# Noise-like fluctuations drive shifts in 1/ f α power-law dynamics associated with changes in brain states

Axel Hutt, Matteus Mcculloch, Aref Pariz, Jeremie Lefebvre

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Axel Hutt, Matteus Mcculloch, Aref Pariz, Jeremie Lefebvre. Noise-like fluctuations drive shifts in 1/ f α power-law dynamics associated with changes in brain states. 2025. hal-05367956

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# Noise-like fluctuations drive shifts in $1/f^\alpha$ power-law dynamics associated with changes in brain states

Axel Hutt¹, Matteus McCulloch², Aref Pariz³,⁴,⁵, and Jérémie Lefebvre²,⁴,⁵,⁶,*

¹MLMS, NECTARINE, Université de Strasbourg, CNRS, Inria, ICube, 67000 Strasbourg, France

²Department of Cellular and Molecular Medicine, University of Ottawa, Ottawa, ON K1N 6N5, Canada

³Institute of Mental Health Research at The Royal, Ottawa, ON, Canada

⁴Department of Biology, University of Ottawa, Ottawa, ON K1N 6N5, Canada

⁵Krembil Brain Institute, University Health Network, Toronto, ON M5T 0S8, Canada

⁶Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada

*Corresponding author: jeremie.lefebvre@uottawa.ca

November 17, 2025

## Abstract

The aperiodic, broadband components of many electrophysiological recordings display a characteristic $1/f^\alpha$ power-law scaling. This highly debated feature of the power spectral density (PSD), observed in many neural systems, exhibits substantial variability across behavioral states, levels of arousal, and pharmacological interventions. Fluctuations in the critical exponent $\alpha$ have been linked to transitions between distinct brain states, ranging from synchronized, slow-wave dynamics during sleep to desynchronized activity during wakefulness, for instance. We investigated the origins and variability of this $1/f^\alpha$ power-law scaling using large-scale, recurrent neural networks model with sparse, balanced and random connectivity, well suited to capture the diversity in dynamics, structure and scale characteristic of neural systems exhibiting such critical-like behavior. By integrating recent advances in random matrix theory, we develop an approach to determine critical features of the PSD by accounting for both nonlinear and stochastic contributions. These proved to be essential in demonstrating that variability in power-law scaling is a generic feature of random, nonlinear networks exposed to changes in additive noise. These results provide insight as to the mechanisms shaping macroscopic spectral signatures of varying brain states.
# 1 INTRODUCTION

The spectral characteristics of the so-called broadband brain activity –specifically the aperiodic component of elec-
trophysiological recordings [1–3] – have been repetitively shown to conform to a 1/f<sup>α</sup> power-law scaling, where α
 corresponds to an exponent parameterizing the decay of the signal amplitude at larger frequencies [1,2,4,5]. For instance,
electrophysiological recordings with a high value of α are dominated by large-amplitude, slow-wave synchronous
fluctuations, and are commonly observed during sleep and anesthesia [6–9]. In contrast, a decrease in the value of
α, reflecting a less steep power spectrum, suggests more desynchronized brain activity, as seen during arousal and
wakefulness [6, 8–10].

Features of the power spectrum – such as frequency and amplitude – show considerable variability with behavior,
arousal, and pharmacological interventions (see [2] and references therein). As such, shifts in the exponent α have
emerged as an index of brain state [11], following variations in arousal [8], anesthesia [8], sleep stages [6, 7], cognitive
load [12], task demands [13], and aging [14]. This variability has been linked to fluctuations in the collective, distributed,
and heterogeneous activity of large neuronal assemblies [1]. These reflect highly variable and dynamic physiological
processes occuring in the cortex, and shaped by the discrete nature of neural discharges, correlations in neural responses,
filtering of neural signals through cortical tissue, and the interplay between excitation and inhibition [1, 2, 15, 16].

