# Additive noise shapes the state and stability of random networks

Axel Hutt

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Axel Hutt. Additive noise shapes the state and stability of random networks. The European Physical Journal. Special Topics, 2025, 10.1140/epjs/s11734-025-01789-4. hal-04967876

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Submitted on 26 Feb 2025

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# The European Physical Journal Special Topics

## Additive noise shapes the state and stability of random networks

--Manuscript Draft--

<table><tr><td>Manuscript Number:</td><td></td></tr><tr><td>Full Title:</td><td>Additive noise shapes the state and stability of random networks</td></tr><tr><td>Article Type:</td><td>Regular Article</td></tr><tr><td>Funding Information:</td><td></td></tr><tr><td>Abstract:</td><td>Natural systems are open to their environment and thus subjected to external perturbations. Some of such systems may be described as large randomly connected networks. The present work investigates how external additive random perturbations affect the state and stability of such finite-size random networks. We observe analytically an additive noise-induced system evolution (ANISE), whose stationary state depends on the additive noise level. In a first pilot study, the principal analytical approach is demonstrated by application to a nonlinear Erdos-Renyi network. In a second more detailed study, a nonlinear random network of excitatory and inhibitory sub-networks describes successfully Event-Related Desynchronization and Synchronization (ERD/ERS) observed in experimental brain signals. In sum, we find that additive noise impacts on the system's stationary state and in turn also affects the system's stability and hence its spectral properties.</td></tr><tr><td>Corresponding Author:</td><td>Axel Hutt, DR<br>Inria Nancy: Inria Centre de Recherche Nancy Grand Est<br>FRANCE</td></tr><tr><td>Corresponding Author Secondary Information:</td><td></td></tr><tr><td>Corresponding Author's Institution:</td><td>Inria Nancy: Inria Centre de Recherche Nancy Grand Est</td></tr><tr><td>Corresponding Author's Secondary Institution:</td><td></td></tr><tr><td>First Author:</td><td>Axel Hutt, DR</td></tr><tr><td>First Author Secondary Information:</td><td></td></tr><tr><td>Order of Authors:</td><td>Axel Hutt, DR</td></tr><tr><td>Order of Authors Secondary Information:</td><td></td></tr><tr><td>Author Comments:</td><td></td></tr></table>
# Additive noise shapes the state and stability of random networks

Axel Hutt¹,²*

¹*MLMS, iCube, University of Strasbourg, Strasbourg, 67000, France.

²*MIMESIS, INRIA Centre at University of Lorraine, Strasbourg, 67000, France.

Corresponding author(s). E-mail(s): axel.hutt@inria.fr;

## Abstract

Natural systems are open to their environment and thus subjected to external perturbations. Some of such systems may be described as large randomly connected networks. The present work investigates how external additive random perturbations affect the state and stability of such finite-size random networks. We observe analytically an additive noise-induced system evolution (ANISE), whose stationary state depends on the additive noise level. In a first pilot study, the principal analytical approach is demonstrated by application to a nonlinear Erdos-Renyi network. In a second more detailed study, a nonlinear random network of excitatory and inhibitory sub-networks describes successfully Event-Related Desynchronization and Synchronization (ERD/ERS) observed in experimental brain signals. In sum, we find that additive noise impacts on the system's stationary state and in turn also affects the system's stability and hence its spectral properties.

**Keywords:** quasi-cycle, Event-Related Desynchronization, Event-Related Synchronization, mode interaction

## 1 Introduction

Complex systems are omnipresent in nature and exhibit a hierarchy of temporal and/or spatial scales, such as in fluids, biological systems, cities and even the financial market [1]. Such systems may represent complex networks [2], which are present in our everyday life, e.g. as technological networks (internet and telephone networks) [3] and social networks [4]. Typically, complex networks exhibit a random topology [5, 6].
Such random networks have attracted increasing attention in the last decades [7–9] and random input to these networks has been shown to be beneficial [10], e.g., in training artificial neural networks [11], to control the resilience of biological systems [12], to assist controlling the level of consciousness in humans [13] or to stabilize the brain in transcranial electric stimulation [14, 15]. Such additive noise-induced system evolution (ANISE) has been studied theoretically in the last decades comprising various effects, such as stochastic resonance, coherence resonance or stochastic bifurcations [16].

Recently, in a first study we proposed that modification of additive noise in random neural networks may explain task-dependent power decreases and increases in electroencephalographic data (EEG) [17]. These phenomena are called Event-Related Desynchronization (ERD) and Event-Related Synchronization (ERS), respectively. They are two important phenomena observed during the processing of sensory events [18]. Figure 1 demonstrates the possible impact of additive noise on the oscillatory mean-field dynamics of a random network involving excitatory and inhibitory sub-networks. Additive noise at excitatory network nodes may destroy oscillatory activity in a certain frequency band (Fig. 1A). Such an observation is known as ERD [18–21]. We point out that the neural origin of ERD is not known and we suggest that it may result from an underlying modification of excitatory intrinsic noise. In a different network configuration (Fig. 1B), additive noise at inhibitory nodes may induce oscillatory activity. Such an emerging spectral power surge has also been observed in EEG during cognitive experiments and has been termed ERS [18–21]. We propose to describe it by a transient increase in inhibitory additive noise level. The present work provides a detailed analysis of the additive noise impact on a biologically realistic nonlinear random network and explains in detail how the network’s nonlinear dynamics leads to ERD and ERS. Finite-size effects are discussed.

