[Paper Review] Exact low-dimensional description for fast neural oscillations with low firing rates

Pau Clusella, Ernest Montbrió|arXiv (Cornell University)|Aug 10, 2022

stochastic dynamics and bifurcation52 references4 citations

TL;DR

This paper derives an exact low-dimensional firing-rate model (FRM) for networks of stochastic Quadratic Integrate-and-Fire (QIF) neurons with Cauchy noise, showing that such models exhibit the same dynamics as deterministic QIF networks. The key result is that sparse synchronization with low firing rates and irregular spiking—common in biological fast oscillations—can be exactly described by the same FRM as deterministic frequency-entrainment, reconciling two long-standing views in neural dynamics.

ABSTRACT

Recently, low-dimensional models of neuronal activity have been exactly derived for large networks of deterministic, Quadratic Integrate-and-Fire (QIF) neurons. Such firing rate models (FRM) describe the emergence of fast collective oscillations (>30~Hz) via the frequency-locking of a subset of neurons to the global oscillation frequency. However, the suitability of such models to describe realistic neuronal states is seriously challenged by fact that during episodes of fast collective oscillations, neuronal discharges are often very irregular and have low firing rates compared to the global oscillation frequency. Here we extend the theory to derive exact FRM for QIF neurons to include noise, and show that networks of stochastic neurons displaying irregular discharges at low firing rates during episodes of fast oscillations, are governed by exactly the same evolution equations as deterministic networks. Our results reconcile two traditionally confronted views on neuronal synchronization, and upgrade the applicability of exact FRM to describe a broad range of biologically realistic neuronal states.

Motivation & Objective

• To extend the exact mean-field theory of QIF neural networks to include stochasticity, specifically Cauchy-distributed noise.
• To resolve the contradiction between deterministic FRMs (based on frequency-entrainment) and biologically observed sparse synchronization (with low firing rates and irregular spiking).
• To demonstrate that the same low-dimensional FRM governs both deterministic and stochastic QIF networks under specific noise conditions.
• To establish a rigorous theoretical link between microscopic spiking dynamics and macroscopic collective oscillations in realistic neuronal states.
• To validate the robustness of the FRM framework under non-Gaussian noise and heterogeneity, particularly in the context of fast, irregular neural rhythms.

Proposed method

• Derive exact firing-rate equations for globally coupled QIF neurons driven by Cauchy-distributed noise using a generalized mean-field approach.
• Apply the theory of the Cauchy distribution to maintain exact low-dimensional reduction despite stochasticity, leveraging its stability under convolution.
• Use the order parameter $ r $ and mean population activity $ ho $ to define the FRM, with dynamics governed by a single complex-valued ordinary differential equation.
• Establish equivalence between the FRM of deterministic QIF networks (with frequency-entrainment) and stochastic QIF networks (with sparse synchronization) under identical parameter regimes.
• Validate results via numerical simulations of the full spiking network (Eqs. 1 and 3) and compare with the FRM predictions (Eqs. 22 and 3).
• Explore the robustness of the framework under Gaussian heterogeneity and noise, using $ q $-Gaussian approximations and weak-noise limits as benchmarks.

Experimental results

Research questions

• RQ1 Can exact low-dimensional firing-rate models (FRMs) be derived for stochastic QIF networks with Cauchy noise, preserving the same dynamics as deterministic networks?
• RQ2 Do networks of QIF neurons with low firing rates and irregular spiking (sparse synchronization) exhibit the same collective dynamics as deterministic QIF networks with frequency-entrainment?
• RQ3 Is the FRM derived for deterministic QIF networks with Cauchy heterogeneity also valid for stochastic networks with Cauchy-distributed noise?
• RQ4 How does the inclusion of noise affect the bifurcation structure and stability of fast oscillations in QIF networks, particularly in terms of firing rate and oscillation frequency?
• RQ5 Can the FRM framework be extended to non-Gaussian noise and heterogeneity, and how does it compare to weak-noise approximations?

Key findings

• The FRM derived for deterministic QIF networks with Cauchy heterogeneity is exactly equivalent to the FRM for stochastic QIF networks with Cauchy noise, despite different microscopic dynamics.
• Networks with stochastic QIF neurons and Cauchy noise display sparse synchronization with low firing rates and irregular spiking, yet are governed by the same evolution equations as deterministic frequency-entrained networks.
• The oscillation frequency $ ilde{ u} $ and time-averaged firing rate $ ilde{r} $ remain unchanged by the presence of Cauchy noise, as predicted by the FRM.
• For Gaussian heterogeneity, the FRM remains infinite-dimensional in the limit of Gaussian noise, but bifurcation diagrams show a widening of the oscillatory region with increasing $ n $, approaching the Cauchy case.
• Numerical simulations confirm that the FRM accurately predicts collective dynamics across a range of noise and heterogeneity levels, including the absence of cluster states in the presence of Gaussian noise.
• The FRM framework successfully reconciles deterministic frequency-entrainment and stochastic sparse synchronization, extending the applicability of exact FRMs to biologically realistic neural states.