While its origin remains highly debated [9, 17–21], the presence of 1/f<sup>α</sup> power-law scaling, also called scale-free
dynamics, is deeply intertwined with the notion of dynamical stability in complex systems. Indeed, the exponent α has
been directly associated with the distance to bifurcation points, and a general disposition for pattern formation and
self-organized criticality [9, 17, 20–22] (but see [18, 19]). As such, it has been reported in a variety of physical [9, 17]
and neuroscience [15] phenomena, ranging from long-range temporal correlations in neuronal oscillations [1, 2, 4, 5] to
the size and duration of neuronal avalanches [23, 24].

The variability in 1/f<sup>α</sup> is consistent with the phenomenology of nonlinear systems undergoing phase transitions: brain dynamics shift between distinct states, accompanied by a pronounced change in coherent activity [25–27]. Such transitions have repeatedly been linked with the reduction in noise-like, seemingly random fluctuations in brain activity characterizing the awake, aroused state of the cortex [25, 28–37]. These irregular fluctuations reflect elevated neural activity, originating from sensory stimuli, ascending inputs from brainstem (e.g., the Ascending Reticular Arousal System), and inter-area communication. It has been well established that these fluctuations, ubiquitous at all scales of the nervous system [38], represent an important contributor for the function and stability of brain networks [26, 28, 34, 35, 37, 39–41].