The study of ERD and ERS reveals how additive noise tunes the spectral properties of the random network under study. Before addressing this aspect, we perform a pilot study in section 2 on a general random network subjected to additive noise, which demonstrates that additive noise predominantly determines the network’s stationary state. We show analytically, that additive noise affects nonlinearly coupled system modes, whose interaction impacts the system dynamics as known from multiplicative noise interactions. Then section 3 details the mean-field analysis of a finite-size random network and explains ERD and ERS. In this context, we clarify the conditions when the employed mean-field theory is valid.

## 2 A pilot study

For illustration of a possible effect of additive noise on random networks, we start with a simple nonlinear network model of size $N$

$$ \frac{du}{dt} = Kf(u) + \xi(t) \quad (1) $$
<figure><img src="image_2.png" /><figcaption>Fig. 1 Simulation of random network activity under additive noise input. (A) The excitatory noise level jumps from $D_e = 0.3$ to $D_e = 0.8$ at time instant 0.75s and back to $D_e = 0.3$ at time instant 2.25s. The inhibitory noise level was chosen to $D_i = 0.2$. (B) The inhibitory noise level jumps from $D_i = 0.2$ to $D_i = 0.4$ at time instant 0.75s and back to $D_i = 0.2$ at time instant 2.25s. The excitatory noise level was chosen to $D_e = 0.3$. Other parameters are $N = 100$, $J_1^e = 1.1$, $J_1^i = 0.4$, $\tau_e = 0.005$s, $\tau_i = 0.02$s, $F_0 = 2.17$, $M_0 = 3.87$, $S_o = 1.7$.</figcaption></figure>

with $u$, $f$, $\xi \in \mathbb{R}$, $K \in \mathbb{R}^{N \times N}$, written equivalently in components

$$ \frac{du_i(t)}{dt} = \sum_{j=1}^{N} K_{ij}(f_1 u_j(t) + f_2 u_j^2(t)) + \xi_i(t) \quad , \quad i = 1, \dots, N \qquad (2) $$

with $(u)_i = u_i(t)$, $(f(u))_i = f_1 u_i + f_2 u_i^2$ and $(\xi(t))_i = \xi_i(t)$, the node activity $u_i$ at node $i$ at time $t$, a linear and nonlinear node interaction with constant $f_1 \in \mathbb{R}$ and $f_2 \in \mathbb{R}$, respectively, and the interaction weight matrix elements $K_{ij} \in \mathbb{R}$. The network is driven by additive time-dependent Gaussian noise $\xi_i(t)$ at node $i$, which is spectral-white in time and uncorrelated between network nodes, i.e.

$$ \langle \xi_i(t) = 0 \rangle \quad , \quad \langle \xi_i(t) \xi_j(t) \rangle = D \delta_{ij} \delta(t-T) \qquad (3) $$

with the ensemble average $\langle \cdot \rangle$ and the Kronecker symbol $\delta_{ij}$ and the delta-distribution $\delta(t)$. In the forthcoming paragraphs, we will also assume, that the network mean of the additive noise vanishes, i.e.

$$ \frac{1}{N} \sum_{i=1}^{N} \xi_i(t) = 0 . \qquad (4) $$

However, it is important to mention that this additional assumption will be used for better illustration only, but does not affect the general derivation.
<figure><img src="image_3.png" /><figcaption>Fig. 2 Network activity under additive noise. (A) The stationary probability density of network activity $u_1(t)$ at node $i=1$ at low ($D=0.001$) and high ($D=0.2$) noise level. The broader distribution of $u_1$ for $D=0.2$ is obvious, whereas a slight but poorly detectable shift of the distribution mean is present. Panel (B) shows the distribution of corresponding network mean $\sum_{n=1}^{N} u_n/N$ and we observe a clear shift of the stationary state by additive noise for larger noise level.</figcaption></figure>

The network connections are random, Gaussian distributed and non-directional with symmetric weights $K_{ij} = K_{ji}$, $i \neq j$. These non-diagonal weights have the mean $\mu = K_0/N \neq 0$ and standard deviation $\sigma = \sigma_0/N$ with $K_0, \sigma_0 > 0 \in \mathbb{R}$. Moreover, the self-interaction is given by $K_{ii} = \nu$. Figure 2 presents the numerically estimated probability densities of the networks stationary state. Increasing the level of additive noise seem to render the stationary state distribution of $u_1$ being asymmetric (Fig. 2(A)). Estimating the network mean stationary state (Fig. 2(B)) reveals a clear shift of the stationary distribution by additive noise. The subsequent paragraphs demonstrate how this effect can be understood analytically.

