兴奋-抑制非线性神经元模型的爆破、定态和长时间行为
Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models
摘要：
神经网络常用激励和抑制非线性、含噪、漏积分和激发模型来描述。最近，新的数学结果提供了对它们的更好理解。已经证明了一个完全兴奋的网络可以在有限时间内爆破，而一个完全抑制的网络在任何初始数据下都有一个全局解。一个纯粹的兴奋性或抑制性网络的稳态的一般描述也已给出。我们将这项研究扩展到由一个兴奋性种群和一个抑制性种群组成的系统。我们证明了该系统也可以在有限时间内爆破，并分析了其稳态和长时间行为。此外，我们用一些数值结果来说明我们的解析描述。用于实现我们目标的主要工具是：控制爆破结果的指数矩，这是一种比Cáceres，Carrillo和Perthame（2011）中考虑的更复杂的策略，用于研究稳态的数量，熵方法与Poincaré不等式结合用于长时间行为，最后，高阶数值格式和并行计算技术，以获得我们的数值结果。
英文摘要：
Excitatory and inhibitory nonlinear noisy leaky integrate and fire models are often used to describe neural networks. Recently, new mathematical results have provided a better understanding of them. It has been proved that a fully excitatory network can blow-up in finite time, while a fully inhibitory network has a global in time solution for any initial data. A general description of the steady states of a purely excitatory or inhibitory network has been also given. We extend this study to the system composed of an excitatory population and an inhibitory one. We prove that this system can also blow-up in finite time and analyse its steady states and long time behaviour. Besides, we illustrate our analytical description with some numerical results. The main tools used to reach our aims are: the control of an exponential moment for the blow-up results, a more complicate strategy than that considered in Cáceres, Carrillo and Perthame (2011), for studying the number of steady states, entropy methods combined with Poincaré inequalities for the long time behaviour and, finally, high order numerical schemes together with parallel computation techniques in order to obtain our numerical results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