To better understand the origins and variability of this 1/f<sup>α</sup> power-law scaling, we analyzed the power spectral
properties of a widely used class of recurrent neural network (RNN), commonly used to study interacting neural
units and/or populations at a variety of spatial and temporal scales [22, 40–48]. Although this framework may at first
appear narrow in scope, we argue that, to the contrary, neural networks with random connectivity provide a robust
and versatile means of capturing the diversity, inherent variability, scale and structural complexity characteristic of
neural systems in which critical, power-law like behavior have been reported (see [9, 23, 25] and references therein). We
leveraged this framework combined with recent advances in random neural networks [49] to examine the influence of
state-dependent fluctuations - therein modeled as additive noise of different amplitudes - on the variability in 1/f<sup>α</sup>
power-law scaling experienced across the transition between different brain states [40–48]. Using the theory of random
matrices, we developed a method that enables the derivation of the mean power spectral density (PSD) taking into
account nonlinear, stochastic contributions. Our results show that state-dependent fluctuations in neural activity can
generically induce shifts in power-law scaling and critical dynamics [40, 41, 44], mirroring those seen during transitions
between wakefulness and anesthesia, and reflected in changes in the critical exponent.
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Example of state-dependent variability in power-law dynamics a. Response of the balanced, nonlinear neural network in Eq. (1) to random fluctuations whose amplitude is allowed to vary in time. Driven by changes in noise amplitude (D; blue line; top), the system exhibits intermittent shifts in dynamics (network average; black line; middle). These are marked by high-amplitude, oscillatory, and synchronous activity (green box), alternating with epochs of high-frequency, irregular, and asynchronous patterns (red box), visible from the time-frequency spectrum (bottom). b. Power spectral density of the mean network activity over an epoch of low noise intensity (ΔT = [22–27] sec), exhibiting strong low frequency power (gray shaded area) and steep 1/f^α decay at high frequencies. The green line represents linear regression performed over the [10–30]Hz frequency range. c. Power spectral density of the mean network activity over an epoch of high noise intensity (ΔT = [10–15] sec), where low frequency power has been suppressed (gray shaded area), and the slope at higher frequency is increased, indicating a shift in critical exponent (α) caused by additive state-dependent fluctuations. The red line represents linear regression performed over the [10–30]Hz frequency range. Parameters are N = 250, ρ = 0.05, r = 0.8, β = 50, μ<sub>e</sub> = 5 × 10<sup>-4</sup>, μ<sub>i</sub> = -r(1-r)<sup>-1</sup>μ<sub>e</sub>, σ<sub>e</sub><sup>2</sup> = σ<sub>i</sub><sup>2</sup> = 2 × 10<sup>-4</sup> and I<sub>i</sub> = 0. Numerical integration was performed over a trial of duration T = 50s with a time step of Δt = 1ms. 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.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>**Figure 2.** State-dependent fluctuations drive shifts in stability. **a.** Probability density function (PDF) of the network activity, computed numerically for different noise amplitudes: $D = 1 \times 10^{-5}$ (bright red line), $D = 1 \times 10^{-4}$ (red line) and $D = 1 \times 10^{-3}$ (dark red line). These adopt a Gaussian profile. **b.** The variance of the PDF computed numerically (dark red line) scales linearly with noise amplitude (D; dashed black line). **c.** Spectral disk of the corresponding Jacobian $J̄$, cf. Eq. (11), delimiting the eigenvalues (gray circles) in the complex plane. Increasing the amplitude 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); $D = 1 \times 10^{-5}$ (bright red circle); $D = 1 \times 10^{-4}$ (red circle) and $D = 1 \times 10^{-3}$ (dark red circle)). This reinforces linear behavior and hence stability, pushing the network eigenspectra away from the May-Wigner instability threshold ($\Gamma_{crit} = 1$; vertical dashed black line). Parameters are $N = 250$, $\rho = 0.05$, $r = 0.8$, $\beta = 50$, $\mu_e = 5 \times 10^{-4}$, $\mu_i = -r(1-r)^{-1}\mu_e$, $\sigma_e^2 = \sigma_i^2 = 9 \times 10^{-5}$ and $I_i = 0$. Numerical integration was performed over a trial of duration $T = 5$s using $\Delta t = 1$ms.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Stochastic analysis captures the power-law scaling of the power spectrum. The effective mean power spectral density $\tilde{S}_{\text{eff}}(\omega)$ (see Eq. (17)) is shown for increasing noise variance (i.e., $D = 1 \times 10^{-3}$, $D = 1 \times 10^{-4}$, $D = 1 \times 10^{-5}$, and $D = 1 \times 10^{-6}$, 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 amplitudes. Dots correspond to numerical simulations, averaged over independent network realizations, whereas dashed lines represent the theoretical prediction taking into account nonlinear, stochastic effects (i.e., Eq. (17)). The bold lines represent the prediction ignoring noise contributions (i.e., Eq. (8)). Across each panel, the synaptic weight variance is increased, i.e., **a.** $\sigma_w^2 = 0$ (blue); **b.** $\sigma_w^2 = 4 \times 10^{-5}$ (orange); **c.** $\sigma_w^2 = 6.5 \times 10^{-5}$ (green). The shaded areas denote the error range corresponding to the standard deviation. Other parameters are $N = 300$, $\rho = 0.05$, $r = 0.8$, and $\beta = 50$. Results were averaged over 10 independent trials of duration $T = 30$ s with a time step of $\Delta t = 1$ ms.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Figure 4.** State-dependent fluctuations suppress the critical exponent. Larger noise amplitude ($D$) suppresses the critical exponent ($\alpha$) of the mean power spectral density $\tilde{S}_{\text{eff}}(\omega)$, as shown for increasing connectivity variance ($\sigma_w^2 = 0$ (blue), $2 \times 10^{-5}$ (orange) and $4 \times 10^{-5}$ (green)). The theoretical predictions based on noise-induced, nonlinear effects are shown as dashed lines, whereas horizontal lines represent the prediction ignoring noise contributions, at a reference frequency of $\omega_o = 12\text{Hz}$. The dots depict the critical exponent, numerically determined using linear regression performed over the [11–18]Hz frequency range, and averaged over independent network realizations. The shaded area denotes the error range corresponding to the standard deviation. Results were averaged over 10 independent trials of duration $T = 30$ sec with a time step of $\Delta t = 1$ms. Other parameters are $N = 300$, $\rho = 0.05$, $r = 0.8$, and $\beta = 50$.</figcaption></figure>
## 2 RESULTS

We here analyzed the collective activity $\mathbf{u}(t)$ of a network of neural populations of number $N$, whose dynamics obeys

$$ \frac{d}{dt} \mathbf{u} = L[\mathbf{u}] + \mathbf{Wf}[\mathbf{u}] + \mathbf{I} + \eta(t) \quad (1) $$

with $u \in \mathbb{R}^N$, where $L[\mathbf{u}]$ denotes a linear relaxation operator, and the vector $\mathbf{f}[\mathbf{u}] = (f[u_1], f[u_2], ..., f[u_N])^T$ is a sigmoidal activation function and $\mathbf{I}$ is a constant input. The matrix $\mathbf{W}$ denotes the network connectivity, which is sparse and balanced with connection probability $c$: excitatory and inhibitory synaptic weights are drawn from normal distributions while their sum at each node is zero [44,47,48]. To emulate changes in a brain state, 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., $\eta(t) = \sqrt{2D}(\xi_1(t), \xi_2(t), ..., \xi_N(t))^T$ with $\xi_n \sim \mathcal{N}(0, 1)$ (see section 4.1 for additional details).