## Mode projection

It is known that random matrices $K$ as in Eq. (1) exhibit an edge spectrum and a bulk spectrum, whose elements are the matrix' eigenvalues $\lambda_k$ defined by

$$ K\phi_k = \lambda_k \phi_k, \quad k = 1, \dots, N. \qquad (5) $$

By virtue of the matrix symmetry, the eigenvalues $\lambda_k \in \mathbb{R}$ are real-valued and the corresponding eigenvectors $\phi_k \in \mathbb{R}^N$ are real-valued and orthogonal with

$$ \phi_k^\dagger \phi_l = \delta_{kl}, \qquad (6) $$

here and in the following $\dagger$ indicates the transpose. The edge spectrum includes the maximum eigenvalues of matrix $K$, which are random variables with expectation $(N-1)\mu + \nu + \sigma^2/\mu$ and variance $2\sigma^2$ [22]. Hence, for $N \to \infty$, for our choice of $K$
<figure><img src="image_4.png" /><figcaption>Fig. 3 The spectrum of matrix K for different number of nodes N. The edge spectrum (on the right) includes a single eigenvalue $\lambda_1$ with expectation $K_0(N-1)/N + \nu + \sigma_0^2/K_0N$ and variance $2\sigma_0^2/N^2$ and the bulk spectrum (on the left) obeys the semi-circle law with radius $2\sigma_0/\sqrt{N}$ about $\nu$. For $N \to \infty$, Eq. (7) and (8) hold.</figcaption></figure>

it is $\sigma^2 \to 0$ and the edge spectrum collapses to a single eigenvalue

$$ \lambda_1 \to K_0 + \nu. \qquad (7) $$

The bulk spectrum includes the rest of the spectrum with eigenvalues $\lambda_2, \dots, \lambda_N$, which are distributed according to the semi-circle law in the interval $\nu - 2\sigma\sqrt{N} < \lambda_i < \nu + 2\sigma\sqrt{N}, i = 2, \dots, N$. For $N \to \infty$, the bulk spectrum shrinks to

$$ \lambda_{i>1} \to \nu. \qquad (8) $$

Figure 3 illustrates these properties.

The eigenvector of the edge spectrum is [23]

$$ \phi_1 = (1/\sqrt{N}, \dots, 1/\sqrt{N}) + \epsilon_N, \qquad (9) $$

where $\epsilon_N \to 0$ for $N \to \infty$, see [22]. Moreover, the components of the bulk spectrum's eigenvectors $\phi_{i>1}$ are random and uniformly distributed.

Now interpreting the matrix eigenvectors as network modes $\phi_k$, the network activity decomposes into

$$ \mathbf{u}(t) = \sum_{k=1}^{N} x_k(t) \phi_k \qquad (10) $$

with mode amplitudes $x_k(t)$. A first consequence of this decomposition is the identity

$$ \frac{1}{N} \sum_{n=1}^{N} u_n(t) = \frac{1}{\sqrt{N}} x_1(t). \qquad (11) $$

In other words, the network mean is proportional to mode $x_1(t)$.

Then, projecting the network activity onto the network modes yields

$$ \frac{dx_k}{dt} = f_1\lambda_k x_k + f_2\lambda_k \sum_{l,n=1}^{N} \Phi_{kln}x_l x_n + \phi_k^\dagger \xi(t) \qquad (12) $$
with $\Phi_{kln} = \sum_{j=1}^{N}(\phi_k)_j(\phi_l)_j(\phi_n)_j$. Since the network modes $\{\phi_k\}$ represent an orthonormal basis, for $N \to \infty$ it is

$$ \Phi_{1ln} = \frac{\delta_{ln}}{\sqrt{N}}, \quad \Phi_{k1n} = \frac{\delta_{kn}}{\sqrt{N}}, \quad \Phi_{kln} = 0 \text{ otherwise.} \qquad (13) $$

Inserting these expressions into (12)

$$ \frac{dx_1}{dt} = f_1 \lambda_1 x_1 + \frac{f_2 \lambda_1}{\sqrt{N}} \sum_{n=1}^{N} x_n^2 + \phi_1^\dagger \xi(t) \qquad (14) $$

$$ \frac{dx_k}{dt} = f_1 \lambda_k x_l + \frac{f_2 \lambda_k}{\sqrt{N}} x_1 x_k + \phi_k^\dagger \xi(t), \quad k = 2, \dots, N \qquad (15) $$

with $\lambda_1 \approx \nu + K_0$, $\lambda_k \approx \nu$.

The modes $x_1(t)$ and $x_{k>1}(t)$ are driven by additive noise and interact with other scaled by factor $f_2$. Since all modes are stochastic, their nonlinear interaction represents multiplicative noise coupling and one expects a noise-induced stationary state and corresponding stability shift [24].

## The order parameter

At first neglecting additive noise, i.e. $D = 0$, we find a single asymptotically stable fixed point $\mathbf{x}^o = \mathbf{0}$ if

$$ \begin{aligned} & f_1 < 0, \nu > 0, f_2 > 0, -\frac{f_1}{f_2}\sqrt{N} > x_1 \\ & f_2 < 0, \frac{f_1}{|f_2|}\sqrt{N} < x_1 \\ & f_1 > 0, \nu < 0, f_2 > 0, -\frac{f_1}{f_2}\sqrt{N} < x_1 \\ & f_2 < 0, \frac{f_1}{|f_2|}\sqrt{N} > x_1. \end{aligned} \qquad (16) $$

