Neural network dynamics emerge from the interaction of spiking cells. One way to formulate the problem is through a theoretical framework inspired by ideas coming from statistical physics, the so-called mean-field theory. In this document, we investigate different issues related to the mean-field description of an excitatory network made up of leaky integrate-and-fire neurons. The description is written in the form a nonlinear partial differential equation which is known to blow up in finite time when the network is strongly connected. We prove that in a moderate coupling regime the equation is globally well-posed in the space of measures, and that there exist stationary solutions. In the case of weak connectivity we also demonstrate the uniqueness of the steady state and its global exponential stability. The method to show those mathematical results relies on a contraction argument of Doeblin's type in the linear case, which corresponds to a population of non-interacting units.

中文翻译：

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泄漏积分激发神经网络的平均场方程：测量解和稳态

神经网络动力学源于尖峰细胞的相互作用。提出问题的一种方法是通过一个受统计物理学思想启发的理论框架，即所谓的平均场理论。在本文档中，我们研究了与由泄漏积分和激发神经元组成的兴奋性网络的平均场描述相关的不同问题。该描述以非线性偏微分方程的形式编写，已知当网络强连接时，该方程会在有限时间内爆炸。我们证明，在中等耦合状态下，方程在测度空间中是全局适定的，并且存在平稳解。在弱连通性的情况下，我们还证明了稳态的唯一性及其全局指数稳定性。