A representative example of the resulting dynamics can be seen in Fig 1a. Additive noise, whose variance was allowed to vary randomly in time, was applied to each node of the network while the power spectral density of the system was examined. State-dependent fluctuations cause intermittent shifts in dynamics: epochs, where noise variance is low, are characterized by high amplitude, oscillatory and synchronous activity. In contrast, these dynamics were replaced by high frequency, irregular and asynchronous activity in epochs whenever the noise variance increases. These transitions suggest noise-induced changes in network stability, where state-dependent fluctuations can engage with and disrupt the network's intrinsic dynamics, thereby facilitating qualitatively distinct dynamical regimes [39, 41, 47, 48]. These regimes are notably characterized by distinct power spectral density (PSD) profiles. As shown in Fig. 1b,c), increasing the noise variance fluctuations causes a suppression of low-frequency contributions and an enhancement of higher frequencies leading to a net reduction in the critical exponent, as observed experimentally [8, 22].

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 [49, 50] to compute the PSD. However, such approaches offer limited insight: they are difficult to perform by matrix inversion in large networks [49, 50], and further fail at capturing nonlinear effects. Indeed, analyses relying on such deterministic premises preclude any noise-induced change in the corresponding critical PSD exponent $\alpha$. 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 $\bar{S}(\omega) = 2D(1 - \Gamma^2 + \omega^2)^{-1}$, where $\Gamma$ corresponds to the noise-independent spectral radius of the network [49], cf. section 4.2. This distributions possesses a $1/f^\alpha$ power-law scaling with fixed critical exponent $\alpha = 2$ whenever $\omega$ is large, irrespective of the value of $D$. However, this does not agree with experimental findings, see also Fig. 1.
To bypass these limitations and obtain further insight, 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 (MSF) to determine stability, not by linearizing about deterministic equilibria, but about the stochastic system evolution assuming the stationary probability density function of **u** of Gaussian shape. This assumption is motivated by the uniqueness of the (noiseless) fixed point **u**<sup>o</sup> [45, 47, 48] and further confirmed by extensive numerical simulations supporting this approximation (see Fig. 2a-b).

This enabled us to determine an *effective* spectral radius $\Gamma_{\text{eff}}$, 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 amplitude of state-dependent fluctuations. Indeed, as illustrated in Fig. 2c, increasing $D$ contracts the spectral radius, thereby amplifying linear behavior and steering the system away from instability. We note that these results are fully consistent with the predictions made about the stabilizing role of noise in networks with similar architectures [34, 37].

To confront these theoretical findings, we revisited the expression for the mean PSD, but now using the effective spectral radius (i.e. $\Gamma_{\text{eff}}$, see Section 4.3). The resulting effective mean PSD, $\bar{S}_{\text{eff}}(\omega)$, possesses an explicit, nonlinear dependence on the amplitude of the state-dependent fluctuations (i.e., $D$). As can be seen in Fig. 3, this approach accurately captured the variability in power-law scaling induced by state-dependent fluctuations. This prediction was further supported by numerical simulations, averaged across multiple, independent realizations of initial conditions and for different network synaptic connectivity (i.e., $\sigma_w^2$, see Eq. (3)), and preserved across noise amplitudes over multiple orders of magnitude (Fig. 3a-c). This alignment provides strong support for the theoretical analysis and its underlying assumptions (see Section 4.3). Importantly, incorporating noise into the analysis reveals that the system is far more stable than predicted by standard linearization techniques: the mean PSD is notably less steep. Significant deviations emerge between the actual mean PSD and that predicted by linearization 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 $\sigma_w^2$ with $\Gamma \rightarrow \Gamma_{\text{crit}}$) and for larger noise amplitudes, where deviations between the noise-corrected PSD and deterministic predictions are largest (see Fig. 3c). As the noise amplitude is decreased, these nonlinear corrections to the PSD vanish (Fig. 3a-b).