Now we employ a time scale separation of mode $x_1$ and the modes $x_{k>1}$ assuming that $\lambda_1 = |\nu + K_0|$ is small and $|\nu|$ is large with $\lambda_1 < 0$ $\nu < 0$. This means network nodes inhibit themselves reflecting local inhibition and the lateral interaction $K_0 > 0$ is large, i.e. strong lateral excitation. According to the stability conditions (16), this stipulates $f_1 > 0$. Then Eq. (14) and (15) read

$$ \frac{dx_1}{dt} = f_1(\nu + K_0)x_1 + \frac{f_2(\nu + K_0)}{\sqrt{N}} x_1^2 + \frac{f_2(\nu + K_0)}{\sqrt{N}} \sum_{n=2}^{N} x_n^2 + \phi_1^\dagger \xi(t) \qquad (17) $$

$$ \frac{dx_k}{dt} = \nu \left( f_1 + \frac{f_2}{\sqrt{N}} x_1 \right) x_k + \phi_k^\dagger \xi(t) \qquad (18) $$
implying $\alpha = \nu(f_1 + f_2x_1/\sqrt{N}) < 0$. In the vicinity of the fixed point $x^o = 0$, it is $f_1(\nu + K_0) \gg \alpha$ and thus the mode amplitude $x_1$ evolves much slower than the fast modes $x_{k>1}$ and represents the systems order parameter. Hence, we can treat $x_1$ in Eq. (18) as a constant $x_1 \approx \text{const}$ compared to $x_{k>1}$. This represents an adiabatic elimination and one may say that the order parameter $x_1$ determines the dynamics $x_{k>2}$ and thus enslave them [25].

Then $x_{k>1}(t)$ is an Ornstein-Uhlenbeck process with

$$x_k(t) \approx \int_{-\infty}^{t} e^{\alpha(t-t')} \phi_k^\dagger \xi(t') dt' \quad (19)$$

leading to

$$\begin{aligned} \sum_{k=2}^{N} x_k^2(t) &= \int_{-\infty}^{t} \int_{-\infty}^{t} e^{\alpha(2t-t'-t'')} \xi^\dagger(t'') \left( \sum_{k=2}^{N} \phi_k \phi_k^\dagger \right) \xi(t') dt' dt'' \\ &= \int_{-\infty}^{t} \int_{-\infty}^{t} e^{\alpha(2t-t'-t'')} \xi^\dagger(t'') (I - \phi_1 \phi_1^\dagger) \xi(t') dt' dt'' \\ &= \int_{-\infty}^{t} \int_{-\infty}^{t} e^{\alpha(2t-t'-t'')} \sum_{i=1}^{N} \xi_i(t'') \xi_i(t') dt' dt'' \end{aligned}$$

utilizing the assumed vanishing noise mean $\phi_1^\dagger \xi = 0$, cf. Eq. (4). It is

$$\sum_{i=1}^{N} \xi_i(t'')^t \xi_i(t') = D_o(t') \delta(t' - t'') \quad (20)$$

and $D_o$ is a $\chi^2$-distributed variable with mean ND and variance $2ND$. For simplicity, we approximate $D_o \approx ND = \text{const}$. We keep in mind that this is a strong assumption, but it provides a reasonably well description for small noise levels as can be seen later. Of course a more detailed analysis is possible here and in the subsequent paragraphs, but it would exceed the aim of this section.

Taking into account the last results

$$\begin{aligned} \sum_{k=2}^{N} x_k^2(t) &\approx ND \int_{-\infty}^{t} \int_{-\infty}^{t} e^{\alpha(2t-t'-t'')} \delta(t'' - t') dt' dt'' \quad (21) \\ &= \frac{ND}{2|\alpha|} \end{aligned}$$

and Eq. (17) reads

$$\frac{dx_1}{dt} = f_1(\nu + K_0)x_1 + \frac{f_2(\nu + K_0)}{\sqrt{N}} x_1^2 - f_2(\nu + K_0) \frac{\sqrt{N}D}{2\nu(f_1 + f_2x_1/\sqrt{N})} \quad (22)$$
<figure><img src="image_5.png" /><figcaption>Fig. 4 Analytical results on additive noise-induced shift of the systems stationary state.
(A) Simulated time series $\phi_1 u(t)$ (top panel, black) and $x_1(t)$ (top panel, red) and $x_2(t)$ (bottom panel, green) from Eqs. (17), (18). (B) Numerically estimated probability density of $\phi_1 u$, $x_1$ and the stationary solution (blue) of the order parameter equation (22). (C) Comparison of numerically estimated mean of $\phi_1 u$ and the analytically derived stationary order parameter $x_1$. Parameters are $N = 500$, $f_1 = 1.0$, $f_2 = 1.0$, $\nu = -10.0$, $K_0 = 9.0$, $\sigma_0 = 1.0$, solutions were computed employing a Euler-Maryuama method with time step $\Delta t = 0.01$ and 25000 time steps.</figcaption></figure>

utilizing again the assumption $\phi_1^\dagger \xi = 0$. Equation (22) is deterministic and describes the mean field activity, cf. Eq. (11).

Figure 4(A, top panel) compares the solution time series of $\phi_1^\dagger u$ (computed from the full system (2)) and the mode $x_1(t)$ as solution of Eqs. (17), (18) for network size $N = 500$. Here, the difference between both time series is small and results from the infinite-size network assumption, which implies an eigenvalue spectrum shrunk to point clouds. We point out that increasing $N$ further moves $x_1$ closer to $\phi_1^\dagger u$ (not shown). Moreover, it is interesting to note that the slaved modes $x_k(t)$, $k > 1$, are much larger in magnitude than the order parameter $x_1$, as can be seen in Fig. 4(A, bottom panel).

