Metastability in networks of nonlinear stochastic integrate-and-fire neurons

TL;DR

This work links single-neuron nonlinearities to macroscopic metastable population states in recurrent networks of stochastic integrate-and-fire neurons using a field-theoretic MSR framework. By comparing threshold-power-law and exponential nonlinearities, it derives mean-field phase diagrams and employs one-loop fluctuations and renewal theory to quantify how spike resets and nonlinear transfer shape mean activity, revealing metastable high/low firing-rate states and state-dependent modulation of membrane potential. The analysis spans homogeneous and excitatory-inhibitory networks, with explicit results showing that superlinear power laws can produce two coexisting active states, while exponentials tend to bistability between low and high activity; fluctuations can either promote or suppress activity depending on concavity and regime, with clear surfaces where corrections vanish. Together, these results provide a principled link between single-neuron nonlinearities and population-level metastable dynamics, offering a framework to understand up-down-like transitions and flexible cortical computation.

Abstract

Neurons in the brain continuously process the barrage of sensory inputs they receive from the environment. A wide array of experimental work has shown that the collective activity of neural populations encodes and processes this constant bombardment of information. How these collective patterns of activity depend on single-neuron properties is often unclear. Single-neuron recordings have shown that individual neurons' responses to inputs are nonlinear, which prevents a straightforward extrapolation from single neuron features to emergent collective states. Here, we use a field-theoretic formulation of a stochastic leaky integrate-and-fire model to study the impact of single-neuron nonlinearities on macroscopic network activity. In this model, a neuron integrates spiking output from other neurons in its membrane voltage and emits spikes stochastically with an intensity depending on the membrane voltage, after which the voltage resets. We show that the interplay between nonlinear spike intensity functions and membrane potential resets can i) give rise to metastable active firing rate states in recurrent networks, and ii) can enhance or suppress mean firing rates and membrane potentials in the same or paradoxically opposite directions.

Figures (5)

• Figure 1: a) (Top) Voltage trace showcasing multiple hard-resets after a spike is emitted by a neuron in the model. (Bottom) Voltage trace of an example neuron from the cell electrophysiological recording showcasing the full spike generation and post spike hyper-polarization. The model differs from the true dynamics of the cell since the details of the spike generation and hyper-polarization are replaced by the hard-reset. b) Mean membrane potential vs firing rate data for an example cell (empty red circles), with the power law (pink) and exponential (dark blue) fits. The text shows the estimated parameters for both fits. c) The average estimated exponent of the threshold power law fit for different cell types (as identified by Cre line labeling). The inset shows the estimated gain for the fit. d) Same as c but for the exponential fit showing the gain. Cell type 4 is Vip+ neurons, a type of inhibitory cell; all other cell types here are excitatory neurons (Table \ref{['table0']}). e) Same as c but showing the threshold and the normalized threshold (inset) for the power law fit. f) Same as c but showing the threshold and the normalized threshold (inset) for the exponential fit. See the main text for the definition of the normalized threshold.
• Figure 2: a) Schematic for the homogeneous network. The network has a coupling strength $J$ and receives an external current $\mathcal{E}$. b) Mean-field (MF) phase diagram for the homogeneous network with $\alpha=1$ in the $\mathcal{E}{-}J$ plane, separating the bistable (B--H, Q) high firing rate and quiescent region from the monostable high (H) firing rate and quiescent (Q) regions. c) Same as b for $\alpha=2$ separating the bistable (B--H, Q) high firing rate and quiescent, bistable (B--H, L) high and low firing rate regions from the monostable high (H) firing rate and quiescent (Q) regions. d) Same as c for $\alpha=3$. e) Raster plot for the stochastic spiking network ($\alpha=2$) at the parameter marked with cross in panel c ($J=3.2$, $\mathcal{E}=1.05$), illustrating input driven transition between the two active states. f) Raster plot for the stochastic spiking network ($\alpha=2$) at the parameter marked with square in panel c ($J=3.0$, $\mathcal{E}=1.07$), illustrating stochastic transition between the two active states. g) MF (green), 1-loop (cyan) and renewal theory (dark blue) firing rate predictions compared to simulations (brown) for fixed $J=3.0$ ($\alpha=2$). Inset highlights the low firing rate state in the black square. h) MF phase diagram that depicts that the critical point continuously increases with the exponent of the intensity and reaches a limiting value for arbitrarily large $\alpha$. Violet: $\alpha = 1$, Blue: $\alpha=2$, Yellow: $\alpha=3$, Red: $\alpha \rightarrow \infty$.
• Figure 3: a) MF phase diagram for the exponential intensity function separating the bistable (B) regime from the monostable (M) regime for various values of $\theta \in [0, 5]$b) Raster plot for the stochastic spiking network ($\theta=1$) at the parameter marked with square in panel a ($J=4.0$, $\mathcal{E}=-0.75$), illustrating the monostable states. c) Raster plot for the stochastic spiking network ($\theta=1$) at the parameter marked with cross in panel a ($J=4.0$, $\mathcal{E}=-2.0$), illustrating input driven transition between the the two active states. d) MF (green), 1-loop (cyan) and renewal theory (dark blue) firing rate predictions compared to simulations (brown) for fixed $J=6.0$ ($\theta=3$). Inset highlights the low firing rate state in the main plot.
• Figure 4: a) Schematic for the excitatory-inhibitory (EI) network (E: Orange, I: Blue). The excitatory cluster has a coupling strength $J$, the relative strength of inhibition is $g$, and both populations receive an external current $\mathcal{E}$. b) MF phase diagram for threshold power law intensity in the $\mathcal{E}{-}g$ plane for $\alpha=1$ (dashed), $\alpha=2$ (dot dashed) and $\alpha=3$ (solid). The phase are same as labeled in Fig. \ref{['Figure2']}. c) Raster plot for the stochastic spiking network ($\alpha=2$) at $J=5.0$, $g=0.4$, and $\mathcal{E}=1.07$ illustrating stochastic transition between the two active states. d) MF (green), 1-loop (cyan) and renewal theory (dark blue) firing rate predictions compared to simulations (brown) for fixed $J=5.0$, $g=0.4$ ($\alpha=3$). Inset highlights the low firing rate state in the main plot.
• Figure 5: a) MF phase diagram for the exponential intensity in the $\mathcal{E}{-}g$ plane for various values of $\theta \in [0, 5]$. The phase are same as labelled in Fig. \ref{['Figure3']}. b) Same as a but in the $J{-}g$ plane for $\theta=1$. c) Raster plot for the stochastic spiking network ($\theta=1$) at the parameter marked with cross in panel a ($J=8.0$, $g=0.4$, and $\mathcal{E}=-2.5$), illustrating input driven transition between the the two active states. d) MF (green), 1-loop (cyan) and renewal theory (dark blue) firing rate predictions compared to simulations (brown) for fixed $J=8.0$ and $g=0.4$ ($\theta=1$). The inset shows the low firing rate state in the main figure.