Collectively, these findings suggest that, within the constraints of our model, state-dependent fluctuations - modeled here as additive noise - impact the stability of the network under study, an effect that should be immediately visible in the critical exponent calculations that followed. We thus confronted the value of $\alpha$ determined using theory (see Section 4.4) to numerical simulations, by performing linear regression on the PSD over a set frequency range. As can be seen in Fig 4, for both the theoretical derivation and simulations, state-dependent fluctuations suppress the value of $\alpha$, the signature of an increasingly flat, desynchronized power spectrum. This was not captured by standard approaches
neglecting noise. We note that our analytical derivations consistently underestimated the decrease in $\alpha$ observed numerically across the range of noise amplitudes explored. This discrepancy results from the use of approximations in our derivations. Taken together, these results allow us to conclude that critical exponent variability appears to be a generic feature of random, nonlinear networks exposed to changes in additive noise amplitude.

# 3 DISCUSSION

We used a large-scale, recurrent neural networks model with sparse, balanced and random connectivity to investigate the variability of $1/f^\alpha$ characterizing the broadband components of various types electrophysiological recordings. Changes in the critical exponent $\alpha$ have repeatedly been shown to accompany transitions brain states, following shifts in behavior, arousal, or as a result of different pharmacological interventions. Our joint mathematical and computational analyses demonstrate that this variability spectral properties correspond to a generic feature of random networks in response to state-dependent fluctuations - here modeled as additive noise - of varying amplitudes. Utilizing previous results provided by Krumbeck and colleagues [49] 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 may capture the power-law scaling as well as other features of the broadband power distribution measured experimentally [9, 15, 25]. These have further suggested that changes in spatially-dependent connectivity profiles could be key in generating changes in $1/f^\alpha$ scaling that resemble those observed across different brain states [8]. 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 [41]. This view also echoes findings made in neural mass models of interacting excitatory and inhibitory pools [16, 41], highlighting the critical importance of noisy stimuli and excitation-inhibition balance in the genesis of critical-like activity.

Our model deliberately balances biological relevance and mathematical tractability to the benefit of interpretability. It thus understandably suffers from limitations. First and foremost, neural networks are not random like the Erdős–Rényi graph we assumed here, but instead possess complex, hierarchical and/or small-world like architectures [51, 52]. While our theoretical framework could be extended to other, more physiologically grounded topologies [53], its impact on the stability analysis we performed, and the $1/f^\alpha$ scaling that followed remain to be further explored. Although oversimplified, we 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. The results we provide here rely heavily on the use of approximations: First, the Gaussian assumption concerning the stationary probability density function $\rho_S(u)$, inspired by previous work from the authors [34, 36, 37], 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 [47, 48]) 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 [54] chosen for analytical convenience.

# 4 METHODS

## 4.1 Neural Network Model

Our analysis is motivated by a well-known form of recurrent neural network (RNN) [22, 41, 42, 44, 45, 47, 48] with $N$ number of nodes in which the activity $u_j$ of a neuron $j$ obeys the set of coupled evolution equations

$$ \frac{d}{dt} u_i(t) = L[u_i] + \sum_{j=1}^{N} w_{ij} f[u_j] + I_i + \sqrt{2D} \xi_i(t) \quad (2) $$