The additional approximation in Eq. (21) permits to derive the final deterministic order parameter equation (22), whose stationary probability density is given in Fig. 4(B) together with the stationary probability distribution of $\phi_1^\dagger u$ and $x_1$ for a large noise level. We observe a small difference between the probability densities, but all are clearly shifted away from the trivial state. Hence, the order parameter equation
(22) describes qualitatively that additive noise moves the network's stationary state. This is confirmed further by Fig. 4(C) comparing the average stationary state of $\phi_1^\dagger u$ and the stationary order parameter from Eq. (22). We observe a very good accordance validating the previous analysis steps.

## 3 The neuronal network model

After illustrating the analysis steps and the underlying mechanism how additive noise tunes the system's stationary state, we employ the analytical methods to a biologically more realistic network driven by additive noise in order to explain the noise impact observed in Fig. 1.

The network under study has a number of nodes $N$ and each node represents an ensemble of microscopic neurons, a so-called micro-column as observed in mammalian cortex [26–29]. The network of micro-columns comprises two interacting sub-networks of excitatory and inhibitory nodes. Mathematically, the activity of excitatory and inhibitory nodes are collected in the vectors $V \in \mathbb{R}^N$ and $W \in \mathbb{R}^N$, respectively, which represent mean dendritic currents of the corresponding micro-column. The associated dynamics of these micro-columns obey

$$ 
\begin{aligned}
\tau_e dV &= (-V + S_o FS[V] - MS[W] + eI_e)dt + dY_e(t) \\
\tau_i dW &= (-W + S_o MS[V] - FS[W] + eI_i)dt + dY_i(t)
\end{aligned}
\qquad (23) $$

with the $N$-dimensional identity vector $e = (1, \dots, 1)^\dagger$, the intra-subnetwork connectivity matrix $F$, the inter-subnetwork connectivity matrix $M$ and the synaptic time scales $\tau_e$ and $\tau_i$. The network nodes are randomly connected, the connection weights may depend on the direction (different to the network in section 2) and thus $F$ and $M$ are $N \times N$-dimensional non-symmetric random matrices.

For intra-subnetwork connections $F \in \mathbb{R}^{N \times N}$, $F_{ii} = 0$ and elements $F_{ij} = F_{ji} = X_{ij}K_o/cN \ \forall i \neq j$, $K_o > 0$ are Bernoulli-distributed with probability $c$, constant $K_o$ and the Bernoulli random variable $X_{ij}$. Then the mean connectivity is $\mu_F = F_0/N$ with variance $\sigma_F^2 = F_o^2(1-c)/cN^2$. Equivalent expressions hold for inter-subnetwork connections with the identical connection probability, i.e. $\mu_M = M_0/N$, $\sigma_M^2 = M_o^2(1-c)/cN^2$.

The external stochastic input is a $N$-dimensional Wiener process $Y_{e,i} \in \mathbb{R}^N$ with

$$ 
\begin{aligned}
\langle (dY_{e,i})_n(t) \rangle &= 0 \\
\langle (dY_{e,i})_n(t) (dY_{e,i})_m(t') \rangle &= 2D_{e,i}\delta_{nm}\delta(t-t')dt dt'
\end{aligned}
$$

for $n, m = 1, \dots, N$ and the noise variance $D_{e,i}$. Hence, each network node is driven by noise, which is uncorrelated in network space and time.

Nodes in the network exhibit a nonlinear transfer function $S(\cdot)$ for excitatory and inhibitory connections with the scaling factor $S_0$. Such a transfer function at a node corresponds to the population firing rate in the respective micro-column. We assume a McCullogh-Pitts neuron model [30, 31] with $S(x) = \Theta(x)$ and $\Theta(x)$ is the Heaviside function $\Theta(x) = 0 \forall x < 0$, $\Theta(x) = 1 \forall x \ge 0$. The function vector $S[x] \in \mathbb{R}^N$ is defined
by $(S[x])_n = S(x_n)$. This neuron model implies identical thresholds for all micro-columns. Moreover, we point out that the McCullogh-Pitts model neuron is one of the simplest mathematical models, but it may be extended easily to more biologically realistic firing rate models, such as the Poisson-model [32].

## Order parameter equation: the mean-field

Similar to section 2, the key idea of our mean-field approach is that the network activity fluctuates about the network mean activity, which is determined by the mean connection strength. Considering the Galerkin approach, a projection of the network activity on the matrices' eigenmodes yields a separation between mean-field activity and fluctuating stochastic modes. Assuming the basis composition

$$V(t) = \sum_{n=1}^{N} x_n(t) \phi_n^e, \quad W(t) = \sum_{n=1}^{N} y_n(t) \phi_n^i \quad (24)$$

with the vector basis {$\phi_n^e$}, {$\phi_n^i$} and $\phi_n^e, \phi_n^i \in \mathbb{C}^N$ and a bi-orthonormal basis {$\psi_k^e$}, {$\psi_k^i$} and $\psi_k^e, \psi_k^i \in \mathbb{C}^N$ with

$$\psi_k^{e,\dagger} \phi_l^e = \delta_{kl}, \quad \psi_k^{i,\dagger} \phi_l^i = \delta_{kl},$$

$$\psi_k^{x,\dagger} \phi_l^y = 0 \quad \forall x, y = e, i,$$

and [17] $\phi_1^e = \phi_1^i = e, \psi_1^e = \psi_1^i = e/N$. Moreover, we define the eigenvalue problems

