussion

We used a large-scale, recurrent neural networks model with sparse, balanced and random connectivity to investigate the variability of broadband spectral features characterizing various types of electrophysiological recordings. Changes in the spectral slope through the exponent have repeatedly been shown to accompany transitions in brain states, following shifts in behavior, arousal, or as a result of different pharmacological interventions. Our joint mathematical and computational analyses show that such variability can arise from a generic nonlinear mechanism driven by state-dependent fluctuations—modeled here as additive noise—of varying intensity, which primarily act by modulating the effective neuronal timescale of the network. This modulation is directly reflected in shifts in the spectral knee, which reorganize the frequency ranges over which power-law scaling is observed and, consequently, influence estimates of the critical exponent. Utilizing previous results provided by Krumbeck and colleagues [58] enabled us to compute the mean PSD analytically. Introducing noise-dependent, nonlinear corrections proved essential in the alignment between theory and numerical simulations, and found to subsist across a wide parameter range.

Previous studies have shown that random networks operating in near-critical dynamical regimes capture the power-law scaling as well as other features of the broadband electrophysiological power distribution measured experimentally [11,17,19,30]. These have further suggested that changes in spatially-dependent connectivity profiles could be key in generating changes in scaling that resemble those observed across different brain states [10]. These results were also shown to rely on modifications of the eigenvalue spectra of the corresponding network connectivity matrices. Our work complements these findings by showing that variations in the statistics of endogenous neural activity could on its own be responsible for such an effect, without requiring any change in connectivity. This conclusion adds to the growing body of evidence indicating that temporal variability in neural activity - oftentimes modeled as noise - represents an important control parameter for the organization and stability of neural circuits operation [48,59]. This view also echos findings made in neural mass models of interacting excitatory and inhibitory pools [20,44,48], highlighting the critical importance of noisy stimuli and excitation-inhibition balance in the genesis of critical-like activity.

The variability in the spectral knee and associated scaling laws likely arises from multiple, dynamic and potentially interacting mechanisms. In addition to the nonlinear network effects discussed here, processes such as tissue filtering of extracellular fields [3,23,24] and synaptic filtering [7,66] can independently shape spectral structure, particularly across distinct frequency ranges and measurement modalities. However, these mechanisms alone do not account for all observed instances of 1/f scaling, for example in spiking activity recorded from multi-unit recordings [67] or optical imaging data [68], suggesting that additional dynamical mechanisms are required. Accordingly, we do not view these mechanisms as mutually exclusive, but rather as complementary: our results highlight that these mechanisms may jointly account for the diversity of spectral features observed across experimental conditions.

Our model deliberately balances biological relevance and mathematical tractability to the benefit of interpretability. It thus understandably suffers from limitations. First and foremost, biological neural networks are not random like the Erdős–Rényi graph we assumed here, but instead exhibit complex, hierarchical and/or small-world like architectures [69,70]. One important biological constraint is that connectivity of neural networks has to conform to Dale’s law, by which neurons or populations make either purely excitatory or purely inhibitory projections towards their target(s) [55,71], which is not satisfied in all analysis parts in the present work. Previous studies have shown that networks conforming to Dale’s law possess distinct stability properties, notably non-uniform eigenvalue distributions within the spectral disk (see [72] for a thorough discussion). The impact of these features on the stability analysis and resulting noise-induced spectral shifts remain to be further explored in detail. Yet, preliminary numerical simulations reveal that our conclusions hold qualitatively in network conforming to Dale’s law (see S2 Fig, S2 Text). Another important avenue for future work is to replace the random connectivity used here with empirically derived structural connectivity (e.g., from diffusion MRI). Empirical connectomes typically contain non-negative weights, and hence do not explicitly represent long-range inhibition. This represents an important limitation for the implementation of the excitation–inhibition balance that is known to critically shape network dynamics and associated spectral scaling [53,71]. Although oversimplified, we nonetheless believe random network models provide a robust and flexible framework for reconciling observations about power-law scaling made across systems exhibiting significant heterogeneity in scale and structural features. Moreover, the predefined frequency ranges to estimate the exponent are chosen for the animal data and the simulation model individually to reflect best their corresponding power spectrum. Despite these differences, enhanced intrinsic noise levels render consistently the power spectrum more flat in both natural and theoretical system.

