Macroscopic dynamics of quadratic integrate-and-fire neurons subject to correlated noise

TL;DR

The paper investigates how correlated Gaussian noise shapes the macroscopic dynamics of a large network of quadratic integrate-and-fire neurons. Using a cumulant-expansion-based mean-field reduction, it derives a two-dimensional system for the population firing rate and mean membrane potential, valid across a range of noise correlation levels. The authors uncover two main results: increasing noise correlation can suppress overall activity (correlated-noise-inhibited spiking) and, in a bistable parameter regime, produces metastable dynamics with noise-driven transitions between high- and low-activity states. This work provides a practical framework for reducing stochastic neural networks and sheds light on how shared fluctuations can govern state transitions in neural circuits.

Abstract

The presence of correlated noise, arising from a mixture of independent fluctuations and a common noisy input shared across the neural population, is a ubiquitous feature of neural circuits, yet its impact on collective network dynamics remains poorly understood. We analyze a network of quadratic integrate-and-fire neurons driven by Gaussian noise with a tunable degree of correlation. Using the cumulant expansion method, we derive a reduced set of effective mean-field equations that accurately describe the evolution of the population's mean firing rate and membrane potential. Our analysis reveals a counterintuitive phenomenon: increasing the noise correlation strength suppresses the mean network activity, an effect we term correlated-noise-inhibited spiking. Furthermore, within a specific parameter regime, the network exhibits metastability, manifesting itself as spontaneous, noise-driven transitions between distinct high- and low-activity states. These results provide a theoretical framework for reducing the dynamics of complex stochastic networks and demonstrate how correlated noise can fundamentally regulate macroscopic neural activity, with implications for understanding state transitions in biological systems.

Figures (3)

• Figure 1: Transient dynamics of an ensemble of $10^{4}$ coupled QIF neurons driven solely by independent noise ($c=0$). (a) and (b) depict the population-averaged firing rate $r(t)$ and the second cumulant $q_2(t)$ for $\Delta=1$, respectively. External step current input is set as $I_0=0.5$ for $t\in[0,20]$ and $I_0=0$ otherwise. (c) and (d) are the population-averaged firing rate $r(t)$ and the second cumulant $q_2(t)$ for $\Delta=0.5$, respectively. External step current input in this case is set as $I_0=5$ for $t\in[0,20]$ and $I_0=0$ otherwise. The solid black curves represent microscopic simulations, while the green and orange curves correspond to the present truncation ansatz Eq. \ref{['Eq:Langevin_r_v']} and the truncation proposed by Goldobin et al. goldobin2021reduction, respectively. Other common system parameters used in (a)-(d): coupling strength $J=15$ and Lorentzian distribution parameters $\bar{\eta}=-3.94$. The integration time step is $dt=10^{-4}$.
• Figure 2: (a) Phase diagrams for uncorrelated ($c=0$; red dashed boundaries) and fully correlated ($c=1$; black solid boundaries) noise. Dynamical regimes are color-coded. (b) Bifurcation diagram of the mean firing rate $r$ versus the median excitability $\bar{\eta}$. Theory (lines) agrees with microscopic QIF neuron simulations (dots). Both are obtained from the deterministic part of Eq. \ref{['Eq:Langevin_r_v']}, where the explicit noise term $\sqrt{c}\xi_c(t)$ is omitted, while the coefficients $q_2$ and $p_2$ retain their dependence on the correlation coefficient $c$ and noise intensity $D$. (c, d) Dependence of the (c) mean firing rate and (d) mean membrane potential on the correlation strength $c$. Theory (solid curves) and simulation (symbols) are shown for parameters selected in (a) (matched by color/shape). Fixed parameters: $\Delta = 1$, $I_0 = 0.5$, $D=1$ and $dt=10^{-4}$ for simulations.
• Figure 3: This figure illustrates the changes in (a) firing rate and (b) mean membrane potential over time, as well as (c) the raster plots for the time interval $t\in[1500, 2000]$. The red dashed vertical line at $t=1500$ in panels (a) and (b) indicates the beginning of this interval. (d) Stationary probability density $\rho(r,v)$ obtained numerically from the two-dimensional Fokker–Planck equation Eq. \ref{['Eq:FKE_corr_noise']}. The color scale indicates the density magnitude, and white contours highlight regions of equal probability. (e) Deterministic phase portrait of the corresponding system. (f) and (g) give the stationary probability distribution of the mean firing rate and mean membrane potential of the whole neuron network, respectively. The histograms depict microscopic simulations involving $10^4$ neurons over $2000$ seconds, while the theoretical results are illustrated by the red dashed lines. Parameters are set as $\overline{\eta} = -4.04$, $\Delta = 1$, $J = 15$, $I_0 = 0.5$, $c = 0.1$ and $D=1$.