with $i = 1, \dots, N$ and where $L[u] = -u$ is a linear relaxation operator. The variable $I_i$ represents an external input constant in time and $f[u] = \frac{1}{2}(1 + \text{erf}[\beta u])$ is a nonlinear sigmoidal activation function with gain $\beta$ setting the individual excitability of neurons. The connectivity matrix $W$ in Eq. (2), where each element $[W]_{ij} = w_{ij}$ denotes the synaptic coupling from neuron $j$ to neuron $i$, represents a randomly connected excitatory-inhibitory network [44] with connection probability $c$. The weights $w_{ij}$ are normally distributed with mean $\mu_W = E_W[w_{ij}]$ and variance $\sigma_W^2 = \text{Var}_W[w_{ij}] > 0$, defined over the ensemble of matrix realizations, and in numerical simulations they scale as $w_{ij} \sim 1/\sqrt{N}$ without loss of generality. The strength of these excitatory and inhibitory synaptic connections are individually Gaussian-distributed with mean $\mu_e$ and $\mu_i$, variance $\sigma_e^2$ and $\sigma_i^2$ and with probability density functions $p_e(w)$ and $p_i(w)$, respectively. We ensure that there are no self-connections i.e. $w_{ii} = 0 \forall i$. In addition, we parametrize the relative density of excitatory versus inhibitory connections by a coefficient $r$, $0 \le r \le 1$. Consequently, the probability density function of synaptic weights $w_{ij}$ may be written as $p(w) = crp_e(w) - c(1-r)p_i(w)$. Moreover, we choose a balanced connection connectivity with $\sum_{j=1}^N w_{ij} = 0$ and $\mu_W = 0$, i.e. the sum over excitatory and inhibitory synaptic connections and the ensemble mean over network realizations vanish at each node.

Given these constraints, the mean connectivity of the network is given by $\mu_W = cr\mu_e + c(1-r)\mu_i = 0$. Renaming $\mu_e = \mu$, we have $\mu_i = r\mu_e/(r-1) = r\mu/(r-1)$ and the variance of the synaptic connectivity becomes [44]

$$ \sigma_W^2 = \text{Var}[w_{ij}] = c(r\sigma_e^2 + (1-r)\sigma_i^2 + r\mu^2/(1-r)). \quad (3) $$
This connectivity motif corresponds to a weighted Erdös–Rényi random graph. We emphasize, however, that the following results may be easily extended to other topologies, e.g. see the work of Yan and colleagues [53].

The system in Eq. (2) is further subjected to additive random fluctuations in the form of N independent Gaussian white noise processes of zero mean and variance D, i.e., $\eta(t) = \sqrt{2D}\xi(t)$ with $\xi(t) = (\xi_1(t), \xi_2(t), ..., \xi_N(t))^T$, such that $\langle \xi_i(s)\xi_j(t) \rangle = \delta(t-s)\delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta. The variance D scales the net, collective variance of these fluctuations, and determines the amplitude 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

$$ \frac{d}{dt} \mathbf{y} = J[\mathbf{0}]\mathbf{y} + \mathbf{B}\xi(t) \quad (4) $$

where $[\mathbf{B}]_{ij} = \delta_{ij}\sqrt{2D}$ is a constant diagonal matrix with zero-valued non-diagonal entries. The matrix

$$ \mathbf{J}[\mathbf{u}^o] = \mathcal{D}\mathcal{L}[\mathbf{u}^o] + \mathbf{W}\mathcal{D}f[\mathbf{u}^o] $$

refers to the Jacobian associated with the deterministic, unique and trivial fixed point $\mathbf{u}^o = \mathbf{0}$ [45, 47, 48] computed by setting $D = 0$ in Eq. (2). This is ensured by choosing the constant input to $I_i = -f(0)\sum_{j=1}^{N} K_{ij}$. We mention that $I_i \approx 0$ for small noise variances since the network is balanced.

Here $\mathcal{D}\mathcal{L}$ and $\mathcal{D}f$ denote the Jacobians of L and F, respectively. The entries of the Jacobian are given by $J_{ij}[\mathbf{0}] = -\delta_{ij} + w_{ij}\frac{\partial f}{\partial u_j}|_{u_j^o=0} = -\delta_{ij} + \frac{w_{ij}\beta}{\sqrt{\pi}}$ [47, 48]. The circular law states [53, 55] that the complex eigenvalues associated with the Jacobian $\mathbf{J}[\mathbf{0}]$ are confined within a disk of radius $\Gamma$ given by [47, 48, 56–60]

$$ \Gamma = \sqrt{(N-1)\mathrm{Var}_W[J_{ij}[\mathbf{0}]]} = \sqrt{\frac{(N-1)\beta^2\sigma_w^2}{\pi}}, \quad (5) $$

where we recall that $\sigma_w^2$ corresponds to the variance of the synaptic weights as per Eq. (3).