The results we provide here rely heavily on the use of approximations: First, the Gaussian assumption concerning the stationary probability density function , inspired by previous work from the authors [40,42,43], remains an ansatz. While yielding good agreement with simulations, many of our results rely on the shape of this function (most specifically on its variance, also shown elsewhere [56,57]) and as such, further work is needed to derive its expression analytically. Second, state-dependent fluctuations are here modeled using additive, uncorrelated Gaussian white noise, an oversimplification [73] chosen for analytical convenience. Preliminary simulations nonetheless suggest that the phenomenon we characterize here generalizes to colored noise over a wide range of correlation time scales (see S2 Fig, S2 Text). Qualitatively, the magnitude of effect was found to decline as the noise autocorrelation time increases. We hypothesize that the autocorrelation time of the state-dependent fluctuations may be a key parameter in supporting the spectral shifts reported here. Further work is warranted to align our theoretical framework with these numerical predictions.

4. Methods

4.1. Neural network model

Our analysis is motivated by a well-known form of random neural network (RNN) [48,51,53,54,56,57] which have been shown to capture both the power-law scaling and spectral knee features observed in the broadband PSD of human EEG/ECoG [10,17]. In this network, each node corresponds to a single neural population with activity and the dynamics of N nodes obey the set of coupled evolution equations

(2)

with and where is a linear relaxation operator. The variable represents a mesoscopic population activity, corresponding to a coarse-grained aggregate of synaptic or membrane potentials over ensembles of neurons and across time [60]. The relaxation operator L reflects a mean synaptic response function of first order with re-scaled unit time constant. The variable represents an external input constant in time and is a nonlinear sigmoidal activation function with gain setting the individual excitability of neural populations. The connectivity matrix in Eq. (2), where each element denotes the connectivity from neuron population j to population i, represents a large-scale randomly connected excitatory–inhibitory network [17,53,57] with connection probability c. The weights are random, independent and normally distributed with mean and variance , defined over the ensemble of matrix realizations, and in numerical simulations they scale as without loss of generality. The strength of these excitatory and inhibitory connections are individually Gaussian-distributed with mean and , variance and and with probability density functions and , respectively. In addition to this network topology, a more biologically realistic setup may consider network nodes with either purely excitatory or purely inhibitory connections (Dale’s law, see, e.g., [55,71]). Such a topology is considered in addition in the current work and corresponding results are presented in S1 Text, confirming our simpler setup. Moreover, we assume that there are no self-connections, or that those have been absorbed in the relaxation operator L, so that all diagonal values of the connectivity matrix are zero . We note that this condition can be relaxed [21,63,74]. In addition, we parametrize the relative density of excitatory versus inhibitory connections by a coefficient r, . Consequently, the probability density function of connectivity weights may be written as . Moreover, we choose a balanced connection connectivity with and , i.e., the sum over excitatory and inhibitory connections and the ensemble mean over network realizations vanish at each node.

Given these constraints, the mean connectivity of the network is given by . Renaming , we have and the variance of the connectivity becomes [53]

(3)

This connectivity motif corresponds to a weighted Erdös–Rényi random graph, whose sparseness results from the low connection probability c. We emphasize, however, that the following results may be easily extended to other topologies, e.g., see the work of Yan and colleagues [75].

The system in Eq. (2) is further subjected to additive, state-dependent fluctuations here modeled as N independent Gaussian white noise processes of zero mean and variance D, i.e., with , such that , where is the Kronecker delta. The variance D scales the net, collective intensity of these fluctuations, and determines the scale of state-dependent fluctuations in the network.

4.2. Power spectral density

The classical approach to compute the power spectral density of complex systems such as those in Eq. (2) is to consider the linearized problem

(4)

where is a constant diagonal matrix with zero-valued non-diagonal entries (see Section 4.1). The matrix

refers to the Jacobian associated with the deterministic, unique and trivial fixed point [54,56,57] computed by setting D = 0 in Eq. (2). This is ensured by choosing the constant input to and thus assuming a balanced network.

Here and denote the Jacobians of L and F, respectively. The entries of the Jacobian are given by [56,57]. The circular law [75,76] states that the complex eigenvalues associated with the Jacobian are confined within a disk of radius given by [56,57,62,63,77–79]

(5)

where we recall that corresponds to the variance of the weights as per Eq. (3).