$$\psi_k^{e,i,\dagger} \mathbf{F} = \lambda_k^F \psi_k^{e,i,\dagger}, \quad \psi_k^{e,i,\dagger} \mathbf{M} = \lambda_k^M \psi_k^{e,i,\dagger} \quad (25)$$

for $k = 1, \dots, N$. Then, projections on the vector basis and utilizing Eqs. (25) yields

$$\tau_e dx_1 = \left( -x_1 + \frac{\lambda_1^F}{N} e^\dagger S_0 \mathbf{S}[\mathbf{V}] - \frac{\lambda_1^M}{N} e^\dagger \mathbf{S}[\mathbf{W}] + I_e \right) dt + d\eta^e(t) \quad (26)$$

$$\tau_i dy_1 = \left( -y_1 + \frac{\lambda_1^M}{N} e^\dagger S_0 \mathbf{S}[\mathbf{V}] - \frac{\lambda_1^F}{N} e^\dagger \mathbf{S}[\mathbf{W}] + I_i \right) dt + d\eta^i(t)$$

with $d\eta^{e,i}(t) = e^\dagger d\mathbf{Y}_{e,i}(t)/N$ and the remaining modes obey

$$\tau_e dx_k(t) = \left( -x_k(t) + \lambda_k^F \psi_k^{e\dagger} S_0 \mathbf{S}[\mathbf{V}] - \lambda_k^M \psi_k^{e\dagger} \mathbf{S}[\mathbf{W}] + \psi_k^{e\dagger} e I_e \right) dt + d\eta_k^e(t) \quad (27)$$

$$\tau_i dy_k(t) = \left( -y_k(t) + \lambda_k^M \psi_k^{i\dagger} S_0 \mathbf{S}[\mathbf{V}] - \lambda_k^F \psi_k^{i\dagger} \mathbf{S}[\mathbf{W}] + \psi_k^{i\dagger} e I_i \right) dt + d\eta_k^i(t)$$

for $k > 1$ with $d\eta_k^{e,i}(t) = \psi_k^{e,i,\dagger} d\mathbf{Y}_{e,i}(t)$, cf. also section 2. The mean-field modes serve as order parameters, obey Eqs. (26) and slaves the stochastic modes in Eqs. (27), which in turn affects the order parameters' evolution.
We observe that all modes are stochastic and interact with each other. This stipulates an additive noise-induced shift of stationary state and corresponding stability similar to section 2.

Similar to section 2, for $N \to \infty$ the eigenvalue spectrum shrinks $\lambda_{k>1}^M$, $\lambda_{k>1}^F \to 0$ and the enslaved modes $x_{k>1}, y_{k>1}$ obey an Ornstein-Uhlenbeck process. Then, for the case of the excitatory population, one may re-write $V(t)$ in Eq. (24) as [17]

$$V \approx (x_1 - Z(t))e + w_e(t) \quad (28)$$

with

$$w_e(t) = \frac{1}{\tau} \int_{-\infty}^{t} e^{-(t-t')/\tau} dY_e(t') \quad , \quad Z(t) = \frac{1}{\tau} \int_{-\infty}^{t} e^{-(t-t')/\tau} dR^e(t').$$

The finite size-fluctuations $R^e(t)$ are Wiener processes with $dR^e \sim N(0, D_e/N)dt$, the finite size-fluctuations $Z(t)$ obey an Ornstein-Uhlenbeck process which represents red-coloured noise.

To compute the terms $S[V]$, $S[W]$ in Eqs. (26), we assume for the moment a general function vector $S[u] = (\Theta(u_1), \dots, \Theta(u_N))^{\dagger} \in \mathbb{R}^N$ with Heaviside function $\Theta(\cdot)$ and any vector $u = xe + w \in \mathbb{R}^N$ with $x \in \mathbb{R}$ and random vector $w \in \mathbb{R}^N$, $w_i \sim N(0, D_e/\tau_e)$. Then one finds [17]

$$\frac{\mathbf{e}^T}{N} S[u] \approx \int_{-\infty}^{\infty} S(x+w)p_e(w)dw = \frac{1}{2} \left( 1 + \operatorname{erf} \left( \frac{x}{\sqrt{2(D_e/\tau_e)}} \right) \right) . \quad (29)$$

The dynamics of $W(t)$ can also be written similarly and the derivation resembles closely to the one of $V(t)$.

The derivation of the mode equations (26) and (27) and the approximation (29) is heavily based on the homogeneity assumption of the network: the network degree at each node is very similar across nodes, and identical in infinite networks, i.e. the degree distribution is unimodal with a narrow peak. This results into a homogeneous eigenvector in the matrices' edge spectrum i.e., $\phi_1^e \approx e$, and consequently the impact of the transfer function is identical at each node. Hence, the nonlinear interaction and the network decouple yielding the term $(x_1 - Z_1)e$. If the network topology does not exhibit such a narrow degree distribution (e.g. in scale-free networks or small-world networks), the node connectivity is not identical in all nodes. Hence, the eigenvector becomes $\phi_1^e \neq e$ and Eq. (26) and (29) does not hold anymore. This implies that the impact of the nonlinear transfer function is different at each node, the nonlinear interactions and network do not decouple and $V$ does not have the form (28).
Inserting these latter results into Eqs. (26) yields finally