The power spectral density matrix $\mathbf{S}(\omega)$ 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 [50]

$$ \mathbf{S}(\omega) = \frac{1}{2\pi} (J[\mathbf{0}] + i\omega \mathbb{I}_N)^{-1} \mathbf{B}\mathbf{B}^T (J[\mathbf{0}]^T - i\omega \mathbb{I}_N)^{-1} \quad (6) $$
where $\mathbb{I}_N$ stands as the identity matrix with the angular frequency $\omega = 2\pi f$ and the system's frequency $f$. However, as rightfully noted in [49], Eq. (6) is generally intractable due to the presence of matrix inversion, especially for large $N$. However, Krumbeck and colleagues [49] 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.

$$ \bar{S}(w) = \mathbb{E}_W[S_{ii}(\omega)] \qquad (7) $$

with $S_{ii} = (S)_{ii}$. For the random Erdös-Renyi connectivity matrix considered here, the $\bar{S}(w)$ can be shown to simplify dramatically to the Lorentzian [49]

$$ \bar{S}(\omega) = \frac{2D}{1 - \Gamma^2 + \omega^2} \qquad (8) $$

which possesses a $1/f^\alpha$ power-law scaling with critical exponent $\alpha = 2$ whenever $\omega$ is large. Immediately identifiable in the denominator is the May-Wigner instability threshold $\Gamma_{crit} = 1$, at which point slow frequency fluctuations in the network become infinitely large [49]. Maximum stability is achieved whenever $\Gamma \to 0$, in which case the network's properties become identical to the one obtained from a system of uncoupled Ornstein-Uhlenbeck processes, and $\bar{S}(\omega) = 2D(1 + \omega^2)^{-1}$ [50].

## 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. Indeed, it suggests that the critical exponent $\alpha$ is independent of the noise intensity $D$. This is a direct consequence of the linearization performed in Eq. (4) about the deterministic equilibrium $u^o$, which discards potential non-linear contributions and noise-induced phenomena [39, 61, 62].

To circumvent this limitation, one may characterize the effect of additive noise through a variational approach inspired by the master stability function (MSF) and Floquet theory [63, 64]. Decomposing the activity $u(t)$ into the sum of a linear stochastic part $y(t)$ and deviations $\delta u(t)$ with the local node dynamics taken from Eq. (4) yields the variational problem

$$ \frac{d}{dt} \delta u = \tilde{\mathbf{J}}[y(t)] \delta u. \qquad (9) $$

Here, $\delta u$ is assumed to be small compared to $y$ and the Jacobian $\tilde{\mathbf{J}}$ is

$$ \tilde{J}_{ij} = -\delta_{ij} + w_{ij} \frac{\partial f}{\partial u_j} \Big|_{y(t)} \qquad (10) $$

$$ = -\delta_{ij} + \frac{w_{ij}\beta}{\sqrt{\pi}} e^{-\beta^2 y_j^2(t)}. \qquad (11) $$
The probability density function of $y_i$ is well approximated by a zero-mean Gaussian with variance $D$, i.e. $\rho(y) \sim N(0, D)$ due to the network's balanced nature.

By virtue to the system's stationarity, Eq. (9) states that the Jacobian matrix elements a random variables with a certain time-independent probability density function. We express this by defining the random matrix $\tilde{J}_y \sim \tilde{J}[y(t)]$. Here, $A \sim B$ means two matrices $A, B$ share their statistical properties. Then, with small deviations $\delta u(t)$ and temporally constant $\tilde{J}_y$, we may write

$$ \begin{aligned} \frac{du}{dt} &= \frac{dy}{dt} + \frac{d}{dt}\delta u(t) \\ &= \mathbf{J}[0]y + \tilde{\mathbf{J}}_y \delta u + B\xi(t) \end{aligned} $$

which can be split up into two independent evolution equations

$$ \frac{dy}{dt} = \mathbf{J}[0]y \qquad (12) $$

$$ \frac{d}{dt} \delta u(t) = \tilde{\mathbf{J}}_y \delta u + B \xi(t). \qquad (13) $$

Since the trivial stationary state is stable, $|y(t)| \to 0$.