The power spectral density matrix associated with the system’s linearized dynamics about its fixed point, cf. in Eq. (4), can be directly obtained by computing the Fourier transform of the autocorrelation function, yielding the well-known expression for coupled Ornstein-Uhlenbeck processes [61]

(6)

where stands as the identity matrix with the angular frequency and the system’s frequency f. However, as rightfully noted in [58], Eq. (6) is generally intractable due to the presence of matrix inversion, especially for large N. However, Krumbeck and colleagues [58] have recently shown that given our model assumptions, it is possible to compute the mean power spectral density (PSD) associated with Eq. (2) by the means of complex Gaussian integrals. The mean PSD is defined here by the ensemble average of the auto-spectra, i.e.,

(7)

with . For the random Erdös-Renyi connectivity matrix considered here, the can be shown to simplify dramatically to the Lorentzian [58]

(8)

which possesses a power-law scaling with critical exponent whenever is large. Immediately identifiable in the denominator is the May-Wigner instability threshold , at which point slow frequency fluctuations in the network become infinitely large [58]. Maximum stability is achieved whenever , in which case the network’s properties become identical to the one obtained from a system of uncoupled Ornstein-Uhlenbeck processes, and [61].

4.3. Stochastic spectra

Equation (8) describes linear fluctuations about the system’s trivial fixed point. However, it fails to capture non-linear, noise-induced effects on the PSD as seen in Fig 1. Indeed, it suggests that the spectral slope and associated exponent are independent of the noise intensity D. This is a direct consequence of the linearization performed in Eq. (4) about the deterministic equilibrium , which discards potential non-linear contributions and noise-induced phenomena [46,59,80].

To circumvent this limitation, one may characterize the effect of additive noise through a variational approach inspired by the master stability function and Floquet theory [81,82]. Decomposing the activity into the sum of a linear stochastic part and deviations with the local node dynamics taken from Eq. (4) yields the variational problem

(9)

Here, is assumed to be small compared to and the Jacobian is

(10)(11)

The probability density function of is well approximated by a zero-mean Gaussian with variance D, i.e., due to the network’s balanced nature. We point out that previous studies linearize the transfer function f about the stationary state, in which case the nonlinear second term in Eq. (11) is not a function of . In contrast, we consider here the transfer function’s full nonlinearity, which yields the nonlinear effect observed in Fig 1.

By virtue to the system’s stationarity, Eq. (9) states that the Jacobian matrix elements are random variables with a certain time-independent probability density function. We express this by defining the random matrix . Here, means two matrices share their statistical properties. Then, by virtue of a stationary solution , we replace by the temporally constant , and may write

which can be split up into two independent evolution equations

(12)(13)

with . Since the trivial stationary state is stable, one has .

We point out that the assumption of small is motivated by i) the uniqueness of the (noiseless) trivial fixed point [56,57,83]; ii) previous studies performed on similar networks supporting the use of this ansat1z [40,42,43]; and iii) numerical observations suggesting the use of this approximation (i.e., Fig 2a-b). Consequently, the effective spectral radius reads for D > 0

(14)

with variance

(15)(16)

and . We point out that yields from Eq. (5). Moreover, we mention that previous studies linearize the transfer function about the stationary state, which makes the exponential function under the integral in Eq. (15) vanish yielding , i.e., independent of D. Hence, the presence of the nonlinearity renders the spectral radius dependent on the noise variance D.

Following the steps above, the spectral radius of the network activity about the stationary state obeying the linear node dynamics becomes a function of the noise intensity, i.e., . Motivated by Eq. (8), we may express the effective mean power spectral density as

(17)(18)

An important corollary of these calculation is a manifest shift in the May-Wigner instability threshold: Indeed, for , one obtains to first order , and the instability is shifted to a higher value of the spectral radius, i.e., . This implies that increased additive noise stabilizes the network, delaying the onset of instability as observed in other nonlinear high-dimensional systems [47,48,53,84].

4.4. Spectral knee

The slope of the broadband PSD commonly exhibits a characteristic bend that reflects the dominant temporal timescale of neural activity. This so called spectral knee indexes the frequency at which the aperiodic power law changes slope, marking a transition between different scaling regimes [6,7,17]. The spectral knee shifts the frequency range over which reliable estimates of the spectral slope and/or critical exponent can be obtained. The position of the spectral knee corresponds to the frequency where the spectrum bends away from a single power law [4], that is whenever

(19)

We mention, that this corresponds to the frequency where the exponent is , cf. Eq. (19) in the subsequent section. The equation above implies that the position of the spectral knee changes with the noise variance D: essentially the contraction of the spectral radius resulting from state-dependent fluctuations reduces the effective dominant neuronal timescale, and suppresses low-frequency activity.

4.5. Spectral alope and critical exponent

To determine the spectral slope and associated exponent , we computed the slope of the mean power spectral density in Eq. (17) in the log-log domain at a reference frequency , which upon the use of the chain rule yields

(20)

This result shows, that for small variances D and consequently the PSD flattens with increasing noise variance.