$$
\begin{align}
\tau_e dx_1 &= (-x_1 + \lambda_1^F G_e [x_1 - Z_e(t), t] - \lambda_1^M G_i [y_1 - Z_i(t), t] + I_e) dt + dR^e(t) \tag{30} \\
\tau_i dy_1 &= (-y_1 + \lambda_1^M G_e [x_1 - Z_e(t), t] - \lambda_1^F G_i [y_1 - Z_i(t), t] + I_i) dt + dR^i(t) \nonumber
\end{align}
$$

with

$$
G_{\alpha}(x) = \frac{S_{\alpha}}{2} \left( 1 + \operatorname{erf} \left( \frac{x}{\sqrt{2(D_{\alpha}/\tau_{\alpha})}} \right) \right) , \quad \alpha = e, i, \qquad (31)
$$

$S_e = S_0, S_i = 1$ and with finite-size fluctuations $dR^\alpha = \xi_\alpha dt, \xi_\alpha \sim \mathcal{N}(0, D_\alpha/N)$

$$
Z_{\alpha}(t) = \frac{1}{\tau} \int_{-\infty}^{t} e^{-(t-t')/\tau} dR^{\alpha}(t') , \alpha = e, i .
$$

Simulations

In the following, we consider large but finite-size networks obeying Eqs. (30). The terms $G_{e,i}$ are effective nonlinear transfer functions of the order parameter, i.e. the mean-field, and its evolution involves state-dependent finite-size fluctuations $Z(t)$, $W(t)$. The fluctuation strength of $Z(t)$ and $W(t)$ are proportional to $\sqrt{D_e}$ and $\sqrt{D_i}$, respectively, while the nonlinear gain of the transfer functions $G_e$ and $G_i$ (their slope) decrease nonlinearly with $\sqrt{D_e}$ and $\sqrt{D_i}$, respectively. Consequently, stronger additive noise enhances the finite-size fluctuations $Z_\alpha(t)$ but diminishes the response sensitivity (the nonlinear gain) of the system in accordance to previous studies [33, 34].

To obtain some first insight into the systems topography, Fig. 5 shows the station-
ary states for which $dx_j = dy_j = 0$, $j = 1, 2$ for different excitatory and inhibitory
noise levels $D_e$ and $D_i$ for infinite network sizes $N \to \infty$, i.e. $dR_{1,2}^{e,i} \to 0$. In the top
panels of (A) and (B), we observe multi-stability comprising a stable node and focus
in addition to a saddle-node. Linearizing Eq. (30) about the stationary state provides
the eigenvalues $\Lambda_{1,2}$ of the corresponding Jacobian matrix. The maximum of the eigen-
value's real part $\lambda_m = \max(\mathrm{Re}(\Lambda_1), \mathrm{Re}(\Lambda_2))$ and the corresponding eigenfrequency
$\nu = \mathrm{Im}(\Lambda_{1,2})$ (Fig. 5, bottom panels) reflect well the stationary state's dynamical
properties. They explain well ERD and ERS shown in Fig. 1. In Fig. 5(A), at $D_e = 0.3$
the lower fixed point branch (top panel) is an unstable focus with an eigenfrequency
of about 45Hz. In finite-size networks, $dR_{1,2}^{e,i}(t)$, $Z(t)$, $W(t) \neq 0$, the mean-field is
driven by noise and the system oscillates in a limit cycle due to the present nonlinear
interactions. A jump of the excitatory noise level to $D_e = 0.8$ (cf. Fig. 1(A)) stabilizes
the focus, the system performs a quasi-cycle [35–37] and diminishes the magnitude
of the mean-field reflecting ERD.

In Fig. 5(B), at $D_i = 0.2$ the lower fixed point branch (cf. top panel) is a stable
focus with an eigenfrequency of about 40Hz and performs a quasi-cycle in the finite-
size network. The system is a damped oscillator with low magnitude. A jump to the
higher inhibitory noise level $D_i = 0.4$ (cf. Fig. 1(B)) destabilizes the focus, renders it
a nonlinear limit cycle and enhances the magnitude of the mean-field, which reflects
ERS.
<figure><img src="image_6.png" /><figcaption>**Fig. 5 Stationary state $\bar{x}_1$ computed from Eq. (30).** (A) Top panel: the stationary mean-field state $\bar{x}_1$ depends on the excitatory noise level $D_e$ and may represent a stable node (dotted line), a stable focus (solid line) or a saddle-node (dashed line) for low $D_e$. The multi-stability vanishes for larger excitatory noise levels $D_e$. Here, $D_i = 0.2\tau_i$. Lower panel: (B) $D_e = 0.3\tau_e$. Other parameters are given in Fig. 1.</figcaption></figure>

To validate the meaning of the maximum eigenvalue real part $\lambda_m$ and the eigen-frequency $\nu$, Fig. 6 (A) and (B) present the maximum real-part and eigenfrequency of the fixed point eigenvalues, respectively. Their dependence from the noise levels $D_e$ and $D_i$ resemble well the spectral peak power (panel C) and principal peak frequency (panel D) of the respective spectral power distribution. Interestingly, we find a maximum principal frequency for $D_i > 0.35\tau_i$ at $D_e/\tau_e \approx 0.35$.

