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Sequence of pseudoequilibria describes the long-time behavior of the nonlinear noisy leaky integrate-and-fire model with large delay
Analysis of PDEs · Submitted 1 Mar 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | There is a wide range of mathematical models that describe populations of large numbers of neurons. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s1 | 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. | background 0.91 | 0.21 | 0.03 | asserted 0.95 |
| s2 | 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. | contribution 1.00 | 0.95 | 0.52 | hedged possibility 0.49 |
| s3 | 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. | method 0.47 | 0.89 | 0.10 | asserted 1.00 |
| s4 | 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. | method 0.65 | 0.87 | 0.31 | asserted 0.68 |
| s5 | Unlike direct entropy dissipation methods, this technique allows us to see how a large delay favors convergence. | interpretation 0.77 | 0.78 | 0.12 | asserted 0.97 |
| s6 | We present a detailed numerical study to support our results. | method 0.60 | 0.93 | 0.03 | asserted 0.97 |
| s7 | This study helps us understand, among other phenomena, the appearance of periodic solutions in strongly inhibitory networks. | interpretation 0.94 | 0.78 | 0.09 | asserted 0.76 |
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The paper
Original 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.
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2 arXiv papers cite this in 7 sentences
What it cites · 4 sentences without a confident label
Compares with 1 paper
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This could also be compared to a similar result in [ref] .
On the asymptotic behavior of the NNLIF neuron model for general connectivity strength · Convergence to equilibrium along the pseudo equilibria sequence for weakly connected networks · 1.00
Cites as background 1 paper
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It was numerically analysed in [ref] , and behaviour after the explosion was studied in [ref] by a time change of variable which dilates the blow-up time.
Dilating blow-up time: A generalized solution of the NNLIF neuron model and its global well-posedness · Introduction · 0.88
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