Sequence of pseudoequilibria describes the long-time behavior of the nonlinear noisy leaky integrate-and-fire model with large delay

TL;DR

This work investigates the long-time behavior of the nonlinear noisy leaky integrate-and-fire PDE with large synaptic delay by introducing a discrete firing-rate map $N_{k+1,\infty}=1/I(N_{k,\infty})$ and an associated sequence of pseudo-equilibria $p_{k,\infty}$. The authors prove exponential convergence to equilibrium for weakly connected networks (small $|b|$) by following the pseudo-equilibria, and provide extensive numerical evidence that the discrete sequence captures the PDE dynamics even for larger $|b|$, including plateau formation and periodic oscillations in strongly inhibitory systems. They analyze how the number and stability of fixed points of the firing-rate map depend on the connectivity parameter $b$, identifying regimes with unique equilibria, bi-stability, or 2-cycles. The approach offers a practical, parameter-driven framework to predict long-time dynamics, linking a tractable discrete system to the full continuum PDE and shedding light on phenomena such as plateaus and delay-induced oscillations in neural populations.

Abstract

There is a wide range of mathematical models that describe populations of large numbers of neurons. In this article, we focus on nonlinear noisy leaky integrate-and-fire (NNLIF) models that describe neuronal activity at the level of the membrane potential. We introduce a sequence of states, which we call pseudoequilibria, and give evidence of their defining role in the behavior of the NNLIF system when a significant synaptic delay is considered. The advantage is that these states are determined solely by the system's parameters and are derived from a sequence of firing rates that result from solving a recurrence equation. We propose a strategy to show convergence to an equilibrium for a weakly connected system with large transmission delay, based on following the sequence of pseudoequilibria. Unlike direct entropy dissipation methods, this technique allows us to see how a large delay favors convergence. We present a detailed numerical study to support our results. This study helps us understand, among other phenomena, the appearance of periodic solutions in strongly inhibitory networks.

Figures (13)

• Figure 1: Comparison of "plateau" distributions with profiles \ref{['eq:profile-plateau']}. Graphs taken from CR-L. Left: $b=1.5$. Right: $b=2.2$.
• Figure 2: Function $\boldsymbol{\frac{1}{I(N)}}$ (see \ref{['eq: I']}) for different values of the connectivity parameter $\boldsymbol{b}$.
• Figure 3: Function $\boldsymbol{F:=f\circ f}$ with $\boldsymbol{f(N)=1/I(N)}$ (see \ref{['eq: I']}) for different (negative) values of the connectivity parameter $\boldsymbol{b}$.
• Figure 4: Schematic representation of the solution to the Cauchy problem \ref{['eq: large-delta']} in time through the sequence $\boldsymbol{p_{k}(v,t)}$ given by \ref{['eq:p']}.
• Figure 5: Firing rate sequence $\boldsymbol{N_{k,\infty}}$\ref{['eq:sequence-N']} with $\boldsymbol{b=1.5}$ and different values of initial condition $\boldsymbol{N_{0,\infty}}$. Solid and dashed straight horizontal lines represent equilibria $N_1^*$ and $N_2^*$ respectively.

Theorems & Definitions (25)

• Definition 2.1: Firing rate sequence
• Definition 2.2: Pseudo-equilibria sequence
• Theorem 2.3: Monotonicity of the firing rate sequence $\left\{N_{k,\infty}\right\}_{k\ge 0}$ for excitatory networks
• proof
• Remark 2.4
• Theorem 2.5: Monotonicity of the firing rate sequence $\left\{N_{k,\infty}\right\}_{k\ge 0}$ for inhibitory networks
• proof
• Remark 2.6
• Remark 2.7
• Remark 2.8