The previous paragraphs describe oscillations in frequencies at about 40Hz. To understand better why oscillations emerge in this specific frequency band, we modified the mean connection weights $M_0$ and $F_0$, cf. Fig. 7. It turns out that different choices of mean connection weights yield different frequency bands. For instance, Fig. 7(A) predicts limit cycle oscillations with large values of $\lambda_m$ at $F_0 = 2.3$, $M_0 = 3.5$, which corresponds to lower frequencies $\nu < 30$Hz (see Fig. 7(B)). In EEG, this band is known to be part of the upper $\beta$-band, which plays an important role in ERD and ERS in human motor cortex [38].

These latter results imply the validity of the mean-field description implying $N \to \infty$. Since numerical simulations can not achieve this limit, the question arises how large the network should be chosen in order to achieve a sufficient mean-field approximation. For instance, finite-size errors and thus additional fluctuations emerge if the sum in
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 6 Power and principal frequency dependent of excitatory and inhibitory additive noise at the lower fixed point branch. (A) The maximum real part $λ_m$ computed by deterministic linear analysis about the stationary state. (B) Eigenfrequency $ν$ computed by deterministic linear analysis about the corresponding stationary state. (C) Spectral power value at the peak of maximum power computed numerically from simulations. (D) Frequency of power peak computed numerically from simulations. Parameters are taken from Fig. 1 and the spectral power and principal frequency has been computed from a simulated time series of duration 20s, integration time step $Δt = 0.5$ms by employing the Bartlett-Welch method.</figcaption></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 7 Spectral properties of stationary states for various intra-area and inter-area connection weights. (A) The maximum eigenvalue real part $λ_m$ of fixed points. (B) The eigen-frequency $ν$ of fixed points. Simulation results were obtained with parameters from Fig. 1 with $D_e = 0.3τ_e$, $D_i = 0.2τ_i$. The white point denotes the parameter pair used in Figs. 1-6.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 8 Finite-size error Δ in approximation (29). The error is defined in Eq. (32). Parameters are $D_e = 0.3\tau_e$, $D_i = 0.2\tau_i$ and others are taken from Fig. 1.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 9 Validity of mean-field analysis by description of ERD and ERS. Comparison of the estimated power spectral density of the numerically simulated network average activity $\bar{V}(t) = \sum_{n=1}^{N_1} V_n(t)/N_1$ (solid line) and the mean-field activity $x_1(t)$ for $D_e = 0.3\tau_e$, $D_i = 0.2\tau_i$ (left panel), $D_e = 0.8\tau_e$, $D_i = 0.2\tau_i$ (center panel) and $D_e = 0.3\tau_e$, $D_i = 0.4\tau_i$ (right panel). The network simulations have been performed with $N = 1000$, integration time step $\Delta t = 0.5$ms, simulation duration 40s and the power spectra have been estimated with the Bartlett-Welch method. Other parameters have been taken from Fig. 1.</figcaption></figure>

Eq. (29) is replaced by the integral. Figure 8 shows the corresponding error

$$ \Delta = \frac{1}{N} \sum_{n=1}^{N} \Theta(V_n) - \frac{1}{2} \left( 1 + \operatorname{erf} \left( \frac{\bar{V}}{\sqrt{2D_e/\tau_e}} \right) \right) \quad (32) $$

computed for an evolving network over time for different number of network nodes $N$. We observe that the fluctuations' magnitude decreases with increasing number of nodes $N$ and $N > 1000$ already yields a good approximation.

Finally, to affirm the analytical finding's validity Fig. 9 compares the power spectrum distribution of numerical solutions of the network mean and the mean-field computed analytically for three parameter sets. Here we assumed $N = 1000$. These results confirm the induced desynchronisation by excitatory noise enhancement and induced synchronisation by enhanced inhibitory noise.
# 4 Discussion

The present work demonstrates how to imply additive noise in the mathematical description of stochastically driven random networks. The pilot study in section 2 illustrates that additive noise affects the system's stationary state by a nonlinear coupling between interacting stochastically driven modes. Since multiplicative noise is well-known to modify a system's stationary state and stability [24, 39], this impact can be interpreted as multiplicative noise through the back door: the nonlinear coupling of modes each driven by additive noise represents multiplicative noise in the system.

The analytical approach demonstrated in the pilot study was applied to a biologically more realistic and more complex network (cf. section 3) comprising interacting excitatory and inhibitory sub-networks. The presented analysis describes experimentally observed ERD and ERS by respective increases of additive noise level at excitatory and inhibitory network nodes. Brief comparison studies of numerical simulations and analytical results proof the validity of the analytical results for large random networks.

The present work considers random networks with equally distributed connection weights of the Erdos-Renyi type and thus comprising a uni-modal network degree distribution. More realistic networks involve more complex topologies, e.g., with exponentially distributed network degrees or heterogeneous network structures [40]. Moreover, the random networks in the present study exhibit a non-vanishing connection mean yielding spectrally separated edge and bulk spectrum, which permits to describe a mean-field by projecting on the edge spectrum eigenmodes. However, this approach fails in case of balanced networks with vanishing mean weights. Such networks are supposed to play an important role in neuro-biological systems. The analysis of such networks request a different analysis approach and will be the objective of future work.

# Data availability statement

Data sets generated during the current study are available from the corresponding author on reasonable request.
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