论文标题
对非线性噪声泄漏和火神经元模型的物理解决方案和爆炸现象的理解
An understanding of the physical solutions and the blow-up phenomenon for Nonlinear Noisy Leaky Integrate and Fire neuronal models
论文作者
论文摘要
非线性噪声泄漏整合和火神经元模型是描述神经网络活性的数学模型。使用随机微分方程以及使用Fokker-Planck类型方程的平均场限制，使用随机微分方程和介观/宏观水平研究了这些模型。本文的目的是使用对粒子系统的数值研究来提高他们的理解。我们深入分析随机微分方程的经典和物理解的行为，并将其与已经知道的Fokker-Planck方程进行了比较。这使我们能够更好地理解当有限时间发生爆炸时神经网络中发生的情况。同时发射所有神经元后，如果系统弱连接，神经网络将朝着其独特的稳态收敛。否则，它的行为更加复杂，因为它可以倾向于固定状态或“平稳”分布。
The Nonlinear Noisy Leaky Integrate and Fire neuronal models are mathematical models that describe the activity of neural networks. These models have been studied at a microscopic level, using Stochastic Differential Equations, and at a mesoscopic/macroscopic level, through the mean field limits using Fokker-Planck type equations. The aim of this paper is to improve their understanding, using a numerical study of their particle systems. We analyse in depth the behaviour of the classical and physical solutions of the Stochastic Differential Equations and, we compare it with what is already known about the Fokker-Planck equation. This allows us to better understand what happens in the neural network when an explosion occurs in finite time. After firing all neurons at the same time, if the system is weakly connected, the neural network converges towards its unique steady state. Otherwise, its behaviour is more complex, because it can tend towards a stationary state or a "plateau" distribution.
