论文标题
扩张爆炸时间：NNLIF神经元模型及其全局良好的通用解决方案
Dilating blow-up time: A generalized solution of the NNLIF neuron model and its global well-posedness
论文作者
论文摘要
非线性噪声泄漏的集成和火力（NNLIF）模型是对大量相互作用神经元的流行均值描述，该模型吸引了数学家从各个方面进行研究。该模型的核心特性是发射速率的有限时间爆炸，科学上与神经元网络的同步相对应，并且在数学上可以防止存在全球经典解决方案。在这项工作中，我们提出了一种新的广义解决方案，基于对PDE模型进行重新设计的特定变化时间的重新设计。引入了触发速率依赖性时间尺度，其中即使在爆破的情况下，转换后的方程也可以在全球范围内适合任何连接性参数。然后，通过时间尺度的向后更改来定义广义解决方案，并且当射击速率爆炸时可能会跳跃。我们建立了广义解决方案的特性，包括对原始时间尺度中的爆破表征和全球辅助性。广义解决方案提供了一种新的观点，可以理解射击速率爆炸以及爆炸后解决方案的延续时的动态。
The nonlinear noisy leaky integrate-and-fire (NNLIF) model is a popular mean-field description of a large number of interacting neurons, which has attracted mathematicians to study from various aspects. A core property of this model is the finite time blow-up of the firing rate, which scientifically corresponds to the synchronization of a neuron network, and mathematically prevents the existence of a global classical solution. In this work, we propose a new generalized solution based on reformulating the PDE model with a specific change of variable in time. A firing rate dependent timescale is introduced, in which the transformed equation can be shown to be globally well-posed for any connectivity parameter even in the event of the blow-up. The generalized solution is then defined via the backward change of timescale, and it may have a jump when the firing rate blows up. We establish properties of the generalized solution including the characterization of blow-ups and the global well-posedness in the original timescale. The generalized solution provides a new perspective to understand the dynamics when the firing rate blows up as well as the continuation of the solution after a blow-up.