4.6. Numerics

For the present study, network simulations rely on the use of the Euler-Maruyama scheme with a time step of , implemented in Python and using the Numba package to speed up integration. The duration of the simulations, as well as the specific value of the parameters used in each case are specified in the figure panels, whenever appropriate. Our code was tested for numerical stability for various values of the time step , and its specified value was selected for convenience, guaranteeing numerical stability while limiting simulation time. Spectral properties were estimated by computing the power spectrum using the Discrete Fourier Transform (DFT) function in NumPy, applied either to individual node time series or to their population average. The time-frequency spectrum was estimated by applying the Short Time Fourier Transform implemented as spectrogram in the scipy module. To calculate the experimental spectral exponent at the given reference frequency , we performed simple linear regression over the log-log domain of the power spectral density on a frequency range centered around , i.e., . Further details can be found in the custom codes accompanying this publication.

4.7. Experimental data and statistics

We analyze previously recorded local field potentials (LFPs) [85,86] obtained using a multi-electrode array implanted at the border between the posterior parietal association cortex (PtA) and secondary visual cortex in adult male Sprague–Dawley rats. All experimental procedures and protocols were approved by the Institutional Animal Care and Use Committee of the Medical College of Wisconsin (Milwaukee, Wisconsin), they conformed to the Guiding Principles in the Care and Use of Animals of the American Physiologic Society and were in accordance with the Guide for the Care and Use of Laboratory Animals (National Academy Press, Washington, DC, USA, 1996), for more details see [86].

Our study examines data from seven rats in a resting state at different levels of the anaesthetic desflurane in the absence of any stimulation. Anesthetic concentrations of 3.5% and 6% are referred to as light and deep anesthesia, respectively. The time series under study have a duration of 12s observed under a sampling rate of 1kHz and the observed signal has been band-pass filtered between 0.3Hz and 500Hz. Subsequently the signal was down-sampled to 500Hz. The power spectral density (PSD) was computed with the Welch method with sliding time window of 5s and overlap rate 0.8 yielding a frequency resolution of 0.2Hz. The linear function fits assumed the frequency range [5Hz;50Hz]. Moreover, we applied a paired bootstrap test with 105 surrogate trials and the significance threshold is chosen to 0.05.

Supporting information

S1 Text. Extension to connectivities conforming to Dale’s law.

Description of the construction and normalization of sparse excitatory–inhibitory networks satisfying Dale’s law, used to assess the robustness of noise-induced spectral shifts to biologically constrained connectivity.

https://doi.org/10.1371/journal.pcsy.0000133.s001

(PDF)

S2 Text. Extension to temporally correlated Ornstein–Uhlenbeck noise.

Description of simulations using Ornstein–Uhlenbeck noise to assess whether noise-induced shifts in the spectral knee and critical exponent persist in the presence of temporally correlated fluctuations.

https://doi.org/10.1371/journal.pcsy.0000133.s002

(PDF)

S1 Fig. Noise-induced spectral shifts in networks conforming to Dale’s law.

The effective mean power spectral density (see Eq. (18); normalized) is shown for increasing noise variance (D = 10–3, 10–4, 10–5, and 10–6, from pale to dark). Scatter points correspond to numerical simulations of Eq. (2), averaged over independent network realizations. Across panels the variance of synaptic weights is increased: a. (blue); b. (orange); c. (green). Data points denote the mean SED (standard error of the deviation) computed across independent trials. Connectivity matrices satisfy Dale’s law such that each neural population makes either purely excitatory or purely inhibitory projections. A fraction r = 0.8 of populations are excitatory and inhibitory. Parameters are otherwise identical to Fig 3 in the main text: N = 300, , c = 0.05 and . Results were averaged over 10 independent trials of duration T = 30 s with time step ms.

https://doi.org/10.1371/journal.pcsy.0000133.s003

(TIF)

S2 Fig. Spectral shifts induced by colored noise with increasing correlation time.

The normalized mean power spectral density is shown for increasing variance ( ,, and , from pale to dark). Note that the values of D have been increased compared to the results in the main text to compensate for the smaller effective variance of the processes . The noise correlation time is increased across each row. Across columns, the variance of synaptic weights is increased: a,d,g,j. (blue); b,e,h,k. (orange); c,f,i,l. (green). Scatter points correspond to numerical simulations of Eq. (2), averaged over independent network realizations. Data points denote the mean SED (standard error of the deviation) computed across independent trials. Parameters are otherwise identical to Fig 3 in the main text: N = 300, , c = 0.05 and . Results were averaged over 10 independent trials of duration T = 30 s with time step ms.

https://doi.org/10.1371/journal.pcsy.0000133.s004

(TIF)

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