We point out that the assumption of small $|\delta u|$ is motivated by i) the uniqueness of the (noiseless) trivial fixed point $\mathbf{u}^o = \mathbf{0}$ [47, 48, 65]; ii) previous studies performed on similar networks supporting the use of this ansatz [34, 36, 37]; and iii) numerical observations suggesting the use of this approximation (i.e., Fig 2a-b). Consequently, the effective spectral radius $\Gamma_{\text{eff}}$ reads for $D > 0$

$$ \Gamma_{\text{eff}} = \sqrt{(N-1)\text{Var}_W[(\tilde{\mathbf{J}}_y)_{ij}]} \qquad (14) $$

with variance

$$ \text{Var}_W [(\tilde{\mathbf{J}}_y)_{ij}] = \frac{\sigma_w^2 \beta^2}{\pi} \int_{-\infty}^{\infty} e^{-2\beta^2 y^2} \rho_y(y) dy \qquad (15) $$

$$ = \frac{\sigma_w^2 \beta^2}{\pi \sqrt{\gamma}}, \qquad (16) $$

and $\gamma = 1 + 4\beta^2 D$. We point out that $D \to 0$ yields $\Gamma_{\text{eff}} \to \Gamma$ from Eq. (5).

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., $\Gamma_{\text{eff}} = \Gamma_{\text{eff}}(D)$. Motivated by Eq. (8), we may express the effective mean power spectral density as
$$ \begin{aligned} \bar{S}_{\text{eff}}(\omega, D) &= \frac{2D}{1 - \Gamma_{\text{eff}}^2(D) + \omega^2} \\ &= \frac{2D}{1 - (N - 1)\sigma_W^2\beta^2/\pi\sqrt{1 + 4\beta^2D} + \omega^2} \end{aligned} \qquad (17) $$

An important corollary of these calculation is a manifest shift in the May-Wigner instability threshold: Indeed, for $D \ll 1$, one obtains to first order $\Gamma_{\text{eff}} = (1 - \beta^2 D)\Gamma + \mathcal{O}(2)$, and the instability is shifted to a higher value of the spectral radius i.e., $\Gamma_{\text{crit}} = (1 - \beta^2 D)^{-1}$. This implies that increased additive noise stabilizes the network, delaying the onset of instability as observed in other nonlinear high-dimensional systems [40, 41, 44, 66].

## 4.4 Critical exponent

To determine the critical exponent, we computed the slope of the mean power spectral density $\bar{S}(\omega)$ in Eq.(17) in the log-log domain at a reference frequency $\omega_o$, which upon the use of the chain rule yields

$$ \alpha = \frac{d \log[\bar{S}(\omega)]}{d \log[\omega]} \Big|_{\omega=\omega_o} = - \frac{2\omega_o^2}{1 - \Gamma_{\text{eff}}(D)^2 + \omega_o^2}. \qquad (18) $$

This result shows, that $d\alpha/dD > 0$ for small variances $D$ and consequently the PSD flattens with increasing noise variance.

## 4.5 Numerics

For the present study, network simulations rely on the use of the Euler-Maruyama scheme with a time step of $\Delta t$, 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 $\Delta t$, 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 $\omega_o$, we performed simple linear regression over the log-log domain of the power spectral density on a frequency range $\Delta\omega$ centered around $\omega_o$ i.e., $\Delta\omega = [\omega_o - \delta\omega, \omega_o + \delta\omega]$. Further details can be found in the custom codes accompanying this publication.
**Author Contributions**

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JL designed the study, performed the analysis and wrote the manuscript. MM provided extensive support for the simulations. AP performed extra simulations for result validation and reviewed the manuscript. AH designed the study, performed parts of the analysis and wrote the manuscript.

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**Data Availability Statement**

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The data that support the findings of this study are available from the corresponding author upon reasonable request.
Custom codes developed for this study will be made publicly available upon acceptance.

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**Acknowledgments**

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We thank the National Sciences and Engineering Research Council of Canada (NSERC Grants RGPIN-2017-06662 to JL a) for support of this research.

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